Subgroups — #13092384

Order 1472.

158 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#41×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#67267	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₂₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²⋊(C₂₃⋊C₈)	C₂	1	4	yes	maximal, complemented
5	#41×C₂₃	—	2	2	no	—
6	#41×C₂₃	—	2	2	no	—
7	C₂×Q₈×C₂₃	C₄	1	1	yes	characteristic
8	C₂×D₄×C₂₃	C₄	1	2	yes	—
9	C₂×D₄×C₂₃	C₄	1	1	yes	characteristic
10	C₂²×C₄×C₂₃	C₂²	1	2	yes	—
11	C₂²×C₄×C₂₃	C₂²	1	1	yes	characteristic
12	#67267	—	2	2	no	—
13	C₂×C₂₃⋊C₈	—	2	2	no	—
14	C₂×C₂₃⋊C₈	C₂²	1	2	yes	—
15	C₂×C₂₃⋊C₈	C₂²	1	2	yes	—
16	C₂³×C₂₃	C₂×C₄	1	2	yes	—
17	C₂³×C₂₃	C₂×C₄	1	1	yes	characteristic
18	Q₈×C₂₃	—	2	2	no	—
19	D₄×C₂₃	—	2	4	no	—
20	D₄×C₂₃	—	2	2	no	—
21	C₂×C₄×C₂₃	—	2	2	no	—
22	C₂×C₄×C₂₃	—	2	2	no	—
23	C₂×C₄×C₂₃	C₂×C₄	1	2	yes	—
24	C₂×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×C₂₃	D₄	1	2	yes	—
26	C₂×C₄×C₂₃	D₄	1	2	yes	—
27	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
28	C₂₃⋊C₈	—	2	2	no	—
29	C₂₃⋊C₈	—	2	2	no	—
30	C₂²×C₂₃	—	2	2	no	—
31	C₂²×C₂₃	—	2	2	no	—
32	C₂²×C₂₃	—	2	4	no	—
33	C₂²×C₂₃	C₂²⋊C₄	1	2	yes	—
34	C₂²×C₂₃	C₂²×C₄	1	1	yes	characteristic
35	C₄×C₂₃	—	2	2	no	—
36	C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
37	C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
38	C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
39	#408	—	23	1	no	maximal, Sylow
40	C₂×C₂₃	—	2	2	no	—
41	C₂×C₂₃	—	4	2	no	—
42	C₂×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₂₃	#131	1	2	yes	—
44	C₂×#41	D₂₃	1	1	yes	characteristic
45	C₂×M₁₆	—	23	1	no	—
46	C₂²×C₈	—	23	1	no	—
47	C₂²⋊C₈	—	23	4	no	—
48	C₂₃	#408	1	1	yes	characteristic, complemented, Sylow
49	#41	—	2	2	no	—
50	#41	—	2	2	no	—
51	C₂×Q₈	Dic₂₃	1	1	yes	characteristic
52	C₂×D₄	Dic₂₃	1	2	yes	—
53	C₂×D₄	Dic₂₃	1	1	yes	characteristic
54	C₂²×C₄	C₂×D₂₃	1	2	yes	—
55	C₂²×C₄	C₂×D₂₃	1	1	yes	characteristic
56	M₁₆	—	46	2	no	—
57	C₂×C₈	—	23	2	no	—
58	C₂×C₈	—	23	2	no	—
59	C₂×C₈	—	46	2	no	—
60	C₂³	C₂×Dic₂₃	1	2	yes	—
61	C₂³	C₂×Dic₂₃	1	1	yes	characteristic
62	Q₈	—	2	2	no	—
63	D₄	—	2	4	no	—
64	D₄	—	2	2	no	—
65	C₂×C₄	—	2	2	no	—
66	C₂×C₄	—	2	2	no	—
67	C₂×C₄	C₂×Dic₂₃	1	2	yes	—
68	C₂×C₄	C₂×Dic₂₃	1	1	yes	characteristic
69	C₂×C₄	C₂²⋊D₂₃	1	2	yes	—
70	C₂×C₄	C₂²⋊D₂₃	1	2	yes	—
71	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
72	C₈	—	46	2	no	—
73	C₈	—	46	2	no	—
74	C₂²	—	2	2	no	—
75	C₂²	—	2	4	no	—
76	C₂²	—	2	2	no	—
77	C₂²	C₂²⋊Dic₂₃	1	2	yes	—
78	C₂²	C₂²×Dic₂₃	1	1	yes	characteristic
79	C₄	—	2	2	no	—
80	C₄	C₂²⋊Dic₂₃	1	2	yes	—
81	C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
82	C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
83	C₂	—	2	2	no	—
84	C₂	—	4	2	no	—
85	C₂	#10625119	1	2	yes	—
86	C₂	C₂×C₂²⋊Dic₂₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₂₃
commutator, socle	C₂²×C₂₃	C₂²×C₄
Fitting, prime radical	C₂×#41×C₂₃	C₂
cosocle	C₂×C₄×C₂₃	C₂³
hypercenter	C₂×#41	D₂₃
hypocenter	C₂₃	#408
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#41

Complements

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

N	H ≅ G/N	type
2. C₂×#67267	C₂	semidirect
3. C₂²×C₂₃⋊C₈	C₂	semidirect
4. C₂²⋊(C₂₃⋊C₈)	C₂	semidirect
48. C₂₃	#408	semidirect

Maximal subgroups

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

subgroup	index
C₂×#41×C₂₃	2
C₂×#67267	2
C₂²×C₂₃⋊C₈	2
C₂²⋊(C₂₃⋊C₈)	2
#408	23