Subgroups — #13092324

Order 1472.

186 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊Dic₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×Dic₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂₃⋊#95	C₂	1	4	yes	maximal
5	C₂×Q₈×C₂₃	—	2	2	no	—
6	C₂×Q₈×C₂₃	—	2	2	no	—
7	C₂×Q₈×C₂₃	C₄	1	4	yes	complemented
8	C₂²×C₄×C₂₃	C₂²	1	2	yes	—
9	C₂²×C₄×C₂₃	C₂²	1	1	yes	characteristic
10	C₂²×Dic₂₃	C₂²	1	2	yes	—
11	C₂²×Dic₂₃	C₂²	1	2	yes	—
12	C₄⋊Dic₂₃	—	2	2	no	—
13	C₄×Dic₂₃	—	2	2	no	—
14	C₂³×C₂₃	C₂³	1	1	yes	characteristic
15	Q₈×C₂₃	—	2	8	no	—
16	C₂×C₄×C₂₃	—	2	2	no	—
17	C₂×C₄×C₂₃	—	2	2	no	—
18	C₂×C₄×C₂₃	C₂×C₄	1	4	yes	—
19	C₂×C₄×C₂₃	C₂×C₄	1	2	yes	—
20	C₂×C₄×C₂₃	D₄	1	2	yes	—
21	C₂×C₄×C₂₃	D₄	1	2	yes	—
22	C₂×Dic₂₃	—	2	2	no	—
23	C₂×Dic₂₃	—	2	2	no	—
24	C₂×Dic₂₃	—	2	2	no	—
25	C₂×Dic₂₃	—	2	4	no	—
26	C₂×Dic₂₃	Q₈	1	4	yes	complemented
27	C₂²×C₂₃	C₂²×C₄	1	1	yes	characteristic
28	C₂²×C₂₃	C₂×D₄	1	1	yes	characteristic
29	C₂²×C₂₃	C₂×D₄	1	1	yes	characteristic
30	C₂²×C₂₃	C₂×Q₈	1	2	yes	—
31	C₂²×C₂₃	#41	1	2	yes	—
32	C₄×C₂₃	—	2	4	no	—
33	C₄×C₂₃	C₂²⋊C₄	1	4	yes	—
34	Dic₂₃	—	2	4	no	—
35	Dic₂₃	—	4	2	no	—
36	#391	—	23	1	no	maximal, Sylow
37	C₂×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₂₃	C₄×Q₈	1	2	yes	—
39	C₂×C₂₃	C₂²⋊Q₈	1	2	yes	—
40	C₂×C₂₃	#124	1	1	yes	characteristic
41	C₂×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
42	C₂²×Q₈	D₂₃	1	1	yes	characteristic
43	C₂×C₄⋊C₄	—	23	1	no	—
44	C₂×C₄²	—	23	1	no	—
45	#95	—	23	4	no	—
46	C₂₃	#391	1	1	yes	characteristic, complemented, Sylow
47	C₂×Q₈	—	2	2	no	—
48	C₂×Q₈	—	2	2	no	—
49	C₂×Q₈	Dic₂₃	1	4	yes	complemented
50	C₂²×C₄	—	23	2	no	—
51	C₂²×C₄	—	23	2	no	—
52	C₂²×C₄	C₂×D₂₃	1	2	yes	—
53	C₂²×C₄	C₂×D₂₃	1	1	yes	characteristic
54	C₄⋊C₄	—	46	2	no	—
55	C₄²	—	46	2	no	—
56	C₂³	C₂²×D₂₃	1	1	yes	characteristic
57	Q₈	—	2	8	no	—
58	C₂×C₄	—	2	2	no	—
59	C₂×C₄	—	2	2	no	—
60	C₂×C₄	—	23	4	no	—
61	C₂×C₄	—	46	2	no	—
62	C₂×C₄	—	46	2	no	—
63	C₂×C₄	—	46	4	no	—
64	C₂×C₄	—	46	2	no	—
65	C₂×C₄	C₂×Dic₂₃	1	4	yes	—
66	C₂×C₄	C₂×Dic₂₃	1	2	yes	—
67	C₂×C₄	C₂²⋊D₂₃	1	2	yes	—
68	C₂×C₄	C₂²⋊D₂₃	1	2	yes	—
69	C₂²	Q₈×D₂₃	1	2	yes	—
70	C₂²	#67292	1	2	yes	—
71	C₂²	C₂²×Dic₂₃	1	1	yes	characteristic
72	C₂²	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
73	C₂²	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
74	C₄	—	2	4	no	—
75	C₄	—	46	4	no	—
76	C₄	—	92	2	no	—
77	C₄	C₂²⋊Dic₂₃	1	4	yes	—
78	C₂	#10625115	1	1	yes	characteristic
79	C₂	Q₈×Dic₂₃	1	2	yes	—
80	C₂	#10625117	1	2	yes	—
81	C₂	#10625118	1	1	yes	characteristic
82	C₂	C₂×C₂²⋊Dic₂₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₂₃
commutator	C₂²×C₂₃	C₂²×C₄
Fitting	C₂²×Q₈×C₂₃	C₂
socle, prime radical, cosocle	C₂³×C₂₃	C₂³
hypercenter	C₂²×Q₈	D₂₃
hypocenter	C₂₃	#391
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₂²×Q₈

Complements

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

N	H ≅ G/N	type
7. C₂×Q₈×C₂₃	C₄	semidirect
26. C₂×Dic₂₃	Q₈	semidirect
46. C₂₃	#391	semidirect
49. C₂×Q₈	Dic₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Q₈×C₂₃	2
C₂×C₄⋊Dic₂₃	2
C₂×C₄×Dic₂₃	2
C₂₃⋊#95	2
#391	23