Subgroups — #28383564

Order 14272.

150 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16813154	C₂	1	2	yes	maximal, complemented
2	#20297223	C₂	1	2	yes	maximal, complemented
3	#20297221	C₂	1	2	yes	maximal, complemented
4	C₂×#41×C₂₂₃	C₂	1	1	yes	maximal, characteristic
5	#16813154	—	2	2	no	—
6	#16813154	C₂²	1	4	yes	—
7	C₂×C₂₂₃⋊C₈	—	2	2	no	—
8	#41×C₂₂₃	—	2	2	no	—
9	#41×C₂₂₃	—	2	2	no	—
10	C₂×Q₈×C₂₂₃	C₂²	1	1	yes	characteristic
11	C₂×D₄×C₂₂₃	C₄	1	2	yes	—
12	C₂×D₄×C₂₂₃	C₂²	1	1	yes	characteristic
13	C₂²×C₄×C₂₂₃	C₄	1	2	yes	—
14	C₂²×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
15	C₂₂₃⋊C₈	—	2	4	no	—
16	C₂³×C₂₂₃	C₂×C₄	1	2	yes	—
17	C₂³×C₂₂₃	C₂×C₄	1	1	yes	characteristic
18	Q₈×C₂₂₃	—	2	2	no	—
19	D₄×C₂₂₃	—	2	4	no	—
20	D₄×C₂₂₃	—	2	2	no	—
21	C₂×C₄×C₂₂₃	—	2	2	no	—
22	C₂×C₄×C₂₂₃	—	2	2	no	—
23	C₂×C₄×C₂₂₃	C₂×C₄	1	2	yes	—
24	C₂×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×C₂₂₃	D₄	1	2	yes	—
26	C₂×C₄×C₂₂₃	D₄	1	2	yes	—
27	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
28	C₂²×C₂₂₃	—	2	2	no	—
29	C₂²×C₂₂₃	—	4	1	no	—
30	C₂²×C₂₂₃	—	4	2	no	—
31	C₂²×C₂₂₃	C₂²⋊C₄	1	2	yes	—
32	C₂²×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
33	C₄×C₂₂₃	—	2	2	no	—
34	C₄×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
35	C₄×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
36	C₄×C₂₂₃	C₂×D₄	1	2	yes	—
37	C₂×C₂₂₃	—	2	1	no	—
38	C₂×C₂₂₃	—	2	2	no	—
39	C₂×C₂₂₃	—	4	2	no	—
40	C₂×C₂₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
41	C₂₂₃	#413	1	1	yes	characteristic, complemented, Sylow
42	#413	—	223	1	no	maximal, Sylow
43	C₂×#41	D₂₂₃	1	1	yes	characteristic
44	C₂×M₁₆	—	223	2	no	—
45	#101	—	223	2	no	—
46	#100	—	223	2	no	—
47	#41	—	2	2	no	—
48	#41	—	2	2	no	—
49	C₂×Q₈	C₂×D₂₂₃	1	1	yes	characteristic
50	C₂×D₄	Dic₂₂₃	1	2	yes	—
51	C₂×D₄	C₂×D₂₂₃	1	1	yes	characteristic
52	C₂²×C₄	Dic₂₂₃	1	2	yes	—
53	C₂²×C₄	C₂×D₂₂₃	1	1	yes	characteristic
54	M₁₆	—	446	2	no	—
55	M₁₆	—	223	4	no	—
56	C₂×C₈	—	446	2	no	—
57	C₂³	C₂×Dic₂₂₃	1	2	yes	—
58	C₂³	C₂×Dic₂₂₃	1	1	yes	characteristic
59	Q₈	—	2	2	no	—
60	D₄	—	2	4	no	—
61	D₄	—	2	2	no	—
62	C₂×C₄	—	2	2	no	—
63	C₂×C₄	—	2	2	no	—
64	C₂×C₄	C₂×Dic₂₂₃	1	2	yes	—
65	C₂×C₄	C₂×Dic₂₂₃	1	1	yes	characteristic
66	C₂×C₄	C₂²⋊D₂₂₃	1	2	yes	—
67	C₂×C₄	C₂²⋊D₂₂₃	1	2	yes	—
68	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
69	C₈	—	446	4	no	—
70	C₂²	—	2	2	no	—
71	C₂²	—	4	1	no	—
72	C₂²	—	4	2	no	—
73	C₂²	C₂²⋊Dic₂₂₃	1	2	yes	—
74	C₂²	C₂²×Dic₂₂₃	1	1	yes	characteristic
75	C₄	—	2	2	no	—
76	C₄	C₂²⋊Dic₂₂₃	1	1	yes	characteristic
77	C₄	C₂²⋊Dic₂₂₃	1	1	yes	characteristic
78	C₄	C₂×C₂²⋊D₂₂₃	1	2	yes	—
79	C₂	—	2	1	no	—
80	C₂	—	2	2	no	—
81	C₂	—	4	2	no	—
82	C₂	C₂×C₂²⋊Dic₂₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²⋊Dic₂₂₃
commutator	C₂²×C₂₂₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₂₂₃
Fitting, prime radical	C₂×#41×C₂₂₃	C₂
socle	C₂×C₂₂₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄×C₂₂₃	C₂³
hypercenter	C₂×#41	D₂₂₃
hypocenter	C₂₂₃	#413
LCS	G ▷ C₂²×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#41

Complements

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

N	H ≅ G/N	type
1. C₂×#16813154	C₂	semidirect
2. #20297223	C₂	semidirect
3. #20297221	C₂	semidirect
41. C₂₂₃	#413	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16813154	2
#20297223	2
#20297221	2
C₂×#41×C₂₂₃	2
#413	223