Subgroups — #13093142

Order 1472.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	Q₈×Dic₂₃	C₂	1	2	yes	maximal
3	#10625115	C₂	1	2	yes	maximal
4	#10625061	C₂	1	2	yes	maximal
5	#10625060	C₂	1	4	yes	maximal
6	#10625058	C₂	1	2	yes	maximal
7	C₄×Dic₄₆	C₂	1	1	yes	maximal, characteristic
8	C₄×Dic₄₆	C₂	1	1	yes	maximal, characteristic
9	C₂×Q₈×C₂₃	C₂²	1	2	yes	—
10	C₂×Dic₄₆	C₂²	1	1	yes	characteristic
11	C₂×Dic₄₆	C₂²	1	1	yes	characteristic
12	C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
13	C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
14	C₄²×C₂₃	C₂²	1	1	yes	characteristic
15	C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
16	C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
17	C₄⋊Dic₂₃	C₂²	1	4	yes	—
18	Dic₂₃⋊C₄	C₂²	1	2	yes	—
19	Dic₂₃⋊C₄	C₂²	1	2	yes	—
20	Dic₂₃⋊C₄	C₂²	1	4	yes	—
21	Dic₂₃⋊C₄	C₂²	1	4	yes	—
22	C₄×Dic₂₃	C₂²	1	2	yes	—
23	C₄×Dic₂₃	C₂²	1	2	yes	—
24	C₄×Dic₂₃	C₂²	1	4	yes	—
25	Q₈×C₂₃	—	2	4	no	—
26	C₂×C₄×C₂₃	C₂³	1	4	yes	—
27	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
28	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
29	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
30	C₂×Dic₂₃	C₂³	1	2	yes	—
31	C₂×Dic₂₃	C₂³	1	2	yes	—
32	C₂×Dic₂₃	C₂³	1	2	yes	—
33	C₂×Dic₂₃	C₂³	1	2	yes	—
34	Dic₄₆	—	2	2	no	—
35	Dic₄₆	Q₈	1	4	yes	complemented
36	C₂²×C₂₃	C₂⁴	1	1	yes	characteristic
37	C₄×C₂₃	—	2	4	no	—
38	C₄×C₂₃	—	2	1	no	—
39	C₄×C₂₃	C₂×Q₈	1	2	yes	—
40	C₄×C₂₃	#41	1	2	yes	—
41	Dic₂₃	—	2	2	no	—
42	Dic₂₃	—	2	2	no	—
43	Dic₂₃	—	2	2	no	—
44	Dic₂₃	C₂×Q₈	1	4	yes	—
45	#557	—	23	1	no	maximal, Sylow
46	C₂×C₂₃	C₂²×Q₈	1	1	yes	characteristic
47	C₂×C₂₃	C₂×#41	1	1	yes	characteristic
48	C₂×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
49	C₄⋊Q₈	—	23	2	no	—
50	C₄⋊Q₈	D₂₃	1	1	yes	characteristic
51	#125	—	23	4	no	—
52	#125	—	23	2	no	—
53	C₄×Q₈	—	23	2	no	—
54	C₄×Q₈	—	23	2	no	—
55	C₄×Q₈	—	23	1	no	—
56	C₄×Q₈	—	23	1	no	—
57	C₂₃	#557	1	1	yes	characteristic, complemented, Sylow
58	C₂×Q₈	—	23	1	no	—
59	C₂×Q₈	—	23	1	no	—
60	C₂×Q₈	C₂×D₂₃	1	2	yes	—
61	C₄⋊C₄	—	23	1	no	—
62	C₄⋊C₄	—	23	2	no	—
63	C₄⋊C₄	—	23	2	no	—
64	C₄⋊C₄	—	23	4	no	—
65	C₄⋊C₄	—	23	1	no	—
66	C₄⋊C₄	—	23	4	no	—
67	C₄⋊C₄	—	23	4	no	—
68	C₄⋊C₄	C₂×D₂₃	1	2	yes	—
69	C₄⋊C₄	C₂×D₂₃	1	2	yes	—
70	C₄²	—	23	4	no	—
71	C₄²	—	23	2	no	—
72	C₄²	—	23	2	no	—
73	C₄²	C₂×D₂₃	1	1	yes	characteristic
74	Q₈	—	2	4	no	—
75	Q₈	—	23	4	no	—
76	Q₈	—	46	2	no	—
77	C₂×C₄	—	23	2	no	—
78	C₂×C₄	—	23	2	no	—
79	C₂×C₄	—	23	2	no	—
80	C₂×C₄	—	23	2	no	—
81	C₂×C₄	C₂²×D₂₃	1	4	yes	—
82	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
83	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
84	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
85	C₂²	C₂³×D₂₃	1	1	yes	characteristic
86	C₄	—	2	4	no	—
87	C₄	—	2	1	no	—
88	C₄	—	23	4	no	—
89	C₄	—	46	2	no	—
90	C₄	—	46	2	no	—
91	C₄	—	46	2	no	—
92	C₄	#67290	1	2	yes	—
93	C₄	Q₈×D₂₃	1	2	yes	—
94	C₂	C₂×#67290	1	1	yes	characteristic
95	C₂	C₂×Q₈×D₂₃	1	1	yes	characteristic
96	C₂	#10625158	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₃
commutator, socle, prime radical, cosocle	C₂²×C₂₃	C₂⁴
Fitting	C₂₃×C₄⋊Q₈	C₂
hypercenter	C₄⋊Q₈	D₂₃
hypocenter	C₂₃	#557
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
35. Dic₄₆	Q₈	semidirect
57. C₂₃	#557	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×C₄⋊Q₈	2
Q₈×Dic₂₃	2
#10625115	2
#10625061	2
#10625060	2
#10625058	2
C₄×Dic₄₆	2
#557	23