Subgroups — #28383314

Order 14272.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²⋊(C₂₂₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₂₃⋊#95	C₂	1	1	yes	maximal, characteristic, complemented
3	#121×C₂₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×Dic₂₂₃	—	2	1	no	—
5	C₂×C₂₂₃⋊C₈	—	2	1	no	—
6	C₂×D₄×C₂₂₃	—	2	1	no	—
7	C₂×D₄×C₂₂₃	C₄	1	1	yes	characteristic, complemented
8	C₂²×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
9	C₄⋊C₄×C₂₂₃	C₄	1	1	yes	characteristic, complemented
10	C₂²⋊C₄×C₂₂₃	—	2	1	no	—
11	C₂×Dic₂₂₃	—	2	2	no	—
12	C₂×Dic₂₂₃	—	4	1	no	—
13	C₂×Dic₂₂₃	—	4	1	no	—
14	C₂₂₃⋊C₈	—	4	1	no	—
15	C₂³×C₂₂₃	—	2	1	no	—
16	C₂³×C₂₂₃	D₄	1	1	yes	characteristic
17	D₄×C₂₂₃	—	4	1	no	—
18	D₄×C₂₂₃	—	4	1	no	—
19	D₄×C₂₂₃	—	4	1	no	—
20	C₂×C₄×C₂₂₃	—	2	1	no	—
21	C₂×C₄×C₂₂₃	—	4	1	no	—
22	C₂×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×C₂₂₃	D₄	1	1	yes	characteristic
24	C₂²×C₂₂₃	—	2	1	no	—
25	C₂²×C₂₂₃	—	2	1	no	—
26	C₂²×C₂₂₃	—	4	1	no	—
27	C₂²×C₂₂₃	—	4	1	no	—
28	C₂²×C₂₂₃	—	4	1	no	—
29	C₂²×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
30	C₂²×C₂₂₃	D₈	1	1	yes	characteristic
31	C₂²×C₂₂₃	SD₁₆	1	1	yes	characteristic
32	C₄×C₂₂₃	—	2	1	no	—
33	C₄×C₂₂₃	—	2	1	no	—
34	C₄×C₂₂₃	—	4	1	no	—
35	Dic₂₂₃	—	4	2	no	—
36	C₂×C₂₂₃	—	2	1	no	—
37	C₂×C₂₂₃	—	2	1	no	—
38	C₂×C₂₂₃	—	8	1	no	—
39	C₂×C₂₂₃	#99	1	1	yes	characteristic
40	C₂×C₂₂₃	#102	1	1	yes	characteristic
41	C₂×C₂₂₃	#104	1	1	yes	characteristic
42	C₂₂₃	#327	1	1	yes	characteristic, complemented, Sylow
43	#327	—	223	1	no	maximal, Sylow
44	#121	D₂₂₃	1	1	yes	characteristic
45	C₂²⋊C₈	—	223	1	no	—
46	#95	—	223	1	no	—
47	C₂×D₄	—	2	1	no	—
48	C₂×D₄	Dic₂₂₃	1	1	yes	characteristic, complemented
49	C₂²×C₄	—	446	1	no	—
50	C₂²×C₄	C₂×D₂₂₃	1	1	yes	characteristic
51	C₂×C₈	—	446	1	no	—
52	C₄⋊C₄	Dic₂₂₃	1	1	yes	characteristic, complemented
53	C₂²⋊C₄	—	2	1	no	—
54	C₂³	—	2	1	no	—
55	C₂³	C₂²⋊D₂₂₃	1	1	yes	characteristic
56	D₄	—	4	1	no	—
57	D₄	—	4	1	no	—
58	D₄	—	4	1	no	—
59	C₂×C₄	—	2	1	no	—
60	C₂×C₄	—	4	1	no	—
61	C₂×C₄	—	892	1	no	—
62	C₂×C₄	—	892	1	no	—
63	C₂×C₄	—	446	2	no	—
64	C₂×C₄	C₂×Dic₂₂₃	1	1	yes	characteristic
65	C₂×C₄	C₂²⋊D₂₂₃	1	1	yes	characteristic
66	C₈	—	892	1	no	—
67	C₂²	—	2	1	no	—
68	C₂²	—	2	1	no	—
69	C₂²	—	4	1	no	—
70	C₂²	—	4	1	no	—
71	C₂²	—	4	1	no	—
72	C₂²	C₂²⋊Dic₂₂₃	1	1	yes	characteristic
73	C₂²	#16813155	1	1	yes	characteristic
74	C₂²	#16813158	1	1	yes	characteristic
75	C₄	—	2	1	no	—
76	C₄	—	2	1	no	—
77	C₄	—	4	1	no	—
78	C₄	—	892	2	no	—
79	C₂	—	2	1	no	—
80	C₂	—	2	1	no	—
81	C₂	—	8	1	no	—
82	C₂	#20297219	1	1	yes	characteristic
83	C₂	#20297225	1	1	yes	characteristic
84	C₂	#20297231	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊Dic₂₂₃
commutator	C₂×C₄×C₂₂₃	C₂×C₄
Frattini	C₂²×C₄	C₂×D₂₂₃
Fitting, prime radical	#121×C₂₂₃	C₂
socle	C₂²×C₂₂₃	C₂²⋊C₄
cosocle	C₂²×C₄×C₂₂₃	C₂²
hypercenter	#121	D₂₂₃
hypocenter	C₂₂₃	#327
LCS	G ▷ C₂×C₄×C₂₂₃ ▷ C₂²×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #121

Complements

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

N	H ≅ G/N	type
1. C₂²⋊(C₂₂₃⋊C₈)	C₂	semidirect
2. C₂₂₃⋊#95	C₂	semidirect
7. C₂×D₄×C₂₂₃	C₄	semidirect
9. C₄⋊C₄×C₂₂₃	C₄	semidirect
42. C₂₂₃	#327	semidirect
48. C₂×D₄	Dic₂₂₃	semidirect
52. C₄⋊C₄	Dic₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²⋊(C₂₂₃⋊C₈)	2
C₂₂₃⋊#95	2
#121×C₂₂₃	2
#327	223