Subgroups — #31771657

Order 17216.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22575520	C₂	1	1	yes	maximal, characteristic
2	#22575520	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#14524137	C₂	1	1	yes	maximal, characteristic
4	#22575508	C₂	1	1	yes	maximal, characteristic
5	#22575508	C₂	1	1	yes	maximal, characteristic
6	C₄×#14524137	C₂	1	1	yes	maximal, characteristic
7	#22575418	C₂	1	1	yes	maximal, characteristic
8	C₂×#14524137	C₂²	1	1	yes	characteristic
9	C₂×#14524137	C₂²	1	1	yes	characteristic
10	C₂×#14524137	C₂²	1	1	yes	characteristic
11	C₂×#14524137	C₂²	1	1	yes	characteristic
12	C₂×Dic₅₃₈	C₄	1	1	yes	characteristic
13	#16864381	C₄	1	1	yes	characteristic
14	#16864381	C₄	1	1	yes	characteristic
15	C₄×Dic₂₆₉	C₂²	1	1	yes	characteristic
16	C₄×Dic₂₆₉	C₂²	1	1	yes	characteristic
17	C₄×Dic₂₆₉	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×C₂₆₉	C₄	1	1	yes	characteristic
19	#14524137	—	2	1	no	—
20	#14524137	—	2	1	no	—
21	#14524137	—	2	1	no	—
22	#14524137	Q₈	1	2	yes	complemented
23	Dic₅₃₈	—	2	2	no	—
24	C₂×Dic₂₆₉	C₂×C₄	1	1	yes	characteristic
25	C₂×Dic₂₆₉	C₂×C₄	1	1	yes	characteristic
26	C₂×Dic₂₆₉	C₂×C₄	1	1	yes	characteristic
27	C₂×Dic₂₆₉	C₂³	1	1	yes	characteristic
28	C₂×C₄×C₂₆₉	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×C₂₆₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×C₂₆₉	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
32	C₄×C₂₆₉	—	2	1	no	—
33	C₄×C₂₆₉	—	2	1	no	—
34	C₄×C₂₆₉	—	2	1	no	—
35	Dic₂₆₉	—	2	1	no	—
36	Dic₂₆₉	—	2	1	no	—
37	Dic₂₆₉	M₁₆	1	2	yes	—
38	Dic₂₆₉	C₂×Q₈	1	1	yes	characteristic
39	Dic₂₆₉	#41	1	1	yes	characteristic
40	C₂×C₂₆₉	C₄×Q₈	1	1	yes	characteristic
41	C₂×C₂₆₉	C₂×M₁₆	1	1	yes	characteristic
42	C₂×C₂₆₉	#131	1	1	yes	characteristic
43	C₂₆₉	#446	1	1	yes	characteristic, complemented, Sylow
44	#446	—	269	1	no	maximal, Sylow
45	C₄×Q₈	—	269	1	no	—
46	C₄⋊C₈	—	269	1	no	—
47	C₄⋊C₈	—	269	1	no	—
48	C₄⋊C₈	—	269	1	no	—
49	#97	—	269	1	no	—
50	#97	—	269	1	no	—
51	C₄×C₈	—	269	1	no	—
52	C₂×Q₈	—	269	1	no	—
53	C₂×C₈	—	269	1	no	—
54	C₂×C₈	—	269	1	no	—
55	C₂×C₈	—	269	1	no	—
56	C₂×C₈	—	269	1	no	—
57	C₄⋊C₄	—	269	1	no	—
58	C₄⋊C₄	—	269	1	no	—
59	C₄⋊C₄	#11761640	1	1	yes	characteristic
60	C₄²	—	269	1	no	—
61	C₄²	—	269	1	no	—
62	C₄²	—	269	1	no	—
63	Q₈	—	538	2	no	—
64	C₂×C₄	—	269	1	no	—
65	C₂×C₄	—	269	1	no	—
66	C₂×C₄	—	269	1	no	—
67	C₂×C₄	—	269	1	no	—
68	C₂×C₄	C₂×#11761640	1	1	yes	characteristic
69	C₂×C₄	C₂×#11761640	1	1	yes	characteristic
70	C₂×C₄	C₂×#11761640	1	1	yes	characteristic
71	C₈	—	269	2	no	—
72	C₈	—	538	1	no	—
73	C₈	—	538	1	no	—
74	C₈	—	538	1	no	—
75	C₂²	C₂²×#11761640	1	1	yes	characteristic
76	C₄	—	2	1	no	—
77	C₄	—	2	1	no	—
78	C₄	—	2	1	no	—
79	C₄	—	269	1	no	—
80	C₄	—	269	1	no	—
81	C₄	—	269	2	no	—
82	C₄	—	538	1	no	—
83	C₄	—	538	1	no	—
84	C₂	Q₈×#11761640	1	1	yes	characteristic
85	C₂	D₂₆₉⋊M₁₆	1	1	yes	characteristic
86	C₂	#22575540	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×#11761640
commutator, socle, prime radical	C₂²×C₂₆₉	C₂²×C₄
Fitting	C₄⋊C₄×C₂₆₉	C₄
cosocle	C₂×Dic₂₆₉	C₂³
hypercenter	C₄⋊C₄	#11761640
hypocenter	C₂₆₉	#446
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
22. #14524137	Q₈	semidirect
43. C₂₆₉	#446	semidirect

Maximal subgroups

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

subgroup	index
#22575520	2
C₄⋊#14524137	2
#22575508	2
C₄×#14524137	2
#22575418	2
#446	269