Subgroups — #28407829

Order 14464.

360 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20299720	C₂	1	8	yes	maximal, complemented
2	#20299717	C₂	1	6	yes	maximal, complemented
3	C₂×#16813710	C₂	1	1	yes	maximal, characteristic
4	#16813719	C₂²	1	12	yes	complemented
5	#16813718	C₂²	1	4	yes	—
6	#16813717	C₂²	1	12	yes	complemented
7	#16813710	C₄	1	4	yes	—
8	#16813710	C₂²	1	4	yes	—
9	C₂×#14268657	C₄	1	3	yes	—
10	C₂×#14268655	C₂²	1	3	yes	—
11	C₂×#14268648	C₄	1	1	yes	characteristic
12	#14268659	C₂³	1	8	yes	—
13	#14268657	C₂×C₄	1	12	yes	—
14	#14268655	C₂×C₄	1	6	yes	—
15	#14268655	C₂³	1	6	yes	—
16	C₂×#11744753	C₂×C₄	1	3	yes	—
17	C₂×#11744753	C₂³	1	1	yes	characteristic
18	#14268648	C₈	1	4	yes	—
19	#14268648	C₂×C₄	1	4	yes	—
20	C₂×D₄₅₂	C₈	1	3	yes	—
21	C₂×C₄×D₁₁₃	C₂×C₄	1	3	yes	—
22	C₂×Q₈×C₁₁₃	C₈	1	1	yes	characteristic
23	C₂²×D₁₁₃	C₂×C₈	1	3	yes	—
24	Q₈×C₁₁₃	C₂×C₈	1	4	yes	—
25	C₂×C₄×C₁₁₃	C₂×C₈	1	3	yes	—
26	C₂×Dic₁₁₃	C₂²×C₄	1	1	yes	characteristic
27	D₄₅₂	C₂×C₈	1	12	yes	—
28	C₄×D₁₁₃	C₂×C₈	1	6	yes	—
29	C₄×D₁₁₃	C₂²×C₄	1	6	yes	—
30	#11744753	C₂²×C₄	1	6	yes	—
31	#11744753	C₂²×C₄	1	1	yes	characteristic
32	#11744753	C₂⁴	1	1	yes	characteristic
33	C₂²×C₁₁₃	C₂²×C₈	1	1	yes	characteristic
34	C₂×D₁₁₃	—	4	3	no	—
35	C₂×D₁₁₃	C₂²×C₈	1	6	yes	—
36	C₄×C₁₁₃	C₂²×C₈	1	6	yes	—
37	Dic₁₁₃	C₂²×C₈	1	1	yes	characteristic
38	Dic₁₁₃	C₂³×C₄	1	1	yes	characteristic
39	C₂×C₁₁₃	—	2	1	no	—
40	C₂×C₁₁₃	C₂³×C₈	1	1	yes	characteristic
41	D₁₁₃	—	2	6	no	—
42	#3407	—	113	1	no	maximal, Sylow
43	C₁₁₃	#3407	1	1	yes	characteristic, complemented, Sylow
44	C₂×#131	—	113	1	no	—
45	#504	—	113	8	no	—
46	C₂×M₃₂	—	113	6	no	—
47	C₂×#41	—	113	1	no	—
48	#131	—	113	4	no	—
49	#131	—	113	4	no	—
50	C₂×M₁₆	—	113	3	no	—
51	C₂²×C₈	—	113	3	no	—
52	M₃₂	—	113	4	no	—
53	M₃₂	—	113	12	no	—
54	C₂×C₁₆	—	113	12	no	—
55	#41	—	113	4	no	—
56	#41	—	113	4	no	—
57	C₂×Q₈	#11744762	1	1	yes	characteristic
58	C₂×D₄	—	113	3	no	—
59	C₂²×C₄	—	113	3	no	—
60	M₁₆	—	113	12	no	—
61	C₂×C₈	—	113	6	no	—
62	C₂×C₈	—	113	1	no	—
63	C₂×C₈	—	113	3	no	—
64	C₂×C₈	—	113	6	no	—
65	C₁₆	—	113	8	no	—
66	C₂³	—	113	3	no	—
67	Q₈	C₂×#11744762	1	4	yes	—
68	D₄	—	113	12	no	—
69	C₂×C₄	—	113	1	no	—
70	C₂×C₄	—	113	6	no	—
71	C₂×C₄	—	113	6	no	—
72	C₂×C₄	C₂×#11744762	1	3	yes	—
73	C₈	—	113	1	no	—
74	C₈	—	113	1	no	—
75	C₈	—	113	6	no	—
76	C₂²	—	113	6	no	—
77	C₂²	—	452	3	no	—
78	C₂²	C₂²×#11744762	1	1	yes	characteristic
79	C₄	—	113	1	no	—
80	C₄	—	113	1	no	—
81	C₄	C₂²×#11744762	1	6	yes	—
82	C₂	—	2	1	no	—
83	C₂	—	226	6	no	—
84	C₂	C₂³×#11744762	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂³×#11744762
commutator, socle	C₂×C₁₁₃	C₂³×C₈
Fitting	C₂×Q₈×C₁₁₃	C₈
prime radical	C₂×#14268648	C₄
cosocle	#11744753	C₂⁴
hypercenter	C₂×Q₈	#11744762
hypocenter	C₁₁₃	#3407
LCS	G ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂ ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
1. #20299720	C₂	semidirect
2. #20299717	C₂	semidirect
4. #16813719	C₂²	semidirect
6. #16813717	C₂²	semidirect
43. C₁₁₃	#3407	semidirect

Maximal subgroups

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

subgroup	index
#20299720	2
#20299717	2
C₂×#16813710	2
#3407	113