Subgroups — #37535608

Order 23168.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26041711	C₂	1	1	yes	maximal, characteristic
2	#26041397	C₂	1	2	yes	maximal, complemented
3	#26041109	C₂	1	1	yes	maximal, characteristic, complemented
4	#26041087	C₂	1	2	yes	maximal, complemented
5	#346×C₁₈₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#19153838	C₂²	1	2	yes	—
7	#19153822	—	2	2	no	—
8	Dic₁₄₄₈	C₄	1	1	yes	characteristic, complemented
9	#19153761	C₄	1	2	yes	complemented
10	D₂₈₉₆	C₄	1	1	yes	characteristic, complemented
11	#19153752	C₂²	1	2	yes	—
12	#19153744	—	2	2	no	—
13	C₂×C₁₆×C₁₈₁	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×C₁₈₁	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₁₈₁	C₂²	1	1	yes	characteristic
16	#15668449	—	2	2	no	—
17	Dic₇₂₄	C₂×C₄	1	2	yes	—
18	#15668438	—	2	2	no	—
19	D₁₄₄₈	C₂×C₄	1	2	yes	—
20	#15668433	—	2	2	no	—
21	#15668432	—	4	2	no	—
22	C₈×D₁₈₁	—	4	2	no	—
23	C₂×C₈×C₁₈₁	D₄	1	1	yes	characteristic
24	C₂×C₈×C₁₈₁	C₂³	1	1	yes	characteristic
25	C₄²×C₁₈₁	D₄	1	1	yes	characteristic
26	C₁₆×C₁₈₁	—	2	1	no	—
27	C₁₆×C₁₈₁	C₂×C₄	1	1	yes	characteristic
28	C₁₆×C₁₈₁	C₂×C₄	1	1	yes	characteristic
29	Dic₃₆₂	—	2	2	no	—
30	D₇₂₄	—	2	2	no	—
31	C₂²⋊D₁₈₁	—	4	2	no	—
32	C₄×D₁₈₁	—	4	2	no	—
33	#13089678	—	4	2	no	—
34	C₂×C₄×C₁₈₁	—	2	1	no	—
35	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
36	C₈×C₁₈₁	D₈	1	2	yes	—
37	C₈×C₁₈₁	C₂²×C₄	1	1	yes	characteristic
38	C₈×C₁₈₁	#41	1	1	yes	characteristic
39	C₂²×C₁₈₁	#135	1	1	yes	characteristic
40	C₂×D₁₈₁	—	4	2	no	—
41	C₄×C₁₈₁	—	4	1	no	—
42	C₄×C₁₈₁	C₄×D₄	1	1	yes	characteristic
43	C₄×C₁₈₁	C₂×D₈	1	1	yes	characteristic
44	Dic₁₈₁	—	4	2	no	—
45	C₂×C₁₈₁	—	2	1	no	—
46	C₂×C₁₈₁	C₄×D₈	1	1	yes	characteristic
47	D₁₈₁	—	8	2	no	—
48	C₁₈₁	#2179	1	1	yes	characteristic, complemented, Sylow
49	#2179	—	181	1	no	maximal, Sylow
50	#508	—	181	1	no	—
51	#443	—	181	2	no	—
52	#368	—	181	1	no	—
53	#359	—	181	2	no	—
54	#346	D₁₈₁	1	1	yes	characteristic, complemented
55	#135	—	181	2	no	—
56	#131	—	362	2	no	—
57	Q₃₂	—	181	1	no	—
58	SD₃₂	—	181	2	no	—
59	D₁₆	—	181	1	no	—
60	C₂×C₁₆	C₂×D₁₈₁	1	1	yes	characteristic
61	C₂×C₁₆	C₂×D₁₈₁	1	1	yes	characteristic
62	#108	—	181	2	no	—
63	#104	—	362	2	no	—
64	C₄×C₈	C₂×D₁₈₁	1	1	yes	characteristic
65	#41	—	362	2	no	—
66	Q₁₆	—	181	2	no	—
67	SD₁₆	—	362	2	no	—
68	D₈	—	181	2	no	—
69	M₁₆	—	362	2	no	—
70	M₁₆	—	724	2	no	—
71	C₂×C₈	—	724	2	no	—
72	C₂×C₈	D₇₂₄	1	1	yes	characteristic
73	C₂×C₈	C₂²×D₁₈₁	1	1	yes	characteristic
74	C₄²	D₇₂₄	1	1	yes	characteristic
75	C₁₆	—	2	1	no	—
76	C₁₆	C₄×D₁₈₁	1	1	yes	characteristic
77	C₁₆	C₄×D₁₈₁	1	1	yes	characteristic
78	Q₈	—	362	2	no	—
79	D₄	—	362	2	no	—
80	D₄	—	724	2	no	—
81	C₂×C₄	—	2	1	no	—
82	C₂×C₄	—	724	2	no	—
83	C₂×C₄	C₂×D₇₂₄	1	1	yes	characteristic
84	C₈	—	724	2	no	—
85	C₈	D₁₄₄₈	1	2	yes	—
86	C₈	C₂×C₄×D₁₈₁	1	1	yes	characteristic
87	C₈	#15668449	1	1	yes	characteristic
88	C₂²	—	724	2	no	—
89	C₂²	#19153838	1	1	yes	characteristic
90	C₄	—	4	1	no	—
91	C₄	—	724	2	no	—
92	C₄	C₄×D₇₂₄	1	1	yes	characteristic
93	C₄	C₂×D₁₄₄₈	1	1	yes	characteristic
94	C₂	—	2	1	no	—
95	C₂	—	1448	2	no	—
96	C₂	C₄×D₁₄₄₈	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₁₄₄₈
commutator	C₈×C₁₈₁	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₈₁
Fitting	#346×C₁₈₁	C₂
socle	C₂×C₁₈₁	C₄×D₈
prime radical	#26041711	C₂
cosocle	C₂×C₈×C₁₈₁	C₂³
hypercenter	#346	D₁₈₁
hypocenter	C₁₈₁	#2179
LCS	G ▷ C₈×C₁₈₁ ▷ C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
2. #26041397	C₂	semidirect
3. #26041109	C₂	semidirect
4. #26041087	C₂	semidirect
5. #346×C₁₈₁	C₂	semidirect
8. Dic₁₄₄₈	C₄	semidirect
9. #19153761	C₄	semidirect
10. D₂₈₉₆	C₄	semidirect
48. C₁₈₁	#2179	semidirect
54. #346	D₁₈₁	semidirect

Maximal subgroups

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

subgroup	index
#26041711	2
#26041397	2
#26041109	2
#26041087	2
#346×C₁₈₁	2
#2179	181