Subgroups — #13092899

Order 1472.

294 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#67292	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₈×Dic₂₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#10625064	C₂	1	3	yes	maximal, complemented
5	C₄⋊C₄×D₂₃	C₂	1	3	yes	maximal, complemented
6	C₄×D₉₂	C₂	1	3	yes	maximal, complemented
7	#10625043	C₂	1	3	yes	maximal, complemented
8	C₂×Q₈×C₂₃	C₂²	1	1	yes	characteristic
9	#67292	C₄	1	8	yes	complemented
10	C₂×D₉₂	C₂²	1	3	yes	—
11	C₂×C₄×D₂₃	C₂²	1	6	yes	—
12	C₂×C₄×D₂₃	C₂²	1	3	yes	—
13	C₄⋊C₄×C₂₃	C₂²	1	3	yes	—
14	C₄²×C₂₃	C₂²	1	3	yes	—
15	(C₂×D₂₃)⋊C₄	C₂²	1	3	yes	—
16	(C₂×D₂₃)⋊C₄	C₂²	1	3	yes	—
17	C₄⋊Dic₂₃	C₂²	1	3	yes	—
18	Dic₂₃⋊C₄	C₂²	1	1	yes	characteristic
19	Dic₂₃⋊C₄	C₂²	1	3	yes	—
20	C₄×Dic₂₃	C₂²	1	3	yes	—
21	C₂²×D₂₃	C₂³	1	3	yes	—
22	Q₈×C₂₃	C₂×C₄	1	4	yes	complemented
23	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
24	C₂×C₄×C₂₃	C₂³	1	3	yes	—
25	C₂×C₄×C₂₃	C₂³	1	3	yes	—
26	C₂×Dic₂₃	C₂³	1	1	yes	characteristic
27	C₂×Dic₂₃	C₂³	1	3	yes	—
28	C₂×Dic₂₃	C₂³	1	1	yes	characteristic
29	D₉₂	C₂×C₄	1	12	yes	complemented
30	C₄×D₂₃	—	4	6	no	—
31	C₄×D₂₃	C₂×C₄	1	12	yes	complemented
32	C₂²×C₂₃	C₂⁴	1	1	yes	characteristic
33	C₂×D₂₃	—	2	3	no	—
34	C₂×D₂₃	—	2	3	no	—
35	C₂×D₂₃	C₂²×C₄	1	6	yes	—
36	C₄×C₂₃	—	2	3	no	—
37	C₄×C₂₃	—	2	1	no	—
38	C₄×C₂₃	C₂²×C₄	1	6	yes	—
39	Dic₂₃	—	2	3	no	—
40	Dic₂₃	—	2	1	no	—
41	Dic₂₃	C₂²×C₄	1	2	yes	—
42	#520	—	23	1	no	maximal, Sylow
43	C₂×C₂₃	C₂³×C₄	1	1	yes	characteristic
44	C₂×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
45	C₂×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
46	D₂₃	—	2	6	no	—
47	C₂×#41	—	23	1	no	—
48	C₄×Q₈	—	23	1	no	—
49	C₄×Q₈	D₂₃	1	1	yes	characteristic, complemented
50	C₄×D₄	—	23	3	no	—
51	C₄×D₄	—	23	3	no	—
52	#117	—	23	3	no	—
53	C₂×C₄⋊C₄	—	23	3	no	—
54	C₂₃	#520	1	1	yes	characteristic, complemented, Sylow
55	#41	—	23	8	no	—
56	C₂×Q₈	C₂×D₂₃	1	1	yes	characteristic
57	C₂×D₄	—	23	3	no	—
58	C₂²×C₄	—	23	3	no	—
59	C₂²×C₄	—	23	6	no	—
60	C₄⋊C₄	—	23	3	no	—
61	C₄⋊C₄	—	23	3	no	—
62	C₄⋊C₄	—	23	1	no	—
63	C₄⋊C₄	C₂×D₂₃	1	3	yes	—
64	C₂²⋊C₄	—	23	3	no	—
65	C₂²⋊C₄	—	23	3	no	—
66	C₄²	—	23	3	no	—
67	C₄²	C₂×D₂₃	1	3	yes	—
68	C₂³	—	23	3	no	—
69	Q₈	C₄×D₂₃	1	4	yes	complemented
70	D₄	—	23	12	no	—
71	C₂×C₄	—	23	1	no	—
72	C₂×C₄	—	23	12	no	—
73	C₂×C₄	—	23	1	no	—
74	C₂×C₄	—	23	3	no	—
75	C₂×C₄	—	92	6	no	—
76	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
77	C₂×C₄	C₂²×D₂₃	1	3	yes	—
78	C₂×C₄	C₂²×D₂₃	1	3	yes	—
79	C₂²	—	23	6	no	—
80	C₂²	—	46	3	no	—
81	C₂²	—	46	3	no	—
82	C₂²	C₂³×D₂₃	1	1	yes	characteristic
83	C₄	—	2	1	no	—
84	C₄	—	2	3	no	—
85	C₄	—	23	2	no	—
86	C₄	—	46	1	no	—
87	C₄	—	46	3	no	—
88	C₄	C₂×C₄×D₂₃	1	6	yes	—
89	C₂	—	46	6	no	—
90	C₂	C₂²×C₄×D₂₃	1	1	yes	characteristic
91	C₂	#10625158	1	1	yes	characteristic
92	C₂	#10625160	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₃
commutator	C₂×C₂₃	C₂³×C₄
Fitting	C₄×Q₈×C₂₃	C₂
socle, cosocle	C₂²×C₂₃	C₂⁴
prime radical	C₂×#67292	C₂
hypercenter	C₄×Q₈	D₂₃
hypocenter	C₂₃	#520
LCS	G ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
2. C₄×Q₈×C₂₃	C₂	semidirect
3. Q₈×Dic₂₃	C₂	semidirect
4. #10625064	C₂	semidirect
5. C₄⋊C₄×D₂₃	C₂	semidirect
6. C₄×D₉₂	C₂	semidirect
7. #10625043	C₂	semidirect
9. #67292	C₄	semidirect
22. Q₈×C₂₃	C₂×C₄	semidirect
29. D₉₂	C₂×C₄	semidirect
31. C₄×D₂₃	C₂×C₄	semidirect
49. C₄×Q₈	D₂₃	semidirect
54. C₂₃	#520	semidirect
69. Q₈	C₄×D₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#67292	2
C₄×Q₈×C₂₃	2
Q₈×Dic₂₃	2
#10625064	2
C₄⋊C₄×D₂₃	2
C₄×D₉₂	2
#10625043	2
#520	23