Subgroups — #27156655

Order 12416.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261624	C₂	1	1	yes	maximal, characteristic
2	#20261576	C₂	1	1	yes	maximal, characteristic
3	#20260510	C₂	1	1	yes	maximal, characteristic
4	C₂×#13095514	C₄	1	2	yes	—
5	C₄×#11715897	C₄	1	1	yes	characteristic
6	C₄×#11715897	C₄	1	1	yes	characteristic
7	C₂×#13095486	C₄	1	1	yes	characteristic
8	C₈×Dic₉₇	C₂²	1	1	yes	characteristic
9	C₂×C₁₆×C₉₇	C₄	1	1	yes	characteristic
10	C₂×#11715897	—	2	1	no	—
11	C₂×#11715897	—	2	1	no	—
12	#13095514	C₈	1	4	yes	complemented
13	C₂×#11715895	C₂×C₄	1	1	yes	characteristic
14	C₄×Dic₉₇	C₂×C₄	1	1	yes	characteristic
15	#13095486	—	2	1	no	—
16	C₂×C₈×C₉₇	C₂×C₄	1	1	yes	characteristic
17	C₁₆×C₉₇	—	2	1	no	—
18	C₂×C₄×C₉₇	C₄²	1	1	yes	characteristic
19	C₂×Dic₉₇	M₁₆	1	1	yes	characteristic
20	C₂×Dic₉₇	M₁₆	1	1	yes	characteristic
21	#11715897	—	4	1	no	—
22	#11715897	—	4	1	no	—
23	C₈×C₉₇	M₁₆	1	1	yes	characteristic
24	C₈×C₉₇	M₁₆	1	1	yes	characteristic
25	#11715895	C₂×C₈	1	2	yes	—
26	C₂²×C₉₇	#97	1	1	yes	characteristic
27	C₄×C₉₇	C₄×C₈	1	1	yes	characteristic
28	C₄×C₉₇	#97	1	1	yes	characteristic
29	Dic₉₇	—	2	1	no	—
30	Dic₉₇	—	2	1	no	—
31	C₂×C₉₇	#322	1	1	yes	characteristic
32	C₂×C₉₇	#347	1	1	yes	characteristic
33	C₂×C₉₇	#347	1	1	yes	characteristic
34	#1313	—	97	1	no	maximal, Sylow
35	C₉₇	#1313	1	1	yes	characteristic, complemented, Sylow
36	#346	—	97	1	no	—
37	#346	—	97	1	no	—
38	#322	—	97	1	no	—
39	C₂×C₁₆	—	97	2	no	—
40	C₂×C₁₆	—	97	1	no	—
41	C₂×C₁₆	#87604	1	1	yes	characteristic
42	C₄×C₈	—	97	1	no	—
43	C₄×C₈	—	97	1	no	—
44	C₄×C₈	—	97	1	no	—
45	C₂×C₈	—	97	1	no	—
46	C₂×C₈	—	194	1	no	—
47	C₂×C₈	—	194	1	no	—
48	C₂×C₈	C₂×#87604	1	1	yes	characteristic
49	C₄²	—	97	1	no	—
50	C₁₆	—	2	1	no	—
51	C₁₆	—	97	4	no	—
52	C₁₆	—	194	1	no	—
53	C₂×C₄	—	97	1	no	—
54	C₂×C₄	—	97	1	no	—
55	C₂×C₄	C₄×#87604	1	1	yes	characteristic
56	C₈	—	97	2	no	—
57	C₈	—	388	1	no	—
58	C₈	—	388	1	no	—
59	C₈	#13095520	1	1	yes	characteristic
60	C₈	#13095521	1	1	yes	characteristic
61	C₂²	#15690864	1	1	yes	characteristic
62	C₄	—	194	1	no	—
63	C₄	—	194	1	no	—
64	C₄	C₈×#87604	1	1	yes	characteristic
65	C₄	#15690865	1	1	yes	characteristic
66	C₂	#20261577	1	1	yes	characteristic
67	C₂	#20261625	1	1	yes	characteristic
68	C₂	#20261626	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₄×#87604
commutator	C₄×C₉₇	C₄×C₈
Frattini	C₂×C₈	C₂×#87604
Fitting	C₂×C₁₆×C₉₇	C₄
socle, prime radical	C₂²×C₉₇	#97
cosocle	C₈×Dic₉₇	C₂²
hypercenter	C₂×C₁₆	#87604
hypocenter	C₉₇	#1313
LCS	G ▷ C₄×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
12. #13095514	C₈	semidirect
35. C₉₇	#1313	semidirect

Maximal subgroups

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

subgroup	index
#20261624	2
#20261576	2
#20260510	2
#1313	97