Subgroups — #28383662

Order 14272.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₇₈₄	C₂	1	1	yes	maximal, characteristic
2	#20297266	C₂	1	2	yes	maximal, complemented
3	#20297224	C₂	1	2	yes	maximal, complemented
4	C₈×Dic₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
5	#107×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₈₉₂	C₂²	1	2	yes	—
7	C₂×C₄×D₂₂₃	—	2	2	no	—
8	D₁₇₈₄	C₄	1	4	yes	complemented
9	C₂×C₂₂₃⋊C₈	C₂²	1	1	yes	characteristic
10	(C₂×D₂₂₃)⋊C₄	—	2	2	no	—
11	C₄×Dic₂₂₃	C₂²	1	1	yes	characteristic
12	C₂×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×C₂₂₃	C₂²	1	2	yes	—
14	C₂²×D₂₂₃	—	2	2	no	—
15	D₈₉₂	—	2	2	no	—
16	D₈₉₂	C₂×C₄	1	4	yes	complemented
17	C₂×Dic₂₂₃	D₄	1	1	yes	characteristic
18	C₂×Dic₂₂₃	D₄	1	1	yes	characteristic
19	C₄×D₂₂₃	—	4	4	no	—
20	C₂₂₃⋊C₈	—	2	1	no	—
21	C₂×C₄×C₂₂₃	—	2	2	no	—
22	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
23	C₈×C₂₂₃	C₂×C₄	1	2	yes	complemented
24	C₂²×C₂₂₃	C₂×D₄	1	1	yes	characteristic
25	C₂×D₂₂₃	—	2	4	no	—
26	C₂×D₂₂₃	—	4	2	no	—
27	C₂×D₂₂₃	—	4	2	no	—
28	C₄×C₂₂₃	—	4	2	no	—
29	C₄×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
30	C₄×C₂₂₃	#41	1	1	yes	characteristic
31	Dic₂₂₃	—	2	1	no	—
32	Dic₂₂₃	D₈	1	2	yes	complemented
33	C₂×C₂₂₃	C₄×D₄	1	1	yes	characteristic
34	C₂×C₂₂₃	C₂×D₈	1	1	yes	characteristic
35	C₂×C₂₂₃	#135	1	1	yes	characteristic
36	D₂₂₃	—	4	4	no	—
37	C₂₂₃	C₄×D₈	1	1	yes	characteristic, complemented, Sylow
38	C₄×D₈	—	223	1	no	maximal, Sylow
39	C₂×D₈	—	223	1	no	—
40	C₄×D₄	—	223	2	no	—
41	#107	D₂₂₃	1	1	yes	characteristic, complemented
42	#102	—	223	2	no	—
43	C₄×C₈	—	223	1	no	—
44	C₂×D₄	—	223	2	no	—
45	C₂²×C₄	—	446	2	no	—
46	D₈	—	223	4	no	—
47	C₂×C₈	—	223	1	no	—
48	C₂×C₈	C₂×D₂₂₃	1	1	yes	characteristic
49	C₄⋊C₄	C₂×D₂₂₃	1	2	yes	—
50	C₂²⋊C₄	—	446	2	no	—
51	C₄²	—	223	1	no	—
52	C₂³	—	446	2	no	—
53	D₄	—	446	2	no	—
54	D₄	—	223	4	no	—
55	C₂×C₄	—	2	2	no	—
56	C₂×C₄	—	892	4	no	—
57	C₂×C₄	—	223	1	no	—
58	C₂×C₄	—	223	1	no	—
59	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
60	C₈	—	446	1	no	—
61	C₈	C₄×D₂₂₃	1	2	yes	complemented
62	C₂²	—	892	2	no	—
63	C₂²	—	892	2	no	—
64	C₂²	—	446	4	no	—
65	C₂²	D₄×D₂₂₃	1	1	yes	characteristic
66	C₄	—	4	2	no	—
67	C₄	—	446	1	no	—
68	C₄	—	223	2	no	—
69	C₄	C₂×C₄×D₂₂₃	1	1	yes	characteristic
70	C₄	#16813171	1	1	yes	characteristic
71	C₂	—	892	4	no	—
72	C₂	#20297266	1	1	yes	characteristic
73	C₂	D₈×D₂₂₃	1	1	yes	characteristic
74	C₂	#20297327	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₂₂₃
commutator	C₄×C₂₂₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₂₂₃
Fitting	#107×C₂₂₃	C₂
socle	C₂²×C₂₂₃	C₂×D₄
prime radical	C₂×D₁₇₈₄	C₂
cosocle	C₂×C₄×C₂₂₃	C₂³
hypercenter	#107	D₂₂₃
hypocenter	C₂₂₃	C₄×D₈
LCS	G ▷ C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

Complements

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

N	H ≅ G/N	type
2. #20297266	C₂	semidirect
3. #20297224	C₂	semidirect
4. C₈×Dic₂₂₃	C₂	semidirect
5. #107×C₂₂₃	C₂	semidirect
8. D₁₇₈₄	C₄	semidirect
16. D₈₉₂	C₂×C₄	semidirect
23. C₈×C₂₂₃	C₂×C₄	semidirect
32. Dic₂₂₃	D₈	semidirect
37. C₂₂₃	C₄×D₈	semidirect
41. #107	D₂₂₃	semidirect
61. C₈	C₄×D₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₁₇₈₄	2
#20297266	2
#20297224	2
C₈×Dic₂₂₃	2
#107×C₂₂₃	2
C₄×D₈	223