Subgroups — #31770569

Order 17216.

190 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16864386	C₂	1	2	yes	maximal, complemented
2	C₂²⋊#14524130	C₂	1	4	yes	maximal, complemented
3	C₂³×C₄×C₂₆₉	C₂	1	1	yes	maximal, characteristic
4	#16864386	—	2	4	no	—
5	C₂×#14524130	C₂²	1	4	yes	complemented
6	C₂⁴×C₂₆₉	C₄	1	1	yes	characteristic
7	C₂²×C₄×C₂₆₉	—	2	4	no	—
8	C₂²×C₄×C₂₆₉	C₄	1	1	yes	characteristic
9	C₂²×C₄×C₂₆₉	C₄	1	2	yes	—
10	C₂²×C₄×C₂₆₉	C₂²	1	2	yes	—
11	C₂²×C₄×C₂₆₉	C₂²	1	1	yes	characteristic
12	#14524130	—	2	4	no	—
13	C₂³×C₂₆₉	—	2	2	no	—
14	C₂³×C₂₆₉	—	2	4	no	—
15	C₂³×C₂₆₉	C₂×C₄	1	1	yes	characteristic
16	C₂³×C₂₆₉	C₂×C₄	1	2	yes	—
17	C₂×C₄×C₂₆₉	—	2	2	no	—
18	C₂×C₄×C₂₆₉	—	2	4	no	—
19	C₂×C₄×C₂₆₉	—	2	4	no	—
20	C₂×C₄×C₂₆₉	C₂×C₄	1	2	yes	—
21	C₂×C₄×C₂₆₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×C₂₆₉	D₄	1	4	yes	—
23	C₂×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
24	C₂²×C₂₆₉	—	2	8	no	—
25	C₂²×C₂₆₉	—	2	2	no	—
26	C₂²×C₂₆₉	—	2	2	no	—
27	C₂²×C₂₆₉	—	2	2	no	—
28	C₂²×C₂₆₉	C₂²⋊C₄	1	2	yes	—
29	C₂²×C₂₆₉	M₁₆	1	4	yes	complemented
30	C₂²×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
31	C₄×C₂₆₉	—	2	2	no	—
32	C₄×C₂₆₉	C₂²⋊C₄	1	2	yes	—
33	C₄×C₂₆₉	C₂×D₄	1	2	yes	—
34	C₂×C₂₆₉	—	2	2	no	—
35	C₂×C₂₆₉	—	2	4	no	—
36	C₂×C₂₆₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₂₆₉	C₂×M₁₆	1	2	yes	—
38	C₂₆₉	#407	1	1	yes	characteristic, complemented, Sylow
39	#407	—	269	1	no	maximal, Sylow
40	C₂³×C₄	D₂₆₉	1	1	yes	characteristic
41	C₂×M₁₆	—	269	2	no	—
42	C₂²⋊C₈	—	269	4	no	—
43	C₂⁴	Dic₂₆₉	1	1	yes	characteristic
44	C₂²×C₄	—	2	4	no	—
45	C₂²×C₄	Dic₂₆₉	1	1	yes	characteristic
46	C₂²×C₄	Dic₂₆₉	1	2	yes	—
47	C₂²×C₄	C₂×D₂₆₉	1	2	yes	—
48	C₂²×C₄	C₂×D₂₆₉	1	1	yes	characteristic
49	M₁₆	—	538	4	no	—
50	C₂×C₈	—	269	4	no	—
51	C₂³	—	2	2	no	—
52	C₂³	—	2	4	no	—
53	C₂³	C₂×Dic₂₆₉	1	1	yes	characteristic
54	C₂³	C₂×Dic₂₆₉	1	2	yes	—
55	C₂×C₄	—	2	2	no	—
56	C₂×C₄	—	2	4	no	—
57	C₂×C₄	—	2	4	no	—
58	C₂×C₄	C₂×Dic₂₆₉	1	2	yes	—
59	C₂×C₄	C₂×Dic₂₆₉	1	1	yes	characteristic
60	C₂×C₄	C₂²⋊D₂₆₉	1	4	yes	—
61	C₂×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
62	C₈	—	538	4	no	—
63	C₂²	—	2	8	no	—
64	C₂²	—	2	2	no	—
65	C₂²	—	2	2	no	—
66	C₂²	—	2	2	no	—
67	C₂²	C₂²⋊Dic₂₆₉	1	2	yes	—
68	C₂²	#16864386	1	4	yes	complemented
69	C₂²	C₂²×Dic₂₆₉	1	1	yes	characteristic
70	C₄	—	2	2	no	—
71	C₄	C₂²⋊Dic₂₆₉	1	2	yes	—
72	C₄	C₂×C₂²⋊D₂₆₉	1	2	yes	—
73	C₂	—	2	2	no	—
74	C₂	—	2	4	no	—
75	C₂	C₂×C₂²⋊Dic₂₆₉	1	1	yes	characteristic
76	C₂	C₂×#16864386	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₂₆₉
commutator, socle	C₂²×C₂₆₉	C₂²×C₄
Fitting	C₂³×C₄×C₂₆₉	C₂
prime radical	C₂⁴×C₂₆₉	C₄
cosocle	C₂×C₄×C₂₆₉	C₂³
hypercenter	C₂³×C₄	D₂₆₉
hypocenter	C₂₆₉	#407
LCS	G ▷ C₂²×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄

Complements

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

N	H ≅ G/N	type
1. C₂×#16864386	C₂	semidirect
2. C₂²⋊#14524130	C₂	semidirect
5. C₂×#14524130	C₂²	semidirect
29. C₂²×C₂₆₉	M₁₆	semidirect
38. C₂₆₉	#407	semidirect
68. C₂²	#16864386	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16864386	2
C₂²⋊#14524130	2
C₂³×C₄×C₂₆₉	2
#407	269