Subgroups — #13092413

Order 1472.

158 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#41×C₂₃	C₂	1	1	yes	maximal, characteristic
2	#10625097	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₂₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#10625017	C₂	1	2	yes	maximal, complemented
5	#10625014	C₂	1	2	yes	maximal, complemented
6	#41×C₂₃	—	2	2	no	—
7	#41×C₂₃	C₄	1	4	yes	complemented
8	C₂×Q₈×C₂₃	C₂²	1	1	yes	characteristic
9	C₂×D₄×C₂₃	—	2	1	no	—
10	C₂×D₄×C₂₃	C₂²	1	1	yes	characteristic
11	C₂²×C₄×C₂₃	—	2	1	no	—
12	C₂²×C₄×C₂₃	C₂²	1	1	yes	characteristic
13	C₂²⋊Dic₂₃	—	2	1	no	—
14	C₄⋊Dic₂₃	C₂²	1	2	yes	—
15	C₄×Dic₂₃	—	2	1	no	—
16	C₂×C₂₃⋊C₈	—	2	2	no	—
17	C₂×C₂₃⋊C₈	C₂²	1	2	yes	—
18	C₂³×C₂₃	—	2	1	no	—
19	C₂³×C₂₃	D₄	1	1	yes	characteristic
20	Q₈×C₂₃	—	2	1	no	—
21	Q₈×C₂₃	C₂×C₄	1	2	yes	—
22	D₄×C₂₃	—	2	4	no	—
23	D₄×C₂₃	—	2	1	no	—
24	D₄×C₂₃	C₂×C₄	1	2	yes	—
25	C₂×C₄×C₂₃	—	2	1	no	—
26	C₂×C₄×C₂₃	—	2	4	no	—
27	C₂×C₄×C₂₃	C₂×C₄	1	2	yes	—
28	C₂×C₄×C₂₃	D₄	1	2	yes	—
29	C₂×C₄×C₂₃	D₄	1	1	yes	characteristic
30	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
31	C₂×Dic₂₃	—	2	2	no	—
32	C₂₃⋊C₈	—	2	2	no	—
33	C₂²×C₂₃	—	2	2	no	—
34	C₂²×C₂₃	—	2	2	no	—
35	C₂²×C₂₃	—	4	2	no	—
36	C₂²×C₂₃	C₂²⋊C₄	1	2	yes	—
37	C₂²×C₂₃	C₂×D₄	1	1	yes	characteristic
38	C₄×C₂₃	—	2	2	no	—
39	C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
40	C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
41	C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
42	Dic₂₃	—	4	2	no	—
43	#416	—	23	1	no	maximal, Sylow
44	C₂×C₂₃	—	2	2	no	—
45	C₂×C₂₃	—	4	2	no	—
46	C₂×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₂₃	#135	1	2	yes	—
48	C₂×#41	D₂₃	1	1	yes	characteristic
49	C₂²×C₈	—	23	1	no	—
50	#117	—	23	1	no	—
51	#103	—	23	2	no	—
52	#102	—	23	2	no	—
53	C₂₃	#416	1	1	yes	characteristic, complemented, Sylow
54	#41	—	2	2	no	—
55	#41	Dic₂₃	1	4	yes	complemented
56	C₂×Q₈	C₂×D₂₃	1	1	yes	characteristic
57	C₂×D₄	—	2	1	no	—
58	C₂×D₄	C₂×D₂₃	1	1	yes	characteristic
59	C₂²×C₄	—	2	1	no	—
60	C₂²×C₄	C₂×D₂₃	1	1	yes	characteristic
61	C₂×C₈	—	23	2	no	—
62	C₂×C₈	—	46	2	no	—
63	C₄⋊C₄	—	23	2	no	—
64	C₂²⋊C₄	—	46	1	no	—
65	C₄²	—	46	1	no	—
66	C₂³	—	2	1	no	—
67	C₂³	C₂²⋊D₂₃	1	1	yes	characteristic
68	Q₈	—	2	1	no	—
69	Q₈	C₂×Dic₂₃	1	2	yes	—
70	D₄	—	2	4	no	—
71	D₄	—	2	1	no	—
72	D₄	C₂×Dic₂₃	1	2	yes	—
73	C₂×C₄	—	2	1	no	—
74	C₂×C₄	—	2	4	no	—
75	C₂×C₄	—	46	2	no	—
76	C₂×C₄	C₂×Dic₂₃	1	2	yes	—
77	C₂×C₄	C₂²⋊D₂₃	1	2	yes	—
78	C₂×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
79	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
80	C₈	—	46	2	no	—
81	C₂²	—	2	2	no	—
82	C₂²	—	2	2	no	—
83	C₂²	—	4	2	no	—
84	C₂²	C₂²⋊Dic₂₃	1	2	yes	—
85	C₂²	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
86	C₄	—	2	2	no	—
87	C₄	—	92	2	no	—
88	C₄	C₂²⋊Dic₂₃	1	2	yes	—
89	C₄	C₂²×Dic₂₃	1	1	yes	characteristic
90	C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
91	C₂	—	2	2	no	—
92	C₂	—	4	2	no	—
93	C₂	#10625121	1	2	yes	—
94	C₂	C₂×C₂²⋊Dic₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊D₂₃
commutator	C₄×C₂₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₂₃
Fitting, prime radical	C₂×#41×C₂₃	C₂
socle	C₂²×C₂₃	C₂×D₄
cosocle	C₂×C₄×C₂₃	C₂³
hypercenter	C₂×#41	D₂₃
hypocenter	C₂₃	#416
LCS	G ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#41

Complements

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

N	H ≅ G/N	type
2. #10625097	C₂	semidirect
3. C₂²×C₂₃⋊C₈	C₂	semidirect
4. #10625017	C₂	semidirect
5. #10625014	C₂	semidirect
7. #41×C₂₃	C₄	semidirect
53. C₂₃	#416	semidirect
55. #41	Dic₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#41×C₂₃	2
#10625097	2
C₂²×C₂₃⋊C₈	2
#10625017	2
#10625014	2
#416	23