Subgroups — #13093153

Order 1472.

236 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#127	C₂	1	1	yes	maximal, characteristic
2	#10625107	C₂	1	4	yes	maximal, complemented
3	#10625106	C₂	1	4	yes	maximal, complemented
4	D₄×Dic₂₃	C₂	1	4	yes	maximal, complemented
5	C₄×Dic₄₆	C₂	1	2	yes	maximal, complemented
6	C₂×D₄×C₂₃	C₂²	1	4	yes	—
7	C₂×D₄×C₂₃	C₂²	1	2	yes	—
8	C₂²×Dic₂₃	C₂²	1	4	yes	—
9	C₂×Dic₄₆	C₂²	1	2	yes	—
10	C₄²×C₂₃	C₂²	1	1	yes	characteristic
11	C₂²⋊Dic₂₃	C₂²	1	8	yes	—
12	C₂²⋊Dic₂₃	C₂²	1	4	yes	—
13	C₄⋊Dic₂₃	C₂²	1	2	yes	—
14	Dic₂₃⋊C₄	C₂²	1	4	yes	—
15	C₄×Dic₂₃	C₂²	1	4	yes	—
16	C₂³×C₂₃	C₂³	1	4	yes	—
17	D₄×C₂₃	—	2	8	no	—
18	D₄×C₂₃	—	4	2	no	—
19	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
20	C₂×C₄×C₂₃	C₂³	1	2	yes	—
21	C₂×Dic₂₃	—	4	4	no	—
22	C₂×Dic₂₃	C₂³	1	4	yes	—
23	C₂×Dic₂₃	C₂³	1	4	yes	—
24	Dic₄₆	—	2	4	no	—
25	C₂²×C₂₃	—	2	8	no	—
26	C₂²×C₂₃	—	2	4	no	—
27	C₂²×C₂₃	C₂⁴	1	1	yes	characteristic
28	C₄×C₂₃	—	2	1	no	—
29	C₄×C₂₃	#41	1	4	yes	—
30	Dic₂₃	—	2	4	no	—
31	Dic₂₃	—	2	4	no	—
32	#559	—	23	1	no	maximal, Sylow
33	C₂×C₂₃	—	4	4	no	—
34	C₂×C₂₃	C₂×#41	1	2	yes	—
35	C₂×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
36	#127	D₂₃	1	1	yes	characteristic
37	#124	—	23	4	no	—
38	#123	—	23	4	no	—
39	C₄×Q₈	—	23	2	no	—
40	C₄×D₄	—	23	4	no	—
41	C₂₃	#559	1	1	yes	characteristic, complemented, Sylow
42	C₂×Q₈	—	23	2	no	—
43	C₂×D₄	C₂×D₂₃	1	4	yes	—
44	C₂×D₄	C₂×D₂₃	1	2	yes	—
45	C₂²×C₄	—	23	4	no	—
46	C₄⋊C₄	—	23	2	no	—
47	C₄⋊C₄	—	23	4	no	—
48	C₂²⋊C₄	—	23	8	no	—
49	C₂²⋊C₄	—	23	4	no	—
50	C₄²	—	23	4	no	—
51	C₄²	C₂×D₂₃	1	1	yes	characteristic
52	C₂³	C₂²×D₂₃	1	4	yes	—
53	Q₈	—	46	4	no	—
54	D₄	—	2	8	no	—
55	D₄	—	4	2	no	—
56	C₂×C₄	—	23	4	no	—
57	C₂×C₄	—	23	4	no	—
58	C₂×C₄	—	92	4	no	—
59	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
60	C₂×C₄	C₂²×D₂₃	1	2	yes	—
61	C₂²	—	2	8	no	—
62	C₂²	—	2	4	no	—
63	C₂²	C₂³×D₂₃	1	1	yes	characteristic
64	C₄	—	2	1	no	—
65	C₄	—	46	4	no	—
66	C₄	—	46	4	no	—
67	C₄	#67290	1	4	yes	—
68	C₂	—	4	4	no	—
69	C₂	C₂×#67290	1	2	yes	—
70	C₂	#10625155	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₃
commutator, socle, cosocle	C₂²×C₂₃	C₂⁴
Fitting, prime radical	C₂₃×#127	C₂
hypercenter	#127	D₂₃
hypocenter	C₂₃	#559
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ #127

Complements

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

N	H ≅ G/N	type
2. #10625107	C₂	semidirect
3. #10625106	C₂	semidirect
4. D₄×Dic₂₃	C₂	semidirect
5. C₄×Dic₄₆	C₂	semidirect
41. C₂₃	#559	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#127	2
#10625107	2
#10625106	2
D₄×Dic₂₃	2
C₄×Dic₄₆	2
#559	23