Subgroups — C₂×C₃×#65978 #14480774

Order 1920.

2672 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×#4093	C₂	1	2	yes	maximal, complemented
3	C₂×C₃×#4076	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₃×#685	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#65978	C₂	1	16	yes	maximal, direct product factor
6	C₂×C₃×#4044	C₂	1	2	yes	maximal, complemented
7	C₂×C₃×#4041	C₂	1	2	yes	maximal, complemented
8	C₂×C₃×#4040	C₂	1	2	yes	maximal, complemented
9	C₂×C₃×#4028	C₂	1	2	yes	maximal, complemented
10	C₂×C₃×#4027	C₂	1	2	yes	maximal, complemented
11	C₂×#65978	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
12	C₂⁴×C₃×D₅	C₂²	1	1	yes	characteristic
13	C₂²×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic, complemented
14	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
15	C₂²×C₃×C₂²⋊D₅	C₂²	1	2	yes	—
16	C₂³×C₃×Dic₅	C₂²	1	1	yes	characteristic
17	C₂²×C₃×D₂₀	C₂²	1	2	yes	—
18	C₂²×C₃×D₂₀	C₂²	1	2	yes	—
19	C₂²×C₃×D₂₀	C₂²	1	1	yes	characteristic, complemented
20	C₂²×C₃×C₄×D₅	C₂²	1	2	yes	complemented
21	C₃×C₅×C₂²⋊Q₈	C₂²	1	8	yes	complemented
22	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	complemented
23	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
24	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
25	C₃×#4093	C₂²	1	16	yes	complemented
26	C₃×#4076	C₂²	1	8	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	8	yes	complemented
32	C₂×C₃×#685	C₂²	1	2	yes	complemented
33	C₂×C₃×#685	C₂²	1	2	yes	complemented
34	C₂×C₃×C₄⋊Dic₅	C₂²	1	1	yes	characteristic, complemented
35	C₂×C₃×#683	C₂²	1	2	yes	complemented
36	C₂×C₃×C₄×Dic₅	C₂²	1	2	yes	complemented
37	C₃×#4044	C₂²	1	16	yes	complemented
38	C₃×#4041	C₂²	1	16	yes	complemented
39	C₃×#4040	C₂²	1	16	yes	complemented
40	C₃×#4028	C₂²	1	16	yes	complemented
41	C₃×#4027	C₂²	1	16	yes	complemented
42	C₂×C₃×#538	—	5	1	no	maximal, Hall
43	C₂×C₅×C₂²⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
44	C₂×#4093	C₂×C₃	1	2	yes	complemented
45	C₂×#4076	C₂×C₃	1	1	yes	characteristic, complemented
46	C₂²×#685	C₂×C₃	1	1	yes	characteristic, complemented
47	#65978	C₂×C₃	1	16	yes	direct product factor
48	C₂×#4044	C₂×C₃	1	2	yes	complemented
49	C₂×#4041	C₂×C₃	1	2	yes	complemented
50	C₂×#4040	C₂×C₃	1	2	yes	complemented
51	C₂×#4028	C₂×C₃	1	2	yes	complemented
52	C₂×#4027	C₂×C₃	1	2	yes	complemented
53	C₂⁴×C₃×C₅	C₂³	1	1	yes	characteristic
54	C₂³×C₃×D₅	—	2	4	no	—
55	C₂³×C₃×D₅	—	4	2	no	—
56	C₂³×C₃×D₅	—	4	2	no	—
57	C₂³×C₃×D₅	C₂³	1	2	yes	—
58	C₂³×C₃×D₅	C₂³	1	2	yes	—
59	C₂³×C₃×D₅	C₂³	1	4	yes	—
60	C₂×C₃×C₅×Q₈	—	4	2	no	—
61	C₂×C₃×C₅×Q₈	C₂³	1	4	yes	complemented
62	C₂²×C₃×C₄×C₅	—	4	2	no	—
63	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
64	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
65	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
66	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
67	C₂²×C₃×C₄×C₅	C₂³	1	4	yes	—
68	C₂×C₃×C₂²⋊D₅	—	2	8	no	—
69	C₂×C₃×C₂²⋊D₅	C₂³	1	8	yes	—
70	C₂²×C₃×Dic₅	—	4	2	no	—
71	C₂²×C₃×Dic₅	C₂³	1	2	yes	—
72	C₂²×C₃×Dic₅	C₂³	1	2	yes	—
73	C₂²×C₃×Dic₅	C₂³	1	4	yes	—
74	C₂×C₃×D₂₀	—	4	4	no	—
75	C₂×C₃×D₂₀	—	4	4	no	—
76	C₂×C₃×D₂₀	—	4	2	no	—
77	C₂×C₃×D₂₀	C₂³	1	8	yes	—
78	C₂×C₃×D₂₀	C₂³	1	8	yes	—
79	C₂×C₃×D₂₀	C₂³	1	4	yes	complemented
80	C₂×C₃×C₄×D₅	—	4	4	no	—
81	C₂×C₃×C₄×D₅	C₂³	1	8	yes	complemented
82	C₃×C₅×C₄⋊C₄	C₂³	1	4	yes	complemented
83	C₃×C₅×C₄⋊C₄	C₂³	1	8	yes	complemented
84	C₃×C₅×C₂²⋊C₄	C₂³	1	8	yes	—
85	C₃×#685	C₂³	1	8	yes	—
86	C₃×#685	C₂³	1	8	yes	—
87	C₃×#685	C₂³	1	8	yes	—
88	C₃×#685	C₂³	1	8	yes	—
89	C₃×#685	C₂³	1	8	yes	complemented
90	C₃×#685	C₂³	1	8	yes	complemented
91	C₃×C₄⋊Dic₅	C₂³	1	4	yes	complemented
92	C₃×#683	C₂³	1	8	yes	complemented
93	C₃×C₄×Dic₅	C₂³	1	8	yes	complemented
94	C₃×#538	—	5	16	no	—
95	C₂×C₃×#126	—	5	2	no	—
96	C₂×C₃×#124	—	5	2	no	—
97	C₂×C₃×#123	—	5	2	no	—
98	C₂×C₃×C₂²⋊Q₈	D₅	1	1	yes	characteristic, complemented
99	C₂×C₃×#121	—	5	1	no	—
100	C₂×C₃×#121	—	5	2	no	—
101	C₂×C₃×#120	—	5	2	no	—
102	C₂×C₃×C₄×D₄	—	5	2	no	—
103	C₂²×C₃×C₂²⋊C₄	—	5	1	no	—
104	C₂⁴×D₅	C₂²×C₃	1	1	yes	characteristic
105	C₂²×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic, complemented
106	C₂³×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
107	C₂²×C₂²⋊D₅	C₂²×C₃	1	2	yes	—
108	C₂³×Dic₅	C₂²×C₃	1	1	yes	characteristic
109	C₂²×D₂₀	C₂²×C₃	1	2	yes	—
110	C₂²×D₂₀	C₂²×C₃	1	2	yes	—
111	C₂²×D₂₀	C₂²×C₃	1	1	yes	characteristic, complemented
112	C₂²×C₄×D₅	C₂²×C₃	1	2	yes	complemented
113	C₅×C₂²⋊Q₈	C₂²×C₃	1	8	yes	complemented
114	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic, complemented
115	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	complemented
116	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
117	#4093	C₂²×C₃	1	16	yes	complemented
118	#4076	C₂²×C₃	1	8	yes	complemented
119	C₂×#685	C₂²×C₃	1	2	yes	—
120	C₂×#685	C₂²×C₃	1	2	yes	—
121	C₂×#685	C₂²×C₃	1	2	yes	—
122	C₂×#685	C₂²×C₃	1	2	yes	—
123	C₂×#685	C₂²×C₃	1	8	yes	complemented
124	C₂×#685	C₂²×C₃	1	2	yes	complemented
125	C₂×#685	C₂²×C₃	1	2	yes	complemented
126	C₂×C₄⋊Dic₅	C₂²×C₃	1	1	yes	characteristic, complemented
127	C₂×#683	C₂²×C₃	1	2	yes	complemented
128	C₂×C₄×Dic₅	C₂²×C₃	1	2	yes	complemented
129	#4044	C₂²×C₃	1	16	yes	complemented
130	#4041	C₂²×C₃	1	16	yes	complemented
131	#4040	C₂²×C₃	1	16	yes	complemented
132	#4028	C₂²×C₃	1	16	yes	complemented
133	#4027	C₂²×C₃	1	16	yes	complemented
134	C₂×#538	—	5	1	no	Sylow
135	C₂³×C₃×C₅	—	2	2	no	—
136	C₂³×C₃×C₅	—	2	2	no	—
137	C₂³×C₃×C₅	#41	1	4	yes	—
138	C₂³×C₃×C₅	C₂⁴	1	2	yes	—
139	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
140	C₂²×C₃×D₅	—	2	4	no	—
141	C₂²×C₃×D₅	—	2	8	no	—
142	C₂²×C₃×D₅	—	2	4	no	—
143	C₂²×C₃×D₅	—	2	4	no	—
144	C₂²×C₃×D₅	—	2	4	no	—
145	C₂²×C₃×D₅	—	2	4	no	—
146	C₂²×C₃×D₅	—	2	4	no	—
147	C₂²×C₃×D₅	—	4	4	no	—
148	C₂²×C₃×D₅	—	4	4	no	—
149	C₂²×C₃×D₅	—	4	16	no	—
150	C₂²×C₃×D₅	C₂⁴	1	4	yes	—
151	C₂²×C₃×D₅	C₂⁴	1	4	yes	—
152	C₃×C₅×Q₈	—	4	4	no	—
153	C₂×C₃×C₄×C₅	—	2	4	no	—
154	C₂×C₃×C₄×C₅	—	2	2	no	—
155	C₂×C₃×C₄×C₅	—	2	4	no	—
156	C₂×C₃×C₄×C₅	—	2	2	no	—
157	C₂×C₃×C₄×C₅	—	4	4	no	—
158	C₂×C₃×C₄×C₅	C₂⁴	1	4	yes	—
159	C₂×C₃×C₄×C₅	C₂⁴	1	4	yes	—
160	C₂×C₃×C₄×C₅	C₂⁴	1	2	yes	—
161	C₂×C₃×C₄×C₅	C₂⁴	1	2	yes	—
162	C₃×C₂²⋊D₅	—	2	16	no	—
163	C₂×C₃×Dic₅	—	2	4	no	—
164	C₂×C₃×Dic₅	—	2	4	no	—
165	C₂×C₃×Dic₅	—	4	4	no	—
166	C₂×C₃×Dic₅	C₂⁴	1	4	yes	—
167	C₂×C₃×Dic₅	C₂⁴	1	4	yes	—
168	C₃×D₂₀	—	4	8	no	—
169	C₃×D₂₀	—	4	8	no	—
170	C₃×D₂₀	—	4	4	no	—
171	C₃×C₄×D₅	—	4	8	no	—
172	C₂⁵×C₃	—	5	1	no	—
173	C₂²×C₃×Q₈	C₂×D₅	1	1	yes	characteristic, complemented
174	C₂²×C₃×D₄	—	5	2	no	—
175	C₂²×C₃×D₄	—	5	2	no	—
176	C₂²×C₃×D₄	—	5	2	no	—
177	C₂²×C₃×D₄	—	5	1	no	—
178	C₂³×C₃×C₄	—	5	1	no	—
179	C₂³×C₃×C₄	—	5	2	no	—
180	C₂³×C₃×C₄	C₂×D₅	1	1	yes	characteristic
181	C₃×#126	—	5	16	no	—
182	C₃×#124	—	5	16	no	—
183	C₃×#123	—	5	16	no	—
184	C₃×C₂²⋊Q₈	C₂×D₅	1	8	yes	complemented
185	C₃×#121	—	5	8	no	—
186	C₃×#121	—	5	16	no	—
187	C₃×#120	—	5	16	no	—
188	C₃×C₄×D₄	—	5	16	no	—
189	C₂×C₃×C₄⋊C₄	—	5	2	no	—
190	C₂×C₃×C₄⋊C₄	—	5	1	no	—
191	C₂×C₃×C₄⋊C₄	C₂×D₅	1	2	yes	complemented
192	C₂×C₃×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic, complemented
193	C₂×C₃×C₂²⋊C₄	—	5	2	no	—
194	C₂×C₃×C₂²⋊C₄	—	5	2	no	—
195	C₂×C₃×C₂²⋊C₄	—	5	8	no	—
196	C₂×C₃×C₂²⋊C₄	—	5	2	no	—
197	C₂×C₃×C₂²⋊C₄	—	5	2	no	—
198	C₂×C₃×C₂²⋊C₄	—	5	2	no	—
199	C₂×C₃×C₂²⋊C₄	—	5	2	no	—
200	C₂×C₃×C₂²⋊C₄	C₂×D₅	1	2	yes	—
201	C₂×C₃×C₄²	—	5	2	no	—
202	C₂⁴×C₅	C₂³×C₃	1	1	yes	characteristic
203	C₂³×D₅	—	2	4	no	—
204	C₂³×D₅	—	4	2	no	—
205	C₂³×D₅	—	4	2	no	—
206	C₂³×D₅	C₂³×C₃	1	2	yes	—
207	C₂³×D₅	C₂³×C₃	1	2	yes	—
208	C₂³×D₅	C₂³×C₃	1	4	yes	—
209	C₂×C₅×Q₈	—	4	2	no	—
210	C₂×C₅×Q₈	C₂³×C₃	1	4	yes	complemented
211	C₂²×C₄×C₅	—	4	2	no	—
212	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
213	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
214	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
215	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
216	C₂²×C₄×C₅	C₂³×C₃	1	4	yes	—
217	C₂×C₂²⋊D₅	—	2	8	no	—
218	C₂×C₂²⋊D₅	C₂³×C₃	1	8	yes	—
219	C₂²×Dic₅	—	4	2	no	—
220	C₂²×Dic₅	C₂³×C₃	1	2	yes	—
221	C₂²×Dic₅	C₂³×C₃	1	2	yes	—
222	C₂²×Dic₅	C₂³×C₃	1	4	yes	—
223	C₂×D₂₀	—	4	4	no	—
224	C₂×D₂₀	—	4	4	no	—
225	C₂×D₂₀	—	4	2	no	—
226	C₂×D₂₀	C₂³×C₃	1	8	yes	—
227	C₂×D₂₀	C₂³×C₃	1	8	yes	—
228	C₂×D₂₀	C₂³×C₃	1	4	yes	complemented
229	C₂×C₄×D₅	—	4	4	no	—
230	C₂×C₄×D₅	C₂³×C₃	1	8	yes	complemented
231	C₅×C₄⋊C₄	C₂³×C₃	1	4	yes	complemented
232	C₅×C₄⋊C₄	C₂³×C₃	1	8	yes	complemented
233	C₅×C₂²⋊C₄	C₂³×C₃	1	8	yes	—
234	#685	C₂³×C₃	1	8	yes	—
235	#685	C₂³×C₃	1	8	yes	—
236	#685	C₂³×C₃	1	8	yes	—
237	#685	C₂³×C₃	1	8	yes	—
238	#685	C₂³×C₃	1	8	yes	complemented
239	#685	C₂³×C₃	1	8	yes	complemented
240	C₄⋊Dic₅	C₂³×C₃	1	4	yes	complemented
241	#683	C₂³×C₃	1	8	yes	complemented
242	C₄×Dic₅	C₂³×C₃	1	8	yes	complemented
243	#538	—	5	16	no	—
244	C₂×#126	—	5	2	no	—
245	C₂×#124	—	5	2	no	—
246	C₂×#123	—	5	2	no	—
247	C₂×C₂²⋊Q₈	C₃×D₅	1	1	yes	characteristic, complemented
248	C₂×#121	—	5	1	no	—
249	C₂×#121	—	5	2	no	—
250	C₂×#120	—	5	2	no	—
251	C₂×C₄×D₄	—	5	2	no	—
252	C₂²×C₂²⋊C₄	—	5	1	no	—
253	C₂²×C₃×C₅	—	2	8	no	—
254	C₂²×C₃×C₅	—	2	2	no	—
255	C₂²×C₃×C₅	—	2	2	no	—
256	C₂²×C₃×C₅	C₂×#41	1	4	yes	—
257	C₂²×C₃×C₅	C₂×#41	1	2	yes	—
258	C₂²×C₃×C₅	H(5,𝔽₂)	1	2	yes	—
259	C₂²×C₃×C₅	H(5,𝔽₂)	1	2	yes	—
260	C₂²×C₃×C₅	C₂⁵	1	1	yes	characteristic
261	C₂×C₃×D₅	—	2	4	no	—
262	C₂×C₃×D₅	—	2	4	no	—
263	C₂×C₃×D₅	—	2	4	no	—
264	C₂×C₃×D₅	—	2	4	no	—
265	C₂×C₃×D₅	—	2	4	no	—
266	C₂×C₃×D₅	—	2	4	no	—
267	C₂×C₃×D₅	—	4	16	no	—
268	C₂×C₃×D₅	—	4	8	no	—
269	C₂×C₃×D₅	—	4	8	no	—
270	C₃×C₄×C₅	—	2	4	no	—
271	C₃×C₄×C₅	—	2	2	no	—
272	C₃×C₄×C₅	—	2	4	no	—
273	C₃×C₄×C₅	—	2	2	no	—
274	C₃×Dic₅	—	2	4	no	—
275	C₃×Dic₅	—	2	4	no	—
276	C₂⁴×C₃	—	5	2	no	—
277	C₂⁴×C₃	—	5	2	no	—
278	C₂⁴×C₃	—	5	4	no	—
279	C₂⁴×C₃	—	10	4	no	—
280	C₂⁴×C₃	—	20	2	no	—
281	C₂⁴×C₃	—	20	2	no	—
282	C₂⁴×C₃	C₂²×D₅	1	1	yes	characteristic
283	C₂×C₃×Q₈	—	4	2	no	—
284	C₂×C₃×Q₈	C₂²×D₅	1	4	yes	complemented
285	C₂×C₃×D₄	—	5	8	no	—
286	C₂×C₃×D₄	—	5	8	no	—
287	C₂×C₃×D₄	—	5	8	no	—
288	C₂×C₃×D₄	—	5	4	no	—
289	C₂×C₃×D₄	—	10	8	no	—
290	C₂×C₃×D₄	—	20	2	no	—
291	C₂×C₃×D₄	—	20	4	no	—
292	C₂×C₃×D₄	—	20	4	no	—
293	C₂²×C₃×C₄	—	4	2	no	—
294	C₂²×C₃×C₄	—	5	2	no	—
295	C₂²×C₃×C₄	—	5	2	no	—
296	C₂²×C₃×C₄	—	5	4	no	—
297	C₂²×C₃×C₄	—	5	8	no	—
298	C₂²×C₃×C₄	—	20	4	no	—
299	C₂²×C₃×C₄	—	20	2	no	—
300	C₂²×C₃×C₄	C₂²×D₅	1	2	yes	—
301	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
302	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
303	C₂²×C₃×C₄	C₂²×D₅	1	2	yes	—
304	C₂²×C₃×C₄	C₂²×D₅	1	4	yes	—
305	C₃×C₄⋊C₄	—	5	8	no	—
306	C₃×C₄⋊C₄	—	5	4	no	—
307	C₃×C₄⋊C₄	C₂²×D₅	1	8	yes	complemented
308	C₃×C₄⋊C₄	C₂²×D₅	1	4	yes	complemented
309	C₃×C₂²⋊C₄	—	5	8	no	—
310	C₃×C₂²⋊C₄	—	5	8	no	—
311	C₃×C₂²⋊C₄	—	5	8	no	—
312	C₃×C₂²⋊C₄	—	5	8	no	—
313	C₃×C₂²⋊C₄	—	5	8	no	—
314	C₃×C₂²⋊C₄	—	5	8	no	—
315	C₃×C₂²⋊C₄	C₂²×D₅	1	8	yes	—
316	C₃×C₄²	—	5	8	no	—
317	C₂³×C₅	—	2	2	no	—
318	C₂³×C₅	—	2	2	no	—
319	C₂³×C₅	C₃×#41	1	4	yes	—
320	C₂³×C₅	C₂⁴×C₃	1	2	yes	—
321	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
322	C₂²×D₅	—	2	4	no	—
323	C₂²×D₅	—	2	8	no	—
324	C₂²×D₅	—	2	4	no	—
325	C₂²×D₅	—	2	4	no	—
326	C₂²×D₅	—	2	4	no	—
327	C₂²×D₅	—	2	4	no	—
328	C₂²×D₅	—	2	4	no	—
329	C₂²×D₅	—	4	4	no	—
330	C₂²×D₅	—	4	4	no	—
331	C₂²×D₅	—	4	16	no	—
332	C₂²×D₅	C₂⁴×C₃	1	4	yes	—
333	C₂²×D₅	C₂⁴×C₃	1	4	yes	—
334	C₅×Q₈	—	4	4	no	—
335	C₂×C₄×C₅	—	2	4	no	—
336	C₂×C₄×C₅	—	2	2	no	—
337	C₂×C₄×C₅	—	2	4	no	—
338	C₂×C₄×C₅	—	2	2	no	—
339	C₂×C₄×C₅	—	4	4	no	—
340	C₂×C₄×C₅	C₂⁴×C₃	1	4	yes	—
341	C₂×C₄×C₅	C₂⁴×C₃	1	4	yes	—
342	C₂×C₄×C₅	C₂⁴×C₃	1	2	yes	—
343	C₂×C₄×C₅	C₂⁴×C₃	1	2	yes	—
344	C₂²⋊D₅	—	2	16	no	—
345	C₂×Dic₅	—	2	4	no	—
346	C₂×Dic₅	—	2	4	no	—
347	C₂×Dic₅	—	4	4	no	—
348	C₂×Dic₅	C₂⁴×C₃	1	4	yes	—
349	C₂×Dic₅	C₂⁴×C₃	1	4	yes	—
350	D₂₀	—	4	8	no	—
351	D₂₀	—	4	8	no	—
352	D₂₀	—	4	4	no	—
353	C₄×D₅	—	4	8	no	—
354	C₂⁵	—	5	1	no	—
355	C₂²×Q₈	C₂×C₃×D₅	1	1	yes	characteristic, complemented
356	C₂²×D₄	—	5	2	no	—
357	C₂²×D₄	—	5	2	no	—
358	C₂²×D₄	—	5	2	no	—
359	C₂²×D₄	—	5	1	no	—
360	C₂³×C₄	—	5	1	no	—
361	C₂³×C₄	—	5	2	no	—
362	C₂³×C₄	C₂×C₃×D₅	1	1	yes	characteristic
363	#126	—	5	16	no	—
364	#124	—	5	16	no	—
365	#123	—	5	16	no	—
366	C₂²⋊Q₈	C₂×C₃×D₅	1	8	yes	complemented
367	#121	—	5	8	no	—
368	#121	—	5	16	no	—
369	#120	—	5	16	no	—
370	C₄×D₄	—	5	16	no	—
371	C₂×C₄⋊C₄	—	5	2	no	—
372	C₂×C₄⋊C₄	—	5	1	no	—
373	C₂×C₄⋊C₄	C₂×C₃×D₅	1	2	yes	complemented
374	C₂×C₄⋊C₄	C₂×C₃×D₅	1	1	yes	characteristic, complemented
375	C₂×C₂²⋊C₄	—	5	2	no	—
376	C₂×C₂²⋊C₄	—	5	2	no	—
377	C₂×C₂²⋊C₄	—	5	8	no	—
378	C₂×C₂²⋊C₄	—	5	2	no	—
379	C₂×C₂²⋊C₄	—	5	2	no	—
380	C₂×C₂²⋊C₄	—	5	2	no	—
381	C₂×C₂²⋊C₄	—	5	2	no	—
382	C₂×C₂²⋊C₄	C₂×C₃×D₅	1	2	yes	—
383	C₂×C₄²	—	5	2	no	—
384	C₂×C₃×C₅	—	2	4	no	—
385	C₂×C₃×C₅	#538	1	4	yes	complemented
386	C₂×C₃×C₅	C₂²×#41	1	1	yes	characteristic
387	C₂×C₃×C₅	C₂×H(5,𝔽₂)	1	1	yes	characteristic
388	C₂×C₃×C₅	C₂×H(5,𝔽₂)	1	1	yes	characteristic
389	C₃×D₅	—	4	4	no	—
390	C₃×D₅	—	4	4	no	—
391	C₂³×C₃	—	2	2	no	—
392	C₂³×C₃	—	2	2	no	—
393	C₂³×C₃	—	5	4	no	—
394	C₂³×C₃	—	5	4	no	—
395	C₂³×C₃	—	10	4	no	—
396	C₂³×C₃	—	10	4	no	—
397	C₂³×C₃	—	10	4	no	—
398	C₂³×C₃	—	10	8	no	—
399	C₂³×C₃	—	10	4	no	—
400	C₂³×C₃	—	10	4	no	—
401	C₂³×C₃	—	10	4	no	—
402	C₂³×C₃	—	20	4	no	—
403	C₂³×C₃	—	20	4	no	—
404	C₂³×C₃	—	20	16	no	—
405	C₂³×C₃	#713	1	4	yes	—
406	C₂³×C₃	C₂³×D₅	1	2	yes	—
407	C₂³×C₃	C₂³×D₅	1	1	yes	characteristic
408	C₃×Q₈	—	4	4	no	—
409	C₃×D₄	—	10	16	no	—
410	C₃×D₄	—	20	4	no	—
411	C₃×D₄	—	20	8	no	—
412	C₃×D₄	—	20	8	no	—
413	C₂×C₃×C₄	—	2	2	no	—
414	C₂×C₃×C₄	—	2	2	no	—
415	C₂×C₃×C₄	—	2	4	no	—
416	C₂×C₃×C₄	—	2	4	no	—
417	C₂×C₃×C₄	—	4	4	no	—
418	C₂×C₃×C₄	—	5	4	no	—
419	C₂×C₃×C₄	—	5	4	no	—
420	C₂×C₃×C₄	—	10	4	no	—
421	C₂×C₃×C₄	—	10	4	no	—
422	C₂×C₃×C₄	—	20	8	no	—
423	C₂×C₃×C₄	—	20	4	no	—
424	C₂×C₃×C₄	C₂³×D₅	1	4	yes	—
425	C₂×C₃×C₄	C₂³×D₅	1	2	yes	—
426	C₂×C₃×C₄	C₂³×D₅	1	2	yes	—
427	C₂×C₃×C₄	C₂³×D₅	1	4	yes	—
428	C₂²×C₅	—	2	2	no	—
429	C₂²×C₅	—	2	2	no	—
430	C₂²×C₅	—	2	8	no	—
431	C₂²×C₅	C₂×C₃×#41	1	4	yes	—
432	C₂²×C₅	C₂×C₃×#41	1	2	yes	—
433	C₂²×C₅	C₃×H(5,𝔽₂)	1	2	yes	—
434	C₂²×C₅	C₃×H(5,𝔽₂)	1	2	yes	—
435	C₂²×C₅	C₂⁵×C₃	1	1	yes	characteristic
436	C₂×D₅	—	2	4	no	—
437	C₂×D₅	—	2	4	no	—
438	C₂×D₅	—	2	4	no	—
439	C₂×D₅	—	2	4	no	—
440	C₂×D₅	—	2	4	no	—
441	C₂×D₅	—	2	4	no	—
442	C₂×D₅	—	4	16	no	—
443	C₂×D₅	—	4	8	no	—
444	C₂×D₅	—	4	8	no	—
445	C₄×C₅	—	2	4	no	—
446	C₄×C₅	—	2	2	no	—
447	C₄×C₅	—	2	4	no	—
448	C₄×C₅	—	2	2	no	—
449	Dic₅	—	2	4	no	—
450	Dic₅	—	2	4	no	—
451	C₂⁴	—	5	2	no	—
452	C₂⁴	—	5	2	no	—
453	C₂⁴	—	5	4	no	—
454	C₂⁴	—	10	4	no	—
455	C₂⁴	—	20	2	no	—
456	C₂⁴	—	20	2	no	—
457	C₂⁴	C₂²×C₃×D₅	1	1	yes	characteristic
458	C₂×Q₈	—	4	2	no	—
459	C₂×Q₈	C₂²×C₃×D₅	1	4	yes	complemented
460	C₂×D₄	—	5	8	no	—
461	C₂×D₄	—	5	8	no	—
462	C₂×D₄	—	5	8	no	—
463	C₂×D₄	—	5	4	no	—
464	C₂×D₄	—	10	8	no	—
465	C₂×D₄	—	20	2	no	—
466	C₂×D₄	—	20	4	no	—
467	C₂×D₄	—	20	4	no	—
468	C₂²×C₄	—	4	2	no	—
469	C₂²×C₄	—	5	2	no	—
470	C₂²×C₄	—	5	2	no	—
471	C₂²×C₄	—	5	4	no	—
472	C₂²×C₄	—	5	8	no	—
473	C₂²×C₄	—	20	4	no	—
474	C₂²×C₄	—	20	2	no	—
475	C₂²×C₄	C₂²×C₃×D₅	1	2	yes	—
476	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
477	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
478	C₂²×C₄	C₂²×C₃×D₅	1	2	yes	—
479	C₂²×C₄	C₂²×C₃×D₅	1	4	yes	—
480	C₄⋊C₄	—	5	8	no	—
481	C₄⋊C₄	—	5	4	no	—
482	C₄⋊C₄	C₂²×C₃×D₅	1	8	yes	complemented
483	C₄⋊C₄	C₂²×C₃×D₅	1	4	yes	complemented
484	C₂²⋊C₄	—	5	8	no	—
485	C₂²⋊C₄	—	5	8	no	—
486	C₂²⋊C₄	—	5	8	no	—
487	C₂²⋊C₄	—	5	8	no	—
488	C₂²⋊C₄	—	5	8	no	—
489	C₂²⋊C₄	—	5	8	no	—
490	C₂²⋊C₄	C₂²×C₃×D₅	1	8	yes	—
491	C₄²	—	5	8	no	—
492	C₃×C₅	C₂×#538	1	1	yes	characteristic, complemented, Hall
493	C₂²×C₃	—	2	8	no	—
494	C₂²×C₃	—	2	2	no	—
495	C₂²×C₃	—	2	2	no	—
496	C₂²×C₃	—	10	4	no	—
497	C₂²×C₃	—	10	4	no	—
498	C₂²×C₃	—	10	4	no	—
499	C₂²×C₃	—	10	4	no	—
500	C₂²×C₃	—	10	4	no	—
501	C₂²×C₃	—	10	4	no	—
502	C₂²×C₃	—	20	16	no	—
503	C₂²×C₃	—	20	8	no	—
504	C₂²×C₃	—	20	8	no	—
505	C₂²×C₃	#4144	1	2	yes	—
506	C₂²×C₃	C₂×#713	1	4	yes	—
507	C₂²×C₃	C₂×#713	1	2	yes	—
508	C₂²×C₃	#4149	1	2	yes	—
509	C₂²×C₃	C₂⁴×D₅	1	1	yes	characteristic
510	C₃×C₄	—	2	2	no	—
511	C₃×C₄	—	2	2	no	—
512	C₃×C₄	—	2	4	no	—
513	C₃×C₄	—	2	4	no	—
514	C₃×C₄	—	10	4	no	—
515	C₃×C₄	—	10	4	no	—
516	C₂×C₅	—	2	4	no	—
517	C₂×C₅	C₃×#538	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×H(5,𝔽₂)	1	1	yes	characteristic
521	D₅	—	4	4	no	—
522	D₅	—	4	4	no	—
523	C₂³	—	2	2	no	—
524	C₂³	—	2	2	no	—
525	C₂³	—	5	4	no	—
526	C₂³	—	5	4	no	—
527	C₂³	—	10	4	no	—
528	C₂³	—	10	4	no	—
529	C₂³	—	10	4	no	—
530	C₂³	—	10	8	no	—
531	C₂³	—	10	4	no	—
532	C₂³	—	10	4	no	—
533	C₂³	—	10	4	no	—
534	C₂³	—	20	4	no	—
535	C₂³	—	20	4	no	—
536	C₂³	—	20	16	no	—
537	C₂³	C₃×#713	1	4	yes	—
538	C₂³	C₂³×C₃×D₅	1	2	yes	—
539	C₂³	C₂³×C₃×D₅	1	1	yes	characteristic
540	Q₈	—	4	4	no	—
541	D₄	—	10	16	no	—
542	D₄	—	20	4	no	—
543	D₄	—	20	8	no	—
544	D₄	—	20	8	no	—
545	C₂×C₄	—	2	2	no	—
546	C₂×C₄	—	2	2	no	—
547	C₂×C₄	—	2	4	no	—
548	C₂×C₄	—	2	4	no	—
549	C₂×C₄	—	4	4	no	—
550	C₂×C₄	—	5	4	no	—
551	C₂×C₄	—	5	4	no	—
552	C₂×C₄	—	10	4	no	—
553	C₂×C₄	—	10	4	no	—
554	C₂×C₄	—	20	8	no	—
555	C₂×C₄	—	20	4	no	—
556	C₂×C₄	C₂³×C₃×D₅	1	4	yes	—
557	C₂×C₄	C₂³×C₃×D₅	1	2	yes	—
558	C₂×C₄	C₂³×C₃×D₅	1	2	yes	—
559	C₂×C₄	C₂³×C₃×D₅	1	4	yes	—
560	C₂×C₃	—	2	4	no	—
561	C₂×C₃	—	20	4	no	—
562	C₂×C₃	—	20	4	no	—
563	C₂×C₃	#65978	1	4	yes	direct product factor
564	C₂×C₃	C₂×#4144	1	1	yes	characteristic
565	C₂×C₃	C₂²×#713	1	1	yes	characteristic
566	C₂×C₃	C₂×#4149	1	1	yes	characteristic
567	C₅	C₂×C₃×#538	1	1	yes	characteristic, complemented, Sylow
568	C₂²	—	2	2	no	—
569	C₂²	—	2	2	no	—
570	C₂²	—	2	8	no	—
571	C₂²	—	10	4	no	—
572	C₂²	—	10	4	no	—
573	C₂²	—	10	4	no	—
574	C₂²	—	10	4	no	—
575	C₂²	—	10	4	no	—
576	C₂²	—	10	4	no	—
577	C₂²	—	20	16	no	—
578	C₂²	—	20	8	no	—
579	C₂²	—	20	8	no	—
580	C₂²	C₃×#4144	1	2	yes	—
581	C₂²	C₂×C₃×#713	1	4	yes	—
582	C₂²	C₂×C₃×#713	1	2	yes	—
583	C₂²	C₃×#4149	1	2	yes	—
584	C₂²	C₂⁴×C₃×D₅	1	1	yes	characteristic
585	C₄	—	2	2	no	—
586	C₄	—	2	2	no	—
587	C₄	—	2	4	no	—
588	C₄	—	2	4	no	—
589	C₄	—	10	4	no	—
590	C₄	—	10	4	no	—
591	C₃	C₂×#65978	1	1	yes	characteristic, direct product factor, Sylow
592	C₂	—	2	4	no	—
593	C₂	—	20	4	no	—
594	C₂	—	20	4	no	—
595	C₂	C₃×#65978	1	4	yes	direct product factor
596	C₂	C₂×C₃×#4144	1	1	yes	characteristic
597	C₂	C₂²×C₃×#713	1	1	yes	characteristic
598	C₂	C₂×C₃×#4149	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×#538
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. C₂×C₃×C₅×C₂²⋊Q₈	C₂	semidirect
2. C₂×C₃×#4093	C₂	semidirect
3. C₂×C₃×#4076	C₂	semidirect
4. C₂²×C₃×#685	C₂	semidirect
5. C₃×#65978	C₂	direct (N × H)
6. C₂×C₃×#4044	C₂	semidirect
7. C₂×C₃×#4041	C₂	semidirect
8. C₂×C₃×#4040	C₂	semidirect
9. C₂×C₃×#4028	C₂	semidirect
10. C₂×C₃×#4027	C₂	semidirect
11. C₂×#65978	C₃	direct (N × H)
13. C₂²×C₃×C₅×Q₈	C₂²	semidirect
19. C₂²×C₃×D₂₀	C₂²	semidirect
20. C₂²×C₃×C₄×D₅	C₂²	semidirect
21. C₃×C₅×C₂²⋊Q₈	C₂²	semidirect
22. C₂×C₃×C₅×C₄⋊C₄	C₂²	semidirect
23. C₂×C₃×C₅×C₄⋊C₄	C₂²	semidirect
25. C₃×#4093	C₂²	semidirect
26. C₃×#4076	C₂²	semidirect
31. C₂×C₃×#685	C₂²	semidirect
32. C₂×C₃×#685	C₂²	semidirect
33. C₂×C₃×#685	C₂²	semidirect
34. C₂×C₃×C₄⋊Dic₅	C₂²	semidirect
35. C₂×C₃×#683	C₂²	semidirect
36. C₂×C₃×C₄×Dic₅	C₂²	semidirect
37. C₃×#4044	C₂²	semidirect
38. C₃×#4041	C₂²	semidirect
39. C₃×#4040	C₂²	semidirect
40. C₃×#4028	C₂²	semidirect
41. C₃×#4027	C₂²	semidirect
43. C₂×C₅×C₂²⋊Q₈	C₂×C₃	semidirect
44. C₂×#4093	C₂×C₃	semidirect
45. C₂×#4076	C₂×C₃	semidirect
46. C₂²×#685	C₂×C₃	semidirect
47. #65978	C₂×C₃	direct (N × H)
48. C₂×#4044	C₂×C₃	semidirect
49. C₂×#4041	C₂×C₃	semidirect
50. C₂×#4040	C₂×C₃	semidirect
51. C₂×#4028	C₂×C₃	semidirect
52. C₂×#4027	C₂×C₃	semidirect
61. C₂×C₃×C₅×Q₈	C₂³	semidirect
79. C₂×C₃×D₂₀	C₂³	semidirect
81. C₂×C₃×C₄×D₅	C₂³	semidirect
82. C₃×C₅×C₄⋊C₄	C₂³	semidirect
83. C₃×C₅×C₄⋊C₄	C₂³	semidirect
89. C₃×#685	C₂³	semidirect
90. C₃×#685	C₂³	semidirect
91. C₃×C₄⋊Dic₅	C₂³	semidirect
92. C₃×#683	C₂³	semidirect
93. C₃×C₄×Dic₅	C₂³	semidirect
98. C₂×C₃×C₂²⋊Q₈	D₅	semidirect
105. C₂²×C₅×Q₈	C₂²×C₃	semidirect
111. C₂²×D₂₀	C₂²×C₃	semidirect
112. C₂²×C₄×D₅	C₂²×C₃	semidirect
113. C₅×C₂²⋊Q₈	C₂²×C₃	semidirect
114. C₂×C₅×C₄⋊C₄	C₂²×C₃	semidirect
115. C₂×C₅×C₄⋊C₄	C₂²×C₃	semidirect
117. #4093	C₂²×C₃	semidirect
118. #4076	C₂²×C₃	semidirect
123. C₂×#685	C₂²×C₃	semidirect
124. C₂×#685	C₂²×C₃	semidirect
125. C₂×#685	C₂²×C₃	semidirect
126. C₂×C₄⋊Dic₅	C₂²×C₃	semidirect
127. C₂×#683	C₂²×C₃	semidirect
128. C₂×C₄×Dic₅	C₂²×C₃	semidirect
129. #4044	C₂²×C₃	semidirect
130. #4041	C₂²×C₃	semidirect
131. #4040	C₂²×C₃	semidirect
132. #4028	C₂²×C₃	semidirect
133. #4027	C₂²×C₃	semidirect
173. C₂²×C₃×Q₈	C₂×D₅	semidirect
184. C₃×C₂²⋊Q₈	C₂×D₅	semidirect
191. C₂×C₃×C₄⋊C₄	C₂×D₅	semidirect
192. C₂×C₃×C₄⋊C₄	C₂×D₅	semidirect
210. C₂×C₅×Q₈	C₂³×C₃	semidirect
228. C₂×D₂₀	C₂³×C₃	semidirect
230. C₂×C₄×D₅	C₂³×C₃	semidirect
231. C₅×C₄⋊C₄	C₂³×C₃	semidirect
232. C₅×C₄⋊C₄	C₂³×C₃	semidirect
238. #685	C₂³×C₃	semidirect
239. #685	C₂³×C₃	semidirect
240. C₄⋊Dic₅	C₂³×C₃	semidirect
241. #683	C₂³×C₃	semidirect
242. C₄×Dic₅	C₂³×C₃	semidirect
247. C₂×C₂²⋊Q₈	C₃×D₅	semidirect
284. C₂×C₃×Q₈	C₂²×D₅	semidirect
307. C₃×C₄⋊C₄	C₂²×D₅	semidirect
308. C₃×C₄⋊C₄	C₂²×D₅	semidirect
355. C₂²×Q₈	C₂×C₃×D₅	semidirect
366. C₂²⋊Q₈	C₂×C₃×D₅	semidirect
373. C₂×C₄⋊C₄	C₂×C₃×D₅	semidirect
374. C₂×C₄⋊C₄	C₂×C₃×D₅	semidirect
385. C₂×C₃×C₅	#538	semidirect
459. C₂×Q₈	C₂²×C₃×D₅	semidirect
482. C₄⋊C₄	C₂²×C₃×D₅	semidirect
483. C₄⋊C₄	C₂²×C₃×D₅	semidirect
492. C₃×C₅	C₂×#538	semidirect
517. C₂×C₅	C₃×#538	semidirect
563. C₂×C₃	#65978	direct (N × H)
567. C₅	C₂×C₃×#538	semidirect
591. C₃	C₂×#65978	direct (N × H)
595. C₂	C₃×#65978	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×C₂²⋊Q₈	2
C₂×C₃×#4093	2
C₂×C₃×#4076	2
C₂²×C₃×#685	2
C₃×#65978	2
C₂×C₃×#4044	2
C₂×C₃×#4041	2
C₂×C₃×#4040	2
C₂×C₃×#4028	2
C₂×C₃×#4027	2
C₂×#65978	3
C₂×C₃×#538	5

Subgroups — C2×C3×#65978 #14480774

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#65978 #14480774