Group #27181629 — subgroups

Subgroups — C₂×#20265573 #27181629

Order 12928.

424 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 197 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×#15692255	C₂	1	1	yes	maximal, characteristic
2	#20265573	C₂	1	8	yes	maximal, direct product factor
3	C₂×#15692186	C₂	1	4	yes	maximal
4	C₂×C₄×#11717579	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Q₈×C₁₀₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₀₂	—	2	1	no	—
7	C₂²×#11717579	C₂²	1	2	yes	—
8	#15692255	—	2	2	no	—
9	#15692255	C₂²	1	4	yes	—
10	C₂×C₄⋊Dic₁₀₁	—	2	1	no	—
11	C₂×C₄⋊Dic₁₀₁	C₂²	1	2	yes	—
12	#15692186	C₂²	1	16	yes	—
13	C₄×#11717579	C₂²	1	4	yes	—
14	C₂²×Q₈×C₁₀₁	—	2	1	no	—
15	C₄⋊Q₈×C₁₀₁	—	2	2	no	—
16	C₄⋊Q₈×C₁₀₁	C₂²	1	4	yes	—
17	C₂×C₄⋊C₄×C₁₀₁	—	2	1	no	—
18	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	2	yes	—
19	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
20	C₂×Dic₂₀₂	—	2	4	no	—
21	C₂×Dic₂₀₂	—	4	4	no	—
22	C₂²×Dic₁₀₁	—	2	2	no	—
23	C₂×#11717579	Q₈	1	8	yes	complemented
24	C₂×#11717579	C₂³	1	4	yes	—
25	C₄⋊Dic₁₀₁	—	2	4	no	—
26	C₄⋊Dic₁₀₁	—	2	2	no	—
27	C₄⋊Dic₁₀₁	C₂³	1	4	yes	—
28	C₂×Q₈×C₁₀₁	—	2	4	no	—
29	C₂×Q₈×C₁₀₁	—	4	4	no	—
30	C₂²×C₄×C₁₀₁	—	2	2	no	—
31	C₂²×C₄×C₁₀₁	D₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₀₁	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
34	C₄⋊C₄×C₁₀₁	—	2	4	no	—
35	C₄⋊C₄×C₁₀₁	—	2	2	no	—
36	C₄⋊C₄×C₁₀₁	C₂³	1	4	yes	—
37	C₄²×C₁₀₁	D₄	1	2	yes	—
38	C₄²×C₁₀₁	C₂³	1	2	yes	—
39	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
40	Q₈×C₁₀₁	—	4	8	no	—
41	C₂×C₄×C₁₀₁	—	2	4	no	—
42	C₂×C₄×C₁₀₁	—	4	4	no	—
43	C₂×C₄×C₁₀₁	SD₁₆	1	4	yes	—
44	C₂×C₄×C₁₀₁	SD₁₆	1	4	yes	—
45	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
47	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₁₀₁	C₂×Q₈	1	4	yes	—
49	C₂×C₄×C₁₀₁	C₂⁴	1	1	yes	characteristic
50	C₂×Dic₁₀₁	—	2	4	no	—
51	C₂×Dic₁₀₁	—	4	4	no	—
52	Dic₂₀₂	—	4	8	no	—
53	#11717579	C₂×Q₈	1	8	yes	complemented
54	C₂²×C₁₀₁	C₄⋊Q₈	1	2	yes	—
55	C₂²×C₁₀₁	C₂×SD₁₆	1	2	yes	—
56	C₂²×C₁₀₁	C₂×SD₁₆	1	2	yes	—
57	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
58	C₄×C₁₀₁	—	4	4	no	—
59	C₄×C₁₀₁	C₄⋊Q₈	1	2	yes	—
60	C₄×C₁₀₁	C₂×SD₁₆	1	4	yes	—
61	C₄×C₁₀₁	C₂×SD₁₆	1	4	yes	—
62	C₄×C₁₀₁	C₂²×Q₈	1	2	yes	—
63	Dic₁₀₁	—	4	4	no	—
64	C₂×C₁₀₁	#498	1	4	yes	complemented
65	C₂×C₁₀₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
66	C₂×C₁₀₁	C₂²×SD₁₆	1	1	yes	characteristic
67	C₂×C₁₀₁	C₂²×SD₁₆	1	1	yes	characteristic
68	C₂×#498	—	101	1	no	maximal, Sylow
69	C₁₀₁	C₂×#498	1	1	yes	characteristic, complemented, Sylow
70	C₂×C₄⋊Q₈	—	101	1	no	—
71	C₂×C₄⋊Q₈	D₁₀₁	1	1	yes	characteristic
72	#498	—	101	8	no	—
73	C₂×#106	—	101	4	no	—
74	C₂×C₄×C₈	—	101	1	no	—
75	C₂²×Q₈	—	2	1	no	—
76	C₂²×Q₈	—	202	1	no	—
77	C₂²×C₈	—	101	2	no	—
78	C₄⋊Q₈	—	2	2	no	—
79	C₄⋊Q₈	—	101	4	no	—
80	C₄⋊Q₈	—	202	2	no	—
81	C₄⋊Q₈	C₂×D₁₀₁	1	4	yes	—
82	C₂×C₄⋊C₄	—	2	1	no	—
83	C₂×C₄⋊C₄	—	101	2	no	—
84	C₂×C₄⋊C₄	—	202	1	no	—
85	C₂×C₄⋊C₄	C₂×D₁₀₁	1	2	yes	—
86	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
87	#106	—	101	16	no	—
88	C₄×C₈	—	101	4	no	—
89	C₂×Q₈	—	2	4	no	—
90	C₂×Q₈	—	4	4	no	—
91	C₂×Q₈	—	404	4	no	—
92	C₂×Q₈	—	202	4	no	—
93	C₂²×C₄	—	2	2	no	—
94	C₂²×C₄	—	202	2	no	—
95	C₂²×C₄	C₂²⋊D₁₀₁	1	1	yes	characteristic
96	C₂²×C₄	C₂²⋊D₁₀₁	1	1	yes	characteristic
97	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
98	C₂×C₈	—	101	8	no	—
99	C₂×C₈	—	101	4	no	—
100	C₄⋊C₄	—	2	4	no	—
101	C₄⋊C₄	—	2	2	no	—
102	C₄⋊C₄	—	101	4	no	—
103	C₄⋊C₄	—	202	2	no	—
104	C₄⋊C₄	—	202	4	no	—
105	C₄⋊C₄	C₂²×D₁₀₁	1	4	yes	—
106	C₄²	C₂²⋊D₁₀₁	1	2	yes	—
107	C₄²	C₂²×D₁₀₁	1	2	yes	—
108	C₂³	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
109	Q₈	—	4	8	no	—
110	Q₈	—	404	8	no	—
111	C₂×C₄	—	2	4	no	—
112	C₂×C₄	—	4	4	no	—
113	C₂×C₄	—	404	4	no	—
114	C₂×C₄	—	202	4	no	—
115	C₂×C₄	#13108061	1	4	yes	—
116	C₂×C₄	#13108062	1	4	yes	—
117	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
118	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
119	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
120	C₂×C₄	Q₈×D₁₀₁	1	4	yes	—
121	C₂×C₄	C₂³×D₁₀₁	1	1	yes	characteristic
122	C₈	—	101	8	no	—
123	C₂²	#15692253	1	2	yes	—
124	C₂²	C₂×#13108061	1	2	yes	—
125	C₂²	C₂×#13108062	1	2	yes	—
126	C₂²	C₂²×C₂²⋊D₁₀₁	1	1	yes	characteristic
127	C₄	—	4	4	no	—
128	C₄	—	404	4	no	—
129	C₄	#15692253	1	2	yes	—
130	C₄	C₂×#13108061	1	4	yes	—
131	C₄	C₂×#13108062	1	4	yes	—
132	C₄	C₂×Q₈×D₁₀₁	1	2	yes	—
133	C₂	#20265573	1	4	yes	direct product factor
134	C₂	C₂×#15692253	1	1	yes	characteristic
135	C₂	C₂²×#13108061	1	1	yes	characteristic
136	C₂	C₂²×#13108062	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15692255

C₂

yes

maximal, characteristic

#20265573

C₂

yes

maximal, direct product factor

C₂×#15692186

C₂

yes

maximal

C₂×C₄×#11717579

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₁₀₁

C₂

yes

maximal, characteristic

C₂²×Dic₂₀₂

—

C₂²×#11717579

C₂²

yes

—

#15692255

—

#15692255

C₂²

yes

—

C₂×C₄⋊Dic₁₀₁

—

C₂×C₄⋊Dic₁₀₁

C₂²

yes

—

#15692186

C₂²

yes

—

C₄×#11717579

C₂²

yes

—

C₂²×Q₈×C₁₀₁

—

C₄⋊Q₈×C₁₀₁

—

C₄⋊Q₈×C₁₀₁

C₂²

yes

—

C₂×C₄⋊C₄×C₁₀₁

—

C₂×C₄⋊C₄×C₁₀₁

C₂²

yes

—

C₂×C₄²×C₁₀₁

C₂²

yes

characteristic

C₂×Dic₂₀₂

—

C₂×Dic₂₀₂

—

C₂²×Dic₁₀₁

—

C₂×#11717579

Q₈

yes

complemented

C₂×#11717579

C₂³

yes

—

C₄⋊Dic₁₀₁

—

C₄⋊Dic₁₀₁

—

C₄⋊Dic₁₀₁

C₂³

yes

—

C₂×Q₈×C₁₀₁

—

C₂×Q₈×C₁₀₁

—

C₂²×C₄×C₁₀₁

—

C₂²×C₄×C₁₀₁

D₄

yes

characteristic

C₂²×C₄×C₁₀₁

D₄

yes

characteristic

C₂²×C₄×C₁₀₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₁

—

C₄⋊C₄×C₁₀₁

—

C₄⋊C₄×C₁₀₁

C₂³

yes

—

C₄²×C₁₀₁

D₄

yes

—

C₄²×C₁₀₁

C₂³

yes

—

C₂³×C₁₀₁

C₂×D₄

yes

characteristic

Q₈×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

SD₁₆

yes

—

C₂×C₄×C₁₀₁

SD₁₆

yes

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₁

C₂×D₄

yes

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

—

C₂×C₄×C₁₀₁

C₂×Q₈

yes

—

C₂×C₄×C₁₀₁

C₂⁴

yes

characteristic

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

Dic₂₀₂

—

#11717579

C₂×Q₈

yes

complemented

C₂²×C₁₀₁

C₄⋊Q₈

yes

—

C₂²×C₁₀₁

C₂×SD₁₆

yes

—

C₂²×C₁₀₁

C₂×SD₁₆

yes

—

C₂²×C₁₀₁

C₂²×D₄

yes

characteristic

C₄×C₁₀₁

—

C₄×C₁₀₁

C₄⋊Q₈

yes

—

C₄×C₁₀₁

C₂×SD₁₆

yes

—

C₄×C₁₀₁

C₂×SD₁₆

yes

—

C₄×C₁₀₁

C₂²×Q₈

yes

—

Dic₁₀₁

—

C₂×C₁₀₁

#498

yes

complemented

C₂×C₁₀₁

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₁₀₁

C₂²×SD₁₆

yes

characteristic

C₂×C₁₀₁

C₂²×SD₁₆

yes

characteristic

C₂×#498

—

101

maximal, Sylow

C₁₀₁

C₂×#498

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

101

—

C₂×C₄⋊Q₈

D₁₀₁

yes

characteristic

#498

—

101

—

C₂×#106

—

101

—

C₂×C₄×C₈

—

101

—

C₂²×Q₈

—

C₂²×Q₈

—

202

—

C₂²×C₈

—

101

—

C₄⋊Q₈

—

C₄⋊Q₈

—

101

—

C₄⋊Q₈

—

202

—

C₄⋊Q₈

C₂×D₁₀₁

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

101

—

C₂×C₄⋊C₄

—

202

—

C₂×C₄⋊C₄

C₂×D₁₀₁

yes

—

C₂×C₄²

C₂×D₁₀₁

yes

characteristic

#106

—

101

—

C₄×C₈

—

101

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

404

—

C₂×Q₈

—

202

—

C₂²×C₄

—

C₂²×C₄

—

202

—

C₂²×C₄

C₂²⋊D₁₀₁

yes

characteristic

C₂²×C₄

C₂²⋊D₁₀₁

yes

characteristic

C₂²×C₄

C₂²×D₁₀₁

yes

characteristic

C₂×C₈

—

101

—

C₂×C₈

—

101

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

101

—

103

C₄⋊C₄

—

202

—

104

C₄⋊C₄

—

202

—

105

C₄⋊C₄

C₂²×D₁₀₁

yes

—

106

C₄²

C₂²⋊D₁₀₁

yes

—

107

C₄²

C₂²×D₁₀₁

yes

—

108

C₂³

C₂×C₂²⋊D₁₀₁

yes

characteristic

109

Q₈

—

110

Q₈

—

404

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

404

—

114

C₂×C₄

—

202

—

115

C₂×C₄

#13108061

yes

—

116

C₂×C₄

#13108062

yes

—

117

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

characteristic

118

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

—

119

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

—

120

C₂×C₄

Q₈×D₁₀₁

yes

—

121

C₂×C₄

C₂³×D₁₀₁

yes

characteristic

122

C₈

—

101

—

123

C₂²

#15692253

yes

—

124

C₂²

C₂×#13108061

yes

—

125

C₂²

C₂×#13108062

yes

—

126

C₂²

C₂²×C₂²⋊D₁₀₁

yes

characteristic

127

C₄

—

128

C₄

—

404

—

129

C₄

#15692253

yes

—

130

C₄

C₂×#13108061

yes

—

131

C₄

C₂×#13108062

yes

—

132

C₄

C₂×Q₈×D₁₀₁

yes

—

133

C₂

#20265573

yes

direct product factor

134

C₂

C₂×#15692253

yes

characteristic

135

C₂

C₂²×#13108061

yes

characteristic

136

C₂

C₂²×#13108062

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₀₁
commutator, cosocle	C₂×C₄×C₁₀₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₁
Fitting	C₂×C₄⋊Q₈×C₁₀₁	C₂
socle, prime radical	C₂³×C₁₀₁	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	D₁₀₁
hypocenter	C₁₀₁	C₂×#498
LCS	G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₀₁

commutator, cosocle

C₂×C₄×C₁₀₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₀₁

Fitting

C₂×C₄⋊Q₈×C₁₀₁

C₂

socle, prime radical

C₂³×C₁₀₁

C₂×D₄

hypercenter

C₂×C₄⋊Q₈

D₁₀₁

hypocenter

C₁₀₁

C₂×#498

LCS

G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
2. #20265573	C₂	direct (N × H)
23. C₂×#11717579	Q₈	semidirect
53. #11717579	C₂×Q₈	semidirect
64. C₂×C₁₀₁	#498	semidirect
69. C₁₀₁	C₂×#498	semidirect
133. C₂	#20265573	direct (N × H)

H ≅ G/N

type

2. #20265573

C₂

direct (N × H)

23. C₂×#11717579

Q₈

semidirect

53. #11717579

C₂×Q₈

semidirect

64. C₂×C₁₀₁

#498

semidirect

69. C₁₀₁

C₂×#498

semidirect

133. C₂

#20265573

direct (N × H)

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

subgroup	index
C₂×#15692255	2
#20265573	2
C₂×#15692186	2
C₂×C₄×#11717579	2
C₂×C₄⋊Q₈×C₁₀₁	2
C₂×#498	101

subgroup

index

C₂×C₄⋊Q₈×C₁₀₁

101

Subgroups — C₂×#20265573 #27181629

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#20265573 #27181629

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20265573 #27181629