Subgroups — #72323210

Order 4096.

At least 10759 conjugacy classes of subgroups; only the 301 normal subgroups are shown.

The subgroup lattice has 211 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72317935	C₂	1	1	yes	maximal, characteristic, complemented
2	#72315164	C₂	1	1	yes	maximal, characteristic, complemented
3	#72312205	C₂	1	1	yes	maximal, characteristic, complemented
4	#72310380	C₂	1	1	yes	maximal, characteristic, complemented
5	#72310339	C₂	1	2	yes	maximal, complemented
6	#72310332	C₂	1	1	yes	maximal, characteristic, complemented
7	#72309548	C₂	1	2	yes	maximal, complemented
8	#72266073	C₂²	1	1	yes	characteristic
9	#72264793	C₂	1	2	yes	maximal, complemented
10	#72262330	C₂	1	1	yes	maximal, characteristic, complemented
11	#72238345	C₂	1	2	yes	maximal, complemented
12	#72237591	C₂	1	1	yes	maximal, characteristic, complemented
13	#72237400	C₂²	1	1	yes	characteristic
14	#72234668	C₂²	1	1	yes	characteristic
15	#72234661	C₂²	1	1	yes	characteristic
16	#72225667	C₂²	1	1	yes	characteristic
17	#72225650	C₂²	1	2	yes	—
18	#72225619	C₂²	1	1	yes	characteristic
19	#72225617	C₂²	1	2	yes	—
20	C₂×#115348	C₂²	1	1	yes	characteristic
21	#72219418	C₂²	1	1	yes	characteristic
22	#72219418	C₂²	1	1	yes	characteristic
23	#72219418	C₂²	1	2	yes	—
24	#72219311	C₂²	1	2	yes	—
25	#72219248	C₂²	1	2	yes	—
26	#72219237	C₂²	1	2	yes	—
27	#72219229	C₂²	1	1	yes	characteristic
28	C₂²×#8534	C₂²	1	1	yes	characteristic
29	#72218381	C₂²	1	2	yes	—
30	#72218381	C₂²	1	2	yes	—
31	#72218057	C₂²	1	2	yes	—
32	#72218057	C₂²	1	2	yes	—
33	#72213862	C₂²	1	2	yes	—
34	#72210555	C₂²	1	2	yes	—
35	C₂²×#3484	C₂³	1	1	yes	characteristic
36	#7630939	C₂³	1	2	yes	—
37	C₂×#34980	C₂³	1	1	yes	characteristic
38	#7626202	D₄	1	1	yes	characteristic, complemented
39	C₂×#32873	D₄	1	1	yes	characteristic, complemented
40	#6440460	D₄	1	2	yes	complemented
41	C₂×#23895	D₄	1	1	yes	characteristic, complemented
42	#6437621	D₄	1	2	yes	complemented
43	C₂×#21776	D₄	1	2	yes	complemented
44	C₂×#21746	D₄	1	1	yes	characteristic, complemented
45	C₂²×#1781	C₂³	1	1	yes	characteristic
46	C₂²×#1781	C₂³	1	1	yes	characteristic
47	C₂²×#1781	C₂³	1	1	yes	characteristic
48	C₂²×#1781	C₂³	1	1	yes	characteristic
49	C₂²×#1438	C₂³	1	1	yes	characteristic
50	#392109	C₂³	1	2	yes	—
51	#392109	C₂³	1	2	yes	—
52	#392109	C₂³	1	2	yes	—
53	#392030	D₄	1	1	yes	characteristic, complemented
54	#392030	D₄	1	2	yes	complemented
55	#382544	D₄	1	2	yes	complemented
56	#382434	D₄	1	2	yes	complemented
57	#373637	D₄	1	1	yes	characteristic, complemented
58	#359811	D₄	1	2	yes	complemented
59	#359549	D₄	1	2	yes	complemented
60	#359442	D₄	1	2	yes	complemented
61	#355913	D₄	1	1	yes	characteristic, complemented
62	#355913	D₄	1	1	yes	characteristic, complemented
63	#355096	D₄	1	2	yes	complemented
64	C₂³×H(5,𝔽₂)	#41	1	1	yes	characteristic, complemented
65	C₂⁵×D₄	C₂⁴	1	1	yes	characteristic
66	C₂×#3484	C₂×D₄	1	2	yes	—
67	C₂×#3484	C₂×D₄	1	2	yes	—
68	C₂×#3484	C₂×D₄	1	2	yes	—
69	C₂²×#534	C₂×D₄	1	2	yes	—
70	C₂³×#120	C₂×D₄	1	1	yes	characteristic
71	C₂²×#518	#41	1	1	yes	characteristic, complemented
72	#34171	C₂×D₄	1	2	yes	—
73	#34171	C₂×D₄	1	2	yes	—
74	#34171	C₂×D₄	1	2	yes	—
75	C₂×#2308	C₂×D₄	1	2	yes	—
76	C₂²×#379	C₂×D₄	1	1	yes	characteristic
77	C₂²×#379	C₂×D₄	1	1	yes	characteristic
78	C₂²×#379	C₂×D₄	1	1	yes	characteristic
79	C₂²×#379	C₂×D₄	1	1	yes	characteristic
80	C₂²×#379	C₂×D₄	1	1	yes	characteristic
81	C₂×#1781	C₂×D₄	1	2	yes	—
82	C₂×#1781	C₂×D₄	1	1	yes	characteristic
83	C₂×#1781	C₂×D₄	1	1	yes	characteristic
84	C₂×#1781	C₂×D₄	1	2	yes	—
85	C₂×#1781	C₂×D₄	1	1	yes	characteristic
86	C₂×#1781	C₂×D₄	1	1	yes	characteristic
87	C₂×#1781	C₂×D₄	1	2	yes	—
88	C₂×#1781	C₂×D₄	1	2	yes	—
89	C₂×#1781	C₂×D₄	1	2	yes	—
90	C₂×#1781	C₂×D₄	1	2	yes	—
91	C₂×#1781	C₂×D₄	1	2	yes	—
92	C₂×#1781	C₂×D₄	1	2	yes	—
93	C₂×#1781	#41	1	2	yes	complemented
94	C₂×#1781	#41	1	4	yes	complemented
95	C₂⁷	C₂²×D₄	1	1	yes	characteristic
96	C₂⁷	H(5,𝔽₂)	1	1	yes	characteristic
97	C₂²×H(5,𝔽₂)	#120	1	2	yes	—
98	C₂⁴×D₄	#120	1	1	yes	characteristic
99	C₂⁴×D₄	C₂²×D₄	1	2	yes	—
100	C₂⁴×D₄	C₂²×D₄	1	1	yes	characteristic
101	C₂⁴×D₄	C₂²×D₄	1	2	yes	—
102	C₂⁴×D₄	C₂²×D₄	1	1	yes	characteristic
103	C₂⁴×D₄	C₂×#41	1	2	yes	—
104	C₂⁴×D₄	C₂×#41	1	1	yes	characteristic
105	C₂⁴×D₄	C₂×#41	1	1	yes	characteristic
106	C₂⁴×D₄	H(5,𝔽₂)	1	2	yes	—
107	C₂⁵×C₄	2₋¹⁺⁴	1	1	yes	characteristic
108	C₂×#518	#120	1	2	yes	—
109	C₂×#518	#120	1	2	yes	—
110	C₂×#518	#120	1	2	yes	—
111	C₂³×C₂²⋊C₄	#120	1	2	yes	—
112	C₂³×C₂²⋊C₄	#120	1	1	yes	characteristic
113	C₂²×#99	#120	1	2	yes	—
114	C₂²×#99	#120	1	2	yes	—
115	C₂²×#99	#120	1	2	yes	—
116	C₂²×#99	#120	1	2	yes	—
117	C₂×#379	#120	1	1	yes	characteristic
118	C₂×#379	#120	1	1	yes	characteristic
119	C₂×#379	#120	1	1	yes	characteristic
120	C₂×#379	#120	1	1	yes	characteristic
121	C₂×#379	#120	1	2	yes	—
122	C₂×#379	#120	1	1	yes	characteristic
123	C₂×#379	#120	1	1	yes	characteristic
124	C₂⁶	B(4,𝔽₂)	1	1	yes	characteristic
125	C₂⁶	C₂×#120	1	1	yes	characteristic
126	C₂⁶	C₂×#120	1	1	yes	characteristic
127	C₂⁶	C₂×#120	1	1	yes	characteristic
128	C₂⁶	#525	1	1	yes	characteristic
129	C₂⁶	#532	1	1	yes	characteristic
130	C₂⁶	#535	1	1	yes	characteristic
131	C₂⁶	#535	1	2	yes	—
132	C₂⁶	#537	1	1	yes	characteristic
133	C₂⁶	#546	1	1	yes	characteristic
134	C₂⁶	#546	1	2	yes	—
135	C₂⁶	#550	1	1	yes	characteristic
136	C₂³×D₄	C₂×#120	1	2	yes	—
137	C₂³×D₄	C₂×#120	1	2	yes	—
138	C₂³×D₄	#525	1	2	yes	—
139	C₂³×D₄	#532	1	2	yes	—
140	C₂³×D₄	#546	1	2	yes	—
141	C₂³×D₄	#546	1	2	yes	—
142	C₂³×D₄	#546	1	2	yes	—
143	C₂³×D₄	#550	1	2	yes	—
144	C₂⁴×C₄	B(4,𝔽₂)	1	1	yes	characteristic
145	C₂⁴×C₄	#537	1	2	yes	—
146	C₂⁴×C₄	#547	1	2	yes	—
147	C₂⁴×C₄	#547	1	1	yes	characteristic
148	C₂⁴×C₄	#548	1	1	yes	characteristic
149	C₂²×C₂²⋊C₄	B(4,𝔽₂)	1	2	yes	—
150	#379	B(4,𝔽₂)	1	2	yes	—
151	#379	B(4,𝔽₂)	1	2	yes	—
152	C₂⁵	#2388	1	1	yes	characteristic
153	C₂⁵	#2404	1	1	yes	characteristic
154	C₂⁵	#2404	1	2	yes	—
155	C₂⁵	#2407	1	1	yes	characteristic
156	C₂⁵	#2430	1	1	yes	characteristic
157	C₂⁵	#2431	1	2	yes	—
158	C₂⁵	#2431	1	1	yes	characteristic
159	C₂⁵	#2615	1	1	yes	characteristic
160	C₂⁵	#2846	1	1	yes	characteristic
161	C₂⁵	C₂×B(4,𝔽₂)	1	1	yes	characteristic
162	C₂⁵	C₂×B(4,𝔽₂)	1	1	yes	characteristic
163	C₂⁵	C₂×B(4,𝔽₂)	1	1	yes	characteristic
164	C₂⁵	C₂×B(4,𝔽₂)	1	1	yes	characteristic
165	C₂⁵	#3025	1	1	yes	characteristic
166	C₂⁵	#3025	1	1	yes	characteristic
167	C₂⁵	#3026	1	1	yes	characteristic
168	C₂⁵	#3026	1	1	yes	characteristic
169	C₂⁵	#3026	1	1	yes	characteristic
170	C₂⁵	#3026	1	1	yes	characteristic
171	C₂⁵	#3027	1	1	yes	characteristic
172	C₂⁵	#3027	1	1	yes	characteristic
173	C₂³×C₄	#2407	1	2	yes	—
174	C₂³×C₄	#2430	1	2	yes	—
175	C₂⁴	#14885	1	1	yes	characteristic
176	C₂⁴	#23595	1	1	yes	characteristic
177	C₂⁴	#23597	1	1	yes	characteristic
178	C₂⁴	#23984	1	1	yes	characteristic
179	C₂⁴	#23987	1	1	yes	characteristic
180	C₂⁴	#23996	1	4	yes	—
181	C₂⁴	#24140	1	1	yes	characteristic
182	C₂⁴	#24142	1	1	yes	characteristic
183	C₂⁴	#25563	1	1	yes	characteristic
184	C₂⁴	#25563	1	1	yes	characteristic
185	C₂⁴	#25585	1	1	yes	characteristic
186	C₂⁴	#25587	1	1	yes	characteristic
187	C₂⁴	#29684	1	1	yes	characteristic
188	C₂⁴	#29686	1	1	yes	characteristic
189	C₂⁴	#32898	1	1	yes	characteristic
190	C₂⁴	#32898	1	1	yes	characteristic
191	C₂⁴	#32898	1	1	yes	characteristic
192	C₂⁴	#32898	1	1	yes	characteristic
193	C₂⁴	#32899	1	1	yes	characteristic
194	C₂⁴	#32899	1	1	yes	characteristic
195	C₂³	#347042	1	1	yes	characteristic
196	C₂³	#347043	1	1	yes	characteristic
197	C₂³	#347043	1	1	yes	characteristic
198	C₂³	#347044	1	1	yes	characteristic
199	C₂³	#403939	1	1	yes	characteristic
200	C₂³	#403941	1	1	yes	characteristic
201	C₂³	#403943	1	2	yes	—
202	C₂³	#403952	1	1	yes	characteristic
203	C₂³	#404015	1	2	yes	—
204	C₂²	#72232805	1	1	yes	characteristic
205	C₂²	#72234956	1	2	yes	—
206	C₂²	#72234958	1	2	yes	—
207	C₂²	#72258395	1	1	yes	characteristic
208	C₂²	#72265945	1	1	yes	characteristic
209	C₂	#72266256	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72266256
commutator, Frattini, cosocle	C₂⁵×D₄	C₂⁴
LCS	G ▷ C₂⁵×D₄ ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁴ ◁ C₂⁵×D₄ ◁ 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. #72317935	C₂	semidirect
2. #72315164	C₂	semidirect
3. #72312205	C₂	semidirect
4. #72310380	C₂	semidirect
5. #72310339	C₂	semidirect
6. #72310332	C₂	semidirect
7. #72309548	C₂	semidirect
9. #72264793	C₂	semidirect
10. #72262330	C₂	semidirect
11. #72238345	C₂	semidirect
12. #72237591	C₂	semidirect
38. #7626202	D₄	semidirect
39. C₂×#32873	D₄	semidirect
40. #6440460	D₄	semidirect
41. C₂×#23895	D₄	semidirect
42. #6437621	D₄	semidirect
43. C₂×#21776	D₄	semidirect
44. C₂×#21746	D₄	semidirect
53. #392030	D₄	semidirect
54. #392030	D₄	semidirect
55. #382544	D₄	semidirect
56. #382434	D₄	semidirect
57. #373637	D₄	semidirect
58. #359811	D₄	semidirect
59. #359549	D₄	semidirect
60. #359442	D₄	semidirect
61. #355913	D₄	semidirect
62. #355913	D₄	semidirect
63. #355096	D₄	semidirect
64. C₂³×H(5,𝔽₂)	#41	semidirect
71. C₂²×#518	#41	semidirect
93. C₂×#1781	#41	semidirect
94. C₂×#1781	#41	semidirect

Maximal subgroups

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

subgroup	index
#72317935	2
#72315164	2
#72312205	2
#72310380	2
#72310339	2
#72310332	2
#72309548	2
#72264793	2
#72262330	2
#72238345	2
#72237591	2