Subgroups — #72309986

Order 4096.

At least 7764 conjugacy classes of subgroups; only the 489 normal subgroups are shown.

The subgroup lattice has 210 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72309929	C₂	1	1	yes	maximal, characteristic, complemented
2	#72309848	C₂	1	1	yes	maximal, characteristic, complemented
3	#72309737	C₂	1	1	yes	maximal, characteristic, complemented
4	#72309539	C₂	1	4	yes	maximal, complemented
5	#72266039	C₂	1	1	yes	maximal, characteristic, complemented
6	#72266028	C₂²	1	2	yes	complemented
7	#72264854	C₂	1	4	yes	maximal, complemented
8	#72259008	C₂²	1	1	yes	characteristic
9	#72258968	C₂	1	1	yes	maximal, characteristic
10	#72238614	C₂	1	1	yes	maximal, characteristic, complemented
11	#72238595	C₄	1	4	yes	complemented
12	#72237588	C₂	1	1	yes	maximal, characteristic, complemented
13	#72234668	C₂²	1	1	yes	characteristic
14	#72225713	C₂²	1	1	yes	characteristic
15	#72225617	C₂²	1	4	yes	—
16	#72224136	C₂²	1	1	yes	characteristic
17	#72219414	C₂²	1	4	yes	complemented
18	#72219311	C₂²	1	1	yes	characteristic
19	#72219310	C₂²	1	1	yes	characteristic
20	#72219254	C₂²	1	1	yes	characteristic
21	C₂²×#8541	C₂²	1	2	yes	complemented
22	#72218381	C₂²	1	4	yes	complemented
23	#72218198	C₂²	1	2	yes	complemented
24	#72218180	C₂²	1	2	yes	—
25	#72218069	C₂²	1	4	yes	—
26	#72215858	C₂²	1	2	yes	—
27	#72215179	C₂²	1	2	yes	complemented
28	#72209716	C₄	1	4	yes	complemented
29	#10586404	D₄	1	1	yes	characteristic
30	C₂⁴×#120	C₂³	1	1	yes	characteristic
31	#7630939	C₂³	1	2	yes	—
32	C₂×#34980	C₂³	1	1	yes	characteristic
33	#7626202	D₄	1	1	yes	characteristic, complemented
34	#6499870	D₄	1	4	yes	—
35	#6499867	C₂×C₄	1	4	yes	—
36	#6374251	D₄	1	1	yes	characteristic, complemented
37	#6374210	D₄	1	2	yes	—
38	C₂×#21776	D₄	1	4	yes	complemented
39	#6374128	D₄	1	1	yes	characteristic
40	#6374128	D₄	1	1	yes	characteristic, complemented
41	C₂×#21746	D₄	1	1	yes	characteristic, complemented
42	C₂²×#1781	C₂³	1	2	yes	—
43	C₂²×#1781	C₂³	1	2	yes	—
44	#6342371	C₂×C₄	1	2	yes	complemented
45	#6342371	C₂×C₄	1	4	yes	complemented
46	#4308128	C₂×C₄	1	2	yes	complemented
47	#517411	C₂³	1	1	yes	characteristic
48	#392675	C₂×C₄	1	4	yes	complemented
49	#392113	C₂³	1	2	yes	—
50	#392109	C₂³	1	2	yes	—
51	#392109	C₂³	1	2	yes	—
52	#382707	D₄	1	2	yes	complemented
53	#382544	D₄	1	4	yes	complemented
54	#382434	D₄	1	4	yes	complemented
55	#355913	D₄	1	2	yes	complemented
56	#355096	D₄	1	4	yes	complemented
57	#115690	C₂×C₄	1	8	yes	complemented
58	#115690	C₂×C₄	1	4	yes	complemented
59	C₂⁸	C₂²×C₄	1	1	yes	characteristic, complemented
60	C₂⁵×D₄	C₂⁴	1	1	yes	characteristic
61	#62817	C₂²⋊C₄	1	4	yes	—
62	C₂×#3484	C₂×D₄	1	2	yes	—
63	C₂×#3484	C₂×D₄	1	4	yes	—
64	C₂²×#534	C₂×D₄	1	1	yes	characteristic
65	C₂²×#534	C₂×D₄	1	1	yes	characteristic
66	C₂³×#120	C₂×D₄	1	1	yes	characteristic
67	C₂⁴×C₂²⋊C₄	C₂²×C₄	1	1	yes	characteristic, complemented
68	C₂×#2849	#41	1	2	yes	—
69	C₂×#2846	C₂²×C₄	1	4	yes	—
70	C₂×#2846	C₂²×C₄	1	2	yes	—
71	#34980	C₂×D₄	1	4	yes	—
72	#34980	#41	1	8	yes	—
73	C₂×#2613	C₂×D₄	1	2	yes	—
74	C₂×#2613	#41	1	2	yes	—
75	#34171	C₂×D₄	1	4	yes	—
76	#34171	C₂×D₄	1	2	yes	—
77	C₂×#2308	C₂×D₄	1	1	yes	characteristic
78	C₂×#2308	C₂×D₄	1	1	yes	characteristic
79	C₂²×#379	C₂×D₄	1	2	yes	—
80	C₂²×#379	C₂×D₄	1	2	yes	—
81	C₂²×#379	C₂×D₄	1	1	yes	characteristic
82	#21746	C₂²⋊C₄	1	8	yes	—
83	#21746	C₂²⋊C₄	1	4	yes	—
84	C₂×#1781	C₂×D₄	1	4	yes	—
85	C₂×#1781	C₂×D₄	1	4	yes	—
86	C₂×#1781	C₂×D₄	1	4	yes	—
87	C₂×#1781	C₂×D₄	1	4	yes	—
88	C₂×#1781	C₂×D₄	1	2	yes	—
89	C₂×#1781	C₂×D₄	1	2	yes	—
90	#14679	C₂²×C₄	1	2	yes	complemented
91	#14679	C₂²×C₄	1	4	yes	complemented
92	C₂⁷	C₂³×C₄	1	1	yes	characteristic
93	C₂⁷	C₂²×D₄	1	1	yes	characteristic
94	C₂²×H(5,𝔽₂)	#120	1	2	yes	—
95	C₂⁴×D₄	#120	1	1	yes	characteristic
96	C₂⁴×D₄	C₂²×D₄	1	1	yes	characteristic
97	C₂⁴×D₄	C₂²×D₄	1	1	yes	characteristic
98	C₂⁴×D₄	C₂²×D₄	1	1	yes	characteristic
99	C₂⁴×D₄	C₂²×D₄	1	4	yes	—
100	C₂⁴×D₄	C₂×#41	1	1	yes	characteristic
101	C₂⁴×D₄	C₂×#41	1	4	yes	—
102	C₂⁵×C₄	C₂×#41	1	1	yes	characteristic
103	C₂×#534	#121	1	8	yes	—
104	C₂×#534	#123	1	4	yes	—
105	C₂²×#123	#121	1	2	yes	—
106	C₂²×#123	#123	1	2	yes	—
107	C₂²×#120	C₂×C₂²⋊C₄	1	4	yes	—
108	C₂²×#120	C₂×C₂²⋊C₄	1	2	yes	—
109	C₂²×#120	C₄×D₄	1	2	yes	—
110	C₂²×#120	C₄×D₄	1	8	yes	—
111	C₂²×#120	C₄×D₄	1	2	yes	—
112	C₂²×#120	#121	1	2	yes	—
113	C₂×#518	#120	1	2	yes	—
114	C₂×#518	#120	1	4	yes	—
115	C₂³×C₂²⋊C₄	#120	1	1	yes	characteristic
116	C₂³×C₂²⋊C₄	#120	1	1	yes	characteristic
117	C₂³×C₂²⋊C₄	#120	1	1	yes	characteristic
118	C₂²×#99	#120	1	2	yes	—
119	C₂²×#99	#120	1	4	yes	—
120	C₂²×#99	#120	1	2	yes	—
121	#2308	#121	1	8	yes	—
122	#2308	#123	1	4	yes	—
123	C₂×#381	#121	1	2	yes	—
124	C₂×#381	#123	1	2	yes	—
125	C₂×#379	C₂×C₂²⋊C₄	1	4	yes	—
126	C₂×#379	C₂×C₂²⋊C₄	1	2	yes	—
127	C₂×#379	C₄×D₄	1	2	yes	—
128	C₂×#379	C₄×D₄	1	8	yes	—
129	C₂×#379	C₄×D₄	1	2	yes	—
130	C₂×#379	#120	1	2	yes	—
131	C₂×#379	#120	1	4	yes	—
132	C₂×#379	#120	1	2	yes	—
133	C₂×#379	#121	1	2	yes	—
134	C₂⁶	B(4,𝔽₂)	1	1	yes	characteristic
135	C₂⁶	C₂²×C₂²⋊C₄	1	1	yes	characteristic
136	C₂⁶	C₂×C₄×D₄	1	1	yes	characteristic
137	C₂⁶	C₂×C₄×D₄	1	4	yes	—
138	C₂⁶	C₂×C₄×D₄	1	1	yes	characteristic
139	C₂⁶	C₂×#120	1	2	yes	—
140	C₂⁶	C₂×#120	1	1	yes	characteristic
141	C₂⁶	C₂×#121	1	1	yes	characteristic
142	C₂⁶	#525	1	2	yes	—
143	C₂³×D₄	#386	1	2	yes	—
144	C₂³×D₄	#386	1	4	yes	—
145	C₂³×D₄	B(4,𝔽₂)	1	2	yes	—
146	C₂³×D₄	C₂×#120	1	2	yes	—
147	C₂³×D₄	C₂×#120	1	2	yes	—
148	C₂³×D₄	C₂×#121	1	4	yes	—
149	C₂³×D₄	C₂×#123	1	2	yes	—
150	C₂³×D₄	#525	1	2	yes	—
151	C₂³×D₄	#525	1	4	yes	—
152	C₂⁴×C₄	B(4,𝔽₂)	1	1	yes	characteristic
153	C₂⁴×C₄	C₂×#121	1	1	yes	characteristic
154	C₂⁴×C₄	C₂×#123	1	1	yes	characteristic
155	C₂⁴×C₄	#525	1	4	yes	—
156	C₂²×C₂²⋊C₄	#386	1	4	yes	—
157	C₂²×C₂²⋊C₄	#386	1	8	yes	—
158	C₂²×C₂²⋊C₄	#386	1	2	yes	—
159	C₂²×C₂²⋊C₄	#386	1	2	yes	—
160	C₂²×C₂²⋊C₄	#386	1	2	yes	—
161	#379	B(4,𝔽₂)	1	4	yes	complemented
162	C₂⁵	#1889	1	2	yes	—
163	C₂⁵	#1913	1	4	yes	complemented
164	C₂⁵	C₂×#386	1	1	yes	characteristic
165	C₂⁵	C₂×#386	1	2	yes	—
166	C₂⁵	C₄×#120	1	2	yes	—
167	C₂⁵	C₄×#120	1	2	yes	—
168	C₂⁵	#2846	1	1	yes	characteristic
169	C₂⁵	#2847	1	2	yes	—
170	C₂⁵	#2849	1	1	yes	characteristic
171	C₂⁵	C₂×B(4,𝔽₂)	1	2	yes	—
172	C₂⁵	C₂×B(4,𝔽₂)	1	1	yes	characteristic
173	C₂⁵	C₂×B(4,𝔽₂)	1	1	yes	characteristic
174	C₂⁵	#3025	1	2	yes	—
175	C₂⁵	#3026	1	2	yes	—
176	C₂⁵	#3026	1	1	yes	characteristic
177	C₂⁵	#3026	1	1	yes	characteristic
178	C₂⁵	#3027	1	2	yes	—
179	C₂³×C₄	#1889	1	2	yes	—
180	C₂³×C₄	#1913	1	4	yes	complemented
181	C₂³×C₄	#2847	1	2	yes	—
182	C₂³×C₄	#2849	1	2	yes	—
183	C₂⁴	#14694	1	1	yes	characteristic
184	C₂⁴	C₂×#1889	1	1	yes	characteristic
185	C₂⁴	#20960	1	1	yes	characteristic
186	C₂⁴	C₂×#1913	1	4	yes	—
187	C₂⁴	C₄×B(4,𝔽₂)	1	2	yes	—
188	C₂⁴	#21482	1	2	yes	—
189	C₂⁴	#32898	1	2	yes	—
190	C₂⁴	#32898	1	1	yes	characteristic
191	C₂⁴	#32898	1	1	yes	characteristic
192	C₂⁴	#32899	1	2	yes	—
193	C₂⁴	#32926	1	2	yes	—
194	C₂⁴	#32928	1	2	yes	—
195	C₂⁴	#33020	1	1	yes	characteristic
196	C₂⁴	#33020	1	1	yes	characteristic
197	C₂³	#347042	1	1	yes	characteristic
198	C₂³	#347043	1	2	yes	—
199	C₂³	#347044	1	1	yes	characteristic
200	C₂³	#392658	1	1	yes	characteristic
201	C₂³	#392663	1	2	yes	—
202	C₂³	#392699	1	2	yes	—
203	C₂³	#392719	1	2	yes	—
204	C₂²	#72219234	1	1	yes	characteristic
205	C₂²	#72234596	1	1	yes	characteristic
206	C₂²	#72258512	1	1	yes	characteristic
207	C₂²	#72261568	1	4	yes	—
208	C₂	#72266138	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72266138
commutator	C₂⁷	C₂³×C₄
Frattini, cosocle	C₂⁵×D₄	C₂⁴
prime radical	#72258968	C₂
LCS	G ▷ C₂⁷ ▷ 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. #72309929	C₂	semidirect
2. #72309848	C₂	semidirect
3. #72309737	C₂	semidirect
4. #72309539	C₂	semidirect
5. #72266039	C₂	semidirect
6. #72266028	C₂²	semidirect
7. #72264854	C₂	semidirect
10. #72238614	C₂	semidirect
11. #72238595	C₄	semidirect
12. #72237588	C₂	semidirect
17. #72219414	C₂²	semidirect
21. C₂²×#8541	C₂²	semidirect
22. #72218381	C₂²	semidirect
23. #72218198	C₂²	semidirect
27. #72215179	C₂²	semidirect
28. #72209716	C₄	semidirect
33. #7626202	D₄	semidirect
36. #6374251	D₄	semidirect
38. C₂×#21776	D₄	semidirect
40. #6374128	D₄	semidirect
41. C₂×#21746	D₄	semidirect
44. #6342371	C₂×C₄	semidirect
45. #6342371	C₂×C₄	semidirect
46. #4308128	C₂×C₄	semidirect
48. #392675	C₂×C₄	semidirect
52. #382707	D₄	semidirect
53. #382544	D₄	semidirect
54. #382434	D₄	semidirect
55. #355913	D₄	semidirect
56. #355096	D₄	semidirect
57. #115690	C₂×C₄	semidirect
58. #115690	C₂×C₄	semidirect
59. C₂⁸	C₂²×C₄	semidirect
67. C₂⁴×C₂²⋊C₄	C₂²×C₄	semidirect
90. #14679	C₂²×C₄	semidirect
91. #14679	C₂²×C₄	semidirect
161. #379	B(4,𝔽₂)	semidirect
163. C₂⁵	#1913	semidirect
180. C₂³×C₄	#1913	semidirect

Maximal subgroups

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

subgroup	index
#72309929	2
#72309848	2
#72309737	2
#72309539	2
#72266039	2
#72264854	2
#72258968	2
#72238614	2
#72237588	2