Subgroups — #15675274

Order 2944.

172 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₃₆₈	C₂	1	1	yes	maximal, characteristic
2	C₄×D₁₈₄	C₂	1	2	yes	maximal, complemented
3	#13092245	C₂	1	1	yes	maximal, characteristic, complemented
4	#13092219	C₂	1	2	yes	maximal, complemented
5	C₂₃×#346	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₈₄	C₂²	1	2	yes	—
7	C₄×D₉₂	—	2	2	no	—
8	C₂×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
9	C₂×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
10	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
11	#10625003	—	2	2	no	—
12	#10625000	C₂²	1	2	yes	—
13	D₃₆₈	C₄	1	4	yes	complemented
14	C₂×D₉₂	—	2	2	no	—
15	C₂×C₄×D₂₃	—	4	2	no	—
16	C₂×C₈×C₂₃	D₄	1	1	yes	characteristic
17	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
18	C₄²×C₂₃	D₄	1	1	yes	characteristic
19	(C₂×D₂₃)⋊C₄	—	4	2	no	—
20	C₄⋊Dic₂₃	—	2	2	no	—
21	D₁₈₄	—	2	2	no	—
22	D₁₈₄	C₂×C₄	1	4	yes	complemented
23	C₁₆×C₂₃	—	2	1	no	—
24	C₁₆×C₂₃	C₂×C₄	1	2	yes	complemented
25	C₂²×D₂₃	—	4	2	no	—
26	C₂×C₄×C₂₃	D₈	1	2	yes	—
27	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
28	C₂×Dic₂₃	—	4	2	no	—
29	D₉₂	—	2	4	no	—
30	D₉₂	—	4	2	no	—
31	C₄×D₂₃	—	8	4	no	—
32	C₈×C₂₃	—	2	1	no	—
33	C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
34	C₈×C₂₃	#41	1	1	yes	characteristic
35	#2177	—	23	1	no	maximal, Sylow
36	C₂²×C₂₃	C₂×D₈	1	1	yes	characteristic
37	C₂×D₂₃	—	4	4	no	—
38	C₂×D₂₃	—	8	2	no	—
39	C₂×D₂₃	—	8	2	no	—
40	C₄×C₂₃	—	2	2	no	—
41	C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
42	C₄×C₂₃	#135	1	1	yes	characteristic
43	Dic₂₃	—	8	2	no	—
44	C₂×D₁₆	—	23	1	no	—
45	C₄×D₈	—	23	2	no	—
46	#367	—	23	1	no	—
47	#357	—	23	2	no	—
48	#346	D₂₃	1	1	yes	characteristic, complemented
49	C₂×C₂₃	C₄×D₈	1	1	yes	characteristic
50	C₂×C₂₃	#509	1	2	yes	—
51	D₂₃	—	8	4	no	—
52	C₂×D₈	—	23	2	no	—
53	C₄×D₄	—	46	2	no	—
54	D₁₆	—	23	4	no	—
55	C₂×C₁₆	C₂×D₂₃	1	1	yes	characteristic
56	C₂×C₁₆	C₂×D₂₃	1	1	yes	characteristic
57	#107	—	23	2	no	—
58	#102	—	46	2	no	—
59	C₄×C₈	C₂×D₂₃	1	1	yes	characteristic
60	C₂₃	#2177	1	1	yes	characteristic, complemented, Sylow
61	C₂×D₄	—	46	2	no	—
62	C₂²×C₄	—	92	2	no	—
63	D₈	—	23	4	no	—
64	D₈	—	46	2	no	—
65	C₂×C₈	D₉₂	1	1	yes	characteristic
66	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
67	C₄⋊C₄	—	46	2	no	—
68	C₂²⋊C₄	—	92	2	no	—
69	C₄²	D₉₂	1	1	yes	characteristic
70	C₁₆	—	2	1	no	—
71	C₁₆	C₄×D₂₃	1	2	yes	complemented
72	C₂³	—	92	2	no	—
73	D₄	—	46	4	no	—
74	D₄	—	92	2	no	—
75	C₂×C₄	—	92	2	no	—
76	C₂×C₄	—	184	4	no	—
77	C₂×C₄	D₁₈₄	1	2	yes	—
78	C₂×C₄	C₂×D₉₂	1	1	yes	characteristic
79	C₈	—	2	1	no	—
80	C₈	C₂×C₄×D₂₃	1	1	yes	characteristic
81	C₈	#67288	1	1	yes	characteristic
82	C₂²	—	92	4	no	—
83	C₂²	—	184	2	no	—
84	C₂²	—	184	2	no	—
85	C₂²	C₂×D₁₈₄	1	1	yes	characteristic
86	C₄	—	2	2	no	—
87	C₄	—	184	2	no	—
88	C₄	C₄×D₉₂	1	1	yes	characteristic
89	C₄	#10625075	1	1	yes	characteristic
90	C₂	—	184	4	no	—
91	C₂	C₄×D₁₈₄	1	1	yes	characteristic
92	C₂	#13092855	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₁₈₄
commutator	C₈×C₂₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₂₃
Fitting	C₂₃×#346	C₂
socle	C₂²×C₂₃	C₂×D₈
prime radical	C₂×D₃₆₈	C₂
cosocle	C₂×C₈×C₂₃	C₂³
hypercenter	#346	D₂₃
hypocenter	C₂₃	#2177
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. C₄×D₁₈₄	C₂	semidirect
3. #13092245	C₂	semidirect
4. #13092219	C₂	semidirect
5. C₂₃×#346	C₂	semidirect
13. D₃₆₈	C₄	semidirect
22. D₁₈₄	C₂×C₄	semidirect
24. C₁₆×C₂₃	C₂×C₄	semidirect
48. #346	D₂₃	semidirect
60. C₂₃	#2177	semidirect
71. C₁₆	C₄×D₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₃₆₈	2
C₄×D₁₈₄	2
#13092245	2
#13092219	2
C₂₃×#346	2
#2177	23