Subgroups — #31771640

Order 17216.

122 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16864411	C₂	1	2	yes	maximal, complemented
2	C₄⋊#14524137	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²⋊#14524137	C₂	1	2	yes	maximal, complemented
4	C₄×#14524137	C₂	1	1	yes	maximal, characteristic, complemented
5	D₄×Dic₂₆₉	C₂	1	1	yes	maximal, characteristic
6	#16864411	—	2	4	no	—
7	C₂×#14524137	C₂²	1	1	yes	characteristic
8	C₂×#14524137	C₂²	1	2	yes	—
9	C₂×#14524137	C₂²	1	1	yes	characteristic
10	C₂²×Dic₂₆₉	C₂²	1	2	yes	—
11	C₄⋊Dic₂₆₉	C₄	1	1	yes	characteristic
12	C₂²⋊Dic₂₆₉	C₄	1	2	yes	—
13	C₄×Dic₂₆₉	C₂²	1	1	yes	characteristic
14	C₂×D₄×C₂₆₉	C₄	1	1	yes	characteristic
15	#14524137	—	2	2	no	—
16	#14524137	—	2	1	no	—
17	#14524137	D₄	1	2	yes	complemented
18	C₂×Dic₂₆₉	—	2	2	no	—
19	C₂×Dic₂₆₉	—	2	2	no	—
20	C₂×Dic₂₆₉	C₂×C₄	1	1	yes	characteristic
21	C₂×Dic₂₆₉	C₂×C₄	1	2	yes	—
22	C₂×Dic₂₆₉	C₂³	1	1	yes	characteristic
23	C₂³×C₂₆₉	C₂×C₄	1	2	yes	—
24	D₄×C₂₆₉	—	2	2	no	—
25	C₂×C₄×C₂₆₉	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₂₆₉	—	2	2	no	—
27	C₂²×C₂₆₉	—	2	2	no	—
28	C₂²×C₂₆₉	—	2	2	no	—
29	C₂²×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
30	C₄×C₂₆₉	M₁₆	1	2	yes	complemented
31	Dic₂₆₉	—	2	2	no	—
32	Dic₂₆₉	—	2	1	no	—
33	Dic₂₆₉	C₂×D₄	1	1	yes	characteristic
34	Dic₂₆₉	#41	1	1	yes	characteristic
35	C₂×C₂₆₉	—	4	2	no	—
36	C₂×C₂₆₉	C₄×D₄	1	1	yes	characteristic
37	C₂×C₂₆₉	C₂×M₁₆	1	1	yes	characteristic
38	C₂×C₂₆₉	#131	1	1	yes	characteristic
39	C₂₆₉	#436	1	1	yes	characteristic, complemented, Sylow
40	#436	—	269	1	no	maximal, Sylow
41	C₂×M₁₆	—	269	2	no	—
42	C₄×D₄	—	269	1	no	—
43	C₄⋊C₈	—	269	1	no	—
44	C₂²⋊C₈	—	269	2	no	—
45	C₄×C₈	—	269	1	no	—
46	C₂×D₄	#11761640	1	1	yes	characteristic
47	C₂²×C₄	—	269	2	no	—
48	M₁₆	—	538	4	no	—
49	C₂×C₈	—	269	1	no	—
50	C₂×C₈	—	269	1	no	—
51	C₂×C₈	—	269	2	no	—
52	C₄⋊C₄	—	269	1	no	—
53	C₂²⋊C₄	—	269	2	no	—
54	C₄²	—	269	1	no	—
55	C₂³	C₂×#11761640	1	2	yes	—
56	D₄	—	2	2	no	—
57	C₂×C₄	—	269	1	no	—
58	C₂×C₄	—	269	1	no	—
59	C₂×C₄	—	269	2	no	—
60	C₂×C₄	—	538	2	no	—
61	C₂×C₄	—	538	2	no	—
62	C₂×C₄	C₂×#11761640	1	1	yes	characteristic
63	C₈	—	269	2	no	—
64	C₈	—	538	2	no	—
65	C₈	—	538	1	no	—
66	C₂²	—	2	2	no	—
67	C₂²	—	2	2	no	—
68	C₂²	—	2	2	no	—
69	C₂²	C₂²×#11761640	1	1	yes	characteristic
70	C₄	—	269	1	no	—
71	C₄	—	269	1	no	—
72	C₄	—	538	2	no	—
73	C₄	—	538	1	no	—
74	C₄	#16864411	1	2	yes	complemented
75	C₂	—	4	2	no	—
76	C₂	D₄×#11761640	1	1	yes	characteristic
77	C₂	C₂×#16864411	1	1	yes	characteristic
78	C₂	#22575540	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×#11761640
commutator, socle	C₂²×C₂₆₉	C₂²×C₄
Fitting, prime radical	C₂×D₄×C₂₆₉	C₄
cosocle	C₂×Dic₂₆₉	C₂³
hypercenter	C₂×D₄	#11761640
hypocenter	C₂₆₉	#436
LCS	G ▷ C₂²×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂² ◁ C₂×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×#16864411	C₂	semidirect
2. C₄⋊#14524137	C₂	semidirect
3. C₂²⋊#14524137	C₂	semidirect
4. C₄×#14524137	C₂	semidirect
17. #14524137	D₄	semidirect
30. C₄×C₂₆₉	M₁₆	semidirect
39. C₂₆₉	#436	semidirect
74. C₄	#16864411	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16864411	2
C₄⋊#14524137	2
C₂²⋊#14524137	2
C₄×#14524137	2
D₄×Dic₂₆₉	2
#436	269