Subgroups — #46826032

Order 34432.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#31771514	C₂	1	1	yes	maximal, characteristic
2	#31771512	C₂	1	1	yes	maximal, characteristic
3	#31770960	C₂	1	1	yes	maximal, characteristic
4	#22575514	—	2	1	no	—
5	#22575512	—	2	1	no	—
6	#22575445	C₄	1	1	yes	characteristic, complemented
7	#22575441	C₂²	1	1	yes	characteristic
8	#22575378	—	2	1	no	—
9	#22575375	—	2	1	no	—
10	C₄×C₈×C₂₆₉	C₄	1	1	yes	characteristic, complemented
11	#16864411	—	4	1	no	—
12	C₂²⋊#11761640	—	4	1	no	—
13	C₂×Dic₅₃₈	D₄	1	1	yes	characteristic
14	C₂×D₁₀₇₆	D₄	1	1	yes	characteristic
15	C₄⋊Dic₂₆₉	—	2	1	no	—
16	#16864381	—	4	1	no	—
17	#16864379	—	4	1	no	—
18	C₂×C₈×C₂₆₉	—	2	1	no	—
19	C₄²×C₂₆₉	C₂×C₄	1	1	yes	characteristic
20	C₂×#11761640	—	8	1	no	—
21	#14524137	—	8	1	no	—
22	C₂²×D₂₆₉	—	2	1	no	—
23	Dic₅₃₈	—	4	1	no	—
24	D₁₀₇₆	—	4	1	no	—
25	C₂×Dic₂₆₉	—	2	1	no	—
26	C₂×Dic₂₆₉	—	4	1	no	—
27	C₂×C₄×C₂₆₉	—	2	1	no	—
28	C₂×C₄×C₂₆₉	C₂²⋊C₄	1	1	yes	characteristic
29	C₈×C₂₆₉	—	4	1	no	—
30	C₂²×C₂₆₉	#99	1	1	yes	characteristic
31	C₂×D₂₆₉	—	4	1	no	—
32	C₂×D₂₆₉	—	8	1	no	—
33	#11761640	—	16	1	no	—
34	C₄×C₂₆₉	—	2	1	no	—
35	C₄×C₂₆₉	—	4	1	no	—
36	Dic₂₆₉	—	4	1	no	—
37	Dic₂₆₉	—	8	1	no	—
38	C₂×C₂₆₉	—	2	1	no	—
39	C₂×C₂₆₉	#354	1	1	yes	characteristic
40	D₂₆₉	—	8	1	no	—
41	C₂₆₉	#1414	1	1	yes	characteristic, complemented, Sylow
42	#1414	—	269	1	no	maximal, Sylow
43	#488	—	269	1	no	—
44	#355	—	269	1	no	—
45	#354	—	269	1	no	—
46	#125	—	269	1	no	—
47	#124	—	269	1	no	—
48	#103	—	538	1	no	—
49	#102	—	538	1	no	—
50	#101	—	538	1	no	—
51	#99	—	538	1	no	—
52	C₄×C₈	#11761640	1	1	yes	characteristic, complemented
53	C₂×Q₈	—	269	1	no	—
54	C₂×D₄	—	269	1	no	—
55	M₁₆	—	1076	1	no	—
56	C₂×C₈	—	2	1	no	—
57	C₄⋊C₄	—	538	1	no	—
58	C₄⋊C₄	—	1076	1	no	—
59	C₂²⋊C₄	—	1076	1	no	—
60	C₂²⋊C₄	—	1076	1	no	—
61	C₄²	C₂×#11761640	1	1	yes	characteristic
62	C₂³	—	538	1	no	—
63	Q₈	—	1076	1	no	—
64	D₄	—	1076	1	no	—
65	C₂×C₄	—	2	1	no	—
66	C₂×C₄	—	538	1	no	—
67	C₂×C₄	—	1076	1	no	—
68	C₂×C₄	—	2152	1	no	—
69	C₂×C₄	C₂²⋊#11761640	1	1	yes	characteristic
70	C₈	—	4	1	no	—
71	C₈	—	2152	1	no	—
72	C₂²	—	1076	1	no	—
73	C₂²	—	2152	1	no	—
74	C₂²	#22575512	1	1	yes	characteristic
75	C₄	—	2	1	no	—
76	C₄	—	4	1	no	—
77	C₄	—	1076	1	no	—
78	C₄	—	2152	1	no	—
79	C₄	—	4304	1	no	—
80	C₂	—	2	1	no	—
81	C₂	—	2152	1	no	—
82	C₂	#31771512	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#31771512
commutator	C₄²×C₂₆₉	C₂×C₄
Frattini	C₄²	C₂×#11761640
Fitting	C₄×C₈×C₂₆₉	C₄
socle	C₂×C₂₆₉	#354
prime radical	C₂×D₁₀₇₆	D₄
cosocle	#22575441	C₂²
hypercenter	C₄×C₈	#11761640
hypocenter	C₂₆₉	#1414
LCS	G ▷ C₄²×C₂₆₉ ▷ C₂×C₄×C₂₆₉ ▷ C₂²×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
6. #22575445	C₄	semidirect
10. C₄×C₈×C₂₆₉	C₄	semidirect
41. C₂₆₉	#1414	semidirect
52. C₄×C₈	#11761640	semidirect

Maximal subgroups

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

subgroup	index
#31771514	2
#31771512	2
#31770960	2
#1414	269