Subgroups — #72323208

Order 4096.

At least 17303 conjugacy classes of subgroups; only the 431 normal subgroups are shown.

The subgroup lattice has 213 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72313138	C₂	1	1	yes	maximal, characteristic, complemented
2	#72310419	C₂	1	1	yes	maximal, characteristic, complemented
3	#72309574	C₂	1	4	yes	maximal, complemented
4	#72265763	C₂	1	1	yes	maximal, characteristic, complemented
5	#72264799	C₂	1	4	yes	maximal, complemented
6	#72235213	C₂	1	1	yes	maximal, characteristic, complemented
7	#72234600	C₂	1	1	yes	maximal, characteristic, complemented
8	#72225676	C₂	1	1	yes	maximal, characteristic, complemented
9	#72225619	C₂²	1	4	yes	—
10	#72219414	C₂²	1	4	yes	complemented
11	#72219247	C₂	1	1	yes	maximal, characteristic, complemented
12	#72219243	C₂²	1	4	yes	complemented
13	#72219237	C₂²	1	4	yes	complemented
14	#72219228	C₂²	1	2	yes	complemented
15	#72219227	C₂²	1	1	yes	characteristic
16	C₂²×#8534	C₂²	1	2	yes	complemented
17	C₂×#359428	C₂²	1	1	yes	characteristic
18	#72218518	C₂²	1	1	yes	characteristic
19	#72218455	C₂²	1	2	yes	complemented
20	#72218291	C₂²	1	1	yes	characteristic
21	#72218180	C₂²	1	2	yes	complemented
22	#72217875	C₂²	1	1	yes	characteristic
23	#72217511	C₂²	1	1	yes	characteristic
24	#72217511	C₂²	1	1	yes	characteristic
25	#72213862	C₂²	1	2	yes	complemented
26	#72210511	C₂²	1	2	yes	complemented
27	C₂×#62817	D₄	1	1	yes	characteristic
28	C₂×#62817	D₄	1	1	yes	characteristic, complemented
29	C₂²×#3484	C₂³	1	1	yes	characteristic
30	C₂⁴×#120	C₂³	1	1	yes	characteristic
31	#7642244	D₄	1	2	yes	—
32	#7642244	D₄	1	2	yes	complemented
33	#7630939	C₂³	1	2	yes	—
34	#7630939	C₂³	1	2	yes	—
35	C₂×#35346	D₄	1	2	yes	complemented
36	#7626202	D₄	1	1	yes	characteristic
37	#7626202	D₄	1	1	yes	characteristic, complemented
38	C₂³×#379	D₄	1	1	yes	characteristic, complemented
39	C₂×#23895	D₄	1	4	yes	complemented
40	C₂²×#1781	C₂³	1	2	yes	—
41	C₂×#14679	D₄	1	1	yes	characteristic, complemented
42	#517411	C₂³	1	1	yes	characteristic
43	#515285	D₄	1	2	yes	complemented
44	C₂²×#1438	C₂³	1	2	yes	—
45	#392113	C₂³	1	2	yes	—
46	#392109	D₄	1	4	yes	complemented
47	#392109	C₂³	1	2	yes	—
48	#373637	D₄	1	4	yes	complemented
49	#359549	D₄	1	4	yes	complemented
50	#359549	D₄	1	4	yes	complemented
51	C₂×#10725	D₄	1	2	yes	complemented
52	#184773	D₄	1	4	yes	complemented
53	C₂⁸	C₂×D₄	1	1	yes	characteristic
54	C₂³×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic, complemented
55	C₂⁵×D₄	C₂×D₄	1	2	yes	—
56	C₂⁵×D₄	C₂⁴	1	1	yes	characteristic
57	C₂×#3484	C₂×D₄	1	2	yes	—
58	C₂×#3484	C₂×D₄	1	2	yes	—
59	C₂×#3484	C₂×D₄	1	4	yes	—
60	C₂×#3484	C₂×D₄	1	4	yes	—
61	C₂³×#120	C₂×D₄	1	1	yes	characteristic
62	C₂³×#120	C₂×D₄	1	1	yes	characteristic
63	C₂³×#120	C₂×D₄	1	2	yes	—
64	C₂³×#120	C₂×D₄	1	1	yes	characteristic
65	C₂³×#120	C₂×D₄	1	1	yes	characteristic
66	C₂³×#120	#41	1	2	yes	—
67	C₂²×#518	C₂×D₄	1	1	yes	characteristic, complemented
68	C₂⁴×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
69	C₂⁴×C₂²⋊C₄	C₂×D₄	1	2	yes	—
70	#34171	C₂×D₄	1	2	yes	—
71	#34171	C₂×D₄	1	2	yes	—
72	#34171	C₂×D₄	1	4	yes	—
73	#34171	C₂×D₄	1	4	yes	—
74	C₂×#2308	C₂×D₄	1	2	yes	complemented
75	C₂²×#379	C₂×D₄	1	2	yes	—
76	C₂²×#379	C₂×D₄	1	2	yes	—
77	C₂×#1781	C₂×D₄	1	4	yes	—
78	C₂×#1781	C₂×D₄	1	4	yes	—
79	#20579	C₂×D₄	1	4	yes	complemented
80	#14679	C₂×D₄	1	4	yes	complemented
81	C₂⁷	#120	1	1	yes	characteristic
82	C₂⁷	#120	1	1	yes	characteristic
83	C₂⁷	#121	1	2	yes	—
84	C₂⁷	#127	1	2	yes	—
85	C₂⁷	C₂²×D₄	1	1	yes	characteristic
86	C₂⁷	C₂²×D₄	1	1	yes	characteristic
87	C₂²×H(5,𝔽₂)	#120	1	2	yes	—
88	C₂²×H(5,𝔽₂)	#120	1	2	yes	—
89	C₂⁴×D₄	#120	1	2	yes	—
90	C₂⁴×D₄	#120	1	2	yes	—
91	C₂⁴×D₄	#121	1	2	yes	—
92	C₂⁴×D₄	#121	1	2	yes	—
93	C₂⁴×D₄	C₂²×D₄	1	1	yes	characteristic
94	C₂⁴×D₄	C₂²×D₄	1	1	yes	characteristic
95	C₂⁴×D₄	C₂²×D₄	1	1	yes	characteristic
96	C₂⁴×D₄	C₂²×D₄	1	4	yes	—
97	C₂⁴×D₄	C₂×#41	1	1	yes	characteristic
98	C₂⁴×D₄	H(5,𝔽₂)	1	4	yes	—
99	C₂⁵×C₄	#127	1	2	yes	—
100	C₂⁵×C₄	C₂²×D₄	1	1	yes	characteristic
101	C₂×#518	#120	1	2	yes	—
102	C₂×#518	#120	1	2	yes	—
103	C₂×#518	#120	1	4	yes	—
104	C₂×#518	#120	1	4	yes	—
105	C₂³×C₂²⋊C₄	#120	1	1	yes	characteristic
106	C₂³×C₂²⋊C₄	#120	1	1	yes	characteristic
107	C₂³×C₂²⋊C₄	#120	1	2	yes	—
108	C₂³×C₂²⋊C₄	#120	1	2	yes	—
109	C₂³×C₂²⋊C₄	#121	1	2	yes	—
110	C₂³×C₂²⋊C₄	#121	1	2	yes	—
111	C₂³×C₂²⋊C₄	#121	1	2	yes	—
112	C₂×#379	#120	1	4	yes	—
113	C₂×#379	#120	1	2	yes	—
114	C₂×#379	#120	1	2	yes	—
115	C₂⁶	#392	1	2	yes	—
116	C₂⁶	#392	1	2	yes	—
117	C₂⁶	B(4,𝔽₂)	1	1	yes	characteristic
118	C₂⁶	B(4,𝔽₂)	1	2	yes	—
119	C₂⁶	B(4,𝔽₂)	1	1	yes	characteristic
120	C₂⁶	C₂×#120	1	1	yes	characteristic
121	C₂⁶	C₂×#120	1	2	yes	—
122	C₂⁶	C₂×#120	1	1	yes	characteristic
123	C₂⁶	C₂×#121	1	1	yes	characteristic
124	C₂⁶	C₂×#121	1	1	yes	characteristic
125	C₂⁶	C₂×#127	1	1	yes	characteristic
126	C₂⁶	#535	1	2	yes	—
127	C₂⁶	D₄²	1	4	yes	—
128	C₂³×D₄	B(4,𝔽₂)	1	4	yes	—
129	C₂³×D₄	B(4,𝔽₂)	1	4	yes	—
130	C₂³×D₄	C₂×#120	1	2	yes	—
131	C₂³×D₄	C₂×#120	1	2	yes	—
132	C₂³×D₄	#534	1	2	yes	—
133	C₂³×D₄	#534	1	2	yes	—
134	C₂³×D₄	D₄²	1	4	yes	—
135	C₂³×D₄	#546	1	4	yes	—
136	C₂⁴×C₄	#392	1	2	yes	—
137	C₂⁴×C₄	#392	1	2	yes	—
138	C₂⁴×C₄	C₂×#120	1	1	yes	characteristic
139	C₂⁴×C₄	C₂×#121	1	1	yes	characteristic
140	C₂⁴×C₄	D₄²	1	4	yes	—
141	C₂²×C₂²⋊C₄	B(4,𝔽₂)	1	4	yes	—
142	C₂²×C₂²⋊C₄	B(4,𝔽₂)	1	4	yes	—
143	#379	B(4,𝔽₂)	1	4	yes	complemented
144	C₂⁵	#2021	1	2	yes	—
145	C₂⁵	#2021	1	2	yes	—
146	C₂⁵	#2021	1	2	yes	—
147	C₂⁵	#2021	1	2	yes	—
148	C₂⁵	#2021	1	2	yes	—
149	C₂⁵	#2021	1	2	yes	—
150	C₂⁵	C₂×#392	1	1	yes	characteristic
151	C₂⁵	#2403	1	2	yes	—
152	C₂⁵	#2408	1	2	yes	—
153	C₂⁵	#2410	1	2	yes	—
154	C₂⁵	#2433	1	2	yes	—
155	C₂⁵	#2613	1	1	yes	characteristic
156	C₂⁵	#2846	1	1	yes	characteristic
157	C₂⁵	C₂×B(4,𝔽₂)	1	1	yes	characteristic
158	C₂⁵	C₂×B(4,𝔽₂)	1	1	yes	characteristic
159	C₂⁵	C₂×B(4,𝔽₂)	1	4	yes	—
160	C₂⁵	C₂×B(4,𝔽₂)	1	2	yes	—
161	C₂⁵	#3025	1	2	yes	—
162	C₂⁵	#3026	1	1	yes	characteristic
163	C₂⁵	#3026	1	1	yes	characteristic
164	C₂⁵	#3026	1	2	yes	—
165	C₂⁵	#3027	1	2	yes	—
166	C₂³×C₄	#2021	1	4	yes	—
167	C₂³×C₄	#2021	1	4	yes	—
168	C₂³×C₄	#2403	1	2	yes	—
169	C₂³×C₄	#2410	1	2	yes	—
170	C₂³×C₄	C₂×B(4,𝔽₂)	1	4	yes	—
171	C₂⁴	#10336	1	2	yes	—
172	C₂⁴	#10336	1	2	yes	—
173	C₂⁴	#10336	1	2	yes	—
174	C₂⁴	#10336	1	2	yes	—
175	C₂⁴	#14715	1	1	yes	characteristic
176	C₂⁴	C₂×#2021	1	1	yes	characteristic
177	C₂⁴	C₂×#2021	1	2	yes	—
178	C₂⁴	C₂×#2021	1	2	yes	—
179	C₂⁴	#23444	1	1	yes	characteristic
180	C₂⁴	#23900	1	4	yes	—
181	C₂⁴	#24260	1	2	yes	—
182	C₂⁴	#24262	1	2	yes	—
183	C₂⁴	#25771	1	2	yes	—
184	C₂⁴	#25773	1	2	yes	—
185	C₂⁴	#29634	1	1	yes	characteristic
186	C₂⁴	#29634	1	1	yes	characteristic
187	C₂⁴	#32898	1	1	yes	characteristic
188	C₂⁴	#32898	1	1	yes	characteristic
189	C₂⁴	#32898	1	2	yes	—
190	C₂⁴	#32899	1	2	yes	—
191	C₂²×C₄	#23900	1	4	yes	—
192	C₂³	#144364	1	4	yes	complemented
193	C₂³	C₂×#10336	1	2	yes	—
194	C₂³	#233541	1	2	yes	—
195	C₂³	#347042	1	1	yes	characteristic
196	C₂³	#347043	1	2	yes	—
197	C₂³	#347044	1	1	yes	characteristic
198	C₂³	#393929	1	1	yes	characteristic
199	C₂³	#393933	1	2	yes	—
200	C₂³	#393934	1	2	yes	—
201	C₂³	#394058	1	1	yes	characteristic
202	C₂³	#394058	1	1	yes	characteristic
203	C₂²	#72214549	1	1	yes	characteristic
204	C₂²	C₂×#144364	1	2	yes	—
205	C₂²	#72225661	1	1	yes	characteristic
206	C₂²	#72234155	1	2	yes	—
207	C₂²	#72234155	1	2	yes	—
208	C₂²	#72236631	1	2	yes	—
209	C₂²	#72258515	1	1	yes	characteristic
210	C₂	#72258111	1	2	yes	—
211	C₂	#72266377	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#72214549
commutator, Frattini, cosocle	C₂⁵×D₄	C₂⁴
LCS	G ▷ C₂⁵×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. #72313138	C₂	semidirect
2. #72310419	C₂	semidirect
3. #72309574	C₂	semidirect
4. #72265763	C₂	semidirect
5. #72264799	C₂	semidirect
6. #72235213	C₂	semidirect
7. #72234600	C₂	semidirect
8. #72225676	C₂	semidirect
10. #72219414	C₂²	semidirect
11. #72219247	C₂	semidirect
12. #72219243	C₂²	semidirect
13. #72219237	C₂²	semidirect
14. #72219228	C₂²	semidirect
16. C₂²×#8534	C₂²	semidirect
19. #72218455	C₂²	semidirect
21. #72218180	C₂²	semidirect
25. #72213862	C₂²	semidirect
26. #72210511	C₂²	semidirect
28. C₂×#62817	D₄	semidirect
32. #7642244	D₄	semidirect
35. C₂×#35346	D₄	semidirect
37. #7626202	D₄	semidirect
38. C₂³×#379	D₄	semidirect
39. C₂×#23895	D₄	semidirect
41. C₂×#14679	D₄	semidirect
43. #515285	D₄	semidirect
46. #392109	D₄	semidirect
48. #373637	D₄	semidirect
49. #359549	D₄	semidirect
50. #359549	D₄	semidirect
51. C₂×#10725	D₄	semidirect
52. #184773	D₄	semidirect
54. C₂³×H(5,𝔽₂)	C₂×D₄	semidirect
67. C₂²×#518	C₂×D₄	semidirect
74. C₂×#2308	C₂×D₄	semidirect
79. #20579	C₂×D₄	semidirect
80. #14679	C₂×D₄	semidirect
143. #379	B(4,𝔽₂)	semidirect
192. C₂³	#144364	semidirect

Maximal subgroups

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

subgroup	index
#72313138	2
#72310419	2
#72309574	2
#72265763	2
#72264799	2
#72235213	2
#72234600	2
#72225676	2
#72219247	2