Subgroups — #28383776

Order 14272.

140 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₇₈₄	C₂	1	2	yes	maximal, complemented
2	#20297301	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×D₈₉₂	C₂	1	1	yes	maximal, characteristic, complemented
4	#20297226	C₂	1	2	yes	maximal, complemented
5	C₄⋊C₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₈₉₂	—	2	1	no	—
7	C₂×D₈₉₂	—	2	1	no	—
8	C₂×D₈₉₂	C₂²	1	2	yes	—
9	C₂×D₈₉₂	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₂₂₃	—	2	1	no	—
11	D₁₇₈₄	—	2	4	no	—
12	C₄⋊Dic₂₂₃	C₂²	1	1	yes	characteristic
13	(C₂×D₂₂₃)⋊C₄	—	2	1	no	—
14	C₂×C₈×C₂₂₃	C₂²	1	2	yes	—
15	C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
16	C₂²×D₂₂₃	—	2	1	no	—
17	C₂²×D₂₂₃	—	2	2	no	—
18	D₈₉₂	—	2	2	no	—
19	D₈₉₂	—	2	1	no	—
20	D₈₉₂	—	2	2	no	—
21	D₈₉₂	—	4	2	no	—
22	D₈₉₂	—	4	2	no	—
23	D₈₉₂	D₄	1	2	yes	complemented
24	C₂×Dic₂₂₃	—	2	1	no	—
25	C₄×D₂₂₃	—	4	2	no	—
26	C₂×C₄×C₂₂₃	D₄	1	1	yes	characteristic
27	C₂×C₄×C₂₂₃	D₄	1	1	yes	characteristic
28	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
29	C₈×C₂₂₃	—	2	2	no	—
30	C₂²×C₂₂₃	C₂×D₄	1	1	yes	characteristic
31	C₂×D₂₂₃	—	2	2	no	—
32	C₂×D₂₂₃	—	4	1	no	—
33	C₂×D₂₂₃	—	4	1	no	—
34	C₂×D₂₂₃	—	4	2	no	—
35	C₂×D₂₂₃	—	4	2	no	—
36	C₂×D₂₂₃	—	4	2	no	—
37	C₄×C₂₂₃	—	2	1	no	—
38	C₄×C₂₂₃	D₈	1	2	yes	complemented
39	C₄×C₂₂₃	C₂×D₄	1	1	yes	characteristic
40	C₄×C₂₂₃	#41	1	1	yes	characteristic
41	Dic₂₂₃	—	4	1	no	—
42	C₂×C₂₂₃	#121	1	1	yes	characteristic
43	C₂×C₂₂₃	C₂×D₈	1	1	yes	characteristic
44	C₂×C₂₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
45	D₂₂₃	—	4	2	no	—
46	D₂₂₃	—	8	2	no	—
47	C₂₂₃	#459	1	1	yes	characteristic, complemented, Sylow
48	#459	—	223	1	no	maximal, Sylow
49	C₂×D₈	—	223	2	no	—
50	#127	—	223	1	no	—
51	C₄×D₄	—	223	1	no	—
52	C₄⋊C₈	D₂₂₃	1	1	yes	characteristic, complemented
53	#102	—	223	2	no	—
54	C₂×D₄	—	446	1	no	—
55	C₂×D₄	—	446	1	no	—
56	C₂×D₄	—	223	2	no	—
57	C₂×D₄	—	223	1	no	—
58	C₂²×C₄	—	446	1	no	—
59	D₈	—	446	4	no	—
60	C₂×C₈	C₂×D₂₂₃	1	2	yes	—
61	C₄⋊C₄	—	223	1	no	—
62	C₂²⋊C₄	—	446	1	no	—
63	C₄²	C₂×D₂₂₃	1	1	yes	characteristic
64	C₂³	—	446	2	no	—
65	C₂³	—	446	1	no	—
66	D₄	—	892	2	no	—
67	D₄	—	892	2	no	—
68	D₄	—	446	2	no	—
69	D₄	—	446	2	no	—
70	D₄	—	446	1	no	—
71	D₄	—	223	2	no	—
72	C₂×C₄	—	892	2	no	—
73	C₂×C₄	—	446	1	no	—
74	C₂×C₄	D₈₉₂	1	1	yes	characteristic
75	C₂×C₄	D₈₉₂	1	1	yes	characteristic
76	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
77	C₈	—	2	2	no	—
78	C₂²	—	892	1	no	—
79	C₂²	—	892	1	no	—
80	C₂²	—	892	2	no	—
81	C₂²	—	892	2	no	—
82	C₂²	—	892	2	no	—
83	C₂²	—	446	2	no	—
84	C₂²	C₂×D₈₉₂	1	1	yes	characteristic
85	C₄	—	2	1	no	—
86	C₄	—	892	1	no	—
87	C₄	D₁₇₈₄	1	2	yes	complemented
88	C₄	D₄×D₂₂₃	1	1	yes	characteristic
89	C₄	#16813171	1	1	yes	characteristic
90	C₂	—	892	2	no	—
91	C₂	—	1784	2	no	—
92	C₂	#20297279	1	1	yes	characteristic
93	C₂	C₂×D₁₇₈₄	1	1	yes	characteristic
94	C₂	#20297332	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈₉₂
commutator, cosocle	C₂×C₄×C₂₂₃	C₂³
Frattini	C₂×C₄	C₂²×D₂₂₃
Fitting	C₄⋊C₈×C₂₂₃	C₂
socle	C₂²×C₂₂₃	C₂×D₄
hypercenter	C₄⋊C₈	D₂₂₃
hypocenter	C₂₂₃	#459
LCS	G ▷ C₂×C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
1. C₂×D₁₇₈₄	C₂	semidirect
2. #20297301	C₂	semidirect
3. C₄×D₈₉₂	C₂	semidirect
4. #20297226	C₂	semidirect
5. C₄⋊C₈×C₂₂₃	C₂	semidirect
23. D₈₉₂	D₄	semidirect
38. C₄×C₂₂₃	D₈	semidirect
47. C₂₂₃	#459	semidirect
52. C₄⋊C₈	D₂₂₃	semidirect
87. C₄	D₁₇₈₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₁₇₈₄	2
#20297301	2
C₄×D₈₉₂	2
#20297226	2
C₄⋊C₈×C₂₂₃	2
#459	223