Subgroups — #38740781

Order 25216.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#15691333	C₂	1	1	yes	maximal, characteristic
2	C₈×#13096802	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₁₆×C₁₉₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#15691333	—	2	1	no	—
5	C₄×#13096802	C₄	1	1	yes	characteristic
6	C₄×#13096802	C₄	1	1	yes	characteristic
7	C₂×C₁₆×C₁₉₇	—	2	1	no	—
8	C₄×C₈×C₁₉₇	C₂²	1	1	yes	characteristic
9	C₂×#13096802	—	2	1	no	—
10	C₂×#13096802	C₈	1	1	yes	characteristic
11	C₂×#13096802	C₈	1	1	yes	characteristic
12	#15691333	—	4	1	no	—
13	C₂×C₈×C₁₉₇	D₄	1	1	yes	characteristic
14	C₂×C₈×C₁₉₇	Q₈	1	1	yes	characteristic
15	C₄²×C₁₉₇	C₂×C₄	1	1	yes	characteristic
16	C₁₆×C₁₉₇	—	4	1	no	—
17	#13096802	—	2	2	no	—
18	#13096802	—	2	1	no	—
19	#13096802	C₁₆	1	2	yes	complemented
20	C₂×C₄×C₁₉₇	C₄⋊C₄	1	1	yes	characteristic
21	C₂×C₄×C₁₉₇	C₂×C₈	1	1	yes	characteristic
22	C₂×C₄×C₁₉₇	M₁₆	1	1	yes	characteristic
23	C₈×C₁₉₇	—	2	1	no	—
24	C₈×C₁₉₇	D₈	1	1	yes	characteristic
25	C₈×C₁₉₇	Q₁₆	1	1	yes	characteristic
26	C₂²×C₁₉₇	C₄⋊C₈	1	1	yes	characteristic
27	C₄×C₁₉₇	—	2	1	no	—
28	C₄×C₁₉₇	#107	1	1	yes	characteristic
29	C₄×C₁₉₇	#108	1	1	yes	characteristic
30	C₄×C₁₉₇	C₂×C₁₆	1	1	yes	characteristic
31	C₄×C₁₉₇	M₃₂	1	1	yes	characteristic
32	C₂×C₁₉₇	#335	1	1	yes	characteristic
33	C₂×C₁₉₇	C₄⋊C₁₆	1	1	yes	characteristic
34	C₂×C₁₉₇	#364	1	1	yes	characteristic
35	C₁₉₇	#1370	1	1	yes	characteristic, complemented, Sylow
36	#1370	—	197	1	no	maximal, Sylow
37	C₄⋊C₁₆	—	197	1	no	—
38	C₄⋊C₁₆	D₁₉₇	1	1	yes	characteristic
39	C₈²	—	197	1	no	—
40	C₂×C₁₆	—	2	1	no	—
41	C₂×C₁₆	—	394	1	no	—
42	C₄×C₈	—	197	1	no	—
43	C₄×C₈	—	197	1	no	—
44	C₄×C₈	C₂×D₁₉₇	1	1	yes	characteristic
45	C₂×C₈	—	394	1	no	—
46	C₂×C₈	—	197	1	no	—
47	C₂×C₈	—	197	1	no	—
48	C₂×C₈	C₂²⋊D₁₉₇	1	1	yes	characteristic
49	C₂×C₈	Dic₃₉₄	1	1	yes	characteristic
50	C₄²	C₄×D₁₉₇	1	1	yes	characteristic
51	C₁₆	—	4	1	no	—
52	C₁₆	—	788	1	no	—
53	C₂×C₄	#15691337	1	1	yes	characteristic
54	C₂×C₄	C₈×D₁₉₇	1	1	yes	characteristic
55	C₂×C₄	#15691341	1	1	yes	characteristic
56	C₈	—	2	1	no	—
57	C₈	—	394	2	no	—
58	C₈	—	394	1	no	—
59	C₈	—	197	2	no	—
60	C₈	#15691343	1	1	yes	characteristic
61	C₈	#15691348	1	1	yes	characteristic
62	C₂²	#20263255	1	1	yes	characteristic
63	C₄	—	2	1	no	—
64	C₄	#20263259	1	1	yes	characteristic
65	C₄	#20263261	1	1	yes	characteristic
66	C₄	C₁₆×D₁₉₇	1	1	yes	characteristic
67	C₄	#20263265	1	1	yes	characteristic
68	C₂	#27160831	1	1	yes	characteristic
69	C₂	#27160897	1	1	yes	characteristic
70	C₂	#27160899	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²⋊D₁₉₇
commutator	C₄×C₁₉₇	C₂×C₁₆
Frattini	C₄×C₈	C₂×D₁₉₇
Fitting	C₄⋊C₁₆×C₁₉₇	C₂
socle, prime radical	C₂²×C₁₉₇	C₄⋊C₈
cosocle	C₄×C₈×C₁₉₇	C₂²
hypercenter	C₄⋊C₁₆	D₁₉₇
hypocenter	C₁₉₇	#1370
LCS	G ▷ C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
19. #13096802	C₁₆	semidirect
35. C₁₉₇	#1370	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊#15691333	2
C₈×#13096802	2
C₄⋊C₁₆×C₁₉₇	2
#1370	197