Group #27141532 — subgroups

Subgroups — C₄×#15690729 #27141532

Order 12416.

296 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 129 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×#15690729	C₂	1	2	yes	maximal
2	C₄×#13095495	C₂	1	2	yes	maximal
3	#20260504	C₂	1	2	yes	maximal
4	C₂×C₄×Q₈×C₉₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#13095495	C₂²	1	4	yes	—
6	#15690729	C₄	1	8	yes	direct product factor
7	#15690721	—	2	2	no	—
8	C₄×#11715895	—	2	2	no	—
9	C₂²×Q₈×C₉₇	C₂²	1	1	yes	characteristic
10	C₄×Q₈×C₉₇	—	2	2	no	—
11	C₄×Q₈×C₉₇	—	2	2	no	—
12	C₂×C₄⋊C₄×C₉₇	C₄	1	2	yes	—
13	C₂×C₄⋊C₄×C₉₇	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₉₇	C₄	1	2	yes	—
15	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
16	#13095495	—	2	4	no	—
17	#13095495	C₂×C₄	1	8	yes	—
18	C₂×#11715895	—	2	4	no	—
19	C₂×Q₈×C₉₇	—	2	2	no	—
20	C₂×Q₈×C₉₇	—	2	2	no	—
21	C₂×Q₈×C₉₇	C₂×C₄	1	4	yes	—
22	C₂²×C₄×C₉₇	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₉₇	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
27	C₄⋊C₄×C₉₇	—	2	4	no	—
28	C₄⋊C₄×C₉₇	—	2	2	no	—
29	C₄²×C₉₇	—	2	2	no	—
30	C₄²×C₉₇	—	2	2	no	—
31	C₄²×C₉₇	D₄	1	2	yes	—
32	C₄²×C₉₇	D₄	1	2	yes	—
33	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
34	Q₈×C₉₇	—	2	8	no	—
35	C₂×C₄×C₉₇	—	2	2	no	—
36	C₂×C₄×C₉₇	—	2	2	no	—
37	C₂×C₄×C₉₇	—	2	2	no	—
38	C₂×C₄×C₉₇	—	2	2	no	—
39	C₂×C₄×C₉₇	—	4	1	no	—
40	C₂×C₄×C₉₇	—	4	2	no	—
41	C₂×C₄×C₉₇	C₄²	1	4	yes	—
42	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₉₇	C₂²⋊C₄	1	2	yes	—
44	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
46	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
47	C₂×C₄×C₉₇	#41	1	2	yes	—
48	#11715895	—	2	8	no	—
49	C₂²×C₉₇	—	2	1	no	—
50	C₂²×C₉₇	—	2	1	no	—
51	C₂²×C₉₇	C₂×C₄²	1	1	yes	characteristic
52	C₂²×C₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₉₇	#117	1	1	yes	characteristic
54	C₄×C₉₇	—	2	4	no	—
55	C₄×C₉₇	—	2	1	no	—
56	C₄×C₉₇	—	2	2	no	—
57	C₄×C₉₇	—	4	2	no	—
58	C₄×C₉₇	#101	1	2	yes	complemented
59	C₄×C₉₇	C₄×D₄	1	4	yes	—
60	C₂×C₉₇	—	2	1	no	—
61	C₂×C₉₇	—	2	1	no	—
62	C₂×C₉₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₉₇	C₂×#101	1	1	yes	characteristic
64	C₂×C₉₇	#413	1	1	yes	characteristic
65	C₄×#101	—	97	1	no	maximal, Sylow
66	C₉₇	C₄×#101	1	1	yes	characteristic, complemented, Sylow
67	C₂×C₄×Q₈	D₉₇	1	1	yes	characteristic
68	C₂×#101	—	97	2	no	—
69	C₄×M₁₆	—	97	2	no	—
70	#343	—	97	2	no	—
71	C₂²×Q₈	C₂×D₉₇	1	1	yes	characteristic
72	C₂×M₁₆	—	97	4	no	—
73	C₄×Q₈	—	2	2	no	—
74	C₄×Q₈	—	2	2	no	—
75	C₂×C₄⋊C₄	Dic₉₇	1	2	yes	—
76	C₂×C₄⋊C₄	C₂×D₉₇	1	1	yes	characteristic
77	C₂×C₄²	Dic₉₇	1	2	yes	—
78	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
79	#101	—	97	8	no	—
80	#97	—	194	2	no	—
81	C₄×C₈	—	194	2	no	—
82	C₂×Q₈	—	2	2	no	—
83	C₂×Q₈	—	2	2	no	—
84	C₂×Q₈	C₄×D₉₇	1	4	yes	—
85	C₂²×C₄	C₂×Dic₉₇	1	2	yes	—
86	C₂²×C₄	C₂×Dic₉₇	1	1	yes	characteristic
87	C₂²×C₄	C₂×Dic₉₇	1	2	yes	—
88	C₂²×C₄	C₂×Dic₉₇	1	1	yes	characteristic
89	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
90	M₁₆	—	97	8	no	—
91	M₁₆	—	194	4	no	—
92	C₂×C₈	—	194	4	no	—
93	C₄⋊C₄	—	2	4	no	—
94	C₄⋊C₄	—	2	2	no	—
95	C₄²	—	2	2	no	—
96	C₄²	—	2	2	no	—
97	C₄²	C₂²⋊D₉₇	1	2	yes	—
98	C₄²	C₂²⋊D₉₇	1	2	yes	—
99	C₂³	C₂²×Dic₉₇	1	1	yes	characteristic
100	Q₈	—	2	8	no	—
101	C₂×C₄	—	2	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	4	1	no	—
106	C₂×C₄	—	4	2	no	—
107	C₂×C₄	C₄×Dic₉₇	1	4	yes	—
108	C₂×C₄	C₂²⋊Dic₉₇	1	1	yes	characteristic
109	C₂×C₄	C₂²⋊Dic₉₇	1	2	yes	—
110	C₂×C₄	C₂²⋊Dic₉₇	1	1	yes	characteristic
111	C₂×C₄	C₂×C₄×D₉₇	1	2	yes	—
112	C₂×C₄	C₂×C₂²⋊D₉₇	1	2	yes	—
113	C₂×C₄	#13095511	1	2	yes	—
114	C₈	—	194	8	no	—
115	C₂²	—	2	1	no	—
116	C₂²	—	2	1	no	—
117	C₂²	C₂×C₄×Dic₉₇	1	1	yes	characteristic
118	C₂²	C₂×C₂²⋊Dic₉₇	1	1	yes	characteristic
119	C₂²	#15690769	1	1	yes	characteristic
120	C₄	—	2	1	no	—
121	C₄	—	2	2	no	—
122	C₄	—	2	4	no	—
123	C₄	—	4	2	no	—
124	C₄	#15690729	1	2	yes	direct product factor
125	C₄	C₄×C₂²⋊D₉₇	1	4	yes	—
126	C₂	—	2	1	no	—
127	C₂	—	2	1	no	—
128	C₂	C₄×C₂²⋊Dic₉₇	1	1	yes	characteristic
129	C₂	C₂×#15690729	1	1	yes	characteristic
130	C₂	#20260719	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15690729

C₂

yes

maximal

C₄×#13095495

C₂

yes

maximal

#20260504

C₂

yes

maximal

C₂×C₄×Q₈×C₉₇

C₂

yes

maximal, characteristic

C₂×#13095495

C₂²

yes

—

#15690729

C₄

yes

direct product factor

#15690721

—

C₄×#11715895

—

C₂²×Q₈×C₉₇

C₂²

yes

characteristic

C₄×Q₈×C₉₇

—

C₄×Q₈×C₉₇

—

C₂×C₄⋊C₄×C₉₇

C₄

yes

—

C₂×C₄⋊C₄×C₉₇

C₂²

yes

characteristic

C₂×C₄²×C₉₇

C₄

yes

—

C₂×C₄²×C₉₇

C₂²

yes

characteristic

#13095495

—

#13095495

C₂×C₄

yes

—

C₂×#11715895

—

C₂×Q₈×C₉₇

—

C₂×Q₈×C₉₇

—

C₂×Q₈×C₉₇

C₂×C₄

yes

—

C₂²×C₄×C₉₇

C₂×C₄

yes

—

C₂²×C₄×C₉₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₉₇

C₂×C₄

yes

—

C₂²×C₄×C₉₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₉₇

C₂³

yes

characteristic

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

—

C₄²×C₉₇

—

C₄²×C₉₇

—

C₄²×C₉₇

D₄

yes

—

C₄²×C₉₇

D₄

yes

—

C₂³×C₉₇

C₂²×C₄

yes

characteristic

Q₈×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₄²

yes

—

C₂×C₄×C₉₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₉₇

C₂²⋊C₄

yes

—

C₂×C₄×C₉₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₉₇

C₂²×C₄

yes

—

C₂×C₄×C₉₇

C₂×D₄

yes

—

C₂×C₄×C₉₇

#41

yes

—

#11715895

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

C₂×C₄²

yes

characteristic

C₂²×C₉₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₉₇

#117

yes

characteristic

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

#101

yes

complemented

C₄×C₉₇

C₄×D₄

yes

—

C₂×C₉₇

—

C₂×C₉₇

—

C₂×C₉₇

C₄×C₂²⋊C₄

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

—

C₉₇

yes

characteristic, complemented, Sylow

C₂×C₄×Q₈

D₉₇

yes

characteristic

C₂×#101

—

C₄×M₁₆

—

#343

—

C₂²×Q₈

C₂×D₉₇

yes

characteristic

C₂×M₁₆

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

Dic₉₇

yes

—

C₂×C₄⋊C₄

C₂×D₉₇

yes

characteristic

C₂×C₄²

Dic₉₇

yes

—

C₂×C₄²

C₂×D₉₇

yes

characteristic

#101

—

#97

—

194

—

C₄×C₈

—

194

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₄×D₉₇

yes

—

C₂²×C₄

C₂×Dic₉₇

yes

—

C₂²×C₄

C₂×Dic₉₇

yes

characteristic

C₂²×C₄

C₂×Dic₉₇

yes

—

C₂²×C₄

C₂×Dic₉₇

yes

characteristic

C₂²×C₄

C₂²×D₉₇

yes

characteristic

M₁₆

—

M₁₆

—

194

—

C₂×C₈

—

194

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂²⋊D₉₇

yes

—

C₄²

C₂²⋊D₉₇

yes

—

C₂³

C₂²×Dic₉₇

yes

characteristic

100

Q₈

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

C₄×Dic₉₇

yes

—

108

C₂×C₄

C₂²⋊Dic₉₇

yes

characteristic

109

C₂×C₄

C₂²⋊Dic₉₇

yes

—

110

C₂×C₄

C₂²⋊Dic₉₇

yes

characteristic

111

C₂×C₄

C₂×C₄×D₉₇

yes

—

112

C₂×C₄

C₂×C₂²⋊D₉₇

yes

—

113

C₂×C₄

#13095511

yes

—

114

C₈

—

194

—

115

C₂²

—

116

C₂²

—

117

C₂²

C₂×C₄×Dic₉₇

yes

characteristic

118

C₂²

C₂×C₂²⋊Dic₉₇

yes

characteristic

119

C₂²

#15690769

yes

characteristic

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

#15690729

yes

direct product factor

125

C₄

C₄×C₂²⋊D₉₇

yes

—

126

C₂

—

127

C₂

—

128

C₂

C₄×C₂²⋊Dic₉₇

yes

characteristic

129

C₂

C₂×#15690729

yes

characteristic

130

C₂

#20260719

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊Dic₉₇
commutator	C₂²×C₉₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₉₇
Fitting	C₂×C₄×Q₈×C₉₇	C₂
socle	C₂²×C₉₇	C₂×C₂²⋊C₄
prime radical	C₂³×C₉₇	C₂²×C₄
cosocle	C₂²×C₄×C₉₇	C₂³
hypercenter	C₂×C₄×Q₈	D₉₇
hypocenter	C₉₇	C₄×#101
LCS	G ▷ C₂²×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×Q₈

subgroup

G/H

center

C₂×C₄

C₂²⋊Dic₉₇

commutator

C₂²×C₉₇

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₉₇

Fitting

C₂×C₄×Q₈×C₉₇

C₂

socle

C₂²×C₉₇

C₂×C₂²⋊C₄

prime radical

C₂³×C₉₇

C₂²×C₄

cosocle

C₂²×C₄×C₉₇

C₂³

hypercenter

C₂×C₄×Q₈

D₉₇

hypocenter

C₉₇

C₄×#101

LCS

G ▷ C₂²×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
6. #15690729	C₄	direct (N × H)
58. C₄×C₉₇	#101	semidirect
66. C₉₇	C₄×#101	semidirect
124. C₄	#15690729	direct (N × H)

H ≅ G/N

type

6. #15690729

C₄

direct (N × H)

58. C₄×C₉₇

#101

semidirect

66. C₉₇

C₄×#101

semidirect

124. C₄

#15690729

direct (N × H)

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

subgroup	index
C₂×#15690729	2
C₄×#13095495	2
#20260504	2
C₂×C₄×Q₈×C₉₇	2
C₄×#101	97

subgroup

index

C₂×C₄×Q₈×C₉₇

Subgroups — C₄×#15690729 #27141532

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C4×#15690729 #27141532

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#15690729 #27141532