Subgroups — #27143964

Order 12416.

140 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261140	C₂	1	1	yes	maximal, characteristic
2	#20261115	C₂	1	1	yes	maximal, characteristic
3	#20260540	C₂	1	4	yes	maximal
4	C₄×C₁₆×C₉₇	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₃₈₈	—	2	1	no	—
6	C₂×Dic₃₈₈	C₂²	1	2	yes	—
7	#15690810	—	2	1	no	—
8	#15690810	—	2	1	no	—
9	#15690744	—	2	1	no	—
10	#15690744	C₂²	1	2	yes	—
11	C₂×C₁₆×C₉₇	C₂²	1	2	yes	—
12	C₄×C₈×C₉₇	C₂²	1	1	yes	characteristic
13	C₂×Dic₁₉₄	—	2	2	no	—
14	C₂×Dic₁₉₄	—	4	1	no	—
15	Dic₃₈₈	—	2	4	no	—
16	Dic₃₈₈	—	4	2	no	—
17	C₄⋊Dic₉₇	—	2	2	no	—
18	C₄⋊Dic₉₇	—	4	1	no	—
19	C₄⋊Dic₉₇	—	4	1	no	—
20	C₄⋊Dic₉₇	—	4	1	no	—
21	C₂×C₈×C₉₇	D₄	1	1	yes	characteristic
22	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
23	C₄²×C₉₇	D₄	1	1	yes	characteristic
24	C₁₆×C₉₇	—	2	2	no	—
25	C₂×C₄×C₉₇	D₈	1	1	yes	characteristic
26	C₂×C₄×C₉₇	D₈	1	1	yes	characteristic
27	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
28	C₂×Dic₉₇	—	4	2	no	—
29	C₂×Dic₉₇	—	4	2	no	—
30	Dic₁₉₄	—	4	2	no	—
31	Dic₁₉₄	—	4	2	no	—
32	Dic₁₉₄	—	8	2	no	—
33	C₈×C₉₇	SD₁₆	1	2	yes	—
34	C₈×C₉₇	#41	1	2	yes	—
35	C₂²×C₉₇	C₂×D₈	1	1	yes	characteristic
36	C₄×C₉₇	SD₃₂	1	2	yes	—
37	C₄×C₉₇	Q₃₂	1	2	yes	—
38	C₄×C₉₇	#124	1	1	yes	characteristic
39	C₄×C₉₇	C₂×SD₁₆	1	1	yes	characteristic
40	Dic₉₇	—	8	2	no	—
41	Dic₉₇	—	8	2	no	—
42	C₂×C₉₇	#486	1	1	yes	characteristic
43	C₂×C₉₇	C₂×SD₃₂	1	1	yes	characteristic
44	C₂×C₉₇	C₂×Q₃₂	1	1	yes	characteristic
45	#2241	—	97	1	no	maximal, Sylow
46	C₉₇	#2241	1	1	yes	characteristic, complemented, Sylow
47	#500	—	97	1	no	—
48	#494	—	97	1	no	—
49	#358	—	97	4	no	—
50	C₄×C₁₆	D₉₇	1	1	yes	characteristic
51	C₂×Q₁₆	—	97	2	no	—
52	C₂×Q₁₆	—	194	1	no	—
53	C₄⋊Q₈	—	194	1	no	—
54	C₄⋊Q₈	—	194	1	no	—
55	C₂×C₁₆	C₂×D₉₇	1	2	yes	—
56	#107	—	97	2	no	—
57	#107	—	194	1	no	—
58	C₄×C₈	C₂×D₉₇	1	1	yes	characteristic
59	C₂×Q₈	—	388	1	no	—
60	C₂×Q₈	—	194	2	no	—
61	Q₁₆	—	388	2	no	—
62	Q₁₆	—	194	4	no	—
63	C₂×C₈	D₃₈₈	1	1	yes	characteristic
64	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
65	C₄⋊C₄	—	388	1	no	—
66	C₄⋊C₄	—	388	1	no	—
67	C₄⋊C₄	—	388	1	no	—
68	C₄⋊C₄	—	194	2	no	—
69	C₄²	D₃₈₈	1	1	yes	characteristic
70	C₁₆	—	2	2	no	—
71	Q₈	—	776	2	no	—
72	Q₈	—	388	2	no	—
73	Q₈	—	388	2	no	—
74	C₂×C₄	—	388	2	no	—
75	C₂×C₄	—	388	2	no	—
76	C₂×C₄	D₇₇₆	1	1	yes	characteristic
77	C₂×C₄	D₇₇₆	1	1	yes	characteristic
78	C₂×C₄	C₂×D₃₈₈	1	1	yes	characteristic
79	C₈	#13095500	1	2	yes	—
80	C₈	#13095511	1	2	yes	—
81	C₂²	C₂×D₇₇₆	1	1	yes	characteristic
82	C₄	—	776	2	no	—
83	C₄	—	776	2	no	—
84	C₄	#15690755	1	2	yes	—
85	C₄	Dic₇₇₆	1	2	yes	—
86	C₄	#15690798	1	1	yes	characteristic
87	C₄	C₂×#13095500	1	1	yes	characteristic
88	C₂	#20261082	1	1	yes	characteristic
89	C₂	C₂×#15690755	1	1	yes	characteristic
90	C₂	C₂×Dic₇₇₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₇₇₆
commutator, cosocle	C₂×C₈×C₉₇	C₂³
Frattini	C₂×C₈	C₂²×D₉₇
Fitting	C₄×C₁₆×C₉₇	C₂
socle, prime radical	C₂²×C₉₇	C₂×D₈
hypercenter	C₄×C₁₆	D₉₇
hypocenter	C₉₇	#2241
LCS	G ▷ C₂×C₈×C₉₇ ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
46. C₉₇	#2241	semidirect

Maximal subgroups

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

subgroup	index
#20261140	2
#20261115	2
#20260540	2
C₄×C₁₆×C₉₇	2
#2241	97