Subgroups — #13093167

Order 1472.

196 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#124	C₂	1	1	yes	maximal, characteristic
2	#10625115	C₂	1	2	yes	maximal, complemented
3	#10625106	C₂	1	2	yes	maximal
4	#10625050	C₂	1	4	yes	maximal, complemented
5	#10625049	C₂	1	4	yes	maximal, complemented
6	#10625041	C₂	1	2	yes	maximal, complemented
7	C₂×Q₈×C₂₃	C₂²	1	1	yes	characteristic
8	C₂×D₄×C₂₃	C₂²	1	1	yes	characteristic
9	C₂²×Dic₂₃	C₂²	1	2	yes	—
10	C₂×Dic₄₆	C₂²	1	2	yes	—
11	C₂²⋊C₄×C₂₃	C₂²	1	4	yes	—
12	C₄²×C₂₃	C₂²	1	1	yes	characteristic
13	C₂²⋊Dic₂₃	C₂²	1	4	yes	—
14	C₂²⋊Dic₂₃	C₂²	1	2	yes	—
15	C₄⋊Dic₂₃	C₂²	1	2	yes	—
16	C₄⋊Dic₂₃	C₂²	1	2	yes	—
17	Dic₂₃⋊C₄	C₂²	1	4	yes	—
18	Dic₂₃⋊C₄	C₂²	1	4	yes	—
19	Dic₂₃⋊C₄	C₂²	1	4	yes	—
20	C₄×Dic₂₃	C₂²	1	2	yes	—
21	C₂³×C₂₃	C₂³	1	2	yes	—
22	Q₈×C₂₃	—	4	1	no	—
23	D₄×C₂₃	—	4	1	no	—
24	C₂×C₄×C₂₃	C₂³	1	2	yes	—
25	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
26	C₂×C₄×C₂₃	C₂³	1	2	yes	—
27	C₂×Dic₂₃	—	4	2	no	—
28	C₂×Dic₂₃	C₂³	1	4	yes	—
29	C₂×Dic₂₃	C₂³	1	4	yes	—
30	Dic₄₆	—	4	2	no	—
31	C₂²×C₂₃	—	2	2	no	—
32	C₂²×C₂₃	—	2	4	no	—
33	C₂²×C₂₃	C₂⁴	1	1	yes	characteristic
34	C₄×C₂₃	—	2	2	no	—
35	C₄×C₂₃	—	2	1	no	—
36	C₄×C₂₃	—	2	2	no	—
37	Dic₂₃	—	2	4	no	—
38	Dic₂₃	—	2	4	no	—
39	#563	—	23	1	no	maximal, Sylow
40	C₂×C₂₃	—	4	2	no	—
41	C₂×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
42	C₂×C₂₃	2₋¹⁺⁴	1	2	yes	—
43	C₄⋊Q₈	—	23	2	no	—
44	#126	—	23	4	no	—
45	#125	—	23	2	no	—
46	#124	D₂₃	1	1	yes	characteristic
47	#123	—	23	2	no	—
48	C₂²⋊Q₈	—	23	4	no	—
49	C₂₃	#563	1	1	yes	characteristic, complemented, Sylow
50	C₂×Q₈	—	23	2	no	—
51	C₂×Q₈	C₂×D₂₃	1	1	yes	characteristic
52	C₂×D₄	C₂×D₂₃	1	1	yes	characteristic
53	C₂²×C₄	—	23	2	no	—
54	C₄⋊C₄	—	23	2	no	—
55	C₄⋊C₄	—	23	4	no	—
56	C₄⋊C₄	—	23	4	no	—
57	C₄⋊C₄	—	23	4	no	—
58	C₄⋊C₄	—	23	2	no	—
59	C₂²⋊C₄	—	23	4	no	—
60	C₂²⋊C₄	—	23	2	no	—
61	C₂²⋊C₄	C₂×D₂₃	1	4	yes	—
62	C₄²	—	23	2	no	—
63	C₄²	C₂×D₂₃	1	1	yes	characteristic
64	C₂³	C₂²×D₂₃	1	2	yes	—
65	Q₈	—	4	1	no	—
66	Q₈	—	92	2	no	—
67	D₄	—	4	1	no	—
68	C₂×C₄	—	23	4	no	—
69	C₂×C₄	—	23	4	no	—
70	C₂×C₄	—	92	2	no	—
71	C₂×C₄	C₂²×D₂₃	1	2	yes	—
72	C₂×C₄	C₂²×D₂₃	1	2	yes	—
73	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
74	C₂²	—	2	2	no	—
75	C₂²	—	2	4	no	—
76	C₂²	C₂³×D₂₃	1	1	yes	characteristic
77	C₄	—	2	1	no	—
78	C₄	—	2	2	no	—
79	C₄	—	2	2	no	—
80	C₄	—	46	4	no	—
81	C₄	—	46	4	no	—
82	C₂	—	4	2	no	—
83	C₂	#10625155	1	1	yes	characteristic
84	C₂	#10625161	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₃
commutator, socle, cosocle	C₂²×C₂₃	C₂⁴
Fitting	C₂₃×#124	C₂
prime radical	C₂×D₄×C₂₃	C₂²
hypercenter	#124	D₂₃
hypocenter	C₂₃	#563
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ #124

Complements

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

N	H ≅ G/N	type
2. #10625115	C₂	semidirect
4. #10625050	C₂	semidirect
5. #10625049	C₂	semidirect
6. #10625041	C₂	semidirect
49. C₂₃	#563	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#124	2
#10625115	2
#10625106	2
#10625050	2
#10625049	2
#10625041	2
#563	23