Subgroups — #27139828

Order 12416.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15690744	C₂	1	2	yes	maximal
2	C₂²×C₁₆×C₉₇	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊Dic₉₇	—	2	2	no	—
4	#15690744	—	2	2	no	—
5	#15690744	C₄	1	4	yes	complemented
6	C₂²×C₈×C₉₇	C₂²	1	1	yes	characteristic
7	C₂×C₁₆×C₉₇	—	2	2	no	—
8	C₂×C₁₆×C₉₇	C₄	1	2	yes	complemented
9	C₂²×Dic₉₇	—	4	2	no	—
10	C₄⋊Dic₉₇	—	2	4	no	—
11	C₄⋊Dic₉₇	—	4	2	no	—
12	C₂²×C₄×C₉₇	D₄	1	1	yes	characteristic
13	C₂×C₈×C₉₇	C₂×C₄	1	2	yes	—
14	C₂×C₈×C₉₇	C₂×C₄	1	1	yes	characteristic
15	C₂×C₈×C₉₇	D₄	1	2	yes	—
16	C₂×C₈×C₉₇	Q₈	1	1	yes	characteristic
17	C₁₆×C₉₇	—	2	2	no	—
18	C₂³×C₉₇	D₈	1	1	yes	characteristic
19	C₂×C₄×C₉₇	C₂²⋊C₄	1	2	yes	—
20	C₂×C₄×C₉₇	C₄⋊C₄	1	1	yes	characteristic
21	C₂×C₄×C₉₇	SD₁₆	1	2	yes	—
22	C₂×C₄×C₉₇	Q₁₆	1	1	yes	characteristic
23	C₂×Dic₉₇	—	4	4	no	—
24	C₂×Dic₉₇	—	8	2	no	—
25	C₂×Dic₉₇	—	8	2	no	—
26	C₈×C₉₇	C₄²	1	1	yes	characteristic
27	C₈×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
28	C₈×C₉₇	C₄⋊C₄	1	2	yes	—
29	C₂²×C₉₇	#102	1	2	yes	—
30	C₂²×C₉₇	#107	1	1	yes	characteristic
31	C₂²×C₉₇	D₁₆	1	1	yes	characteristic
32	C₂²×C₉₇	SD₃₂	1	2	yes	—
33	C₂²×C₉₇	Q₃₂	1	1	yes	characteristic
34	C₄×C₉₇	#95	1	1	yes	characteristic
35	C₄×C₉₇	#103	1	2	yes	—
36	C₄×C₉₇	#106	1	1	yes	characteristic
37	Dic₉₇	—	8	4	no	—
38	C₂×C₉₇	#340	1	1	yes	characteristic
39	C₂×C₉₇	#357	1	2	yes	—
40	C₂×C₉₇	#358	1	2	yes	—
41	C₂×C₉₇	#366	1	1	yes	characteristic
42	C₂×C₉₇	#367	1	1	yes	characteristic
43	#1380	—	97	1	no	maximal, Sylow
44	C₉₇	#1380	1	1	yes	characteristic, complemented, Sylow
45	C₂²×C₁₆	D₉₇	1	1	yes	characteristic
46	C₂×#107	—	97	2	no	—
47	C₂²×C₈	C₂×D₉₇	1	1	yes	characteristic
48	C₂×C₄⋊C₄	—	194	2	no	—
49	C₂×C₁₆	—	2	2	no	—
50	C₂×C₁₆	Dic₉₇	1	2	yes	complemented
51	#107	—	97	4	no	—
52	#107	—	194	2	no	—
53	C₂²×C₄	—	388	2	no	—
54	C₂²×C₄	D₃₈₈	1	1	yes	characteristic
55	C₂×C₈	Dic₁₉₄	1	1	yes	characteristic
56	C₂×C₈	C₄×D₉₇	1	2	yes	—
57	C₂×C₈	C₂×Dic₉₇	1	1	yes	characteristic
58	C₂×C₈	C₂²⋊D₉₇	1	2	yes	—
59	C₄⋊C₄	—	388	2	no	—
60	C₄⋊C₄	—	194	4	no	—
61	C₁₆	—	2	2	no	—
62	C₂³	D₇₇₆	1	1	yes	characteristic
63	C₂×C₄	—	776	2	no	—
64	C₂×C₄	—	776	2	no	—
65	C₂×C₄	—	388	4	no	—
66	C₂×C₄	#13095488	1	2	yes	—
67	C₂×C₄	C₄⋊Dic₉₇	1	1	yes	characteristic
68	C₂×C₄	#13095500	1	2	yes	—
69	C₂×C₄	Dic₃₈₈	1	1	yes	characteristic
70	C₈	C₄×Dic₉₇	1	1	yes	characteristic
71	C₈	C₂²⋊Dic₉₇	1	1	yes	characteristic
72	C₈	#13095490	1	2	yes	—
73	C₂²	#15690732	1	2	yes	—
74	C₂²	#15690744	1	1	yes	characteristic
75	C₂²	D₁₅₅₂	1	1	yes	characteristic
76	C₂²	#15690755	1	2	yes	—
77	C₂²	Dic₇₇₆	1	1	yes	characteristic
78	C₄	—	776	4	no	—
79	C₄	#15690717	1	1	yes	characteristic
80	C₄	#15690735	1	2	yes	—
81	C₄	#15690742	1	1	yes	characteristic
82	C₂	#20260498	1	1	yes	characteristic
83	C₂	#20260537	1	2	yes	—
84	C₂	#20260540	1	2	yes	—
85	C₂	#20260561	1	1	yes	characteristic
86	C₂	#20260563	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₇₇₆
commutator	C₈×C₉₇	C₄²
Frattini	C₂²×C₈	C₂×D₉₇
Fitting	C₂²×C₁₆×C₉₇	C₂
socle, prime radical	C₂³×C₉₇	D₈
cosocle	C₂²×C₈×C₉₇	C₂²
hypercenter	C₂²×C₁₆	D₉₇
hypocenter	C₉₇	#1380
LCS	G ▷ C₈×C₉₇ ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
5. #15690744	C₄	semidirect
8. C₂×C₁₆×C₉₇	C₄	semidirect
44. C₉₇	#1380	semidirect
50. C₂×C₁₆	Dic₉₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15690744	2
C₂²×C₁₆×C₉₇	2
#1380	97