Group #28373344 — subgroups

Subgroups — C₂×#20295378 #28373344

Order 13952.

424 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 181 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×#16812699	C₂	1	1	yes	maximal, characteristic
2	#20295378	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16812622	C₂	1	4	yes	maximal
4	C₂×C₄×#11724784	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Q₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₁₈	C₂²	1	2	yes	—
7	C₂²×#11724784	C₂²	1	2	yes	—
8	#16812699	—	2	2	no	—
9	#16812699	C₂²	1	4	yes	—
10	C₂×C₄⋊Dic₁₀₉	—	2	1	no	—
11	C₂×C₄⋊Dic₁₀₉	—	2	1	no	—
12	#16812622	C₂²	1	16	yes	—
13	C₄×#11724784	C₂²	1	4	yes	—
14	C₂²×Q₈×C₁₀₉	—	2	1	no	—
15	C₄⋊Q₈×C₁₀₉	—	2	2	no	—
16	C₄⋊Q₈×C₁₀₉	C₂²	1	4	yes	—
17	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
18	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
19	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
20	C₂×Dic₂₁₈	—	2	8	no	—
21	C₂×Dic₂₁₈	—	2	2	no	—
22	C₂×Dic₂₁₈	C₂³	1	4	yes	—
23	C₂²×Dic₁₀₉	—	2	2	no	—
24	C₂×#11724784	—	2	4	no	—
25	C₂×#11724784	C₂³	1	4	yes	—
26	C₄⋊Dic₁₀₉	—	2	4	no	—
27	C₄⋊Dic₁₀₉	—	2	4	no	—
28	C₂×Q₈×C₁₀₉	—	2	4	no	—
29	C₂×Q₈×C₁₀₉	—	4	4	no	—
30	C₂²×C₄×C₁₀₉	—	2	2	no	—
31	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
34	C₄⋊C₄×C₁₀₉	—	2	4	no	—
35	C₄⋊C₄×C₁₀₉	—	2	2	no	—
36	C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
37	C₄²×C₁₀₉	D₄	1	2	yes	—
38	C₄²×C₁₀₉	C₂³	1	2	yes	—
39	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
40	Q₈×C₁₀₉	—	4	8	no	—
41	C₂×C₄×C₁₀₉	—	2	4	no	—
42	C₂×C₄×C₁₀₉	—	4	4	no	—
43	C₂×C₄×C₁₀₉	SD₁₆	1	4	yes	—
44	C₂×C₄×C₁₀₉	Q₁₆	1	4	yes	—
45	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
47	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₁₀₉	#41	1	4	yes	—
49	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
50	C₂×Dic₁₀₉	—	2	4	no	—
51	C₂×Dic₁₀₉	—	4	4	no	—
52	Dic₂₁₈	—	2	8	no	—
53	Dic₂₁₈	—	4	4	no	—
54	#11724784	—	2	4	no	—
55	C₂²×C₁₀₉	#124	1	2	yes	—
56	C₂²×C₁₀₉	C₂×SD₁₆	1	2	yes	—
57	C₂²×C₁₀₉	C₂×Q₁₆	1	2	yes	—
58	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
59	C₄×C₁₀₉	—	4	4	no	—
60	C₄×C₁₀₉	#124	1	2	yes	—
61	C₄×C₁₀₉	C₂×SD₁₆	1	4	yes	—
62	C₄×C₁₀₉	C₂×Q₁₆	1	4	yes	—
63	C₄×C₁₀₉	C₂×#41	1	2	yes	—
64	Dic₁₀₉	—	4	4	no	—
65	C₂×C₁₀₉	#487	1	4	yes	complemented
66	C₂×C₁₀₉	C₂×#124	1	1	yes	characteristic
67	C₂×C₁₀₉	C₂²×SD₁₆	1	1	yes	characteristic
68	C₂×C₁₀₉	C₂²×Q₁₆	1	1	yes	characteristic
69	C₂×#487	—	109	1	no	maximal, Sylow
70	C₁₀₉	C₂×#487	1	1	yes	characteristic, complemented, Sylow
71	C₂×C₄⋊Q₈	—	109	1	no	—
72	C₂×C₄⋊Q₈	D₁₀₉	1	1	yes	characteristic
73	#487	—	109	8	no	—
74	C₂×#103	—	109	4	no	—
75	C₂×C₄×C₈	—	109	1	no	—
76	C₂²×Q₈	—	2	1	no	—
77	C₂²×Q₈	—	109	2	no	—
78	C₂²×C₈	—	109	2	no	—
79	C₄⋊Q₈	—	2	2	no	—
80	C₄⋊Q₈	—	109	4	no	—
81	C₄⋊Q₈	—	218	2	no	—
82	C₄⋊Q₈	C₂×D₁₀₉	1	4	yes	—
83	C₂×C₄⋊C₄	—	2	1	no	—
84	C₂×C₄⋊C₄	—	218	1	no	—
85	C₂×C₄⋊C₄	—	218	1	no	—
86	C₂×C₄⋊C₄	C₂×D₁₀₉	1	2	yes	—
87	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
88	#103	—	109	16	no	—
89	C₄×C₈	—	109	4	no	—
90	C₂×Q₈	—	2	4	no	—
91	C₂×Q₈	—	4	4	no	—
92	C₂×Q₈	—	109	4	no	—
93	C₂×Q₈	—	218	8	no	—
94	C₂×Q₈	—	218	2	no	—
95	C₂²×C₄	—	2	2	no	—
96	C₂²×C₄	—	218	2	no	—
97	C₂²×C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
98	C₂²×C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
99	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
100	C₂×C₈	—	109	4	no	—
101	C₂×C₈	—	218	4	no	—
102	C₄⋊C₄	—	2	4	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	218	4	no	—
105	C₄⋊C₄	—	218	4	no	—
106	C₄⋊C₄	C₂²×D₁₀₉	1	4	yes	—
107	C₄²	C₂²⋊D₁₀₉	1	2	yes	—
108	C₄²	C₂²×D₁₀₉	1	2	yes	—
109	C₂³	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
110	Q₈	—	4	8	no	—
111	Q₈	—	436	4	no	—
112	Q₈	—	218	8	no	—
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	4	4	no	—
115	C₂×C₄	—	436	4	no	—
116	C₂×C₄	—	218	4	no	—
117	C₂×C₄	#13182017	1	4	yes	—
118	C₂×C₄	#13182019	1	4	yes	—
119	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
120	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	2	yes	—
121	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	2	yes	—
122	C₂×C₄	#13182030	1	4	yes	—
123	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
124	C₈	—	218	4	no	—
125	C₂²	#16812688	1	2	yes	—
126	C₂²	C₂×#13182017	1	2	yes	—
127	C₂²	C₂×#13182019	1	2	yes	—
128	C₂²	C₂²×C₂²⋊D₁₀₉	1	1	yes	characteristic
129	C₄	—	4	4	no	—
130	C₄	—	436	4	no	—
131	C₄	#16812688	1	2	yes	—
132	C₄	C₂×#13182017	1	4	yes	—
133	C₄	C₂×#13182019	1	4	yes	—
134	C₄	C₂×#13182030	1	2	yes	—
135	C₂	#20295378	1	4	yes	direct product factor
136	C₂	C₂×#16812688	1	1	yes	characteristic
137	C₂	C₂²×#13182017	1	1	yes	characteristic
138	C₂	C₂²×#13182019	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812699

C₂

yes

maximal, characteristic

#20295378

C₂

yes

maximal, direct product factor

C₂×#16812622

C₂

yes

maximal

C₂×C₄×#11724784

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₁₀₉

C₂

yes

maximal, characteristic

C₂²×Dic₂₁₈

C₂²

yes

—

C₂²×#11724784

C₂²

yes

—

#16812699

—

#16812699

C₂²

yes

—

C₂×C₄⋊Dic₁₀₉

—

C₂×C₄⋊Dic₁₀₉

—

#16812622

C₂²

yes

—

C₄×#11724784

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

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

C₂³

yes

—

C₂²×Dic₁₀₉

—

C₂×#11724784

—

C₂×#11724784

C₂³

yes

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

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

SD₁₆

yes

—

C₂×C₄×C₁₀₉

Q₁₆

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

#41

yes

—

C₂×C₄×C₁₀₉

C₂⁴

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

Dic₂₁₈

—

Dic₂₁₈

—

#11724784

—

C₂²×C₁₀₉

#124

yes

—

C₂²×C₁₀₉

C₂×SD₁₆

yes

—

C₂²×C₁₀₉

C₂×Q₁₆

yes

—

C₂²×C₁₀₉

C₂²×D₄

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

#124

yes

—

C₄×C₁₀₉

C₂×SD₁₆

yes

—

C₄×C₁₀₉

C₂×Q₁₆

yes

—

C₄×C₁₀₉

C₂×#41

yes

—

Dic₁₀₉

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

C₂²×SD₁₆

yes

characteristic

C₂×C₁₀₉

C₂²×Q₁₆

yes

characteristic

C₂×#487

—

109

maximal, Sylow

C₁₀₉

C₂×#487

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

109

—

C₂×C₄⋊Q₈

D₁₀₉

yes

characteristic

#487

—

109

—

C₂×#103

—

109

—

C₂×C₄×C₈

—

109

—

C₂²×Q₈

—

C₂²×Q₈

—

109

—

C₂²×C₈

—

109

—

C₄⋊Q₈

—

C₄⋊Q₈

—

109

—

C₄⋊Q₈

—

218

—

C₄⋊Q₈

C₂×D₁₀₉

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

218

—

C₂×C₄⋊C₄

—

218

—

C₂×C₄⋊C₄

C₂×D₁₀₉

yes

—

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

#103

—

109

—

C₄×C₈

—

109

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

109

—

C₂×Q₈

—

218

—

C₂×Q₈

—

218

—

C₂²×C₄

—

C₂²×C₄

—

218

—

C₂²×C₄

C₂²⋊D₁₀₉

yes

characteristic

C₂²×C₄

C₂²⋊D₁₀₉

yes

characteristic

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

100

C₂×C₈

—

109

—

101

C₂×C₈

—

218

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

218

—

105

C₄⋊C₄

—

218

—

106

C₄⋊C₄

C₂²×D₁₀₉

yes

—

107

C₄²

C₂²⋊D₁₀₉

yes

—

108

C₄²

C₂²×D₁₀₉

yes

—

109

C₂³

C₂×C₂²⋊D₁₀₉

yes

characteristic

110

Q₈

—

111

Q₈

—

436

—

112

Q₈

—

218

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

436

—

116

C₂×C₄

—

218

—

117

C₂×C₄

#13182017

yes

—

118

C₂×C₄

#13182019

yes

—

119

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

120

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

—

121

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

—

122

C₂×C₄

#13182030

yes

—

123

C₂×C₄

C₂³×D₁₀₉

yes

characteristic

124

C₈

—

218

—

125

C₂²

#16812688

yes

—

126

C₂²

C₂×#13182017

yes

—

127

C₂²

C₂×#13182019

yes

—

128

C₂²

C₂²×C₂²⋊D₁₀₉

yes

characteristic

129

C₄

—

130

C₄

—

436

—

131

C₄

#16812688

yes

—

132

C₄

C₂×#13182017

yes

—

133

C₄

C₂×#13182019

yes

—

134

C₄

C₂×#13182030

yes

—

135

C₂

#20295378

yes

direct product factor

136

C₂

C₂×#16812688

yes

characteristic

137

C₂

C₂²×#13182017

yes

characteristic

138

C₂

C₂²×#13182019

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×#487
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×#487

LCS

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
2. #20295378	C₂	direct (N × H)
65. C₂×C₁₀₉	#487	semidirect
70. C₁₀₉	C₂×#487	semidirect
135. C₂	#20295378	direct (N × H)

H ≅ G/N

type

2. #20295378

C₂

direct (N × H)

65. C₂×C₁₀₉

#487

semidirect

70. C₁₀₉

C₂×#487

semidirect

135. C₂

#20295378

direct (N × H)

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

subgroup	index
C₂×#16812699	2
#20295378	2
C₂×#16812622	2
C₂×C₄×#11724784	2
C₂×C₄⋊Q₈×C₁₀₉	2
C₂×#487	109

subgroup

index

C₂×C₄⋊Q₈×C₁₀₉

109

Subgroups — C₂×#20295378 #28373344

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#20295378 #28373344

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20295378 #28373344