Subgroups — #27139768

Order 12416.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261100	C₂	1	1	yes	maximal, characteristic
2	#20260513	C₂	1	1	yes	maximal, characteristic, complemented
3	#347×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#15690807	C₄	1	1	yes	characteristic
5	#15690802	C₄	1	1	yes	characteristic
6	#15690750	—	2	1	no	—
7	#15690732	—	2	1	no	—
8	#15690732	—	2	1	no	—
9	M₃₂×C₉₇	—	2	1	no	—
10	#97×C₉₇	C₂²	1	1	yes	characteristic
11	C₂×D₃₈₈	—	2	1	no	—
12	C₂×D₃₈₈	—	4	1	no	—
13	C₄⋊Dic₉₇	—	2	1	no	—
14	#13095490	—	4	1	no	—
15	#13095486	—	2	2	no	—
16	C₂×C₈×C₉₇	D₄	1	1	yes	characteristic
17	C₂×C₈×C₉₇	D₄	1	1	yes	characteristic
18	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
19	C₁₆×C₉₇	—	2	2	no	—
20	C₂²×D₉₇	—	4	1	no	—
21	C₂×C₄×C₉₇	—	2	1	no	—
22	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
23	C₂×Dic₉₇	—	4	1	no	—
24	D₃₈₈	—	4	1	no	—
25	D₃₈₈	—	4	1	no	—
26	D₃₈₈	—	8	2	no	—
27	C₈×C₉₇	—	2	1	no	—
28	C₈×C₉₇	—	2	1	no	—
29	C₂²×C₉₇	#100	1	1	yes	characteristic
30	C₂×D₉₇	—	8	2	no	—
31	C₂×D₉₇	—	8	1	no	—
32	C₄×C₉₇	—	4	1	no	—
33	C₄×C₉₇	#104	1	1	yes	characteristic
34	C₄×C₉₇	#104	1	1	yes	characteristic
35	Dic₉₇	—	8	1	no	—
36	C₂×C₉₇	—	2	1	no	—
37	C₂×C₉₇	#329	1	1	yes	characteristic
38	D₉₇	—	16	1	no	—
39	#1355	—	97	1	no	maximal, Sylow
40	C₉₇	#1355	1	1	yes	characteristic, complemented, Sylow
41	#490	—	97	1	no	—
42	#347	—	97	1	no	—
43	#347	D₉₇	1	1	yes	characteristic, complemented
44	#127	—	97	1	no	—
45	#125	—	97	1	no	—
46	M₃₂	—	2	1	no	—
47	M₃₂	—	194	1	no	—
48	#102	—	194	1	no	—
49	#102	—	194	1	no	—
50	#97	C₂×D₉₇	1	1	yes	characteristic
51	C₂×D₄	—	388	1	no	—
52	C₂×D₄	—	194	1	no	—
53	C₂×C₈	D₃₈₈	1	1	yes	characteristic
54	C₂×C₈	C₂²⋊D₉₇	1	1	yes	characteristic
55	C₄⋊C₄	—	388	1	no	—
56	C₄⋊C₄	—	194	1	no	—
57	C₄²	C₄×D₉₇	1	1	yes	characteristic
58	C₁₆	—	2	2	no	—
59	C₁₆	—	194	2	no	—
60	C₂³	—	388	1	no	—
61	D₄	—	776	2	no	—
62	D₄	—	388	1	no	—
63	D₄	—	388	1	no	—
64	C₂×C₄	—	2	1	no	—
65	C₂×C₄	—	388	1	no	—
66	C₂×C₄	#13095488	1	1	yes	characteristic
67	C₈	—	2	1	no	—
68	C₈	—	2	1	no	—
69	C₂²	—	776	2	no	—
70	C₂²	—	776	1	no	—
71	C₂²	#15690726	1	1	yes	characteristic
72	C₄	—	4	1	no	—
73	C₄	—	776	1	no	—
74	C₄	#15690736	1	1	yes	characteristic
75	C₄	#15690738	1	1	yes	characteristic
76	C₂	—	2	1	no	—
77	C₂	—	1552	1	no	—
78	C₂	#20260474	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20260474
commutator	C₄²×C₉₇	C₂×C₄
Frattini	#97	C₂×D₉₇
Fitting	#347×C₉₇	C₂
socle	C₂×C₉₇	#329
prime radical	#15690807	C₄
cosocle	#97×C₉₇	C₂²
hypercenter	#347	D₉₇
hypocenter	C₉₇	#1355
LCS	G ▷ C₄²×C₉₇ ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #97 ◁ #347

Complements

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

N	H ≅ G/N	type
2. #20260513	C₂	semidirect
3. #347×C₉₇	C₂	semidirect
40. C₉₇	#1355	semidirect
43. #347	D₉₇	semidirect

Maximal subgroups

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

subgroup	index
#20261100	2
#20260513	2
#347×C₉₇	2
#1355	97