Subgroups — C₂₃×#2658 #15670277

Order 2944.

612 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×C₄⋊Q₈	C₂	1	3	yes	maximal, complemented
2	C₂×C₂₃×#124	C₂	1	3	yes	maximal
3	C₂₃×#399	C₂	1	8	yes	maximal, complemented
4	C₂₃×#376	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×Q₈×C₂₃	C₂²	1	3	yes	—
6	C₂²×D₄×C₂₃	C₂²	1	1	yes	characteristic
7	C₂₃×C₄⋊Q₈	—	2	12	no	—
8	C₂₃×#124	—	2	12	no	—
9	C₂×C₄⋊C₄×C₂₃	C₂²	1	12	yes	—
10	C₂×C₂²⋊C₄×C₂₃	C₂²	1	12	yes	—
11	C₂×C₄²×C₂₃	C₂²	1	3	yes	—
12	C₂₃×#95	C₂²	1	4	yes	—
13	C₂⁴×C₂₃	C₂³	1	2	yes	—
14	C₂×Q₈×C₂₃	—	2	6	no	—
15	C₂×Q₈×C₂₃	—	2	12	no	—
16	C₂×D₄×C₂₃	—	2	6	no	—
17	C₂²×C₄×C₂₃	C₂³	1	6	yes	—
18	C₂²×C₄×C₂₃	C₂³	1	6	yes	—
19	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
20	C₄⋊C₄×C₂₃	—	2	24	no	—
21	C₂²⋊C₄×C₂₃	—	2	24	no	—
22	C₄²×C₂₃	D₄	1	12	yes	—
23	C₂³×C₂₃	—	2	6	no	—
24	C₂³×C₂₃	—	2	8	no	—
25	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
26	Q₈×C₂₃	—	4	12	no	—
27	D₄×C₂₃	—	4	4	no	—
28	C₂×C₄×C₂₃	—	2	12	no	—
29	C₂×C₄×C₂₃	—	2	6	no	—
30	C₂×C₄×C₂₃	—	2	12	no	—
31	C₂×C₄×C₂₃	C₂×D₄	1	12	yes	—
32	C₂×C₄×C₂₃	C₂×D₄	1	6	yes	—
33	#2658	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
34	C₂²×C₂₃	—	4	2	no	—
35	C₂²×C₂₃	—	4	12	no	—
36	C₂²×C₂₃	C₂²×D₄	1	3	yes	—
37	C₂²×C₂₃	2₋¹⁺⁴	1	4	yes	—
38	C₄×C₂₃	—	2	12	no	—
39	C₄×C₂₃	—	4	6	no	—
40	C₄×C₂₃	#127	1	4	yes	—
41	C₂×C₄⋊Q₈	C₂×C₂₃	1	3	yes	complemented
42	C₂×#124	C₂×C₂₃	1	3	yes	—
43	#399	C₂×C₂₃	1	8	yes	complemented
44	#376	C₂×C₂₃	1	1	yes	characteristic, complemented
45	C₂×C₂₃	—	8	2	no	—
46	C₂×C₂₃	C₂×#127	1	1	yes	characteristic
47	C₂×C₂₃	#536	1	6	yes	—
48	C₂²×Q₈	C₂²×C₂₃	1	3	yes	—
49	C₂²×D₄	C₂²×C₂₃	1	1	yes	characteristic
50	C₄⋊Q₈	—	2	12	no	—
51	#124	—	2	12	no	—
52	C₂×C₄⋊C₄	C₂²×C₂₃	1	12	yes	—
53	C₂×C₂²⋊C₄	C₂²×C₂₃	1	12	yes	—
54	C₂×C₄²	C₂²×C₂₃	1	3	yes	—
55	#95	C₂²×C₂₃	1	4	yes	—
56	C₂₃	#2658	1	1	yes	characteristic, direct product factor, Sylow
57	C₂⁴	C₂³×C₂₃	1	2	yes	—
58	C₂×Q₈	—	2	6	no	—
59	C₂×Q₈	—	2	12	no	—
60	C₂×D₄	—	2	6	no	—
61	C₂²×C₄	C₂³×C₂₃	1	6	yes	—
62	C₂²×C₄	C₂³×C₂₃	1	6	yes	—
63	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
64	C₄⋊C₄	—	2	24	no	—
65	C₂²⋊C₄	—	2	24	no	—
66	C₄²	D₄×C₂₃	1	12	yes	—
67	C₂³	—	2	6	no	—
68	C₂³	—	2	8	no	—
69	C₂³	C₂⁴×C₂₃	1	1	yes	characteristic
70	Q₈	—	4	12	no	—
71	D₄	—	4	4	no	—
72	C₂×C₄	—	2	12	no	—
73	C₂×C₄	—	2	6	no	—
74	C₂×C₄	—	2	12	no	—
75	C₂×C₄	C₂×D₄×C₂₃	1	12	yes	—
76	C₂×C₄	C₂×D₄×C₂₃	1	6	yes	—
77	C₂²	—	4	2	no	—
78	C₂²	—	4	12	no	—
79	C₂²	C₂²×D₄×C₂₃	1	3	yes	—
80	C₂²	C₂₃×2₋¹⁺⁴	1	4	yes	—
81	C₄	—	2	12	no	—
82	C₄	—	4	6	no	—
83	C₄	C₂₃×#127	1	4	yes	—
84	C₂	—	8	2	no	—
85	C₂	C₂×C₂₃×#127	1	1	yes	characteristic
86	C₂	C₂₃×#536	1	6	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₂₃
prime radical	C₂²×D₄×C₂₃	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×C₄⋊Q₈	C₂	semidirect
3. C₂₃×#399	C₂	semidirect
4. C₂₃×#376	C₂	semidirect
33. #2658	C₂₃	direct (N × H)
41. C₂×C₄⋊Q₈	C₂×C₂₃	semidirect
43. #399	C₂×C₂₃	semidirect
44. #376	C₂×C₂₃	semidirect
56. C₂₃	#2658	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₂₃×C₄⋊Q₈	2
C₂×C₂₃×#124	2
C₂₃×#399	2
C₂₃×#376	2
#2658	23

Subgroups — C23×#2658 #15670277

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2658 #15670277