Group #27141889 — subgroups

Subgroups — #27141889

Order 12416.

256 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 117 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	C₂×C₄⋊#11715895	C₂	1	1	yes	maximal, characteristic
2	C₂×#15690739	C₂	1	4	yes	maximal
3	#20260581	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊C₈×C₉₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11715895	C₂²	1	2	yes	—
6	C₂×C₄×Dic₉₇	C₂²	1	2	yes	—
7	C₄⋊#11715895	—	2	2	no	—
8	#15690739	—	2	8	no	—
9	#15690717	C₄	1	2	yes	—
10	#15690717	C₄	1	2	yes	—
11	C₂²×C₈×C₉₇	C₂²	1	2	yes	—
12	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
13	C₄⋊C₈×C₉₇	—	2	2	no	—
14	C₂²×Dic₉₇	C₂×C₄	1	4	yes	—
15	C₂×#11715895	—	2	4	no	—
16	C₂×#11715895	—	2	2	no	—
17	C₄×Dic₉₇	D₄	1	4	yes	—
18	C₄×Dic₉₇	Q₈	1	4	yes	—
19	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
22	C₂×C₈×C₉₇	—	2	2	no	—
23	C₂×C₈×C₉₇	—	2	4	no	—
24	C₄²×C₉₇	D₄	1	2	yes	—
25	C₄²×C₉₇	Q₈	1	2	yes	—
26	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
27	C₂×C₄×C₉₇	—	2	2	no	—
28	C₂×C₄×C₉₇	—	2	2	no	—
29	C₂×C₄×C₉₇	C₄⋊C₄	1	2	yes	—
30	C₂×C₄×C₉₇	C₄⋊C₄	1	2	yes	—
31	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
32	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₉₇	C₂×Q₈	1	2	yes	—
34	C₂×C₄×C₉₇	C₂×Q₈	1	1	yes	characteristic
35	C₂×Dic₉₇	—	2	4	no	—
36	C₂×Dic₉₇	—	2	4	no	—
37	C₂×Dic₉₇	C₄⋊C₄	1	8	yes	—
38	C₈×C₉₇	—	4	2	no	—
39	#11715895	—	4	2	no	—
40	C₂²×C₉₇	C₂×C₄⋊C₄	1	2	yes	—
41	C₂²×C₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
42	C₂²×C₉₇	#131	1	2	yes	—
43	C₂²×C₉₇	#131	1	2	yes	—
44	C₄×C₉₇	—	2	2	no	—
45	C₄×C₉₇	—	2	2	no	—
46	C₄×C₉₇	#127	1	1	yes	characteristic
47	C₄×C₉₇	C₄⋊Q₈	1	1	yes	characteristic
48	C₄×C₉₇	C₄⋊Q₈	1	2	yes	—
49	Dic₉₇	—	2	8	no	—
50	C₂×C₉₇	#384	1	1	yes	characteristic
51	C₂×C₉₇	#424	1	4	yes	—
52	C₂×C₉₇	#424	1	1	yes	characteristic
53	C₂×C₉₇	#424	1	1	yes	characteristic
54	#1843	—	97	1	no	maximal, Sylow
55	C₉₇	#1843	1	1	yes	characteristic, complemented, Sylow
56	C₂×C₄⋊C₈	—	97	4	no	—
57	C₂×C₄⋊C₈	—	97	1	no	—
58	C₂×C₄⋊C₈	D₉₇	1	1	yes	characteristic
59	#376	—	97	1	no	—
60	C₂²×C₈	—	97	2	no	—
61	C₂²×C₈	C₂×D₉₇	1	2	yes	—
62	C₂×C₄²	—	97	2	no	—
63	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
64	C₄⋊C₈	—	2	2	no	—
65	C₄⋊C₈	—	194	8	no	—
66	C₄⋊C₈	—	194	2	no	—
67	#95	—	97	2	no	—
68	#95	—	97	2	no	—
69	C₂²×C₄	—	97	4	no	—
70	C₂²×C₄	C₄×D₉₇	1	1	yes	characteristic
71	C₂²×C₄	C₄×D₉₇	1	1	yes	characteristic
72	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
73	C₂×C₈	—	2	4	no	—
74	C₂×C₈	—	2	2	no	—
75	C₂×C₈	—	194	4	no	—
76	C₂×C₈	—	194	2	no	—
77	C₄²	—	97	4	no	—
78	C₄²	—	97	4	no	—
79	C₄²	Dic₁₉₄	1	2	yes	—
80	C₄²	C₂²⋊D₉₇	1	2	yes	—
81	C₂³	C₂×C₄×D₉₇	1	1	yes	characteristic
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	2	2	no	—
84	C₂×C₄	—	97	8	no	—
85	C₂×C₄	—	194	4	no	—
86	C₂×C₄	—	194	4	no	—
87	C₂×C₄	#13095490	1	2	yes	—
88	C₂×C₄	#13095490	1	2	yes	—
89	C₂×C₄	D₄×D₉₇	1	2	yes	—
90	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
91	C₂×C₄	Q₈×D₉₇	1	2	yes	—
92	C₂×C₄	C₂×Dic₁₉₄	1	1	yes	characteristic
93	C₈	—	4	2	no	—
94	C₈	—	388	2	no	—
95	C₂²	C₄⋊C₄×D₉₇	1	2	yes	—
96	C₂²	C₂×#13095490	1	1	yes	characteristic
97	C₂²	#15690816	1	2	yes	—
98	C₂²	#15690818	1	2	yes	—
99	C₄	—	2	2	no	—
100	C₄	—	2	2	no	—
101	C₄	—	194	8	no	—
102	C₄	#15690806	1	1	yes	characteristic
103	C₄	#15690808	1	1	yes	characteristic
104	C₄	#15690809	1	2	yes	—
105	C₂	#20260604	1	1	yes	characteristic
106	C₂	#20260760	1	4	yes	—
107	C₂	#20260761	1	1	yes	characteristic
108	C₂	#20260763	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#11715895

C₂

yes

maximal, characteristic

C₂×#15690739

C₂

yes

maximal

#20260581

C₂

yes

maximal, characteristic

C₂×C₄⋊C₈×C₉₇

C₂

yes

maximal, characteristic

C₂²×#11715895

C₂²

yes

—

C₂×C₄×Dic₉₇

C₂²

yes

—

C₄⋊#11715895

—

#15690739

—

#15690717

C₄

yes

—

#15690717

C₄

yes

—

C₂²×C₈×C₉₇

C₂²

yes

—

C₂×C₄²×C₉₇

C₂²

yes

characteristic

C₄⋊C₈×C₉₇

—

C₂²×Dic₉₇

C₂×C₄

yes

—

C₂×#11715895

—

C₂×#11715895

—

C₄×Dic₉₇

D₄

yes

—

C₄×Dic₉₇

Q₈

yes

—

C₂²×C₄×C₉₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₉₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₉₇

C₂³

yes

characteristic

C₂×C₈×C₉₇

—

C₂×C₈×C₉₇

—

C₄²×C₉₇

D₄

yes

—

C₄²×C₉₇

Q₈

yes

—

C₂³×C₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₄⋊C₄

yes

—

C₂×C₄×C₉₇

C₄⋊C₄

yes

—

C₂×C₄×C₉₇

C₂×D₄

yes

—

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

C₂×Q₈

yes

—

C₂×C₄×C₉₇

C₂×Q₈

yes

characteristic

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

C₄⋊C₄

yes

—

C₈×C₉₇

—

#11715895

—

C₂²×C₉₇

C₂×C₄⋊C₄

yes

—

C₂²×C₉₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₉₇

#131

yes

—

C₂²×C₉₇

#131

yes

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

#127

yes

characteristic

C₄×C₉₇

C₄⋊Q₈

yes

characteristic

C₄×C₉₇

C₄⋊Q₈

yes

—

Dic₉₇

—

C₂×C₉₇

#384

yes

characteristic

C₂×C₉₇

#424

yes

—

C₂×C₉₇

yes

C₂×C₉₇

yes

—

C₉₇

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

D₉₇

yes

characteristic

#376

—

C₂²×C₈

—

C₂²×C₈

C₂×D₉₇

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₉₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

194

—

C₄⋊C₈

—

194

—

#95

—

#95

—

C₂²×C₄

—

C₂²×C₄

C₄×D₉₇

yes

characteristic

C₂²×C₄

C₄×D₉₇

yes

characteristic

C₂²×C₄

C₂²×D₉₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

194

—

C₂×C₈

—

194

—

C₄²

—

C₄²

—

C₄²

Dic₁₉₄

yes

—

C₄²

C₂²⋊D₉₇

yes

—

C₂³

C₂×C₄×D₉₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

194

—

C₂×C₄

—

194

—

C₂×C₄

#13095490

yes

—

C₂×C₄

#13095490

yes

—

C₂×C₄

D₄×D₉₇

yes

—

C₂×C₄

C₂×C₂²⋊D₉₇

yes

characteristic

C₂×C₄

Q₈×D₉₇

yes

—

C₂×C₄

C₂×Dic₁₉₄

yes

characteristic

C₈

—

C₈

—

388

—

C₂²

C₄⋊C₄×D₉₇

yes

—

C₂²

C₂×#13095490

yes

characteristic

C₂²

#15690816

yes

—

C₂²

#15690818

yes

—

C₄

—

100

C₄

—

101

C₄

—

194

—

102

C₄

#15690806

yes

characteristic

103

C₄

#15690808

yes

characteristic

104

C₄

#15690809

yes

—

105

C₂

#20260604

yes

characteristic

106

C₂

#20260760

yes

—

107

C₂

#20260761

yes

characteristic

108

C₂

#20260763

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₉₇
commutator, socle, prime radical	C₂³×C₉₇	C₂²×C₄
Fitting	C₂×C₄⋊C₈×C₉₇	C₂
cosocle	C₂²×C₄×C₉₇	C₂³
hypercenter	C₂×C₄⋊C₈	D₉₇
hypocenter	C₉₇	#1843
LCS	G ▷ C₂³×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₉₇

commutator, socle, prime radical

C₂³×C₉₇

C₂²×C₄

Fitting

C₂×C₄⋊C₈×C₉₇

C₂

cosocle

C₂²×C₄×C₉₇

C₂³

hypercenter

C₂×C₄⋊C₈

D₉₇

C₉₇

G ▷ C₂³×C₉₇ ▷ C₉₇

C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
55. C₉₇	#1843	semidirect

H ≅ G/N

type

55. C₉₇

#1843

semidirect

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

subgroup	index
C₂×C₄⋊#11715895	2
C₂×#15690739	2
#20260581	2
C₂×C₄⋊C₈×C₉₇	2
#1843	97

subgroup

index

C₂×C₄⋊C₈×C₉₇

Subgroups — #27141889

Conjugacy classes of subgroups

Complements

Maximal subgroups