Subgroups — #28383748

Order 14272.

168 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20297329	C₂	1	2	yes	maximal, complemented
2	#20297300	C₂	1	1	yes	maximal, characteristic, complemented
3	#20297231	C₂	1	2	yes	maximal, complemented
4	#20297221	C₂	1	1	yes	maximal, characteristic, complemented
5	H(5,𝔽₂)×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₈₉₂	C₂²	1	1	yes	characteristic, complemented
7	C₂×C₂²⋊D₂₂₃	—	2	2	no	—
8	#16813157	—	2	2	no	—
9	#16813155	—	2	2	no	—
10	#16813154	C₂²	1	2	yes	complemented
11	C₄×Dic₂₂₃	C₂²	1	1	yes	characteristic, complemented
12	#41×C₂₂₃	—	2	2	no	—
13	#41×C₂₂₃	C₂²	1	2	yes	—
14	C₂×D₄×C₂₂₃	—	2	2	no	—
15	C₂×D₄×C₂₂₃	—	2	2	no	—
16	C₂×D₄×C₂₂₃	C₂²	1	1	yes	characteristic
17	C₂²×D₂₂₃	—	2	1	no	—
18	D₈₉₂	—	2	2	no	—
19	C₂²⋊D₂₂₃	—	4	4	no	—
20	C₂×Dic₂₂₃	—	2	1	no	—
21	C₂₂₃⋊C₈	—	2	2	no	—
22	C₂³×C₂₂₃	—	2	2	no	—
23	C₂³×C₂₂₃	D₄	1	2	yes	—
24	Q₈×C₂₂₃	D₄	1	2	yes	complemented
25	D₄×C₂₂₃	—	2	4	no	—
26	D₄×C₂₂₃	—	2	2	no	—
27	D₄×C₂₂₃	—	2	2	no	—
28	D₄×C₂₂₃	D₄	1	2	yes	complemented
29	C₂×C₄×C₂₂₃	—	2	2	no	—
30	C₂×C₄×C₂₂₃	—	2	2	no	—
31	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
32	C₂²×C₂₂₃	—	2	2	no	—
33	C₂²×C₂₂₃	—	2	2	no	—
34	C₂²×C₂₂₃	—	4	2	no	—
35	C₂²×C₂₂₃	—	8	2	no	—
36	C₂²×C₂₂₃	C₂×D₄	1	1	yes	characteristic
37	C₂×D₂₂₃	—	4	2	no	—
38	C₂×D₂₂₃	—	4	1	no	—
39	C₄×C₂₂₃	—	2	2	no	—
40	C₄×C₂₂₃	C₂×D₄	1	2	yes	—
41	Dic₂₂₃	—	2	2	no	—
42	C₂×C₂₂₃	—	2	1	no	—
43	C₂×C₂₂₃	—	4	2	no	—
44	C₂×C₂₂₃	—	4	2	no	—
45	C₂×C₂₂₃	#120	1	1	yes	characteristic
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)	D₂₂₃	1	1	yes	characteristic, complemented
50	AGL(1,ℤ/8)	—	223	2	no	—
51	#127	—	223	1	no	—
52	#104	—	223	2	no	—
53	#100	—	223	1	no	—
54	#41	—	2	2	no	—
55	#41	C₂×D₂₂₃	1	2	yes	—
56	C₂×D₄	—	2	2	no	—
57	C₂×D₄	—	2	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	—	223	2	no	—
64	C₄²	—	223	1	no	—
65	C₂³	—	2	2	no	—
66	C₂³	—	446	1	no	—
67	C₂³	C₂²⋊D₂₂₃	1	2	yes	—
68	Q₈	C₂²⋊D₂₂₃	1	2	yes	complemented
69	D₄	—	2	4	no	—
70	D₄	—	2	2	no	—
71	D₄	—	2	2	no	—
72	D₄	—	892	4	no	—
73	D₄	—	446	2	no	—
74	D₄	C₂²⋊D₂₂₃	1	2	yes	complemented
75	C₂×C₄	—	2	2	no	—
76	C₂×C₄	—	2	2	no	—
77	C₂×C₄	—	446	1	no	—
78	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
79	C₈	—	446	2	no	—
80	C₂²	—	2	2	no	—
81	C₂²	—	2	2	no	—
82	C₂²	—	4	2	no	—
83	C₂²	—	8	2	no	—
84	C₂²	—	892	2	no	—
85	C₂²	—	892	1	no	—
86	C₂²	C₂×C₂²⋊D₂₂₃	1	1	yes	characteristic
87	C₄	—	2	2	no	—
88	C₄	—	446	2	no	—
89	C₄	C₂×C₂²⋊D₂₂₃	1	2	yes	—
90	C₂	—	2	1	no	—
91	C₂	—	4	2	no	—
92	C₂	—	4	2	no	—
93	C₂	—	1784	1	no	—
94	C₂	#20297273	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20297273
commutator, cosocle	C₂×C₄×C₂₂₃	C₂³
Frattini	C₂×C₄	C₂²×D₂₂₃
Fitting	H(5,𝔽₂)×C₂₂₃	C₂
socle	C₂×C₂₂₃	#120
hypercenter	H(5,𝔽₂)	D₂₂₃
hypocenter	C₂₂₃	#453
LCS	G ▷ C₂×C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ H(5,𝔽₂)

Complements

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

N	H ≅ G/N	type
1. #20297329	C₂	semidirect
2. #20297300	C₂	semidirect
3. #20297231	C₂	semidirect
4. #20297221	C₂	semidirect
5. H(5,𝔽₂)×C₂₂₃	C₂	semidirect
6. C₂×D₈₉₂	C₂²	semidirect
10. #16813154	C₂²	semidirect
11. C₄×Dic₂₂₃	C₂²	semidirect
24. Q₈×C₂₂₃	D₄	semidirect
28. D₄×C₂₂₃	D₄	semidirect
47. C₂₂₃	#453	semidirect
49. H(5,𝔽₂)	D₂₂₃	semidirect
68. Q₈	C₂²⋊D₂₂₃	semidirect
74. D₄	C₂²⋊D₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
#20297329	2
#20297300	2
#20297231	2
#20297221	2
H(5,𝔽₂)×C₂₂₃	2
#453	223