Subgroups — C₂×C₃×#65967 #14482809

Order 1920.

3240 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×Q₈×D₅	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃×#709	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×C₅×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃×#4093	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃×C₄×C₂²⋊D₅	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#65967	C₂	1	16	yes	maximal, direct product factor
7	C₂×C₃×#4042	C₂	1	2	yes	maximal, complemented
8	C₂×C₃×#4041	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₃×#4039	C₂	1	2	yes	maximal, complemented
10	C₂×C₃×#4033	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₃×#4031	C₂	1	2	yes	maximal, complemented
12	C₂×C₃×#4030	C₂	1	2	yes	maximal, complemented
13	C₂×#65967	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
14	C₂²×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
15	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
16	C₂²×C₃×C₂²⋊D₅	C₂²	1	1	yes	characteristic
17	C₂²×C₃×C₂²⋊D₅	C₂²	1	2	yes	—
18	C₂×C₃×Q₈×D₅	—	2	8	no	—
19	C₂×C₃×Q₈×D₅	C₂²	1	8	yes	complemented
20	C₂×C₃×#709	—	2	8	no	—
21	C₂×C₃×#709	C₂²	1	8	yes	complemented
22	C₂²×C₃×D₂₀	C₂²	1	1	yes	characteristic
23	C₂²×C₃×D₂₀	C₂²	1	2	yes	—
24	C₂²×C₃×C₄×D₅	C₂²	1	1	yes	characteristic
25	C₂²×C₃×C₄×D₅	C₂²	1	2	yes	—
26	C₂²×C₃×C₄×D₅	C₂²	1	2	yes	—
27	C₂²×C₃×Dic₁₀	C₂²	1	2	yes	—
28	C₂²×C₃×Dic₁₀	C₂²	1	1	yes	characteristic
29	C₃×C₅×C₂²⋊Q₈	C₂²	1	8	yes	complemented
30	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
31	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
32	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
33	C₂×C₃×C₂²⋊Dic₅	C₂²	1	1	yes	characteristic
34	C₃×#4093	C₂²	1	8	yes	complemented
35	C₃×C₄×C₂²⋊D₅	C₂²	1	8	yes	complemented
36	C₂×C₃×#685	C₂²	1	1	yes	characteristic
37	C₂×C₃×#685	C₂²	1	2	yes	—
38	C₂×C₃×#685	C₂²	1	2	yes	—
39	C₂×C₃×#685	C₂²	1	2	yes	—
40	C₂×C₃×#683	C₂²	1	1	yes	characteristic
41	C₂×C₃×#683	C₂²	1	2	yes	—
42	C₂×C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
43	C₂×C₃×C₄×Dic₅	C₂²	1	2	yes	—
44	C₃×#4042	C₂²	1	16	yes	complemented
45	C₃×#4041	C₂²	1	8	yes	complemented
46	C₃×#4039	C₂²	1	16	yes	complemented
47	C₃×#4033	C₂²	1	8	yes	complemented
48	C₃×#4031	C₂²	1	16	yes	complemented
49	C₃×#4030	C₂²	1	16	yes	complemented
50	C₂×C₃×#548	—	5	1	no	maximal, Hall
51	C₂²×Q₈×D₅	C₂×C₃	1	1	yes	characteristic, complemented
52	C₂²×#709	C₂×C₃	1	1	yes	characteristic, complemented
53	C₂×C₅×C₂²⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
54	C₂×#4093	C₂×C₃	1	1	yes	characteristic, complemented
55	C₂×C₄×C₂²⋊D₅	C₂×C₃	1	1	yes	characteristic, complemented
56	#65967	C₂×C₃	1	16	yes	direct product factor
57	C₂×#4042	C₂×C₃	1	2	yes	complemented
58	C₂×#4041	C₂×C₃	1	1	yes	characteristic, complemented
59	C₂×#4039	C₂×C₃	1	2	yes	complemented
60	C₂×#4033	C₂×C₃	1	1	yes	characteristic, complemented
61	C₂×#4031	C₂×C₃	1	2	yes	complemented
62	C₂×#4030	C₂×C₃	1	2	yes	complemented
63	C₂⁴×C₃×C₅	C₂³	1	1	yes	characteristic
64	C₂³×C₃×D₅	C₂³	1	2	yes	—
65	C₂³×C₃×D₅	C₂³	1	1	yes	characteristic
66	C₂×C₃×C₅×Q₈	—	2	4	no	—
67	C₂×C₃×C₅×Q₈	C₂³	1	4	yes	—
68	C₂²×C₃×C₄×C₅	—	2	4	no	—
69	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
70	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
71	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
72	C₂²×C₃×C₄×C₅	C₂³	1	4	yes	—
73	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
74	C₂×C₃×C₂²⋊D₅	—	2	4	no	—
75	C₂×C₃×C₂²⋊D₅	—	2	8	no	—
76	C₂×C₃×C₂²⋊D₅	C₂³	1	8	yes	—
77	C₂×C₃×C₂²⋊D₅	C₂³	1	4	yes	—
78	C₂²×C₃×Dic₅	C₂³	1	1	yes	characteristic
79	C₂²×C₃×Dic₅	C₂³	1	1	yes	characteristic
80	C₂²×C₃×Dic₅	C₂³	1	1	yes	characteristic
81	C₂²×C₃×Dic₅	C₂³	1	2	yes	—
82	C₃×Q₈×D₅	—	2	32	no	—
83	C₃×#709	—	2	32	no	—
84	C₂×C₃×D₂₀	—	2	8	no	—
85	C₂×C₃×D₂₀	—	2	4	no	—
86	C₂×C₃×D₂₀	C₂³	1	4	yes	—
87	C₂×C₃×D₂₀	C₂³	1	8	yes	—
88	C₂×C₃×C₄×D₅	—	2	4	no	—
89	C₂×C₃×C₄×D₅	—	2	8	no	—
90	C₂×C₃×C₄×D₅	—	2	8	no	—
91	C₂×C₃×C₄×D₅	C₂³	1	8	yes	—
92	C₂×C₃×C₄×D₅	C₂³	1	8	yes	—
93	C₂×C₃×C₄×D₅	C₂³	1	4	yes	—
94	C₂×C₃×Dic₁₀	—	2	8	no	—
95	C₂×C₃×Dic₁₀	D₄	1	8	yes	complemented
96	C₂×C₃×Dic₁₀	C₂³	1	8	yes	—
97	C₂×C₃×Dic₁₀	C₂³	1	4	yes	—
98	C₃×C₅×C₄⋊C₄	C₂³	1	4	yes	—
99	C₃×C₅×C₄⋊C₄	C₂³	1	8	yes	—
100	C₃×C₅×C₂²⋊C₄	C₂³	1	8	yes	—
101	C₃×C₂²⋊Dic₅	C₂³	1	4	yes	—
102	C₃×#685	C₂³	1	8	yes	—
103	C₃×#685	C₂³	1	8	yes	—
104	C₃×#685	C₂³	1	8	yes	—
105	C₃×#685	C₂³	1	4	yes	—
106	C₃×#683	C₂³	1	8	yes	—
107	C₃×#683	C₂³	1	4	yes	—
108	C₃×C₄×Dic₅	C₂³	1	4	yes	—
109	C₃×C₄×Dic₅	C₂³	1	8	yes	—
110	C₂²×C₃×#41	—	5	1	no	—
111	C₂³×C₃×Q₈	—	5	1	no	—
112	C₃×#548	—	5	16	no	—
113	C₂×C₃×#124	—	5	1	no	—
114	C₂×C₃×#124	—	5	2	no	—
115	C₂×C₃×C₂²⋊Q₈	—	5	2	no	—
116	C₂×C₃×C₂²⋊Q₈	D₅	1	1	yes	characteristic, complemented
117	C₂×C₃×#121	—	5	1	no	—
118	C₂×C₃×#121	—	5	2	no	—
119	C₂×C₃×C₄×Q₈	—	5	1	no	—
120	C₂×C₃×C₄×D₄	—	5	1	no	—
121	C₂×C₃×C₄×D₄	—	5	2	no	—
122	C₂²×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
123	C₂³×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
124	C₂²×C₂²⋊D₅	C₂²×C₃	1	1	yes	characteristic
125	C₂²×C₂²⋊D₅	C₂²×C₃	1	2	yes	—
126	C₂×Q₈×D₅	—	2	8	no	—
127	C₂×Q₈×D₅	C₂²×C₃	1	8	yes	complemented
128	C₂×#709	—	2	8	no	—
129	C₂×#709	C₂²×C₃	1	8	yes	complemented
130	C₂²×D₂₀	C₂²×C₃	1	2	yes	—
131	C₂²×D₂₀	C₂²×C₃	1	1	yes	characteristic
132	C₂²×C₄×D₅	C₂²×C₃	1	2	yes	—
133	C₂²×C₄×D₅	C₂²×C₃	1	2	yes	—
134	C₂²×C₄×D₅	C₂²×C₃	1	1	yes	characteristic
135	C₂²×Dic₁₀	C₂²×C₃	1	2	yes	—
136	C₂²×Dic₁₀	C₂²×C₃	1	1	yes	characteristic
137	C₅×C₂²⋊Q₈	C₂²×C₃	1	8	yes	complemented
138	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
139	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
140	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
141	C₂×C₂²⋊Dic₅	C₂²×C₃	1	1	yes	characteristic
142	#4093	C₂²×C₃	1	8	yes	complemented
143	C₄×C₂²⋊D₅	C₂²×C₃	1	8	yes	complemented
144	C₂×#685	C₂²×C₃	1	2	yes	—
145	C₂×#685	C₂²×C₃	1	2	yes	—
146	C₂×#685	C₂²×C₃	1	2	yes	—
147	C₂×#685	C₂²×C₃	1	1	yes	characteristic
148	C₂×#683	C₂²×C₃	1	2	yes	—
149	C₂×#683	C₂²×C₃	1	1	yes	characteristic
150	C₂×C₄×Dic₅	C₂²×C₃	1	2	yes	—
151	C₂×C₄×Dic₅	C₂²×C₃	1	1	yes	characteristic
152	#4042	C₂²×C₃	1	16	yes	complemented
153	#4041	C₂²×C₃	1	8	yes	complemented
154	#4039	C₂²×C₃	1	16	yes	complemented
155	#4033	C₂²×C₃	1	8	yes	complemented
156	#4031	C₂²×C₃	1	16	yes	complemented
157	#4030	C₂²×C₃	1	16	yes	complemented
158	C₂×#548	—	5	1	no	Sylow
159	C₂³×C₃×C₅	—	2	2	no	—
160	C₂³×C₃×C₅	—	2	2	no	—
161	C₂³×C₃×C₅	—	2	2	no	—
162	C₂³×C₃×C₅	C₂⁴	1	2	yes	—
163	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
164	C₂²×C₃×D₅	—	2	2	no	—
165	C₂²×C₃×D₅	—	2	2	no	—
166	C₂²×C₃×D₅	—	2	4	no	—
167	C₂²×C₃×D₅	—	2	4	no	—
168	C₂²×C₃×D₅	—	2	4	no	—
169	C₂²×C₃×D₅	#41	1	4	yes	complemented
170	C₂²×C₃×D₅	C₂⁴	1	4	yes	—
171	C₂²×C₃×D₅	C₂⁴	1	2	yes	—
172	C₃×C₅×Q₈	—	2	8	no	—
173	C₂×C₃×C₄×C₅	—	2	4	no	—
174	C₂×C₃×C₄×C₅	—	2	2	no	—
175	C₂×C₃×C₄×C₅	—	2	4	no	—
176	C₂×C₃×C₄×C₅	—	2	8	no	—
177	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
178	C₂×C₃×C₄×C₅	C₂⁴	1	4	yes	—
179	C₂×C₃×C₄×C₅	C₂⁴	1	2	yes	—
180	C₂×C₃×C₄×C₅	C₂⁴	1	4	yes	—
181	C₂×C₃×C₄×C₅	C₂⁴	1	2	yes	—
182	C₃×C₂²⋊D₅	—	2	8	no	—
183	C₃×C₂²⋊D₅	—	2	16	no	—
184	C₂×C₃×Dic₅	—	2	2	no	—
185	C₂×C₃×Dic₅	—	2	2	no	—
186	C₂×C₃×Dic₅	—	2	2	no	—
187	C₂×C₃×Dic₅	C₂×D₄	1	8	yes	—
188	C₂×C₃×Dic₅	C₂⁴	1	2	yes	—
189	C₂×C₃×Dic₅	C₂⁴	1	2	yes	—
190	C₂×C₃×Dic₅	C₂⁴	1	2	yes	—
191	C₂×C₃×Dic₅	C₂⁴	1	4	yes	—
192	C₃×D₂₀	—	2	16	no	—
193	C₃×D₂₀	—	2	8	no	—
194	C₃×C₄×D₅	—	2	8	no	—
195	C₃×C₄×D₅	—	2	16	no	—
196	C₃×C₄×D₅	—	2	16	no	—
197	C₃×Dic₁₀	—	2	16	no	—
198	C₃×Dic₁₀	C₂×D₄	1	16	yes	complemented
199	C₂×C₃×#41	—	5	8	no	—
200	C₂×C₃×#41	—	10	8	no	—
201	C₂²×C₃×Q₈	—	5	8	no	—
202	C₂²×C₃×Q₈	—	5	1	no	—
203	C₂²×C₃×Q₈	—	5	2	no	—
204	C₂²×C₃×Q₈	—	10	8	no	—
205	C₂²×C₃×Q₈	C₂×D₅	1	1	yes	characteristic
206	C₂²×C₃×D₄	—	5	2	no	—
207	C₂²×C₃×D₄	—	5	1	no	—
208	C₂²×C₃×D₄	—	5	1	no	—
209	C₂²×C₃×D₄	—	5	2	no	—
210	C₂³×C₃×C₄	—	5	2	no	—
211	C₂³×C₃×C₄	—	5	1	no	—
212	C₂³×C₃×C₄	—	5	2	no	—
213	C₂³×C₃×C₄	C₂×D₅	1	1	yes	characteristic
214	C₃×#124	—	5	8	no	—
215	C₃×#124	—	5	16	no	—
216	C₃×C₂²⋊Q₈	—	5	16	no	—
217	C₃×C₂²⋊Q₈	C₂×D₅	1	8	yes	complemented
218	C₃×#121	—	5	8	no	—
219	C₃×#121	—	5	16	no	—
220	C₃×C₄×Q₈	—	5	8	no	—
221	C₃×C₄×D₄	—	5	8	no	—
222	C₃×C₄×D₄	—	5	16	no	—
223	C₂×C₃×C₄⋊C₄	—	5	1	no	—
224	C₂×C₃×C₄⋊C₄	—	5	2	no	—
225	C₂×C₃×C₄⋊C₄	C₂×D₅	1	2	yes	—
226	C₂×C₃×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
227	C₂×C₃×C₂²⋊C₄	—	5	1	no	—
228	C₂×C₃×C₂²⋊C₄	—	5	1	no	—
229	C₂×C₃×C₂²⋊C₄	—	5	2	no	—
230	C₂×C₃×C₂²⋊C₄	—	5	2	no	—
231	C₂×C₃×C₂²⋊C₄	—	5	2	no	—
232	C₂×C₃×C₂²⋊C₄	C₂×D₅	1	2	yes	—
233	C₂×C₃×C₄²	—	5	2	no	—
234	C₂×C₃×C₄²	—	5	1	no	—
235	C₂⁴×C₅	C₂³×C₃	1	1	yes	characteristic
236	C₂³×D₅	C₂³×C₃	1	2	yes	—
237	C₂³×D₅	C₂³×C₃	1	1	yes	characteristic
238	C₂×C₅×Q₈	—	2	4	no	—
239	C₂×C₅×Q₈	C₂³×C₃	1	4	yes	—
240	C₂²×C₄×C₅	—	2	4	no	—
241	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
242	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
243	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
244	C₂²×C₄×C₅	C₂³×C₃	1	4	yes	—
245	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
246	C₂×C₂²⋊D₅	—	2	4	no	—
247	C₂×C₂²⋊D₅	—	2	8	no	—
248	C₂×C₂²⋊D₅	C₂³×C₃	1	8	yes	—
249	C₂×C₂²⋊D₅	C₂³×C₃	1	4	yes	—
250	C₂²×Dic₅	C₂³×C₃	1	1	yes	characteristic
251	C₂²×Dic₅	C₂³×C₃	1	1	yes	characteristic
252	C₂²×Dic₅	C₂³×C₃	1	1	yes	characteristic
253	C₂²×Dic₅	C₂³×C₃	1	2	yes	—
254	Q₈×D₅	—	2	32	no	—
255	#709	—	2	32	no	—
256	C₂×D₂₀	—	2	8	no	—
257	C₂×D₂₀	—	2	4	no	—
258	C₂×D₂₀	C₂³×C₃	1	4	yes	—
259	C₂×D₂₀	C₂³×C₃	1	8	yes	—
260	C₂×C₄×D₅	—	2	4	no	—
261	C₂×C₄×D₅	—	2	8	no	—
262	C₂×C₄×D₅	—	2	8	no	—
263	C₂×C₄×D₅	C₂³×C₃	1	8	yes	—
264	C₂×C₄×D₅	C₂³×C₃	1	8	yes	—
265	C₂×C₄×D₅	C₂³×C₃	1	4	yes	—
266	C₂×Dic₁₀	—	2	8	no	—
267	C₂×Dic₁₀	C₃×D₄	1	8	yes	complemented
268	C₂×Dic₁₀	C₂³×C₃	1	8	yes	—
269	C₂×Dic₁₀	C₂³×C₃	1	4	yes	—
270	C₅×C₄⋊C₄	C₂³×C₃	1	4	yes	—
271	C₅×C₄⋊C₄	C₂³×C₃	1	8	yes	—
272	C₅×C₂²⋊C₄	C₂³×C₃	1	8	yes	—
273	C₂²⋊Dic₅	C₂³×C₃	1	4	yes	—
274	#685	C₂³×C₃	1	8	yes	—
275	#685	C₂³×C₃	1	8	yes	—
276	#685	C₂³×C₃	1	8	yes	—
277	#685	C₂³×C₃	1	4	yes	—
278	#683	C₂³×C₃	1	8	yes	—
279	#683	C₂³×C₃	1	4	yes	—
280	C₄×Dic₅	C₂³×C₃	1	8	yes	—
281	C₄×Dic₅	C₂³×C₃	1	4	yes	—
282	C₂²×#41	—	5	1	no	—
283	C₂³×Q₈	—	5	1	no	—
284	#548	—	5	16	no	—
285	C₂×#124	—	5	1	no	—
286	C₂×#124	—	5	2	no	—
287	C₂×C₂²⋊Q₈	—	5	2	no	—
288	C₂×C₂²⋊Q₈	C₃×D₅	1	1	yes	characteristic, complemented
289	C₂×#121	—	5	1	no	—
290	C₂×#121	—	5	2	no	—
291	C₂×C₄×Q₈	—	5	1	no	—
292	C₂×C₄×D₄	—	5	1	no	—
293	C₂×C₄×D₄	—	5	2	no	—
294	C₂²×C₃×C₅	—	2	2	no	—
295	C₂²×C₃×C₅	—	2	2	no	—
296	C₂²×C₃×C₅	—	2	2	no	—
297	C₂²×C₃×C₅	—	4	4	no	—
298	C₂²×C₃×C₅	C₂²×D₄	1	2	yes	—
299	C₂²×C₃×C₅	C₂×#41	1	2	yes	—
300	C₂²×C₃×C₅	2₋¹⁺⁴	1	2	yes	—
301	C₂²×C₃×C₅	C₂⁵	1	1	yes	characteristic
302	C₂×C₃×D₅	—	2	8	no	—
303	C₂×C₃×D₅	—	2	2	no	—
304	C₂×C₃×D₅	—	2	2	no	—
305	C₂×C₃×D₅	—	2	4	no	—
306	C₂×C₃×D₅	—	2	4	no	—
307	C₂×C₃×D₅	—	2	4	no	—
308	C₂×C₃×D₅	—	4	8	no	—
309	C₂×C₃×D₅	C₂×#41	1	4	yes	complemented
310	C₃×C₄×C₅	—	2	4	no	—
311	C₃×C₄×C₅	—	2	2	no	—
312	C₃×C₄×C₅	—	2	4	no	—
313	C₃×C₄×C₅	C₂²×D₄	1	4	yes	—
314	C₃×Dic₅	—	2	2	no	—
315	C₃×Dic₅	—	2	2	no	—
316	C₃×Dic₅	—	2	2	no	—
317	C₃×Dic₅	C₂²×D₄	1	8	yes	—
318	C₂⁴×C₃	—	5	1	no	—
319	C₂⁴×C₃	—	5	2	no	—
320	C₂⁴×C₃	C₂²×D₅	1	1	yes	characteristic
321	C₃×#41	—	10	32	no	—
322	C₂×C₃×Q₈	—	2	4	no	—
323	C₂×C₃×Q₈	—	5	8	no	—
324	C₂×C₃×Q₈	—	5	4	no	—
325	C₂×C₃×Q₈	—	5	8	no	—
326	C₂×C₃×Q₈	—	10	8	no	—
327	C₂×C₃×Q₈	—	10	32	no	—
328	C₂×C₃×Q₈	C₂²×D₅	1	4	yes	—
329	C₂×C₃×D₄	—	5	4	no	—
330	C₂×C₃×D₄	—	5	8	no	—
331	C₂×C₃×D₄	—	5	8	no	—
332	C₂×C₃×D₄	—	5	4	no	—
333	C₂×C₃×D₄	—	10	8	no	—
334	C₂×C₃×D₄	—	10	8	no	—
335	C₂×C₃×D₄	—	10	4	no	—
336	C₂×C₃×D₄	—	10	4	no	—
337	C₂²×C₃×C₄	—	2	4	no	—
338	C₂²×C₃×C₄	—	5	2	no	—
339	C₂²×C₃×C₄	—	5	1	no	—
340	C₂²×C₃×C₄	—	5	1	no	—
341	C₂²×C₃×C₄	—	5	1	no	—
342	C₂²×C₃×C₄	—	5	8	no	—
343	C₂²×C₃×C₄	—	5	4	no	—
344	C₂²×C₃×C₄	—	5	8	no	—
345	C₂²×C₃×C₄	—	10	8	no	—
346	C₂²×C₃×C₄	—	10	8	no	—
347	C₂²×C₃×C₄	—	10	4	no	—
348	C₂²×C₃×C₄	C₂²×D₅	1	4	yes	—
349	C₂²×C₃×C₄	C₂²×D₅	1	2	yes	—
350	C₂²×C₃×C₄	C₂²×D₅	1	2	yes	—
351	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
352	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
353	C₃×C₄⋊C₄	—	5	4	no	—
354	C₃×C₄⋊C₄	—	5	8	no	—
355	C₃×C₄⋊C₄	C₂²×D₅	1	8	yes	—
356	C₃×C₄⋊C₄	C₂²×D₅	1	4	yes	—
357	C₃×C₂²⋊C₄	—	5	4	no	—
358	C₃×C₂²⋊C₄	—	5	4	no	—
359	C₃×C₂²⋊C₄	—	5	8	no	—
360	C₃×C₂²⋊C₄	—	5	8	no	—
361	C₃×C₂²⋊C₄	—	5	8	no	—
362	C₃×C₂²⋊C₄	C₂²×D₅	1	8	yes	—
363	C₃×C₄²	—	5	8	no	—
364	C₃×C₄²	—	5	4	no	—
365	C₂³×C₅	—	2	2	no	—
366	C₂³×C₅	—	2	2	no	—
367	C₂³×C₅	—	2	2	no	—
368	C₂³×C₅	C₂⁴×C₃	1	2	yes	—
369	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
370	C₂²×D₅	—	2	2	no	—
371	C₂²×D₅	—	2	2	no	—
372	C₂²×D₅	—	2	4	no	—
373	C₂²×D₅	—	2	4	no	—
374	C₂²×D₅	—	2	4	no	—
375	C₂²×D₅	C₃×#41	1	4	yes	complemented
376	C₂²×D₅	C₂⁴×C₃	1	4	yes	—
377	C₂²×D₅	C₂⁴×C₃	1	2	yes	—
378	C₅×Q₈	—	2	8	no	—
379	C₂×C₄×C₅	—	2	4	no	—
380	C₂×C₄×C₅	—	2	2	no	—
381	C₂×C₄×C₅	—	2	4	no	—
382	C₂×C₄×C₅	—	2	8	no	—
383	C₂×C₄×C₅	C₂×C₃×D₄	1	4	yes	—
384	C₂×C₄×C₅	C₂⁴×C₃	1	4	yes	—
385	C₂×C₄×C₅	C₂⁴×C₃	1	2	yes	—
386	C₂×C₄×C₅	C₂⁴×C₃	1	4	yes	—
387	C₂×C₄×C₅	C₂⁴×C₃	1	2	yes	—
388	C₂²⋊D₅	—	2	8	no	—
389	C₂²⋊D₅	—	2	16	no	—
390	C₂×Dic₅	—	2	2	no	—
391	C₂×Dic₅	—	2	2	no	—
392	C₂×Dic₅	—	2	2	no	—
393	C₂×Dic₅	C₂×C₃×D₄	1	8	yes	—
394	C₂×Dic₅	C₂⁴×C₃	1	2	yes	—
395	C₂×Dic₅	C₂⁴×C₃	1	2	yes	—
396	C₂×Dic₅	C₂⁴×C₃	1	2	yes	—
397	C₂×Dic₅	C₂⁴×C₃	1	4	yes	—
398	D₂₀	—	2	16	no	—
399	D₂₀	—	2	8	no	—
400	C₄×D₅	—	2	8	no	—
401	C₄×D₅	—	2	16	no	—
402	C₄×D₅	—	2	16	no	—
403	Dic₁₀	—	2	16	no	—
404	Dic₁₀	C₂×C₃×D₄	1	16	yes	complemented
405	C₂×#41	—	5	8	no	—
406	C₂×#41	—	10	8	no	—
407	C₂²×Q₈	—	5	8	no	—
408	C₂²×Q₈	—	5	1	no	—
409	C₂²×Q₈	—	5	2	no	—
410	C₂²×Q₈	—	10	8	no	—
411	C₂²×Q₈	C₂×C₃×D₅	1	1	yes	characteristic
412	C₂²×D₄	—	5	1	no	—
413	C₂²×D₄	—	5	2	no	—
414	C₂²×D₄	—	5	2	no	—
415	C₂²×D₄	—	5	1	no	—
416	C₂³×C₄	—	5	2	no	—
417	C₂³×C₄	—	5	1	no	—
418	C₂³×C₄	—	5	2	no	—
419	C₂³×C₄	C₂×C₃×D₅	1	1	yes	characteristic
420	#124	—	5	8	no	—
421	#124	—	5	16	no	—
422	C₂²⋊Q₈	—	5	16	no	—
423	C₂²⋊Q₈	C₂×C₃×D₅	1	8	yes	complemented
424	#121	—	5	8	no	—
425	#121	—	5	16	no	—
426	C₄×Q₈	—	5	8	no	—
427	C₄×D₄	—	5	8	no	—
428	C₄×D₄	—	5	16	no	—
429	C₂×C₄⋊C₄	—	5	1	no	—
430	C₂×C₄⋊C₄	—	5	2	no	—
431	C₂×C₄⋊C₄	C₂×C₃×D₅	1	2	yes	—
432	C₂×C₄⋊C₄	C₂×C₃×D₅	1	1	yes	characteristic
433	C₂×C₂²⋊C₄	—	5	1	no	—
434	C₂×C₂²⋊C₄	—	5	1	no	—
435	C₂×C₂²⋊C₄	—	5	2	no	—
436	C₂×C₂²⋊C₄	—	5	2	no	—
437	C₂×C₂²⋊C₄	—	5	2	no	—
438	C₂×C₂²⋊C₄	C₂×C₃×D₅	1	2	yes	—
439	C₂×C₄²	—	5	2	no	—
440	C₂×C₄²	—	5	1	no	—
441	C₂×C₃×C₅	—	4	2	no	—
442	C₂×C₃×C₅	#548	1	4	yes	complemented
443	C₂×C₃×C₅	C₂³×D₄	1	1	yes	characteristic
444	C₂×C₃×C₅	C₂²×#41	1	1	yes	characteristic
445	C₂×C₃×C₅	C₂×2₋¹⁺⁴	1	1	yes	characteristic
446	C₃×D₅	—	2	4	no	—
447	C₃×D₅	—	4	4	no	—
448	C₂³×C₃	—	2	2	no	—
449	C₂³×C₃	—	2	2	no	—
450	C₂³×C₃	—	2	2	no	—
451	C₂³×C₃	—	5	4	no	—
452	C₂³×C₃	—	5	2	no	—
453	C₂³×C₃	—	5	4	no	—
454	C₂³×C₃	—	10	2	no	—
455	C₂³×C₃	—	10	2	no	—
456	C₂³×C₃	—	10	4	no	—
457	C₂³×C₃	—	10	4	no	—
458	C₂³×C₃	—	10	4	no	—
459	C₂³×C₃	C₂³×D₅	1	2	yes	—
460	C₂³×C₃	C₂³×D₅	1	1	yes	characteristic
461	C₃×Q₈	—	2	8	no	—
462	C₃×Q₈	—	5	16	no	—
463	C₃×Q₈	—	10	16	no	—
464	C₃×D₄	—	10	16	no	—
465	C₃×D₄	—	10	16	no	—
466	C₃×D₄	—	10	8	no	—
467	C₃×D₄	—	10	8	no	—
468	C₂×C₃×C₄	—	2	2	no	—
469	C₂×C₃×C₄	—	2	4	no	—
470	C₂×C₃×C₄	—	2	4	no	—
471	C₂×C₃×C₄	—	2	8	no	—
472	C₂×C₃×C₄	—	5	8	no	—
473	C₂×C₃×C₄	—	5	4	no	—
474	C₂×C₃×C₄	—	5	2	no	—
475	C₂×C₃×C₄	—	5	2	no	—
476	C₂×C₃×C₄	—	5	2	no	—
477	C₂×C₃×C₄	—	10	16	no	—
478	C₂×C₃×C₄	—	10	16	no	—
479	C₂×C₃×C₄	—	10	2	no	—
480	C₂×C₃×C₄	—	10	2	no	—
481	C₂×C₃×C₄	—	10	2	no	—
482	C₂×C₃×C₄	—	10	8	no	—
483	C₂×C₃×C₄	D₄×D₅	1	4	yes	—
484	C₂×C₃×C₄	C₂³×D₅	1	4	yes	—
485	C₂×C₃×C₄	C₂³×D₅	1	4	yes	—
486	C₂×C₃×C₄	C₂³×D₅	1	2	yes	—
487	C₂×C₃×C₄	C₂³×D₅	1	2	yes	—
488	C₂²×C₅	—	2	2	no	—
489	C₂²×C₅	—	2	2	no	—
490	C₂²×C₅	—	2	2	no	—
491	C₂²×C₅	—	4	4	no	—
492	C₂²×C₅	C₂²×C₃×D₄	1	2	yes	—
493	C₂²×C₅	C₂×C₃×#41	1	2	yes	—
494	C₂²×C₅	C₃×2₋¹⁺⁴	1	2	yes	—
495	C₂²×C₅	C₂⁵×C₃	1	1	yes	characteristic
496	C₂×D₅	—	2	2	no	—
497	C₂×D₅	—	2	2	no	—
498	C₂×D₅	—	2	8	no	—
499	C₂×D₅	—	2	4	no	—
500	C₂×D₅	—	2	4	no	—
501	C₂×D₅	—	2	4	no	—
502	C₂×D₅	—	4	8	no	—
503	C₂×D₅	C₂×C₃×#41	1	4	yes	complemented
504	C₄×C₅	—	2	4	no	—
505	C₄×C₅	—	2	2	no	—
506	C₄×C₅	—	2	4	no	—
507	C₄×C₅	C₂²×C₃×D₄	1	4	yes	—
508	Dic₅	—	2	2	no	—
509	Dic₅	—	2	2	no	—
510	Dic₅	—	2	2	no	—
511	Dic₅	C₂²×C₃×D₄	1	8	yes	—
512	C₂⁴	—	5	1	no	—
513	C₂⁴	—	5	2	no	—
514	C₂⁴	C₂²×C₃×D₅	1	1	yes	characteristic
515	#41	—	10	32	no	—
516	C₂×Q₈	—	2	4	no	—
517	C₂×Q₈	—	5	8	no	—
518	C₂×Q₈	—	5	4	no	—
519	C₂×Q₈	—	5	8	no	—
520	C₂×Q₈	—	10	8	no	—
521	C₂×Q₈	—	10	32	no	—
522	C₂×Q₈	C₂²×C₃×D₅	1	4	yes	—
523	C₂×D₄	—	5	4	no	—
524	C₂×D₄	—	5	8	no	—
525	C₂×D₄	—	5	8	no	—
526	C₂×D₄	—	5	4	no	—
527	C₂×D₄	—	10	8	no	—
528	C₂×D₄	—	10	8	no	—
529	C₂×D₄	—	10	4	no	—
530	C₂×D₄	—	10	4	no	—
531	C₂²×C₄	—	2	4	no	—
532	C₂²×C₄	—	5	2	no	—
533	C₂²×C₄	—	5	1	no	—
534	C₂²×C₄	—	5	1	no	—
535	C₂²×C₄	—	5	1	no	—
536	C₂²×C₄	—	5	8	no	—
537	C₂²×C₄	—	5	4	no	—
538	C₂²×C₄	—	5	8	no	—
539	C₂²×C₄	—	10	8	no	—
540	C₂²×C₄	—	10	8	no	—
541	C₂²×C₄	—	10	4	no	—
542	C₂²×C₄	C₂²×C₃×D₅	1	2	yes	—
543	C₂²×C₄	C₂²×C₃×D₅	1	2	yes	—
544	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
545	C₂²×C₄	C₂²×C₃×D₅	1	4	yes	—
546	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
547	C₄⋊C₄	—	5	4	no	—
548	C₄⋊C₄	—	5	8	no	—
549	C₄⋊C₄	C₂²×C₃×D₅	1	8	yes	—
550	C₄⋊C₄	C₂²×C₃×D₅	1	4	yes	—
551	C₂²⋊C₄	—	5	4	no	—
552	C₂²⋊C₄	—	5	4	no	—
553	C₂²⋊C₄	—	5	8	no	—
554	C₂²⋊C₄	—	5	8	no	—
555	C₂²⋊C₄	—	5	8	no	—
556	C₂²⋊C₄	C₂²×C₃×D₅	1	8	yes	—
557	C₄²	—	5	8	no	—
558	C₄²	—	5	4	no	—
559	C₃×C₅	C₂×#548	1	1	yes	characteristic, complemented, Hall
560	C₂²×C₃	—	2	2	no	—
561	C₂²×C₃	—	2	2	no	—
562	C₂²×C₃	—	2	2	no	—
563	C₂²×C₃	—	4	4	no	—
564	C₂²×C₃	—	5	4	no	—
565	C₂²×C₃	—	10	8	no	—
566	C₂²×C₃	—	10	2	no	—
567	C₂²×C₃	—	10	2	no	—
568	C₂²×C₃	—	10	4	no	—
569	C₂²×C₃	—	10	4	no	—
570	C₂²×C₃	—	10	4	no	—
571	C₂²×C₃	—	20	8	no	—
572	C₂²×C₃	C₂×D₄×D₅	1	2	yes	—
573	C₂²×C₃	#4147	1	2	yes	—
574	C₂²×C₃	D₅×#41	1	2	yes	—
575	C₂²×C₃	C₂⁴×D₅	1	1	yes	characteristic
576	C₃×C₄	—	2	2	no	—
577	C₃×C₄	—	2	4	no	—
578	C₃×C₄	—	2	4	no	—
579	C₃×C₄	—	5	8	no	—
580	C₃×C₄	—	10	2	no	—
581	C₃×C₄	—	10	2	no	—
582	C₃×C₄	—	10	2	no	—
583	C₃×C₄	C₂×D₄×D₅	1	4	yes	—
584	C₂×C₅	—	4	2	no	—
585	C₂×C₅	C₃×#548	1	4	yes	complemented
586	C₂×C₅	C₂³×C₃×D₄	1	1	yes	characteristic
587	C₂×C₅	C₂²×C₃×#41	1	1	yes	characteristic
588	C₂×C₅	C₂×C₃×2₋¹⁺⁴	1	1	yes	characteristic
589	D₅	—	2	4	no	—
590	D₅	—	4	4	no	—
591	C₂³	—	2	2	no	—
592	C₂³	—	2	2	no	—
593	C₂³	—	2	2	no	—
594	C₂³	—	5	4	no	—
595	C₂³	—	5	2	no	—
596	C₂³	—	5	4	no	—
597	C₂³	—	10	2	no	—
598	C₂³	—	10	2	no	—
599	C₂³	—	10	4	no	—
600	C₂³	—	10	4	no	—
601	C₂³	—	10	4	no	—
602	C₂³	C₂³×C₃×D₅	1	2	yes	—
603	C₂³	C₂³×C₃×D₅	1	1	yes	characteristic
604	Q₈	—	2	8	no	—
605	Q₈	—	5	16	no	—
606	Q₈	—	10	16	no	—
607	D₄	—	10	16	no	—
608	D₄	—	10	16	no	—
609	D₄	—	10	8	no	—
610	D₄	—	10	8	no	—
611	C₂×C₄	—	2	2	no	—
612	C₂×C₄	—	2	4	no	—
613	C₂×C₄	—	2	4	no	—
614	C₂×C₄	—	2	8	no	—
615	C₂×C₄	—	5	8	no	—
616	C₂×C₄	—	5	4	no	—
617	C₂×C₄	—	5	2	no	—
618	C₂×C₄	—	5	2	no	—
619	C₂×C₄	—	5	2	no	—
620	C₂×C₄	—	10	16	no	—
621	C₂×C₄	—	10	16	no	—
622	C₂×C₄	—	10	2	no	—
623	C₂×C₄	—	10	2	no	—
624	C₂×C₄	—	10	2	no	—
625	C₂×C₄	—	10	8	no	—
626	C₂×C₄	C₃×D₄×D₅	1	4	yes	—
627	C₂×C₄	C₂³×C₃×D₅	1	4	yes	—
628	C₂×C₄	C₂³×C₃×D₅	1	4	yes	—
629	C₂×C₄	C₂³×C₃×D₅	1	2	yes	—
630	C₂×C₄	C₂³×C₃×D₅	1	2	yes	—
631	C₂×C₃	—	4	2	no	—
632	C₂×C₃	—	10	4	no	—
633	C₂×C₃	—	20	4	no	—
634	C₂×C₃	#65967	1	4	yes	direct product factor
635	C₂×C₃	C₂²×D₄×D₅	1	1	yes	characteristic
636	C₂×C₃	C₂×#4147	1	1	yes	characteristic
637	C₂×C₃	C₂×D₅×#41	1	1	yes	characteristic
638	C₅	C₂×C₃×#548	1	1	yes	characteristic, complemented, Sylow
639	C₂²	—	2	2	no	—
640	C₂²	—	2	2	no	—
641	C₂²	—	2	2	no	—
642	C₂²	—	4	4	no	—
643	C₂²	—	5	4	no	—
644	C₂²	—	10	2	no	—
645	C₂²	—	10	2	no	—
646	C₂²	—	10	8	no	—
647	C₂²	—	10	4	no	—
648	C₂²	—	10	4	no	—
649	C₂²	—	10	4	no	—
650	C₂²	—	20	8	no	—
651	C₂²	C₂×C₃×D₄×D₅	1	2	yes	—
652	C₂²	C₃×#4147	1	2	yes	—
653	C₂²	C₃×D₅×#41	1	2	yes	—
654	C₂²	C₂⁴×C₃×D₅	1	1	yes	characteristic
655	C₄	—	2	2	no	—
656	C₄	—	2	4	no	—
657	C₄	—	2	4	no	—
658	C₄	—	5	8	no	—
659	C₄	—	10	2	no	—
660	C₄	—	10	2	no	—
661	C₄	—	10	2	no	—
662	C₄	C₂×C₃×D₄×D₅	1	4	yes	—
663	C₃	C₂×#65967	1	1	yes	characteristic, direct product factor, Sylow
664	C₂	—	4	2	no	—
665	C₂	—	10	4	no	—
666	C₂	—	20	4	no	—
667	C₂	C₃×#65967	1	4	yes	direct product factor
668	C₂	C₂²×C₃×D₄×D₅	1	1	yes	characteristic
669	C₂	C₂×C₃×#4147	1	1	yes	characteristic
670	C₂	C₂×C₃×D₅×#41	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₃	C₂³×D₅
commutator, cosocle	C₂²×C₅	C₂⁵×C₃
Frattini	C₂²	C₂⁴×C₃×D₅
Fitting	C₂×C₃×C₅×C₂²⋊Q₈	C₂
socle	C₂³×C₃×C₅	C₂⁴
hypercenter	C₂×C₃×C₂²⋊Q₈	D₅
hypocenter	C₅	C₂×C₃×#548
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₂²⋊Q₈

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂²×C₃×Q₈×D₅	C₂	semidirect
2. C₂²×C₃×#709	C₂	semidirect
3. C₂×C₃×C₅×C₂²⋊Q₈	C₂	semidirect
4. C₂×C₃×#4093	C₂	semidirect
5. C₂×C₃×C₄×C₂²⋊D₅	C₂	semidirect
6. C₃×#65967	C₂	direct (N × H)
7. C₂×C₃×#4042	C₂	semidirect
8. C₂×C₃×#4041	C₂	semidirect
9. C₂×C₃×#4039	C₂	semidirect
10. C₂×C₃×#4033	C₂	semidirect
11. C₂×C₃×#4031	C₂	semidirect
12. C₂×C₃×#4030	C₂	semidirect
13. C₂×#65967	C₃	direct (N × H)
19. C₂×C₃×Q₈×D₅	C₂²	semidirect
21. C₂×C₃×#709	C₂²	semidirect
29. C₃×C₅×C₂²⋊Q₈	C₂²	semidirect
34. C₃×#4093	C₂²	semidirect
35. C₃×C₄×C₂²⋊D₅	C₂²	semidirect
44. C₃×#4042	C₂²	semidirect
45. C₃×#4041	C₂²	semidirect
46. C₃×#4039	C₂²	semidirect
47. C₃×#4033	C₂²	semidirect
48. C₃×#4031	C₂²	semidirect
49. C₃×#4030	C₂²	semidirect
51. C₂²×Q₈×D₅	C₂×C₃	semidirect
52. C₂²×#709	C₂×C₃	semidirect
53. C₂×C₅×C₂²⋊Q₈	C₂×C₃	semidirect
54. C₂×#4093	C₂×C₃	semidirect
55. C₂×C₄×C₂²⋊D₅	C₂×C₃	semidirect
56. #65967	C₂×C₃	direct (N × H)
57. C₂×#4042	C₂×C₃	semidirect
58. C₂×#4041	C₂×C₃	semidirect
59. C₂×#4039	C₂×C₃	semidirect
60. C₂×#4033	C₂×C₃	semidirect
61. C₂×#4031	C₂×C₃	semidirect
62. C₂×#4030	C₂×C₃	semidirect
95. C₂×C₃×Dic₁₀	D₄	semidirect
116. C₂×C₃×C₂²⋊Q₈	D₅	semidirect
127. C₂×Q₈×D₅	C₂²×C₃	semidirect
129. C₂×#709	C₂²×C₃	semidirect
137. C₅×C₂²⋊Q₈	C₂²×C₃	semidirect
142. #4093	C₂²×C₃	semidirect
143. C₄×C₂²⋊D₅	C₂²×C₃	semidirect
152. #4042	C₂²×C₃	semidirect
153. #4041	C₂²×C₃	semidirect
154. #4039	C₂²×C₃	semidirect
155. #4033	C₂²×C₃	semidirect
156. #4031	C₂²×C₃	semidirect
157. #4030	C₂²×C₃	semidirect
169. C₂²×C₃×D₅	#41	semidirect
198. C₃×Dic₁₀	C₂×D₄	semidirect
217. C₃×C₂²⋊Q₈	C₂×D₅	semidirect
267. C₂×Dic₁₀	C₃×D₄	semidirect
288. C₂×C₂²⋊Q₈	C₃×D₅	semidirect
309. C₂×C₃×D₅	C₂×#41	semidirect
375. C₂²×D₅	C₃×#41	semidirect
404. Dic₁₀	C₂×C₃×D₄	semidirect
423. C₂²⋊Q₈	C₂×C₃×D₅	semidirect
442. C₂×C₃×C₅	#548	semidirect
503. C₂×D₅	C₂×C₃×#41	semidirect
559. C₃×C₅	C₂×#548	semidirect
585. C₂×C₅	C₃×#548	semidirect
634. C₂×C₃	#65967	direct (N × H)
638. C₅	C₂×C₃×#548	semidirect
663. C₃	C₂×#65967	direct (N × H)
667. C₂	C₃×#65967	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₃×Q₈×D₅	2
C₂²×C₃×#709	2
C₂×C₃×C₅×C₂²⋊Q₈	2
C₂×C₃×#4093	2
C₂×C₃×C₄×C₂²⋊D₅	2
C₃×#65967	2
C₂×C₃×#4042	2
C₂×C₃×#4041	2
C₂×C₃×#4039	2
C₂×C₃×#4033	2
C₂×C₃×#4031	2
C₂×C₃×#4030	2
C₂×#65967	3
C₂×C₃×#548	5

Subgroups — C2×C3×#65967 #14482809

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#65967 #14482809