Subgroups — #31770884

Order 17216.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22575452	C₂	1	1	yes	maximal, characteristic, complemented
2	#22575415	C₂	1	1	yes	maximal, characteristic
3	#22575386	C₂	1	1	yes	maximal, characteristic, complemented
4	#22575382	C₂	1	1	yes	maximal, characteristic, complemented
5	#22575375	C₂	1	1	yes	maximal, characteristic
6	#22575373	C₂	1	1	yes	maximal, characteristic
7	#107×C₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₅₃₈	—	2	1	no	—
9	C₂×D₁₀₇₆	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₂₆₉	—	2	1	no	—
11	C₂×#14524130	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₂₆₉	C₂²	1	1	yes	characteristic
13	#16864381	—	2	1	no	—
14	#16864379	—	2	1	no	—
15	C₄×Dic₂₆₉	C₂²	1	1	yes	characteristic
16	C₂×C₈×C₂₆₉	C₂²	1	1	yes	characteristic
17	C₄⋊C₄×C₂₆₉	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×C₂₆₉	C₂²	1	1	yes	characteristic
19	C₂²×D₂₆₉	—	2	1	no	—
20	Dic₅₃₈	—	4	2	no	—
21	D₁₀₇₆	—	2	1	no	—
22	D₁₀₇₆	Q₈	1	2	yes	complemented
23	C₂×Dic₂₆₉	—	2	1	no	—
24	C₂×Dic₂₆₉	D₄	1	1	yes	characteristic
25	C₂×Dic₂₆₉	D₄	1	1	yes	characteristic
26	C₄×D₂₆₉	—	4	2	no	—
27	#14524130	—	2	1	no	—
28	C₂×C₄×C₂₆₉	—	2	1	no	—
29	C₂×C₄×C₂₆₉	—	2	1	no	—
30	C₂×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
31	C₈×C₂₆₉	—	2	1	no	—
32	C₂²×C₂₆₉	C₂×D₄	1	1	yes	characteristic
33	C₂×D₂₆₉	—	2	2	no	—
34	C₂×D₂₆₉	—	4	1	no	—
35	C₂×D₂₆₉	—	4	1	no	—
36	C₄×C₂₆₉	—	4	1	no	—
37	C₄×C₂₆₉	—	4	1	no	—
38	C₄×C₂₆₉	C₂×Q₈	1	1	yes	characteristic
39	C₄×C₂₆₉	#41	1	1	yes	characteristic
40	Dic₂₆₉	—	2	1	no	—
41	Dic₂₆₉	—	4	1	no	—
42	Dic₂₆₉	D₈	1	2	yes	—
43	C₂×C₂₆₉	C₂²⋊Q₈	1	1	yes	characteristic
44	C₂×C₂₆₉	C₂×D₈	1	1	yes	characteristic
45	C₂×C₂₆₉	#137	1	1	yes	characteristic
46	D₂₆₉	—	4	2	no	—
47	C₂₆₉	#474	1	1	yes	characteristic, complemented, Sylow
48	#474	—	269	1	no	maximal, Sylow
49	C₄⋊Q₈	—	269	1	no	—
50	C₄×D₄	—	269	1	no	—
51	#107	—	269	1	no	—
52	#107	D₂₆₉	1	1	yes	characteristic, complemented
53	C₄⋊C₈	—	269	1	no	—
54	#102	—	269	1	no	—
55	#102	—	269	1	no	—
56	C₂×Q₈	—	538	1	no	—
57	C₂×D₄	—	269	1	no	—
58	C₂²×C₄	—	538	1	no	—
59	C₂×C₈	—	269	1	no	—
60	C₂×C₈	C₂×D₂₆₉	1	1	yes	characteristic
61	C₄⋊C₄	—	269	1	no	—
62	C₄⋊C₄	—	538	1	no	—
63	C₄⋊C₄	C₂×D₂₆₉	1	1	yes	characteristic
64	C₄⋊C₄	C₂×D₂₆₉	1	1	yes	characteristic
65	C₂²⋊C₄	—	538	1	no	—
66	C₄²	—	269	1	no	—
67	C₂³	—	538	1	no	—
68	Q₈	—	1076	2	no	—
69	D₄	—	269	2	no	—
70	D₄	—	538	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	2	1	no	—
73	C₂×C₄	—	269	1	no	—
74	C₂×C₄	—	269	1	no	—
75	C₂×C₄	—	538	1	no	—
76	C₂×C₄	—	1076	2	no	—
77	C₂×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
78	C₈	—	2	1	no	—
79	C₈	—	538	1	no	—
80	C₂²	—	538	2	no	—
81	C₂²	—	1076	1	no	—
82	C₂²	—	1076	1	no	—
83	C₂²	D₄×D₂₆₉	1	1	yes	characteristic
84	C₄	—	4	1	no	—
85	C₄	—	4	1	no	—
86	C₄	—	269	2	no	—
87	C₄	—	538	1	no	—
88	C₄	—	1076	1	no	—
89	C₄	Q₈×D₂₆₉	1	1	yes	characteristic
90	C₄	#16864402	1	1	yes	characteristic
91	C₂	—	1076	2	no	—
92	C₂	#22575431	1	1	yes	characteristic
93	C₂	D₈×D₂₆₉	1	1	yes	characteristic
94	C₂	#22575485	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₂₆₉
commutator, cosocle	C₂×C₄×C₂₆₉	C₂³
Frattini	C₂×C₄	C₂²×D₂₆₉
Fitting	#107×C₂₆₉	C₂
socle	C₂²×C₂₆₉	C₂×D₄
prime radical	C₂×D₁₀₇₆	C₂²
hypercenter	#107	D₂₆₉
hypocenter	C₂₆₉	#474
LCS	G ▷ C₂×C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

Complements

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

N	H ≅ G/N	type
1. #22575452	C₂	semidirect
3. #22575386	C₂	semidirect
4. #22575382	C₂	semidirect
7. #107×C₂₆₉	C₂	semidirect
22. D₁₀₇₆	Q₈	semidirect
47. C₂₆₉	#474	semidirect
52. #107	D₂₆₉	semidirect

Maximal subgroups

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

subgroup	index
#22575452	2
#22575415	2
#22575386	2
#22575382	2
#22575375	2
#22575373	2
#107×C₂₆₉	2
#474	269