Subgroups — #72238372

Order 4096.

At least 8306 conjugacy classes of subgroups; only the 171 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	#72238371	C₂	1	1	yes	maximal, characteristic, complemented
2	#72238364	C₂	1	1	yes	maximal, characteristic, complemented
3	#72237483	C₂	1	1	yes	maximal, characteristic, complemented
4	#72237480	C₂	1	1	yes	maximal, characteristic, complemented
5	#72237436	C₂	1	1	yes	maximal, characteristic, complemented
6	#72237391	C₂	1	1	yes	maximal, characteristic, complemented
7	#72235718	C₂²	1	1	yes	characteristic
8	#72234717	C₂	1	1	yes	maximal, characteristic
9	#72234716	C₂²	1	1	yes	characteristic
10	#72234715	C₄	1	2	yes	complemented
11	#72234714	C₄	1	2	yes	complemented
12	#72234668	C₂²	1	1	yes	characteristic
13	#72225870	C₂²	1	1	yes	characteristic
14	#72225870	C₂²	1	1	yes	characteristic
15	#72224136	C₂²	1	1	yes	characteristic
16	#72219314	C₂²	1	1	yes	characteristic
17	#10586404	D₄	1	1	yes	characteristic
18	C₂×#47031	D₄	1	1	yes	characteristic
19	C₂×#34980	C₂³	1	1	yes	characteristic
20	#7626202	D₄	1	1	yes	characteristic
21	#6499871	D₄	1	2	yes	—
22	#6499869	C₂×C₄	1	2	yes	—
23	#6374128	D₄	1	1	yes	characteristic
24	C₂×#21746	D₄	1	1	yes	characteristic
25	#5025772	D₄	1	1	yes	characteristic
26	#5020747	C₂×C₄	1	1	yes	characteristic
27	#4435227	D₄	1	1	yes	characteristic
28	#686193	C₂×C₄	1	1	yes	characteristic
29	#576189	D₄	1	1	yes	characteristic
30	#480250	D₄	1	2	yes	—
31	C₂×#11499	D₄	1	2	yes	—
32	#188109	D₄	1	1	yes	characteristic
33	#188049	D₄	1	1	yes	characteristic
34	#134035	C₂×C₄	1	2	yes	—
35	#134035	D₄	1	2	yes	—
36	#134033	D₄	1	2	yes	—
37	C₂⁵×D₄	#41	1	1	yes	characteristic
38	#62820	C₂²⋊C₄	1	2	yes	—
39	C₂²×#534	C₂×D₄	1	1	yes	characteristic
40	C₂²×#534	C₂×D₄	1	1	yes	characteristic
41	C₂³×#120	C₂×D₄	1	1	yes	characteristic
42	C₂×#2815	C₂²⋊C₄	1	1	yes	characteristic, complemented
43	C₂×#2849	C₂²×C₄	1	1	yes	characteristic
44	C₂×#2849	C₂×D₄	1	1	yes	characteristic
45	C₂×#2846	C₂×D₄	1	1	yes	characteristic
46	C₂×#2846	#41	1	1	yes	characteristic
47	C₂×#2607	C₂²⋊C₄	1	1	yes	characteristic, complemented
48	C₂×#2613	C₂×D₄	1	1	yes	characteristic
49	C₂×#2613	C₂×D₄	1	1	yes	characteristic
50	C₂×#2613	#41	1	1	yes	characteristic
51	C₂×#2613	#41	1	1	yes	characteristic
52	C₂×#2308	C₂×D₄	1	1	yes	characteristic
53	C₂×#2308	C₂×D₄	1	1	yes	characteristic
54	C₂²×#379	C₂×D₄	1	1	yes	characteristic
55	#21747	C₂²⋊C₄	1	2	yes	—
56	#13828	C₂²⋊C₄	1	1	yes	characteristic, complemented
57	#13807	C₂²⋊C₄	1	1	yes	characteristic, complemented
58	#11499	C₂²⋊C₄	1	4	yes	complemented
59	C₂⁷	#121	1	1	yes	characteristic
60	C₂⁷	#124	1	1	yes	characteristic
61	C₂⁴×D₄	#120	1	1	yes	characteristic
62	C₂⁴×D₄	#121	1	1	yes	characteristic
63	C₂⁴×D₄	#121	1	1	yes	characteristic
64	C₂⁴×D₄	#123	1	1	yes	characteristic
65	C₂⁴×D₄	#123	1	1	yes	characteristic
66	C₂⁵×C₄	C₄×D₄	1	1	yes	characteristic
67	C₂²×#123	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×#123	C₄×D₄	1	1	yes	characteristic
69	C₂²×#123	#120	1	1	yes	characteristic
70	C₂²×#123	#121	1	1	yes	characteristic
71	C₂²×#120	C₄×D₄	1	1	yes	characteristic
72	C₂²×#120	#120	1	1	yes	characteristic
73	C₂²×#120	#121	1	1	yes	characteristic
74	C₂²×#120	#121	1	1	yes	characteristic
75	C₂²×#120	#121	1	1	yes	characteristic
76	C₂²×#120	#121	1	1	yes	characteristic
77	C₂²×#120	#123	1	1	yes	characteristic
78	C₂²×#120	#127	1	1	yes	characteristic
79	C₂³×C₂²⋊C₄	#120	1	1	yes	characteristic
80	C₂³×C₂²⋊C₄	#120	1	1	yes	characteristic
81	C₂³×C₂²⋊C₄	#127	1	1	yes	characteristic
82	C₂×#381	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₂×#381	C₄×D₄	1	1	yes	characteristic
84	C₂×#381	#120	1	1	yes	characteristic
85	C₂×#381	#121	1	1	yes	characteristic
86	C₂×#379	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	C₂×#379	#120	1	1	yes	characteristic
88	C₂×#379	#120	1	1	yes	characteristic
89	C₂×#379	#121	1	1	yes	characteristic
90	C₂×#379	#121	1	1	yes	characteristic
91	C₂×#379	#121	1	1	yes	characteristic
92	C₂×#379	#123	1	1	yes	characteristic
93	C₂×#379	#124	1	1	yes	characteristic
94	C₂⁶	#390	1	1	yes	characteristic
95	C₂⁶	#392	1	1	yes	characteristic
96	C₂⁶	#394	1	1	yes	characteristic
97	C₂⁶	#394	1	1	yes	characteristic
98	C₂⁶	#394	1	1	yes	characteristic
99	C₂⁶	B(4,𝔽₂)	1	1	yes	characteristic
100	C₂³×D₄	#386	1	1	yes	characteristic
101	C₂³×D₄	#392	1	1	yes	characteristic
102	C₂³×D₄	B(4,𝔽₂)	1	1	yes	characteristic
103	C₂³×D₄	B(4,𝔽₂)	1	1	yes	characteristic
104	C₂⁴×C₄	#386	1	1	yes	characteristic
105	C₂⁴×C₄	#386	1	1	yes	characteristic
106	C₂⁴×C₄	B(4,𝔽₂)	1	1	yes	characteristic
107	C₂²×C₂²⋊C₄	#379	1	1	yes	characteristic
108	C₂²×C₂²⋊C₄	#386	1	1	yes	characteristic
109	C₂²×C₂²⋊C₄	#390	1	1	yes	characteristic
110	C₂²×C₂²⋊C₄	#392	1	1	yes	characteristic
111	C₂²×C₂²⋊C₄	#392	1	1	yes	characteristic
112	C₂²×C₂²⋊C₄	#394	1	1	yes	characteristic
113	C₂²×C₂²⋊C₄	B(4,𝔽₂)	1	2	yes	—
114	C₂²×C₂²⋊C₄	B(4,𝔽₂)	1	2	yes	—
115	C₂²×C₂²⋊C₄	B(4,𝔽₂)	1	2	yes	—
116	C₂²×C₂²⋊C₄	#458	1	2	yes	—
117	C₂⁵	#1438	1	1	yes	characteristic
118	C₂⁵	#1889	1	1	yes	characteristic
119	C₂⁵	#2021	1	1	yes	characteristic
120	C₂⁵	#2021	1	1	yes	characteristic
121	C₂⁵	#2021	1	1	yes	characteristic
122	C₂⁵	#2021	1	1	yes	characteristic
123	C₂⁵	#2021	1	1	yes	characteristic
124	C₂⁵	#2021	1	1	yes	characteristic
125	C₂⁵	#2022	1	1	yes	characteristic
126	C₂³×C₄	#1889	1	1	yes	characteristic
127	C₂³×C₄	#2021	1	1	yes	characteristic
128	C₂×C₂²⋊C₄	#1790	1	2	yes	—
129	C₂×C₂²⋊C₄	#1889	1	2	yes	—
130	C₂×C₂²⋊C₄	#2021	1	2	yes	—
131	C₂×C₂²⋊C₄	#2044	1	2	yes	—
132	C₂⁴	#10336	1	1	yes	characteristic
133	C₂⁴	#10336	1	1	yes	characteristic
134	C₂⁴	#10337	1	1	yes	characteristic
135	C₂⁴	#10337	1	1	yes	characteristic
136	C₂⁴	#10991	1	1	yes	characteristic
137	C₂⁴	#10991	1	1	yes	characteristic
138	C₂⁴	#10995	1	1	yes	characteristic
139	C₂³	#111458	1	1	yes	characteristic
140	C₂³	#144382	1	2	yes	—
141	C₂³	#144382	1	1	yes	characteristic
142	C₂³	#144383	1	1	yes	characteristic
143	C₂²	#72218317	1	1	yes	characteristic
144	C₂²	#72236523	1	1	yes	characteristic
145	C₂²	#72236678	1	1	yes	characteristic
146	C₂	#72236923	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72236923
commutator	C₂×#2849	C₂²×C₄
Frattini, cosocle	C₂×#34980	C₂³
prime radical	#72234717	C₂
LCS	G ▷ C₂×#2849 ▷ 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. #72238371	C₂	semidirect
2. #72238364	C₂	semidirect
3. #72237483	C₂	semidirect
4. #72237480	C₂	semidirect
5. #72237436	C₂	semidirect
6. #72237391	C₂	semidirect
10. #72234715	C₄	semidirect
11. #72234714	C₄	semidirect
42. C₂×#2815	C₂²⋊C₄	semidirect
47. C₂×#2607	C₂²⋊C₄	semidirect
56. #13828	C₂²⋊C₄	semidirect
57. #13807	C₂²⋊C₄	semidirect
58. #11499	C₂²⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
#72238371	2
#72238364	2
#72237483	2
#72237480	2
#72237436	2
#72237391	2
#72234717	2