Subgroups — #28383303

Order 14272.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×(C₂×D₂₂₃)⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₂²⋊(C₂₂₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²⋊C₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×D₂₂₃	C₄	1	1	yes	characteristic
5	C₂²×Dic₂₂₃	C₄	1	1	yes	characteristic
6	C₂×C₂₂₃⋊C₈	—	2	1	no	—
7	(C₂×D₂₂₃)⋊C₄	—	2	1	no	—
8	(C₂×D₂₂₃)⋊C₄	—	2	1	no	—
9	C₂²×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
10	C₂×C₈×C₂₂₃	—	2	1	no	—
11	C₂²×D₂₂₃	—	2	1	no	—
12	C₂²×D₂₂₃	—	2	1	no	—
13	C₂²×D₂₂₃	—	4	1	no	—
14	C₂²×D₂₂₃	—	4	1	no	—
15	C₂²×D₂₂₃	C₈	1	2	yes	complemented
16	C₂×Dic₂₂₃	—	2	1	no	—
17	C₂×Dic₂₂₃	—	4	1	no	—
18	C₂₂₃⋊C₈	—	4	1	no	—
19	C₂³×C₂₂₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₂₂₃	—	4	1	no	—
21	C₂×C₄×C₂₂₃	D₄	1	1	yes	characteristic
22	C₂×C₄×C₂₂₃	D₄	1	1	yes	characteristic
23	C₈×C₂₂₃	—	4	1	no	—
24	C₂²×C₂₂₃	—	2	1	no	—
25	C₂²×C₂₂₃	—	2	1	no	—
26	C₂²×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
27	C₂²×C₂₂₃	C₂×C₈	1	1	yes	characteristic
28	C₂²×C₂₂₃	M₁₆	1	1	yes	characteristic
29	C₂×D₂₂₃	—	2	2	no	—
30	C₂×D₂₂₃	—	4	2	no	—
31	C₂×D₂₂₃	—	4	1	no	—
32	C₂×D₂₂₃	—	4	1	no	—
33	C₂×D₂₂₃	—	4	2	no	—
34	C₄×C₂₂₃	—	2	1	no	—
35	C₄×C₂₂₃	—	2	1	no	—
36	Dic₂₂₃	—	4	1	no	—
37	C₂×C₂₂₃	—	2	1	no	—
38	C₂×C₂₂₃	—	2	1	no	—
39	C₂×C₂₂₃	C₂²⋊C₈	1	1	yes	characteristic
40	C₂×C₂₂₃	#99	1	1	yes	characteristic
41	C₂×C₂₂₃	#100	1	1	yes	characteristic
42	D₂₂₃	—	4	2	no	—
43	C₂₂₃	#323	1	1	yes	characteristic, complemented, Sylow
44	#323	—	223	1	no	maximal, Sylow
45	C₂×C₂²⋊C₄	—	223	1	no	—
46	C₂²⋊C₈	—	223	1	no	—
47	C₂²⋊C₈	D₂₂₃	1	1	yes	characteristic, complemented
48	C₂⁴	—	223	1	no	—
49	C₂²×C₄	—	223	1	no	—
50	C₂²×C₄	C₂×D₂₂₃	1	1	yes	characteristic
51	C₂×C₈	—	2	1	no	—
52	C₂×C₈	—	446	1	no	—
53	C₂²⋊C₄	—	446	1	no	—
54	C₂²⋊C₄	—	446	1	no	—
55	C₂³	—	892	1	no	—
56	C₂³	—	892	1	no	—
57	C₂³	—	446	1	no	—
58	C₂³	—	446	1	no	—
59	C₂³	—	223	2	no	—
60	C₂³	C₄×D₂₂₃	1	1	yes	characteristic
61	C₂×C₄	—	4	1	no	—
62	C₂×C₄	—	892	1	no	—
63	C₂×C₄	—	446	1	no	—
64	C₂×C₄	C₂²⋊D₂₂₃	1	1	yes	characteristic
65	C₂×C₄	D₈₉₂	1	1	yes	characteristic
66	C₈	—	4	1	no	—
67	C₈	—	892	1	no	—
68	C₂²	—	2	1	no	—
69	C₂²	—	2	1	no	—
70	C₂²	—	892	2	no	—
71	C₂²	—	892	2	no	—
72	C₂²	—	892	1	no	—
73	C₂²	—	892	1	no	—
74	C₂²	—	446	2	no	—
75	C₂²	(C₂×D₂₂₃)⋊C₄	1	1	yes	characteristic
76	C₂²	C₈×D₂₂₃	1	1	yes	characteristic
77	C₂²	#16813153	1	1	yes	characteristic
78	C₄	—	2	1	no	—
79	C₄	—	2	1	no	—
80	C₄	—	892	1	no	—
81	C₂	—	2	1	no	—
82	C₂	—	2	1	no	—
83	C₂	—	892	2	no	—
84	C₂	(C₂×D₂₂₃)⋊C₈	1	1	yes	characteristic
85	C₂	#20297218	1	1	yes	characteristic
86	C₂	#20297220	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	(C₂×D₂₂₃)⋊C₄
commutator	C₂²×C₂₂₃	C₂×C₈
Frattini	C₂²×C₄	C₂×D₂₂₃
Fitting	C₂²⋊C₈×C₂₂₃	C₂
socle	C₂²×C₂₂₃	C₂²⋊C₄
prime radical	C₂³×D₂₂₃	C₄
cosocle	C₂²×C₄×C₂₂₃	C₂²
hypercenter	C₂²⋊C₈	D₂₂₃
hypocenter	C₂₂₃	#323
LCS	G ▷ C₂²×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₂²⋊(C₂₂₃⋊C₈)	C₂	semidirect
3. C₂²⋊C₈×C₂₂₃	C₂	semidirect
15. C₂²×D₂₂₃	C₈	semidirect
43. C₂₂₃	#323	semidirect
47. C₂²⋊C₈	D₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×(C₂×D₂₂₃)⋊C₄	2
C₂²⋊(C₂₂₃⋊C₈)	2
C₂²⋊C₈×C₂₂₃	2
#323	223