Subgroups — #72312471

Order 4096.

At least 12902 conjugacy classes of subgroups; only the 365 normal subgroups are shown.

The subgroup lattice has 232 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72312430	C₂	1	1	yes	maximal, characteristic, complemented
2	#72312241	C₂	1	1	yes	maximal, characteristic, complemented
3	#72312205	C₂	1	1	yes	maximal, characteristic, complemented
4	#72310388	C₂	1	2	yes	maximal, complemented
5	#72310380	C₂	1	1	yes	maximal, characteristic, complemented
6	#72309519	C₂	1	2	yes	maximal, complemented
7	#72266831	C₂	1	2	yes	maximal, complemented
8	#72266073	C₂²	1	1	yes	characteristic, complemented
9	#72264799	C₂	1	2	yes	maximal, complemented
10	#72262330	C₂	1	1	yes	maximal, characteristic, complemented
11	#72251494	C₂	1	1	yes	maximal, characteristic, complemented
12	#72238598	C₂	1	1	yes	maximal, characteristic, complemented
13	#72237400	C₂²	1	1	yes	characteristic, complemented
14	#72234668	C₂²	1	1	yes	characteristic
15	#72234661	C₂²	1	1	yes	characteristic
16	#72225619	C₂²	1	2	yes	—
17	#72225619	C₂²	1	1	yes	characteristic
18	#72225619	C₂²	1	1	yes	characteristic
19	C₂×#115348	C₂²	1	1	yes	characteristic
20	#72224136	C₂²	1	2	yes	—
21	#72219418	C₂²	1	1	yes	characteristic, complemented
22	#72219415	C₂²	1	1	yes	characteristic, complemented
23	#72219414	C₂²	1	2	yes	complemented
24	#72219300	C₂²	1	2	yes	—
25	#72219243	C₂²	1	2	yes	complemented
26	#72219237	C₂²	1	2	yes	complemented
27	#72219229	C₂²	1	1	yes	characteristic, complemented
28	C₂²×#8534	C₂²	1	1	yes	characteristic, complemented
29	#72218069	C₂²	1	2	yes	complemented
30	#72218069	C₂²	1	2	yes	complemented
31	#72218057	C₂²	1	2	yes	complemented
32	#72218057	C₂²	1	2	yes	complemented
33	#72213862	C₂²	1	2	yes	complemented
34	#72210511	C₂²	1	2	yes	complemented
35	#10586404	D₄	1	2	yes	complemented
36	C₂²×#3484	C₂³	1	1	yes	characteristic
37	#7630939	C₂³	1	2	yes	—
38	#7630939	C₂³	1	2	yes	—
39	C₂×#34980	C₂³	1	1	yes	characteristic
40	#7626202	D₄	1	1	yes	characteristic, complemented
41	#6629258	D₄	1	2	yes	complemented
42	#6628892	D₄	1	2	yes	complemented
43	C₂×#23895	D₄	1	1	yes	characteristic, complemented
44	C₂×#23895	D₄	1	1	yes	characteristic, complemented
45	C₂×#23895	D₄	1	2	yes	complemented
46	#6374210	D₄	1	2	yes	complemented
47	#6374210	D₄	1	2	yes	complemented
48	#6374128	D₄	1	2	yes	complemented
49	C₂×#21746	D₄	1	1	yes	characteristic, complemented
50	C₂²×#1781	C₂³	1	1	yes	characteristic
51	C₂²×#1781	C₂³	1	1	yes	characteristic
52	C₂²×#1781	C₂³	1	1	yes	characteristic
53	C₂²×#1781	C₂³	1	1	yes	characteristic
54	C₂²×#1438	C₂³	1	1	yes	characteristic
55	#392113	C₂³	1	2	yes	—
56	#392109	C₂³	1	2	yes	—
57	#373637	D₄	1	1	yes	characteristic, complemented
58	#373637	D₄	1	2	yes	complemented
59	#373637	D₄	1	1	yes	characteristic, complemented
60	#359549	D₄	1	2	yes	complemented
61	#359549	D₄	1	2	yes	complemented
62	#359549	D₄	1	2	yes	complemented
63	#359549	D₄	1	2	yes	complemented
64	#355913	D₄	1	1	yes	characteristic, complemented
65	#355913	D₄	1	1	yes	characteristic, complemented
66	#147567	D₄	1	8	yes	complemented
67	C₂³×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic, complemented
68	C₂⁵×D₄	C₂⁴	1	1	yes	characteristic
69	C₂×#3484	C₂×D₄	1	2	yes	—
70	C₂×#3484	C₂×D₄	1	2	yes	—
71	C₂×#3484	C₂×D₄	1	2	yes	—
72	C₂×#3484	C₂×D₄	1	2	yes	—
73	C₂×#3484	C₂×D₄	1	2	yes	—
74	C₂×#3484	C₂×D₄	1	2	yes	—
75	C₂²×#534	C₂×D₄	1	2	yes	—
76	C₂³×#120	C₂×D₄	1	1	yes	characteristic
77	C₂²×#518	C₂×D₄	1	1	yes	characteristic, complemented
78	#34980	C₂×D₄	1	2	yes	—
79	#34980	C₂×D₄	1	4	yes	—
80	#34980	#41	1	2	yes	—
81	#34171	C₂×D₄	1	2	yes	—
82	#34171	C₂×D₄	1	2	yes	—
83	#34171	C₂×D₄	1	2	yes	—
84	#34171	C₂×D₄	1	2	yes	—
85	#34171	C₂×D₄	1	2	yes	—
86	#34171	C₂×D₄	1	2	yes	—
87	C₂×#2308	C₂×D₄	1	2	yes	—
88	C₂²×#379	C₂×D₄	1	1	yes	characteristic
89	C₂²×#379	C₂×D₄	1	1	yes	characteristic
90	C₂²×#379	C₂×D₄	1	1	yes	characteristic
91	C₂²×#379	C₂×D₄	1	1	yes	characteristic
92	C₂²×#379	C₂×D₄	1	1	yes	characteristic
93	C₂×#1783	C₂×D₄	1	4	yes	complemented
94	C₂×#1781	C₂×D₄	1	1	yes	characteristic
95	C₂×#1781	C₂×D₄	1	1	yes	characteristic
96	C₂×#1781	C₂×D₄	1	2	yes	—
97	C₂×#1781	C₂×D₄	1	1	yes	characteristic
98	C₂×#1781	C₂×D₄	1	1	yes	characteristic
99	C₂×#1781	C₂×D₄	1	2	yes	—
100	C₂×#1781	C₂×D₄	1	2	yes	complemented
101	C₂×#1781	C₂×D₄	1	4	yes	complemented
102	C₂⁷	C₂²×D₄	1	1	yes	characteristic
103	C₂⁷	H(5,𝔽₂)	1	1	yes	characteristic
104	C₂²×H(5,𝔽₂)	#120	1	2	yes	—
105	C₂²×H(5,𝔽₂)	#120	1	2	yes	—
106	C₂⁴×D₄	#120	1	1	yes	characteristic
107	C₂⁴×D₄	C₂²×D₄	1	2	yes	—
108	C₂⁴×D₄	C₂²×D₄	1	1	yes	characteristic
109	C₂⁴×D₄	C₂²×D₄	1	2	yes	—
110	C₂⁴×D₄	C₂²×D₄	1	1	yes	characteristic
111	C₂⁴×D₄	C₂²×D₄	1	2	yes	—
112	C₂⁴×D₄	C₂²×D₄	1	1	yes	characteristic
113	C₂⁴×D₄	C₂×#41	1	1	yes	characteristic
114	C₂⁴×D₄	H(5,𝔽₂)	1	2	yes	—
115	C₂⁵×C₄	H(5,𝔽₂)	1	1	yes	characteristic
116	C₂×#534	#120	1	4	yes	—
117	C₂×#534	#121	1	4	yes	—
118	C₂²×#120	#121	1	2	yes	—
119	C₂²×#120	#127	1	2	yes	—
120	C₂×#518	#120	1	2	yes	—
121	C₂×#518	#120	1	2	yes	—
122	C₂×#518	#120	1	2	yes	—
123	C₂×#518	#120	1	2	yes	—
124	C₂×#518	#120	1	2	yes	—
125	C₂×#518	#120	1	2	yes	—
126	C₂³×C₂²⋊C₄	#120	1	1	yes	characteristic
127	C₂³×C₂²⋊C₄	#120	1	2	yes	—
128	#2308	#120	1	4	yes	—
129	#2308	#121	1	4	yes	—
130	C₂×#379	#120	1	1	yes	characteristic
131	C₂×#379	#120	1	1	yes	characteristic
132	C₂×#379	#120	1	2	yes	—
133	C₂×#379	#120	1	1	yes	characteristic
134	C₂×#379	#120	1	1	yes	characteristic
135	C₂×#379	#120	1	1	yes	characteristic
136	C₂×#379	#120	1	1	yes	characteristic
137	C₂×#379	#121	1	2	yes	—
138	C₂×#379	#127	1	2	yes	—
139	C₂⁶	B(4,𝔽₂)	1	1	yes	characteristic
140	C₂⁶	C₂×#120	1	1	yes	characteristic
141	C₂⁶	C₂×#120	1	1	yes	characteristic
142	C₂⁶	C₂×#120	1	1	yes	characteristic
143	C₂⁶	C₂×#121	1	1	yes	characteristic
144	C₂⁶	C₂×#127	1	1	yes	characteristic
145	C₂⁶	#534	1	2	yes	—
146	C₂⁶	#535	1	1	yes	characteristic
147	C₂⁶	#535	1	1	yes	characteristic
148	C₂⁶	D₄²	1	1	yes	characteristic
149	C₂⁶	D₄²	1	2	yes	—
150	C₂⁶	#546	1	1	yes	characteristic
151	C₂³×D₄	#392	1	2	yes	—
152	C₂³×D₄	B(4,𝔽₂)	1	2	yes	—
153	C₂³×D₄	C₂×#120	1	2	yes	—
154	C₂³×D₄	C₂×#120	1	2	yes	—
155	C₂³×D₄	C₂×#120	1	2	yes	—
156	C₂³×D₄	C₂×#121	1	2	yes	—
157	C₂³×D₄	D₄²	1	2	yes	—
158	C₂³×D₄	D₄²	1	2	yes	—
159	C₂³×D₄	D₄²	1	2	yes	—
160	C₂³×D₄	#546	1	2	yes	—
161	C₂⁴×C₄	B(4,𝔽₂)	1	1	yes	characteristic
162	C₂⁴×C₄	#534	1	2	yes	—
163	C₂⁴×C₄	D₄²	1	1	yes	characteristic
164	C₂⁴×C₄	D₄²	1	2	yes	—
165	C₂⁴×C₄	#546	1	1	yes	characteristic
166	C₂²×C₂²⋊C₄	#392	1	2	yes	—
167	C₂²×C₂²⋊C₄	#392	1	4	yes	—
168	#379	B(4,𝔽₂)	1	2	yes	—
169	#379	B(4,𝔽₂)	1	2	yes	—
170	C₂⁵	#2021	1	2	yes	—
171	C₂⁵	C₂×#392	1	1	yes	characteristic
172	C₂⁵	#2403	1	2	yes	—
173	C₂⁵	#2408	1	1	yes	characteristic
174	C₂⁵	#2408	1	1	yes	characteristic
175	C₂⁵	#2410	1	2	yes	—
176	C₂⁵	#2433	1	1	yes	characteristic
177	C₂⁵	#2433	1	1	yes	characteristic
178	C₂⁵	#2613	1	1	yes	characteristic
179	C₂⁵	#2846	1	1	yes	characteristic
180	C₂⁵	C₂×B(4,𝔽₂)	1	1	yes	characteristic
181	C₂⁵	C₂×B(4,𝔽₂)	1	1	yes	characteristic
182	C₂⁵	C₂×B(4,𝔽₂)	1	1	yes	characteristic
183	C₂⁵	C₂×B(4,𝔽₂)	1	1	yes	characteristic
184	C₂⁵	#3025	1	1	yes	characteristic
185	C₂⁵	#3025	1	1	yes	characteristic
186	C₂⁵	#3026	1	1	yes	characteristic
187	C₂⁵	#3026	1	1	yes	characteristic
188	C₂⁵	#3026	1	1	yes	characteristic
189	C₂⁵	#3026	1	1	yes	characteristic
190	C₂⁵	#3027	1	1	yes	characteristic
191	C₂⁵	#3027	1	1	yes	characteristic
192	C₂³×C₄	#2021	1	2	yes	—
193	C₂³×C₄	#2403	1	2	yes	—
194	C₂³×C₄	#2410	1	2	yes	—
195	C₂⁴	#14715	1	1	yes	characteristic
196	C₂⁴	C₂×#2021	1	1	yes	characteristic
197	C₂⁴	#23444	1	1	yes	characteristic
198	C₂⁴	#23895	1	4	yes	—
199	C₂⁴	#24260	1	1	yes	characteristic
200	C₂⁴	#24260	1	1	yes	characteristic
201	C₂⁴	#24262	1	1	yes	characteristic
202	C₂⁴	#24262	1	1	yes	characteristic
203	C₂⁴	#25771	1	1	yes	characteristic
204	C₂⁴	#25771	1	1	yes	characteristic
205	C₂⁴	#25773	1	1	yes	characteristic
206	C₂⁴	#25773	1	1	yes	characteristic
207	C₂⁴	#29634	1	1	yes	characteristic
208	C₂⁴	#29634	1	1	yes	characteristic
209	C₂⁴	#32898	1	1	yes	characteristic
210	C₂⁴	#32898	1	1	yes	characteristic
211	C₂⁴	#32898	1	1	yes	characteristic
212	C₂⁴	#32898	1	1	yes	characteristic
213	C₂⁴	#32899	1	1	yes	characteristic
214	C₂⁴	#32899	1	1	yes	characteristic
215	C₂²×C₄	#23895	1	4	yes	—
216	C₂³	#347042	1	1	yes	characteristic
217	C₂³	#347043	1	1	yes	characteristic
218	C₂³	#347043	1	1	yes	characteristic
219	C₂³	#347044	1	1	yes	characteristic
220	C₂³	#393929	1	1	yes	characteristic
221	C₂³	#393934	1	1	yes	characteristic
222	C₂³	#393934	1	1	yes	characteristic
223	C₂³	#394064	1	2	yes	—
224	C₂³	#394064	1	2	yes	—
225	C₂²	#72225610	1	2	yes	—
226	C₂²	#72225661	1	1	yes	characteristic
227	C₂²	#72233127	1	1	yes	characteristic
228	C₂²	#72233127	1	2	yes	—
229	C₂²	#72233127	1	1	yes	characteristic
230	C₂	#72266142	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72266142
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. #72312430	C₂	semidirect
2. #72312241	C₂	semidirect
3. #72312205	C₂	semidirect
4. #72310388	C₂	semidirect
5. #72310380	C₂	semidirect
6. #72309519	C₂	semidirect
7. #72266831	C₂	semidirect
8. #72266073	C₂²	semidirect
9. #72264799	C₂	semidirect
10. #72262330	C₂	semidirect
11. #72251494	C₂	semidirect
12. #72238598	C₂	semidirect
13. #72237400	C₂²	semidirect
21. #72219418	C₂²	semidirect
22. #72219415	C₂²	semidirect
23. #72219414	C₂²	semidirect
25. #72219243	C₂²	semidirect
26. #72219237	C₂²	semidirect
27. #72219229	C₂²	semidirect
28. C₂²×#8534	C₂²	semidirect
29. #72218069	C₂²	semidirect
30. #72218069	C₂²	semidirect
31. #72218057	C₂²	semidirect
32. #72218057	C₂²	semidirect
33. #72213862	C₂²	semidirect
34. #72210511	C₂²	semidirect
35. #10586404	D₄	semidirect
40. #7626202	D₄	semidirect
41. #6629258	D₄	semidirect
42. #6628892	D₄	semidirect
43. C₂×#23895	D₄	semidirect
44. C₂×#23895	D₄	semidirect
45. C₂×#23895	D₄	semidirect
46. #6374210	D₄	semidirect
47. #6374210	D₄	semidirect
48. #6374128	D₄	semidirect
49. C₂×#21746	D₄	semidirect
57. #373637	D₄	semidirect
58. #373637	D₄	semidirect
59. #373637	D₄	semidirect
60. #359549	D₄	semidirect
61. #359549	D₄	semidirect
62. #359549	D₄	semidirect
63. #359549	D₄	semidirect
64. #355913	D₄	semidirect
65. #355913	D₄	semidirect
66. #147567	D₄	semidirect
67. C₂³×H(5,𝔽₂)	C₂×D₄	semidirect
77. C₂²×#518	C₂×D₄	semidirect
93. C₂×#1783	C₂×D₄	semidirect
100. C₂×#1781	C₂×D₄	semidirect
101. C₂×#1781	C₂×D₄	semidirect

Maximal subgroups

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

subgroup	index
#72312430	2
#72312241	2
#72312205	2
#72310388	2
#72310380	2
#72309519	2
#72266831	2
#72264799	2
#72262330	2
#72251494	2
#72238598	2