Subgroups — #13092282

Order 1472.

154 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊C₄×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	C₂²×C₄×C₂₃	C₂²	1	2	yes	—
6	C₂²×C₄×C₂₃	C₂²	1	1	yes	characteristic
7	C₂²×Dic₂₃	C₂²	1	2	yes	—
8	C₂²×Dic₂₃	C₂²	1	2	yes	—
9	C₄⋊C₄×C₂₃	—	2	2	no	—
10	C₄⋊Dic₂₃	—	2	2	no	—
11	C₄×Dic₂₃	C₄	1	4	yes	complemented
12	C₂³×C₂₃	C₂³	1	1	yes	characteristic
13	C₂×C₄×C₂₃	—	2	2	no	—
14	C₂×C₄×C₂₃	—	2	4	no	—
15	C₂×C₄×C₂₃	C₂×C₄	1	2	yes	—
16	C₂×C₄×C₂₃	D₄	1	2	yes	—
17	C₂×C₄×C₂₃	Q₈	1	2	yes	—
18	C₂×Dic₂₃	—	2	2	no	—
19	C₂×Dic₂₃	—	2	2	no	—
20	C₂×Dic₂₃	—	2	2	no	—
21	C₂×Dic₂₃	—	2	4	no	—
22	C₂×Dic₂₃	C₂×C₄	1	4	yes	—
23	C₂²×C₂₃	C₂²×C₄	1	1	yes	characteristic
24	C₂²×C₂₃	C₂×D₄	1	1	yes	characteristic
25	C₂²×C₂₃	C₂×Q₈	1	1	yes	characteristic
26	C₂²×C₂₃	#41	1	2	yes	—
27	C₂²×C₂₃	#41	1	2	yes	—
28	C₄×C₂₃	—	4	2	no	—
29	C₄×C₂₃	C₄⋊C₄	1	4	yes	—
30	Dic₂₃	—	2	4	no	—
31	Dic₂₃	—	4	2	no	—
32	#382	—	23	1	no	maximal, Sylow
33	C₂×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₂₃	#117	1	2	yes	—
35	C₂×C₂₃	#124	1	1	yes	characteristic
36	C₂×C₂₃	#124	1	1	yes	characteristic
37	C₂×C₂₃	#125	1	2	yes	—
38	C₂×C₄⋊C₄	—	23	1	no	—
39	C₂×C₄⋊C₄	D₂₃	1	1	yes	characteristic
40	C₂×C₄²	—	23	1	no	—
41	#95	—	23	4	no	—
42	C₂₃	#382	1	1	yes	characteristic, complemented, Sylow
43	C₂²×C₄	—	23	2	no	—
44	C₂²×C₄	—	23	2	no	—
45	C₂²×C₄	C₂×D₂₃	1	2	yes	—
46	C₂²×C₄	C₂×D₂₃	1	1	yes	characteristic
47	C₄⋊C₄	—	2	2	no	—
48	C₄⋊C₄	—	46	2	no	—
49	C₄²	—	23	4	no	—
50	C₂³	C₂²×D₂₃	1	1	yes	characteristic
51	C₂×C₄	—	2	4	no	—
52	C₂×C₄	—	2	2	no	—
53	C₂×C₄	—	23	4	no	—
54	C₂×C₄	—	46	2	no	—
55	C₂×C₄	—	46	2	no	—
56	C₂×C₄	—	46	4	no	—
57	C₂×C₄	—	46	2	no	—
58	C₂×C₄	Dic₄₆	1	2	yes	—
59	C₂×C₄	C₄×D₂₃	1	2	yes	—
60	C₂×C₄	C₂²⋊D₂₃	1	2	yes	—
61	C₂²	C₂×Dic₄₆	1	1	yes	characteristic
62	C₂²	C₂×C₄×D₂₃	1	1	yes	characteristic
63	C₂²	#67290	1	2	yes	—
64	C₂²	#67292	1	2	yes	—
65	C₂²	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
66	C₄	—	4	2	no	—
67	C₄	—	46	4	no	—
68	C₄	—	92	2	no	—
69	C₄	Dic₂₃⋊C₄	1	4	yes	—
70	C₂	#10625061	1	2	yes	—
71	C₂	#10625063	1	2	yes	—
72	C₂	C₂×Dic₂₃⋊C₄	1	1	yes	characteristic
73	C₂	#10625107	1	1	yes	characteristic
74	C₂	#10625118	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₂₃
commutator	C₂²×C₂₃	C₂²×C₄
Fitting	C₂×C₄⋊C₄×C₂₃	C₂
socle, prime radical, cosocle	C₂³×C₂₃	C₂³
hypercenter	C₂×C₄⋊C₄	D₂₃
hypocenter	C₂₃	#382
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
11. C₄×Dic₂₃	C₄	semidirect
42. C₂₃	#382	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊C₄×C₂₃	2
C₂×C₄⋊Dic₂₃	2
C₂×C₄×Dic₂₃	2
C₂₃⋊#95	2
#382	23