Subgroups — #28384330

Order 14272.

196 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20297302	C₂	1	2	yes	maximal, complemented
2	#20297297	C₂	1	4	yes	maximal, complemented
3	#20297296	C₂	1	2	yes	maximal, complemented
4	#20297285	C₂	1	2	yes	maximal
5	#20297281	C₂	1	4	yes	maximal, complemented
6	#124×C₂₂₃	C₂	1	1	yes	maximal, characteristic
7	C₂×Dic₄₄₆	C₂²	1	2	yes	—
8	C₂²×Dic₂₂₃	C₂²	1	2	yes	—
9	C₄⋊Dic₂₂₃	C₂²	1	2	yes	—
10	C₄⋊Dic₂₂₃	C₂²	1	2	yes	—
11	Dic₂₂₃⋊C₄	C₂²	1	4	yes	—
12	Dic₂₂₃⋊C₄	C₂²	1	4	yes	—
13	Dic₂₂₃⋊C₄	C₂²	1	4	yes	—
14	C₂²⋊Dic₂₂₃	C₂²	1	4	yes	—
15	C₂²⋊Dic₂₂₃	C₂²	1	2	yes	—
16	C₄×Dic₂₂₃	C₂²	1	2	yes	—
17	C₂×Q₈×C₂₂₃	C₂²	1	1	yes	characteristic
18	C₂×D₄×C₂₂₃	C₂²	1	1	yes	characteristic
19	C₂²⋊C₄×C₂₂₃	C₂²	1	4	yes	—
20	C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
21	Dic₄₄₆	—	4	2	no	—
22	C₂×Dic₂₂₃	—	4	2	no	—
23	C₂×Dic₂₂₃	C₂³	1	4	yes	—
24	C₂×Dic₂₂₃	C₂³	1	4	yes	—
25	C₂³×C₂₂₃	C₂³	1	2	yes	—
26	Q₈×C₂₂₃	—	4	1	no	—
27	D₄×C₂₂₃	—	4	1	no	—
28	C₂×C₄×C₂₂₃	C₂³	1	2	yes	—
29	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
30	C₂×C₄×C₂₂₃	C₂³	1	2	yes	—
31	C₂²×C₂₂₃	—	2	2	no	—
32	C₂²×C₂₂₃	—	2	4	no	—
33	C₂²×C₂₂₃	C₂⁴	1	1	yes	characteristic
34	C₄×C₂₂₃	—	2	1	no	—
35	C₄×C₂₂₃	—	2	2	no	—
36	C₄×C₂₂₃	—	2	2	no	—
37	Dic₂₂₃	—	2	4	no	—
38	Dic₂₂₃	—	2	4	no	—
39	C₂×C₂₂₃	—	4	2	no	—
40	C₂×C₂₂₃	H(5,𝔽₂)	1	1	yes	characteristic
41	C₂×C₂₂₃	2₋¹⁺⁴	1	2	yes	—
42	C₂₂₃	#563	1	1	yes	characteristic, complemented, Sylow
43	#563	—	223	1	no	maximal, Sylow
44	C₄⋊Q₈	—	223	2	no	—
45	#126	—	223	4	no	—
46	#125	—	223	2	no	—
47	#124	D₂₂₃	1	1	yes	characteristic
48	#123	—	223	2	no	—
49	C₂²⋊Q₈	—	223	4	no	—
50	C₂×Q₈	—	223	2	no	—
51	C₂×Q₈	C₂×D₂₂₃	1	1	yes	characteristic
52	C₂×D₄	C₂×D₂₂₃	1	1	yes	characteristic
53	C₂²×C₄	—	223	2	no	—
54	C₄⋊C₄	—	223	4	no	—
55	C₄⋊C₄	—	223	4	no	—
56	C₄⋊C₄	—	223	4	no	—
57	C₄⋊C₄	—	223	2	no	—
58	C₄⋊C₄	—	223	2	no	—
59	C₂²⋊C₄	—	223	4	no	—
60	C₂²⋊C₄	—	223	2	no	—
61	C₂²⋊C₄	C₂×D₂₂₃	1	4	yes	—
62	C₄²	—	223	2	no	—
63	C₄²	C₂×D₂₂₃	1	1	yes	characteristic
64	C₂³	C₂²×D₂₂₃	1	2	yes	—
65	Q₈	—	4	1	no	—
66	Q₈	—	892	2	no	—
67	D₄	—	4	1	no	—
68	C₂×C₄	—	892	2	no	—
69	C₂×C₄	—	223	4	no	—
70	C₂×C₄	—	223	4	no	—
71	C₂×C₄	C₂²×D₂₂₃	1	2	yes	—
72	C₂×C₄	C₂²×D₂₂₃	1	2	yes	—
73	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
74	C₂²	—	2	2	no	—
75	C₂²	—	2	4	no	—
76	C₂²	C₂³×D₂₂₃	1	1	yes	characteristic
77	C₄	—	2	1	no	—
78	C₄	—	2	2	no	—
79	C₄	—	2	2	no	—
80	C₄	—	446	4	no	—
81	C₄	—	446	4	no	—
82	C₂	—	4	2	no	—
83	C₂	#20297349	1	1	yes	characteristic
84	C₂	#20297351	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₂₃
commutator, socle, cosocle	C₂²×C₂₂₃	C₂⁴
Fitting	#124×C₂₂₃	C₂
prime radical	C₂×D₄×C₂₂₃	C₂²
hypercenter	#124	D₂₂₃
hypocenter	C₂₂₃	#563
LCS	G ▷ C₂²×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂² ◁ #124

Complements

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

N	H ≅ G/N	type
1. #20297302	C₂	semidirect
2. #20297297	C₂	semidirect
3. #20297296	C₂	semidirect
5. #20297281	C₂	semidirect
42. C₂₂₃	#563	semidirect

Maximal subgroups

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

subgroup	index
#20297302	2
#20297297	2
#20297296	2
#20297285	2
#20297281	2
#124×C₂₂₃	2
#563	223