Subgroups — #27139794

Order 12416.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#13095486	C₂	1	1	yes	maximal, characteristic
2	#20260488	C₂	1	1	yes	maximal, characteristic
3	#500×C₉₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#13095486	—	2	1	no	—
5	C₄⋊#11715895	—	2	1	no	—
6	C₄⋊Q₈×C₉₇	—	2	1	no	—
7	#107×C₉₇	—	2	1	no	—
8	#107×C₉₇	C₄	1	2	yes	—
9	C₄×C₈×C₉₇	C₂²	1	1	yes	characteristic
10	C₂×#11715895	—	4	1	no	—
11	#13095486	—	4	1	no	—
12	C₂×Q₈×C₉₇	—	4	1	no	—
13	C₂×C₈×C₉₇	C₂×C₄	1	1	yes	characteristic
14	C₂×C₈×C₉₇	D₄	1	1	yes	characteristic
15	C₄⋊C₄×C₉₇	—	2	2	no	—
16	C₄⋊C₄×C₉₇	—	4	1	no	—
17	C₄²×C₉₇	D₄	1	1	yes	characteristic
18	Q₈×C₉₇	—	8	2	no	—
19	C₂×C₄×C₉₇	—	4	2	no	—
20	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
21	C₂×C₄×C₉₇	D₈	1	1	yes	characteristic
22	C₂×C₄×C₉₇	SD₁₆	1	1	yes	characteristic
23	C₈×C₉₇	—	2	1	no	—
24	C₈×C₉₇	SD₁₆	1	1	yes	characteristic
25	C₈×C₉₇	Q₁₆	1	1	yes	characteristic
26	#11715895	—	8	1	no	—
27	C₂²×C₉₇	#102	1	1	yes	characteristic
28	C₄×C₉₇	—	2	1	no	—
29	C₄×C₉₇	—	8	2	no	—
30	C₄×C₉₇	#101	1	1	yes	characteristic
31	C₄×C₉₇	#103	1	1	yes	characteristic
32	C₄×C₉₇	SD₃₂	1	1	yes	characteristic
33	C₄×C₉₇	Q₃₂	1	1	yes	characteristic
34	C₂×C₉₇	#332	1	1	yes	characteristic
35	C₂×C₉₇	#358	1	1	yes	characteristic
36	C₂×C₉₇	#361	1	1	yes	characteristic
37	#1365	—	97	1	no	maximal, Sylow
38	C₉₇	#1365	1	1	yes	characteristic, complemented, Sylow
39	#500	D₉₇	1	1	yes	characteristic
40	C₄⋊C₁₆	—	97	1	no	—
41	#335	—	97	1	no	—
42	C₄⋊Q₈	—	2	1	no	—
43	C₂×C₁₆	—	194	1	no	—
44	#107	—	2	1	no	—
45	#107	Dic₉₇	1	2	yes	—
46	C₄⋊C₈	—	194	1	no	—
47	C₄×C₈	C₂×D₉₇	1	1	yes	characteristic
48	C₂×Q₈	—	4	1	no	—
49	C₂×C₈	—	388	1	no	—
50	C₂×C₈	C₂×Dic₉₇	1	1	yes	characteristic
51	C₂×C₈	C₂²⋊D₉₇	1	1	yes	characteristic
52	C₄⋊C₄	—	2	2	no	—
53	C₄⋊C₄	—	4	1	no	—
54	C₄²	C₂²⋊D₉₇	1	1	yes	characteristic
55	C₁₆	—	388	1	no	—
56	Q₈	—	8	2	no	—
57	C₂×C₄	—	4	2	no	—
58	C₂×C₄	C₂²⋊Dic₉₇	1	1	yes	characteristic
59	C₂×C₄	#13095496	1	1	yes	characteristic
60	C₂×C₄	#13095499	1	1	yes	characteristic
61	C₈	—	2	1	no	—
62	C₈	—	776	1	no	—
63	C₈	#13095498	1	1	yes	characteristic
64	C₈	#13095501	1	1	yes	characteristic
65	C₂²	#15690731	1	1	yes	characteristic
66	C₄	—	2	1	no	—
67	C₄	—	8	2	no	—
68	C₄	#15690729	1	1	yes	characteristic
69	C₄	#15690734	1	1	yes	characteristic
70	C₄	#15690753	1	1	yes	characteristic
71	C₄	#15690756	1	1	yes	characteristic
72	C₂	#20260481	1	1	yes	characteristic
73	C₂	#20260539	1	1	yes	characteristic
74	C₂	#20260548	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#15690731
commutator	C₂×C₈×C₉₇	C₂×C₄
Frattini	C₄×C₈	C₂×D₉₇
Fitting	#500×C₉₇	C₂
socle, prime radical	C₂²×C₉₇	#102
cosocle	C₄×C₈×C₉₇	C₂²
hypercenter	#500	D₉₇
hypocenter	C₉₇	#1365
LCS	G ▷ C₂×C₈×C₉₇ ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #500

Complements

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

N	H ≅ G/N	type
38. C₉₇	#1365	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊#13095486	2
#20260488	2
#500×C₉₇	2
#1365	97