Subgroups — #31771612

Order 17216.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16864412	C₂	1	1	yes	maximal, characteristic
2	C₂×D₂₆₉⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#22575520	C₂	1	2	yes	maximal, complemented
4	C₄⋊#14524137	C₂	1	2	yes	maximal, complemented
5	#22575411	C₂	1	1	yes	maximal, characteristic
6	#16864412	—	2	2	no	—
7	D₂₆₉⋊C₈	—	2	2	no	—
8	C₂×#14524137	C₂²	1	1	yes	characteristic
9	C₂×#14524137	C₂²	1	2	yes	—
10	C₂×#14524137	C₂²	1	1	yes	characteristic
11	C₂×C₄×D₂₆₉	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₂₆₉	C₄	1	1	yes	characteristic
13	#16864379	C₄	1	2	yes	—
14	C₄×Dic₂₆₉	C₂²	1	2	yes	—
15	C₄⋊C₄×C₂₆₉	C₄	1	1	yes	characteristic
16	#14524137	—	2	1	no	—
17	#14524137	—	2	1	no	—
18	#14524137	—	2	2	no	—
19	C₂²×D₂₆₉	C₂×C₄	1	1	yes	characteristic
20	C₂×Dic₂₆₉	C₂×C₄	1	2	yes	—
21	C₂×Dic₂₆₉	C₂³	1	1	yes	characteristic
22	C₄×D₂₆₉	D₄	1	2	yes	—
23	C₄×D₂₆₉	Q₈	1	2	yes	—
24	C₂×C₄×C₂₆₉	C₂×C₄	1	2	yes	—
25	C₂×C₄×C₂₆₉	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
27	C₂×D₂₆₉	—	2	1	no	—
28	C₂×D₂₆₉	—	2	1	no	—
29	C₂×D₂₆₉	C₄⋊C₄	1	2	yes	—
30	C₄×C₂₆₉	—	2	2	no	—
31	C₄×C₂₆₉	C₄⋊C₄	1	2	yes	—
32	Dic₂₆₉	—	2	2	no	—
33	Dic₂₆₉	C₂×D₄	1	1	yes	characteristic
34	Dic₂₆₉	C₂×Q₈	1	1	yes	characteristic
35	C₂×C₂₆₉	C₂×C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₂₆₉	#131	1	1	yes	characteristic
37	C₂×C₂₆₉	#131	1	1	yes	characteristic
38	D₂₆₉	—	2	2	no	—
39	C₂₆₉	#424	1	1	yes	characteristic, complemented, Sylow
40	#424	—	269	1	no	maximal, Sylow
41	C₂×M₁₆	—	269	1	no	—
42	C₂²×C₈	—	269	1	no	—
43	#117	—	269	1	no	—
44	C₄⋊C₈	—	269	2	no	—
45	C₄⋊C₈	—	269	2	no	—
46	C₂²×C₄	—	269	1	no	—
47	M₁₆	—	538	2	no	—
48	C₂×C₈	—	269	1	no	—
49	C₂×C₈	—	269	1	no	—
50	C₂×C₈	—	269	2	no	—
51	C₂×C₈	—	538	2	no	—
52	C₄⋊C₄	—	269	1	no	—
53	C₄⋊C₄	#11761640	1	1	yes	characteristic
54	C₂²⋊C₄	—	269	2	no	—
55	C₄²	—	269	2	no	—
56	C₂³	—	269	1	no	—
57	C₂×C₄	—	269	1	no	—
58	C₂×C₄	—	269	2	no	—
59	C₂×C₄	—	269	2	no	—
60	C₂×C₄	—	269	2	no	—
61	C₂×C₄	C₂×#11761640	1	2	yes	—
62	C₂×C₄	C₂×#11761640	1	1	yes	characteristic
63	C₈	—	538	1	no	—
64	C₈	—	538	1	no	—
65	C₈	—	538	2	no	—
66	C₂²	—	269	2	no	—
67	C₂²	—	538	1	no	—
68	C₂²	—	538	1	no	—
69	C₂²	C₂²×#11761640	1	1	yes	characteristic
70	C₄	—	2	2	no	—
71	C₄	—	269	1	no	—
72	C₄	—	269	1	no	—
73	C₄	—	538	2	no	—
74	C₄	C₄⋊#11761640	1	2	yes	—
75	C₂	—	538	2	no	—
76	C₂	C₂×C₄⋊#11761640	1	1	yes	characteristic
77	C₂	#22575540	1	1	yes	characteristic
78	C₂	#22575541	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×#11761640
commutator, socle	C₂²×C₂₆₉	C₂²×C₄
Fitting	C₄⋊C₄×C₂₆₉	C₄
prime radical	C₂²×D₂₆₉	C₂×C₄
cosocle	C₂×Dic₂₆₉	C₂³
hypercenter	C₄⋊C₄	#11761640
hypocenter	C₂₆₉	#424
LCS	G ▷ C₂²×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂² ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
3. #22575520	C₂	semidirect
4. C₄⋊#14524137	C₂	semidirect
39. C₂₆₉	#424	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16864412	2
C₂×D₂₆₉⋊C₈	2
#22575520	2
C₄⋊#14524137	2
#22575411	2
#424	269