Subgroups — #38740816

Order 25216.

124 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20263262	C₂	1	2	yes	maximal
2	C₂×M₃₂×C₁₉₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#15691342	—	2	2	no	—
4	#20263262	—	2	2	no	—
5	#20263262	C₄	1	4	yes	—
6	C₂²×C₈×C₁₉₇	C₂²	1	1	yes	characteristic
7	M₃₂×C₁₉₇	—	2	2	no	—
8	C₂×C₁₆×C₁₉₇	C₄	1	2	yes	—
9	#15691342	—	2	4	no	—
10	#15691342	—	4	2	no	—
11	C₂×#13096802	—	4	2	no	—
12	C₂²×C₄×C₁₉₇	D₄	1	1	yes	characteristic
13	C₂×C₈×C₁₉₇	C₂×C₄	1	2	yes	—
14	C₂×C₈×C₁₉₇	C₂×C₄	1	1	yes	characteristic
15	C₂×C₈×C₁₉₇	D₄	1	2	yes	—
16	C₂×C₈×C₁₉₇	Q₈	1	1	yes	characteristic
17	C₁₆×C₁₉₇	—	2	2	no	—
18	#13096802	—	4	4	no	—
19	C₂³×C₁₉₇	Q₁₆	1	1	yes	characteristic
20	C₂×C₄×C₁₉₇	C₂²⋊C₄	1	2	yes	—
21	C₂×C₄×C₁₉₇	C₄⋊C₄	1	1	yes	characteristic
22	C₂×C₄×C₁₉₇	D₈	1	1	yes	characteristic
23	C₂×C₄×C₁₉₇	SD₁₆	1	2	yes	—
24	C₈×C₁₉₇	C₄²	1	1	yes	characteristic
25	C₈×C₁₉₇	C₂²⋊C₄	1	1	yes	characteristic
26	C₈×C₁₉₇	C₄⋊C₄	1	2	yes	—
27	C₂²×C₁₉₇	—	4	1	no	—
28	C₂²×C₁₉₇	#103	1	2	yes	—
29	C₂²×C₁₉₇	#107	1	1	yes	characteristic
30	C₄×C₁₉₇	#95	1	1	yes	characteristic
31	C₄×C₁₉₇	#102	1	2	yes	—
32	C₄×C₁₉₇	#106	1	1	yes	characteristic
33	C₂×C₁₉₇	—	2	2	no	—
34	C₂×C₁₉₇	—	2	1	no	—
35	C₂×C₁₉₇	#340	1	1	yes	characteristic
36	C₁₉₇	#1385	1	1	yes	characteristic, complemented, Sylow
37	#1385	—	197	1	no	maximal, Sylow
38	C₂×M₃₂	D₁₉₇	1	1	yes	characteristic
39	C₂×#108	—	197	2	no	—
40	C₂×M₁₆	—	394	2	no	—
41	C₂²×C₈	C₂×D₁₉₇	1	1	yes	characteristic
42	M₃₂	—	2	2	no	—
43	C₂×C₁₆	Dic₁₉₇	1	2	yes	—
44	#108	—	394	2	no	—
45	#108	—	197	4	no	—
46	C₂²×C₄	D₇₈₈	1	1	yes	characteristic
47	M₁₆	—	788	2	no	—
48	M₁₆	—	394	4	no	—
49	C₂×C₈	—	788	2	no	—
50	C₂×C₈	C₄×D₁₉₇	1	2	yes	—
51	C₂×C₈	C₂×Dic₁₉₇	1	1	yes	characteristic
52	C₂×C₈	C₂²⋊D₁₉₇	1	2	yes	—
53	C₂×C₈	Dic₃₉₄	1	1	yes	characteristic
54	C₁₆	—	2	2	no	—
55	C₂³	Dic₇₈₈	1	1	yes	characteristic
56	C₂×C₄	#15691335	1	2	yes	—
57	C₂×C₄	C₄⋊Dic₁₉₇	1	1	yes	characteristic
58	C₂×C₄	D₁₅₇₆	1	1	yes	characteristic
59	C₂×C₄	#15691347	1	2	yes	—
60	C₈	—	788	4	no	—
61	C₈	C₄×Dic₁₉₇	1	1	yes	characteristic
62	C₈	C₂²⋊Dic₁₉₇	1	1	yes	characteristic
63	C₈	#15691337	1	2	yes	—
64	C₂²	—	4	1	no	—
65	C₂²	#20263251	1	2	yes	—
66	C₂²	#20263260	1	1	yes	characteristic
67	C₄	#20263233	1	1	yes	characteristic
68	C₄	#20263248	1	2	yes	—
69	C₄	#20263258	1	1	yes	characteristic
70	C₂	—	2	2	no	—
71	C₂	—	2	1	no	—
72	C₂	#27160842	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#20263258
commutator	C₈×C₁₉₇	C₄²
Frattini	C₂²×C₈	C₂×D₁₉₇
Fitting	C₂×M₃₂×C₁₉₇	C₂
socle	C₂×C₁₉₇	#340
prime radical	C₂³×C₁₉₇	Q₁₆
cosocle	C₂²×C₈×C₁₉₇	C₂²
hypercenter	C₂×M₃₂	D₁₉₇
hypocenter	C₁₉₇	#1385
LCS	G ▷ C₈×C₁₉₇ ▷ C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

Complements

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

N	H ≅ G/N	type
36. C₁₉₇	#1385	semidirect

Maximal subgroups

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

subgroup	index
C₂×#20263262	2
C₂×M₃₂×C₁₉₇	2
#1385	197