Subgroups — #15671563

Order 2944.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13092791	C₂	1	1	yes	maximal, characteristic
2	#13092216	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#356	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
5	C₂×#67274	—	2	1	no	—
6	C₂×#67274	—	2	1	no	—
7	#10625045	C₄	1	1	yes	characteristic
8	C₂₃×#101	—	2	1	no	—
9	#10625018	—	2	1	no	—
10	C₄×C₂₃⋊C₈	C₄	1	1	yes	characteristic
11	C₂×Q₈×C₂₃	D₄	1	1	yes	characteristic
12	C₂×Q₈×C₂₃	D₄	1	1	yes	characteristic
13	C₂×D₉₂	—	2	1	no	—
14	C₂×D₉₂	—	4	1	no	—
15	M₁₆×C₂₃	—	4	1	no	—
16	C₄⋊C₄×C₂₃	—	4	1	no	—
17	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
18	#67274	—	4	2	no	—
19	#67274	—	4	2	no	—
20	#67267	—	4	1	no	—
21	C₂×C₂₃⋊C₈	—	2	1	no	—
22	C₂²×D₂₃	—	4	1	no	—
23	Q₈×C₂₃	—	4	1	no	—
24	Q₈×C₂₃	—	4	1	no	—
25	C₂×C₄×C₂₃	—	2	1	no	—
26	C₂×C₄×C₂₃	—	2	1	no	—
27	C₂×C₄×C₂₃	—	2	1	no	—
28	C₂×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
29	D₉₂	—	4	1	no	—
30	D₉₂	—	4	1	no	—
31	D₉₂	—	8	2	no	—
32	C₈×C₂₃	—	8	1	no	—
33	C₂₃⋊C₈	—	2	2	no	—
34	C₂₃⋊C₈	—	8	1	no	—
35	#1410	—	23	1	no	maximal, Sylow
36	C₂²×C₂₃	#99	1	1	yes	characteristic
37	C₂×D₂₃	—	8	2	no	—
38	C₂×D₂₃	—	8	1	no	—
39	C₄×C₂₃	—	2	2	no	—
40	C₄×C₂₃	—	2	1	no	—
41	C₄×C₂₃	—	4	1	no	—
42	C₄×C₂₃	—	4	1	no	—
43	#492	—	23	1	no	—
44	#356	—	23	1	no	—
45	#356	D₂₃	1	1	yes	characteristic, complemented
46	C₂×C₂₃	—	2	1	no	—
47	C₂×C₂₃	#353	1	1	yes	characteristic
48	D₂₃	—	16	1	no	—
49	C₂×SD₁₆	—	46	1	no	—
50	C₂×SD₁₆	—	46	1	no	—
51	C₄⋊Q₈	C₂×D₂₃	1	1	yes	characteristic
52	#127	—	23	1	no	—
53	#101	—	2	1	no	—
54	#101	—	46	1	no	—
55	C₄×C₈	—	23	1	no	—
56	C₂₃	#1410	1	1	yes	characteristic, complemented, Sylow
57	C₂×Q₈	D₉₂	1	1	yes	characteristic
58	C₂×Q₈	C₂²⋊D₂₃	1	1	yes	characteristic
59	C₂×D₄	—	46	1	no	—
60	C₂×D₄	—	92	1	no	—
61	SD₁₆	—	92	2	no	—
62	SD₁₆	—	92	2	no	—
63	M₁₆	—	4	1	no	—
64	M₁₆	—	92	1	no	—
65	C₂×C₈	—	46	1	no	—
66	C₄⋊C₄	—	4	1	no	—
67	C₄²	C₄×D₂₃	1	1	yes	characteristic
68	C₂³	—	92	1	no	—
69	Q₈	—	4	1	no	—
70	Q₈	—	4	1	no	—
71	D₄	—	92	1	no	—
72	D₄	—	92	1	no	—
73	D₄	—	184	2	no	—
74	C₂×C₄	—	2	1	no	—
75	C₂×C₄	—	2	1	no	—
76	C₂×C₄	—	2	1	no	—
77	C₂×C₄	(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
78	C₈	—	8	1	no	—
79	C₈	—	46	2	no	—
80	C₈	—	184	1	no	—
81	C₂²	—	184	2	no	—
82	C₂²	—	184	1	no	—
83	C₂²	#10624988	1	1	yes	characteristic
84	C₄	—	2	2	no	—
85	C₄	—	2	1	no	—
86	C₄	—	4	1	no	—
87	C₄	—	4	1	no	—
88	C₂	—	2	1	no	—
89	C₂	—	368	1	no	—
90	C₂	#13092209	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#13092209
commutator	C₄²×C₂₃	C₂×C₄
Frattini	C₄⋊Q₈	C₂×D₂₃
Fitting	C₂₃×#356	C₂
socle	C₂×C₂₃	#353
prime radical	#10625045	C₄
cosocle	C₂₃×C₄⋊Q₈	C₂²
hypercenter	#356	D₂₃
hypocenter	C₂₃	#1410
LCS	G ▷ C₄²×C₂₃ ▷ C₂×C₄×C₂₃ ▷ C₂²×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊Q₈ ◁ #356

Complements

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

N	H ≅ G/N	type
2. #13092216	C₂	semidirect
3. C₂₃×#356	C₂	semidirect
45. #356	D₂₃	semidirect
56. C₂₃	#1410	semidirect

Maximal subgroups

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

subgroup	index
#13092791	2
#13092216	2
C₂₃×#356	2
#1410	23