Subgroups — #15671365

Order 2944.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×D₁₈₄	C₂	1	1	yes	maximal, characteristic
2	#13092168	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#346	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₁₈₄	C₄	1	1	yes	characteristic
5	C₄×D₉₂	—	2	1	no	—
6	C₂×C₁₆×C₂₃	—	2	1	no	—
7	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
8	#10625003	—	2	1	no	—
9	#10625000	C₄	1	1	yes	characteristic
10	C₄⋊(C₂₃⋊C₈)	—	2	1	no	—
11	C₂×D₉₂	—	2	1	no	—
12	C₂×C₄×D₂₃	—	4	1	no	—
13	C₂×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
14	C₂×C₈×C₂₃	D₄	1	1	yes	characteristic
15	C₄²×C₂₃	D₄	1	1	yes	characteristic
16	(C₂×D₂₃)⋊C₄	—	4	1	no	—
17	C₄⋊Dic₂₃	—	2	1	no	—
18	C₂×C₂₃⋊C₈	—	4	1	no	—
19	D₁₈₄	—	2	1	no	—
20	D₁₈₄	C₈	1	2	yes	complemented
21	C₁₆×C₂₃	—	2	2	no	—
22	C₂²×D₂₃	—	4	1	no	—
23	C₂×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
24	C₂×C₄×C₂₃	D₈	1	1	yes	characteristic
25	C₂×C₄×C₂₃	SD₁₆	1	1	yes	characteristic
26	C₂×Dic₂₃	—	4	1	no	—
27	D₉₂	—	2	2	no	—
28	D₉₂	—	4	1	no	—
29	C₄×D₂₃	—	8	2	no	—
30	C₈×C₂₃	—	2	1	no	—
31	C₈×C₂₃	C₂×C₈	1	1	yes	characteristic
32	C₈×C₂₃	M₁₆	1	1	yes	characteristic
33	C₂₃⋊C₈	—	8	1	no	—
34	#1333	—	23	1	no	maximal, Sylow
35	C₂²×C₂₃	#102	1	1	yes	characteristic
36	C₂×D₂₃	—	4	2	no	—
37	C₂×D₂₃	—	8	1	no	—
38	C₂×D₂₃	—	8	1	no	—
39	C₄×C₂₃	—	2	1	no	—
40	C₄×C₂₃	—	2	1	no	—
41	C₄×C₂₃	C₂²⋊C₈	1	1	yes	characteristic
42	C₄×C₂₃	#104	1	1	yes	characteristic
43	Dic₂₃	—	8	1	no	—
44	C₄×D₈	—	23	1	no	—
45	#346	D₂₃	1	1	yes	characteristic, complemented
46	#334	—	23	1	no	—
47	C₂×C₂₃	D₄⋊C₈	1	1	yes	characteristic
48	C₂×C₂₃	#360	1	1	yes	characteristic
49	C₂×C₂₃	#361	1	1	yes	characteristic
50	D₂₃	—	8	2	no	—
51	C₂×D₈	—	23	1	no	—
52	C₄×D₄	—	46	1	no	—
53	C₂×C₁₆	—	2	1	no	—
54	#107	—	23	1	no	—
55	C₄⋊C₈	—	46	1	no	—
56	#102	—	46	1	no	—
57	C₄×C₈	C₂×D₂₃	1	1	yes	characteristic
58	C₂₃	#1333	1	1	yes	characteristic, complemented, Sylow
59	C₂×D₄	—	46	1	no	—
60	C₂²×C₄	—	92	1	no	—
61	D₈	—	23	2	no	—
62	D₈	—	46	1	no	—
63	C₂×C₈	—	92	1	no	—
64	C₂×C₈	C₄×D₂₃	1	1	yes	characteristic
65	C₂×C₈	C₂²⋊D₂₃	1	1	yes	characteristic
66	C₄⋊C₄	—	46	1	no	—
67	C₂²⋊C₄	—	92	1	no	—
68	C₄²	D₉₂	1	1	yes	characteristic
69	C₁₆	—	2	2	no	—
70	C₂³	—	92	1	no	—
71	D₄	—	46	2	no	—
72	D₄	—	92	1	no	—
73	C₂×C₄	—	92	1	no	—
74	C₂×C₄	—	184	2	no	—
75	C₂×C₄	#67263	1	1	yes	characteristic
76	C₂×C₄	D₁₈₄	1	1	yes	characteristic
77	C₂×C₄	(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
78	C₈	—	2	1	no	—
79	C₈	—	184	1	no	—
80	C₈	C₈×D₂₃	1	1	yes	characteristic
81	C₈	#67262	1	1	yes	characteristic
82	C₂²	—	92	2	no	—
83	C₂²	—	184	1	no	—
84	C₂²	—	184	1	no	—
85	C₂²	#10625003	1	1	yes	characteristic
86	C₄	—	2	1	no	—
87	C₄	—	2	1	no	—
88	C₄	—	184	1	no	—
89	C₄	#10624987	1	1	yes	characteristic
90	C₄	(C₂×D₂₃)⋊C₈	1	1	yes	characteristic
91	C₂	—	184	2	no	—
92	C₂	#13092146	1	1	yes	characteristic
93	C₂	#13092228	1	1	yes	characteristic
94	C₂	#13092231	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#10625003
commutator	C₈×C₂₃	C₂×C₈
Frattini	C₄×C₈	C₂×D₂₃
Fitting	C₂₃×#346	C₂
socle	C₂²×C₂₃	#102
prime radical	C₂×D₁₈₄	C₄
cosocle	C₄×C₈×C₂₃	C₂²
hypercenter	#346	D₂₃
hypocenter	C₂₃	#1333
LCS	G ▷ C₈×C₂₃ ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
2. #13092168	C₂	semidirect
3. C₂₃×#346	C₂	semidirect
20. D₁₈₄	C₈	semidirect
45. #346	D₂₃	semidirect
58. C₂₃	#1333	semidirect

Maximal subgroups

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

subgroup	index
C₄×D₁₈₄	2
#13092168	2
C₂₃×#346	2
#1333	23