Subgroups — #13092328

Order 1472.

202 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, complemented
2	C₂×(C₂×D₂₃)⋊C₄	C₂	1	3	yes	maximal
3	C₂₃⋊#95	C₂	1	3	yes	maximal, complemented
4	C₂³×D₂₃	C₂²	1	1	yes	characteristic
5	C₂×Q₈×C₂₃	—	2	6	no	—
6	C₂²×C₄×C₂₃	C₂²	1	3	yes	—
7	C₂²×Dic₂₃	C₂²	1	3	yes	—
8	(C₂×D₂₃)⋊C₄	—	2	6	no	—
9	C₂³×C₂₃	C₂³	1	1	yes	characteristic
10	C₂²×D₂₃	—	2	3	no	—
11	C₂²×D₂₃	—	2	3	no	—
12	C₂²×D₂₃	Q₈	1	2	yes	complemented
13	Q₈×C₂₃	—	2	8	no	—
14	C₂×C₄×C₂₃	—	2	6	no	—
15	C₂×C₄×C₂₃	D₄	1	6	yes	—
16	C₂×Dic₂₃	—	2	3	no	—
17	C₂×Dic₂₃	—	2	6	no	—
18	C₂²×C₂₃	C₂×D₄	1	3	yes	—
19	C₂²×C₂₃	C₂×Q₈	1	1	yes	characteristic
20	C₂²×C₂₃	#41	1	3	yes	—
21	C₂×D₂₃	—	2	6	no	—
22	C₂×D₂₃	—	4	1	no	—
23	C₂×D₂₃	—	4	3	no	—
24	C₄×C₂₃	—	2	6	no	—
25	Dic₂₃	—	4	3	no	—
26	#393	—	23	1	no	maximal, Sylow
27	C₂×C₂₃	#120	1	1	yes	characteristic
28	C₂×C₂₃	C₂²⋊Q₈	1	3	yes	—
29	C₂×C₂₃	#124	1	3	yes	—
30	D₂₃	—	4	2	no	—
31	C₂²×Q₈	D₂₃	1	1	yes	characteristic, complemented
32	C₂×C₂²⋊C₄	—	23	3	no	—
33	#95	—	23	3	no	—
34	C₂₃	#393	1	1	yes	characteristic, complemented, Sylow
35	C₂⁴	—	23	1	no	—
36	C₂×Q₈	—	2	6	no	—
37	C₂²×C₄	—	23	3	no	—
38	C₂²×C₄	C₂×D₂₃	1	3	yes	—
39	C₂²⋊C₄	—	46	6	no	—
40	C₂³	—	23	2	no	—
41	C₂³	—	46	3	no	—
42	C₂³	—	46	3	no	—
43	C₂³	C₂²×D₂₃	1	1	yes	characteristic
44	Q₈	—	2	8	no	—
45	C₂×C₄	—	2	6	no	—
46	C₂×C₄	—	46	6	no	—
47	C₂×C₄	—	46	3	no	—
48	C₂×C₄	C₂²⋊D₂₃	1	6	yes	—
49	C₂²	—	46	6	no	—
50	C₂²	—	92	3	no	—
51	C₂²	—	92	1	no	—
52	C₂²	Q₈×D₂₃	1	1	yes	characteristic
53	C₂²	#67292	1	3	yes	—
54	C₂²	C₂×C₂²⋊D₂₃	1	3	yes	—
55	C₄	—	2	6	no	—
56	C₄	—	92	3	no	—
57	C₂	—	92	2	no	—
58	C₂	#10625117	1	3	yes	—
59	C₂	#10625118	1	3	yes	—
60	C₂	#10625124	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₂₃
commutator, socle, cosocle	C₂³×C₂₃	C₂³
Fitting	C₂²×Q₈×C₂₃	C₂
prime radical	C₂³×D₂₃	C₂²
hypercenter	C₂²×Q₈	D₂₃
hypocenter	C₂₃	#393
LCS	G ▷ C₂³×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₂²×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂²×Q₈×C₂₃	C₂	semidirect
3. C₂₃⋊#95	C₂	semidirect
12. C₂²×D₂₃	Q₈	semidirect
31. C₂²×Q₈	D₂₃	semidirect
34. C₂₃	#393	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×(C₂×D₂₃)⋊C₄	2
C₂₃⋊#95	2
#393	23