Subgroups — #72316962

Order 4096.

At least 15127 conjugacy classes of subgroups; only the 579 normal subgroups are shown.

The subgroup lattice has 238 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72310419	C₂	1	1	yes	maximal, characteristic, complemented
2	#72309452	C₂	1	4	yes	maximal, complemented
3	#72309341	C₂	1	1	yes	maximal, characteristic, complemented
4	#72265678	C₂	1	1	yes	maximal, characteristic, complemented
5	#72264793	C₂	1	4	yes	maximal, complemented
6	#72240489	C₂	1	1	yes	maximal, characteristic, complemented
7	#72234600	C₂	1	1	yes	maximal, characteristic
8	#72233065	C₄	1	4	yes	complemented
9	#72225676	C₂	1	1	yes	maximal, characteristic, complemented
10	#72225617	C₂²	1	4	yes	—
11	#72219418	C₂²	1	4	yes	complemented
12	#72219298	C₂	1	1	yes	maximal, characteristic, complemented
13	#72219248	C₂²	1	4	yes	complemented
14	#72219237	C₂²	1	4	yes	—
15	#72219228	C₂²	1	2	yes	complemented
16	#72219227	C₂²	1	1	yes	characteristic
17	C₂²×#8534	C₂²	1	2	yes	complemented
18	C₂×#359428	C₂²	1	1	yes	characteristic
19	#72219038	C₄	1	4	yes	complemented
20	#72218518	C₂²	1	1	yes	characteristic
21	#72218518	C₂²	1	1	yes	characteristic
22	#72218498	C₂²	1	2	yes	complemented
23	#72218180	C₂²	1	2	yes	—
24	#72217875	C₂²	1	1	yes	characteristic
25	#72217789	C₂²	1	1	yes	characteristic
26	#72217511	C₂²	1	1	yes	characteristic
27	#72213862	C₂²	1	2	yes	—
28	#72210555	C₂²	1	2	yes	complemented
29	C₂×#62817	D₄	1	1	yes	characteristic
30	C₂²×#3484	C₂³	1	1	yes	characteristic
31	C₂⁴×#120	C₂³	1	1	yes	characteristic
32	#7642244	C₂×C₄	1	4	yes	—
33	#7642244	D₄	1	2	yes	—
34	#7630939	D₄	1	4	yes	—
35	#7630939	C₂³	1	2	yes	—
36	C₂×#35346	C₂×C₄	1	2	yes	complemented
37	#7626202	D₄	1	1	yes	characteristic
38	C₂³×#379	D₄	1	1	yes	characteristic, complemented
39	#6628892	C₂×C₄	1	4	yes	complemented
40	C₂×#21776	D₄	1	4	yes	complemented
41	C₂×#21746	D₄	1	1	yes	characteristic, complemented
42	C₂×#21746	D₄	1	1	yes	characteristic, complemented
43	C₂²×#1781	C₂³	1	2	yes	—
44	C₂×#14679	D₄	1	1	yes	characteristic, complemented
45	#517411	C₂³	1	1	yes	characteristic
46	#515285	C₂×C₄	1	2	yes	complemented
47	C₂²×#1438	C₂³	1	2	yes	—
48	#392699	C₂×C₄	1	4	yes	complemented
49	#392109	C₂³	1	2	yes	—
50	#392109	C₂³	1	2	yes	—
51	#392109	C₂³	1	2	yes	—
52	#382544	D₄	1	4	yes	complemented
53	#382434	D₄	1	4	yes	complemented
54	#382422	D₄	1	2	yes	complemented
55	#355096	D₄	1	4	yes	complemented
56	C₂×#10725	D₄	1	2	yes	complemented
57	#201687	C₂×C₄	1	8	yes	complemented
58	C₂×#8541	C₂×C₄	1	4	yes	complemented
59	C₂⁸	C₂×D₄	1	1	yes	characteristic
60	C₂³×H(5,𝔽₂)	C₂²×C₄	1	1	yes	characteristic, complemented
61	C₂⁵×D₄	C₂×D₄	1	2	yes	—
62	C₂⁵×D₄	C₂⁴	1	1	yes	characteristic
63	C₂×#3484	C₂²⋊C₄	1	4	yes	—
64	C₂×#3484	C₂×D₄	1	2	yes	—
65	C₂×#3484	C₂×D₄	1	4	yes	—
66	C₂³×#120	C₂²×C₄	1	2	yes	—
67	C₂³×#120	C₂×D₄	1	1	yes	characteristic
68	C₂³×#120	C₂×D₄	1	1	yes	characteristic
69	C₂³×#120	C₂×D₄	1	1	yes	characteristic
70	C₂³×#120	C₂×D₄	1	1	yes	characteristic
71	C₂³×#120	#41	1	2	yes	—
72	C₂²×#518	C₂²×C₄	1	1	yes	characteristic, complemented
73	C₂⁴×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
74	C₂⁴×C₂²⋊C₄	#41	1	2	yes	—
75	C₂×#2846	C₂×D₄	1	4	yes	—
76	#34980	C₂²⋊C₄	1	8	yes	—
77	#34980	C₂²×C₄	1	4	yes	—
78	#34980	#41	1	8	yes	—
79	#34171	C₂²⋊C₄	1	4	yes	—
80	#34171	C₂×D₄	1	4	yes	—
81	#34171	C₂×D₄	1	2	yes	—
82	C₂×#2308	C₂²×C₄	1	2	yes	complemented
83	C₂²×#379	C₂×D₄	1	2	yes	—
84	C₂²×#379	C₂×D₄	1	2	yes	—
85	C₂×#1781	C₂²×C₄	1	4	yes	complemented
86	C₂×#1781	C₂×D₄	1	4	yes	—
87	C₂×#1781	C₂×D₄	1	4	yes	—
88	C₂×#1781	C₂×D₄	1	4	yes	—
89	C₂×#1781	C₂×D₄	1	4	yes	—
90	C₂×#1781	C₂×D₄	1	2	yes	—
91	C₂×#1781	C₂×D₄	1	2	yes	—
92	C₂⁷	C₄×D₄	1	2	yes	—
93	C₂⁷	#120	1	1	yes	characteristic
94	C₂⁷	#120	1	1	yes	characteristic
95	C₂⁷	#121	1	2	yes	—
96	C₂⁷	C₂²×D₄	1	1	yes	characteristic
97	C₂⁷	C₂²×D₄	1	1	yes	characteristic
98	C₂²×H(5,𝔽₂)	#99	1	4	yes	complemented
99	C₂²×H(5,𝔽₂)	#120	1	2	yes	—
100	C₂⁴×D₄	C₂×C₂²⋊C₄	1	2	yes	—
101	C₂⁴×D₄	C₂×C₂²⋊C₄	1	2	yes	—
102	C₂⁴×D₄	#121	1	2	yes	—
103	C₂⁴×D₄	#121	1	2	yes	—
104	C₂⁴×D₄	C₂³×C₄	1	1	yes	characteristic
105	C₂⁴×D₄	C₂²×D₄	1	1	yes	characteristic
106	C₂⁴×D₄	C₂²×D₄	1	1	yes	characteristic
107	C₂⁴×D₄	C₂²×D₄	1	4	yes	—
108	C₂⁴×D₄	C₂×#41	1	1	yes	characteristic
109	C₂⁴×D₄	C₂×#41	1	4	yes	—
110	C₂⁵×C₄	#121	1	2	yes	—
111	C₂⁵×C₄	C₂×#41	1	1	yes	characteristic
112	C₂×#534	#99	1	8	yes	complemented
113	C₂×#534	C₄×D₄	1	8	yes	—
114	C₂×#534	#123	1	4	yes	—
115	C₂²×#120	C₂×C₂²⋊C₄	1	4	yes	—
116	C₂²×#120	#121	1	8	yes	—
117	C₂×#518	#99	1	4	yes	complemented
118	C₂×#518	#120	1	2	yes	—
119	C₂×#518	#120	1	4	yes	—
120	C₂³×C₂²⋊C₄	C₄×D₄	1	2	yes	—
121	C₂³×C₂²⋊C₄	C₄×D₄	1	2	yes	—
122	C₂³×C₂²⋊C₄	C₄×D₄	1	2	yes	—
123	C₂³×C₂²⋊C₄	#120	1	1	yes	characteristic
124	C₂³×C₂²⋊C₄	#120	1	1	yes	characteristic
125	C₂³×C₂²⋊C₄	#123	1	2	yes	—
126	C₂³×C₂²⋊C₄	#123	1	2	yes	—
127	C₂²×#99	#120	1	2	yes	—
128	C₂²×#99	#120	1	4	yes	—
129	C₂²×#99	#120	1	2	yes	—
130	#2308	C₄×D₄	1	8	yes	—
131	#2308	#123	1	4	yes	—
132	C₂×#379	C₂×C₂²⋊C₄	1	4	yes	—
133	C₂×#379	#120	1	2	yes	—
134	C₂×#379	#120	1	4	yes	—
135	C₂×#379	#120	1	2	yes	—
136	C₂×#379	#121	1	8	yes	—
137	C₂⁶	#386	1	2	yes	—
138	C₂⁶	#386	1	2	yes	—
139	C₂⁶	B(4,𝔽₂)	1	1	yes	characteristic
140	C₂⁶	B(4,𝔽₂)	1	2	yes	—
141	C₂⁶	B(4,𝔽₂)	1	1	yes	characteristic
142	C₂⁶	C₂²×C₂²⋊C₄	1	1	yes	characteristic
143	C₂⁶	C₂×C₄×D₄	1	1	yes	characteristic
144	C₂⁶	C₂×#120	1	1	yes	characteristic
145	C₂⁶	C₂×#120	1	2	yes	—
146	C₂⁶	C₂×#121	1	1	yes	characteristic
147	C₂⁶	C₂×#121	1	1	yes	characteristic
148	C₂⁶	C₂×#121	1	4	yes	—
149	C₂⁶	#525	1	2	yes	—
150	C₂³×D₄	#386	1	4	yes	—
151	C₂³×D₄	C₂×#99	1	4	yes	—
152	C₂³×D₄	C₂×#99	1	4	yes	—
153	C₂³×D₄	C₂×#99	1	8	yes	—
154	C₂³×D₄	C₂×C₄×D₄	1	4	yes	—
155	C₂³×D₄	C₂×#120	1	2	yes	—
156	C₂³×D₄	C₂×#120	1	2	yes	—
157	C₂³×D₄	C₂×#123	1	2	yes	—
158	C₂³×D₄	#525	1	4	yes	—
159	C₂³×D₄	#525	1	2	yes	—
160	C₂⁴×C₄	#386	1	2	yes	—
161	C₂⁴×C₄	#386	1	2	yes	—
162	C₂⁴×C₄	C₂×C₄×D₄	1	1	yes	characteristic
163	C₂⁴×C₄	C₂×#123	1	1	yes	characteristic
164	C₂⁴×C₄	#525	1	4	yes	—
165	C₂²×C₂²⋊C₄	#386	1	8	yes	—
166	C₂²×C₂²⋊C₄	#386	1	4	yes	—
167	C₂²×C₂²⋊C₄	C₂×#99	1	8	yes	—
168	#379	B(4,𝔽₂)	1	4	yes	complemented
169	C₂⁵	#1889	1	2	yes	—
170	C₂⁵	#1889	1	2	yes	—
171	C₂⁵	#1898	1	2	yes	—
172	C₂⁵	#1898	1	2	yes	—
173	C₂⁵	#1898	1	4	yes	complemented
174	C₂⁵	#1913	1	2	yes	—
175	C₂⁵	#1913	1	2	yes	—
176	C₂⁵	C₂×#386	1	2	yes	—
177	C₂⁵	C₂×#386	1	1	yes	characteristic
178	C₂⁵	C₄×#120	1	2	yes	—
179	C₂⁵	#2846	1	1	yes	characteristic
180	C₂⁵	#2847	1	2	yes	—
181	C₂⁵	#2847	1	2	yes	—
182	C₂⁵	#2849	1	1	yes	characteristic
183	C₂⁵	C₂²×#99	1	4	yes	—
184	C₂⁵	C₂×B(4,𝔽₂)	1	1	yes	characteristic
185	C₂⁵	C₂×B(4,𝔽₂)	1	1	yes	characteristic
186	C₂⁵	C₂×B(4,𝔽₂)	1	2	yes	—
187	C₂⁵	#3025	1	2	yes	—
188	C₂⁵	#3026	1	1	yes	characteristic
189	C₂⁵	#3026	1	1	yes	characteristic
190	C₂⁵	#3026	1	2	yes	—
191	C₂⁵	#3027	1	2	yes	—
192	C₂³×C₄	#1898	1	4	yes	—
193	C₂³×C₄	#1898	1	4	yes	complemented
194	C₂³×C₄	#1898	1	8	yes	complemented
195	C₂³×C₄	C₄×#120	1	2	yes	—
196	C₂³×C₄	#2849	1	2	yes	—
197	C₂⁴	#9495	1	2	yes	—
198	C₂⁴	#9495	1	2	yes	—
199	C₂⁴	#9495	1	2	yes	—
200	C₂⁴	#9495	1	2	yes	—
201	C₂⁴	#14694	1	1	yes	characteristic
202	C₂⁴	C₂×#1889	1	1	yes	characteristic
203	C₂⁴	#20960	1	1	yes	characteristic
204	C₂⁴	C₂×#1898	1	2	yes	—
205	C₂⁴	C₂×#1898	1	4	yes	—
206	C₂⁴	C₂×#1913	1	2	yes	—
207	C₂⁴	C₄×B(4,𝔽₂)	1	2	yes	—
208	C₂⁴	#21482	1	2	yes	—
209	C₂⁴	#32898	1	1	yes	characteristic
210	C₂⁴	#32898	1	1	yes	characteristic
211	C₂⁴	#32898	1	2	yes	—
212	C₂⁴	#32899	1	2	yes	—
213	C₂⁴	#32926	1	2	yes	—
214	C₂⁴	#32928	1	2	yes	—
215	C₂⁴	#33020	1	1	yes	characteristic
216	C₂⁴	#33020	1	1	yes	characteristic
217	C₂³	#139743	1	4	yes	complemented
218	C₂³	C₂×#9495	1	2	yes	—
219	C₂³	#201687	1	2	yes	—
220	C₂³	#347042	1	1	yes	characteristic
221	C₂³	#347043	1	2	yes	—
222	C₂³	#347044	1	1	yes	characteristic
223	C₂³	#392658	1	1	yes	characteristic
224	C₂³	#392663	1	2	yes	—
225	C₂³	#392675	1	1	yes	characteristic
226	C₂³	#392699	1	1	yes	characteristic
227	C₂³	#392719	1	2	yes	—
228	C₂²	C₂×#139743	1	2	yes	—
229	C₂²	#72219234	1	1	yes	characteristic
230	C₂²	#72234150	1	2	yes	—
231	C₂²	#72255841	1	2	yes	—
232	C₂²	#72258052	1	1	yes	characteristic
233	C₂²	#72258512	1	1	yes	characteristic
234	C₂²	#72261568	1	2	yes	—
235	C₂	#72258103	1	2	yes	—
236	C₂	#72266364	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#72258052
commutator	C₂⁴×D₄	C₂³×C₄
Frattini, cosocle	C₂⁵×D₄	C₂⁴
prime radical	#72234600	C₂
LCS	G ▷ C₂⁴×D₄ ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ C₂⁴×#120 ◁ G
LECS	G ▷ C₂⁵×D₄ ▷ C₂⁴ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72310419	C₂	semidirect
2. #72309452	C₂	semidirect
3. #72309341	C₂	semidirect
4. #72265678	C₂	semidirect
5. #72264793	C₂	semidirect
6. #72240489	C₂	semidirect
8. #72233065	C₄	semidirect
9. #72225676	C₂	semidirect
11. #72219418	C₂²	semidirect
12. #72219298	C₂	semidirect
13. #72219248	C₂²	semidirect
15. #72219228	C₂²	semidirect
17. C₂²×#8534	C₂²	semidirect
19. #72219038	C₄	semidirect
22. #72218498	C₂²	semidirect
28. #72210555	C₂²	semidirect
36. C₂×#35346	C₂×C₄	semidirect
38. C₂³×#379	D₄	semidirect
39. #6628892	C₂×C₄	semidirect
40. C₂×#21776	D₄	semidirect
41. C₂×#21746	D₄	semidirect
42. C₂×#21746	D₄	semidirect
44. C₂×#14679	D₄	semidirect
46. #515285	C₂×C₄	semidirect
48. #392699	C₂×C₄	semidirect
52. #382544	D₄	semidirect
53. #382434	D₄	semidirect
54. #382422	D₄	semidirect
55. #355096	D₄	semidirect
56. C₂×#10725	D₄	semidirect
57. #201687	C₂×C₄	semidirect
58. C₂×#8541	C₂×C₄	semidirect
60. C₂³×H(5,𝔽₂)	C₂²×C₄	semidirect
72. C₂²×#518	C₂²×C₄	semidirect
82. C₂×#2308	C₂²×C₄	semidirect
85. C₂×#1781	C₂²×C₄	semidirect
98. C₂²×H(5,𝔽₂)	#99	semidirect
112. C₂×#534	#99	semidirect
117. C₂×#518	#99	semidirect
168. #379	B(4,𝔽₂)	semidirect
173. C₂⁵	#1898	semidirect
193. C₂³×C₄	#1898	semidirect
194. C₂³×C₄	#1898	semidirect
217. C₂³	#139743	semidirect

Maximal subgroups

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

subgroup	index
#72310419	2
#72309452	2
#72309341	2
#72265678	2
#72264793	2
#72240489	2
#72234600	2
#72225676	2
#72219298	2