Group #28373548 — subgroups

Subgroups — C₂×#20295432 #28373548

Order 13952.

384 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 165 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×#16812691	C₂	1	1	yes	maximal, characteristic
2	#20295432	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16812632	C₂	1	2	yes	maximal
4	C₂×#16812630	C₂	1	2	yes	maximal
5	C₂×#16812610	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Q₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11724784	C₂²	1	2	yes	—
8	#16812691	—	2	2	no	—
9	#16812691	C₂²	1	4	yes	—
10	C₂×C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
12	C₂×#13182008	—	2	1	no	—
13	C₂×#13182008	—	2	1	no	—
14	#16812632	C₂²	1	8	yes	—
15	#16812630	C₂²	1	8	yes	—
16	#16812610	C₂²	1	4	yes	—
17	C₂²×Q₈×C₁₀₉	—	2	1	no	—
18	C₄⋊Q₈×C₁₀₉	—	2	2	no	—
19	C₄⋊Q₈×C₁₀₉	C₂²	1	4	yes	—
20	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
21	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
22	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
23	C₂²×Dic₁₀₉	—	2	1	no	—
24	C₂²×Dic₁₀₉	—	2	1	no	—
25	C₂×#11724784	Q₈	1	8	yes	complemented
26	C₂×#11724784	C₂³	1	4	yes	—
27	C₄⋊Dic₁₀₉	—	2	1	no	—
28	C₄⋊Dic₁₀₉	—	2	1	no	—
29	C₄⋊Dic₁₀₉	C₂³	1	2	yes	—
30	C₄⋊Dic₁₀₉	C₂³	1	2	yes	—
31	#13182008	—	2	4	no	—
32	#13182008	—	2	4	no	—
33	C₂×Q₈×C₁₀₉	—	2	4	no	—
34	C₂×Q₈×C₁₀₉	—	4	4	no	—
35	C₂²×C₄×C₁₀₉	—	2	2	no	—
36	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
39	C₄⋊C₄×C₁₀₉	—	2	4	no	—
40	C₄⋊C₄×C₁₀₉	—	2	2	no	—
41	C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
42	C₄²×C₁₀₉	D₄	1	2	yes	—
43	C₄²×C₁₀₉	C₂³	1	2	yes	—
44	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
45	Q₈×C₁₀₉	—	4	8	no	—
46	C₂×C₄×C₁₀₉	—	2	2	no	—
47	C₂×C₄×C₁₀₉	—	2	2	no	—
48	C₂×C₄×C₁₀₉	—	2	4	no	—
49	C₂×C₄×C₁₀₉	—	4	4	no	—
50	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₁₀₉	C₂×Q₈	1	4	yes	—
54	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
55	C₂×Dic₁₀₉	—	2	2	no	—
56	C₂×Dic₁₀₉	—	2	2	no	—
57	C₂×Dic₁₀₉	—	4	2	no	—
58	C₂×Dic₁₀₉	—	4	2	no	—
59	#11724784	C₂×Q₈	1	8	yes	complemented
60	C₂²×C₁₀₉	C₄⋊Q₈	1	2	yes	—
61	C₂²×C₁₀₉	AGL(1,ℤ/8)	1	2	yes	—
62	C₂²×C₁₀₉	#137	1	2	yes	—
63	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
64	C₄×C₁₀₉	—	2	2	no	—
65	C₄×C₁₀₉	—	2	2	no	—
66	C₄×C₁₀₉	—	4	4	no	—
67	C₄×C₁₀₉	C₄⋊Q₈	1	2	yes	—
68	C₄×C₁₀₉	C₂²×Q₈	1	2	yes	—
69	Dic₁₀₉	—	4	2	no	—
70	Dic₁₀₉	—	4	2	no	—
71	C₂×C₁₀₉	#501	1	4	yes	complemented
72	C₂×C₁₀₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
73	C₂×C₁₀₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
74	C₂×C₁₀₉	C₂×#137	1	1	yes	characteristic
75	C₂×#501	—	109	1	no	maximal, Sylow
76	C₁₀₉	C₂×#501	1	1	yes	characteristic, complemented, Sylow
77	C₂×C₄⋊Q₈	D₁₀₉	1	1	yes	characteristic
78	C₂×#125	—	109	1	no	—
79	#501	—	109	8	no	—
80	C₂×#107	—	109	2	no	—
81	C₂×#106	—	109	2	no	—
82	C₂×#97	—	109	1	no	—
83	C₂²×Q₈	—	2	1	no	—
84	C₂²×C₈	—	109	2	no	—
85	C₄⋊Q₈	—	2	2	no	—
86	C₄⋊Q₈	C₂×D₁₀₉	1	4	yes	—
87	#125	—	109	4	no	—
88	#125	—	218	2	no	—
89	C₂×C₄⋊C₄	—	2	1	no	—
90	C₂×C₄⋊C₄	—	109	1	no	—
91	C₂×C₄⋊C₄	—	109	1	no	—
92	C₂×C₄⋊C₄	—	218	1	no	—
93	C₂×C₄⋊C₄	—	218	1	no	—
94	C₂×C₄⋊C₄	C₂×D₁₀₉	1	2	yes	—
95	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
96	#107	—	109	8	no	—
97	#106	—	109	8	no	—
98	#97	—	109	4	no	—
99	C₂×Q₈	—	2	4	no	—
100	C₂×Q₈	—	4	4	no	—
101	C₂²×C₄	—	2	2	no	—
102	C₂²×C₄	—	218	1	no	—
103	C₂²×C₄	—	218	1	no	—
104	C₂²×C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
105	C₂²×C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
106	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
107	C₂×C₈	—	109	8	no	—
108	C₂×C₈	—	109	4	no	—
109	C₄⋊C₄	—	2	4	no	—
110	C₄⋊C₄	—	2	2	no	—
111	C₄⋊C₄	—	109	2	no	—
112	C₄⋊C₄	—	109	2	no	—
113	C₄⋊C₄	—	218	4	no	—
114	C₄⋊C₄	—	218	4	no	—
115	C₄⋊C₄	—	218	1	no	—
116	C₄⋊C₄	—	218	1	no	—
117	C₄⋊C₄	C₂²×D₁₀₉	1	4	yes	—
118	C₄²	C₂²⋊D₁₀₉	1	2	yes	—
119	C₄²	C₂²×D₁₀₉	1	2	yes	—
120	C₂³	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
121	Q₈	—	4	8	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	2	4	no	—
125	C₂×C₄	—	4	4	no	—
126	C₂×C₄	—	436	2	no	—
127	C₂×C₄	—	436	2	no	—
128	C₂×C₄	—	218	2	no	—
129	C₂×C₄	—	218	2	no	—
130	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	2	yes	—
131	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	2	yes	—
132	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
133	C₂×C₄	Q₈×D₁₀₉	1	4	yes	—
134	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
135	C₈	—	109	8	no	—
136	C₂²	#16812697	1	2	yes	—
137	C₂²	#16812725	1	2	yes	—
138	C₂²	#16812730	1	2	yes	—
139	C₂²	C₂²×C₂²⋊D₁₀₉	1	1	yes	characteristic
140	C₄	—	2	2	no	—
141	C₄	—	2	2	no	—
142	C₄	—	4	4	no	—
143	C₄	—	436	2	no	—
144	C₄	—	436	2	no	—
145	C₄	#16812697	1	2	yes	—
146	C₄	C₂×Q₈×D₁₀₉	1	2	yes	—
147	C₂	#20295432	1	4	yes	direct product factor
148	C₂	C₂×#16812697	1	1	yes	characteristic
149	C₂	C₂×#16812725	1	1	yes	characteristic
150	C₂	C₂×#16812730	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812691

C₂

yes

maximal, characteristic

#20295432

C₂

yes

maximal, direct product factor

C₂×#16812632

C₂

yes

maximal

C₂×#16812630

C₂

yes

maximal

C₂×#16812610

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₁₀₉

C₂

yes

maximal, characteristic

C₂²×#11724784

C₂²

yes

—

#16812691

—

#16812691

C₂²

yes

—

C₂×C₄⋊Dic₁₀₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₀₉

C₂²

yes

—

—

C₂²

yes

—

#16812630

C₂²

yes

—

#16812610

C₂²

yes

—

C₂²×Q₈×C₁₀₉

—

C₄⋊Q₈×C₁₀₉

—

C₄⋊Q₈×C₁₀₉

C₂²

yes

—

C₂×C₄⋊C₄×C₁₀₉

—

C₂×C₄⋊C₄×C₁₀₉

C₂²

yes

—

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

C₂²×Dic₁₀₉

—

C₂²×Dic₁₀₉

—

C₂×#11724784

Q₈

yes

complemented

C₂×#11724784

C₂³

yes

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

C₂³

yes

—

C₄⋊Dic₁₀₉

C₂³

yes

—

#13182008

—

#13182008

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

D₄

yes

characteristic

C₂²×C₄×C₁₀₉

D₄

yes

characteristic

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

C₂³

yes

—

C₄²×C₁₀₉

D₄

yes

—

C₄²×C₁₀₉

C₂³

yes

—

C₂³×C₁₀₉

C₂×D₄

yes

characteristic

Q₈×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×Q₈

yes

—

C₂×C₄×C₁₀₉

C₂⁴

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

#11724784

C₂×Q₈

yes

complemented

C₂²×C₁₀₉

C₄⋊Q₈

yes

—

C₂²×C₁₀₉

AGL(1,ℤ/8)

yes

—

C₂²×C₁₀₉

#137

yes

—

C₂²×C₁₀₉

C₂²×D₄

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₄⋊Q₈

yes

—

C₄×C₁₀₉

C₂²×Q₈

yes

—

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

#501

yes

complemented

C₂×C₁₀₉

C₂×C₄⋊Q₈

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

109

maximal, Sylow

C₁₀₉

C₂×#501

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

D₁₀₉

yes

characteristic

C₂×#125

—

109

—

#501

—

109

—

C₂×#107

—

109

—

C₂×#106

—

109

—

C₂×#97

—

109

—

C₂²×Q₈

—

C₂²×C₈

—

109

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₁₀₉

yes

—

#125

—

109

—

#125

—

218

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

—

218

—

C₂×C₄⋊C₄

—

218

—

C₂×C₄⋊C₄

C₂×D₁₀₉

yes

—

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

#107

—

109

—

#106

—

109

—

#97

—

109

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂²×C₄

—

102

C₂²×C₄

—

218

—

103

C₂²×C₄

—

218

—

104

C₂²×C₄

C₂²⋊D₁₀₉

yes

characteristic

105

C₂²×C₄

C₂²⋊D₁₀₉

yes

characteristic

106

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

107

C₂×C₈

—

109

—

108

C₂×C₈

—

109

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

109

—

112

C₄⋊C₄

—

109

—

113

C₄⋊C₄

—

218

—

114

C₄⋊C₄

—

218

—

115

C₄⋊C₄

—

218

—

116

C₄⋊C₄

—

218

—

117

C₄⋊C₄

C₂²×D₁₀₉

yes

—

118

C₄²

C₂²⋊D₁₀₉

yes

—

119

C₄²

C₂²×D₁₀₉

yes

—

120

C₂³

C₂×C₂²⋊D₁₀₉

yes

characteristic

121

Q₈

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

436

—

127

C₂×C₄

—

436

—

128

C₂×C₄

—

218

—

129

C₂×C₄

—

218

—

130

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

—

131

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

—

132

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

133

C₂×C₄

Q₈×D₁₀₉

yes

—

134

C₂×C₄

C₂³×D₁₀₉

yes

characteristic

135

C₈

—

109

—

136

C₂²

#16812697

yes

—

137

C₂²

#16812725

yes

—

138

C₂²

#16812730

yes

—

139

C₂²

C₂²×C₂²⋊D₁₀₉

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

436

—

144

C₄

—

436

—

145

C₄

#16812697

yes

—

146

C₄

C₂×Q₈×D₁₀₉

yes

—

147

C₂

#20295432

yes

direct product factor

148

C₂

C₂×#16812697

yes

characteristic

149

C₂

C₂×#16812725

yes

characteristic

150

C₂

C₂×#16812730

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	C₂×C₄⋊Q₈×C₁₀₉	C₂
socle, prime radical	C₂³×C₁₀₉	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	D₁₀₉
hypocenter	C₁₀₉	C₂×#501
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₀₉

commutator, cosocle

C₂×C₄×C₁₀₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₀₉

Fitting

C₂×C₄⋊Q₈×C₁₀₉

C₂

socle, prime radical

C₂³×C₁₀₉

C₂×D₄

hypercenter

C₂×C₄⋊Q₈

D₁₀₉

hypocenter

C₁₀₉

C₂×#501

LCS

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
2. #20295432	C₂	direct (N × H)
25. C₂×#11724784	Q₈	semidirect
59. #11724784	C₂×Q₈	semidirect
71. C₂×C₁₀₉	#501	semidirect
76. C₁₀₉	C₂×#501	semidirect
147. C₂	#20295432	direct (N × H)

H ≅ G/N

type

2. #20295432

C₂

direct (N × H)

25. C₂×#11724784

Q₈

semidirect

59. #11724784

C₂×Q₈

semidirect

71. C₂×C₁₀₉

#501

semidirect

76. C₁₀₉

C₂×#501

semidirect

147. C₂

#20295432

direct (N × H)

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

subgroup	index
C₂×#16812691	2
#20295432	2
C₂×#16812632	2
C₂×#16812630	2
C₂×#16812610	2
C₂×C₄⋊Q₈×C₁₀₉	2
C₂×#501	109

subgroup

index

C₂×C₄⋊Q₈×C₁₀₉

109

Subgroups — C₂×#20295432 #28373548

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#20295432 #28373548

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20295432 #28373548