Subgroups — #27139688

Order 12416.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20260763	C₂	1	1	yes	maximal, characteristic
2	#20260517	C₂	1	1	yes	maximal, characteristic
3	#349×C₉₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11715895	C₄	1	1	yes	characteristic
5	#15690769	C₄	1	1	yes	characteristic
6	#15690750	—	2	1	no	—
7	#15690739	—	2	1	no	—
8	#15690739	—	2	1	no	—
9	C₂×M₁₆×C₉₇	C₂²	1	1	yes	characteristic
10	M₃₂×C₉₇	—	2	1	no	—
11	C₂×#11715895	—	2	1	no	—
12	C₂×#11715895	—	4	1	no	—
13	C₄⋊Dic₉₇	—	2	1	no	—
14	C₂²⋊Dic₉₇	C₈	1	2	yes	—
15	C₄×Dic₉₇	—	2	1	no	—
16	#13095486	—	4	1	no	—
17	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
18	M₁₆×C₉₇	—	4	1	no	—
19	C₂×C₈×C₉₇	D₄	1	1	yes	characteristic
20	C₂×C₈×C₉₇	D₄	1	1	yes	characteristic
21	C₁₆×C₉₇	—	4	1	no	—
22	C₂³×C₉₇	C₂×C₈	1	1	yes	characteristic
23	C₂×C₄×C₉₇	—	2	1	no	—
24	C₂×C₄×C₉₇	—	2	1	no	—
25	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
26	C₂×C₄×C₉₇	M₁₆	1	1	yes	characteristic
27	C₂×Dic₉₇	—	2	2	no	—
28	C₈×C₉₇	—	2	1	no	—
29	C₈×C₉₇	—	2	1	no	—
30	#11715895	—	4	1	no	—
31	C₂²×C₉₇	—	2	1	no	—
32	C₂²×C₉₇	—	4	1	no	—
33	C₂²×C₉₇	C₂²⋊C₈	1	1	yes	characteristic
34	C₄×C₉₇	—	2	1	no	—
35	C₄×C₉₇	#99	1	1	yes	characteristic
36	C₄×C₉₇	#101	1	1	yes	characteristic
37	Dic₉₇	—	4	2	no	—
38	C₂×C₉₇	—	2	1	no	—
39	C₂×C₉₇	—	4	1	no	—
40	C₂×C₉₇	#324	1	1	yes	characteristic
41	#1328	—	97	1	no	maximal, Sylow
42	C₉₇	#1328	1	1	yes	characteristic, complemented, Sylow
43	#424	—	97	1	no	—
44	#349	—	97	1	no	—
45	#349	D₉₇	1	1	yes	characteristic
46	C₂×M₁₆	C₂×D₉₇	1	1	yes	characteristic
47	C₂²×C₈	—	97	1	no	—
48	#117	—	97	1	no	—
49	M₃₂	—	2	1	no	—
50	M₃₂	—	194	1	no	—
51	C₄⋊C₈	—	194	1	no	—
52	C₄⋊C₈	—	194	1	no	—
53	C₂²×C₄	C₄×D₉₇	1	1	yes	characteristic
54	M₁₆	—	4	1	no	—
55	C₂×C₈	—	388	1	no	—
56	C₂×C₈	—	194	1	no	—
57	C₂×C₈	D₃₈₈	1	1	yes	characteristic
58	C₂×C₈	C₂²⋊D₉₇	1	1	yes	characteristic
59	C₄⋊C₄	—	194	1	no	—
60	C₂²⋊C₄	—	97	2	no	—
61	C₄²	—	194	1	no	—
62	C₁₆	—	4	1	no	—
63	C₁₆	—	388	1	no	—
64	C₂³	C₈×D₉₇	1	1	yes	characteristic
65	C₂×C₄	—	2	1	no	—
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	—	194	2	no	—
68	C₂×C₄	#13095488	1	1	yes	characteristic
69	C₂×C₄	#13095494	1	1	yes	characteristic
70	C₈	—	2	1	no	—
71	C₈	—	2	1	no	—
72	C₈	—	388	1	no	—
73	C₂²	—	2	1	no	—
74	C₂²	—	4	1	no	—
75	C₂²	#15690722	1	1	yes	characteristic
76	C₄	—	2	1	no	—
77	C₄	—	388	2	no	—
78	C₄	#15690724	1	1	yes	characteristic
79	C₄	#15690728	1	1	yes	characteristic
80	C₂	—	2	1	no	—
81	C₂	—	4	1	no	—
82	C₂	#20260460	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#15690724
commutator, prime radical	C₂³×C₉₇	C₂×C₈
Frattini	C₂×M₁₆	C₂×D₉₇
Fitting	#349×C₉₇	C₂
socle	C₂×C₉₇	#324
cosocle	C₂×M₁₆×C₉₇	C₂²
hypercenter	#349	D₉₇
hypocenter	C₉₇	#1328
LCS	G ▷ C₂³×C₉₇ ▷ C₂²×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ #349

Complements

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

N	H ≅ G/N	type
42. C₉₇	#1328	semidirect

Maximal subgroups

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

subgroup	index
#20260763	2
#20260517	2
#349×C₉₇	2
#1328	97