Subgroups — #28383719

Order 14272.

158 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16813157	C₂	1	2	yes	maximal, complemented
2	#20297278	C₂	1	1	yes	maximal, characteristic
3	#20297228	C₂	1	2	yes	maximal, complemented
4	C₂²⋊(C₂₂₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×Q₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₈₉₂	C₂²	1	1	yes	characteristic
7	C₂×C₂²⋊D₂₂₃	—	2	1	no	—
8	#16813157	—	2	4	no	—
9	C₂×C₂₂₃⋊C₈	C₂²	1	2	yes	—
10	C₄⋊Dic₂₂₃	C₂²	1	1	yes	characteristic
11	(C₂×D₂₂₃)⋊C₄	—	2	1	no	—
12	C₂×Q₈×C₂₂₃	—	2	4	no	—
13	C₂×Q₈×C₂₂₃	—	2	1	no	—
14	C₂×Q₈×C₂₂₃	C₂²	1	2	yes	—
15	C₂²×C₄×C₂₂₃	—	2	1	no	—
16	C₂²×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
17	C₂²×D₂₂₃	—	2	1	no	—
18	D₈₉₂	—	2	2	no	—
19	C₂²⋊D₂₂₃	—	4	2	no	—
20	C₂×Dic₂₂₃	—	2	1	no	—
21	C₂₂₃⋊C₈	—	2	2	no	—
22	C₂³×C₂₂₃	D₄	1	1	yes	characteristic
23	Q₈×C₂₂₃	—	2	4	no	—
24	Q₈×C₂₂₃	—	2	2	no	—
25	Q₈×C₂₂₃	D₄	1	4	yes	complemented
26	C₂×C₄×C₂₂₃	—	2	4	no	—
27	C₂×C₄×C₂₂₃	—	2	2	no	—
28	C₂×C₄×C₂₂₃	—	2	2	no	—
29	C₂×C₄×C₂₂₃	D₄	1	1	yes	characteristic
30	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
31	C₂²×C₂₂₃	—	2	1	no	—
32	C₂²×C₂₂₃	—	2	1	no	—
33	C₂²×C₂₂₃	SD₁₆	1	2	yes	complemented
34	C₂²×C₂₂₃	C₂×D₄	1	1	yes	characteristic
35	C₂×D₂₂₃	—	4	1	no	—
36	C₂×D₂₂₃	—	4	1	no	—
37	C₂×D₂₂₃	—	4	1	no	—
38	C₄×C₂₂₃	—	2	4	no	—
39	C₄×C₂₂₃	—	2	1	no	—
40	C₄×C₂₂₃	C₂×D₄	1	2	yes	—
41	Dic₂₂₃	—	4	1	no	—
42	C₂×C₂₂₃	—	2	2	no	—
43	C₂×C₂₂₃	#120	1	1	yes	characteristic
44	C₂×C₂₂₃	C₂×SD₁₆	1	1	yes	characteristic
45	C₂×C₂₂₃	#137	1	1	yes	characteristic
46	D₂₂₃	—	8	1	no	—
47	C₂₂₃	#448	1	1	yes	characteristic, complemented, Sylow
48	#448	—	223	1	no	maximal, Sylow
49	C₂²×Q₈	D₂₂₃	1	1	yes	characteristic, complemented
50	C₂×SD₁₆	—	223	2	no	—
51	#121	—	223	1	no	—
52	#103	—	223	2	no	—
53	C₂²⋊C₈	—	223	1	no	—
54	C₂×Q₈	—	2	4	no	—
55	C₂×Q₈	—	2	1	no	—
56	C₂×Q₈	C₂×D₂₂₃	1	2	yes	—
57	C₂×D₄	—	446	1	no	—
58	C₂×D₄	—	223	1	no	—
59	C₂²×C₄	—	2	1	no	—
60	C₂²×C₄	C₂×D₂₂₃	1	1	yes	characteristic
61	SD₁₆	—	446	4	no	—
62	C₂×C₈	—	223	2	no	—
63	C₄⋊C₄	—	223	1	no	—
64	C₂²⋊C₄	—	446	1	no	—
65	C₂³	—	446	1	no	—
66	C₂³	C₂²⋊D₂₂₃	1	1	yes	characteristic
67	Q₈	—	2	4	no	—
68	Q₈	—	2	2	no	—
69	Q₈	C₂²⋊D₂₂₃	1	4	yes	complemented
70	D₄	—	892	2	no	—
71	D₄	—	446	2	no	—
72	C₂×C₄	—	2	4	no	—
73	C₂×C₄	—	2	2	no	—
74	C₂×C₄	—	2	2	no	—
75	C₂×C₄	—	446	1	no	—
76	C₂×C₄	C₂²⋊D₂₂₃	1	1	yes	characteristic
77	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
78	C₈	—	446	2	no	—
79	C₂²	—	2	1	no	—
80	C₂²	—	2	1	no	—
81	C₂²	—	892	1	no	—
82	C₂²	—	892	1	no	—
83	C₂²	—	892	1	no	—
84	C₂²	#16813157	1	2	yes	complemented
85	C₂²	C₂×C₂²⋊D₂₂₃	1	1	yes	characteristic
86	C₄	—	2	4	no	—
87	C₄	—	2	1	no	—
88	C₄	—	892	1	no	—
89	C₄	C₂×C₂²⋊D₂₂₃	1	2	yes	—
90	C₂	—	2	2	no	—
91	C₂	—	1784	1	no	—
92	C₂	#20297273	1	1	yes	characteristic
93	C₂	C₂×#16813157	1	1	yes	characteristic
94	C₂	#20297335	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₂₂₃
commutator, cosocle	C₂×C₄×C₂₂₃	C₂³
Frattini	C₂×C₄	C₂²×D₂₂₃
Fitting	C₂²×Q₈×C₂₂₃	C₂
socle	C₂²×C₂₂₃	C₂×D₄
prime radical	#20297278	C₂
hypercenter	C₂²×Q₈	D₂₂₃
hypocenter	C₂₂₃	#448
LCS	G ▷ C₂×C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×#16813157	C₂	semidirect
3. #20297228	C₂	semidirect
4. C₂²⋊(C₂₂₃⋊C₈)	C₂	semidirect
5. C₂²×Q₈×C₂₂₃	C₂	semidirect
25. Q₈×C₂₂₃	D₄	semidirect
33. C₂²×C₂₂₃	SD₁₆	semidirect
47. C₂₂₃	#448	semidirect
49. C₂²×Q₈	D₂₂₃	semidirect
69. Q₈	C₂²⋊D₂₂₃	semidirect
84. C₂²	#16813157	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16813157	2
#20297278	2
#20297228	2
C₂²⋊(C₂₂₃⋊C₈)	2
C₂²×Q₈×C₂₂₃	2
#448	223