Subgroups — #28384316

Order 14272.

236 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20297291	C₂	1	4	yes	maximal, complemented
2	#20297285	C₂	1	4	yes	maximal, complemented
3	C₄×Dic₄₄₆	C₂	1	2	yes	maximal, complemented
4	D₄×Dic₂₂₃	C₂	1	4	yes	maximal, complemented
5	#127×C₂₂₃	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₄₄₆	C₂²	1	2	yes	—
7	C₂²×Dic₂₂₃	C₂²	1	4	yes	—
8	C₄⋊Dic₂₂₃	C₂²	1	2	yes	—
9	Dic₂₂₃⋊C₄	C₂²	1	4	yes	—
10	C₂²⋊Dic₂₂₃	C₂²	1	8	yes	—
11	C₂²⋊Dic₂₂₃	C₂²	1	4	yes	—
12	C₄×Dic₂₂₃	C₂²	1	4	yes	—
13	C₂×D₄×C₂₂₃	C₂²	1	4	yes	—
14	C₂×D₄×C₂₂₃	C₂²	1	2	yes	—
15	C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
16	Dic₄₄₆	—	2	4	no	—
17	C₂×Dic₂₂₃	—	4	4	no	—
18	C₂×Dic₂₂₃	C₂³	1	4	yes	—
19	C₂×Dic₂₂₃	C₂³	1	4	yes	—
20	C₂³×C₂₂₃	C₂³	1	4	yes	—
21	D₄×C₂₂₃	—	2	8	no	—
22	D₄×C₂₂₃	—	4	2	no	—
23	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
24	C₂×C₄×C₂₂₃	C₂³	1	2	yes	—
25	C₂²×C₂₂₃	—	2	8	no	—
26	C₂²×C₂₂₃	—	2	4	no	—
27	C₂²×C₂₂₃	C₂⁴	1	1	yes	characteristic
28	C₄×C₂₂₃	—	2	1	no	—
29	C₄×C₂₂₃	#41	1	4	yes	—
30	Dic₂₂₃	—	2	4	no	—
31	Dic₂₂₃	—	2	4	no	—
32	C₂×C₂₂₃	—	4	4	no	—
33	C₂×C₂₂₃	C₂×#41	1	2	yes	—
34	C₂×C₂₂₃	H(5,𝔽₂)	1	1	yes	characteristic
35	C₂₂₃	#559	1	1	yes	characteristic, complemented, Sylow
36	#559	—	223	1	no	maximal, Sylow
37	#127	D₂₂₃	1	1	yes	characteristic
38	#124	—	223	4	no	—
39	#123	—	223	4	no	—
40	C₄×Q₈	—	223	2	no	—
41	C₄×D₄	—	223	4	no	—
42	C₂×Q₈	—	223	2	no	—
43	C₂×D₄	C₂×D₂₂₃	1	4	yes	—
44	C₂×D₄	C₂×D₂₂₃	1	2	yes	—
45	C₂²×C₄	—	223	4	no	—
46	C₄⋊C₄	—	223	4	no	—
47	C₄⋊C₄	—	223	2	no	—
48	C₂²⋊C₄	—	223	8	no	—
49	C₂²⋊C₄	—	223	4	no	—
50	C₄²	—	223	4	no	—
51	C₄²	C₂×D₂₂₃	1	1	yes	characteristic
52	C₂³	C₂²×D₂₂₃	1	4	yes	—
53	Q₈	—	446	4	no	—
54	D₄	—	2	8	no	—
55	D₄	—	4	2	no	—
56	C₂×C₄	—	892	4	no	—
57	C₂×C₄	—	223	4	no	—
58	C₂×C₄	—	223	4	no	—
59	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
60	C₂×C₄	C₂²×D₂₂₃	1	2	yes	—
61	C₂²	—	2	8	no	—
62	C₂²	—	2	4	no	—
63	C₂²	C₂³×D₂₂₃	1	1	yes	characteristic
64	C₄	—	2	1	no	—
65	C₄	—	446	4	no	—
66	C₄	—	446	4	no	—
67	C₄	#16813169	1	4	yes	—
68	C₂	—	4	4	no	—
69	C₂	C₂×#16813169	1	2	yes	—
70	C₂	#20297349	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₂₃
commutator, socle, cosocle	C₂²×C₂₂₃	C₂⁴
Fitting, prime radical	#127×C₂₂₃	C₂
hypercenter	#127	D₂₂₃
hypocenter	C₂₂₃	#559
LCS	G ▷ C₂²×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂² ◁ #127

Complements

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

N	H ≅ G/N	type
1. #20297291	C₂	semidirect
2. #20297285	C₂	semidirect
3. C₄×Dic₄₄₆	C₂	semidirect
4. D₄×Dic₂₂₃	C₂	semidirect
35. C₂₂₃	#559	semidirect

Maximal subgroups

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

subgroup	index
#20297291	2
#20297285	2
C₄×Dic₄₄₆	2
D₄×Dic₂₂₃	2
#127×C₂₂₃	2
#559	223