Subgroups — #27156767

Order 12416.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261598	C₂	1	2	yes	maximal
2	C₁₆×Dic₉₇	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#11715897	—	2	2	no	—
4	C₂×#13095486	C₄	1	1	yes	characteristic
5	C₈×Dic₉₇	C₂²	1	1	yes	characteristic
6	C₂×C₁₆×C₉₇	C₄	1	1	yes	characteristic
7	C₂×#11715897	—	4	2	no	—
8	C₂×#11715895	Q₈	1	1	yes	characteristic
9	C₄×Dic₉₇	D₄	1	1	yes	characteristic
10	#13095486	—	2	1	no	—
11	C₂×C₈×C₉₇	C₂×C₄	1	1	yes	characteristic
12	C₁₆×C₉₇	C₈	1	2	yes	complemented
13	C₂×C₄×C₉₇	C₄⋊C₄	1	1	yes	characteristic
14	C₂×Dic₉₇	D₈	1	1	yes	characteristic
15	C₂×Dic₉₇	Q₁₆	1	1	yes	characteristic
16	#11715897	—	8	2	no	—
17	C₈×C₉₇	C₂×C₈	1	1	yes	characteristic
18	C₈×C₉₇	M₁₆	1	1	yes	characteristic
19	#11715895	—	2	1	no	—
20	C₂²×C₉₇	#107	1	1	yes	characteristic
21	C₄×C₉₇	C₄⋊C₈	1	1	yes	characteristic
22	C₄×C₉₇	#108	1	1	yes	characteristic
23	Dic₉₇	—	2	1	no	—
24	Dic₉₇	D₁₆	1	1	yes	characteristic
25	Dic₉₇	Q₃₂	1	1	yes	characteristic
26	C₂×C₉₇	#335	1	1	yes	characteristic
27	C₂×C₉₇	#366	1	1	yes	characteristic
28	C₂×C₉₇	#368	1	1	yes	characteristic
29	#1371	—	97	1	no	maximal, Sylow
30	C₉₇	#1371	1	1	yes	characteristic, complemented, Sylow
31	C₄×C₁₆	—	97	1	no	—
32	#335	—	97	2	no	—
33	C₂×C₁₆	—	97	1	no	—
34	C₂×C₁₆	#87604	1	1	yes	characteristic
35	C₄⋊C₈	—	194	2	no	—
36	C₄×C₈	—	97	1	no	—
37	C₂×C₈	—	97	1	no	—
38	C₂×C₈	—	388	2	no	—
39	C₂×C₈	C₂×#87604	1	1	yes	characteristic
40	C₄²	—	97	1	no	—
41	C₁₆	—	194	1	no	—
42	C₁₆	#11715897	1	2	yes	complemented
43	C₂×C₄	—	97	1	no	—
44	C₂×C₄	—	97	1	no	—
45	C₂×C₄	C₄⋊#87604	1	1	yes	characteristic
46	C₈	—	776	2	no	—
47	C₈	—	194	1	no	—
48	C₈	C₂×#11715897	1	1	yes	characteristic
49	C₈	#13095521	1	1	yes	characteristic
50	C₂²	#15690879	1	1	yes	characteristic
51	C₄	—	97	1	no	—
52	C₄	—	97	1	no	—
53	C₄	—	194	1	no	—
54	C₄	C₄⋊#11715897	1	1	yes	characteristic
55	C₄	#15690881	1	1	yes	characteristic
56	C₂	#20261598	1	1	yes	characteristic
57	C₂	#20261661	1	1	yes	characteristic
58	C₂	#20261663	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#15690879
commutator	C₈×C₉₇	C₂×C₈
Frattini	C₂×C₈	C₂×#87604
Fitting	C₂×C₁₆×C₉₇	C₄
socle, prime radical	C₂²×C₉₇	#107
cosocle	C₈×Dic₉₇	C₂²
hypercenter	C₂×C₁₆	#87604
hypocenter	C₉₇	#1371
LCS	G ▷ C₈×C₉₇ ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
12. C₁₆×C₉₇	C₈	semidirect
30. C₉₇	#1371	semidirect
42. C₁₆	#11715897	semidirect

Maximal subgroups

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

subgroup	index
#20261598	2
C₁₆×Dic₉₇	2
#1371	97