Subgroups — #27139921

Order 12416.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261167	C₂	1	1	yes	maximal, characteristic
2	#20260565	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₂×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#15690832	—	2	1	no	—
5	Dic₇₇₆	C₄	1	1	yes	characteristic
6	#15690755	—	2	1	no	—
7	D₁₅₅₂	C₄	1	1	yes	characteristic
8	#15690746	—	2	1	no	—
9	C₂×C₁₆×C₉₇	C₂²	1	1	yes	characteristic
10	C₃₂×C₉₇	—	2	1	no	—
11	#13095511	—	4	1	no	—
12	Dic₃₈₈	—	2	1	no	—
13	#13095500	—	4	1	no	—
14	D₇₇₆	—	2	1	no	—
15	#13095495	—	4	1	no	—
16	C₂×C₈×C₉₇	D₄	1	1	yes	characteristic
17	C₁₆×C₉₇	C₂×C₄	1	1	yes	characteristic
18	C₁₆×C₉₇	D₄	1	1	yes	characteristic
19	C₂×C₄×C₉₇	D₈	1	1	yes	characteristic
20	C₂²⋊D₉₇	—	8	1	no	—
21	D₃₈₈	—	4	1	no	—
22	C₄×D₉₇	—	8	1	no	—
23	Dic₁₉₄	—	4	1	no	—
24	C₈×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
25	C₈×C₉₇	SD₁₆	1	1	yes	characteristic
26	#11715895	—	8	1	no	—
27	C₂²×C₉₇	SD₃₂	1	1	yes	characteristic
28	C₂×D₉₇	—	8	1	no	—
29	C₄×C₉₇	#102	1	1	yes	characteristic
30	C₄×C₉₇	D₁₆	1	1	yes	characteristic
31	Dic₉₇	—	8	1	no	—
32	C₂×C₉₇	—	2	1	no	—
33	C₂×C₉₇	#357	1	1	yes	characteristic
34	D₉₇	—	16	1	no	—
35	#1417	—	97	1	no	maximal, Sylow
36	C₉₇	#1417	1	1	yes	characteristic, complemented, Sylow
37	#508	—	97	1	no	—
38	C₂×C₃₂	D₉₇	1	1	yes	characteristic, complemented
39	#368	—	97	1	no	—
40	#135	—	194	1	no	—
41	Q₃₂	—	97	1	no	—
42	SD₃₂	—	194	1	no	—
43	D₁₆	—	97	1	no	—
44	C₂×C₁₆	C₂×D₉₇	1	1	yes	characteristic
45	#108	—	194	1	no	—
46	C₃₂	—	2	1	no	—
47	#41	—	388	1	no	—
48	Q₁₆	—	194	1	no	—
49	SD₁₆	—	388	1	no	—
50	D₈	—	194	1	no	—
51	M₁₆	—	388	1	no	—
52	C₂×C₈	D₃₈₈	1	1	yes	characteristic
53	C₁₆	C₄×D₉₇	1	1	yes	characteristic
54	C₁₆	C₂²⋊D₉₇	1	1	yes	characteristic
55	Q₈	—	388	1	no	—
56	D₄	—	776	1	no	—
57	D₄	—	388	1	no	—
58	C₂×C₄	—	776	1	no	—
59	C₂×C₄	D₇₇₆	1	1	yes	characteristic
60	C₈	—	776	1	no	—
61	C₈	#13095488	1	1	yes	characteristic
62	C₈	#13095500	1	1	yes	characteristic
63	C₂²	—	776	1	no	—
64	C₂²	#15690755	1	1	yes	characteristic
65	C₄	—	776	1	no	—
66	C₄	#15690732	1	1	yes	characteristic
67	C₄	D₁₅₅₂	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	1552	1	no	—
70	C₂	#20260537	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₁₅₅₂
commutator	C₁₆×C₉₇	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₉₇
Fitting	C₂×C₃₂×C₉₇	C₂
socle	C₂×C₉₇	#357
prime radical	#20261167	C₂
cosocle	C₂×C₁₆×C₉₇	C₂²
hypercenter	C₂×C₃₂	D₉₇
hypocenter	C₉₇	#1417
LCS	G ▷ C₁₆×C₉₇ ▷ C₈×C₉₇ ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂

Complements

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

N	H ≅ G/N	type
2. #20260565	C₂	semidirect
3. C₂×C₃₂×C₉₇	C₂	semidirect
36. C₉₇	#1417	semidirect
38. C₂×C₃₂	D₉₇	semidirect

Maximal subgroups

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

subgroup	index
#20261167	2
#20260565	2
C₂×C₃₂×C₉₇	2
#1417	97