Subgroups — #46825851

Order 34432.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#16864405	C₂	1	1	yes	maximal, characteristic
2	C₄×#16864405	C₂	1	1	yes	maximal, characteristic
3	Q₈×#14524130	C₂	1	1	yes	maximal, characteristic
4	C₂×#16864405	—	2	1	no	—
5	C₂×#16864405	—	2	1	no	—
6	C₄⋊#14524130	—	2	1	no	—
7	C₄⋊#14524130	C₄	1	1	yes	characteristic
8	C₄×#14524130	—	2	1	no	—
9	C₄×#14524130	C₂²	1	1	yes	characteristic
10	C₄×Q₈×C₂₆₉	C₄	1	1	yes	characteristic
11	#16864405	—	2	2	no	—
12	#16864405	—	4	1	no	—
13	C₂×#14524130	—	2	1	no	—
14	C₂×#14524130	D₄	1	1	yes	characteristic
15	C₂×#14524130	D₄	1	1	yes	characteristic
16	C₂×Q₈×C₂₆₉	C₈	1	1	yes	characteristic
17	C₄⋊C₄×C₂₆₉	—	2	1	no	—
18	C₄⋊C₄×C₂₆₉	C₈	1	1	yes	characteristic
19	C₄²×C₂₆₉	—	2	1	no	—
20	C₄²×C₂₆₉	C₂×C₄	1	1	yes	characteristic
21	#14524130	—	2	1	no	—
22	#14524130	—	4	1	no	—
23	#14524130	SD₁₆	1	1	yes	characteristic
24	#14524130	Q₁₆	1	1	yes	characteristic
25	Q₈×C₂₆₉	—	2	1	no	—
26	Q₈×C₂₆₉	C₁₆	1	2	yes	complemented
27	C₂×C₄×C₂₆₉	—	2	1	no	—
28	C₂×C₄×C₂₆₉	—	2	1	no	—
29	C₂×C₄×C₂₆₉	C₂²⋊C₄	1	1	yes	characteristic
30	C₂×C₄×C₂₆₉	C₂×C₈	1	1	yes	characteristic
31	C₂×C₄×C₂₆₉	M₁₆	1	1	yes	characteristic
32	C₂²×C₂₆₉	C₂²⋊C₈	1	1	yes	characteristic
33	C₄×C₂₆₉	—	2	2	no	—
34	C₄×C₂₆₉	—	2	1	no	—
35	C₄×C₂₆₉	—	4	1	no	—
36	C₄×C₂₆₉	#103	1	1	yes	characteristic
37	C₄×C₂₆₉	#104	1	1	yes	characteristic
38	C₄×C₂₆₉	C₂×C₁₆	1	1	yes	characteristic
39	C₄×C₂₆₉	M₃₂	1	1	yes	characteristic
40	C₂×C₂₆₉	Q₈⋊C₈	1	1	yes	characteristic
41	C₂×C₂₆₉	C₂²⋊C₁₆	1	1	yes	characteristic
42	C₂×C₂₆₉	#350	1	1	yes	characteristic
43	C₂₆₉	Q₈⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
44	Q₈⋊C₁₆	—	269	1	no	maximal, Sylow
45	C₈×Q₈	—	269	1	no	—
46	C₄⋊C₁₆	—	269	1	no	—
47	C₄×C₁₆	—	269	1	no	—
48	C₄×Q₈	#11761640	1	1	yes	characteristic
49	C₂×C₁₆	—	538	1	no	—
50	C₂×C₁₆	—	538	1	no	—
51	C₄⋊C₈	—	269	1	no	—
52	C₄⋊C₈	—	538	1	no	—
53	C₄×C₈	—	269	1	no	—
54	C₄×C₈	—	538	1	no	—
55	C₂×Q₈	#14524137	1	1	yes	characteristic
56	C₂×C₈	—	269	1	no	—
57	C₂×C₈	—	269	1	no	—
58	C₂×C₈	—	538	1	no	—
59	C₄⋊C₄	—	2	1	no	—
60	C₄⋊C₄	#14524137	1	1	yes	characteristic
61	C₄²	—	2	1	no	—
62	C₄²	C₂×#11761640	1	1	yes	characteristic
63	C₁₆	—	538	2	no	—
64	C₁₆	—	1076	1	no	—
65	Q₈	—	2	1	no	—
66	Q₈	#16864405	1	2	yes	complemented
67	C₂×C₄	—	2	1	no	—
68	C₂×C₄	—	2	1	no	—
69	C₂×C₄	C₂²⋊#11761640	1	1	yes	characteristic
70	C₂×C₄	C₂×#14524137	1	1	yes	characteristic
71	C₂×C₄	#16864411	1	1	yes	characteristic
72	C₈	—	269	1	no	—
73	C₈	—	269	1	no	—
74	C₈	—	538	1	no	—
75	C₈	—	1076	1	no	—
76	C₂²	C₂²⋊#14524137	1	1	yes	characteristic
77	C₄	—	2	2	no	—
78	C₄	—	2	1	no	—
79	C₄	—	4	1	no	—
80	C₄	#22575516	1	1	yes	characteristic
81	C₄	#22575518	1	1	yes	characteristic
82	C₄	C₂×#16864405	1	1	yes	characteristic
83	C₄	#22575527	1	1	yes	characteristic
84	C₂	Q₈⋊#14524137	1	1	yes	characteristic
85	C₂	C₂²⋊#16864405	1	1	yes	characteristic
86	C₂	#31771503	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊#11761640
commutator	C₄×C₂₆₉	C₂×C₁₆
Frattini	C₄²	C₂×#11761640
Fitting	C₄×Q₈×C₂₆₉	C₄
socle, prime radical	C₂²×C₂₆₉	C₂²⋊C₈
cosocle	C₄×#14524130	C₂²
hypercenter	C₄×Q₈	#11761640
hypocenter	C₂₆₉	Q₈⋊C₁₆
LCS	G ▷ C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
26. Q₈×C₂₆₉	C₁₆	semidirect
43. C₂₆₉	Q₈⋊C₁₆	semidirect
66. Q₈	#16864405	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊#16864405	2
C₄×#16864405	2
Q₈×#14524130	2
Q₈⋊C₁₆	269