Subgroups — #28383891

Order 14272.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊Dic₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
2	#20297237	C₂	1	2	yes	maximal, complemented
3	#20297225	C₂	1	2	yes	maximal, complemented
4	C₂²⋊(C₂₂₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
5	#121×C₂₂₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₂₃	—	2	1	no	—
7	C₂×C₂₂₃⋊C₈	C₂²	1	2	yes	—
8	C₄⋊Dic₂₂₃	—	2	1	no	—
9	C₄⋊Dic₂₂₃	C₂²	1	2	yes	—
10	C₂×D₄×C₂₂₃	—	2	1	no	—
11	C₂×D₄×C₂₂₃	C₂²	1	1	yes	characteristic
12	C₂²×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×C₂₂₃	C₂²	1	1	yes	characteristic
14	C₂²⋊C₄×C₂₂₃	—	2	1	no	—
15	C₂×Dic₂₂₃	—	2	2	no	—
16	C₂×Dic₂₂₃	—	4	2	no	—
17	C₂₂₃⋊C₈	—	2	2	no	—
18	C₂³×C₂₂₃	—	2	1	no	—
19	C₂³×C₂₂₃	D₄	1	1	yes	characteristic
20	D₄×C₂₂₃	—	2	2	no	—
21	D₄×C₂₂₃	—	4	2	no	—
22	C₂×C₄×C₂₂₃	—	2	2	no	—
23	C₂×C₄×C₂₂₃	—	2	1	no	—
24	C₂×C₄×C₂₂₃	D₄	1	1	yes	characteristic
25	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
26	C₂²×C₂₂₃	—	2	1	no	—
27	C₂²×C₂₂₃	—	2	1	no	—
28	C₂²×C₂₂₃	—	4	1	no	—
29	C₂²×C₂₂₃	—	4	1	no	—
30	C₂²×C₂₂₃	—	4	1	no	—
31	C₂²×C₂₂₃	D₈	1	2	yes	—
32	C₂²×C₂₂₃	C₂×D₄	1	1	yes	characteristic
33	C₄×C₂₂₃	—	2	1	no	—
34	C₄×C₂₂₃	—	4	1	no	—
35	C₄×C₂₂₃	#41	1	2	yes	—
36	Dic₂₂₃	—	4	2	no	—
37	C₂×C₂₂₃	—	2	2	no	—
38	C₂×C₂₂₃	—	8	1	no	—
39	C₂×C₂₂₃	#123	1	1	yes	characteristic
40	C₂×C₂₂₃	C₂×D₈	1	1	yes	characteristic
41	C₂×C₂₂₃	#137	1	1	yes	characteristic
42	C₂₂₃	#480	1	1	yes	characteristic, complemented, Sylow
43	#480	—	223	1	no	maximal, Sylow
44	#121	D₂₂₃	1	1	yes	characteristic
45	C₂×C₄⋊C₄	—	223	1	no	—
46	#107	—	223	2	no	—
47	#102	—	223	2	no	—
48	C₂²⋊C₈	—	223	1	no	—
49	C₂×D₄	—	2	1	no	—
50	C₂×D₄	C₂×D₂₂₃	1	1	yes	characteristic
51	C₂²×C₄	—	446	1	no	—
52	C₂²×C₄	C₂×D₂₂₃	1	1	yes	characteristic
53	C₂×C₈	—	223	2	no	—
54	C₄⋊C₄	—	446	1	no	—
55	C₄⋊C₄	—	223	2	no	—
56	C₄⋊C₄	C₂×D₂₂₃	1	1	yes	characteristic
57	C₂²⋊C₄	—	2	1	no	—
58	C₂³	—	2	1	no	—
59	C₂³	C₂²⋊D₂₂₃	1	1	yes	characteristic
60	D₄	—	2	2	no	—
61	D₄	—	4	2	no	—
62	C₂×C₄	—	2	2	no	—
63	C₂×C₄	—	2	1	no	—
64	C₂×C₄	—	892	2	no	—
65	C₂×C₄	—	446	2	no	—
66	C₂×C₄	C₂²⋊D₂₂₃	1	1	yes	characteristic
67	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
68	C₈	—	446	2	no	—
69	C₂²	—	2	1	no	—
70	C₂²	—	2	1	no	—
71	C₂²	—	4	1	no	—
72	C₂²	—	4	1	no	—
73	C₂²	—	4	1	no	—
74	C₂²	#16813155	1	2	yes	—
75	C₂²	C₂×C₂²⋊D₂₂₃	1	1	yes	characteristic
76	C₄	—	2	1	no	—
77	C₄	—	4	1	no	—
78	C₄	—	892	2	no	—
79	C₄	#16813169	1	2	yes	—
80	C₂	—	2	2	no	—
81	C₂	—	8	1	no	—
82	C₂	#20297285	1	1	yes	characteristic
83	C₂	C₂×#16813155	1	1	yes	characteristic
84	C₂	#20297334	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₂₂₃
commutator, cosocle	C₂×C₄×C₂₂₃	C₂³
Frattini	C₂×C₄	C₂²×D₂₂₃
Fitting, prime radical	#121×C₂₂₃	C₂
socle	C₂²×C₂₂₃	C₂×D₄
hypercenter	#121	D₂₂₃
hypocenter	C₂₂₃	#480
LCS	G ▷ C₂×C₄×C₂₂₃ ▷ C₂×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. C₂×C₄⋊Dic₂₂₃	C₂	semidirect
2. #20297237	C₂	semidirect
3. #20297225	C₂	semidirect
4. C₂²⋊(C₂₂₃⋊C₈)	C₂	semidirect
42. C₂₂₃	#480	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊Dic₂₂₃	2
#20297237	2
#20297225	2
C₂²⋊(C₂₂₃⋊C₈)	2
#121×C₂₂₃	2
#480	223