Subgroups — #27139774

Order 12416.

92 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261115	C₂	1	1	yes	maximal, characteristic
2	#20260556	C₂	1	1	yes	maximal, characteristic
3	#364×C₉₇	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₃₈₈	—	2	1	no	—
5	C₂×Dic₃₈₈	—	2	1	no	—
6	#15690810	C₄	1	2	yes	—
7	#15690750	—	2	1	no	—
8	M₃₂×C₉₇	—	2	1	no	—
9	C₄×C₈×C₉₇	C₂²	1	1	yes	characteristic
10	C₂×Dic₁₉₄	—	2	2	no	—
11	Dic₃₈₈	—	4	1	no	—
12	Dic₃₈₈	—	4	1	no	—
13	Dic₃₈₈	—	4	1	no	—
14	Dic₃₈₈	—	4	1	no	—
15	C₄⋊Dic₉₇	—	4	2	no	—
16	#13095486	—	4	1	no	—
17	C₂×C₈×C₉₇	D₄	1	1	yes	characteristic
18	C₂×C₈×C₉₇	D₄	1	1	yes	characteristic
19	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
20	C₁₆×C₉₇	—	4	1	no	—
21	C₂×C₄×C₉₇	—	2	1	no	—
22	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
23	C₂×Dic₉₇	—	4	2	no	—
24	Dic₁₉₄	—	4	2	no	—
25	Dic₁₉₄	—	4	2	no	—
26	C₈×C₉₇	D₈	1	1	yes	characteristic
27	C₈×C₉₇	D₈	1	1	yes	characteristic
28	C₈×C₉₇	SD₁₆	1	1	yes	characteristic
29	C₈×C₉₇	SD₁₆	1	1	yes	characteristic
30	C₂²×C₉₇	#100	1	1	yes	characteristic
31	C₄×C₉₇	—	2	2	no	—
32	C₄×C₉₇	#102	1	1	yes	characteristic
33	C₄×C₉₇	#102	1	1	yes	characteristic
34	Dic₉₇	—	8	2	no	—
35	C₂×C₉₇	—	2	1	no	—
36	C₂×C₉₇	#331	1	1	yes	characteristic
37	#1358	—	97	1	no	maximal, Sylow
38	C₉₇	#1358	1	1	yes	characteristic, complemented, Sylow
39	#494	—	97	1	no	—
40	#364	—	97	1	no	—
41	#364	D₉₇	1	1	yes	characteristic
42	C₂×Q₁₆	—	194	1	no	—
43	C₂×Q₁₆	—	194	1	no	—
44	C₄⋊Q₈	—	97	2	no	—
45	M₃₂	—	2	1	no	—
46	M₃₂	—	194	1	no	—
47	C₄×C₈	C₂×D₉₇	1	1	yes	characteristic
48	C₂×Q₈	—	194	2	no	—
49	Q₁₆	—	388	1	no	—
50	Q₁₆	—	388	1	no	—
51	Q₁₆	—	388	1	no	—
52	Q₁₆	—	388	1	no	—
53	C₂×C₈	D₃₈₈	1	1	yes	characteristic
54	C₂×C₈	C₂²⋊D₉₇	1	1	yes	characteristic
55	C₄⋊C₄	—	388	2	no	—
56	C₄²	C₄×D₉₇	1	1	yes	characteristic
57	C₁₆	—	4	1	no	—
58	C₁₆	—	388	1	no	—
59	Q₈	—	388	2	no	—
60	Q₈	—	388	2	no	—
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	—	388	2	no	—
63	C₂×C₄	#13095488	1	1	yes	characteristic
64	C₈	#13095496	1	1	yes	characteristic
65	C₈	D₇₇₆	1	1	yes	characteristic
66	C₈	#13095498	1	1	yes	characteristic
67	C₈	#13095500	1	1	yes	characteristic
68	C₂²	#15690726	1	1	yes	characteristic
69	C₄	—	2	2	no	—
70	C₄	—	776	2	no	—
71	C₄	#15690730	1	1	yes	characteristic
72	C₄	#15690732	1	1	yes	characteristic
73	C₂	—	2	1	no	—
74	C₂	#20260478	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20260478
commutator	C₄²×C₉₇	C₂×C₄
Frattini	C₄×C₈	C₂×D₉₇
Fitting	#364×C₉₇	C₂
socle	C₂×C₉₇	#331
prime radical	C₂²×C₉₇	#100
cosocle	C₄×C₈×C₉₇	C₂²
hypercenter	#364	D₉₇
hypocenter	C₉₇	#1358
LCS	G ▷ C₄²×C₉₇ ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

Complements

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

N	H ≅ G/N	type
38. C₉₇	#1358	semidirect

Maximal subgroups

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

subgroup	index
#20261115	2
#20260556	2
#364×C₉₇	2
#1358	97