Subgroups — #46826016

Order 34432.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#31771510	C₂	1	2	yes	maximal, complemented
2	#31770981	C₂	1	1	yes	maximal, characteristic
3	#22575512	—	2	2	no	—
4	C₂×#16864387	—	2	2	no	—
5	#22575450	C₂²	1	1	yes	characteristic
6	C₄×#14524130	C₄	1	1	yes	characteristic, complemented
7	#127×C₂₆₉	C₄	1	1	yes	characteristic, complemented
8	C₂²⋊#11761640	—	4	2	no	—
9	C₂×D₁₀₇₆	—	4	1	no	—
10	C₂×D₁₀₇₆	D₄	1	2	yes	—
11	#16864387	—	4	4	no	—
12	C₂×#14524130	—	2	1	no	—
13	C₂×D₄×C₂₆₉	—	2	1	no	—
14	C₂×D₄×C₂₆₉	—	4	1	no	—
15	C₄²×C₂₆₉	C₂×C₄	1	1	yes	characteristic
16	C₂×#11761640	—	8	2	no	—
17	C₂²×D₂₆₉	—	2	2	no	—
18	D₁₀₇₆	—	4	2	no	—
19	D₁₀₇₆	—	8	2	no	—
20	#14524130	—	2	2	no	—
21	C₂³×C₂₆₉	—	4	1	no	—
22	D₄×C₂₆₉	—	4	2	no	—
23	D₄×C₂₆₉	—	8	2	no	—
24	C₂×C₄×C₂₆₉	—	2	1	no	—
25	C₂×C₄×C₂₆₉	C₂²⋊C₄	1	1	yes	characteristic
26	C₂²×C₂₆₉	—	8	2	no	—
27	C₂²×C₂₆₉	—	8	1	no	—
28	C₂²×C₂₆₉	#99	1	1	yes	characteristic
29	C₂×D₂₆₉	—	4	2	no	—
30	C₂×D₂₆₉	—	8	2	no	—
31	#11761640	—	16	2	no	—
32	C₄×C₂₆₉	—	2	2	no	—
33	C₄×C₂₆₉	—	2	1	no	—
34	C₂×C₂₆₉	—	2	1	no	—
35	C₂×C₂₆₉	—	16	1	no	—
36	C₂×C₂₆₉	#353	1	1	yes	characteristic
37	D₂₆₉	—	8	2	no	—
38	C₂₆₉	#1408	1	1	yes	characteristic, complemented, Sylow
39	#1408	—	269	1	no	maximal, Sylow
40	#493	—	269	1	no	—
41	#353	—	269	2	no	—
42	C₂×D₈	—	538	2	no	—
43	#127	—	269	1	no	—
44	#127	#11761640	1	1	yes	characteristic, complemented
45	#99	—	538	2	no	—
46	C₄×C₈	—	269	1	no	—
47	C₂×D₄	—	2	1	no	—
48	C₂×D₄	—	4	1	no	—
49	C₂×D₄	—	269	2	no	—
50	C₂×D₄	—	1076	1	no	—
51	D₈	—	1076	4	no	—
52	C₂×C₈	—	538	1	no	—
53	C₂²⋊C₄	—	1076	2	no	—
54	C₄²	C₂×#11761640	1	1	yes	characteristic
55	C₂³	—	4	1	no	—
56	C₂³	—	538	2	no	—
57	D₄	—	4	2	no	—
58	D₄	—	8	2	no	—
59	D₄	—	1076	2	no	—
60	D₄	—	2152	2	no	—
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	—	2152	2	no	—
63	C₂×C₄	C₂²⋊#11761640	1	1	yes	characteristic
64	C₈	—	538	2	no	—
65	C₂²	—	8	2	no	—
66	C₂²	—	8	1	no	—
67	C₂²	—	1076	2	no	—
68	C₂²	—	2152	2	no	—
69	C₂²	#22575511	1	1	yes	characteristic
70	C₄	—	2	2	no	—
71	C₄	—	2	1	no	—
72	C₄	—	4304	2	no	—
73	C₂	—	2	1	no	—
74	C₂	—	16	1	no	—
75	C₂	—	2152	2	no	—
76	C₂	#31771509	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#31771509
commutator	C₄²×C₂₆₉	C₂×C₄
Frattini	C₄²	C₂×#11761640
Fitting	#127×C₂₆₉	C₄
socle	C₂×C₂₆₉	#353
prime radical	#31770981	C₂
cosocle	#22575450	C₂²
hypercenter	#127	#11761640
hypocenter	C₂₆₉	#1408
LCS	G ▷ C₄²×C₂₆₉ ▷ C₂×C₄×C₂₆₉ ▷ C₂²×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ #127

Complements

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

N	H ≅ G/N	type
1. #31771510	C₂	semidirect
6. C₄×#14524130	C₄	semidirect
7. #127×C₂₆₉	C₄	semidirect
38. C₂₆₉	#1408	semidirect
44. #127	#11761640	semidirect

Maximal subgroups

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

subgroup	index
#31771510	2
#31770981	2
#1408	269