Subgroups — #13093022

Order 1472.

240 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#117	C₂	1	1	yes	maximal, characteristic, complemented
2	#10625101	C₂	1	2	yes	maximal, complemented
3	#10625065	C₂	1	4	yes	maximal, complemented
4	#10625055	C₂	1	4	yes	maximal, complemented
5	#10625045	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×D₉₂	C₂	1	2	yes	maximal, complemented
7	#10625041	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₄×C₂₃	C₂²	1	1	yes	characteristic
9	C₂×C₂²⋊D₂₃	C₂²	1	4	yes	—
10	C₂×D₉₂	C₂²	1	2	yes	—
11	C₂×D₉₂	C₂²	1	4	yes	—
12	C₂×C₄×D₂₃	C₂²	1	4	yes	—
13	C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
14	C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
15	C₄²×C₂₃	C₂²	1	1	yes	characteristic
16	C₄²×C₂₃	C₂²	1	1	yes	characteristic
17	(C₂×D₂₃)⋊C₄	C₂²	1	4	yes	—
18	(C₂×D₂₃)⋊C₄	C₂²	1	4	yes	—
19	C₄⋊Dic₂₃	C₂²	1	2	yes	—
20	Dic₂₃⋊C₄	C₂²	1	4	yes	—
21	C₂³×C₂₃	C₂³	1	1	yes	characteristic
22	C₂²×D₂₃	C₂³	1	4	yes	—
23	C₂×C₄×C₂₃	—	2	2	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×C₄×C₂₃	C₂³	1	1	yes	characteristic
28	C₂²⋊D₂₃	—	4	4	no	—
29	C₂×Dic₂₃	C₂³	1	4	yes	—
30	D₉₂	—	2	4	no	—
31	D₉₂	—	4	4	no	—
32	C₄×D₂₃	—	4	4	no	—
33	C₂²×C₂₃	—	2	1	no	—
34	C₂²×C₂₃	—	2	2	no	—
35	C₂²×C₂₃	C₂⁴	1	1	yes	characteristic
36	C₂×D₂₃	—	2	4	no	—
37	C₂×D₂₃	—	2	4	no	—
38	C₂×D₂₃	—	2	4	no	—
39	C₄×C₂₃	—	2	1	no	—
40	C₄×C₂₃	—	2	2	no	—
41	C₄×C₂₃	—	2	2	no	—
42	C₄×C₂₃	#41	1	2	yes	—
43	Dic₂₃	—	2	4	no	—
44	#540	—	23	1	no	maximal, Sylow
45	C₂×C₂₃	—	4	1	no	—
46	C₂×C₂₃	C₂×#41	1	1	yes	characteristic
47	C₂×C₂₃	H(5,𝔽₂)	1	2	yes	—
48	D₂₃	—	4	4	no	—
49	#127	—	23	1	no	—
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	D₂₃	1	1	yes	characteristic, complemented
56	C₂₃	#540	1	1	yes	characteristic, complemented, Sylow
57	C₂×D₄	—	23	4	no	—
58	C₂×D₄	—	23	2	no	—
59	C₂×D₄	—	23	4	no	—
60	C₂²×C₄	—	23	4	no	—
61	C₂²×C₄	C₂×D₂₃	1	1	yes	characteristic
62	C₄⋊C₄	—	23	2	no	—
63	C₄⋊C₄	—	23	4	no	—
64	C₄⋊C₄	C₂×D₂₃	1	2	yes	—
65	C₂²⋊C₄	—	23	4	no	—
66	C₂²⋊C₄	—	23	4	no	—
67	C₂²⋊C₄	C₂×D₂₃	1	2	yes	—
68	C₄²	C₂×D₂₃	1	1	yes	characteristic
69	C₄²	C₂×D₂₃	1	1	yes	characteristic
70	C₂³	—	23	4	no	—
71	C₂³	C₂²×D₂₃	1	1	yes	characteristic
72	D₄	—	46	4	no	—
73	D₄	—	92	4	no	—
74	D₄	—	92	4	no	—
75	C₂×C₄	—	2	2	no	—
76	C₂×C₄	—	23	4	no	—
77	C₂×C₄	—	92	4	no	—
78	C₂×C₄	C₂²×D₂₃	1	2	yes	—
79	C₂×C₄	C₂²×D₂₃	1	2	yes	—
80	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
81	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
82	C₂²	—	2	1	no	—
83	C₂²	—	2	2	no	—
84	C₂²	—	46	4	no	—
85	C₂²	—	46	4	no	—
86	C₂²	—	46	4	no	—
87	C₂²	C₂³×D₂₃	1	1	yes	characteristic
88	C₄	—	2	1	no	—
89	C₄	—	2	2	no	—
90	C₄	—	2	2	no	—
91	C₄	—	46	4	no	—
92	C₄	#67288	1	2	yes	—
93	C₂	—	4	1	no	—
94	C₂	—	92	4	no	—
95	C₂	C₂×#67288	1	1	yes	characteristic
96	C₂	#10625160	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₃
commutator, socle, cosocle	C₂²×C₂₃	C₂⁴
Fitting	C₂₃×#117	C₂
hypercenter	#117	D₂₃
hypocenter	C₂₃	#540
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ #117

Complements

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

N	H ≅ G/N	type
1. C₂₃×#117	C₂	semidirect
2. #10625101	C₂	semidirect
3. #10625065	C₂	semidirect
4. #10625055	C₂	semidirect
5. #10625045	C₂	semidirect
6. C₄×D₉₂	C₂	semidirect
7. #10625041	C₂	semidirect
55. #117	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×#117	2
#10625101	2
#10625065	2
#10625055	2
#10625045	2
C₄×D₉₂	2
#10625041	2
#540	23