Subgroups — #27139719

Order 12416.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20260865	C₂	1	1	yes	maximal, characteristic
2	C₄⋊#13095486	C₂	1	1	yes	maximal, characteristic
3	#346×C₉₇	C₂	1	1	yes	maximal, characteristic
4	C₄×Dic₁₉₄	C₄	1	1	yes	characteristic
5	C₂×#13095486	—	2	1	no	—
6	C₄⋊#11715895	C₄	1	1	yes	characteristic
7	#15690739	—	2	1	no	—
8	#15690720	—	2	1	no	—
9	C₂×C₁₆×C₉₇	—	2	1	no	—
10	C₄×C₈×C₉₇	C₂²	1	1	yes	characteristic
11	C₂×Dic₁₉₄	C₈	1	1	yes	characteristic
12	C₂×#11715895	—	2	1	no	—
13	C₄⋊Dic₉₇	C₈	1	1	yes	characteristic
14	#13095490	—	2	1	no	—
15	C₄×Dic₉₇	—	2	1	no	—
16	#13095486	—	4	1	no	—
17	C₂×C₈×C₉₇	D₄	1	1	yes	characteristic
18	C₂×C₈×C₉₇	D₄	1	1	yes	characteristic
19	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
20	C₁₆×C₉₇	—	2	2	no	—
21	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
22	C₂×C₄×C₉₇	C₂×C₈	1	1	yes	characteristic
23	C₂×C₄×C₉₇	M₁₆	1	1	yes	characteristic
24	C₂×Dic₉₇	—	2	1	no	—
25	C₂×Dic₉₇	—	2	1	no	—
26	Dic₁₉₄	—	4	1	no	—
27	C₈×C₉₇	—	2	1	no	—
28	C₈×C₉₇	SD₁₆	1	1	yes	characteristic
29	C₈×C₉₇	Q₁₆	1	1	yes	characteristic
30	#11715895	—	4	1	no	—
31	C₂²×C₉₇	C₂²⋊C₈	1	1	yes	characteristic
32	C₄×C₉₇	—	2	1	no	—
33	C₄×C₉₇	—	2	1	no	—
34	C₄×C₉₇	#103	1	1	yes	characteristic
35	C₄×C₉₇	#104	1	1	yes	characteristic
36	Dic₉₇	—	4	1	no	—
37	Dic₉₇	—	4	1	no	—
38	C₂×C₉₇	Q₈⋊C₈	1	1	yes	characteristic
39	C₂×C₉₇	#349	1	1	yes	characteristic
40	C₂×C₉₇	#350	1	1	yes	characteristic
41	#1338	—	97	1	no	maximal, Sylow
42	C₉₇	#1338	1	1	yes	characteristic, complemented, Sylow
43	#446	—	97	1	no	—
44	C₄⋊C₁₆	—	97	1	no	—
45	#346	D₉₇	1	1	yes	characteristic
46	C₄×Q₈	—	97	1	no	—
47	C₂×C₁₆	—	2	1	no	—
48	C₂×C₁₆	—	194	1	no	—
49	C₄⋊C₈	—	97	1	no	—
50	C₄⋊C₈	—	194	1	no	—
51	#97	—	194	1	no	—
52	C₄×C₈	C₂×D₉₇	1	1	yes	characteristic
53	C₂×Q₈	—	97	1	no	—
54	C₂×C₈	—	194	1	no	—
55	C₂×C₈	D₃₈₈	1	1	yes	characteristic
56	C₂×C₈	C₂²⋊D₉₇	1	1	yes	characteristic
57	C₄⋊C₄	—	97	1	no	—
58	C₄⋊C₄	—	194	1	no	—
59	C₄²	—	194	1	no	—
60	C₄²	C₄×D₉₇	1	1	yes	characteristic
61	C₁₆	—	2	2	no	—
62	C₁₆	—	388	1	no	—
63	Q₈	—	388	1	no	—
64	C₂×C₄	—	194	1	no	—
65	C₂×C₄	—	194	1	no	—
66	C₂×C₄	#13095488	1	1	yes	characteristic
67	C₂×C₄	C₈×D₉₇	1	1	yes	characteristic
68	C₂×C₄	#13095494	1	1	yes	characteristic
69	C₈	—	2	1	no	—
70	C₈	—	388	1	no	—
71	C₈	#13095500	1	1	yes	characteristic
72	C₈	Dic₃₈₈	1	1	yes	characteristic
73	C₂²	#15690722	1	1	yes	characteristic
74	C₄	—	2	1	no	—
75	C₄	—	2	1	no	—
76	C₄	—	388	1	no	—
77	C₄	—	388	1	no	—
78	C₄	#15690735	1	1	yes	characteristic
79	C₄	#15690738	1	1	yes	characteristic
80	C₂	#20260467	1	1	yes	characteristic
81	C₂	#20260516	1	1	yes	characteristic
82	C₂	#20260520	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13095488
commutator	C₂×C₄×C₉₇	C₂×C₈
Frattini	C₄×C₈	C₂×D₉₇
Fitting	#346×C₉₇	C₂
socle, prime radical	C₂²×C₉₇	C₂²⋊C₈
cosocle	C₄×C₈×C₉₇	C₂²
hypercenter	#346	D₉₇
hypocenter	C₉₇	#1338
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
42. C₉₇	#1338	semidirect

Maximal subgroups

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

subgroup	index
#20260865	2
C₄⋊#13095486	2
#346×C₉₇	2
#1338	97