Subgroups — #72226387

Order 4096.

At least 14666 conjugacy classes of subgroups; only the 231 normal subgroups are shown.

The subgroup lattice has 152 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72226174	C₂	1	1	yes	maximal, characteristic, complemented
2	#72225909	C₂	1	2	yes	maximal, complemented
3	#72225836	C₂	1	1	yes	maximal, characteristic, complemented
4	#72225774	C₂	1	2	yes	maximal, complemented
5	#72225730	C₂	1	1	yes	maximal, characteristic
6	#72225675	C₄	1	1	yes	characteristic, complemented
7	#72225651	C₄	1	1	yes	characteristic, complemented
8	#72225624	C₂²	1	1	yes	characteristic
9	#72219393	C₂²	1	2	yes	—
10	C₂×#493474	C₂²	1	2	yes	—
11	#72219286	C₂²	1	1	yes	characteristic
12	#72217730	C₄	1	1	yes	characteristic, complemented
13	C₂×#6499867	C₂²	1	1	yes	characteristic
14	#72210047	C₄	1	1	yes	characteristic, complemented
15	C₂×#62817	D₄	1	1	yes	characteristic
16	C₂×#62817	C₂³	1	1	yes	characteristic
17	#7642897	D₄	1	1	yes	characteristic
18	#7642244	C₂×C₄	1	1	yes	characteristic
19	#7642244	D₄	1	1	yes	characteristic
20	C₂²×#2679	C₂×C₄	1	1	yes	characteristic
21	C₂²×#2846	C₂×C₄	1	1	yes	characteristic
22	C₂²×#2846	D₄	1	2	yes	—
23	#7626264	D₄	1	1	yes	characteristic
24	C₂×#21778	D₄	1	2	yes	complemented
25	C₂×#21776	D₄	1	2	yes	complemented
26	C₂×#21746	C₂×C₄	1	1	yes	characteristic
27	C₂×#21746	D₄	1	2	yes	—
28	#382436	D₄	1	2	yes	complemented
29	#382434	D₄	1	2	yes	complemented
30	#382422	C₂×C₄	1	1	yes	characteristic
31	C₂×#8535	D₄	1	4	yes	—
32	#155622	C₂×C₄	1	1	yes	characteristic
33	C₂⁸	C₂²⋊C₄	1	1	yes	characteristic, complemented
34	C₂×#3490	C₂×D₄	1	2	yes	—
35	C₂×#3490	#41	1	2	yes	—
36	C₂×#3490	#41	1	2	yes	—
37	C₂×#3484	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×#3484	C₂×D₄	1	1	yes	characteristic
39	C₂×#3484	C₂×D₄	1	1	yes	characteristic
40	C₂×#3484	C₂×D₄	1	2	yes	—
41	C₂×#3484	C₂×D₄	1	2	yes	—
42	C₂³×#120	C₂²⋊C₄	1	1	yes	characteristic
43	C₂³×#120	C₂²×C₄	1	1	yes	characteristic
44	C₂³×#120	C₂×D₄	1	2	yes	—
45	#34171	C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×#390	C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×#379	C₂²⋊C₄	1	2	yes	complemented
48	C₂²×#379	C₂²⋊C₄	1	1	yes	characteristic, complemented
49	C₂×#1819	C₂²⋊C₄	1	2	yes	complemented
50	C₂×#1440	C₂²⋊C₄	1	2	yes	complemented
51	C₂⁷	#99	1	1	yes	characteristic, complemented
52	C₂⁷	C₂×C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×H(5,𝔽₂)	#99	1	1	yes	characteristic, complemented
54	C₂²×H(5,𝔽₂)	#120	1	1	yes	characteristic
55	C₂²×H(5,𝔽₂)	#120	1	1	yes	characteristic
56	C₂⁴×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	C₂⁴×D₄	#120	1	2	yes	—
58	C₂×#543	#124	1	2	yes	—
59	C₂×#534	#121	1	2	yes	—
60	C₂×#534	#121	1	2	yes	—
61	C₂×#534	#127	1	2	yes	—
62	C₂²×#120	C₄×D₄	1	2	yes	—
63	C₂²×#120	C₄×D₄	1	2	yes	—
64	C₂²×#120	#120	1	2	yes	—
65	C₂²×#120	#120	1	2	yes	—
66	C₂²×#120	#121	1	2	yes	—
67	C₂²×#120	#121	1	2	yes	—
68	C₂×#518	#99	1	1	yes	characteristic, complemented
69	C₂×#518	#99	1	1	yes	characteristic, complemented
70	C₂×#518	#99	1	1	yes	characteristic, complemented
71	C₂×#518	#121	1	2	yes	—
72	C₂×#518	#121	1	2	yes	—
73	C₂×#518	#123	1	1	yes	characteristic
74	C₂×#518	#123	1	1	yes	characteristic
75	C₂³×C₄⋊C₄	#99	1	1	yes	characteristic, complemented
76	C₂³×C₂²⋊C₄	#99	1	1	yes	characteristic, complemented
77	C₂³×C₂²⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₂³×C₂²⋊C₄	#123	1	2	yes	—
79	C₂³×C₄²	#99	1	1	yes	characteristic, complemented
80	C₂⁶	#379	1	1	yes	characteristic
81	C₂⁶	#386	1	2	yes	—
82	C₂⁶	C₂×#99	1	1	yes	characteristic
83	C₂⁶	B(4,𝔽₂)	1	2	yes	—
84	C₂×H(5,𝔽₂)	B(4,𝔽₂)	1	1	yes	characteristic
85	C₂×H(5,𝔽₂)	B(4,𝔽₂)	1	2	yes	—
86	C₂×H(5,𝔽₂)	B(4,𝔽₂)	1	1	yes	characteristic
87	C₂³×D₄	#386	1	1	yes	characteristic
88	C₂³×D₄	#386	1	1	yes	characteristic
89	C₂³×D₄	#392	1	2	yes	—
90	C₂³×D₄	#392	1	2	yes	—
91	C₂³×D₄	C₂×#99	1	1	yes	characteristic
92	C₂³×D₄	C₂×#99	1	1	yes	characteristic
93	C₂³×D₄	B(4,𝔽₂)	1	2	yes	—
94	C₂³×D₄	B(4,𝔽₂)	1	2	yes	—
95	C₂⁴×C₄	C₂×#99	1	1	yes	characteristic
96	C₂⁴×C₄	B(4,𝔽₂)	1	2	yes	—
97	C₂×#120	B(4,𝔽₂)	1	4	yes	—
98	C₂×#120	#458	1	4	yes	—
99	C₂²×C₂²⋊C₄	#390	1	2	yes	—
100	C₂²×C₂²⋊C₄	#394	1	2	yes	—
101	C₂²×C₂²⋊C₄	#394	1	2	yes	—
102	C₂⁵	#1438	1	1	yes	characteristic
103	C₂⁵	#1781	1	1	yes	characteristic
104	C₂⁵	#1781	1	1	yes	characteristic
105	C₂⁵	#1790	1	4	yes	—
106	C₂⁵	#1889	1	2	yes	—
107	C₂⁵	#1889	1	2	yes	—
108	C₂⁵	#1898	1	1	yes	characteristic
109	C₂⁵	#1898	1	1	yes	characteristic
110	C₂⁵	#2021	1	2	yes	—
111	C₂⁵	#2021	1	2	yes	—
112	C₂²×D₄	#1889	1	1	yes	characteristic
113	C₂²×D₄	#1889	1	1	yes	characteristic
114	C₂²×D₄	#1913	1	1	yes	characteristic
115	C₂²×D₄	#1913	1	1	yes	characteristic
116	C₂²×D₄	#2021	1	2	yes	—
117	C₂²×D₄	#2021	1	2	yes	—
118	C₂²×D₄	#2021	1	2	yes	—
119	C₂²×D₄	#2022	1	2	yes	—
120	C₂³×C₄	#1898	1	1	yes	characteristic
121	C₂³×C₄	#1898	1	1	yes	characteristic
122	C₂³×C₄	#2021	1	2	yes	—
123	C₂³×C₄	#2021	1	2	yes	—
124	C₂×C₂²⋊C₄	#2044	1	4	yes	—
125	C₂⁴	#8534	1	2	yes	—
126	C₂⁴	#8534	1	2	yes	—
127	C₂⁴	#9495	1	1	yes	characteristic
128	C₂⁴	#9495	1	1	yes	characteristic
129	C₂⁴	#10336	1	2	yes	—
130	C₂⁴	#10337	1	2	yes	—
131	C₂⁴	#10965	1	1	yes	characteristic
132	C₂⁴	#10972	1	1	yes	characteristic
133	C₂⁴	#10991	1	2	yes	—
134	C₂⁴	#10991	1	1	yes	characteristic
135	C₂⁴	#10995	1	2	yes	—
136	C₂²×C₄	#9497	1	1	yes	characteristic
137	C₂²×C₄	#9497	1	1	yes	characteristic
138	C₂²×C₄	#10336	1	2	yes	—
139	C₂²×C₄	#10337	1	2	yes	—
140	C₂³	#111458	1	2	yes	—
141	C₂³	#115133	1	1	yes	characteristic
142	C₂³	#115204	1	2	yes	—
143	C₂³	#115348	1	1	yes	characteristic
144	C₂³	#115690	1	1	yes	characteristic
145	C₂²	#72218387	1	1	yes	characteristic
146	C₂²	#72226369	1	1	yes	characteristic
147	C₂²	#72226369	1	1	yes	characteristic
148	C₂	#72226370	1	1	yes	characteristic
149	C₂	#72226377	1	1	yes	characteristic
150	C₂	#72226379	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#72226369
commutator	C₂³×#120	C₂²×C₄
Frattini, cosocle	C₂×#62817	C₂³
prime radical	#72225730	C₂
LCS	G ▷ C₂³×#120 ▷ C₂⁵ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂⁵ ◁ C₂×#62817 ◁ G
LECS	G ▷ C₂×#62817 ▷ C₂⁵ ▷ C₂² ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72226174	C₂	semidirect
2. #72225909	C₂	semidirect
3. #72225836	C₂	semidirect
4. #72225774	C₂	semidirect
6. #72225675	C₄	semidirect
7. #72225651	C₄	semidirect
12. #72217730	C₄	semidirect
14. #72210047	C₄	semidirect
24. C₂×#21778	D₄	semidirect
25. C₂×#21776	D₄	semidirect
28. #382436	D₄	semidirect
29. #382434	D₄	semidirect
33. C₂⁸	C₂²⋊C₄	semidirect
47. C₂²×#379	C₂²⋊C₄	semidirect
48. C₂²×#379	C₂²⋊C₄	semidirect
49. C₂×#1819	C₂²⋊C₄	semidirect
50. C₂×#1440	C₂²⋊C₄	semidirect
51. C₂⁷	#99	semidirect
53. C₂²×H(5,𝔽₂)	#99	semidirect
68. C₂×#518	#99	semidirect
69. C₂×#518	#99	semidirect
70. C₂×#518	#99	semidirect
75. C₂³×C₄⋊C₄	#99	semidirect
76. C₂³×C₂²⋊C₄	#99	semidirect
79. C₂³×C₄²	#99	semidirect

Maximal subgroups

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

subgroup	index
#72226174	2
#72225909	2
#72225836	2
#72225774	2
#72225730	2