Subgroups — #27139807

Order 12416.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13095486	C₂	1	1	yes	maximal, characteristic
2	#20260488	C₂	1	1	yes	maximal, characteristic
3	#335×C₉₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#13095486	C₄	1	1	yes	characteristic
5	C₂×#13095486	C₄	1	1	yes	characteristic
6	C₄⋊#11715895	—	2	1	no	—
7	C₄⋊C₈×C₉₇	—	2	1	no	—
8	C₄×C₈×C₉₇	C₂²	1	1	yes	characteristic
9	C₂×#11715895	—	4	1	no	—
10	#13095486	—	2	1	no	—
11	#13095486	C₈	1	2	yes	complemented
12	C₂×C₈×C₉₇	—	4	1	no	—
13	C₂×C₈×C₉₇	C₂×C₄	1	1	yes	characteristic
14	C₂×C₈×C₉₇	Q₈	1	1	yes	characteristic
15	C₄²×C₉₇	D₄	1	1	yes	characteristic
16	C₂×C₄×C₉₇	C₄⋊C₄	1	1	yes	characteristic
17	C₂×C₄×C₉₇	D₈	1	1	yes	characteristic
18	C₂×C₄×C₉₇	Q₁₆	1	1	yes	characteristic
19	C₈×C₉₇	—	2	1	no	—
20	C₈×C₉₇	—	8	1	no	—
21	C₈×C₉₇	C₂×C₈	1	1	yes	characteristic
22	C₈×C₉₇	M₁₆	1	1	yes	characteristic
23	#11715895	—	8	1	no	—
24	C₂²×C₉₇	#107	1	1	yes	characteristic
25	C₄×C₉₇	—	2	1	no	—
26	C₄×C₉₇	C₄⋊C₈	1	1	yes	characteristic
27	C₄×C₉₇	#108	1	1	yes	characteristic
28	C₄×C₉₇	D₁₆	1	1	yes	characteristic
29	C₄×C₉₇	Q₃₂	1	1	yes	characteristic
30	C₂×C₉₇	#335	1	1	yes	characteristic
31	C₂×C₉₇	#366	1	1	yes	characteristic
32	C₂×C₉₇	#368	1	1	yes	characteristic
33	#1371	—	97	1	no	maximal, Sylow
34	C₉₇	#1371	1	1	yes	characteristic, complemented, Sylow
35	C₄×C₁₆	—	97	1	no	—
36	#335	—	97	1	no	—
37	#335	D₉₇	1	1	yes	characteristic
38	C₂×C₁₆	—	97	1	no	—
39	C₂×C₁₆	—	97	1	no	—
40	C₄⋊C₈	—	2	1	no	—
41	C₄⋊C₈	—	194	1	no	—
42	C₄×C₈	C₂×D₉₇	1	1	yes	characteristic
43	C₂×C₈	—	4	1	no	—
44	C₂×C₈	—	388	1	no	—
45	C₂×C₈	Dic₁₉₄	1	1	yes	characteristic
46	C₂×C₈	C₄×D₉₇	1	1	yes	characteristic
47	C₄²	C₂²⋊D₉₇	1	1	yes	characteristic
48	C₁₆	—	97	2	no	—
49	C₁₆	—	194	1	no	—
50	C₂×C₄	#13095490	1	1	yes	characteristic
51	C₂×C₄	#13095496	1	1	yes	characteristic
52	C₂×C₄	#13095501	1	1	yes	characteristic
53	C₈	—	2	1	no	—
54	C₈	—	8	1	no	—
55	C₈	—	776	1	no	—
56	C₈	C₈×D₉₇	1	1	yes	characteristic
57	C₈	#13095494	1	1	yes	characteristic
58	C₂²	#15690743	1	1	yes	characteristic
59	C₄	—	2	1	no	—
60	C₄	#15690739	1	1	yes	characteristic
61	C₄	#15690745	1	1	yes	characteristic
62	C₄	#15690751	1	1	yes	characteristic
63	C₄	#15690756	1	1	yes	characteristic
64	C₂	#20260487	1	1	yes	characteristic
65	C₂	#20260560	1	1	yes	characteristic
66	C₂	#20260564	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13095496
commutator	C₈×C₉₇	C₂×C₈
Frattini	C₄×C₈	C₂×D₉₇
Fitting	#335×C₉₇	C₂
socle, prime radical	C₂²×C₉₇	#107
cosocle	C₄×C₈×C₉₇	C₂²
hypercenter	#335	D₉₇
hypocenter	C₉₇	#1371
LCS	G ▷ C₈×C₉₇ ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ #335

Complements

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

N	H ≅ G/N	type
11. #13095486	C₈	semidirect
34. C₉₇	#1371	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13095486	2
#20260488	2
#335×C₉₇	2
#1371	97