Subgroups — #13092622

Order 1472.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×D₄×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#67273	C₂	1	2	yes	maximal
3	#10625040	C₂	1	1	yes	maximal, characteristic, complemented
4	#10624992	C₂	1	2	yes	maximal, complemented
5	C₄⋊(C₂₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₄×C₂₃	C₂²	1	1	yes	characteristic
7	C₂²×C₄×C₂₃	—	2	1	no	—
8	C₂×Dic₄₆	C₂²	1	2	yes	—
9	C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
10	C₂²⋊C₄×C₂₃	—	2	1	no	—
11	C₄²×C₂₃	C₂²	1	1	yes	characteristic
12	#67273	—	2	4	no	—
13	C₄⋊Dic₂₃	—	2	1	no	—
14	C₄⋊Dic₂₃	—	2	1	no	—
15	C₂×C₂₃⋊C₈	C₂²	1	2	yes	—
16	C₂³×C₂₃	—	2	1	no	—
17	D₄×C₂₃	—	2	1	no	—
18	D₄×C₂₃	D₄	1	2	yes	—
19	C₂×C₄×C₂₃	—	2	1	no	—
20	C₂×C₄×C₂₃	—	4	2	no	—
21	C₂×C₄×C₂₃	D₄	1	1	yes	characteristic
22	C₂×C₄×C₂₃	D₄	1	1	yes	characteristic
23	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
24	C₂×Dic₂₃	—	2	2	no	—
25	Dic₄₆	—	2	2	no	—
26	Dic₄₆	—	2	2	no	—
27	C₂₃⋊C₈	—	2	2	no	—
28	C₂²×C₂₃	—	2	2	no	—
29	C₂²×C₂₃	—	4	1	no	—
30	C₂²×C₂₃	—	4	1	no	—
31	C₂²×C₂₃	C₂×D₄	1	1	yes	characteristic
32	C₄×C₂₃	—	2	1	no	—
33	C₄×C₂₃	—	4	1	no	—
34	C₄×C₂₃	SD₁₆	1	2	yes	complemented
35	C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
36	C₄×C₂₃	#41	1	1	yes	characteristic
37	Dic₂₃	—	4	2	no	—
38	#461	—	23	1	no	maximal, Sylow
39	C₂×C₂₃	—	4	2	no	—
40	C₂×C₂₃	#121	1	1	yes	characteristic
41	C₂×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
42	C₂×C₂₃	#137	1	1	yes	characteristic
43	C₂×SD₁₆	—	23	2	no	—
44	C₄⋊Q₈	—	23	1	no	—
45	C₄×D₄	D₂₃	1	1	yes	characteristic
46	C₄⋊C₈	—	23	1	no	—
47	#103	—	23	2	no	—
48	C₂₃	#461	1	1	yes	characteristic, complemented, Sylow
49	C₂×Q₈	—	23	2	no	—
50	C₂×D₄	C₂×D₂₃	1	1	yes	characteristic
51	C₂²×C₄	—	2	1	no	—
52	SD₁₆	—	46	4	no	—
53	C₂×C₈	—	23	2	no	—
54	C₄⋊C₄	—	46	1	no	—
55	C₄⋊C₄	—	46	1	no	—
56	C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
57	C₂²⋊C₄	—	2	1	no	—
58	C₄²	C₂×D₂₃	1	1	yes	characteristic
59	C₂³	—	2	1	no	—
60	Q₈	—	46	2	no	—
61	Q₈	—	46	2	no	—
62	D₄	—	2	1	no	—
63	D₄	D₉₂	1	2	yes	—
64	C₂×C₄	—	2	1	no	—
65	C₂×C₄	—	4	2	no	—
66	C₂×C₄	—	46	2	no	—
67	C₂×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
68	C₂×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
69	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
70	C₈	—	46	2	no	—
71	C₂²	—	2	2	no	—
72	C₂²	—	4	1	no	—
73	C₂²	—	4	1	no	—
74	C₂²	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
75	C₄	—	2	1	no	—
76	C₄	—	4	1	no	—
77	C₄	—	92	2	no	—
78	C₄	#67273	1	2	yes	complemented
79	C₄	C₂×D₉₂	1	1	yes	characteristic
80	C₄	#67288	1	1	yes	characteristic
81	C₂	—	4	2	no	—
82	C₂	#10625101	1	1	yes	characteristic
83	C₂	C₂×#67273	1	1	yes	characteristic
84	C₂	#10625122	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₂₃
commutator, cosocle	C₂×C₄×C₂₃	C₂³
Frattini	C₂×C₄	C₂²×D₂₃
Fitting	C₄×D₄×C₂₃	C₂
socle	C₂²×C₂₃	C₂×D₄
prime radical	C₂×D₄×C₂₃	C₂²
hypercenter	C₄×D₄	D₂₃
hypocenter	C₂₃	#461
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
3. #10625040	C₂	semidirect
4. #10624992	C₂	semidirect
5. C₄⋊(C₂₃⋊C₈)	C₂	semidirect
34. C₄×C₂₃	SD₁₆	semidirect
48. C₂₃	#461	semidirect
78. C₄	#67273	semidirect

Maximal subgroups

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

subgroup	index
C₄×D₄×C₂₃	2
C₂×#67273	2
#10625040	2
#10624992	2
C₄⋊(C₂₃⋊C₈)	2
#461	23