Subgroups — #15671356

Order 2944.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13092496	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₂₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#346	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#67262	—	2	1	no	—
5	C₄×D₉₂	C₄	1	1	yes	characteristic
6	C₂×C₁₆×C₂₃	—	2	1	no	—
7	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
8	(C₂×D₂₃)⋊C₈	—	2	1	no	—
9	C₂×C₂₃⋊C₁₆	—	2	1	no	—
10	C₄⋊(C₂₃⋊C₈)	C₄	1	1	yes	characteristic
11	C₂×D₉₂	C₈	1	1	yes	characteristic
12	C₂×C₄×D₂₃	—	2	1	no	—
13	C₂×C₈×C₂₃	D₄	1	1	yes	characteristic
14	C₂×C₈×C₂₃	D₄	1	1	yes	characteristic
15	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
16	(C₂×D₂₃)⋊C₄	—	2	1	no	—
17	C₄⋊Dic₂₃	C₈	1	1	yes	characteristic
18	C₂×C₂₃⋊C₈	—	2	1	no	—
19	#67262	—	4	1	no	—
20	#67262	—	4	1	no	—
21	C₁₆×C₂₃	—	2	2	no	—
22	C₂₃⋊C₁₆	—	4	1	no	—
23	C₂²×D₂₃	—	2	1	no	—
24	C₂×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
25	C₂×C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
26	C₂×C₄×C₂₃	M₁₆	1	1	yes	characteristic
27	C₂×Dic₂₃	—	2	1	no	—
28	D₉₂	—	4	1	no	—
29	C₄×D₂₃	—	4	1	no	—
30	C₄×D₂₃	—	4	1	no	—
31	C₈×C₂₃	—	2	1	no	—
32	C₈×C₂₃	D₈	1	1	yes	characteristic
33	C₈×C₂₃	SD₁₆	1	1	yes	characteristic
34	C₂₃⋊C₈	—	4	1	no	—
35	#1330	—	23	1	no	maximal, Sylow
36	C₂²×C₂₃	C₂²⋊C₈	1	1	yes	characteristic
37	C₂×D₂₃	—	4	1	no	—
38	C₂×D₂₃	—	4	1	no	—
39	C₂×D₂₃	—	4	1	no	—
40	C₄×C₂₃	—	2	1	no	—
41	C₄×C₂₃	—	2	1	no	—
42	C₄×C₂₃	#102	1	1	yes	characteristic
43	C₄×C₂₃	#104	1	1	yes	characteristic
44	Dic₂₃	—	4	1	no	—
45	#436	—	23	1	no	—
46	C₄⋊C₁₆	—	23	1	no	—
47	#346	D₂₃	1	1	yes	characteristic, complemented
48	C₂×C₂₃	D₄⋊C₈	1	1	yes	characteristic
49	C₂×C₂₃	#349	1	1	yes	characteristic
50	C₂×C₂₃	#350	1	1	yes	characteristic
51	D₂₃	—	8	1	no	—
52	C₂×M₁₆	—	46	1	no	—
53	C₄×D₄	—	23	1	no	—
54	C₂×C₁₆	—	2	1	no	—
55	C₂×C₁₆	—	46	1	no	—
56	C₄⋊C₈	—	23	1	no	—
57	C₂²⋊C₈	—	46	1	no	—
58	C₄×C₈	C₂×D₂₃	1	1	yes	characteristic
59	C₂₃	#1330	1	1	yes	characteristic, complemented, Sylow
60	C₂×D₄	—	23	1	no	—
61	C₂²×C₄	—	46	1	no	—
62	M₁₆	—	92	1	no	—
63	M₁₆	—	92	1	no	—
64	C₂×C₈	—	46	1	no	—
65	C₂×C₈	D₉₂	1	1	yes	characteristic
66	C₂×C₈	C₂²⋊D₂₃	1	1	yes	characteristic
67	C₄⋊C₄	—	23	1	no	—
68	C₂²⋊C₄	—	46	1	no	—
69	C₄²	C₄×D₂₃	1	1	yes	characteristic
70	C₁₆	—	2	2	no	—
71	C₁₆	—	92	1	no	—
72	C₂³	—	46	1	no	—
73	D₄	—	92	1	no	—
74	C₂×C₄	—	46	1	no	—
75	C₂×C₄	—	92	1	no	—
76	C₂×C₄	—	92	1	no	—
77	C₂×C₄	C₈×D₂₃	1	1	yes	characteristic
78	C₂×C₄	#67262	1	1	yes	characteristic
79	C₂×C₄	(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
80	C₈	—	2	1	no	—
81	C₈	—	92	1	no	—
82	C₈	#67263	1	1	yes	characteristic
83	C₈	D₁₈₄	1	1	yes	characteristic
84	C₂²	—	92	1	no	—
85	C₂²	—	92	1	no	—
86	C₂²	—	92	1	no	—
87	C₂²	(C₂×D₂₃)⋊C₈	1	1	yes	characteristic
88	C₄	—	2	1	no	—
89	C₄	—	2	1	no	—
90	C₄	—	92	1	no	—
91	C₄	#10624987	1	1	yes	characteristic
92	C₄	#10625003	1	1	yes	characteristic
93	C₂	—	184	1	no	—
94	C₂	#13092146	1	1	yes	characteristic
95	C₂	#13092198	1	1	yes	characteristic
96	C₂	#13092202	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	(C₂×D₂₃)⋊C₄
commutator	C₂×C₄×C₂₃	C₂×C₈
Frattini	C₄×C₈	C₂×D₂₃
Fitting	C₂₃×#346	C₂
socle	C₂²×C₂₃	C₂²⋊C₈
prime radical	C₂×D₉₂	C₈
cosocle	C₄×C₈×C₂₃	C₂²
hypercenter	#346	D₂₃
hypocenter	C₂₃	#1330
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
2. C₄⋊(C₂₃⋊C₁₆)	C₂	semidirect
3. C₂₃×#346	C₂	semidirect
47. #346	D₂₃	semidirect
59. C₂₃	#1330	semidirect

Maximal subgroups

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

subgroup	index
#13092496	2
C₄⋊(C₂₃⋊C₁₆)	2
C₂₃×#346	2
#1330	23