Group #27140874 — subgroups

Subgroups — #27140874

Order 12416.

206 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 57 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261465	C₂	1	1	yes	maximal, characteristic
2	#20261027	C₂	1	2	yes	maximal
3	#20260484	C₂	1	1	yes	maximal, characteristic
4	#20260467	C₂	1	2	yes	maximal
5	#494×C₉₇	C₂	1	1	yes	maximal, characteristic
6	#15690809	—	2	2	no	—
7	#15690808	—	2	2	no	—
8	C₄×Dic₁₉₄	C₂²	1	2	yes	—
9	#15690775	—	2	2	no	—
10	#15690743	—	2	2	no	—
11	C₄⋊#11715895	C₂²	1	2	yes	—
12	#15690733	—	2	2	no	—
13	C₂×Q₁₆×C₉₇	—	2	2	no	—
14	C₄⋊Q₈×C₉₇	C₂²	1	1	yes	characteristic
15	C₄⋊Q₈×C₉₇	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₉₇	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₉₄	—	2	2	no	—
18	C₂×Dic₁₉₄	D₄	1	2	yes	—
19	C₂×#11715895	—	2	2	no	—
20	C₄⋊Dic₉₇	D₄	1	2	yes	—
21	#13095490	—	2	4	no	—
22	#13095490	—	2	2	no	—
23	C₄×Dic₉₇	—	2	2	no	—
24	C₄×Dic₉₇	—	2	2	no	—
25	C₂×Q₈×C₉₇	—	2	1	no	—
26	C₂×Q₈×C₉₇	D₄	1	2	yes	—
27	Q₁₆×C₉₇	—	4	4	no	—
28	C₂×C₈×C₉₇	—	2	1	no	—
29	C₄⋊C₄×C₉₇	—	2	2	no	—
30	C₄⋊C₄×C₉₇	—	2	2	no	—
31	C₄²×C₉₇	C₂³	1	1	yes	characteristic
32	Q₈×C₉₇	—	4	2	no	—
33	Q₈×C₉₇	—	4	2	no	—
34	C₂×C₄×C₉₇	—	2	2	no	—
35	C₂×C₄×C₉₇	—	4	1	no	—
36	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
37	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
38	C₂×Dic₉₇	—	2	2	no	—
39	C₂×Dic₉₇	—	2	2	no	—
40	Dic₁₉₄	—	2	2	no	—
41	Dic₁₉₄	—	8	2	no	—
42	Dic₁₉₄	Q₁₆	1	4	yes	complemented
43	C₈×C₉₇	—	2	2	no	—
44	#11715895	—	4	2	no	—
45	C₂²×C₉₇	#120	1	1	yes	characteristic
46	C₄×C₉₇	—	2	1	no	—
47	C₄×C₉₇	—	4	2	no	—
48	C₄×C₉₇	—	8	1	no	—
49	C₄×C₉₇	C₂×Q₁₆	1	2	yes	—
50	C₄×C₉₇	#137	1	2	yes	—
51	Dic₉₇	—	2	4	no	—
52	Dic₉₇	—	4	2	no	—
53	C₂×C₉₇	#451	1	2	yes	—
54	C₂×C₉₇	#453	1	1	yes	characteristic
55	#1633	—	97	1	no	maximal, Sylow
56	C₉₇	#1633	1	1	yes	characteristic, complemented, Sylow
57	Q₈²	—	97	1	no	—
58	#494	D₉₇	1	1	yes	characteristic
59	#477	—	97	2	no	—
60	#333	—	97	1	no	—
61	Q₈⋊C₈	—	97	2	no	—
62	C₂×Q₁₆	—	2	2	no	—
63	C₄⋊Q₈	—	194	2	no	—
64	C₄⋊Q₈	—	194	2	no	—
65	C₄⋊Q₈	C₂×D₉₇	1	1	yes	characteristic
66	C₄⋊Q₈	C₂×D₉₇	1	1	yes	characteristic
67	C₄×Q₈	—	97	2	no	—
68	C₄×Q₈	—	194	2	no	—
69	#107	—	194	2	no	—
70	C₄⋊C₈	—	97	2	no	—
71	#103	—	194	2	no	—
72	C₄×C₈	C₂×D₉₇	1	1	yes	characteristic
73	C₂×Q₈	—	2	1	no	—
74	C₂×Q₈	—	97	2	no	—
75	C₂×Q₈	—	194	2	no	—
76	C₂×Q₈	C₂²⋊D₉₇	1	2	yes	—
77	Q₁₆	—	4	4	no	—
78	C₂×C₈	—	2	1	no	—
79	C₂×C₈	—	194	2	no	—
80	C₄⋊C₄	—	2	2	no	—
81	C₄⋊C₄	—	2	2	no	—
82	C₄⋊C₄	—	97	2	no	—
83	C₄⋊C₄	—	194	4	no	—
84	C₄⋊C₄	—	194	2	no	—
85	C₄²	—	194	2	no	—
86	C₄²	—	194	2	no	—
87	C₄²	C₂²×D₉₇	1	1	yes	characteristic
88	Q₈	—	4	2	no	—
89	Q₈	—	4	2	no	—
90	Q₈	—	776	2	no	—
91	Q₈	—	97	4	no	—
92	Q₈	—	194	2	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	4	1	no	—
95	C₂×C₄	—	194	2	no	—
96	C₂×C₄	—	194	2	no	—
97	C₂×C₄	D₄×D₉₇	1	2	yes	—
98	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
99	C₈	—	2	2	no	—
100	C₈	—	388	2	no	—
101	C₂²	#15690778	1	1	yes	characteristic
102	C₄	—	2	1	no	—
103	C₄	—	4	2	no	—
104	C₄	—	8	1	no	—
105	C₄	—	388	2	no	—
106	C₄	—	194	4	no	—
107	C₄	Q₁₆×D₉₇	1	2	yes	—
108	C₄	#15690842	1	2	yes	—
109	C₂	#20260889	1	2	yes	—
110	C₂	#20260902	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20261465

C₂

yes

maximal, characteristic

#20261027

C₂

yes

maximal

#20260484

C₂

yes

maximal, characteristic

#20260467

C₂

yes

maximal

#494×C₉₇

C₂

yes

maximal, characteristic

#15690809

—

#15690808

—

C₄×Dic₁₉₄

C₂²

yes

—

#15690775

—

#15690743

—

C₄⋊#11715895

C₂²

yes

—

#15690733

—

C₂×Q₁₆×C₉₇

—

C₄⋊Q₈×C₉₇

C₂²

yes

characteristic

C₄⋊Q₈×C₉₇

C₂²

yes

characteristic

C₄×C₈×C₉₇

C₂²

yes

characteristic

C₂×Dic₁₉₄

—

C₂×Dic₁₉₄

D₄

yes

—

C₂×#11715895

—

C₄⋊Dic₉₇

D₄

yes

—

#13095490

—

#13095490

—

C₄×Dic₉₇

—

C₄×Dic₉₇

—

C₂×Q₈×C₉₇

—

C₂×Q₈×C₉₇

D₄

yes

—

Q₁₆×C₉₇

—

C₂×C₈×C₉₇

—

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

—

C₄²×C₉₇

C₂³

yes

characteristic

Q₈×C₉₇

—

Q₈×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂×D₄

yes

—

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×Dic₉₇

—

C₂×Dic₉₇

—

Dic₁₉₄

—

Dic₁₉₄

—

Dic₁₉₄

Q₁₆

yes

complemented

C₈×C₉₇

—

#11715895

—

C₂²×C₉₇

#120

yes

characteristic

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

C₂×Q₁₆

yes

—

C₄×C₉₇

#137

yes

—

Dic₉₇

—

Dic₉₇

—

C₂×C₉₇

#451

yes

—

C₂×C₉₇

yes

—

C₉₇

yes

characteristic, complemented, Sylow

Q₈²

—

#494

D₉₇

yes

characteristic

#477

—

#333

—

Q₈⋊C₈

—

C₂×Q₁₆

—

C₄⋊Q₈

—

194

—

C₄⋊Q₈

—

194

—

C₄⋊Q₈

C₂×D₉₇

yes

characteristic

C₄⋊Q₈

C₂×D₉₇

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

194

—

#107

—

194

—

C₄⋊C₈

—

#103

—

194

—

C₄×C₈

C₂×D₉₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

194

—

C₂×Q₈

C₂²⋊D₉₇

yes

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

194

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

194

—

C₄⋊C₄

—

194

—

C₄²

—

194

—

C₄²

—

194

—

C₄²

C₂²×D₉₇

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

776

—

Q₈

—

Q₈

—

194

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

194

—

C₂×C₄

—

194

—

C₂×C₄

D₄×D₉₇

yes

—

C₂×C₄

C₂×C₂²⋊D₉₇

yes

characteristic

C₈

—

100

C₈

—

388

—

101

C₂²

#15690778

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

—

388

—

106

C₄

—

194

—

107

C₄

Q₁₆×D₉₇

yes

—

108

C₄

#15690842

yes

—

109

C₂

#20260889

yes

—

110

C₂

#20260902

yes

characteristic

	subgroup	G/H
center	C₂²	#15690778
commutator, cosocle	C₄²×C₉₇	C₂³
Frattini	C₄²	C₂²×D₉₇
Fitting	#494×C₉₇	C₂
socle, prime radical	C₂²×C₉₇	#120
hypercenter	#494	D₉₇
hypocenter	C₉₇	#1633
LCS	G ▷ C₄²×C₉₇ ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

subgroup

G/H

center

C₂²

#15690778

commutator, cosocle

C₄²×C₉₇

C₂³

Frattini

C₄²

C₂²×D₉₇

Fitting

#494×C₉₇

C₂

C₂²×C₉₇

D₉₇

C₉₇

G ▷ C₄²×C₉₇ ▷ C₂²×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #494

N	H ≅ G/N	type
42. Dic₁₉₄	Q₁₆	semidirect
56. C₉₇	#1633	semidirect

H ≅ G/N

type

42. Dic₁₉₄

Q₁₆

semidirect

56. C₉₇

#1633

semidirect

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

subgroup	index
#20261465	2
#20261027	2
#20260484	2
#20260467	2
#494×C₉₇	2
#1633	97

subgroup

index

#494×C₉₇

Subgroups — #27140874

Conjugacy classes of subgroups

Complements

Maximal subgroups