Subgroups — #38740850

Order 25216.

82 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20263232	C₂	1	2	yes	maximal, complemented
2	C₂²×C₁₆×C₁₉₇	C₂	1	1	yes	maximal, characteristic
3	#20263232	—	2	2	no	—
4	C₂²×C₈×C₁₉₇	C₄	1	1	yes	characteristic
5	C₂×C₁₆×C₁₉₇	—	2	2	no	—
6	C₂×C₁₆×C₁₉₇	C₄	1	1	yes	characteristic
7	C₂×C₁₆×C₁₉₇	C₂²	1	1	yes	characteristic
8	C₂²×C₄×C₁₉₇	C₈	1	1	yes	characteristic
9	C₂×C₈×C₁₉₇	—	2	1	no	—
10	C₂×C₈×C₁₉₇	—	2	1	no	—
11	C₂×C₈×C₁₉₇	C₈	1	1	yes	characteristic
12	C₂×C₈×C₁₉₇	C₂×C₄	1	1	yes	characteristic
13	C₁₆×C₁₉₇	—	2	1	no	—
14	C₁₆×C₁₉₇	D₄	1	2	yes	—
15	C₂³×C₁₉₇	C₁₆	1	1	yes	characteristic
16	C₂×C₄×C₁₉₇	—	2	1	no	—
17	C₂×C₄×C₁₉₇	—	2	1	no	—
18	C₂×C₄×C₁₉₇	C₁₆	1	1	yes	characteristic
19	C₂×C₄×C₁₉₇	C₂×C₈	1	1	yes	characteristic
20	C₈×C₁₉₇	—	2	1	no	—
21	C₈×C₁₉₇	C₂²⋊C₄	1	1	yes	characteristic
22	C₈×C₁₉₇	M₁₆	1	1	yes	characteristic
23	C₂²×C₁₉₇	—	2	1	no	—
24	C₂²×C₁₉₇	—	2	1	no	—
25	C₂²×C₁₉₇	C₃₂	1	2	yes	complemented
26	C₂²×C₁₉₇	C₂×C₁₆	1	1	yes	characteristic
27	C₄×C₁₉₇	—	2	1	no	—
28	C₄×C₁₉₇	C₂²⋊C₈	1	1	yes	characteristic
29	C₄×C₁₉₇	M₃₂	1	1	yes	characteristic
30	C₂×C₁₉₇	—	2	2	no	—
31	C₂×C₁₉₇	C₂²⋊C₁₆	1	1	yes	characteristic
32	C₂×C₁₉₇	C₂×C₃₂	1	1	yes	characteristic
33	C₂×C₁₉₇	M₆₄	1	1	yes	characteristic
34	C₁₉₇	C₂²⋊C₃₂	1	1	yes	characteristic, complemented, Sylow
35	C₂²⋊C₃₂	—	197	1	no	maximal, Sylow
36	C₂²×C₁₆	D₁₉₇	1	1	yes	characteristic
37	C₂×C₃₂	—	197	2	no	—
38	C₂²×C₈	Dic₁₉₇	1	1	yes	characteristic
39	C₂×C₁₆	—	2	2	no	—
40	C₂×C₁₆	Dic₁₉₇	1	1	yes	characteristic
41	C₂×C₁₆	C₂×D₁₉₇	1	1	yes	characteristic
42	C₃₂	—	394	2	no	—
43	C₂²×C₄	#13096802	1	1	yes	characteristic
44	C₂×C₈	—	2	1	no	—
45	C₂×C₈	—	2	1	no	—
46	C₂×C₈	#13096802	1	1	yes	characteristic
47	C₂×C₈	C₂×Dic₁₉₇	1	1	yes	characteristic
48	C₁₆	—	2	1	no	—
49	C₁₆	C₂²⋊D₁₉₇	1	2	yes	—
50	C₂³	#15691333	1	1	yes	characteristic
51	C₂×C₄	—	2	1	no	—
52	C₂×C₄	—	2	1	no	—
53	C₂×C₄	#15691333	1	1	yes	characteristic
54	C₂×C₄	C₂×#13096802	1	1	yes	characteristic
55	C₈	—	2	1	no	—
56	C₈	C₂²⋊Dic₁₉₇	1	1	yes	characteristic
57	C₈	#15691342	1	1	yes	characteristic
58	C₂²	—	2	1	no	—
59	C₂²	—	2	1	no	—
60	C₂²	#20263232	1	2	yes	complemented
61	C₂²	C₂×#15691333	1	1	yes	characteristic
62	C₄	—	2	1	no	—
63	C₄	C₂²⋊#13096802	1	1	yes	characteristic
64	C₄	#20263266	1	1	yes	characteristic
65	C₂	—	2	2	no	—
66	C₂	C₂²⋊#15691333	1	1	yes	characteristic
67	C₂	C₂×#20263232	1	1	yes	characteristic
68	C₂	#27160913	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂×D₁₉₇
commutator	C₂×C₁₉₇	C₂×C₃₂
Fitting	C₂²×C₁₆×C₁₉₇	C₂
socle	C₂²×C₁₉₇	C₂×C₁₆
prime radical	C₂³×C₁₉₇	C₁₆
cosocle	C₂×C₁₆×C₁₉₇	C₂²
hypercenter	C₂²×C₁₆	D₁₉₇
hypocenter	C₁₉₇	C₂²⋊C₃₂
LCS	G ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#20263232	C₂	semidirect
25. C₂²×C₁₉₇	C₃₂	semidirect
34. C₁₉₇	C₂²⋊C₃₂	semidirect
60. C₂²	#20263232	semidirect

Maximal subgroups

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

subgroup	index
C₂×#20263232	2
C₂²×C₁₆×C₁₉₇	2
C₂²⋊C₃₂	197