Subgroups — #28383880

Order 14272.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20297296	C₂	1	1	yes	maximal, characteristic, complemented
2	#20297237	C₂	1	1	yes	maximal, characteristic, complemented
3	#20297235	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄⋊(C₂₂₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
5	#20297225	C₂	1	1	yes	maximal, characteristic
6	#20297225	C₂	1	1	yes	maximal, characteristic
7	C₄×D₄×C₂₂₃	C₂	1	1	yes	maximal, characteristic
8	C₂×C₂₂₃⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₂₂₃⋊C₈	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₂₂₃	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₂₂₃	C₂²	1	1	yes	characteristic
12	Dic₂₂₃⋊C₄	—	2	1	no	—
13	Dic₂₂₃⋊C₄	—	2	1	no	—
14	C₂×D₄×C₂₂₃	C₂²	1	1	yes	characteristic
15	C₂²×C₄×C₂₂₃	—	2	1	no	—
16	C₄⋊C₄×C₂₂₃	C₂²	1	1	yes	characteristic
17	C₂²⋊C₄×C₂₂₃	—	2	1	no	—
18	C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
19	C₂×Dic₂₂₃	—	2	1	no	—
20	C₂×Dic₂₂₃	—	2	1	no	—
21	C₂₂₃⋊C₈	—	2	1	no	—
22	C₂₂₃⋊C₈	—	2	1	no	—
23	C₂³×C₂₂₃	—	2	1	no	—
24	D₄×C₂₂₃	—	2	1	no	—
25	D₄×C₂₂₃	Q₈	1	2	yes	—
26	C₂×C₄×C₂₂₃	—	2	1	no	—
27	C₂×C₄×C₂₂₃	—	4	2	no	—
28	C₂×C₄×C₂₂₃	D₄	1	1	yes	characteristic
29	C₂×C₄×C₂₂₃	D₄	1	1	yes	characteristic
30	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
31	C₂²×C₂₂₃	—	2	2	no	—
32	C₂²×C₂₂₃	—	4	1	no	—
33	C₂²×C₂₂₃	—	4	1	no	—
34	C₂²×C₂₂₃	C₂×D₄	1	1	yes	characteristic
35	C₄×C₂₂₃	—	2	1	no	—
36	C₄×C₂₂₃	—	2	1	no	—
37	C₄×C₂₂₃	—	4	1	no	—
38	C₄×C₂₂₃	C₂×Q₈	1	1	yes	characteristic
39	C₄×C₂₂₃	#41	1	1	yes	characteristic
40	Dic₂₂₃	—	4	1	no	—
41	Dic₂₂₃	—	4	1	no	—
42	C₂×C₂₂₃	—	4	2	no	—
43	C₂×C₂₂₃	C₂²⋊Q₈	1	1	yes	characteristic
44	C₂×C₂₂₃	#135	1	1	yes	characteristic
45	C₂×C₂₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
46	C₂₂₃	#478	1	1	yes	characteristic, complemented, Sylow
47	#478	—	223	1	no	maximal, Sylow
48	#125	—	223	1	no	—
49	C₄×D₄	D₂₂₃	1	1	yes	characteristic
50	#107	—	223	1	no	—
51	#106	—	223	1	no	—
52	C₄⋊C₈	—	223	1	no	—
53	#102	—	223	1	no	—
54	#102	—	223	1	no	—
55	C₂×D₄	C₂×D₂₂₃	1	1	yes	characteristic
56	C₂²×C₄	—	2	1	no	—
57	C₂×C₈	—	223	1	no	—
58	C₂×C₈	—	223	1	no	—
59	C₄⋊C₄	—	446	1	no	—
60	C₄⋊C₄	—	446	1	no	—
61	C₄⋊C₄	—	223	1	no	—
62	C₄⋊C₄	—	223	1	no	—
63	C₄⋊C₄	C₂×D₂₂₃	1	1	yes	characteristic
64	C₂²⋊C₄	—	2	1	no	—
65	C₄²	C₂×D₂₂₃	1	1	yes	characteristic
66	C₂³	—	2	1	no	—
67	D₄	—	2	1	no	—
68	D₄	Dic₄₄₆	1	2	yes	—
69	C₂×C₄	—	2	1	no	—
70	C₂×C₄	—	4	2	no	—
71	C₂×C₄	—	446	1	no	—
72	C₂×C₄	—	446	1	no	—
73	C₂×C₄	C₂²⋊D₂₂₃	1	1	yes	characteristic
74	C₂×C₄	C₂²⋊D₂₂₃	1	1	yes	characteristic
75	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
76	C₈	—	446	1	no	—
77	C₈	—	446	1	no	—
78	C₂²	—	2	2	no	—
79	C₂²	—	4	1	no	—
80	C₂²	—	4	1	no	—
81	C₂²	C₂×C₂²⋊D₂₂₃	1	1	yes	characteristic
82	C₄	—	2	1	no	—
83	C₄	—	2	1	no	—
84	C₄	—	4	1	no	—
85	C₄	—	892	1	no	—
86	C₄	—	892	1	no	—
87	C₄	C₂×Dic₄₄₆	1	1	yes	characteristic
88	C₄	#16813170	1	1	yes	characteristic
89	C₂	—	4	2	no	—
90	C₂	#20297283	1	1	yes	characteristic
91	C₂	#20297324	1	1	yes	characteristic
92	C₂	#20297330	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₂₂₃
commutator, cosocle	C₂×C₄×C₂₂₃	C₂³
Frattini	C₂×C₄	C₂²×D₂₂₃
Fitting	C₄×D₄×C₂₂₃	C₂
socle	C₂²×C₂₂₃	C₂×D₄
prime radical	C₂×D₄×C₂₂₃	C₂²
hypercenter	C₄×D₄	D₂₂₃
hypocenter	C₂₂₃	#478
LCS	G ▷ C₂×C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
1. #20297296	C₂	semidirect
2. #20297237	C₂	semidirect
3. #20297235	C₂	semidirect
4. C₄⋊(C₂₂₃⋊C₈)	C₂	semidirect
46. C₂₂₃	#478	semidirect

Maximal subgroups

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

subgroup	index
#20297296	2
#20297237	2
#20297235	2
C₄⋊(C₂₂₃⋊C₈)	2
#20297225	2
C₄×D₄×C₂₂₃	2
#478	223