Subgroups — #72227036

Order 2048.

256 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72225397	C₂	1	2	yes	maximal, complemented
2	#72216263	C₂	1	1	yes	maximal, characteristic
3	#151724	—	2	2	no	—
4	#123403	C₄	1	2	yes	—
5	#94748	—	2	2	no	—
6	C₁₆×C₃₂	C₂²	1	1	yes	characteristic
7	#13725	—	4	2	no	—
8	#13725	—	4	2	no	—
9	#12515	—	2	2	no	—
10	#12515	—	4	2	no	—
11	#7544	—	4	2	no	—
12	C₈×C₃₂	D₄	1	2	yes	—
13	C₁₆²	C₂×C₄	1	1	yes	characteristic
14	C₂×D₃₂	—	8	4	no	—
15	C₂×D₃₂	—	8	2	no	—
16	C₂×D₃₂	—	8	2	no	—
17	#2246	—	4	4	no	—
18	#2246	—	8	4	no	—
19	#1712	—	4	2	no	—
20	M₁₂₈	—	8	2	no	—
21	C₄×C₃₂	D₈	1	2	yes	—
22	C₄×C₃₂	SD₁₆	1	2	yes	—
23	C₈×C₁₆	—	2	1	no	—
24	C₈×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
25	C₂×D₁₆	—	8	4	no	—
26	C₂×D₁₆	—	8	4	no	—
27	C₂×D₁₆	—	16	8	no	—
28	#493	—	8	4	no	—
29	#493	—	8	2	no	—
30	D₃₂	—	16	8	no	—
31	D₃₂	—	16	4	no	—
32	D₃₂	—	16	2	no	—
33	D₃₂	—	16	2	no	—
34	C₂×C₃₂	—	2	2	no	—
35	C₂×C₃₂	D₁₆	1	2	yes	—
36	C₂×C₃₂	SD₃₂	1	2	yes	—
37	C₄×C₁₆	—	2	2	no	—
38	C₄×C₁₆	#102	1	2	yes	—
39	C₈²	#100	1	1	yes	characteristic
40	C₆₄	—	16	2	no	—
41	C₂×D₈	—	16	8	no	—
42	C₂×D₈	—	16	4	no	—
43	#127	—	16	2	no	—
44	D₁₆	—	16	8	no	—
45	D₁₆	—	16	4	no	—
46	D₁₆	—	16	4	no	—
47	D₁₆	—	32	16	no	—
48	C₂×C₁₆	—	2	4	no	—
49	C₂×C₁₆	#357	1	2	yes	—
50	C₂×C₁₆	#361	1	2	yes	—
51	C₄×C₈	—	2	1	no	—
52	C₄×C₈	#331	1	1	yes	characteristic
53	C₃₂	—	2	4	no	—
54	C₃₂	—	2	2	no	—
55	C₃₂	D₃₂	1	2	yes	—
56	C₃₂	SD₆₄	1	2	yes	—
57	C₂×D₄	—	32	4	no	—
58	C₂×D₄	—	32	2	no	—
59	D₈	—	32	16	no	—
60	D₈	—	32	4	no	—
61	D₈	—	32	4	no	—
62	C₂×C₈	—	2	2	no	—
63	C₂×C₈	#1361	1	2	yes	—
64	C₄²	#1357	1	1	yes	characteristic
65	C₁₆	—	2	8	no	—
66	C₁₆	—	2	2	no	—
67	C₁₆	#1415	1	2	yes	—
68	C₁₆	#1419	1	2	yes	—
69	C₂³	—	64	2	no	—
70	D₄	—	64	8	no	—
71	D₄	—	64	4	no	—
72	C₂×C₄	—	2	1	no	—
73	C₂×C₄	#7096	1	1	yes	characteristic
74	C₈	—	2	4	no	—
75	C₈	#7444	1	2	yes	—
76	C₈	#7448	1	2	yes	—
77	C₂²	—	128	4	no	—
78	C₂²	—	128	2	no	—
79	C₂²	#93866	1	1	yes	characteristic
80	C₄	—	2	2	no	—
81	C₄	#93869	1	2	yes	—
82	C₂	—	256	2	no	—
83	C₂	—	2	1	no	—
84	C₂	#72227035	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72227035
commutator	C₁₆²	C₂×C₄
Frattini, cosocle	C₁₆×C₃₂	C₂²
prime radical	#72216263	C₂
LCS	G ▷ C₁₆² ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈×C₁₆ ◁ C₁₆×C₃₂ ◁ G
LECS	G ▷ C₁₆×C₃₂ ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72225397	C₂	semidirect

Maximal subgroups

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

subgroup	index
#72225397	2
#72216263	2