Subgroups — #72262927

Order 4096.

At least 8444 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	#72262621	C₂	1	1	yes	maximal, characteristic, complemented
2	#72238383	C₂	1	1	yes	maximal, characteristic, complemented
3	#72237997	C₂	1	1	yes	maximal, characteristic, complemented
4	#72237886	C₂	1	1	yes	maximal, characteristic, complemented
5	#72237449	C₂	1	1	yes	maximal, characteristic, complemented
6	#72237393	C₂	1	1	yes	maximal, characteristic, complemented
7	#72235725	C₂	1	1	yes	maximal, characteristic
8	#72235718	C₂²	1	1	yes	characteristic
9	#72235677	C₄	1	2	yes	complemented
10	#72234636	C₂²	1	1	yes	characteristic
11	#72234625	C₂²	1	1	yes	characteristic
12	#72233792	C₄	1	2	yes	complemented
13	#72225870	C₂²	1	1	yes	characteristic
14	#72225870	C₂²	1	1	yes	characteristic
15	#72225713	C₂²	1	1	yes	characteristic
16	#72219314	C₂²	1	1	yes	characteristic
17	C₂×#62817	D₄	1	1	yes	characteristic
18	C₂×#46996	D₄	1	1	yes	characteristic
19	C₂×#34980	C₂³	1	1	yes	characteristic
20	#6499870	C₂×C₄	1	2	yes	—
21	#6499870	D₄	1	2	yes	—
22	#6374251	D₄	1	1	yes	characteristic
23	#6374128	D₄	1	1	yes	characteristic
24	C₂×#21746	D₄	1	1	yes	characteristic
25	#5024691	C₂×C₄	1	1	yes	characteristic
26	#5020751	D₄	1	1	yes	characteristic
27	#4430405	C₂×C₄	1	1	yes	characteristic
28	#648360	D₄	1	1	yes	characteristic
29	#576160	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	#134036	C₂×C₄	1	2	yes	—
35	#134036	D₄	1	2	yes	—
36	#134034	D₄	1	2	yes	—
37	C₂⁵×D₄	#41	1	1	yes	characteristic
38	#62817	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×#2840	C₂²⋊C₄	1	1	yes	characteristic, complemented
43	C₂×#2849	C₂×D₄	1	1	yes	characteristic
44	C₂×#2849	#41	1	1	yes	characteristic
45	C₂×#2846	C₂×D₄	1	1	yes	characteristic
46	C₂×#2846	#41	1	1	yes	characteristic
47	C₂×#2403	C₂²⋊C₄	1	1	yes	characteristic, complemented
48	C₂×#2613	C₂²×C₄	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	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	#21746	C₂²⋊C₄	1	2	yes	—
56	#13827	C₂²⋊C₄	1	1	yes	characteristic, complemented
57	C₂²⋊C₄²	C₂²⋊C₄	1	1	yes	characteristic, complemented
58	#11500	C₂²⋊C₄	1	4	yes	complemented
59	C₂⁷	#121	1	1	yes	characteristic
60	C₂⁷	#123	1	1	yes	characteristic
61	C₂⁴×D₄	C₄×D₄	1	1	yes	characteristic
62	C₂⁴×D₄	#120	1	1	yes	characteristic
63	C₂⁴×D₄	#121	1	1	yes	characteristic
64	C₂⁴×D₄	#121	1	1	yes	characteristic
65	C₂⁴×D₄	#123	1	1	yes	characteristic
66	C₂⁵×C₄	#124	1	1	yes	characteristic
67	C₂²×#123	C₄×D₄	1	1	yes	characteristic
68	C₂²×#123	#121	1	1	yes	characteristic
69	C₂²×#123	#121	1	1	yes	characteristic
70	C₂²×#123	#127	1	1	yes	characteristic
71	C₂²×#120	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×#120	C₄×D₄	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	#123	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	#120	1	1	yes	characteristic
84	C₂×#381	#121	1	1	yes	characteristic
85	C₂×#381	#123	1	1	yes	characteristic
86	C₂×#379	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	C₂×#379	C₄×D₄	1	1	yes	characteristic
88	C₂×#379	#120	1	1	yes	characteristic
89	C₂×#379	#120	1	1	yes	characteristic
90	C₂×#379	#121	1	1	yes	characteristic
91	C₂×#379	#121	1	1	yes	characteristic
92	C₂×#379	#121	1	1	yes	characteristic
93	C₂×#379	#124	1	1	yes	characteristic
94	C₂⁶	#386	1	1	yes	characteristic
95	C₂⁶	#386	1	1	yes	characteristic
96	C₂⁶	#392	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	#390	1	1	yes	characteristic
105	C₂⁴×C₄	#394	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	1	yes	characteristic
141	C₂³	#144382	1	1	yes	characteristic
142	C₂³	#144399	1	2	yes	—
143	C₂²	#72218317	1	1	yes	characteristic
144	C₂²	#72236678	1	1	yes	characteristic
145	C₂²	#72251664	1	1	yes	characteristic
146	C₂	#72262389	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72262389
commutator	C₂×#2613	C₂²×C₄
Frattini, cosocle	C₂×#34980	C₂³
prime radical	#72235725	C₂
LCS	G ▷ C₂×#2613 ▷ 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. #72262621	C₂	semidirect
2. #72238383	C₂	semidirect
3. #72237997	C₂	semidirect
4. #72237886	C₂	semidirect
5. #72237449	C₂	semidirect
6. #72237393	C₂	semidirect
9. #72235677	C₄	semidirect
12. #72233792	C₄	semidirect
42. C₂×#2840	C₂²⋊C₄	semidirect
47. C₂×#2403	C₂²⋊C₄	semidirect
56. #13827	C₂²⋊C₄	semidirect
57. C₂²⋊C₄²	C₂²⋊C₄	semidirect
58. #11500	C₂²⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
#72262621	2
#72238383	2
#72237997	2
#72237886	2
#72237449	2
#72237393	2
#72235725	2