Subgroups — #27139927

Order 12416.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₅₅₂	C₂	1	1	yes	maximal, characteristic
2	#20260565	C₂	1	1	yes	maximal, characteristic, complemented
3	M₆₄×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₇₇₆	—	2	1	no	—
5	D₁₅₅₂	—	2	1	no	—
6	D₁₅₅₂	C₄	1	2	yes	—
7	#15690746	—	2	1	no	—
8	C₂×C₁₆×C₉₇	C₂²	1	1	yes	characteristic
9	C₃₂×C₉₇	—	2	1	no	—
10	C₂×D₃₈₈	—	4	1	no	—
11	D₇₇₆	—	2	2	no	—
12	D₇₇₆	—	4	1	no	—
13	#13095495	—	4	1	no	—
14	C₂×C₈×C₉₇	D₄	1	1	yes	characteristic
15	C₁₆×C₉₇	C₂×C₄	1	1	yes	characteristic
16	C₁₆×C₉₇	D₄	1	1	yes	characteristic
17	C₂²×D₉₇	—	8	1	no	—
18	C₂×C₄×C₉₇	D₈	1	1	yes	characteristic
19	D₃₈₈	—	4	2	no	—
20	D₃₈₈	—	8	1	no	—
21	C₈×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
22	C₈×C₉₇	SD₁₆	1	1	yes	characteristic
23	#11715895	—	8	1	no	—
24	C₂²×C₉₇	SD₃₂	1	1	yes	characteristic
25	C₂×D₉₇	—	8	2	no	—
26	C₂×D₉₇	—	16	2	no	—
27	C₄×C₉₇	#102	1	1	yes	characteristic
28	C₄×C₉₇	D₁₆	1	1	yes	characteristic
29	C₂×C₉₇	—	2	1	no	—
30	C₂×C₉₇	#357	1	1	yes	characteristic
31	D₉₇	—	16	2	no	—
32	#1419	—	97	1	no	maximal, Sylow
33	C₉₇	#1419	1	1	yes	characteristic, complemented, Sylow
34	C₂×D₁₆	—	97	1	no	—
35	M₆₄	D₉₇	1	1	yes	characteristic, complemented
36	#368	—	97	1	no	—
37	C₂×D₈	—	194	1	no	—
38	D₁₆	—	97	2	no	—
39	D₁₆	—	194	1	no	—
40	C₂×C₁₆	C₂×D₉₇	1	1	yes	characteristic
41	#108	—	194	1	no	—
42	C₃₂	—	2	1	no	—
43	C₂×D₄	—	388	1	no	—
44	D₈	—	388	1	no	—
45	D₈	—	194	2	no	—
46	M₁₆	—	388	1	no	—
47	C₂×C₈	D₃₈₈	1	1	yes	characteristic
48	C₁₆	C₄×D₉₇	1	1	yes	characteristic
49	C₁₆	C₂²⋊D₉₇	1	1	yes	characteristic
50	C₂³	—	776	1	no	—
51	D₄	—	776	1	no	—
52	D₄	—	388	2	no	—
53	C₂×C₄	D₇₇₆	1	1	yes	characteristic
54	C₈	—	776	1	no	—
55	C₈	#13095488	1	1	yes	characteristic
56	C₈	#13095500	1	1	yes	characteristic
57	C₂²	—	776	2	no	—
58	C₂²	—	1552	2	no	—
59	C₂²	#15690755	1	1	yes	characteristic
60	C₄	#15690732	1	1	yes	characteristic
61	C₄	D₁₅₅₂	1	1	yes	characteristic
62	C₂	—	2	1	no	—
63	C₂	—	1552	2	no	—
64	C₂	#20260537	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20260537
commutator	C₁₆×C₉₇	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₉₇
Fitting	M₆₄×C₉₇	C₂
socle	C₂×C₉₇	#357
prime radical	C₂×D₁₅₅₂	C₂
cosocle	C₂×C₁₆×C₉₇	C₂²
hypercenter	M₆₄	D₉₇
hypocenter	C₉₇	#1419
LCS	G ▷ C₁₆×C₉₇ ▷ C₈×C₉₇ ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
2. #20260565	C₂	semidirect
3. M₆₄×C₉₇	C₂	semidirect
33. C₉₇	#1419	semidirect
35. M₆₄	D₉₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₁₅₅₂	2
#20260565	2
M₆₄×C₉₇	2
#1419	97