Subgroups — #38740851

Order 25216.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20263265	C₂	1	1	yes	maximal, characteristic
2	#27160913	C₂	1	1	yes	maximal, characteristic, complemented
3	M₆₄×C₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₈×D₁₉₇	C₄	1	1	yes	characteristic
5	#20263265	—	2	1	no	—
6	#20263265	—	2	1	no	—
7	C₂×#15691333	C₄	1	1	yes	characteristic
8	#20263232	—	2	1	no	—
9	C₂×C₁₆×C₁₉₇	C₂²	1	1	yes	characteristic
10	C₃₂×C₁₉₇	—	2	1	no	—
11	C₂×C₄×D₁₉₇	C₈	1	1	yes	characteristic
12	C₂×#13096802	C₈	1	1	yes	characteristic
13	C₈×D₁₉₇	—	2	1	no	—
14	C₈×D₁₉₇	—	2	1	no	—
15	#15691333	—	2	1	no	—
16	C₂×C₈×C₁₉₇	C₂×C₄	1	1	yes	characteristic
17	C₁₆×C₁₉₇	D₄	1	1	yes	characteristic
18	C₁₆×C₁₉₇	D₄	1	1	yes	characteristic
19	C₂²×D₁₉₇	C₁₆	1	1	yes	characteristic
20	C₂×Dic₁₉₇	C₁₆	1	1	yes	characteristic
21	C₄×D₁₉₇	—	2	1	no	—
22	C₄×D₁₉₇	—	2	1	no	—
23	#13096802	—	2	1	no	—
24	C₂×C₄×C₁₉₇	C₂×C₈	1	1	yes	characteristic
25	C₈×C₁₉₇	C₂²⋊C₄	1	1	yes	characteristic
26	C₈×C₁₉₇	M₁₆	1	1	yes	characteristic
27	C₂²×C₁₉₇	C₂×C₁₆	1	1	yes	characteristic
28	C₂×D₁₉₇	—	2	1	no	—
29	C₂×D₁₉₇	—	4	1	no	—
30	C₄×C₁₉₇	C₂²⋊C₈	1	1	yes	characteristic
31	C₄×C₁₉₇	M₃₂	1	1	yes	characteristic
32	Dic₁₉₇	—	2	1	no	—
33	C₂×C₁₉₇	—	2	1	no	—
34	C₂×C₁₉₇	C₂²⋊C₁₆	1	1	yes	characteristic
35	D₁₉₇	—	4	1	no	—
36	C₁₉₇	#1400	1	1	yes	characteristic, complemented, Sylow
37	#1400	—	197	1	no	maximal, Sylow
38	C₂×M₃₂	—	197	1	no	—
39	M₆₄	—	197	1	no	—
40	M₆₄	D₁₉₇	1	1	yes	characteristic, complemented
41	C₂²×C₈	—	197	1	no	—
42	M₃₂	—	394	1	no	—
43	M₃₂	—	394	1	no	—
44	C₂×C₁₆	—	197	1	no	—
45	C₂×C₁₆	C₂×D₁₉₇	1	1	yes	characteristic
46	C₃₂	—	2	1	no	—
47	C₃₂	—	394	1	no	—
48	C₂²×C₄	—	197	1	no	—
49	C₂×C₈	—	394	1	no	—
50	C₂×C₈	—	394	1	no	—
51	C₂×C₈	—	197	1	no	—
52	C₂×C₈	C₄×D₁₉₇	1	1	yes	characteristic
53	C₁₆	—	394	1	no	—
54	C₁₆	C₂²⋊D₁₉₇	1	1	yes	characteristic
55	C₁₆	D₇₈₈	1	1	yes	characteristic
56	C₂³	—	197	1	no	—
57	C₂×C₄	—	394	1	no	—
58	C₂×C₄	—	394	1	no	—
59	C₂×C₄	—	197	1	no	—
60	C₂×C₄	C₈×D₁₉₇	1	1	yes	characteristic
61	C₈	—	394	1	no	—
62	C₈	#15691335	1	1	yes	characteristic
63	C₈	#15691341	1	1	yes	characteristic
64	C₂²	—	788	1	no	—
65	C₂²	—	394	1	no	—
66	C₂²	C₁₆×D₁₉₇	1	1	yes	characteristic
67	C₄	—	394	1	no	—
68	C₄	#20263238	1	1	yes	characteristic
69	C₄	#20263265	1	1	yes	characteristic
70	C₂	—	2	1	no	—
71	C₂	—	788	1	no	—
72	C₂	#27160858	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#15691335
commutator	C₂²×C₁₉₇	C₂×C₁₆
Frattini	C₂×C₁₆	C₂×D₁₉₇
Fitting	M₆₄×C₁₉₇	C₂
socle	C₂×C₁₉₇	C₂²⋊C₁₆
prime radical	C₂²×D₁₉₇	C₁₆
cosocle	C₂×C₁₆×C₁₉₇	C₂²
hypercenter	M₆₄	D₁₉₇
hypocenter	C₁₉₇	#1400
LCS	G ▷ C₂²×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₈ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
2. #27160913	C₂	semidirect
3. M₆₄×C₁₉₇	C₂	semidirect
36. C₁₉₇	#1400	semidirect
40. M₆₄	D₁₉₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#20263265	2
#27160913	2
M₆₄×C₁₉₇	2
#1400	197