Group #27142708 — subgroups

Subgroups — #27142708

Order 12416.

276 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 105 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×#15690810	C₂	1	1	yes	maximal, characteristic
2	C₂×#15690721	C₂	1	1	yes	maximal, characteristic
3	#20260497	C₂	1	4	yes	maximal
4	#382×C₉₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₁₉₄	—	2	1	no	—
6	C₂²×#11715895	C₂²	1	2	yes	—
7	#15690810	—	2	2	no	—
8	#15690810	C₄	1	4	yes	complemented
9	C₂×C₄⋊Dic₉₇	—	2	1	no	—
10	C₂×C₄⋊Dic₉₇	C₂²	1	2	yes	—
11	#15690721	—	2	2	no	—
12	C₂×C₄⋊C₄×C₉₇	C₂²	1	2	yes	—
13	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
14	#95×C₉₇	—	2	1	no	—
15	#95×C₉₇	—	2	1	no	—
16	C₂×Dic₁₉₄	—	2	4	no	—
17	C₂×Dic₁₉₄	—	4	4	no	—
18	C₂²×Dic₉₇	—	2	2	no	—
19	C₂×#11715895	—	2	2	no	—
20	C₂×#11715895	—	2	2	no	—
21	C₂×#11715895	Q₈	1	4	yes	complemented
22	C₄⋊Dic₉₇	—	2	4	no	—
23	C₄⋊Dic₉₇	—	2	2	no	—
24	C₄⋊Dic₉₇	C₂×C₄	1	4	yes	—
25	C₂²×C₄×C₉₇	—	2	2	no	—
26	C₂²×C₄×C₉₇	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₉₇	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
29	C₄⋊C₄×C₉₇	—	2	4	no	—
30	C₄²×C₉₇	C₂×C₄	1	2	yes	—
31	C₄²×C₉₇	D₄	1	2	yes	—
32	C₂³×C₉₇	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₉₇	—	2	2	no	—
34	C₂×C₄×C₉₇	—	2	2	no	—
35	C₂×C₄×C₉₇	—	4	2	no	—
36	C₂×C₄×C₉₇	—	4	2	no	—
37	C₂×C₄×C₉₇	—	4	2	no	—
38	C₂×C₄×C₉₇	C₂²⋊C₄	1	2	yes	—
39	C₂×C₄×C₉₇	C₂²⋊C₄	1	2	yes	—
40	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₉₇	C₂×Q₈	1	2	yes	—
43	C₂×C₄×C₉₇	#41	1	2	yes	—
44	C₂×Dic₉₇	—	2	4	no	—
45	C₂×Dic₉₇	—	4	4	no	—
46	Dic₁₉₄	—	4	8	no	—
47	#11715895	—	2	4	no	—
48	C₂²×C₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₉₇	#124	1	1	yes	characteristic
50	C₂²×C₉₇	C₄⋊Q₈	1	1	yes	characteristic
51	C₂²×C₉₇	AGL(1,ℤ/8)	1	2	yes	—
52	C₂²×C₉₇	#137	1	2	yes	—
53	C₄×C₉₇	—	2	2	no	—
54	C₄×C₉₇	—	2	2	no	—
55	C₄×C₉₇	—	8	2	no	—
56	C₄×C₉₇	C₄×Q₈	1	2	yes	—
57	C₄×C₉₇	C₂²⋊Q₈	1	2	yes	—
58	Dic₉₇	—	4	4	no	—
59	C₂×C₉₇	#391	1	1	yes	characteristic
60	C₂×C₉₇	#418	1	1	yes	characteristic
61	C₂×C₉₇	#419	1	1	yes	characteristic
62	C₂×C₉₇	#489	1	2	yes	—
63	C₂×C₉₇	#501	1	2	yes	—
64	#1993	—	97	1	no	maximal, Sylow
65	C₉₇	#1993	1	1	yes	characteristic, complemented, Sylow
66	C₂×C₄⋊Q₈	—	97	1	no	—
67	C₂×#97	—	97	1	no	—
68	#382	D₉₇	1	1	yes	characteristic
69	#340	—	97	4	no	—
70	C₂²×Q₈	—	194	1	no	—
71	C₂²×C₈	—	97	2	no	—
72	C₄⋊Q₈	—	97	4	no	—
73	C₄⋊Q₈	—	194	2	no	—
74	C₂×C₄⋊C₄	—	97	2	no	—
75	C₂×C₄⋊C₄	—	194	1	no	—
76	C₂×C₄⋊C₄	C₂×D₉₇	1	2	yes	—
77	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
78	#97	—	194	2	no	—
79	#95	—	2	1	no	—
80	#95	—	2	1	no	—
81	C₂×Q₈	—	388	4	no	—
82	C₂×Q₈	—	194	4	no	—
83	C₂²×C₄	—	2	2	no	—
84	C₂²×C₄	—	194	2	no	—
85	C₂²×C₄	C₂²⋊D₉₇	1	1	yes	characteristic
86	C₂²×C₄	C₂²⋊D₉₇	1	1	yes	characteristic
87	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
88	C₂×C₈	—	97	4	no	—
89	C₂×C₈	—	194	2	no	—
90	C₂×C₈	—	194	2	no	—
91	C₄⋊C₄	—	2	4	no	—
92	C₄⋊C₄	—	97	4	no	—
93	C₄⋊C₄	—	194	4	no	—
94	C₄⋊C₄	—	194	2	no	—
95	C₄²	C₄×D₉₇	1	2	yes	—
96	C₄²	D₃₈₈	1	2	yes	—
97	C₂³	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
98	Q₈	—	388	8	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	4	2	no	—
102	C₂×C₄	—	4	2	no	—
103	C₂×C₄	—	4	2	no	—
104	C₂×C₄	—	388	4	no	—
105	C₂×C₄	—	194	4	no	—
106	C₂×C₄	#13095488	1	2	yes	—
107	C₂×C₄	#13095488	1	2	yes	—
108	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
109	C₂×C₄	C₂×D₃₈₈	1	1	yes	characteristic
110	C₂×C₄	Q₈×D₉₇	1	2	yes	—
111	C₂×C₄	#13095510	1	2	yes	—
112	C₈	—	194	4	no	—
113	C₂²	C₂×#13095488	1	1	yes	characteristic
114	C₂²	#15690797	1	1	yes	characteristic
115	C₂²	#15690808	1	1	yes	characteristic
116	C₂²	#15690835	1	2	yes	—
117	C₂²	#15690840	1	2	yes	—
118	C₄	—	2	2	no	—
119	C₄	—	2	2	no	—
120	C₄	—	8	2	no	—
121	C₄	—	388	4	no	—
122	C₄	#15690775	1	2	yes	—
123	C₄	#15690790	1	2	yes	—
124	C₂	#20260640	1	1	yes	characteristic
125	C₂	#20260739	1	1	yes	characteristic
126	C₂	#20260743	1	1	yes	characteristic
127	C₂	#20261094	1	2	yes	—
128	C₂	#20261145	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15690810

C₂

yes

maximal, characteristic

C₂×#15690721

C₂

yes

maximal, characteristic

#20260497

C₂

yes

maximal

#382×C₉₇

C₂

yes

maximal, characteristic

C₂²×Dic₁₉₄

—

C₂²×#11715895

C₂²

yes

—

#15690810

—

#15690810

C₄

yes

complemented

C₂×C₄⋊Dic₉₇

—

C₂×C₄⋊Dic₉₇

C₂²

yes

—

#15690721

—

C₂×C₄⋊C₄×C₉₇

C₂²

yes

—

C₂×C₄²×C₉₇

C₂²

yes

characteristic

#95×C₉₇

—

#95×C₉₇

—

C₂×Dic₁₉₄

—

C₂×Dic₁₉₄

—

C₂²×Dic₉₇

—

C₂×#11715895

—

C₂×#11715895

—

C₂×#11715895

Q₈

yes

complemented

C₄⋊Dic₉₇

—

C₄⋊Dic₉₇

—

C₄⋊Dic₉₇

C₂×C₄

yes

—

C₂²×C₄×C₉₇

—

C₂²×C₄×C₉₇

D₄

yes

characteristic

C₂²×C₄×C₉₇

D₄

yes

characteristic

C₂²×C₄×C₉₇

C₂³

yes

characteristic

C₄⋊C₄×C₉₇

—

C₄²×C₉₇

C₂×C₄

yes

—

C₄²×C₉₇

D₄

yes

—

C₂³×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂²⋊C₄

yes

—

C₂×C₄×C₉₇

C₂²⋊C₄

yes

—

C₂×C₄×C₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

C₂×Q₈

yes

—

C₂×C₄×C₉₇

#41

yes

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

Dic₁₉₄

—

#11715895

—

C₂²×C₉₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₉₇

#124

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

—

C₄×C₉₇

C₄×Q₈

yes

—

C₄×C₉₇

C₂²⋊Q₈

yes

—

Dic₉₇

—

C₂×C₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

—

C₂×C₉₇

#501

yes

—

#1993

—

maximal, Sylow

C₉₇

#1993

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

C₂×#97

—

#382

D₉₇

yes

characteristic

#340

—

C₂²×Q₈

—

194

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

194

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

194

—

C₂×C₄⋊C₄

C₂×D₉₇

yes

—

C₂×C₄²

C₂×D₉₇

yes

characteristic

#97

—

194

—

#95

—

#95

—

C₂×Q₈

—

388

—

C₂×Q₈

—

194

—

C₂²×C₄

—

C₂²×C₄

—

194

—

C₂²×C₄

C₂²⋊D₉₇

yes

characteristic

C₂²×C₄

C₂²⋊D₉₇

yes

characteristic

C₂²×C₄

C₂²×D₉₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

194

—

C₂×C₈

—

194

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

194

—

C₄⋊C₄

—

194

—

C₄²

C₄×D₉₇

yes

—

C₄²

D₃₈₈

yes

—

C₂³

C₂×C₂²⋊D₉₇

yes

characteristic

Q₈

—

388

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

388

—

105

C₂×C₄

—

194

—

106

C₂×C₄

#13095488

yes

—

107

C₂×C₄

#13095488

yes

—

108

C₂×C₄

C₂×C₄×D₉₇

yes

characteristic

109

C₂×C₄

C₂×D₃₈₈

yes

characteristic

110

C₂×C₄

Q₈×D₉₇

yes

—

111

C₂×C₄

#13095510

yes

—

112

C₈

—

194

—

113

C₂²

C₂×#13095488

yes

characteristic

114

C₂²

#15690797

yes

characteristic

115

C₂²

#15690808

yes

characteristic

116

C₂²

#15690835

yes

—

117

C₂²

#15690840

yes

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

—

388

—

122

C₄

#15690775

yes

—

123

C₄

#15690790

yes

—

124

C₂

#20260640

yes

characteristic

125

C₂

#20260739

yes

characteristic

126

C₂

#20260743

yes

characteristic

127

C₂

#20261094

yes

—

128

C₂

#20261145

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₉₇
commutator	C₂×C₄×C₉₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₉₇
Fitting	#382×C₉₇	C₂
socle, prime radical	C₂³×C₉₇	C₂×D₄
cosocle	C₂²×C₄×C₉₇	C₂³
hypercenter	#382	D₉₇
hypocenter	C₉₇	#1993
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₉₇

commutator

C₂×C₄×C₉₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₉₇

Fitting

#382×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 ◁ #382

N	H ≅ G/N	type
8. #15690810	C₄	semidirect
21. C₂×#11715895	Q₈	semidirect
65. C₉₇	#1993	semidirect

H ≅ G/N

type

8. #15690810

C₄

semidirect

21. C₂×#11715895

Q₈

semidirect

65. C₉₇

#1993

semidirect

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

subgroup	index
C₂×#15690810	2
C₂×#15690721	2
#20260497	2
#382×C₉₇	2
#1993	97

subgroup

index

#382×C₉₇

Subgroups — #27142708

Conjugacy classes of subgroups

Complements

Maximal subgroups