Subgroups — #28383750

Order 14272.

168 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20297349	C₂	1	1	yes	maximal, characteristic, complemented
2	#20297332	C₂	1	2	yes	maximal, complemented
3	#20297300	C₂	1	1	yes	maximal, characteristic, complemented
4	#20297230	C₂	1	2	yes	maximal, complemented
5	#100×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#16813170	C₂²	1	2	yes	—
7	#16813169	—	2	2	no	—
8	C₂×D₈₉₂	C₂²	1	1	yes	characteristic, complemented
9	C₂×C₂²⋊D₂₂₃	—	2	2	no	—
10	C₂×C₂²⋊D₂₂₃	—	2	2	no	—
11	D₄×D₂₂₃	—	2	2	no	—
12	#16813159	—	2	2	no	—
13	D₁₇₈₄	—	2	2	no	—
14	C₄×Dic₂₂₃	C₂²	1	1	yes	characteristic, complemented
15	C₂×D₄×C₂₂₃	C₂²	1	1	yes	characteristic
16	M₁₆×C₂₂₃	C₂²	1	2	yes	complemented
17	C₂²×D₂₂₃	—	2	2	no	—
18	C₂²×D₂₂₃	—	2	1	no	—
19	Dic₄₄₆	D₄	1	2	yes	complemented
20	D₈₉₂	—	2	2	no	—
21	D₈₉₂	D₄	1	2	yes	complemented
22	C₂²⋊D₂₂₃	—	2	4	no	—
23	C₂²⋊D₂₂₃	—	2	2	no	—
24	C₂²⋊D₂₂₃	—	4	4	no	—
25	C₂×Dic₂₂₃	—	2	2	no	—
26	C₂×Dic₂₂₃	—	2	1	no	—
27	C₄×D₂₂₃	—	2	2	no	—
28	C₂³×C₂₂₃	D₄	1	2	yes	—
29	D₄×C₂₂₃	—	2	2	no	—
30	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
31	C₈×C₂₂₃	—	2	2	no	—
32	C₂²×C₂₂₃	—	2	2	no	—
33	C₂²×C₂₂₃	—	4	2	no	—
34	C₂²×C₂₂₃	C₂×D₄	1	1	yes	characteristic
35	C₂×D₂₂₃	—	2	2	no	—
36	C₂×D₂₂₃	—	4	2	no	—
37	C₂×D₂₂₃	—	4	1	no	—
38	C₂×D₂₂₃	—	8	2	no	—
39	C₄×C₂₂₃	C₂×D₄	1	2	yes	—
40	Dic₂₂₃	—	2	2	no	—
41	Dic₂₂₃	—	2	2	no	—
42	C₂×C₂₂₃	—	2	1	no	—
43	C₂×C₂₂₃	—	4	2	no	—
44	C₂×C₂₂₃	#120	1	1	yes	characteristic
45	D₂₂₃	—	4	2	no	—
46	D₂₂₃	—	8	1	no	—
47	C₂₂₃	#453	1	1	yes	characteristic, complemented, Sylow
48	#453	—	223	1	no	maximal, Sylow
49	H(5,𝔽₂)	—	223	1	no	—
50	AGL(1,ℤ/8)	—	223	2	no	—
51	#127	—	223	1	no	—
52	#104	—	223	2	no	—
53	#100	D₂₂₃	1	1	yes	characteristic, complemented
54	#41	—	446	2	no	—
55	#41	—	223	2	no	—
56	C₂×D₄	—	446	2	no	—
57	C₂×D₄	—	446	2	no	—
58	C₂×D₄	—	446	2	no	—
59	C₂×D₄	—	223	1	no	—
60	C₂×D₄	C₂×D₂₂₃	1	1	yes	characteristic
61	SD₁₆	—	446	2	no	—
62	D₈	—	446	2	no	—
63	M₁₆	C₂×D₂₂₃	1	2	yes	complemented
64	C₄²	—	223	1	no	—
65	C₂³	—	446	2	no	—
66	C₂³	—	446	1	no	—
67	C₂³	D₈₉₂	1	2	yes	—
68	Q₈	—	223	2	no	—
69	D₄	—	2	2	no	—
70	D₄	—	892	4	no	—
71	D₄	—	446	4	no	—
72	D₄	—	446	2	no	—
73	D₄	—	446	2	no	—
74	D₄	—	223	2	no	—
75	C₂×C₄	—	446	2	no	—
76	C₂×C₄	—	446	1	no	—
77	C₂×C₄	—	446	2	no	—
78	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
79	C₈	—	2	2	no	—
80	C₂²	—	2	2	no	—
81	C₂²	—	4	2	no	—
82	C₂²	—	892	2	no	—
83	C₂²	—	892	1	no	—
84	C₂²	—	446	2	no	—
85	C₂²	—	1784	2	no	—
86	C₂²	C₂×D₈₉₂	1	1	yes	characteristic
87	C₄	—	446	2	no	—
88	C₄	—	446	2	no	—
89	C₄	D₄×D₂₂₃	1	2	yes	—
90	C₂	—	2	1	no	—
91	C₂	—	4	2	no	—
92	C₂	—	892	2	no	—
93	C₂	—	1784	1	no	—
94	C₂	#20297274	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20297274
commutator, cosocle	C₂×C₄×C₂₂₃	C₂³
Frattini	C₂×C₄	C₂²×D₂₂₃
Fitting	#100×C₂₂₃	C₂
socle	C₂×C₂₂₃	#120
hypercenter	#100	D₂₂₃
hypocenter	C₂₂₃	#453
LCS	G ▷ C₂×C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #100

Complements

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

N	H ≅ G/N	type
1. #20297349	C₂	semidirect
2. #20297332	C₂	semidirect
3. #20297300	C₂	semidirect
4. #20297230	C₂	semidirect
5. #100×C₂₂₃	C₂	semidirect
8. C₂×D₈₉₂	C₂²	semidirect
14. C₄×Dic₂₂₃	C₂²	semidirect
16. M₁₆×C₂₂₃	C₂²	semidirect
19. Dic₄₄₆	D₄	semidirect
21. D₈₉₂	D₄	semidirect
47. C₂₂₃	#453	semidirect
53. #100	D₂₂₃	semidirect
63. M₁₆	C₂×D₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
#20297349	2
#20297332	2
#20297300	2
#20297230	2
#100×C₂₂₃	2
#453	223