Subgroups — #13093021

Order 1472.

240 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, complemented
2	#10625110	C₂	1	4	yes	maximal, complemented
3	#10625109	C₂	1	2	yes	maximal, complemented
4	#10625106	C₂	1	4	yes	maximal, complemented
5	D₄×Dic₂₃	C₂	1	2	yes	maximal, complemented
6	#10625043	C₂	1	1	yes	maximal, characteristic, complemented
7	#10625041	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₄×C₂₃	C₂²	1	2	yes	—
9	C₂×D₄×C₂₃	C₂²	1	4	yes	—
10	C₂×C₂²⋊D₂₃	C₂²	1	4	yes	—
11	C₂²×Dic₂₃	C₂²	1	4	yes	—
12	C₂×C₄×D₂₃	C₂²	1	1	yes	characteristic
13	C₄²×C₂₃	C₂²	1	1	yes	characteristic
14	C₂²⋊Dic₂₃	C₂²	1	4	yes	—
15	C₂²⋊Dic₂₃	C₂²	1	4	yes	—
16	(C₂×D₂₃)⋊C₄	C₂²	1	2	yes	—
17	C₄⋊Dic₂₃	C₂²	1	2	yes	—
18	Dic₂₃⋊C₄	C₂²	1	4	yes	—
19	Dic₂₃⋊C₄	C₂²	1	2	yes	—
20	C₄×Dic₂₃	C₂²	1	1	yes	characteristic
21	C₂³×C₂₃	C₂³	1	4	yes	—
22	C₂²×D₂₃	C₂³	1	1	yes	characteristic
23	D₄×C₂₃	—	2	4	no	—
24	D₄×C₂₃	—	4	4	no	—
25	C₂×C₄×C₂₃	C₂³	1	2	yes	—
26	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
27	C₂²⋊D₂₃	—	4	4	no	—
28	C₂×Dic₂₃	—	4	4	no	—
29	C₂×Dic₂₃	C₂³	1	2	yes	—
30	C₂×Dic₂₃	C₂³	1	4	yes	—
31	C₂×Dic₂₃	C₂³	1	1	yes	characteristic
32	C₄×D₂₃	—	2	2	no	—
33	C₂²×C₂₃	—	2	4	no	—
34	C₂²×C₂₃	—	2	4	no	—
35	C₂²×C₂₃	—	2	4	no	—
36	C₂²×C₂₃	C₂⁴	1	1	yes	characteristic
37	C₂×D₂₃	—	2	1	no	—
38	C₂×D₂₃	—	2	2	no	—
39	C₄×C₂₃	—	2	2	no	—
40	C₄×C₂₃	#41	1	2	yes	—
41	Dic₂₃	—	2	1	no	—
42	Dic₂₃	—	2	2	no	—
43	Dic₂₃	—	2	4	no	—
44	#540	—	23	1	no	maximal, Sylow
45	C₂×C₂₃	—	4	4	no	—
46	C₂×C₂₃	C₂×#41	1	1	yes	characteristic
47	C₂×C₂₃	H(5,𝔽₂)	1	2	yes	—
48	D₂₃	—	4	1	no	—
49	#127	D₂₃	1	1	yes	characteristic, complemented
50	#125	—	23	1	no	—
51	#123	—	23	4	no	—
52	#121	—	23	2	no	—
53	#121	—	23	4	no	—
54	C₄×D₄	—	23	2	no	—
55	#117	—	23	1	no	—
56	C₂₃	#540	1	1	yes	characteristic, complemented, Sylow
57	C₂×D₄	—	23	4	no	—
58	C₂×D₄	C₂×D₂₃	1	2	yes	—
59	C₂×D₄	C₂×D₂₃	1	4	yes	—
60	C₂²×C₄	—	23	4	no	—
61	C₂²×C₄	—	23	1	no	—
62	C₄⋊C₄	—	23	2	no	—
63	C₄⋊C₄	—	23	2	no	—
64	C₄⋊C₄	—	23	4	no	—
65	C₂²⋊C₄	—	23	4	no	—
66	C₂²⋊C₄	—	23	4	no	—
67	C₂²⋊C₄	—	23	2	no	—
68	C₄²	—	23	1	no	—
69	C₄²	C₂×D₂₃	1	1	yes	characteristic
70	C₂³	—	23	1	no	—
71	C₂³	C₂²×D₂₃	1	4	yes	—
72	D₄	—	2	4	no	—
73	D₄	—	4	4	no	—
74	D₄	—	92	4	no	—
75	C₂×C₄	—	23	2	no	—
76	C₂×C₄	—	23	1	no	—
77	C₂×C₄	—	23	4	no	—
78	C₂×C₄	—	46	2	no	—
79	C₂×C₄	—	92	4	no	—
80	C₂×C₄	C₂²×D₂₃	1	2	yes	—
81	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
82	C₂²	—	2	4	no	—
83	C₂²	—	2	4	no	—
84	C₂²	—	2	4	no	—
85	C₂²	—	46	1	no	—
86	C₂²	—	46	2	no	—
87	C₂²	C₂³×D₂₃	1	1	yes	characteristic
88	C₄	—	2	2	no	—
89	C₄	—	46	1	no	—
90	C₄	—	46	2	no	—
91	C₄	—	46	4	no	—
92	C₄	#67290	1	2	yes	—
93	C₂	—	4	4	no	—
94	C₂	—	92	1	no	—
95	C₂	C₂×#67290	1	1	yes	characteristic
96	C₂	#10625155	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₃
commutator, socle, cosocle	C₂²×C₂₃	C₂⁴
Fitting	C₂₃×#127	C₂
hypercenter	#127	D₂₃
hypocenter	C₂₃	#540
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
1. C₂₃×#127	C₂	semidirect
2. #10625110	C₂	semidirect
3. #10625109	C₂	semidirect
4. #10625106	C₂	semidirect
5. D₄×Dic₂₃	C₂	semidirect
6. #10625043	C₂	semidirect
7. #10625041	C₂	semidirect
49. #127	D₂₃	semidirect
56. C₂₃	#540	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
#10625110	2
#10625109	2
#10625106	2
D₄×Dic₂₃	2
#10625043	2
#10625041	2
#540	23