Subgroups — #72324300

Order 4096.

At least 9147 conjugacy classes of subgroups; only the 175 normal subgroups are shown.

The subgroup lattice has 148 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72323217	C₂	1	1	yes	maximal, characteristic, complemented
2	#72314476	C₂	1	1	yes	maximal, characteristic, complemented
3	#72314165	C₂	1	1	yes	maximal, characteristic, complemented
4	#72265766	C₂²	1	1	yes	characteristic
5	#72258757	C₂	1	1	yes	maximal, characteristic
6	#72237589	C₂	1	1	yes	maximal, characteristic, complemented
7	#72237391	C₂	1	1	yes	maximal, characteristic, complemented
8	#72235718	C₂²	1	1	yes	characteristic
9	#72226018	C₂	1	1	yes	maximal, characteristic, complemented
10	#72225870	C₂²	1	1	yes	characteristic
11	#72224136	C₂²	1	1	yes	characteristic
12	#72219314	C₂²	1	1	yes	characteristic
13	#72219311	C₂²	1	1	yes	characteristic
14	#72219257	C₂²	1	1	yes	characteristic
15	#72219237	C₄	1	2	yes	complemented
16	#72213862	C₄	1	2	yes	complemented
17	#10586404	D₄	1	1	yes	characteristic
18	C₂²×#3484	C₂×C₄	1	1	yes	characteristic
19	#7630939	C₂×C₄	1	2	yes	—
20	C₂×#34980	C₂³	1	1	yes	characteristic
21	#6374210	D₄	1	2	yes	—
22	#6374128	D₄	1	1	yes	characteristic
23	#4930495	D₄	1	1	yes	characteristic
24	#4930495	D₄	1	1	yes	characteristic
25	#4930494	D₄	1	1	yes	characteristic
26	#4930494	D₄	1	1	yes	characteristic
27	C₂²×#1438	C₂×C₄	1	1	yes	characteristic
28	#392109	C₂×C₄	1	2	yes	—
29	#188109	D₄	1	1	yes	characteristic, complemented
30	C₂×#8547	D₄	1	1	yes	characteristic, complemented
31	#188049	D₄	1	1	yes	characteristic, complemented
32	C₂×#8541	D₄	1	1	yes	characteristic, complemented
33	#134036	D₄	1	2	yes	complemented
34	#134035	D₄	1	2	yes	complemented
35	#115690	D₄	1	2	yes	—
36	#115690	D₄	1	2	yes	—
37	C₂³×H(5,𝔽₂)	C₂²⋊C₄	1	1	yes	characteristic, complemented
38	C₂⁵×D₄	C₂²×C₄	1	1	yes	characteristic
39	C₂×#3484	C₂²⋊C₄	1	2	yes	—
40	C₂²×#534	C₂×D₄	1	1	yes	characteristic
41	C₂²×#534	C₂×D₄	1	1	yes	characteristic
42	C₂³×#120	C₂×D₄	1	1	yes	characteristic
43	C₂²×#518	C₂²⋊C₄	1	1	yes	characteristic, complemented
44	C₂×#2849	C₂×D₄	1	1	yes	characteristic
45	C₂×#2849	#41	1	1	yes	characteristic
46	C₂×#2846	C₂×D₄	1	1	yes	characteristic
47	C₂×#2846	C₂×D₄	1	1	yes	characteristic
48	C₂×#2613	C₂×D₄	1	1	yes	characteristic
49	C₂×#2613	C₂×D₄	1	1	yes	characteristic
50	C₂×#2613	C₂×D₄	1	1	yes	characteristic
51	C₂×#2613	#41	1	1	yes	characteristic
52	#34171	C₂²⋊C₄	1	2	yes	—
53	C₂×#2308	C₂×D₄	1	1	yes	characteristic
54	C₂×#2308	#41	1	1	yes	characteristic
55	C₂²×#379	#41	1	1	yes	characteristic
56	#14679	C₂²⋊C₄	1	4	yes	complemented
57	C₂×#1438	C₂²⋊C₄	1	1	yes	characteristic, complemented
58	C₂×#1438	C₂²⋊C₄	1	1	yes	characteristic, complemented
59	C₂⁷	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₂⁷	C₄×D₄	1	1	yes	characteristic
61	C₂²×H(5,𝔽₂)	#99	1	2	yes	—
62	C₂⁴×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₂⁴×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	C₂⁴×D₄	C₄×D₄	1	1	yes	characteristic
65	C₂⁴×D₄	C₄×D₄	1	1	yes	characteristic
66	C₂⁴×D₄	#120	1	1	yes	characteristic
67	C₂⁵×C₄	C₄×D₄	1	1	yes	characteristic
68	C₂²×#123	#120	1	1	yes	characteristic
69	C₂²×#123	#121	1	1	yes	characteristic
70	C₂²×#123	#121	1	1	yes	characteristic
71	C₂²×#123	#121	1	1	yes	characteristic
72	C₂²×#120	#120	1	1	yes	characteristic
73	C₂²×#120	#120	1	1	yes	characteristic
74	C₂²×#120	#120	1	1	yes	characteristic
75	C₂²×#120	#121	1	1	yes	characteristic
76	C₂²×#120	#121	1	1	yes	characteristic
77	C₂²×#120	#121	1	1	yes	characteristic
78	C₂²×#120	#127	1	1	yes	characteristic
79	C₂²×#120	#127	1	1	yes	characteristic
80	C₂×#518	#99	1	2	yes	—
81	C₂³×C₂²⋊C₄	#121	1	1	yes	characteristic
82	C₂³×C₂²⋊C₄	#121	1	1	yes	characteristic
83	C₂³×C₂²⋊C₄	#123	1	1	yes	characteristic
84	#2308	#99	1	4	yes	—
85	C₂×#381	#120	1	1	yes	characteristic
86	C₂×#381	#121	1	1	yes	characteristic
87	C₂×#381	#123	1	1	yes	characteristic
88	C₂×#381	#123	1	1	yes	characteristic
89	C₂×#379	#120	1	1	yes	characteristic
90	C₂×#379	#120	1	1	yes	characteristic
91	C₂×#379	#121	1	1	yes	characteristic
92	C₂×#379	#121	1	1	yes	characteristic
93	C₂×#379	#121	1	1	yes	characteristic
94	C₂×#379	#123	1	1	yes	characteristic
95	C₂×#379	#124	1	1	yes	characteristic
96	C₂×#379	#124	1	1	yes	characteristic
97	C₂⁶	#379	1	1	yes	characteristic
98	C₂⁶	#386	1	1	yes	characteristic
99	C₂⁶	#386	1	1	yes	characteristic
100	C₂⁶	#390	1	1	yes	characteristic
101	C₂⁶	#390	1	1	yes	characteristic
102	C₂⁶	B(4,𝔽₂)	1	1	yes	characteristic
103	C₂³×D₄	#392	1	1	yes	characteristic
104	C₂³×D₄	#392	1	1	yes	characteristic
105	C₂³×D₄	C₂×#99	1	2	yes	—
106	C₂³×D₄	C₂×#99	1	2	yes	—
107	C₂³×D₄	B(4,𝔽₂)	1	1	yes	characteristic
108	C₂³×D₄	B(4,𝔽₂)	1	1	yes	characteristic
109	C₂⁴×C₄	#386	1	1	yes	characteristic
110	C₂⁴×C₄	#386	1	1	yes	characteristic
111	C₂⁴×C₄	B(4,𝔽₂)	1	1	yes	characteristic
112	C₂²×C₂²⋊C₄	#392	1	1	yes	characteristic
113	C₂²×C₂²⋊C₄	#392	1	1	yes	characteristic
114	C₂²×C₂²⋊C₄	#394	1	1	yes	characteristic
115	C₂²×C₂²⋊C₄	#394	1	1	yes	characteristic
116	C₂²×C₂²⋊C₄	#394	1	1	yes	characteristic
117	C₂²×C₂²⋊C₄	#394	1	1	yes	characteristic
118	C₂⁵	#1438	1	1	yes	characteristic
119	C₂⁵	#1781	1	2	yes	—
120	C₂⁵	#1889	1	1	yes	characteristic
121	C₂⁵	#1889	1	1	yes	characteristic
122	C₂⁵	#1898	1	1	yes	characteristic
123	C₂⁵	#1898	1	1	yes	characteristic
124	C₂⁵	#1913	1	1	yes	characteristic
125	C₂⁵	#1913	1	1	yes	characteristic
126	C₂⁵	#2021	1	1	yes	characteristic
127	C₂⁵	#2021	1	1	yes	characteristic
128	C₂³×C₄	#1898	1	2	yes	—
129	C₂³×C₄	#2021	1	1	yes	characteristic
130	C₂³×C₄	#2021	1	1	yes	characteristic
131	C₂⁴	#9495	1	1	yes	characteristic
132	C₂⁴	#9495	1	1	yes	characteristic
133	C₂⁴	#9497	1	1	yes	characteristic
134	C₂⁴	#9497	1	1	yes	characteristic
135	C₂⁴	#10965	1	1	yes	characteristic
136	C₂⁴	#10972	1	1	yes	characteristic
137	C₂⁴	#10991	1	1	yes	characteristic
138	C₂⁴	#12705	1	4	yes	—
139	C₂³	#115133	1	1	yes	characteristic
140	C₂³	#139750	1	1	yes	characteristic
141	C₂³	#139750	1	1	yes	characteristic
142	C₂³	#147567	1	2	yes	—
143	C₂²	#72219158	1	1	yes	characteristic
144	C₂²	#72226366	1	1	yes	characteristic
145	C₂²	#72265789	1	1	yes	characteristic
146	C₂	#72266334	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72266334
commutator	C₂⁵×D₄	C₂²×C₄
Frattini, cosocle	C₂×#34980	C₂³
prime radical	#72258757	C₂
LCS	G ▷ C₂⁵×D₄ ▷ C₂⁵ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂⁶ ◁ C₂×#34980 ◁ G
LECS	G ▷ C₂×#34980 ▷ C₂⁵ ▷ C₂³ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72323217	C₂	semidirect
2. #72314476	C₂	semidirect
3. #72314165	C₂	semidirect
6. #72237589	C₂	semidirect
7. #72237391	C₂	semidirect
9. #72226018	C₂	semidirect
15. #72219237	C₄	semidirect
16. #72213862	C₄	semidirect
29. #188109	D₄	semidirect
30. C₂×#8547	D₄	semidirect
31. #188049	D₄	semidirect
32. C₂×#8541	D₄	semidirect
33. #134036	D₄	semidirect
34. #134035	D₄	semidirect
37. C₂³×H(5,𝔽₂)	C₂²⋊C₄	semidirect
43. C₂²×#518	C₂²⋊C₄	semidirect
56. #14679	C₂²⋊C₄	semidirect
57. C₂×#1438	C₂²⋊C₄	semidirect
58. C₂×#1438	C₂²⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
#72323217	2
#72314476	2
#72314165	2
#72258757	2
#72237589	2
#72237391	2
#72226018	2