Subgroups — #12253739

Order 1280.

710 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₅×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	#10615473	C₂	1	2	yes	maximal, complemented
3	C₂×#65193	C₂	1	1	yes	maximal, characteristic, complemented
4	#10602209	C₂	1	2	yes	maximal, complemented
5	#10602207	C₂	1	4	yes	maximal, complemented
6	#10602206	C₂	1	2	yes	maximal, complemented
7	#10602205	C₂	1	2	yes	maximal, complemented
8	#10602197	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
10	#66125	—	2	2	no	—
11	C₂×#4050	C₂²	1	2	yes	—
12	C₂²×C₅×C₁₆	—	2	4	no	—
13	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
14	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
15	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
16	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
17	#65405	C₂²	1	4	yes	—
18	#65404	C₂²	1	4	yes	—
19	#65403	—	2	4	no	—
20	#65402	—	2	2	no	—
21	#65395	C₂²	1	2	yes	—
22	C₂×Dic₄₀	C₂²	1	1	yes	characteristic, complemented
23	#65193	—	2	4	no	—
24	C₂×#3932	C₂²	1	2	yes	complemented
25	C₂×D₈₀	C₂²	1	1	yes	characteristic, complemented
26	#64729	C₂²	1	4	yes	complemented
27	#64725	C₂²	1	2	yes	complemented
28	#64724	C₂²	1	2	yes	complemented
29	#64723	C₂²	1	4	yes	complemented
30	#33600	—	5	1	no	maximal, Sylow
31	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
32	C₂×#709	—	2	2	no	—
33	C₂²×C₅×C₈	—	2	2	no	—
34	C₂²×C₅×C₈	—	2	2	no	—
35	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
36	C₂²×C₅×C₈	D₄	1	2	yes	—
37	C₂²×C₅×C₈	C₂³	1	2	yes	—
38	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
39	#4099	—	4	2	no	—
40	#4076	—	2	4	no	—
41	#4075	—	4	2	no	—
42	C₄×C₂²⋊D₅	—	4	4	no	—
43	#4072	—	2	2	no	—
44	#4070	—	2	4	no	—
45	C₂×Dic₂₀	C₂³	1	2	yes	—
46	#4050	—	2	4	no	—
47	#4050	—	2	4	no	—
48	C₂×D₄₀	C₂³	1	2	yes	—
49	C₂×#677	—	2	2	no	—
50	C₂×C₅×C₁₆	—	2	2	no	—
51	C₂×C₅×C₁₆	—	2	4	no	—
52	C₂×C₅×C₁₆	—	2	4	no	—
53	C₂×C₅×C₁₆	D₄	1	4	yes	complemented
54	C₂×C₅×C₁₆	C₂³	1	2	yes	—
55	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
56	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
57	#3953	—	2	4	no	—
58	#3950	C₂³	1	4	yes	—
59	#3949	—	2	2	no	—
60	#3948	—	2	4	no	—
61	Dic₄₀	—	2	2	no	—
62	#3932	—	2	4	no	—
63	D₈₀	—	2	2	no	—
64	C₂×#508	—	5	1	no	—
65	C₂³×C₁₆	D₅	1	1	yes	characteristic, complemented
66	#3050	—	5	2	no	—
67	#2217	—	5	4	no	—
68	#2216	—	5	2	no	—
69	#2215	—	5	2	no	—
70	#2158	—	5	1	no	—
71	#2138	—	5	2	no	—
72	C₂⁴×C₅	D₈	1	1	yes	characteristic
73	C₂²×C₄×C₅	—	2	2	no	—
74	C₂²×C₄×C₅	—	2	2	no	—
75	C₂²×C₄×C₅	D₈	1	2	yes	—
76	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
77	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
79	C₂×C₂²⋊D₅	—	4	2	no	—
80	C₂×C₂²⋊D₅	—	4	4	no	—
81	#709	—	4	4	no	—
82	#709	—	4	4	no	—
83	C₂×D₂₀	—	2	2	no	—
84	C₂×C₄×D₅	—	4	2	no	—
85	C₂×Dic₁₀	—	2	2	no	—
86	C₂×C₅×C₈	—	2	2	no	—
87	C₂×C₅×C₈	—	2	4	no	—
88	C₂×C₅×C₈	—	2	2	no	—
89	C₂×C₅×C₈	—	2	2	no	—
90	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
91	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
92	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
93	C₂×C₅×C₈	#41	1	2	yes	—
94	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
95	C₂²⋊Dic₅	—	4	4	no	—
96	C₂²⋊Dic₅	—	4	2	no	—
97	#685	—	4	4	no	—
98	C₄⋊Dic₅	—	2	4	no	—
99	#683	—	4	4	no	—
100	C₄×Dic₅	—	4	2	no	—
101	Dic₂₀	—	2	2	no	—
102	Dic₂₀	—	2	2	no	—
103	D₄₀	—	2	2	no	—
104	D₄₀	—	2	2	no	—
105	#677	—	4	4	no	—
106	C₅×C₁₆	—	2	2	no	—
107	C₅×C₁₆	C₂×D₄	1	2	yes	—
108	C₅×C₁₆	C₂×D₄	1	2	yes	—
109	C₂×#135	—	5	2	no	—
110	C₂³×C₈	C₂×D₅	1	1	yes	characteristic
111	#525	—	10	2	no	—
112	#508	—	10	4	no	—
113	C₂×Q₃₂	—	5	1	no	—
114	C₂×SD₃₂	—	5	2	no	—
115	C₂×D₁₆	—	5	1	no	—
116	C₂²×C₁₆	—	2	4	no	—
117	C₂²×C₁₆	C₂×D₅	1	1	yes	characteristic
118	C₂²×C₁₆	C₂×D₅	1	2	yes	—
119	C₂²×C₁₆	C₂×D₅	1	2	yes	—
120	C₂²×C₁₆	C₂×D₅	1	1	yes	characteristic
121	#467	—	5	4	no	—
122	#466	—	5	4	no	—
123	#465	—	10	4	no	—
124	#427	—	5	2	no	—
125	#416	—	10	2	no	—
126	#367	—	5	2	no	—
127	#366	—	5	2	no	—
128	#358	—	5	4	no	—
129	#357	—	5	4	no	—
130	C₂³×C₅	—	2	2	no	—
131	C₂³×C₅	—	2	4	no	—
132	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
133	C₂³×C₅	C₂×D₈	1	2	yes	—
134	C₂²×D₅	—	4	2	no	—
135	C₂×C₄×C₅	—	2	2	no	—
136	C₂×C₄×C₅	—	2	2	no	—
137	C₂×C₄×C₅	—	2	2	no	—
138	C₂×C₄×C₅	—	2	4	no	—
139	C₂×C₄×C₅	#121	1	2	yes	—
140	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
141	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
142	C₂×C₄×C₅	#135	1	2	yes	—
143	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
144	C₂²⋊D₅	—	8	4	no	—
145	C₂²⋊D₅	—	8	8	no	—
146	C₂×Dic₅	—	4	4	no	—
147	C₂×Dic₅	—	4	2	no	—
148	D₂₀	—	4	2	no	—
149	D₂₀	—	4	2	no	—
150	C₄×D₅	—	8	4	no	—
151	Dic₁₀	—	4	2	no	—
152	Dic₁₀	—	4	2	no	—
153	C₅×C₈	—	2	2	no	—
154	C₅×C₈	#121	1	2	yes	—
155	C₅×C₈	C₂²×D₄	1	1	yes	characteristic
156	C₅×C₈	C₂×#41	1	1	yes	characteristic
157	C₂×#41	—	10	2	no	—
158	C₂³×C₄	D₂₀	1	1	yes	characteristic
159	#135	—	10	4	no	—
160	#135	—	10	4	no	—
161	C₂×Q₁₆	—	5	2	no	—
162	C₂×SD₁₆	—	10	2	no	—
163	C₂×D₈	—	5	2	no	—
164	C₂²×C₈	—	2	2	no	—
165	C₂²×C₈	—	2	2	no	—
166	C₂²×C₈	D₂₀	1	1	yes	characteristic
167	C₂²×C₈	D₂₀	1	2	yes	—
168	C₂²×C₈	C₂²×D₅	1	2	yes	—
169	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
170	#123	—	20	2	no	—
171	C₂²⋊Q₈	—	10	4	no	—
172	#121	—	10	4	no	—
173	#120	—	20	2	no	—
174	C₄×D₄	—	20	4	no	—
175	#117	—	10	2	no	—
176	Q₃₂	—	10	2	no	—
177	SD₃₂	—	10	4	no	—
178	D₁₆	—	10	2	no	—
179	C₂×C₁₆	—	2	2	no	—
180	C₂×C₁₆	—	2	4	no	—
181	C₂×C₁₆	—	2	4	no	—
182	C₂×C₁₆	C₂²⋊D₅	1	4	yes	complemented
183	C₂×C₁₆	C₂²×D₅	1	2	yes	—
184	C₂×C₁₆	C₂²×D₅	1	1	yes	characteristic
185	C₂×C₁₆	C₂²×D₅	1	1	yes	characteristic
186	#107	—	5	4	no	—
187	#106	—	10	2	no	—
188	#103	—	10	4	no	—
189	#102	—	10	4	no	—
190	C₂²×C₅	—	2	8	no	—
191	C₂²×C₅	—	2	2	no	—
192	C₂²×C₅	—	2	2	no	—
193	C₂²×C₅	—	2	2	no	—
194	C₂²×C₅	#466	1	2	yes	—
195	C₂²×C₅	#508	1	4	yes	complemented
196	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
197	C₂×D₅	—	8	4	no	—
198	C₂×D₅	—	8	2	no	—
199	C₄×C₅	—	2	2	no	—
200	C₄×C₅	#466	1	2	yes	—
201	C₄×C₅	C₂×#121	1	1	yes	characteristic
202	C₄×C₅	C₂×#135	1	1	yes	characteristic
203	Dic₅	—	8	4	no	—
204	Dic₅	—	8	2	no	—
205	C₂⁴	D₄₀	1	1	yes	characteristic
206	#41	—	20	4	no	—
207	#41	—	20	4	no	—
208	C₂×Q₈	—	10	2	no	—
209	C₂×D₄	—	10	2	no	—
210	C₂×D₄	—	20	4	no	—
211	C₂×D₄	—	20	2	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	20	2	no	—
215	C₂²×C₄	D₄₀	1	1	yes	characteristic
216	C₂²×C₄	D₄₀	1	2	yes	—
217	C₂²×C₄	C₂×D₂₀	1	2	yes	—
218	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
219	Q₁₆	—	10	2	no	—
220	Q₁₆	—	10	2	no	—
221	SD₁₆	—	20	4	no	—
222	D₈	—	10	2	no	—
223	D₈	—	10	2	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	4	no	—
228	C₂×C₈	C₂×D₂₀	1	2	yes	—
229	C₂×C₈	C₂×D₂₀	1	1	yes	characteristic
230	C₂×C₈	#709	1	2	yes	—
231	C₂×C₈	C₂×C₂²⋊D₅	1	2	yes	—
232	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
233	C₄⋊C₄	—	10	4	no	—
234	C₄⋊C₄	—	20	4	no	—
235	C₂²⋊C₄	—	20	2	no	—
236	C₂²⋊C₄	—	20	4	no	—
237	C₂²⋊C₄	—	20	4	no	—
238	C₄²	—	20	2	no	—
239	C₁₆	—	2	2	no	—
240	C₁₆	C₂×C₂²⋊D₅	1	2	yes	—
241	C₁₆	C₂×C₂²⋊D₅	1	2	yes	—
242	C₂×C₅	—	2	2	no	—
243	C₂×C₅	—	2	4	no	—
244	C₂×C₅	C₂×#466	1	1	yes	characteristic
245	C₂×C₅	C₂×#508	1	2	yes	—
246	D₅	—	16	2	no	—
247	C₂³	—	2	2	no	—
248	C₂³	—	2	4	no	—
249	C₂³	—	20	2	no	—
250	C₂³	C₂×D₄₀	1	1	yes	characteristic
251	C₂³	C₂×D₄₀	1	2	yes	—
252	Q₈	—	20	2	no	—
253	Q₈	—	20	2	no	—
254	D₄	—	20	2	no	—
255	D₄	—	20	2	no	—
256	D₄	—	40	4	no	—
257	D₄	—	40	8	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	20	4	no	—
263	C₂×C₄	—	20	2	no	—
264	C₂×C₄	—	40	4	no	—
265	C₂×C₄	C₂×D₄₀	1	2	yes	—
266	C₂×C₄	C₂×D₄₀	1	1	yes	characteristic
267	C₂×C₄	#4050	1	2	yes	—
268	C₂×C₄	#4076	1	2	yes	—
269	C₂×C₄	C₂²×D₂₀	1	1	yes	characteristic
270	C₈	—	2	2	no	—
271	C₈	#4076	1	2	yes	—
272	C₈	C₂×#709	1	1	yes	characteristic
273	C₈	C₂²×C₂²⋊D₅	1	1	yes	characteristic
274	C₅	#33600	1	1	yes	characteristic, complemented, Sylow
275	C₂²	—	2	8	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	40	2	no	—
280	C₂²	—	40	4	no	—
281	C₂²	#65193	1	4	yes	complemented
282	C₂²	#65404	1	2	yes	—
283	C₂²	C₂²×D₄₀	1	1	yes	characteristic
284	C₄	—	2	2	no	—
285	C₄	—	40	2	no	—
286	C₄	—	40	4	no	—
287	C₄	#65404	1	2	yes	—
288	C₄	C₂×#4050	1	1	yes	characteristic
289	C₄	C₂×#4076	1	1	yes	characteristic
290	C₂	—	2	2	no	—
291	C₂	—	2	4	no	—
292	C₂	—	80	2	no	—
293	C₂	C₂×#65193	1	2	yes	—
294	C₂	C₂×#65404	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄₀
commutator, cosocle	C₂×C₅×C₈	C₂⁴
Frattini	C₂×C₈	C₂³×D₅
Fitting	C₂³×C₅×C₁₆	C₂
socle	C₂²×C₅	C₂²×D₈
hypercenter	C₂³×C₁₆	D₅
hypocenter	C₅	#33600
LCS	G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂³×C₅×C₁₆	C₂	semidirect
2. #10615473	C₂	semidirect
3. C₂×#65193	C₂	semidirect
4. #10602209	C₂	semidirect
5. #10602207	C₂	semidirect
6. #10602206	C₂	semidirect
7. #10602205	C₂	semidirect
8. #10602197	C₂	semidirect
22. C₂×Dic₄₀	C₂²	semidirect
24. C₂×#3932	C₂²	semidirect
25. C₂×D₈₀	C₂²	semidirect
26. #64729	C₂²	semidirect
27. #64725	C₂²	semidirect
28. #64724	C₂²	semidirect
29. #64723	C₂²	semidirect
53. C₂×C₅×C₁₆	D₄	semidirect
65. C₂³×C₁₆	D₅	semidirect
182. C₂×C₁₆	C₂²⋊D₅	semidirect
195. C₂²×C₅	#508	semidirect
274. C₅	#33600	semidirect
281. C₂²	#65193	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₅×C₁₆	2
#10615473	2
C₂×#65193	2
#10602209	2
#10602207	2
#10602206	2
#10602205	2
#10602197	2
#33600	5