Subgroups — #46809057

Order 34432.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#31770973	C₂	1	1	yes	maximal, characteristic
2	#31770404	C₂	1	1	yes	maximal, characteristic
3	#356×C₂₆₉	C₂	1	1	yes	maximal, characteristic
4	#22575445	C₄	1	1	yes	characteristic
5	#22575377	—	2	1	no	—
6	#22575377	—	2	1	no	—
7	#22575372	—	2	1	no	—
8	#22575364	C₄	1	1	yes	characteristic
9	C₄⋊Q₈×C₂₆₉	C₂²	1	1	yes	characteristic
10	#101×C₂₆₉	—	2	1	no	—
11	#16864386	—	4	1	no	—
12	C₂×#14524130	—	2	1	no	—
13	C₄⋊Dic₂₆₉	—	2	1	no	—
14	#16864381	—	4	1	no	—
15	C₂×Q₈×C₂₆₉	D₄	1	1	yes	characteristic
16	C₂×Q₈×C₂₆₉	D₄	1	1	yes	characteristic
17	M₁₆×C₂₆₉	—	4	1	no	—
18	C₄⋊C₄×C₂₆₉	—	4	1	no	—
19	C₄²×C₂₆₉	C₂×C₄	1	1	yes	characteristic
20	C₂×Dic₂₆₉	—	4	1	no	—
21	#14524130	—	4	1	no	—
22	#14524130	—	8	1	no	—
23	Q₈×C₂₆₉	—	4	1	no	—
24	Q₈×C₂₆₉	—	4	1	no	—
25	C₂×C₄×C₂₆₉	—	2	1	no	—
26	C₂×C₄×C₂₆₉	—	2	1	no	—
27	C₂×C₄×C₂₆₉	—	2	1	no	—
28	C₂×C₄×C₂₆₉	C₂²⋊C₄	1	1	yes	characteristic
29	C₈×C₂₆₉	—	8	1	no	—
30	C₂²×C₂₆₉	#99	1	1	yes	characteristic
31	C₄×C₂₆₉	—	2	1	no	—
32	C₄×C₂₆₉	—	4	1	no	—
33	C₄×C₂₆₉	—	4	1	no	—
34	C₄×C₂₆₉	—	4	1	no	—
35	Dic₂₆₉	—	8	1	no	—
36	C₂×C₂₆₉	—	2	1	no	—
37	C₂×C₂₆₉	#354	1	1	yes	characteristic
38	C₂₆₉	#1413	1	1	yes	characteristic, complemented, Sylow
39	#1413	—	269	1	no	maximal, Sylow
40	#491	—	269	1	no	—
41	#356	—	269	1	no	—
42	#356	D₂₆₉	1	1	yes	characteristic
43	C₄⋊Q₈	C₂×D₂₆₉	1	1	yes	characteristic
44	#125	—	269	1	no	—
45	#103	—	538	1	no	—
46	#103	—	538	1	no	—
47	#101	—	2	1	no	—
48	#101	—	538	1	no	—
49	#97	—	269	1	no	—
50	C₂×Q₈	C₂²⋊D₂₆₉	1	1	yes	characteristic
51	C₂×Q₈	D₁₀₇₆	1	1	yes	characteristic
52	M₁₆	—	4	1	no	—
53	M₁₆	—	1076	1	no	—
54	C₂×C₈	—	538	1	no	—
55	C₄⋊C₄	—	4	1	no	—
56	C₄⋊C₄	—	538	1	no	—
57	C₄⋊C₄	—	1076	1	no	—
58	C₄²	C₄×D₂₆₉	1	1	yes	characteristic
59	Q₈	—	4	1	no	—
60	Q₈	—	4	1	no	—
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	—	2	1	no	—
63	C₂×C₄	—	2	1	no	—
64	C₂×C₄	—	1076	1	no	—
65	C₂×C₄	#16864379	1	1	yes	characteristic
66	C₈	—	8	1	no	—
67	C₈	—	1076	1	no	—
68	C₈	—	2152	1	no	—
69	C₂²	#22575367	1	1	yes	characteristic
70	C₄	—	2	1	no	—
71	C₄	—	4	1	no	—
72	C₄	—	4	1	no	—
73	C₄	—	4	1	no	—
74	C₄	—	2152	1	no	—
75	C₂	—	2	1	no	—
76	C₂	#31770399	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#31770399
commutator	C₄²×C₂₆₉	C₂×C₄
Frattini	C₄⋊Q₈	C₂×D₂₆₉
Fitting	#356×C₂₆₉	C₂
socle	C₂×C₂₆₉	#354
prime radical	C₂²×C₂₆₉	#99
cosocle	C₄⋊Q₈×C₂₆₉	C₂²
hypercenter	#356	D₂₆₉
hypocenter	C₂₆₉	#1413
LCS	G ▷ C₄²×C₂₆₉ ▷ C₂×C₄×C₂₆₉ ▷ C₂²×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊Q₈ ◁ #356

Complements

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

N	H ≅ G/N	type
38. C₂₆₉	#1413	semidirect

Maximal subgroups

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

subgroup	index
#31770973	2
#31770404	2
#356×C₂₆₉	2
#1413	269