Subgroups — #38744963

Order 25216.

176 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	C₂×#20263272	C₂	1	2	yes	maximal
3	#27161457	C₂	1	2	yes	maximal
4	C₁₆×Dic₁₉₇	C₂	1	1	yes	maximal, characteristic
5	C₂×Q₃₂×C₁₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₇₈₈	C₂²	1	2	yes	—
7	C₂×#15691348	—	2	2	no	—
8	#20263324	—	2	2	no	—
9	Dic₁₅₇₆	—	2	2	no	—
10	#20263272	—	2	4	no	—
11	C₂×#15691333	C₂²	1	1	yes	characteristic
12	C₈×Dic₁₉₇	C₂²	1	1	yes	characteristic
13	C₂×Q₁₆×C₁₉₇	C₂²	1	2	yes	—
14	Q₃₂×C₁₉₇	—	2	2	no	—
15	C₂×C₁₆×C₁₉₇	C₂²	1	1	yes	characteristic
16	C₂×Dic₃₉₄	—	2	2	no	—
17	Dic₇₈₈	—	2	2	no	—
18	Dic₇₈₈	—	2	2	no	—
19	#15691348	—	4	4	no	—
20	C₂×#13096802	D₄	1	1	yes	characteristic
21	#15691337	—	4	4	no	—
22	C₄×Dic₁₉₇	D₄	1	1	yes	characteristic
23	#15691333	D₄	1	2	yes	—
24	C₂×Q₈×C₁₉₇	—	2	2	no	—
25	Q₁₆×C₁₉₇	—	2	2	no	—
26	Q₁₆×C₁₉₇	—	2	2	no	—
27	C₂×C₈×C₁₉₇	C₂³	1	1	yes	characteristic
28	C₁₆×C₁₉₇	D₄	1	2	yes	—
29	Dic₃₉₄	—	4	2	no	—
30	Dic₃₉₄	—	4	2	no	—
31	C₂×Dic₁₉₇	—	4	2	no	—
32	C₂×Dic₁₉₇	D₈	1	2	yes	—
33	#13096802	D₈	1	2	yes	—
34	Q₈×C₁₉₇	—	4	2	no	—
35	Q₈×C₁₉₇	—	4	2	no	—
36	C₂×C₄×C₁₉₇	—	4	2	no	—
37	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
38	C₈×C₁₉₇	C₂×D₄	1	1	yes	characteristic
39	C₈×C₁₉₇	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₁₉₇	C₂×D₈	1	1	yes	characteristic
41	C₄×C₁₉₇	—	8	2	no	—
42	C₄×C₁₉₇	#127	1	1	yes	characteristic
43	C₄×C₁₉₇	C₂×D₈	1	1	yes	characteristic
44	Dic₁₉₇	—	8	2	no	—
45	Dic₁₉₇	Q₃₂	1	4	yes	complemented
46	C₂×C₁₉₇	#493	1	1	yes	characteristic
47	C₂×C₁₉₇	C₂×Q₃₂	1	2	yes	—
48	C₁₉₇	#2247	1	1	yes	characteristic, complemented, Sylow
49	#2247	—	197	1	no	maximal, Sylow
50	C₂×Q₃₂	—	197	2	no	—
51	C₂×Q₃₂	—	197	1	no	—
52	C₂×Q₃₂	D₁₉₇	1	1	yes	characteristic
53	#494	—	197	2	no	—
54	C₄×C₁₆	—	197	1	no	—
55	C₂×Q₁₆	—	394	2	no	—
56	C₂×Q₁₆	—	197	2	no	—
57	C₂×Q₁₆	C₂×D₁₉₇	1	2	yes	—
58	C₄⋊Q₈	—	394	2	no	—
59	Q₃₂	—	2	2	no	—
60	Q₃₂	—	394	4	no	—
61	Q₃₂	—	394	2	no	—
62	C₂×C₁₆	—	197	1	no	—
63	C₂×C₁₆	C₂×D₁₉₇	1	1	yes	characteristic
64	C₄×C₈	—	197	1	no	—
65	C₂×Q₈	—	2	2	no	—
66	C₂×Q₈	—	394	2	no	—
67	Q₁₆	—	2	2	no	—
68	Q₁₆	—	2	2	no	—
69	Q₁₆	—	788	4	no	—
70	Q₁₆	—	394	2	no	—
71	Q₁₆	—	394	2	no	—
72	C₂×C₈	—	197	1	no	—
73	C₂×C₈	C₂²×D₁₉₇	1	1	yes	characteristic
74	C₄⋊C₄	—	788	4	no	—
75	C₄²	—	197	1	no	—
76	C₁₆	—	197	2	no	—
77	C₁₆	C₂²⋊D₁₉₇	1	2	yes	—
78	Q₈	—	4	2	no	—
79	Q₈	—	4	2	no	—
80	Q₈	—	788	2	no	—
81	Q₈	—	788	2	no	—
82	C₂×C₄	—	4	2	no	—
83	C₂×C₄	—	788	2	no	—
84	C₂×C₄	—	197	2	no	—
85	C₂×C₄	D₄×D₁₉₇	1	1	yes	characteristic
86	C₈	—	197	2	no	—
87	C₈	D₄×D₁₉₇	1	1	yes	characteristic
88	C₈	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
89	C₂²	D₈×D₁₉₇	1	1	yes	characteristic
90	C₄	—	8	2	no	—
91	C₄	—	1576	2	no	—
92	C₄	—	197	4	no	—
93	C₄	#20263322	1	1	yes	characteristic
94	C₄	D₈×D₁₉₇	1	1	yes	characteristic
95	C₂	#27161454	1	1	yes	characteristic
96	C₂	Q₃₂×D₁₉₇	1	2	yes	—

	subgroup	G/H
center	C₂²	D₈×D₁₉₇
commutator, cosocle	C₂×C₈×C₁₉₇	C₂³
Frattini	C₂×C₈	C₂²×D₁₉₇
Fitting	C₂×Q₃₂×C₁₉₇	C₂
socle, prime radical	C₂²×C₁₉₇	C₂×D₈
hypercenter	C₂×Q₃₂	D₁₉₇
hypocenter	C₁₉₇	#2247
LCS	G ▷ C₂×C₈×C₁₉₇ ▷ C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

Complements

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

N	H ≅ G/N	type
45. Dic₁₉₇	Q₃₂	semidirect
48. C₁₉₇	#2247	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
C₂×#20263272	2
#27161457	2
C₁₆×Dic₁₉₇	2
C₂×Q₃₂×C₁₉₇	2
#2247	197