Subgroups — #46810491

Order 34432.

212 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#31770985	C₂	1	1	yes	maximal, characteristic, complemented
2	#31770982	C₂	1	1	yes	maximal, characteristic
3	#31770952	C₂	1	4	yes	maximal, complemented
4	C₈²×C₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×Dic₁₀₇₆	—	2	2	no	—
6	C₂×D₂₁₅₂	—	2	2	no	—
7	#22575453	C₂²	1	2	yes	—
8	#22575450	C₂²	1	2	yes	—
9	#22575375	—	2	8	no	—
10	C₄×C₈×C₂₆₉	C₂²	1	2	yes	—
11	C₄×C₈×C₂₆₉	C₂²	1	1	yes	characteristic
12	C₂×Dic₅₃₈	—	2	2	no	—
13	C₂×D₁₀₇₆	—	2	4	no	—
14	C₂×D₁₀₇₆	—	2	2	no	—
15	Dic₁₀₇₆	—	4	4	no	—
16	D₂₁₅₂	—	4	4	no	—
17	C₄⋊Dic₂₆₉	—	2	4	no	—
18	C₂×C₈×C₂₆₉	D₄	1	4	yes	—
19	C₂×C₈×C₂₆₉	D₄	1	2	yes	—
20	C₄²×C₂₆₉	C₂³	1	1	yes	characteristic
21	C₂²×D₂₆₉	—	4	2	no	—
22	Dic₅₃₈	—	4	4	no	—
23	D₁₀₇₆	—	4	4	no	—
24	D₁₀₇₆	—	4	8	no	—
25	C₂×Dic₂₆₉	—	4	2	no	—
26	C₂×C₄×C₂₆₉	C₂×D₄	1	1	yes	characteristic
27	C₂×C₄×C₂₆₉	C₂×D₄	1	2	yes	—
28	C₈×C₂₆₉	—	2	4	no	—
29	C₈×C₂₆₉	D₈	1	4	yes	—
30	C₂²×C₂₆₉	#127	1	1	yes	characteristic
31	C₂×D₂₆₉	—	8	2	no	—
32	C₂×D₂₆₉	—	8	4	no	—
33	C₄×C₂₆₉	C₂×D₈	1	2	yes	—
34	C₄×C₂₆₉	#135	1	4	yes	—
35	Dic₂₆₉	—	8	2	no	—
36	C₂×C₂₆₉	#493	1	1	yes	characteristic
37	C₂×C₂₆₉	#495	1	2	yes	—
38	D₂₆₉	—	16	2	no	—
39	C₂₆₉	#1709	1	1	yes	characteristic, complemented, Sylow
40	#1709	—	269	1	no	maximal, Sylow
41	#494	—	269	1	no	—
42	#493	—	269	1	no	—
43	#486	—	269	4	no	—
44	C₈²	D₂₆₉	1	1	yes	characteristic, complemented
45	C₂×Q₁₆	—	538	2	no	—
46	C₂×D₈	—	538	2	no	—
47	C₄⋊Q₈	—	269	2	no	—
48	#127	—	269	2	no	—
49	#102	—	538	8	no	—
50	C₄×C₈	C₂×D₂₆₉	1	2	yes	—
51	C₄×C₈	C₂×D₂₆₉	1	1	yes	characteristic
52	C₂×Q₈	—	538	2	no	—
53	C₂×D₄	—	538	4	no	—
54	C₂×D₄	—	538	2	no	—
55	Q₁₆	—	1076	4	no	—
56	D₈	—	1076	4	no	—
57	C₂×C₈	D₁₀₇₆	1	4	yes	—
58	C₂×C₈	D₁₀₇₆	1	2	yes	—
59	C₄⋊C₄	—	538	4	no	—
60	C₄²	C₂²×D₂₆₉	1	1	yes	characteristic
61	C₂³	—	1076	2	no	—
62	Q₈	—	1076	4	no	—
63	D₄	—	1076	4	no	—
64	D₄	—	1076	8	no	—
65	C₂×C₄	—	1076	2	no	—
66	C₂×C₄	C₂×D₁₀₇₆	1	1	yes	characteristic
67	C₂×C₄	C₂×D₁₀₇₆	1	2	yes	—
68	C₈	—	2	4	no	—
69	C₈	D₂₁₅₂	1	4	yes	—
70	C₂²	—	2152	2	no	—
71	C₂²	—	2152	4	no	—
72	C₂²	#22575450	1	1	yes	characteristic
73	C₄	—	2152	2	no	—
74	C₄	C₂×D₂₁₅₂	1	2	yes	—
75	C₄	#22575475	1	4	yes	—
76	C₂	—	4304	2	no	—
77	C₂	#31770982	1	1	yes	characteristic
78	C₂	#31770989	1	2	yes	—

	subgroup	G/H
center	C₂²	#22575450
commutator, cosocle	C₄²×C₂₆₉	C₂³
Frattini	C₄²	C₂²×D₂₆₉
Fitting	C₈²×C₂₆₉	C₂
socle	C₂²×C₂₆₉	#127
prime radical	#31770982	C₂
hypercenter	C₈²	D₂₆₉
hypocenter	C₂₆₉	#1709
LCS	G ▷ C₄²×C₂₆₉ ▷ C₂²×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈²

Complements

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

N	H ≅ G/N	type
1. #31770985	C₂	semidirect
3. #31770952	C₂	semidirect
4. C₈²×C₂₆₉	C₂	semidirect
39. C₂₆₉	#1709	semidirect
44. C₈²	D₂₆₉	semidirect

Maximal subgroups

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

subgroup	index
#31770985	2
#31770982	2
#31770952	2
C₈²×C₂₆₉	2
#1709	269