Subgroups — #15672123

Order 2944.

228 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13092474	C₂	1	2	yes	maximal
2	#13092166	C₂	1	4	yes	maximal, complemented
3	C₂×C₂₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
4	C₂²×Q₈×C₂₃	C₄	1	2	yes	—
5	C₂₃×C₄⋊Q₈	—	2	2	no	—
6	C₂₃×C₄⋊Q₈	C₄	1	2	yes	—
7	C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
8	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
9	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
10	C₂²⋊(C₂₃⋊C₈)	—	2	2	no	—
11	C₄⋊(C₂₃⋊C₈)	C₂²	1	4	yes	—
12	#10624985	—	2	2	no	—
13	C₂×Q₈×C₂₃	—	2	2	no	—
14	C₂×Q₈×C₂₃	—	4	4	no	—
15	C₂×Q₈×C₂₃	C₂×C₄	1	4	yes	—
16	C₂²×C₄×C₂₃	—	2	2	no	—
17	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₂₃	D₄	1	2	yes	—
19	C₄⋊C₄×C₂₃	—	2	4	no	—
20	C₄⋊C₄×C₂₃	—	2	4	no	—
21	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
22	C₄²×C₂₃	D₄	1	2	yes	—
23	C₄²×C₂₃	C₂³	1	1	yes	characteristic
24	C₂×C₂₃⋊C₈	—	2	4	no	—
25	C₂³×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
26	Q₈×C₂₃	—	4	4	no	—
27	Q₈×C₂₃	—	4	4	no	—
28	C₂×C₄×C₂₃	—	2	2	no	—
29	C₂×C₄×C₂₃	—	2	4	no	—
30	C₂×C₄×C₂₃	—	2	4	no	—
31	C₂×C₄×C₂₃	—	4	4	no	—
32	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
33	C₂×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
36	C₂₃⋊C₈	—	4	4	no	—
37	#1558	—	23	1	no	maximal, Sylow
38	C₂²×C₂₃	—	2	2	no	—
39	C₂²×C₂₃	#100	1	2	yes	—
40	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
41	C₄×C₂₃	—	2	1	no	—
42	C₄×C₂₃	—	2	1	no	—
43	C₄×C₂₃	—	2	2	no	—
44	C₄×C₂₃	—	4	4	no	—
45	C₄×C₂₃	#137	1	4	yes	—
46	C₂×C₄⋊Q₈	D₂₃	1	1	yes	characteristic
47	#432	—	23	2	no	—
48	#333	—	23	4	no	—
49	C₂×C₂₃	—	2	2	no	—
50	C₂×C₂₃	C₂×#100	1	1	yes	characteristic
51	C₂×C₂₃	#419	1	2	yes	—
52	C₂²×Q₈	Dic₂₃	1	2	yes	—
53	C₄⋊Q₈	—	2	2	no	—
54	C₄⋊Q₈	Dic₂₃	1	2	yes	—
55	C₄⋊Q₈	C₂×D₂₃	1	2	yes	—
56	C₂×C₄⋊C₄	—	2	2	no	—
57	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
58	C₄⋊C₈	—	23	4	no	—
59	C₂²⋊C₈	—	46	2	no	—
60	#97	—	46	2	no	—
61	C₂₃	#1558	1	1	yes	characteristic, complemented, Sylow
62	C₂×Q₈	—	2	2	no	—
63	C₂×Q₈	—	4	4	no	—
64	C₂×Q₈	C₂×Dic₂₃	1	4	yes	—
65	C₂²×C₄	—	2	2	no	—
66	C₂²×C₄	C₂×Dic₂₃	1	1	yes	characteristic
67	C₂²×C₄	C₂²⋊D₂₃	1	2	yes	—
68	C₂×C₈	—	46	4	no	—
69	C₄⋊C₄	—	2	4	no	—
70	C₄⋊C₄	—	2	4	no	—
71	C₄²	C₂×Dic₂₃	1	1	yes	characteristic
72	C₄²	C₂²⋊D₂₃	1	2	yes	—
73	C₄²	C₂²×D₂₃	1	1	yes	characteristic
74	C₂³	C₂²⋊Dic₂₃	1	1	yes	characteristic
75	Q₈	—	4	4	no	—
76	Q₈	—	4	4	no	—
77	C₂×C₄	—	2	2	no	—
78	C₂×C₄	—	2	4	no	—
79	C₂×C₄	—	2	4	no	—
80	C₂×C₄	—	4	4	no	—
81	C₂×C₄	C₂²⋊Dic₂₃	1	2	yes	—
82	C₂×C₄	C₂²⋊Dic₂₃	1	1	yes	characteristic
83	C₂×C₄	C₂²×Dic₂₃	1	1	yes	characteristic
84	C₂×C₄	C₂×C₂²⋊D₂₃	1	2	yes	—
85	C₈	—	92	4	no	—
86	C₂²	—	2	2	no	—
87	C₂²	#10625015	1	2	yes	—
88	C₂²	C₂×C₂²⋊Dic₂₃	1	1	yes	characteristic
89	C₄	—	2	1	no	—
90	C₄	—	2	1	no	—
91	C₄	—	2	2	no	—
92	C₄	—	4	4	no	—
93	C₄	#10625113	1	4	yes	—
94	C₂	—	2	2	no	—
95	C₂	C₂×#10625015	1	1	yes	characteristic
96	C₂	#13092426	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₂₃
commutator	C₂×C₄×C₂₃	C₂²×C₄
Frattini	C₄²	C₂²×D₂₃
Fitting	C₂×C₂₃×C₄⋊Q₈	C₂
socle	C₂²×C₂₃	C₂×C₂²⋊C₄
prime radical	C₂³×C₂₃	C₂²⋊C₄
cosocle	C₄²×C₂₃	C₂³
hypercenter	C₂×C₄⋊Q₈	D₂₃
hypocenter	C₂₃	#1558
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
2. #13092166	C₂	semidirect
61. C₂₃	#1558	semidirect

Maximal subgroups

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

subgroup	index
#13092474	2
#13092166	2
C₂×C₂₃×C₄⋊Q₈	2
#1558	23