Subgroups — C₂×C₃×#66164 #14480366

Order 1920.

3792 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₅×#41	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₃×D₂₀	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#66164	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₃×#4093	C₂	1	2	yes	maximal, complemented
5	C₂×C₃×#4086	C₂	1	2	yes	maximal, complemented
6	C₂×C₃×#4083	C₂	1	4	yes	maximal, complemented
7	C₂×C₃×#4076	C₂	1	4	yes	maximal, complemented
8	C₂×C₃×#4072	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#66164	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₂⁴×C₃×D₅	C₂²	1	2	yes	—
11	C₂×C₃×C₅×#41	—	2	8	no	—
12	C₂×C₃×C₅×#41	C₂²	1	8	yes	complemented
13	C₂²×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic, complemented
14	C₂²×C₃×C₅×D₄	C₂²	1	2	yes	—
15	C₂²×C₃×C₅×D₄	C₂²	1	1	yes	characteristic, complemented
16	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
17	C₂³×C₃×C₄×C₅	C₂²	1	2	yes	complemented
18	C₂²×C₃×C₂²⋊D₅	C₂²	1	8	yes	complemented
19	C₂²×C₃×D₂₀	—	2	8	no	—
20	C₂²×C₃×D₂₀	C₂²	1	2	yes	—
21	C₂²×C₃×D₂₀	C₂²	1	2	yes	—
22	C₂²×C₃×D₂₀	C₂²	1	8	yes	complemented
23	C₂×C₃×C₂²⋊Dic₅	C₂²	1	2	yes	—
24	C₃×#4093	C₂²	1	16	yes	complemented
25	C₃×#4086	C₂²	1	16	yes	complemented
26	C₃×#4083	C₂²	1	32	yes	complemented
27	C₃×#4076	C₂²	1	32	yes	complemented
28	C₂×C₃×#685	C₂²	1	8	yes	complemented
29	C₃×#4072	C₂²	1	8	yes	complemented
30	C₂×C₃×C₄⋊Dic₅	C₂²	1	2	yes	complemented
31	C₂×C₃×C₄×Dic₅	C₂²	1	2	yes	complemented
32	C₂×C₃×#535	—	5	1	no	maximal, Hall
33	C₂²×C₅×#41	C₂×C₃	1	1	yes	characteristic, complemented
34	C₂³×D₂₀	C₂×C₃	1	1	yes	characteristic, complemented
35	#66164	C₂×C₃	1	16	yes	direct product factor
36	C₂×#4093	C₂×C₃	1	2	yes	complemented
37	C₂×#4086	C₂×C₃	1	2	yes	complemented
38	C₂×#4083	C₂×C₃	1	4	yes	complemented
39	C₂×#4076	C₂×C₃	1	4	yes	complemented
40	C₂×#4072	C₂×C₃	1	1	yes	characteristic, complemented
41	C₂⁴×C₃×C₅	C₂³	1	2	yes	—
42	C₂⁴×C₃×C₅	C₂³	1	1	yes	characteristic
43	C₂³×C₃×D₅	—	2	8	no	—
44	C₂³×C₃×D₅	—	4	8	no	—
45	C₂³×C₃×D₅	C₂³	1	4	yes	—
46	C₂³×C₃×D₅	C₂³	1	8	yes	—
47	C₃×C₅×#41	—	2	32	no	—
48	C₂×C₃×C₅×Q₈	—	2	4	no	—
49	C₂×C₃×C₅×Q₈	C₂³	1	4	yes	complemented
50	C₂×C₃×C₅×D₄	—	2	8	no	—
51	C₂×C₃×C₅×D₄	—	2	4	no	—
52	C₂×C₃×C₅×D₄	C₂³	1	8	yes	—
53	C₂×C₃×C₅×D₄	C₂³	1	4	yes	complemented
54	C₂²×C₃×C₄×C₅	—	2	8	no	—
55	C₂²×C₃×C₄×C₅	D₄	1	8	yes	complemented
56	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
57	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
58	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
59	C₂²×C₃×C₄×C₅	C₂³	1	4	yes	—
60	C₂²×C₃×C₄×C₅	C₂³	1	8	yes	complemented
61	C₂×C₃×C₂²⋊D₅	—	4	16	no	—
62	C₂×C₃×C₂²⋊D₅	C₂³	1	32	yes	complemented
63	C₂²×C₃×Dic₅	C₂³	1	4	yes	—
64	C₂×C₃×D₂₀	—	2	8	no	—
65	C₂×C₃×D₂₀	—	2	8	no	—
66	C₂×C₃×D₂₀	—	2	32	no	—
67	C₂×C₃×D₂₀	C₂³	1	8	yes	—
68	C₂×C₃×D₂₀	C₂³	1	8	yes	—
69	C₃×C₂²⋊Dic₅	C₂³	1	8	yes	—
70	C₃×#685	C₂³	1	32	yes	complemented
71	C₃×C₄⋊Dic₅	C₂³	1	8	yes	complemented
72	C₃×C₄×Dic₅	C₂³	1	8	yes	complemented
73	C₂²×C₃×#41	D₅	1	1	yes	characteristic, complemented
74	C₂³×C₃×D₄	—	5	1	no	—
75	C₃×#535	—	5	16	no	—
76	C₂×C₃×#127	—	5	2	no	—
77	C₂×C₃×#124	—	5	2	no	—
78	C₂×C₃×#121	—	5	4	no	—
79	C₂×C₃×#120	—	5	4	no	—
80	C₂×C₃×#117	—	5	1	no	—
81	C₂⁴×D₅	C₂²×C₃	1	2	yes	—
82	C₂×C₅×#41	—	2	8	no	—
83	C₂×C₅×#41	C₂²×C₃	1	8	yes	complemented
84	C₂²×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic, complemented
85	C₂²×C₅×D₄	C₂²×C₃	1	2	yes	—
86	C₂²×C₅×D₄	C₂²×C₃	1	1	yes	characteristic, complemented
87	C₂³×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
88	C₂³×C₄×C₅	C₂²×C₃	1	2	yes	complemented
89	C₂²×C₂²⋊D₅	C₂²×C₃	1	8	yes	complemented
90	C₂²×D₂₀	—	2	8	no	—
91	C₂²×D₂₀	C₂²×C₃	1	2	yes	—
92	C₂²×D₂₀	C₂²×C₃	1	2	yes	—
93	C₂²×D₂₀	C₂²×C₃	1	8	yes	complemented
94	C₂×C₂²⋊Dic₅	C₂²×C₃	1	2	yes	—
95	#4093	C₂²×C₃	1	16	yes	complemented
96	#4086	C₂²×C₃	1	16	yes	complemented
97	#4083	C₂²×C₃	1	32	yes	complemented
98	#4076	C₂²×C₃	1	32	yes	complemented
99	C₂×#685	C₂²×C₃	1	8	yes	complemented
100	#4072	C₂²×C₃	1	8	yes	complemented
101	C₂×C₄⋊Dic₅	C₂²×C₃	1	2	yes	complemented
102	C₂×C₄×Dic₅	C₂²×C₃	1	2	yes	complemented
103	C₂×#535	—	5	1	no	Sylow
104	C₂³×C₃×C₅	—	2	4	no	—
105	C₂³×C₃×C₅	—	2	4	no	—
106	C₂³×C₃×C₅	—	2	8	no	—
107	C₂³×C₃×C₅	C₂×D₄	1	4	yes	—
108	C₂³×C₃×C₅	C₂⁴	1	4	yes	—
109	C₂³×C₃×C₅	C₂⁴	1	2	yes	—
110	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
111	C₂²×C₃×D₅	—	2	16	no	—
112	C₂²×C₃×D₅	—	2	8	no	—
113	C₂²×C₃×D₅	—	2	16	no	—
114	C₂²×C₃×D₅	—	4	16	no	—
115	C₂²×C₃×D₅	—	4	32	no	—
116	C₂²×C₃×D₅	C₂⁴	1	8	yes	—
117	C₃×C₅×Q₈	—	2	8	no	—
118	C₃×C₅×D₄	—	2	16	no	—
119	C₃×C₅×D₄	—	2	8	no	—
120	C₂×C₃×C₄×C₅	—	2	4	no	—
121	C₂×C₃×C₄×C₅	—	2	16	no	—
122	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
123	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
124	C₂×C₃×C₄×C₅	C₂×D₄	1	16	yes	complemented
125	C₂×C₃×C₄×C₅	C₂⁴	1	4	yes	—
126	C₂×C₃×C₄×C₅	C₂⁴	1	2	yes	—
127	C₂×C₃×C₄×C₅	C₂⁴	1	2	yes	—
128	C₃×C₂²⋊D₅	—	4	32	no	—
129	C₂×C₃×Dic₅	—	2	8	no	—
130	C₂×C₃×Dic₅	C₂⁴	1	8	yes	—
131	C₃×D₂₀	—	2	16	no	—
132	C₃×D₂₀	—	2	16	no	—
133	C₂⁵×C₃	—	5	2	no	—
134	C₂×C₃×#41	—	2	8	no	—
135	C₂×C₃×#41	C₂×D₅	1	8	yes	complemented
136	C₂²×C₃×Q₈	C₂×D₅	1	1	yes	characteristic, complemented
137	C₂²×C₃×D₄	—	5	8	no	—
138	C₂²×C₃×D₄	—	5	2	no	—
139	C₂²×C₃×D₄	—	5	2	no	—
140	C₂²×C₃×D₄	—	5	8	no	—
141	C₂²×C₃×D₄	—	10	8	no	—
142	C₂²×C₃×D₄	C₂×D₅	1	2	yes	—
143	C₂²×C₃×D₄	C₂×D₅	1	1	yes	characteristic, complemented
144	C₂³×C₃×C₄	C₂×D₅	1	1	yes	characteristic
145	C₂³×C₃×C₄	C₂×D₅	1	2	yes	complemented
146	C₃×#127	—	5	16	no	—
147	C₃×#124	—	5	16	no	—
148	C₃×#121	—	5	32	no	—
149	C₃×#120	—	5	32	no	—
150	C₃×#117	—	5	8	no	—
151	C₂×C₃×C₄⋊C₄	—	5	2	no	—
152	C₂×C₃×C₂²⋊C₄	—	5	2	no	—
153	C₂×C₃×C₂²⋊C₄	—	5	8	no	—
154	C₂×C₃×C₄²	—	5	2	no	—
155	C₂⁴×C₅	C₂³×C₃	1	2	yes	—
156	C₂⁴×C₅	C₂³×C₃	1	1	yes	characteristic
157	C₂³×D₅	—	2	8	no	—
158	C₂³×D₅	—	4	8	no	—
159	C₂³×D₅	C₂³×C₃	1	4	yes	—
160	C₂³×D₅	C₂³×C₃	1	8	yes	—
161	C₅×#41	—	2	32	no	—
162	C₂×C₅×Q₈	—	2	4	no	—
163	C₂×C₅×Q₈	C₂³×C₃	1	4	yes	complemented
164	C₂×C₅×D₄	—	2	8	no	—
165	C₂×C₅×D₄	—	2	4	no	—
166	C₂×C₅×D₄	C₂³×C₃	1	8	yes	—
167	C₂×C₅×D₄	C₂³×C₃	1	4	yes	complemented
168	C₂²×C₄×C₅	—	2	8	no	—
169	C₂²×C₄×C₅	C₃×D₄	1	8	yes	complemented
170	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
171	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
172	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
173	C₂²×C₄×C₅	C₂³×C₃	1	4	yes	—
174	C₂²×C₄×C₅	C₂³×C₃	1	8	yes	complemented
175	C₂×C₂²⋊D₅	—	4	16	no	—
176	C₂×C₂²⋊D₅	C₂³×C₃	1	32	yes	complemented
177	C₂²×Dic₅	C₂³×C₃	1	4	yes	—
178	C₂×D₂₀	—	2	8	no	—
179	C₂×D₂₀	—	2	8	no	—
180	C₂×D₂₀	—	2	32	no	—
181	C₂×D₂₀	C₂³×C₃	1	8	yes	—
182	C₂×D₂₀	C₂³×C₃	1	8	yes	—
183	C₂²⋊Dic₅	C₂³×C₃	1	8	yes	—
184	#685	C₂³×C₃	1	32	yes	complemented
185	C₄⋊Dic₅	C₂³×C₃	1	8	yes	complemented
186	C₄×Dic₅	C₂³×C₃	1	8	yes	complemented
187	C₂²×#41	C₃×D₅	1	1	yes	characteristic, complemented
188	C₂³×D₄	—	5	1	no	—
189	#535	—	5	16	no	—
190	C₂×#127	—	5	2	no	—
191	C₂×#124	—	5	2	no	—
192	C₂×#121	—	5	4	no	—
193	C₂×#120	—	5	4	no	—
194	C₂×#117	—	5	1	no	—
195	C₂²×C₃×C₅	—	2	8	no	—
196	C₂²×C₃×C₅	—	2	4	no	—
197	C₂²×C₃×C₅	—	2	4	no	—
198	C₂²×C₃×C₅	—	2	8	no	—
199	C₂²×C₃×C₅	—	4	8	no	—
200	C₂²×C₃×C₅	C₂²×D₄	1	4	yes	—
201	C₂²×C₃×C₅	C₂²×D₄	1	2	yes	—
202	C₂²×C₃×C₅	H(5,𝔽₂)	1	4	yes	—
203	C₂²×C₃×C₅	C₂⁵	1	1	yes	characteristic
204	C₂×C₃×D₅	—	2	8	no	—
205	C₂×C₃×D₅	—	2	16	no	—
206	C₂×C₃×D₅	—	4	32	no	—
207	C₂×C₃×D₅	—	4	16	no	—
208	C₃×C₄×C₅	—	2	4	no	—
209	C₃×C₄×C₅	C₂²×D₄	1	4	yes	—
210	C₃×C₄×C₅	C₂²×D₄	1	4	yes	—
211	C₃×Dic₅	—	2	8	no	—
212	C₂⁴×C₃	—	5	4	no	—
213	C₂⁴×C₃	—	5	8	no	—
214	C₂⁴×C₃	—	10	8	no	—
215	C₂⁴×C₃	—	20	8	no	—
216	C₂⁴×C₃	C₂²×D₅	1	2	yes	—
217	C₂⁴×C₃	C₂²×D₅	1	1	yes	characteristic
218	C₃×#41	—	2	32	no	—
219	C₂×C₃×Q₈	—	2	4	no	—
220	C₂×C₃×Q₈	C₂²×D₅	1	4	yes	complemented
221	C₂×C₃×D₄	—	2	8	no	—
222	C₂×C₃×D₄	—	2	4	no	—
223	C₂×C₃×D₄	—	5	32	no	—
224	C₂×C₃×D₄	—	5	8	no	—
225	C₂×C₃×D₄	—	5	8	no	—
226	C₂×C₃×D₄	—	10	8	no	—
227	C₂×C₃×D₄	—	10	8	no	—
228	C₂×C₃×D₄	—	10	32	no	—
229	C₂×C₃×D₄	—	20	16	no	—
230	C₂×C₃×D₄	C₂²×D₅	1	8	yes	—
231	C₂×C₃×D₄	C₂²×D₅	1	4	yes	complemented
232	C₂²×C₃×C₄	—	2	8	no	—
233	C₂²×C₃×C₄	—	5	4	no	—
234	C₂²×C₃×C₄	C₂²⋊D₅	1	8	yes	complemented
235	C₂²×C₃×C₄	C₂²×D₅	1	2	yes	—
236	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
237	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
238	C₂²×C₃×C₄	C₂²×D₅	1	4	yes	—
239	C₂²×C₃×C₄	C₂²×D₅	1	8	yes	complemented
240	C₃×C₄⋊C₄	—	5	8	no	—
241	C₃×C₂²⋊C₄	—	5	8	no	—
242	C₃×C₂²⋊C₄	—	5	32	no	—
243	C₃×C₄²	—	5	8	no	—
244	C₂³×C₅	—	2	4	no	—
245	C₂³×C₅	—	2	4	no	—
246	C₂³×C₅	—	2	8	no	—
247	C₂³×C₅	C₂×C₃×D₄	1	4	yes	—
248	C₂³×C₅	C₂⁴×C₃	1	4	yes	—
249	C₂³×C₅	C₂⁴×C₃	1	2	yes	—
250	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
251	C₂²×D₅	—	2	16	no	—
252	C₂²×D₅	—	2	8	no	—
253	C₂²×D₅	—	2	16	no	—
254	C₂²×D₅	—	4	16	no	—
255	C₂²×D₅	—	4	32	no	—
256	C₂²×D₅	C₂⁴×C₃	1	8	yes	—
257	C₅×Q₈	—	2	8	no	—
258	C₅×D₄	—	2	16	no	—
259	C₅×D₄	—	2	8	no	—
260	C₂×C₄×C₅	—	2	4	no	—
261	C₂×C₄×C₅	—	2	16	no	—
262	C₂×C₄×C₅	C₂×C₃×D₄	1	4	yes	—
263	C₂×C₄×C₅	C₂×C₃×D₄	1	4	yes	—
264	C₂×C₄×C₅	C₂×C₃×D₄	1	16	yes	complemented
265	C₂×C₄×C₅	C₂⁴×C₃	1	4	yes	—
266	C₂×C₄×C₅	C₂⁴×C₃	1	2	yes	—
267	C₂×C₄×C₅	C₂⁴×C₃	1	2	yes	—
268	C₂²⋊D₅	—	4	32	no	—
269	C₂×Dic₅	—	2	8	no	—
270	C₂×Dic₅	C₂⁴×C₃	1	8	yes	—
271	D₂₀	—	2	16	no	—
272	D₂₀	—	2	16	no	—
273	C₂⁵	—	5	2	no	—
274	C₂×#41	—	2	8	no	—
275	C₂×#41	C₂×C₃×D₅	1	8	yes	complemented
276	C₂²×Q₈	C₂×C₃×D₅	1	1	yes	characteristic, complemented
277	C₂²×D₄	—	5	8	no	—
278	C₂²×D₄	—	5	2	no	—
279	C₂²×D₄	—	5	2	no	—
280	C₂²×D₄	—	5	8	no	—
281	C₂²×D₄	—	10	8	no	—
282	C₂²×D₄	C₂×C₃×D₅	1	2	yes	—
283	C₂²×D₄	C₂×C₃×D₅	1	1	yes	characteristic, complemented
284	C₂³×C₄	C₂×C₃×D₅	1	1	yes	characteristic
285	C₂³×C₄	C₂×C₃×D₅	1	2	yes	complemented
286	#127	—	5	16	no	—
287	#124	—	5	16	no	—
288	#121	—	5	32	no	—
289	#120	—	5	32	no	—
290	#117	—	5	8	no	—
291	C₂×C₄⋊C₄	—	5	2	no	—
292	C₂×C₂²⋊C₄	—	5	2	no	—
293	C₂×C₂²⋊C₄	—	5	8	no	—
294	C₂×C₄²	—	5	2	no	—
295	C₂×C₃×C₅	—	2	4	no	—
296	C₂×C₃×C₅	—	4	4	no	—
297	C₂×C₃×C₅	#535	1	4	yes	complemented
298	C₂×C₃×C₅	C₂³×D₄	1	1	yes	characteristic
299	C₂×C₃×C₅	C₂×H(5,𝔽₂)	1	2	yes	—
300	C₃×D₅	—	4	8	no	—
301	C₂³×C₃	—	2	4	no	—
302	C₂³×C₃	—	2	8	no	—
303	C₂³×C₃	—	2	4	no	—
304	C₂³×C₃	—	5	8	no	—
305	C₂³×C₃	—	10	16	no	—
306	C₂³×C₃	—	10	16	no	—
307	C₂³×C₃	—	10	8	no	—
308	C₂³×C₃	—	20	16	no	—
309	C₂³×C₃	—	20	32	no	—
310	C₂³×C₃	C₂×C₂²⋊D₅	1	4	yes	—
311	C₂³×C₃	C₂³×D₅	1	4	yes	—
312	C₂³×C₃	C₂³×D₅	1	2	yes	—
313	C₂³×C₃	C₂³×D₅	1	1	yes	characteristic
314	C₃×Q₈	—	2	8	no	—
315	C₃×D₄	—	2	16	no	—
316	C₃×D₄	—	2	8	no	—
317	C₃×D₄	—	10	16	no	—
318	C₃×D₄	—	10	16	no	—
319	C₃×D₄	—	20	32	no	—
320	C₂×C₃×C₄	—	2	4	no	—
321	C₂×C₃×C₄	—	2	16	no	—
322	C₂×C₃×C₄	—	5	8	no	—
323	C₂×C₃×C₄	—	10	8	no	—
324	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	4	yes	—
325	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	4	yes	—
326	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	16	yes	complemented
327	C₂×C₃×C₄	C₂³×D₅	1	4	yes	—
328	C₂×C₃×C₄	C₂³×D₅	1	2	yes	—
329	C₂×C₃×C₄	C₂³×D₅	1	2	yes	—
330	C₂²×C₅	—	2	4	no	—
331	C₂²×C₅	—	2	8	no	—
332	C₂²×C₅	—	2	4	no	—
333	C₂²×C₅	—	2	8	no	—
334	C₂²×C₅	—	4	8	no	—
335	C₂²×C₅	C₂²×C₃×D₄	1	4	yes	—
336	C₂²×C₅	C₂²×C₃×D₄	1	2	yes	—
337	C₂²×C₅	C₃×H(5,𝔽₂)	1	4	yes	—
338	C₂²×C₅	C₂⁵×C₃	1	1	yes	characteristic
339	C₂×D₅	—	2	8	no	—
340	C₂×D₅	—	2	16	no	—
341	C₂×D₅	—	4	32	no	—
342	C₂×D₅	—	4	16	no	—
343	C₄×C₅	—	2	4	no	—
344	C₄×C₅	C₂²×C₃×D₄	1	4	yes	—
345	C₄×C₅	C₂²×C₃×D₄	1	4	yes	—
346	Dic₅	—	2	8	no	—
347	C₂⁴	—	5	4	no	—
348	C₂⁴	—	5	8	no	—
349	C₂⁴	—	10	8	no	—
350	C₂⁴	—	20	8	no	—
351	C₂⁴	C₂²×C₃×D₅	1	2	yes	—
352	C₂⁴	C₂²×C₃×D₅	1	1	yes	characteristic
353	#41	—	2	32	no	—
354	C₂×Q₈	—	2	4	no	—
355	C₂×Q₈	C₂²×C₃×D₅	1	4	yes	complemented
356	C₂×D₄	—	2	8	no	—
357	C₂×D₄	—	2	4	no	—
358	C₂×D₄	—	5	32	no	—
359	C₂×D₄	—	5	8	no	—
360	C₂×D₄	—	5	8	no	—
361	C₂×D₄	—	10	8	no	—
362	C₂×D₄	—	10	8	no	—
363	C₂×D₄	—	10	32	no	—
364	C₂×D₄	—	20	16	no	—
365	C₂×D₄	C₂²×C₃×D₅	1	8	yes	—
366	C₂×D₄	C₂²×C₃×D₅	1	4	yes	complemented
367	C₂²×C₄	—	2	8	no	—
368	C₂²×C₄	—	5	4	no	—
369	C₂²×C₄	C₃×C₂²⋊D₅	1	8	yes	complemented
370	C₂²×C₄	C₂²×C₃×D₅	1	2	yes	—
371	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
372	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
373	C₂²×C₄	C₂²×C₃×D₅	1	4	yes	—
374	C₂²×C₄	C₂²×C₃×D₅	1	8	yes	complemented
375	C₄⋊C₄	—	5	8	no	—
376	C₂²⋊C₄	—	5	8	no	—
377	C₂²⋊C₄	—	5	32	no	—
378	C₄²	—	5	8	no	—
379	C₃×C₅	C₂×#535	1	1	yes	characteristic, complemented, Hall
380	C₂²×C₃	—	2	8	no	—
381	C₂²×C₃	—	2	4	no	—
382	C₂²×C₃	—	2	8	no	—
383	C₂²×C₃	—	2	4	no	—
384	C₂²×C₃	—	4	8	no	—
385	C₂²×C₃	—	10	16	no	—
386	C₂²×C₃	—	10	8	no	—
387	C₂²×C₃	—	20	32	no	—
388	C₂²×C₃	—	20	16	no	—
389	C₂²×C₃	#4149	1	4	yes	—
390	C₂²×C₃	C₂²×C₂²⋊D₅	1	4	yes	—
391	C₂²×C₃	C₂²×C₂²⋊D₅	1	2	yes	—
392	C₂²×C₃	C₂⁴×D₅	1	1	yes	characteristic
393	C₃×C₄	—	2	4	no	—
394	C₃×C₄	—	10	8	no	—
395	C₃×C₄	C₂²×C₂²⋊D₅	1	4	yes	—
396	C₃×C₄	C₂²×C₂²⋊D₅	1	4	yes	—
397	C₂×C₅	—	2	4	no	—
398	C₂×C₅	—	4	4	no	—
399	C₂×C₅	C₃×#535	1	4	yes	complemented
400	C₂×C₅	C₂³×C₃×D₄	1	1	yes	characteristic
401	C₂×C₅	C₂×C₃×H(5,𝔽₂)	1	2	yes	—
402	D₅	—	4	8	no	—
403	C₂³	—	2	4	no	—
404	C₂³	—	2	8	no	—
405	C₂³	—	2	4	no	—
406	C₂³	—	5	8	no	—
407	C₂³	—	10	16	no	—
408	C₂³	—	10	16	no	—
409	C₂³	—	10	8	no	—
410	C₂³	—	20	16	no	—
411	C₂³	—	20	32	no	—
412	C₂³	C₂×C₃×C₂²⋊D₅	1	4	yes	—
413	C₂³	C₂³×C₃×D₅	1	4	yes	—
414	C₂³	C₂³×C₃×D₅	1	2	yes	—
415	C₂³	C₂³×C₃×D₅	1	1	yes	characteristic
416	Q₈	—	2	8	no	—
417	D₄	—	2	16	no	—
418	D₄	—	2	8	no	—
419	D₄	—	10	16	no	—
420	D₄	—	10	16	no	—
421	D₄	—	20	32	no	—
422	C₂×C₄	—	2	4	no	—
423	C₂×C₄	—	2	16	no	—
424	C₂×C₄	—	5	8	no	—
425	C₂×C₄	—	10	8	no	—
426	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	4	yes	—
427	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	4	yes	—
428	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	16	yes	complemented
429	C₂×C₄	C₂³×C₃×D₅	1	4	yes	—
430	C₂×C₄	C₂³×C₃×D₅	1	2	yes	—
431	C₂×C₄	C₂³×C₃×D₅	1	2	yes	—
432	C₂×C₃	—	2	4	no	—
433	C₂×C₃	—	4	4	no	—
434	C₂×C₃	—	20	8	no	—
435	C₂×C₃	#66164	1	4	yes	direct product factor
436	C₂×C₃	C₂×#4149	1	2	yes	—
437	C₂×C₃	C₂³×C₂²⋊D₅	1	1	yes	characteristic
438	C₅	C₂×C₃×#535	1	1	yes	characteristic, complemented, Sylow
439	C₂²	—	2	4	no	—
440	C₂²	—	2	8	no	—
441	C₂²	—	2	8	no	—
442	C₂²	—	2	4	no	—
443	C₂²	—	4	8	no	—
444	C₂²	—	10	16	no	—
445	C₂²	—	10	8	no	—
446	C₂²	—	20	32	no	—
447	C₂²	—	20	16	no	—
448	C₂²	C₃×#4149	1	4	yes	—
449	C₂²	C₂²×C₃×C₂²⋊D₅	1	4	yes	—
450	C₂²	C₂²×C₃×C₂²⋊D₅	1	2	yes	—
451	C₂²	C₂⁴×C₃×D₅	1	1	yes	characteristic
452	C₄	—	2	4	no	—
453	C₄	—	10	8	no	—
454	C₄	C₂²×C₃×C₂²⋊D₅	1	4	yes	—
455	C₄	C₂²×C₃×C₂²⋊D₅	1	4	yes	—
456	C₃	C₂×#66164	1	1	yes	characteristic, direct product factor, Sylow
457	C₂	—	2	4	no	—
458	C₂	—	4	4	no	—
459	C₂	—	20	8	no	—
460	C₂	C₃×#66164	1	4	yes	direct product factor
461	C₂	C₂×C₃×#4149	1	2	yes	—
462	C₂	C₂³×C₃×C₂²⋊D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₃	C₂³×D₅
commutator, cosocle	C₂²×C₅	C₂⁵×C₃
Frattini	C₂²	C₂⁴×C₃×D₅
Fitting	C₂²×C₃×C₅×#41	C₂
socle	C₂³×C₃×C₅	C₂⁴
hypercenter	C₂²×C₃×#41	D₅
hypocenter	C₅	C₂×C₃×#535
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₃ ◁ C₂²×C₃×#41

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×C₅×#41	C₂	semidirect
2. C₂³×C₃×D₂₀	C₂	semidirect
3. C₃×#66164	C₂	direct (N × H)
4. C₂×C₃×#4093	C₂	semidirect
5. C₂×C₃×#4086	C₂	semidirect
6. C₂×C₃×#4083	C₂	semidirect
7. C₂×C₃×#4076	C₂	semidirect
8. C₂×C₃×#4072	C₂	semidirect
9. C₂×#66164	C₃	direct (N × H)
12. C₂×C₃×C₅×#41	C₂²	semidirect
13. C₂²×C₃×C₅×Q₈	C₂²	semidirect
15. C₂²×C₃×C₅×D₄	C₂²	semidirect
17. C₂³×C₃×C₄×C₅	C₂²	semidirect
18. C₂²×C₃×C₂²⋊D₅	C₂²	semidirect
22. C₂²×C₃×D₂₀	C₂²	semidirect
24. C₃×#4093	C₂²	semidirect
25. C₃×#4086	C₂²	semidirect
26. C₃×#4083	C₂²	semidirect
27. C₃×#4076	C₂²	semidirect
28. C₂×C₃×#685	C₂²	semidirect
29. C₃×#4072	C₂²	semidirect
30. C₂×C₃×C₄⋊Dic₅	C₂²	semidirect
31. C₂×C₃×C₄×Dic₅	C₂²	semidirect
33. C₂²×C₅×#41	C₂×C₃	semidirect
34. C₂³×D₂₀	C₂×C₃	semidirect
35. #66164	C₂×C₃	direct (N × H)
36. C₂×#4093	C₂×C₃	semidirect
37. C₂×#4086	C₂×C₃	semidirect
38. C₂×#4083	C₂×C₃	semidirect
39. C₂×#4076	C₂×C₃	semidirect
40. C₂×#4072	C₂×C₃	semidirect
49. C₂×C₃×C₅×Q₈	C₂³	semidirect
53. C₂×C₃×C₅×D₄	C₂³	semidirect
55. C₂²×C₃×C₄×C₅	D₄	semidirect
60. C₂²×C₃×C₄×C₅	C₂³	semidirect
62. C₂×C₃×C₂²⋊D₅	C₂³	semidirect
70. C₃×#685	C₂³	semidirect
71. C₃×C₄⋊Dic₅	C₂³	semidirect
72. C₃×C₄×Dic₅	C₂³	semidirect
73. C₂²×C₃×#41	D₅	semidirect
83. C₂×C₅×#41	C₂²×C₃	semidirect
84. C₂²×C₅×Q₈	C₂²×C₃	semidirect
86. C₂²×C₅×D₄	C₂²×C₃	semidirect
88. C₂³×C₄×C₅	C₂²×C₃	semidirect
89. C₂²×C₂²⋊D₅	C₂²×C₃	semidirect
93. C₂²×D₂₀	C₂²×C₃	semidirect
95. #4093	C₂²×C₃	semidirect
96. #4086	C₂²×C₃	semidirect
97. #4083	C₂²×C₃	semidirect
98. #4076	C₂²×C₃	semidirect
99. C₂×#685	C₂²×C₃	semidirect
100. #4072	C₂²×C₃	semidirect
101. C₂×C₄⋊Dic₅	C₂²×C₃	semidirect
102. C₂×C₄×Dic₅	C₂²×C₃	semidirect
124. C₂×C₃×C₄×C₅	C₂×D₄	semidirect
135. C₂×C₃×#41	C₂×D₅	semidirect
136. C₂²×C₃×Q₈	C₂×D₅	semidirect
143. C₂²×C₃×D₄	C₂×D₅	semidirect
145. C₂³×C₃×C₄	C₂×D₅	semidirect
163. C₂×C₅×Q₈	C₂³×C₃	semidirect
167. C₂×C₅×D₄	C₂³×C₃	semidirect
169. C₂²×C₄×C₅	C₃×D₄	semidirect
174. C₂²×C₄×C₅	C₂³×C₃	semidirect
176. C₂×C₂²⋊D₅	C₂³×C₃	semidirect
184. #685	C₂³×C₃	semidirect
185. C₄⋊Dic₅	C₂³×C₃	semidirect
186. C₄×Dic₅	C₂³×C₃	semidirect
187. C₂²×#41	C₃×D₅	semidirect
220. C₂×C₃×Q₈	C₂²×D₅	semidirect
231. C₂×C₃×D₄	C₂²×D₅	semidirect
234. C₂²×C₃×C₄	C₂²⋊D₅	semidirect
239. C₂²×C₃×C₄	C₂²×D₅	semidirect
264. C₂×C₄×C₅	C₂×C₃×D₄	semidirect
275. C₂×#41	C₂×C₃×D₅	semidirect
276. C₂²×Q₈	C₂×C₃×D₅	semidirect
283. C₂²×D₄	C₂×C₃×D₅	semidirect
285. C₂³×C₄	C₂×C₃×D₅	semidirect
297. C₂×C₃×C₅	#535	semidirect
326. C₂×C₃×C₄	C₂×C₂²⋊D₅	semidirect
355. C₂×Q₈	C₂²×C₃×D₅	semidirect
366. C₂×D₄	C₂²×C₃×D₅	semidirect
369. C₂²×C₄	C₃×C₂²⋊D₅	semidirect
374. C₂²×C₄	C₂²×C₃×D₅	semidirect
379. C₃×C₅	C₂×#535	semidirect
399. C₂×C₅	C₃×#535	semidirect
428. C₂×C₄	C₂×C₃×C₂²⋊D₅	semidirect
435. C₂×C₃	#66164	direct (N × H)
438. C₅	C₂×C₃×#535	semidirect
456. C₃	C₂×#66164	direct (N × H)
460. C₂	C₃×#66164	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×C₅×#41	2
C₂³×C₃×D₂₀	2
C₃×#66164	2
C₂×C₃×#4093	2
C₂×C₃×#4086	2
C₂×C₃×#4083	2
C₂×C₃×#4076	2
C₂×C₃×#4072	2
C₂×#66164	3
C₂×C₃×#535	5

Subgroups — C2×C3×#66164 #14480366

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#66164 #14480366