Group #28371750 — subgroups

Subgroups — #28371750

Order 13952.

484 conjugacy classes of subgroups; only the 221 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 221 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×#16812706	C₂	1	2	yes	maximal, complemented
2	#20295098	C₂	1	8	yes	maximal, complemented
3	#20295050	C₂	1	1	yes	maximal, characteristic, complemented
4	#20294993	C₂	1	2	yes	maximal, complemented
5	C₂×#16812610	C₂	1	1	yes	maximal, characteristic, complemented
6	#532×C₁₀₉	C₂	1	1	yes	maximal, characteristic
7	#16812706	—	2	8	no	—
8	C₂×#13182013	C₂²	1	2	yes	—
9	C₂×#13182013	C₂²	1	4	yes	—
10	C₂²×#11724784	C₂²	1	1	yes	characteristic
11	C₂²×#11724784	C₂²	1	4	yes	—
12	C₂²×#11724784	C₂²	1	1	yes	characteristic
13	C₄⋊#11724784	C₂²	1	4	yes	—
14	C₂²⋊#11724784	C₂²	1	8	yes	—
15	#16812610	C₂²	1	4	yes	—
16	C₂×#41×C₁₀₉	C₂²	1	2	yes	—
17	C₄⋊Q₈×C₁₀₉	C₄	1	1	yes	characteristic
18	#127×C₁₀₉	C₄	1	1	yes	characteristic
19	#124×C₁₀₉	C₄	1	2	yes	—
20	#121×C₁₀₉	C₄	1	4	yes	—
21	C₄×D₄×C₁₀₉	C₂²	1	4	yes	—
22	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
23	#13182013	—	2	4	no	—
24	#13182013	—	2	8	no	—
25	C₂×#11724784	—	2	2	no	—
26	C₂×#11724784	—	2	8	no	—
27	C₂×#11724784	D₄	1	4	yes	complemented
28	C₂×#11724784	C₂³	1	2	yes	—
29	C₂×#11724784	C₂³	1	4	yes	—
30	C₂×#11724784	C₂³	1	2	yes	—
31	#41×C₁₀₉	—	2	4	no	—
32	#41×C₁₀₉	—	2	4	no	—
33	C₂×Q₈×C₁₀₉	C₂×C₄	1	2	yes	—
34	C₂×D₄×C₁₀₉	C₂×C₄	1	4	yes	—
35	C₂×D₄×C₁₀₉	C₂×C₄	1	2	yes	—
36	C₂×D₄×C₁₀₉	C₂×C₄	1	4	yes	—
37	C₂²×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
38	C₂²×C₄×C₁₀₉	C₂³	1	4	yes	—
39	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
40	C₄⋊C₄×C₁₀₉	C₂×C₄	1	4	yes	—
41	C₂²⋊C₄×C₁₀₉	C₂×C₄	1	8	yes	—
42	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
43	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
44	C₄²×C₁₀₉	C₂³	1	2	yes	—
45	C₂³×C₁₀₉	C₂²×C₄	1	4	yes	—
46	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
47	Q₈×C₁₀₉	—	2	4	no	—
48	D₄×C₁₀₉	—	2	8	no	—
49	D₄×C₁₀₉	—	2	4	no	—
50	D₄×C₁₀₉	—	4	4	no	—
51	C₂×C₄×C₁₀₉	—	2	4	no	—
52	C₂×C₄×C₁₀₉	—	2	4	no	—
53	C₂×C₄×C₁₀₉	—	4	2	no	—
54	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
55	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
56	C₂×C₄×C₁₀₉	C₂²×C₄	1	4	yes	—
57	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₁₀₉	#41	1	2	yes	—
60	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
61	#11724784	—	2	4	no	—
62	#11724784	—	2	2	no	—
63	#11724784	C₂×D₄	1	4	yes	—
64	C₂²×C₁₀₉	—	2	2	no	—
65	C₂²×C₁₀₉	—	2	4	no	—
66	C₂²×C₁₀₉	—	2	8	no	—
67	C₂²×C₁₀₉	C₄×D₄	1	2	yes	—
68	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
69	C₄×C₁₀₉	—	2	4	no	—
70	C₄×C₁₀₉	—	2	2	no	—
71	C₄×C₁₀₉	—	2	2	no	—
72	C₄×C₁₀₉	C₄×D₄	1	2	yes	—
73	C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
74	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
75	C₂×C₁₀₉	—	2	2	no	—
76	C₂×C₁₀₉	—	4	4	no	—
77	C₂×C₁₀₉	C₂×C₄×D₄	1	1	yes	characteristic
78	C₂×C₁₀₉	#568	1	2	yes	—
79	#2932	—	109	1	no	maximal, Sylow
80	C₁₀₉	#2932	1	1	yes	characteristic, complemented, Sylow
81	C₂×#131	—	109	2	no	—
82	#532	D₁₀₉	1	1	yes	characteristic
83	#435	—	109	8	no	—
84	#423	—	109	1	no	—
85	#408	—	109	2	no	—
86	C₂×#97	—	109	1	no	—
87	C₂×#41	C₂×D₁₀₉	1	2	yes	—
88	#131	—	218	8	no	—
89	C₂×M₁₆	—	109	2	no	—
90	C₂×M₁₆	—	109	4	no	—
91	C₂²×C₈	—	109	1	no	—
92	C₂²×C₈	—	109	4	no	—
93	C₂²×C₈	—	109	1	no	—
94	C₄⋊Q₈	Dic₁₀₉	1	1	yes	characteristic
95	#127	Dic₁₀₉	1	1	yes	characteristic
96	#124	Dic₁₀₉	1	2	yes	—
97	#121	Dic₁₀₉	1	4	yes	—
98	C₄×D₄	C₂×D₁₀₉	1	4	yes	—
99	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
100	C₄⋊C₈	—	109	4	no	—
101	C₂²⋊C₈	—	109	8	no	—
102	#97	—	109	4	no	—
103	#41	—	2	4	no	—
104	#41	—	2	4	no	—
105	C₂×Q₈	C₂×Dic₁₀₉	1	2	yes	—
106	C₂×D₄	C₂×Dic₁₀₉	1	4	yes	—
107	C₂×D₄	C₂×Dic₁₀₉	1	2	yes	—
108	C₂×D₄	C₂×Dic₁₀₉	1	4	yes	—
109	C₂²×C₄	C₂×Dic₁₀₉	1	2	yes	—
110	C₂²×C₄	C₂²×D₁₀₉	1	4	yes	—
111	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
112	M₁₆	—	218	8	no	—
113	M₁₆	—	218	4	no	—
114	C₂×C₈	—	109	4	no	—
115	C₂×C₈	—	109	2	no	—
116	C₂×C₈	—	109	2	no	—
117	C₂×C₈	—	109	4	no	—
118	C₂×C₈	—	218	8	no	—
119	C₂×C₈	—	218	2	no	—
120	C₄⋊C₄	C₂×Dic₁₀₉	1	4	yes	—
121	C₂²⋊C₄	C₂×Dic₁₀₉	1	8	yes	—
122	C₄²	C₂×Dic₁₀₉	1	1	yes	characteristic
123	C₄²	C₂×Dic₁₀₉	1	1	yes	characteristic
124	C₄²	C₂²×D₁₀₉	1	2	yes	—
125	C₂³	C₂²×Dic₁₀₉	1	4	yes	—
126	C₂³	C₂²×Dic₁₀₉	1	1	yes	characteristic
127	Q₈	—	2	4	no	—
128	D₄	—	2	8	no	—
129	D₄	—	2	4	no	—
130	D₄	—	4	4	no	—
131	C₂×C₄	—	2	4	no	—
132	C₂×C₄	—	2	4	no	—
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	C₂²×Dic₁₀₉	1	2	yes	—
135	C₂×C₄	C₂²×Dic₁₀₉	1	2	yes	—
136	C₂×C₄	C₂²×Dic₁₀₉	1	4	yes	—
137	C₂×C₄	C₂²×Dic₁₀₉	1	1	yes	characteristic
138	C₂×C₄	D₄×D₁₀₉	1	2	yes	—
139	C₂×C₄	#13182028	1	2	yes	—
140	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
141	C₈	—	109	4	no	—
142	C₈	—	218	4	no	—
143	C₈	—	218	2	no	—
144	C₂²	—	2	2	no	—
145	C₂²	—	2	4	no	—
146	C₂²	—	2	8	no	—
147	C₂²	D₄×Dic₁₀₉	1	2	yes	—
148	C₂²	C₂³×Dic₁₀₉	1	1	yes	characteristic
149	C₄	—	2	4	no	—
150	C₄	—	2	2	no	—
151	C₄	—	2	2	no	—
152	C₄	D₄×Dic₁₀₉	1	2	yes	—
153	C₄	C₂×D₄×D₁₀₉	1	1	yes	characteristic
154	C₄	C₂×#13182028	1	1	yes	characteristic
155	C₂	—	2	2	no	—
156	C₂	—	4	4	no	—
157	C₂	C₂×D₄×Dic₁₀₉	1	1	yes	characteristic
158	C₂	#20295793	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812706

C₂

yes

maximal, complemented

#20295098

C₂

yes

maximal, complemented

#20295050

C₂

yes

maximal, characteristic, complemented

#20294993

C₂

yes

maximal, complemented

C₂×#16812610

C₂

yes

maximal, characteristic, complemented

#532×C₁₀₉

C₂

yes

maximal, characteristic

#16812706

—

C₂×#13182013

C₂²

yes

—

C₂×#13182013

C₂²

yes

—

C₂²×#11724784

C₂²

yes

characteristic

C₂²×#11724784

C₂²

yes

—

C₂²×#11724784

C₂²

yes

characteristic

C₄⋊#11724784

C₂²

yes

—

C₂²⋊#11724784

C₂²

yes

—

#16812610

C₂²

yes

—

C₂×#41×C₁₀₉

C₂²

yes

—

C₄⋊Q₈×C₁₀₉

C₄

yes

characteristic

#127×C₁₀₉

C₄

yes

characteristic

#124×C₁₀₉

C₄

yes

—

#121×C₁₀₉

C₄

yes

—

C₄×D₄×C₁₀₉

C₂²

yes

—

C₂×C₄²×C₁₀₉

C₂²

yes

—

—

—

—

D₄

yes

complemented

C₂×#11724784

C₂³

yes

—

C₂×#11724784

C₂³

yes

—

C₂×#11724784

C₂³

yes

—

#41×C₁₀₉

—

#41×C₁₀₉

—

C₂×Q₈×C₁₀₉

C₂×C₄

yes

—

C₂×D₄×C₁₀₉

C₂×C₄

yes

—

C₂×D₄×C₁₀₉

C₂×C₄

yes

—

C₂×D₄×C₁₀₉

C₂×C₄

yes

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₉

C₂×C₄

yes

—

C₂²⋊C₄×C₁₀₉

C₂×C₄

yes

—

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₄²×C₁₀₉

C₂³

yes

—

C₂³×C₁₀₉

C₂²×C₄

yes

—

C₂³×C₁₀₉

C₂²×C₄

yes

characteristic

Q₈×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

#41

yes

—

C₂×C₄×C₁₀₉

C₂⁴

yes

—

—

C₂×D₄

yes

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₄×D₄

yes

—

C₂²×C₁₀₉

C₂³×C₄

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₄×D₄

yes

—

C₄×C₁₀₉

C₂²×D₄

yes

characteristic

C₄×C₁₀₉

C₂×#41

yes

characteristic

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

C₂×C₄×D₄

yes

characteristic

C₂×C₁₀₉

#568

yes

—

#2932

—

109

maximal, Sylow

C₁₀₉

#2932

yes

characteristic, complemented, Sylow

C₂×#131

—

109

—

#532

D₁₀₉

yes

characteristic

#435

—

109

—

#423

—

109

—

#408

—

109

—

C₂×#97

—

109

—

C₂×#41

C₂×D₁₀₉

yes

—

#131

—

218

—

C₂×M₁₆

—

109

—

C₂×M₁₆

—

109

—

C₂²×C₈

—

109

—

C₂²×C₈

—

109

—

C₂²×C₈

—

109

—

C₄⋊Q₈

Dic₁₀₉

yes

characteristic

#127

Dic₁₀₉

yes

characteristic

#124

Dic₁₀₉

yes

—

#121

Dic₁₀₉

yes

—

C₄×D₄

C₂×D₁₀₉

yes

—

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

100

C₄⋊C₈

—

109

—

101

C₂²⋊C₈

—

109

—

102

#97

—

109

—

103

#41

—

104

#41

—

105

C₂×Q₈

C₂×Dic₁₀₉

yes

—

106

C₂×D₄

C₂×Dic₁₀₉

yes

—

107

C₂×D₄

C₂×Dic₁₀₉

yes

—

108

C₂×D₄

C₂×Dic₁₀₉

yes

—

109

C₂²×C₄

C₂×Dic₁₀₉

yes

—

110

C₂²×C₄

C₂²×D₁₀₉

yes

—

111

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

112

M₁₆

—

218

—

113

M₁₆

—

218

—

114

C₂×C₈

—

109

—

115

C₂×C₈

—

109

—

116

C₂×C₈

—

109

—

117

C₂×C₈

—

109

—

118

C₂×C₈

—

218

—

119

C₂×C₈

—

218

—

120

C₄⋊C₄

C₂×Dic₁₀₉

yes

—

121

C₂²⋊C₄

C₂×Dic₁₀₉

yes

—

122

C₄²

C₂×Dic₁₀₉

yes

characteristic

123

C₄²

C₂×Dic₁₀₉

yes

characteristic

124

C₄²

C₂²×D₁₀₉

yes

—

125

C₂³

C₂²×Dic₁₀₉

yes

—

126

C₂³

C₂²×Dic₁₀₉

yes

characteristic

127

Q₈

—

128

D₄

—

129

D₄

—

130

D₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

C₂²×Dic₁₀₉

yes

—

135

C₂×C₄

C₂²×Dic₁₀₉

yes

—

136

C₂×C₄

C₂²×Dic₁₀₉

yes

—

137

C₂×C₄

C₂²×Dic₁₀₉

yes

characteristic

138

C₂×C₄

D₄×D₁₀₉

yes

—

139

C₂×C₄

#13182028

yes

—

140

C₂×C₄

C₂³×D₁₀₉

yes

characteristic

141

C₈

—

109

—

142

C₈

—

218

—

143

C₈

—

218

—

144

C₂²

—

145

C₂²

—

146

C₂²

—

147

C₂²

D₄×Dic₁₀₉

yes

—

148

C₂²

C₂³×Dic₁₀₉

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

D₄×Dic₁₀₉

yes

—

153

C₄

C₂×D₄×D₁₀₉

yes

characteristic

154

C₄

C₂×#13182028

yes

characteristic

155

C₂

—

156

C₂

—

157

C₂

C₂×D₄×Dic₁₀₉

yes

characteristic

158

C₂

#20295793

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₀₉
commutator, socle	C₂²×C₁₀₉	C₂³×C₄
Fitting, prime radical	#532×C₁₀₉	C₂
cosocle	C₂×C₄×C₁₀₉	C₂⁴
hypercenter	#532	D₁₀₉
hypocenter	C₁₀₉	#2932
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ #532

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₁₀₉

commutator, socle

C₂²×C₁₀₉

C₂³×C₄

Fitting, prime radical

#532×C₁₀₉

C₂

cosocle

C₂×C₄×C₁₀₉

C₂⁴

D₁₀₉

C₁₀₉

G ▷ C₂²×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂×C₄ ◁ #532

N	H ≅ G/N	type
1. C₂×#16812706	C₂	semidirect
2. #20295098	C₂	semidirect
3. #20295050	C₂	semidirect
4. #20294993	C₂	semidirect
5. C₂×#16812610	C₂	semidirect
27. C₂×#11724784	D₄	semidirect
80. C₁₀₉	#2932	semidirect

H ≅ G/N

type

1. C₂×#16812706

C₂

semidirect

2. #20295098

C₂

semidirect

3. #20295050

C₂

semidirect

4. #20294993

C₂

semidirect

5. C₂×#16812610

C₂

semidirect

27. C₂×#11724784

D₄

semidirect

80. C₁₀₉

#2932

semidirect

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

subgroup	index
C₂×#16812706	2
#20295098	2
#20295050	2
#20294993	2
C₂×#16812610	2
#532×C₁₀₉	2
#2932	109

subgroup

index

#532×C₁₀₉

109

Subgroups — #28371750

Conjugacy classes of subgroups

Complements

Maximal subgroups