Group #27141902 — subgroups

Subgroups — #27141902

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×#15690743	C₂	1	2	yes	maximal
2	C₂×#15690741	C₂	1	2	yes	maximal
3	C₂×#15690721	C₂	1	1	yes	maximal, characteristic
4	#20260601	C₂	1	1	yes	maximal, characteristic
5	#384×C₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11715895	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₉₇	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₉₇	C₂²	1	1	yes	characteristic
9	#15690743	—	2	4	no	—
10	#15690741	—	2	4	no	—
11	#15690721	C₄	1	4	yes	complemented
12	#15690717	—	2	2	no	—
13	C₂×C₄⋊C₄×C₉₇	—	2	2	no	—
14	C₂×C₄⋊C₄×C₉₇	C₂²	1	2	yes	—
15	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
16	C₂²×Dic₉₇	—	2	1	no	—
17	C₂²×Dic₉₇	—	2	1	no	—
18	C₂×#11715895	C₂×C₄	1	4	yes	—
19	C₂×#11715895	D₄	1	4	yes	—
20	C₂×#11715895	Q₈	1	4	yes	—
21	C₄⋊Dic₉₇	—	2	2	no	—
22	C₄⋊Dic₉₇	—	2	2	no	—
23	C₂²×C₄×C₉₇	—	2	2	no	—
24	C₂²×C₄×C₉₇	D₄	1	2	yes	—
25	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
26	C₄⋊C₄×C₉₇	—	2	2	no	—
27	C₄⋊C₄×C₉₇	—	2	2	no	—
28	C₄⋊C₄×C₉₇	—	4	4	no	—
29	C₄²×C₉₇	C₂×C₄	1	2	yes	—
30	C₄²×C₉₇	Q₈	1	2	yes	—
31	C₂³×C₉₇	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₉₇	—	2	2	no	—
33	C₂×C₄×C₉₇	—	2	2	no	—
34	C₂×C₄×C₉₇	—	4	2	no	—
35	C₂×C₄×C₉₇	—	4	4	no	—
36	C₂×C₄×C₉₇	C₄⋊C₄	1	4	yes	—
37	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
39	C₂×C₄×C₉₇	C₂×Q₈	1	2	yes	—
40	C₂×C₄×C₉₇	C₂×Q₈	1	1	yes	characteristic
41	C₂×Dic₉₇	—	4	1	no	—
42	C₂×Dic₉₇	—	4	1	no	—
43	C₂×Dic₉₇	—	4	2	no	—
44	C₂×Dic₉₇	—	4	2	no	—
45	#11715895	C₄⋊C₄	1	8	yes	complemented
46	C₂²×C₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₉₇	#127	1	1	yes	characteristic
48	C₂²×C₉₇	C₄⋊Q₈	1	1	yes	characteristic
49	C₂²×C₉₇	AGL(1,ℤ/8)	1	2	yes	—
50	C₂²×C₉₇	#137	1	2	yes	—
51	C₄×C₉₇	—	2	4	no	—
52	C₄×C₉₇	—	8	2	no	—
53	C₄×C₉₇	C₂×C₄⋊C₄	1	2	yes	—
54	C₄×C₉₇	C₄⋊Q₈	1	2	yes	—
55	Dic₉₇	—	8	1	no	—
56	Dic₉₇	—	8	1	no	—
57	C₂×C₉₇	#384	1	1	yes	characteristic
58	C₂×C₉₇	#428	1	2	yes	—
59	C₂×C₉₇	#496	1	1	yes	characteristic
60	C₂×C₉₇	#497	1	1	yes	characteristic
61	C₂×C₉₇	#501	1	2	yes	—
62	#1847	—	97	1	no	maximal, Sylow
63	C₉₇	#1847	1	1	yes	characteristic, complemented, Sylow
64	C₂×#107	—	97	2	no	—
65	C₂×#106	—	97	2	no	—
66	C₂×#97	—	97	1	no	—
67	#384	D₉₇	1	1	yes	characteristic
68	#382	—	97	1	no	—
69	C₂²×C₈	—	97	2	no	—
70	C₂×C₄⋊C₄	—	2	2	no	—
71	C₂×C₄⋊C₄	—	97	1	no	—
72	C₂×C₄⋊C₄	—	97	1	no	—
73	C₂×C₄⋊C₄	C₂×D₉₇	1	2	yes	—
74	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
75	#107	—	194	4	no	—
76	#106	—	194	4	no	—
77	#97	—	97	4	no	—
78	#95	—	194	2	no	—
79	C₂²×C₄	—	2	2	no	—
80	C₂²×C₄	—	194	1	no	—
81	C₂²×C₄	—	194	1	no	—
82	C₂²×C₄	C₂²⋊D₉₇	1	2	yes	—
83	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
84	C₂×C₈	—	97	4	no	—
85	C₂×C₈	—	97	4	no	—
86	C₂×C₈	—	97	4	no	—
87	C₄⋊C₄	—	2	2	no	—
88	C₄⋊C₄	—	2	2	no	—
89	C₄⋊C₄	—	4	4	no	—
90	C₄⋊C₄	—	194	2	no	—
91	C₄⋊C₄	—	194	2	no	—
92	C₄²	Dic₁₉₄	1	2	yes	—
93	C₄²	C₄×D₉₇	1	2	yes	—
94	C₂³	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	—	4	2	no	—
98	C₂×C₄	—	4	4	no	—
99	C₂×C₄	—	388	1	no	—
100	C₂×C₄	—	388	1	no	—
101	C₂×C₄	—	388	2	no	—
102	C₂×C₄	—	388	2	no	—
103	C₂×C₄	#13095490	1	4	yes	—
104	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
105	C₂×C₄	D₄×D₉₇	1	2	yes	—
106	C₂×C₄	Q₈×D₉₇	1	2	yes	—
107	C₂×C₄	C₂×Dic₁₉₄	1	1	yes	characteristic
108	C₈	—	97	8	no	—
109	C₂²	C₂×#13095490	1	1	yes	characteristic
110	C₂²	#15690806	1	1	yes	characteristic
111	C₂²	#15690808	1	1	yes	characteristic
112	C₂²	#15690836	1	2	yes	—
113	C₂²	#15690841	1	2	yes	—
114	C₄	—	2	4	no	—
115	C₄	—	8	2	no	—
116	C₄	—	776	1	no	—
117	C₄	—	776	1	no	—
118	C₄	C₄⋊C₄×D₉₇	1	2	yes	—
119	C₄	#15690809	1	2	yes	—
120	C₂	#20260604	1	1	yes	characteristic
121	C₂	#20260775	1	2	yes	—
122	C₂	#20261122	1	1	yes	characteristic
123	C₂	#20261127	1	1	yes	characteristic
124	C₂	#20261141	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15690743

C₂

yes

maximal

C₂×#15690741

C₂

yes

maximal

C₂×#15690721

C₂

yes

maximal, characteristic

#20260601

C₂

yes

maximal, characteristic

#384×C₉₇

C₂

yes

maximal, characteristic

C₂²×#11715895

C₂²

yes

—

C₂×C₄⋊Dic₉₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₉₇

C₂²

yes

—

—

C₄

yes

complemented

#15690717

—

C₂×C₄⋊C₄×C₉₇

—

C₂×C₄⋊C₄×C₉₇

C₂²

yes

—

C₂×C₄²×C₉₇

C₂²

yes

characteristic

C₂²×Dic₉₇

—

C₂²×Dic₉₇

—

C₂×#11715895

C₂×C₄

yes

—

C₂×#11715895

D₄

yes

—

C₂×#11715895

Q₈

yes

—

C₄⋊Dic₉₇

—

C₄⋊Dic₉₇

—

C₂²×C₄×C₉₇

—

C₂²×C₄×C₉₇

D₄

yes

—

C₂²×C₄×C₉₇

C₂³

yes

characteristic

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

—

C₄²×C₉₇

C₂×C₄

yes

—

C₄²×C₉₇

Q₈

yes

—

C₂³×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₄⋊C₄

yes

—

C₂×C₄×C₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₉₇

C₂×D₄

yes

—

C₂×C₄×C₉₇

C₂×Q₈

yes

—

C₂×C₄×C₉₇

C₂×Q₈

yes

characteristic

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

#11715895

C₄⋊C₄

yes

complemented

C₂²×C₉₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₉₇

#127

yes

characteristic

C₂²×C₉₇

C₄⋊Q₈

yes

characteristic

C₂²×C₉₇

AGL(1,ℤ/8)

yes

—

C₂²×C₉₇

#137

yes

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

C₂×C₄⋊C₄

yes

—

C₄×C₉₇

C₄⋊Q₈

yes

—

Dic₉₇

—

Dic₉₇

—

C₂×C₉₇

#384

yes

characteristic

C₂×C₉₇

#428

yes

—

C₂×C₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

—

#1847

—

maximal, Sylow

C₉₇

#1847

yes

characteristic, complemented, Sylow

C₂×#107

—

C₂×#106

—

C₂×#97

—

#384

D₉₇

yes

characteristic

#382

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₉₇

yes

—

C₂×C₄²

C₂×D₉₇

yes

characteristic

#107

—

194

—

#106

—

194

—

#97

—

#95

—

194

—

C₂²×C₄

—

C₂²×C₄

—

194

—

C₂²×C₄

—

194

—

C₂²×C₄

C₂²⋊D₉₇

yes

—

C₂²×C₄

C₂²×D₉₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

194

—

C₄⋊C₄

—

194

—

C₄²

Dic₁₉₄

yes

—

C₄²

C₄×D₉₇

yes

—

C₂³

C₂×C₂²⋊D₉₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

388

—

100

C₂×C₄

—

388

—

101

C₂×C₄

—

388

—

102

C₂×C₄

—

388

—

103

C₂×C₄

#13095490

yes

—

104

C₂×C₄

C₂×C₄×D₉₇

yes

characteristic

105

C₂×C₄

D₄×D₉₇

yes

—

106

C₂×C₄

Q₈×D₉₇

yes

—

107

C₂×C₄

C₂×Dic₁₉₄

yes

characteristic

108

C₈

—

109

C₂²

C₂×#13095490

yes

characteristic

110

C₂²

#15690806

yes

characteristic

111

C₂²

#15690808

yes

characteristic

112

C₂²

#15690836

yes

—

113

C₂²

#15690841

yes

—

114

C₄

—

115

C₄

—

116

C₄

—

776

—

117

C₄

—

776

—

118

C₄

C₄⋊C₄×D₉₇

yes

—

119

C₄

#15690809

yes

—

120

C₂

#20260604

yes

characteristic

121

C₂

#20260775

yes

—

122

C₂

#20261122

yes

characteristic

123

C₂

#20261127

yes

characteristic

124

C₂

#20261141

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₉₇
commutator	C₂×C₄×C₉₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₉₇
Fitting	#384×C₉₇	C₂
socle, prime radical	C₂³×C₉₇	C₂×D₄
cosocle	C₂²×C₄×C₉₇	C₂³
hypercenter	#384	D₉₇
hypocenter	C₉₇	#1847
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₉₇

commutator

C₂×C₄×C₉₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₉₇

Fitting

#384×C₉₇

C₂

socle, prime radical

C₂³×C₉₇

C₂×D₄

cosocle

C₂²×C₄×C₉₇

C₂³

D₉₇

C₉₇

G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
11. #15690721	C₄	semidirect
45. #11715895	C₄⋊C₄	semidirect
63. C₉₇	#1847	semidirect

H ≅ G/N

type

11. #15690721

C₄

semidirect

45. #11715895

C₄⋊C₄

semidirect

63. C₉₇

#1847

semidirect

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

subgroup	index
C₂×#15690743	2
C₂×#15690741	2
C₂×#15690721	2
#20260601	2
#384×C₉₇	2
#1847	97

subgroup

index

#384×C₉₇

Subgroups — #27141902

Conjugacy classes of subgroups

Complements

Maximal subgroups