Subgroups — #38744584

Order 25216.

148 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₁₅₇₆	C₂	1	1	yes	maximal, characteristic
2	C₄×Dic₇₈₈	C₂	1	2	yes	maximal
3	#27160907	C₂	1	1	yes	maximal, characteristic
4	#27160884	C₂	1	2	yes	maximal
5	#346×C₁₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₇₈₈	C₂²	1	2	yes	—
7	C₄×Dic₃₉₄	—	2	2	no	—
8	Dic₁₅₇₆	C₄	1	4	yes	complemented
9	#20263260	C₂²	1	2	yes	—
10	#20263251	—	2	2	no	—
11	C₂×C₁₆×C₁₉₇	C₂²	1	1	yes	characteristic
12	C₂×C₁₆×C₁₉₇	C₂²	1	1	yes	characteristic
13	C₄×C₈×C₁₉₇	C₂²	1	1	yes	characteristic
14	C₂×Dic₃₉₄	—	2	2	no	—
15	Dic₇₈₈	—	2	2	no	—
16	Dic₇₈₈	C₂×C₄	1	4	yes	—
17	C₄⋊Dic₁₉₇	—	2	2	no	—
18	#15691337	—	4	2	no	—
19	C₄×Dic₁₉₇	—	4	2	no	—
20	C₂×C₈×C₁₉₇	D₄	1	1	yes	characteristic
21	C₂×C₈×C₁₉₇	C₂³	1	1	yes	characteristic
22	C₄²×C₁₉₇	D₄	1	1	yes	characteristic
23	C₁₆×C₁₉₇	—	2	1	no	—
24	C₁₆×C₁₉₇	C₂×C₄	1	2	yes	—
25	Dic₃₉₄	—	2	4	no	—
26	Dic₃₉₄	—	4	2	no	—
27	C₂×Dic₁₉₇	—	4	2	no	—
28	C₂×Dic₁₉₇	—	4	2	no	—
29	C₂×C₄×C₁₉₇	D₈	1	2	yes	—
30	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
31	C₈×C₁₉₇	—	2	1	no	—
32	C₈×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
33	C₈×C₁₉₇	#41	1	1	yes	characteristic
34	C₂²×C₁₉₇	C₂×D₈	1	1	yes	characteristic
35	C₄×C₁₉₇	—	2	2	no	—
36	C₄×C₁₉₇	C₄×D₄	1	1	yes	characteristic
37	C₄×C₁₉₇	#135	1	1	yes	characteristic
38	Dic₁₉₇	—	4	4	no	—
39	Dic₁₉₇	—	8	2	no	—
40	C₂×C₁₉₇	C₄×D₈	1	1	yes	characteristic
41	C₂×C₁₉₇	#510	1	2	yes	—
42	C₁₉₇	#2176	1	1	yes	characteristic, complemented, Sylow
43	#2176	—	197	1	no	maximal, Sylow
44	C₂×Q₃₂	—	197	1	no	—
45	C₄×Q₁₆	—	197	2	no	—
46	#367	—	197	1	no	—
47	#358	—	197	2	no	—
48	#346	D₁₉₇	1	1	yes	characteristic
49	C₂×Q₁₆	—	197	2	no	—
50	C₄×Q₈	—	394	2	no	—
51	Q₃₂	—	197	4	no	—
52	C₂×C₁₆	C₂×D₁₉₇	1	1	yes	characteristic
53	C₂×C₁₆	C₂×D₁₉₇	1	1	yes	characteristic
54	#107	—	197	2	no	—
55	#103	—	394	2	no	—
56	C₄×C₈	C₂×D₁₉₇	1	1	yes	characteristic
57	C₂×Q₈	—	394	2	no	—
58	Q₁₆	—	394	2	no	—
59	Q₁₆	—	197	4	no	—
60	C₂×C₈	D₇₈₈	1	1	yes	characteristic
61	C₂×C₈	C₂²×D₁₉₇	1	1	yes	characteristic
62	C₄⋊C₄	—	788	2	no	—
63	C₄⋊C₄	—	394	2	no	—
64	C₄²	—	788	2	no	—
65	C₄²	D₇₈₈	1	1	yes	characteristic
66	C₁₆	—	2	1	no	—
67	C₁₆	C₄×D₁₉₇	1	2	yes	—
68	Q₈	—	788	2	no	—
69	Q₈	—	394	4	no	—
70	C₂×C₄	—	788	2	no	—
71	C₂×C₄	—	788	2	no	—
72	C₂×C₄	D₁₅₇₆	1	2	yes	—
73	C₂×C₄	C₂×D₇₈₈	1	1	yes	characteristic
74	C₈	—	2	1	no	—
75	C₈	C₂×C₄×D₁₉₇	1	1	yes	characteristic
76	C₈	#15691358	1	1	yes	characteristic
77	C₂²	C₂×D₁₅₇₆	1	1	yes	characteristic
78	C₄	—	2	2	no	—
79	C₄	—	788	4	no	—
80	C₄	—	1576	2	no	—
81	C₄	C₄×D₇₈₈	1	1	yes	characteristic
82	C₄	#20263348	1	1	yes	characteristic
83	C₂	C₄×D₁₅₇₆	1	1	yes	characteristic
84	C₂	#27161522	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₁₅₇₆
commutator	C₈×C₁₉₇	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₉₇
Fitting	#346×C₁₉₇	C₂
socle, prime radical	C₂²×C₁₉₇	C₂×D₈
cosocle	C₂×C₈×C₁₉₇	C₂³
hypercenter	#346	D₁₉₇
hypocenter	C₁₉₇	#2176
LCS	G ▷ C₈×C₁₉₇ ▷ C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
8. Dic₁₅₇₆	C₄	semidirect
42. C₁₉₇	#2176	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₁₅₇₆	2
C₄×Dic₇₈₈	2
#27160907	2
#27160884	2
#346×C₁₉₇	2
#2176	197