Subgroups — C₂×C₃×#66067 #14481332

Order 1920.

2264 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×#4093	C₂	1	2	yes	maximal, complemented
3	C₂×C₃×#4092	C₂	1	2	yes	maximal, complemented
4	C₃×#66067	C₂	1	16	yes	maximal, direct product factor
5	C₂×C₃×#4044	C₂	1	2	yes	maximal, complemented
6	C₂×C₃×#4043	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃×#4041	C₂	1	1	yes	maximal, characteristic
8	C₂×C₃×#4040	C₂	1	2	yes	maximal, complemented
9	C₂×C₃×#4039	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃×#4033	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₃×#4021	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃×#4018	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#66067	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
14	C₂²×C₃×C₅×Q₈	C₂²	1	2	yes	—
15	C₂²×C₃×D₂₀	C₂²	1	1	yes	characteristic
16	C₂²×C₃×D₂₀	C₂²	1	2	yes	—
17	C₂²×C₃×C₄×D₅	C₂²	1	1	yes	characteristic
18	C₂²×C₃×C₄×D₅	C₂²	1	2	yes	—
19	C₂²×C₃×Dic₁₀	C₂²	1	1	yes	characteristic
20	C₃×C₅×C₄⋊Q₈	C₂²	1	8	yes	complemented
21	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
22	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
23	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
24	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
25	C₃×#4093	C₂²	1	16	yes	complemented
26	C₃×#4092	C₂²	1	16	yes	complemented
27	C₂×C₃×#685	C₂²	1	2	yes	—
28	C₂×C₃×#685	C₂²	1	2	yes	—
29	C₂×C₃×#685	C₂²	1	2	yes	—
30	C₂×C₃×#685	C₂²	1	2	yes	—
31	C₂×C₃×#685	C₂²	1	2	yes	—
32	C₂×C₃×#685	C₂²	1	1	yes	characteristic
33	C₂×C₃×#685	C₂²	1	1	yes	characteristic
34	C₂×C₃×C₄⋊Dic₅	C₂²	1	2	yes	—
35	C₂×C₃×#683	C₂²	1	1	yes	characteristic
36	C₂×C₃×#683	C₂²	1	1	yes	characteristic
37	C₂×C₃×#683	C₂²	1	2	yes	—
38	C₂×C₃×C₄×Dic₅	C₂²	1	2	yes	—
39	C₂×C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
40	C₃×#4044	C₂²	1	16	yes	complemented
41	C₃×#4043	C₂²	1	8	yes	complemented
42	C₃×#4041	C₂²	1	8	yes	—
43	C₃×#4040	C₂²	1	16	yes	complemented
44	C₃×#4039	C₂²	1	8	yes	complemented
45	C₃×#4033	C₂²	1	8	yes	complemented
46	C₃×#4021	C₂²	1	8	yes	complemented
47	C₃×#4018	C₂²	1	8	yes	complemented
48	C₂×C₃×#542	—	5	1	no	maximal, Hall
49	C₂×C₅×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
50	C₂×#4093	C₂×C₃	1	2	yes	complemented
51	C₂×#4092	C₂×C₃	1	2	yes	complemented
52	#66067	C₂×C₃	1	16	yes	direct product factor
53	C₂×#4044	C₂×C₃	1	2	yes	complemented
54	C₂×#4043	C₂×C₃	1	1	yes	characteristic, complemented
55	C₂×#4041	C₂×C₃	1	1	yes	characteristic
56	C₂×#4040	C₂×C₃	1	2	yes	complemented
57	C₂×#4039	C₂×C₃	1	1	yes	characteristic, complemented
58	C₂×#4033	C₂×C₃	1	1	yes	characteristic, complemented
59	C₂×#4021	C₂×C₃	1	1	yes	characteristic, complemented
60	C₂×#4018	C₂×C₃	1	1	yes	characteristic, complemented
61	C₂³×C₃×D₅	C₂³	1	1	yes	characteristic
62	C₂³×C₃×D₅	C₂³	1	2	yes	—
63	C₂×C₃×C₅×Q₈	—	4	4	no	—
64	C₂×C₃×C₅×Q₈	C₂³	1	8	yes	—
65	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
66	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
67	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
68	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
69	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
70	C₂²×C₃×Dic₅	C₂³	1	2	yes	—
71	C₂²×C₃×Dic₅	C₂³	1	1	yes	characteristic
72	C₂²×C₃×Dic₅	C₂³	1	1	yes	characteristic
73	C₂²×C₃×Dic₅	C₂³	1	1	yes	characteristic
74	C₂×C₃×D₂₀	—	2	4	no	—
75	C₂×C₃×D₂₀	—	4	4	no	—
76	C₂×C₃×D₂₀	C₂³	1	4	yes	—
77	C₂×C₃×D₂₀	C₂³	1	8	yes	—
78	C₂×C₃×C₄×D₅	—	2	4	no	—
79	C₂×C₃×C₄×D₅	—	4	4	no	—
80	C₂×C₃×C₄×D₅	C₂³	1	4	yes	—
81	C₂×C₃×C₄×D₅	C₂³	1	8	yes	—
82	C₂×C₃×Dic₁₀	—	2	4	no	—
83	C₂×C₃×Dic₁₀	C₂³	1	4	yes	—
84	C₃×C₅×C₄⋊C₄	C₂³	1	4	yes	—
85	C₃×C₅×C₄⋊C₄	C₂³	1	4	yes	—
86	C₃×C₅×C₄⋊C₄	C₂³	1	8	yes	—
87	C₃×C₄²×C₅	C₂³	1	4	yes	—
88	C₃×#685	C₂³	1	8	yes	—
89	C₃×#685	C₂³	1	8	yes	—
90	C₃×#685	C₂³	1	4	yes	—
91	C₃×#685	C₂³	1	8	yes	—
92	C₃×#685	C₂³	1	8	yes	—
93	C₃×#685	C₂³	1	4	yes	—
94	C₃×#685	C₂³	1	8	yes	—
95	C₃×C₄⋊Dic₅	C₂³	1	8	yes	—
96	C₃×#683	C₂³	1	4	yes	—
97	C₃×#683	C₂³	1	8	yes	—
98	C₃×#683	C₂³	1	4	yes	—
99	C₃×C₄×Dic₅	C₂³	1	8	yes	—
100	C₃×C₄×Dic₅	C₂³	1	4	yes	—
101	C₃×#542	—	5	16	no	—
102	C₂×C₃×C₄⋊Q₈	D₅	1	1	yes	characteristic, complemented
103	C₂×C₃×#126	—	5	2	no	—
104	C₂×C₃×#124	—	5	2	no	—
105	C₂×C₃×#124	—	5	1	no	—
106	C₂×C₃×#123	—	5	2	no	—
107	C₂×C₃×C₂²⋊Q₈	—	5	2	no	—
108	C₂×C₃×C₂²⋊Q₈	—	5	1	no	—
109	C₂×C₃×#121	—	5	1	no	—
110	C₂×C₃×C₄×Q₈	—	5	1	no	—
111	C₂×C₃×C₄×D₄	—	5	1	no	—
112	C₂×C₃×#117	—	5	1	no	—
113	C₂²×C₅×Q₈	C₂²×C₃	1	2	yes	—
114	C₂²×D₂₀	C₂²×C₃	1	1	yes	characteristic
115	C₂²×D₂₀	C₂²×C₃	1	2	yes	—
116	C₂²×C₄×D₅	C₂²×C₃	1	1	yes	characteristic
117	C₂²×C₄×D₅	C₂²×C₃	1	2	yes	—
118	C₂²×Dic₁₀	C₂²×C₃	1	1	yes	characteristic
119	C₅×C₄⋊Q₈	C₂²×C₃	1	8	yes	complemented
120	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
121	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
122	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
123	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
124	#4093	C₂²×C₃	1	16	yes	complemented
125	#4092	C₂²×C₃	1	16	yes	complemented
126	C₂×#685	C₂²×C₃	1	1	yes	characteristic
127	C₂×#685	C₂²×C₃	1	1	yes	characteristic
128	C₂×#685	C₂²×C₃	1	2	yes	—
129	C₂×#685	C₂²×C₃	1	2	yes	—
130	C₂×#685	C₂²×C₃	1	2	yes	—
131	C₂×#685	C₂²×C₃	1	2	yes	—
132	C₂×#685	C₂²×C₃	1	2	yes	—
133	C₂×C₄⋊Dic₅	C₂²×C₃	1	2	yes	—
134	C₂×#683	C₂²×C₃	1	1	yes	characteristic
135	C₂×#683	C₂²×C₃	1	1	yes	characteristic
136	C₂×#683	C₂²×C₃	1	2	yes	—
137	C₂×C₄×Dic₅	C₂²×C₃	1	1	yes	characteristic
138	C₂×C₄×Dic₅	C₂²×C₃	1	2	yes	—
139	#4044	C₂²×C₃	1	16	yes	complemented
140	#4043	C₂²×C₃	1	8	yes	complemented
141	#4041	C₂²×C₃	1	8	yes	—
142	#4040	C₂²×C₃	1	16	yes	complemented
143	#4039	C₂²×C₃	1	8	yes	complemented
144	#4033	C₂²×C₃	1	8	yes	complemented
145	#4021	C₂²×C₃	1	8	yes	complemented
146	#4018	C₂²×C₃	1	8	yes	complemented
147	C₂×#542	—	5	1	no	Sylow
148	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
149	C₂²×C₃×D₅	—	2	2	no	—
150	C₂²×C₃×D₅	—	2	4	no	—
151	C₂²×C₃×D₅	—	2	4	no	—
152	C₂²×C₃×D₅	—	2	2	no	—
153	C₂²×C₃×D₅	—	2	2	no	—
154	C₂²×C₃×D₅	—	2	4	no	—
155	C₂²×C₃×D₅	C₂⁴	1	2	yes	—
156	C₂²×C₃×D₅	C₂⁴	1	4	yes	—
157	C₃×C₅×Q₈	—	4	8	no	—
158	C₂×C₃×C₄×C₅	—	2	4	no	—
159	C₂×C₃×C₄×C₅	—	2	4	no	—
160	C₂×C₃×C₄×C₅	—	2	2	no	—
161	C₂×C₃×C₄×C₅	—	2	2	no	—
162	C₂×C₃×C₄×C₅	#41	1	4	yes	—
163	C₂×C₃×C₄×C₅	C₂⁴	1	4	yes	—
164	C₂×C₃×C₄×C₅	C₂⁴	1	2	yes	—
165	C₂×C₃×C₄×C₅	C₂⁴	1	2	yes	—
166	C₂×C₃×C₄×C₅	C₂⁴	1	4	yes	—
167	C₂×C₃×C₄×C₅	C₂⁴	1	2	yes	—
168	C₂×C₃×Dic₅	—	2	4	no	—
169	C₂×C₃×Dic₅	—	2	2	no	—
170	C₂×C₃×Dic₅	—	2	2	no	—
171	C₂×C₃×Dic₅	—	2	2	no	—
172	C₂×C₃×Dic₅	C₂⁴	1	2	yes	—
173	C₂×C₃×Dic₅	C₂⁴	1	2	yes	—
174	C₂×C₃×Dic₅	C₂⁴	1	4	yes	—
175	C₂×C₃×Dic₅	C₂⁴	1	2	yes	—
176	C₃×D₂₀	—	2	8	no	—
177	C₃×D₂₀	—	4	8	no	—
178	C₃×C₄×D₅	—	2	8	no	—
179	C₃×C₄×D₅	—	4	8	no	—
180	C₃×Dic₁₀	—	2	8	no	—
181	C₂²×C₃×Q₈	—	5	1	no	—
182	C₂²×C₃×Q₈	C₂×D₅	1	2	yes	—
183	C₂²×C₃×D₄	—	5	1	no	—
184	C₂²×C₃×D₄	—	5	2	no	—
185	C₂³×C₃×C₄	—	5	1	no	—
186	C₂³×C₃×C₄	—	5	2	no	—
187	C₃×C₄⋊Q₈	C₂×D₅	1	8	yes	complemented
188	C₃×#126	—	5	16	no	—
189	C₃×#124	—	5	16	no	—
190	C₃×#124	—	5	8	no	—
191	C₃×#123	—	5	16	no	—
192	C₃×C₂²⋊Q₈	—	5	16	no	—
193	C₃×C₂²⋊Q₈	—	5	8	no	—
194	C₃×#121	—	5	8	no	—
195	C₃×C₄×Q₈	—	5	8	no	—
196	C₃×C₄×D₄	—	5	8	no	—
197	C₃×#117	—	5	8	no	—
198	C₂×C₃×C₄⋊C₄	—	5	1	no	—
199	C₂×C₃×C₄⋊C₄	—	5	1	no	—
200	C₂×C₃×C₄⋊C₄	—	5	2	no	—
201	C₂×C₃×C₄⋊C₄	—	5	2	no	—
202	C₂×C₃×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
203	C₂×C₃×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
204	C₂×C₃×C₄⋊C₄	C₂×D₅	1	2	yes	—
205	C₂×C₃×C₂²⋊C₄	—	5	1	no	—
206	C₂×C₃×C₂²⋊C₄	—	5	1	no	—
207	C₂×C₃×C₂²⋊C₄	—	5	2	no	—
208	C₂×C₃×C₂²⋊C₄	—	5	2	no	—
209	C₂×C₃×C₂²⋊C₄	—	5	2	no	—
210	C₂×C₃×C₂²⋊C₄	—	5	2	no	—
211	C₂×C₃×C₂²⋊C₄	—	5	2	no	—
212	C₂×C₃×C₄²	—	5	1	no	—
213	C₂×C₃×C₄²	—	5	2	no	—
214	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
215	C₂³×D₅	C₂³×C₃	1	1	yes	characteristic
216	C₂³×D₅	C₂³×C₃	1	2	yes	—
217	C₂×C₅×Q₈	—	4	4	no	—
218	C₂×C₅×Q₈	C₂³×C₃	1	8	yes	—
219	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
220	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
221	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
222	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
223	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
224	C₂²×Dic₅	C₂³×C₃	1	2	yes	—
225	C₂²×Dic₅	C₂³×C₃	1	1	yes	characteristic
226	C₂²×Dic₅	C₂³×C₃	1	1	yes	characteristic
227	C₂²×Dic₅	C₂³×C₃	1	1	yes	characteristic
228	C₂×D₂₀	—	2	4	no	—
229	C₂×D₂₀	—	4	4	no	—
230	C₂×D₂₀	C₂³×C₃	1	4	yes	—
231	C₂×D₂₀	C₂³×C₃	1	8	yes	—
232	C₂×C₄×D₅	—	2	4	no	—
233	C₂×C₄×D₅	—	4	4	no	—
234	C₂×C₄×D₅	C₂³×C₃	1	4	yes	—
235	C₂×C₄×D₅	C₂³×C₃	1	8	yes	—
236	C₂×Dic₁₀	—	2	4	no	—
237	C₂×Dic₁₀	C₂³×C₃	1	4	yes	—
238	C₅×C₄⋊C₄	C₂³×C₃	1	4	yes	—
239	C₅×C₄⋊C₄	C₂³×C₃	1	4	yes	—
240	C₅×C₄⋊C₄	C₂³×C₃	1	8	yes	—
241	C₄²×C₅	C₂³×C₃	1	4	yes	—
242	#685	C₂³×C₃	1	4	yes	—
243	#685	C₂³×C₃	1	4	yes	—
244	#685	C₂³×C₃	1	8	yes	—
245	#685	C₂³×C₃	1	8	yes	—
246	#685	C₂³×C₃	1	8	yes	—
247	#685	C₂³×C₃	1	8	yes	—
248	#685	C₂³×C₃	1	8	yes	—
249	C₄⋊Dic₅	C₂³×C₃	1	8	yes	—
250	#683	C₂³×C₃	1	4	yes	—
251	#683	C₂³×C₃	1	4	yes	—
252	#683	C₂³×C₃	1	8	yes	—
253	C₄×Dic₅	C₂³×C₃	1	4	yes	—
254	C₄×Dic₅	C₂³×C₃	1	8	yes	—
255	#542	—	5	16	no	—
256	C₂×C₄⋊Q₈	C₃×D₅	1	1	yes	characteristic, complemented
257	C₂×#126	—	5	2	no	—
258	C₂×#124	—	5	2	no	—
259	C₂×#124	—	5	1	no	—
260	C₂×#123	—	5	2	no	—
261	C₂×C₂²⋊Q₈	—	5	2	no	—
262	C₂×C₂²⋊Q₈	—	5	1	no	—
263	C₂×#121	—	5	1	no	—
264	C₂×C₄×Q₈	—	5	1	no	—
265	C₂×C₄×D₄	—	5	1	no	—
266	C₂×#117	—	5	1	no	—
267	C₂²×C₃×C₅	C₂×#41	1	2	yes	—
268	C₂²×C₃×C₅	H(5,𝔽₂)	1	2	yes	—
269	C₂²×C₃×C₅	2₋¹⁺⁴	1	2	yes	—
270	C₂²×C₃×C₅	C₂⁵	1	1	yes	characteristic
271	C₂×C₃×D₅	—	2	2	no	—
272	C₂×C₃×D₅	—	2	4	no	—
273	C₂×C₃×D₅	—	2	4	no	—
274	C₂×C₃×D₅	—	2	2	no	—
275	C₂×C₃×D₅	—	2	2	no	—
276	C₂×C₃×D₅	—	2	4	no	—
277	C₂×C₃×D₅	—	4	4	no	—
278	C₂×C₃×D₅	—	4	8	no	—
279	C₃×C₄×C₅	—	2	4	no	—
280	C₃×C₄×C₅	—	2	4	no	—
281	C₃×C₄×C₅	—	2	2	no	—
282	C₃×C₄×C₅	—	2	2	no	—
283	C₃×C₄×C₅	C₂×#41	1	4	yes	—
284	C₃×Dic₅	—	2	4	no	—
285	C₃×Dic₅	—	2	2	no	—
286	C₃×Dic₅	—	2	2	no	—
287	C₃×Dic₅	—	2	2	no	—
288	C₂⁴×C₃	—	5	2	no	—
289	C₂⁴×C₃	—	5	1	no	—
290	C₂×C₃×Q₈	—	4	4	no	—
291	C₂×C₃×Q₈	—	5	4	no	—
292	C₂×C₃×Q₈	—	10	4	no	—
293	C₂×C₃×Q₈	C₂²×D₅	1	8	yes	—
294	C₂×C₃×D₄	—	5	4	no	—
295	C₂×C₃×D₄	—	5	8	no	—
296	C₂×C₃×D₄	—	10	4	no	—
297	C₂×C₃×D₄	—	20	4	no	—
298	C₂²×C₃×C₄	—	5	1	no	—
299	C₂²×C₃×C₄	—	5	2	no	—
300	C₂²×C₃×C₄	—	5	1	no	—
301	C₂²×C₃×C₄	—	5	1	no	—
302	C₂²×C₃×C₄	—	5	4	no	—
303	C₂²×C₃×C₄	—	5	8	no	—
304	C₂²×C₃×C₄	—	10	4	no	—
305	C₂²×C₃×C₄	—	20	4	no	—
306	C₂²×C₃×C₄	C₂²×D₅	1	2	yes	—
307	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
308	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
309	C₂²×C₃×C₄	C₂²×D₅	1	2	yes	—
310	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
311	C₃×C₄⋊C₄	—	5	4	no	—
312	C₃×C₄⋊C₄	—	5	4	no	—
313	C₃×C₄⋊C₄	—	5	8	no	—
314	C₃×C₄⋊C₄	—	5	8	no	—
315	C₃×C₄⋊C₄	C₂²×D₅	1	4	yes	—
316	C₃×C₄⋊C₄	C₂²×D₅	1	4	yes	—
317	C₃×C₄⋊C₄	C₂²×D₅	1	8	yes	—
318	C₃×C₂²⋊C₄	—	5	4	no	—
319	C₃×C₂²⋊C₄	—	5	4	no	—
320	C₃×C₂²⋊C₄	—	5	8	no	—
321	C₃×C₂²⋊C₄	—	5	8	no	—
322	C₃×C₂²⋊C₄	—	5	8	no	—
323	C₃×C₂²⋊C₄	—	5	8	no	—
324	C₃×C₂²⋊C₄	—	5	8	no	—
325	C₃×C₄²	—	5	4	no	—
326	C₃×C₄²	—	5	8	no	—
327	C₃×C₄²	C₂²×D₅	1	4	yes	—
328	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
329	C₂²×D₅	—	2	2	no	—
330	C₂²×D₅	—	2	4	no	—
331	C₂²×D₅	—	2	4	no	—
332	C₂²×D₅	—	2	2	no	—
333	C₂²×D₅	—	2	2	no	—
334	C₂²×D₅	—	2	4	no	—
335	C₂²×D₅	C₂⁴×C₃	1	2	yes	—
336	C₂²×D₅	C₂⁴×C₃	1	4	yes	—
337	C₅×Q₈	—	4	8	no	—
338	C₂×C₄×C₅	—	2	4	no	—
339	C₂×C₄×C₅	—	2	4	no	—
340	C₂×C₄×C₅	—	2	2	no	—
341	C₂×C₄×C₅	—	2	2	no	—
342	C₂×C₄×C₅	C₃×#41	1	4	yes	—
343	C₂×C₄×C₅	C₂⁴×C₃	1	4	yes	—
344	C₂×C₄×C₅	C₂⁴×C₃	1	2	yes	—
345	C₂×C₄×C₅	C₂⁴×C₃	1	2	yes	—
346	C₂×C₄×C₅	C₂⁴×C₃	1	4	yes	—
347	C₂×C₄×C₅	C₂⁴×C₃	1	2	yes	—
348	C₂×Dic₅	—	2	4	no	—
349	C₂×Dic₅	—	2	2	no	—
350	C₂×Dic₅	—	2	2	no	—
351	C₂×Dic₅	—	2	2	no	—
352	C₂×Dic₅	C₂⁴×C₃	1	2	yes	—
353	C₂×Dic₅	C₂⁴×C₃	1	2	yes	—
354	C₂×Dic₅	C₂⁴×C₃	1	4	yes	—
355	C₂×Dic₅	C₂⁴×C₃	1	2	yes	—
356	D₂₀	—	2	8	no	—
357	D₂₀	—	4	8	no	—
358	C₄×D₅	—	2	8	no	—
359	C₄×D₅	—	4	8	no	—
360	Dic₁₀	—	2	8	no	—
361	C₂²×Q₈	—	5	1	no	—
362	C₂²×Q₈	C₂×C₃×D₅	1	2	yes	—
363	C₂²×D₄	—	5	1	no	—
364	C₂²×D₄	—	5	2	no	—
365	C₂³×C₄	—	5	1	no	—
366	C₂³×C₄	—	5	2	no	—
367	C₄⋊Q₈	C₂×C₃×D₅	1	8	yes	complemented
368	#126	—	5	16	no	—
369	#124	—	5	16	no	—
370	#124	—	5	8	no	—
371	#123	—	5	16	no	—
372	C₂²⋊Q₈	—	5	16	no	—
373	C₂²⋊Q₈	—	5	8	no	—
374	#121	—	5	8	no	—
375	C₄×Q₈	—	5	8	no	—
376	C₄×D₄	—	5	8	no	—
377	#117	—	5	8	no	—
378	C₂×C₄⋊C₄	—	5	1	no	—
379	C₂×C₄⋊C₄	—	5	1	no	—
380	C₂×C₄⋊C₄	—	5	2	no	—
381	C₂×C₄⋊C₄	—	5	2	no	—
382	C₂×C₄⋊C₄	C₂×C₃×D₅	1	1	yes	characteristic
383	C₂×C₄⋊C₄	C₂×C₃×D₅	1	1	yes	characteristic
384	C₂×C₄⋊C₄	C₂×C₃×D₅	1	2	yes	—
385	C₂×C₂²⋊C₄	—	5	1	no	—
386	C₂×C₂²⋊C₄	—	5	1	no	—
387	C₂×C₂²⋊C₄	—	5	2	no	—
388	C₂×C₂²⋊C₄	—	5	2	no	—
389	C₂×C₂²⋊C₄	—	5	2	no	—
390	C₂×C₂²⋊C₄	—	5	2	no	—
391	C₂×C₂²⋊C₄	—	5	2	no	—
392	C₂×C₄²	—	5	1	no	—
393	C₂×C₄²	—	5	2	no	—
394	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
395	C₂×C₃×C₅	#542	1	4	yes	complemented
396	C₂×C₃×C₅	C₂²×#41	1	1	yes	characteristic
397	C₂×C₃×C₅	C₂×H(5,𝔽₂)	1	1	yes	characteristic
398	C₂×C₃×C₅	C₂×2₋¹⁺⁴	1	1	yes	characteristic
399	C₃×D₅	—	4	2	no	—
400	C₃×D₅	—	4	4	no	—
401	C₂³×C₃	—	5	4	no	—
402	C₂³×C₃	—	5	2	no	—
403	C₂³×C₃	—	10	4	no	—
404	C₂³×C₃	—	10	2	no	—
405	C₂³×C₃	—	10	2	no	—
406	C₂³×C₃	—	10	4	no	—
407	C₂³×C₃	—	10	4	no	—
408	C₂³×C₃	—	10	2	no	—
409	C₂³×C₃	C₂³×D₅	1	1	yes	characteristic
410	C₃×Q₈	—	4	8	no	—
411	C₃×Q₈	—	10	8	no	—
412	C₃×D₄	—	10	8	no	—
413	C₃×D₄	—	20	8	no	—
414	C₂×C₃×C₄	—	2	4	no	—
415	C₂×C₃×C₄	—	2	4	no	—
416	C₂×C₃×C₄	—	2	2	no	—
417	C₂×C₃×C₄	—	2	2	no	—
418	C₂×C₃×C₄	—	5	2	no	—
419	C₂×C₃×C₄	—	5	4	no	—
420	C₂×C₃×C₄	—	5	2	no	—
421	C₂×C₃×C₄	—	5	2	no	—
422	C₂×C₃×C₄	—	10	2	no	—
423	C₂×C₃×C₄	—	10	2	no	—
424	C₂×C₃×C₄	—	10	4	no	—
425	C₂×C₃×C₄	—	10	2	no	—
426	C₂×C₃×C₄	—	10	8	no	—
427	C₂×C₃×C₄	—	20	8	no	—
428	C₂×C₃×C₄	#713	1	4	yes	—
429	C₂×C₃×C₄	C₂³×D₅	1	4	yes	—
430	C₂×C₃×C₄	C₂³×D₅	1	2	yes	—
431	C₂×C₃×C₄	C₂³×D₅	1	2	yes	—
432	C₂×C₃×C₄	C₂³×D₅	1	4	yes	—
433	C₂×C₃×C₄	C₂³×D₅	1	2	yes	—
434	C₂²×C₅	C₂×C₃×#41	1	2	yes	—
435	C₂²×C₅	C₃×H(5,𝔽₂)	1	2	yes	—
436	C₂²×C₅	C₃×2₋¹⁺⁴	1	2	yes	—
437	C₂²×C₅	C₂⁵×C₃	1	1	yes	characteristic
438	C₂×D₅	—	2	2	no	—
439	C₂×D₅	—	2	4	no	—
440	C₂×D₅	—	2	4	no	—
441	C₂×D₅	—	2	2	no	—
442	C₂×D₅	—	2	2	no	—
443	C₂×D₅	—	2	4	no	—
444	C₂×D₅	—	4	4	no	—
445	C₂×D₅	—	4	8	no	—
446	C₄×C₅	—	2	4	no	—
447	C₄×C₅	—	2	4	no	—
448	C₄×C₅	—	2	2	no	—
449	C₄×C₅	—	2	2	no	—
450	C₄×C₅	C₂×C₃×#41	1	4	yes	—
451	Dic₅	—	2	4	no	—
452	Dic₅	—	2	2	no	—
453	Dic₅	—	2	2	no	—
454	Dic₅	—	2	2	no	—
455	C₂⁴	—	5	2	no	—
456	C₂⁴	—	5	1	no	—
457	C₂×Q₈	—	4	4	no	—
458	C₂×Q₈	—	5	4	no	—
459	C₂×Q₈	—	10	4	no	—
460	C₂×Q₈	C₂²×C₃×D₅	1	8	yes	—
461	C₂×D₄	—	5	4	no	—
462	C₂×D₄	—	5	8	no	—
463	C₂×D₄	—	10	4	no	—
464	C₂×D₄	—	20	4	no	—
465	C₂²×C₄	—	5	1	no	—
466	C₂²×C₄	—	5	2	no	—
467	C₂²×C₄	—	5	1	no	—
468	C₂²×C₄	—	5	1	no	—
469	C₂²×C₄	—	5	4	no	—
470	C₂²×C₄	—	5	8	no	—
471	C₂²×C₄	—	10	4	no	—
472	C₂²×C₄	—	20	4	no	—
473	C₂²×C₄	C₂²×C₃×D₅	1	2	yes	—
474	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
475	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
476	C₂²×C₄	C₂²×C₃×D₅	1	2	yes	—
477	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
478	C₄⋊C₄	—	5	4	no	—
479	C₄⋊C₄	—	5	4	no	—
480	C₄⋊C₄	—	5	8	no	—
481	C₄⋊C₄	—	5	8	no	—
482	C₄⋊C₄	C₂²×C₃×D₅	1	4	yes	—
483	C₄⋊C₄	C₂²×C₃×D₅	1	4	yes	—
484	C₄⋊C₄	C₂²×C₃×D₅	1	8	yes	—
485	C₂²⋊C₄	—	5	4	no	—
486	C₂²⋊C₄	—	5	4	no	—
487	C₂²⋊C₄	—	5	8	no	—
488	C₂²⋊C₄	—	5	8	no	—
489	C₂²⋊C₄	—	5	8	no	—
490	C₂²⋊C₄	—	5	8	no	—
491	C₂²⋊C₄	—	5	8	no	—
492	C₄²	—	5	4	no	—
493	C₄²	—	5	8	no	—
494	C₄²	C₂²×C₃×D₅	1	4	yes	—
495	C₃×C₅	C₂×#542	1	1	yes	characteristic, complemented, Hall
496	C₂²×C₃	—	10	4	no	—
497	C₂²×C₃	—	10	2	no	—
498	C₂²×C₃	—	10	2	no	—
499	C₂²×C₃	—	10	4	no	—
500	C₂²×C₃	—	10	4	no	—
501	C₂²×C₃	—	10	2	no	—
502	C₂²×C₃	—	20	8	no	—
503	C₂²×C₃	—	20	4	no	—
504	C₂²×C₃	#4144	1	2	yes	—
505	C₂²×C₃	C₂×#713	1	2	yes	—
506	C₂²×C₃	#4147	1	2	yes	—
507	C₂²×C₃	C₂⁴×D₅	1	1	yes	characteristic
508	C₃×C₄	—	2	4	no	—
509	C₃×C₄	—	2	4	no	—
510	C₃×C₄	—	2	2	no	—
511	C₃×C₄	—	2	2	no	—
512	C₃×C₄	—	10	2	no	—
513	C₃×C₄	—	10	2	no	—
514	C₃×C₄	—	10	4	no	—
515	C₃×C₄	—	10	2	no	—
516	C₃×C₄	C₂×#713	1	4	yes	—
517	C₂×C₅	C₃×#542	1	4	yes	complemented
518	C₂×C₅	C₂²×C₃×#41	1	1	yes	characteristic
519	C₂×C₅	C₂×C₃×H(5,𝔽₂)	1	1	yes	characteristic
520	C₂×C₅	C₂×C₃×2₋¹⁺⁴	1	1	yes	characteristic
521	D₅	—	4	2	no	—
522	D₅	—	4	4	no	—
523	C₂³	—	5	4	no	—
524	C₂³	—	5	2	no	—
525	C₂³	—	10	4	no	—
526	C₂³	—	10	2	no	—
527	C₂³	—	10	2	no	—
528	C₂³	—	10	4	no	—
529	C₂³	—	10	4	no	—
530	C₂³	—	10	2	no	—
531	C₂³	C₂³×C₃×D₅	1	1	yes	characteristic
532	Q₈	—	4	8	no	—
533	Q₈	—	10	8	no	—
534	D₄	—	10	8	no	—
535	D₄	—	20	8	no	—
536	C₂×C₄	—	2	4	no	—
537	C₂×C₄	—	2	4	no	—
538	C₂×C₄	—	2	2	no	—
539	C₂×C₄	—	2	2	no	—
540	C₂×C₄	—	5	2	no	—
541	C₂×C₄	—	5	4	no	—
542	C₂×C₄	—	5	2	no	—
543	C₂×C₄	—	5	2	no	—
544	C₂×C₄	—	10	2	no	—
545	C₂×C₄	—	10	2	no	—
546	C₂×C₄	—	10	4	no	—
547	C₂×C₄	—	10	2	no	—
548	C₂×C₄	—	10	8	no	—
549	C₂×C₄	—	20	8	no	—
550	C₂×C₄	C₃×#713	1	4	yes	—
551	C₂×C₄	C₂³×C₃×D₅	1	4	yes	—
552	C₂×C₄	C₂³×C₃×D₅	1	2	yes	—
553	C₂×C₄	C₂³×C₃×D₅	1	2	yes	—
554	C₂×C₄	C₂³×C₃×D₅	1	4	yes	—
555	C₂×C₄	C₂³×C₃×D₅	1	2	yes	—
556	C₂×C₃	—	20	4	no	—
557	C₂×C₃	—	20	2	no	—
558	C₂×C₃	#66067	1	4	yes	direct product factor
559	C₂×C₃	C₂×#4144	1	1	yes	characteristic
560	C₂×C₃	C₂²×#713	1	1	yes	characteristic
561	C₂×C₃	C₂×#4147	1	1	yes	characteristic
562	C₅	C₂×C₃×#542	1	1	yes	characteristic, complemented, Sylow
563	C₂²	—	10	4	no	—
564	C₂²	—	10	2	no	—
565	C₂²	—	10	2	no	—
566	C₂²	—	10	4	no	—
567	C₂²	—	10	4	no	—
568	C₂²	—	10	2	no	—
569	C₂²	—	20	8	no	—
570	C₂²	—	20	4	no	—
571	C₂²	C₃×#4144	1	2	yes	—
572	C₂²	C₂×C₃×#713	1	2	yes	—
573	C₂²	C₃×#4147	1	2	yes	—
574	C₂²	C₂⁴×C₃×D₅	1	1	yes	characteristic
575	C₄	—	2	4	no	—
576	C₄	—	2	4	no	—
577	C₄	—	2	2	no	—
578	C₄	—	2	2	no	—
579	C₄	—	10	2	no	—
580	C₄	—	10	2	no	—
581	C₄	—	10	4	no	—
582	C₄	—	10	2	no	—
583	C₄	C₂×C₃×#713	1	4	yes	—
584	C₃	C₂×#66067	1	1	yes	characteristic, direct product factor, Sylow
585	C₂	—	20	4	no	—
586	C₂	—	20	2	no	—
587	C₂	C₃×#66067	1	4	yes	direct product factor
588	C₂	C₂×C₃×#4144	1	1	yes	characteristic
589	C₂	C₂²×C₃×#713	1	1	yes	characteristic
590	C₂	C₂×C₃×#4147	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×C₄⋊Q₈	C₂
socle	C₂³×C₃×C₅	C₂⁴
prime radical	C₂×C₃×#4041	C₂
hypercenter	C₂×C₃×C₄⋊Q₈	D₅
hypocenter	C₅	C₂×C₃×#542
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₅×C₄⋊Q₈	C₂	semidirect
2. C₂×C₃×#4093	C₂	semidirect
3. C₂×C₃×#4092	C₂	semidirect
4. C₃×#66067	C₂	direct (N × H)
5. C₂×C₃×#4044	C₂	semidirect
6. C₂×C₃×#4043	C₂	semidirect
8. C₂×C₃×#4040	C₂	semidirect
9. C₂×C₃×#4039	C₂	semidirect
10. C₂×C₃×#4033	C₂	semidirect
11. C₂×C₃×#4021	C₂	semidirect
12. C₂×C₃×#4018	C₂	semidirect
13. C₂×#66067	C₃	direct (N × H)
20. C₃×C₅×C₄⋊Q₈	C₂²	semidirect
25. C₃×#4093	C₂²	semidirect
26. C₃×#4092	C₂²	semidirect
40. C₃×#4044	C₂²	semidirect
41. C₃×#4043	C₂²	semidirect
43. C₃×#4040	C₂²	semidirect
44. C₃×#4039	C₂²	semidirect
45. C₃×#4033	C₂²	semidirect
46. C₃×#4021	C₂²	semidirect
47. C₃×#4018	C₂²	semidirect
49. C₂×C₅×C₄⋊Q₈	C₂×C₃	semidirect
50. C₂×#4093	C₂×C₃	semidirect
51. C₂×#4092	C₂×C₃	semidirect
52. #66067	C₂×C₃	direct (N × H)
53. C₂×#4044	C₂×C₃	semidirect
54. C₂×#4043	C₂×C₃	semidirect
56. C₂×#4040	C₂×C₃	semidirect
57. C₂×#4039	C₂×C₃	semidirect
58. C₂×#4033	C₂×C₃	semidirect
59. C₂×#4021	C₂×C₃	semidirect
60. C₂×#4018	C₂×C₃	semidirect
102. C₂×C₃×C₄⋊Q₈	D₅	semidirect
119. C₅×C₄⋊Q₈	C₂²×C₃	semidirect
124. #4093	C₂²×C₃	semidirect
125. #4092	C₂²×C₃	semidirect
139. #4044	C₂²×C₃	semidirect
140. #4043	C₂²×C₃	semidirect
142. #4040	C₂²×C₃	semidirect
143. #4039	C₂²×C₃	semidirect
144. #4033	C₂²×C₃	semidirect
145. #4021	C₂²×C₃	semidirect
146. #4018	C₂²×C₃	semidirect
187. C₃×C₄⋊Q₈	C₂×D₅	semidirect
256. C₂×C₄⋊Q₈	C₃×D₅	semidirect
367. C₄⋊Q₈	C₂×C₃×D₅	semidirect
395. C₂×C₃×C₅	#542	semidirect
495. C₃×C₅	C₂×#542	semidirect
517. C₂×C₅	C₃×#542	semidirect
558. C₂×C₃	#66067	direct (N × H)
562. C₅	C₂×C₃×#542	semidirect
584. C₃	C₂×#66067	direct (N × H)
587. C₂	C₃×#66067	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×C₅×C₄⋊Q₈	2
C₂×C₃×#4093	2
C₂×C₃×#4092	2
C₃×#66067	2
C₂×C₃×#4044	2
C₂×C₃×#4043	2
C₂×C₃×#4041	2
C₂×C₃×#4040	2
C₂×C₃×#4039	2
C₂×C₃×#4033	2
C₂×C₃×#4021	2
C₂×C₃×#4018	2
C₂×#66067	3
C₂×C₃×#542	5

Subgroups — C2×C3×#66067 #14481332

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#66067 #14481332