Subgroups — #28405541

Order 14464.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#14268650	C₂	1	1	yes	maximal, characteristic
2	C₄×#14268650	C₂	1	1	yes	maximal, characteristic
3	#20298722	C₂	1	1	yes	maximal, characteristic
4	C₂×#14268650	—	2	1	no	—
5	C₂×#14268650	—	2	1	no	—
6	C₄×Dic₂₂₆	C₄	1	1	yes	characteristic
7	#16813551	—	2	1	no	—
8	C₄×#11744751	C₂²	1	1	yes	characteristic
9	C₈×Dic₁₁₃	—	2	1	no	—
10	C₄⋊C₈×C₁₁₃	C₄	1	1	yes	characteristic
11	#14268650	—	2	2	no	—
12	#14268650	—	4	1	no	—
13	C₂×Dic₂₂₆	C₈	1	1	yes	characteristic
14	C₂×#11744751	D₄	1	1	yes	characteristic
15	C₂×#11744751	D₄	1	1	yes	characteristic
16	C₄⋊Dic₁₁₃	C₈	1	1	yes	characteristic
17	#14268626	—	2	1	no	—
18	C₄×Dic₁₁₃	—	2	1	no	—
19	C₂×C₈×C₁₁₃	—	2	1	no	—
20	C₄²×C₁₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×C₁₁₃	C₂²⋊C₄	1	1	yes	characteristic
22	C₂×C₄×C₁₁₃	C₂×C₈	1	1	yes	characteristic
23	C₂×C₄×C₁₁₃	M₁₆	1	1	yes	characteristic
24	C₂×Dic₁₁₃	—	2	1	no	—
25	C₂×Dic₁₁₃	—	2	1	no	—
26	Dic₂₂₆	—	2	1	no	—
27	Dic₂₂₆	C₁₆	1	2	yes	complemented
28	C₈×C₁₁₃	—	4	1	no	—
29	#11744751	—	2	1	no	—
30	#11744751	SD₁₆	1	1	yes	characteristic
31	#11744751	Q₁₆	1	1	yes	characteristic
32	C₂²×C₁₁₃	C₂²⋊C₈	1	1	yes	characteristic
33	C₄×C₁₁₃	—	2	1	no	—
34	C₄×C₁₁₃	#103	1	1	yes	characteristic
35	C₄×C₁₁₃	#104	1	1	yes	characteristic
36	C₄×C₁₁₃	C₂×C₁₆	1	1	yes	characteristic
37	C₄×C₁₁₃	M₃₂	1	1	yes	characteristic
38	Dic₁₁₃	—	2	2	no	—
39	Dic₁₁₃	—	4	1	no	—
40	C₂×C₁₁₃	Q₈⋊C₈	1	1	yes	characteristic
41	C₂×C₁₁₃	C₂²⋊C₁₆	1	1	yes	characteristic
42	C₂×C₁₁₃	#350	1	1	yes	characteristic
43	Q₈⋊C₁₆	—	113	1	no	maximal, Sylow
44	C₁₁₃	Q₈⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
45	C₈×Q₈	—	113	1	no	—
46	C₄⋊C₁₆	—	113	1	no	—
47	C₄×C₁₆	—	113	1	no	—
48	C₄×Q₈	—	113	1	no	—
49	C₂×C₁₆	—	226	1	no	—
50	C₂×C₁₆	—	226	1	no	—
51	C₄⋊C₈	—	226	1	no	—
52	C₄⋊C₈	#90673	1	1	yes	characteristic
53	C₄×C₈	—	113	1	no	—
54	C₄×C₈	—	226	1	no	—
55	C₂×Q₈	—	113	1	no	—
56	C₂×C₈	—	2	1	no	—
57	C₂×C₈	—	113	1	no	—
58	C₂×C₈	—	113	1	no	—
59	C₄⋊C₄	—	113	1	no	—
60	C₄⋊C₄	—	226	1	no	—
61	C₄²	—	226	1	no	—
62	C₄²	C₂×#90673	1	1	yes	characteristic
63	C₁₆	—	452	1	no	—
64	C₁₆	—	226	2	no	—
65	Q₈	—	113	2	no	—
66	Q₈	—	226	1	no	—
67	C₂×C₄	—	226	1	no	—
68	C₂×C₄	—	226	1	no	—
69	C₂×C₄	C₂²⋊#90673	1	1	yes	characteristic
70	C₂×C₄	#14268655	1	1	yes	characteristic
71	C₂×C₄	#14268657	1	1	yes	characteristic
72	C₈	—	4	1	no	—
73	C₈	—	113	1	no	—
74	C₈	—	113	1	no	—
75	C₈	—	226	1	no	—
76	C₂²	#16813679	1	1	yes	characteristic
77	C₄	—	2	1	no	—
78	C₄	—	452	1	no	—
79	C₄	—	226	2	no	—
80	C₄	#16813685	1	1	yes	characteristic
81	C₄	#16813686	1	1	yes	characteristic
82	C₄	#16813695	1	1	yes	characteristic
83	C₄	#16813697	1	1	yes	characteristic
84	C₂	#20299448	1	1	yes	characteristic
85	C₂	#20299493	1	1	yes	characteristic
86	C₂	#20299497	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊#90673
commutator	C₄×C₁₁₃	C₂×C₁₆
Frattini	C₄²	C₂×#90673
Fitting	C₄⋊C₈×C₁₁₃	C₄
socle, prime radical	C₂²×C₁₁₃	C₂²⋊C₈
cosocle	C₄×#11744751	C₂²
hypercenter	C₄⋊C₈	#90673
hypocenter	C₁₁₃	Q₈⋊C₁₆
LCS	G ▷ C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
27. Dic₂₂₆	C₁₆	semidirect
44. C₁₁₃	Q₈⋊C₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊#14268650	2
C₄×#14268650	2
#20298722	2
Q₈⋊C₁₆	113