Group #28373380 — subgroups

Subgroups — #28373380

Order 13952.

392 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 133 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	#20295576	C₂	1	1	yes	maximal, characteristic
2	#20295396	C₂	1	2	yes	maximal, complemented
3	#20295392	C₂	1	2	yes	maximal, complemented
4	#20295388	C₂	1	4	yes	maximal, complemented
5	#20295020	C₂	1	4	yes	maximal, complemented
6	C₂×#16812610	C₂	1	1	yes	maximal, characteristic, complemented
7	#533×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13182029	C₂²	1	2	yes	—
9	C₂²×#11724784	C₂²	1	2	yes	—
10	#16812699	C₂²	1	1	yes	characteristic
11	#16812696	C₂²	1	1	yes	characteristic
12	#16812687	C₂²	1	2	yes	—
13	#16812673	—	2	2	no	—
14	C₄×D₄₃₆	—	2	2	no	—
15	#16812622	C₂²	1	8	yes	—
16	#16812619	C₂²	1	8	yes	—
17	#16812610	C₂²	1	2	yes	—
18	#16812610	C₂²	1	2	yes	—
19	C₄⋊Q₈×C₁₀₉	C₂²	1	2	yes	—
20	#125×C₁₀₉	C₂²	1	2	yes	—
21	C₂²⋊Q₈×C₁₀₉	—	2	2	no	—
22	C₄×Q₈×C₁₀₉	—	2	2	no	—
23	#117×C₁₀₉	C₂²	1	2	yes	—
24	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
25	#13182029	—	2	8	no	—
26	C₂×Dic₂₁₈	C₂³	1	2	yes	—
27	C₂×D₄₃₆	—	2	2	no	—
28	C₂×D₄₃₆	C₂³	1	2	yes	—
29	C₂×C₂²⋊D₁₀₉	—	2	2	no	—
30	C₂×C₄×D₁₀₉	—	2	2	no	—
31	C₂×#11724784	—	2	4	no	—
32	C₂×#11724784	C₂³	1	4	yes	—
33	C₄⋊Dic₁₀₉	—	2	2	no	—
34	#13182006	—	2	4	no	—
35	C₂×Q₈×C₁₀₉	—	2	2	no	—
36	C₂²×C₄×C₁₀₉	D₄	1	2	yes	—
37	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
38	C₄⋊C₄×C₁₀₉	—	2	4	no	—
39	C₄⋊C₄×C₁₀₉	—	2	2	no	—
40	C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
41	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
42	C₄²×C₁₀₉	—	2	2	no	—
43	C₄²×C₁₀₉	D₄	1	2	yes	—
44	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
45	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
46	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
47	C₂²×D₁₀₉	—	2	2	no	—
48	Q₈×C₁₀₉	—	4	4	no	—
49	C₂×C₄×C₁₀₉	—	2	4	no	—
50	C₂×C₄×C₁₀₉	—	4	2	no	—
51	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₁₀₉	#41	1	4	yes	—
55	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
56	C₂²⋊D₁₀₉	—	4	4	no	—
57	C₂×Dic₁₀₉	—	2	2	no	—
58	D₄₃₆	—	2	4	no	—
59	D₄₃₆	—	4	4	no	—
60	C₄×D₁₀₉	—	4	4	no	—
61	Dic₂₁₈	—	2	4	no	—
62	#11724784	—	2	4	no	—
63	C₂²×C₁₀₉	—	2	2	no	—
64	C₂²×C₁₀₉	#124	1	2	yes	—
65	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
66	C₂×D₁₀₉	—	4	2	no	—
67	C₂×D₁₀₉	—	4	4	no	—
68	C₄×C₁₀₉	—	2	2	no	—
69	C₄×C₁₀₉	—	2	2	no	—
70	C₄×C₁₀₉	—	4	4	no	—
71	C₄×C₁₀₉	#124	1	2	yes	—
72	C₄×C₁₀₉	C₂×#41	1	2	yes	—
73	Dic₁₀₉	—	4	2	no	—
74	C₂×C₁₀₉	—	2	2	no	—
75	C₂×C₁₀₉	C₂×#124	1	1	yes	characteristic
76	C₂×C₁₀₉	#575	1	2	yes	—
77	D₁₀₉	—	8	2	no	—
78	#3135	—	109	1	no	maximal, Sylow
79	C₁₀₉	#3135	1	1	yes	characteristic, complemented, Sylow
80	#533	D₁₀₉	1	1	yes	characteristic, complemented
81	#532	—	109	1	no	—
82	#491	—	109	2	no	—
83	#490	—	109	2	no	—
84	#489	—	109	4	no	—
85	#416	—	109	4	no	—
86	C₂×#97	—	109	1	no	—
87	C₂×#41	—	109	2	no	—
88	C₂²×C₈	—	109	2	no	—
89	C₄⋊Q₈	—	109	1	no	—
90	C₄⋊Q₈	C₂×D₁₀₉	1	2	yes	—
91	#127	—	109	1	no	—
92	#125	C₂×D₁₀₉	1	2	yes	—
93	#124	—	109	2	no	—
94	C₂²⋊Q₈	—	2	2	no	—
95	#121	—	218	2	no	—
96	C₄×Q₈	—	2	2	no	—
97	C₄×D₄	—	218	2	no	—
98	#117	C₂×D₁₀₉	1	2	yes	—
99	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
100	#103	—	109	8	no	—
101	#102	—	109	8	no	—
102	#97	—	109	2	no	—
103	#97	—	109	2	no	—
104	#41	—	218	8	no	—
105	C₂×Q₈	—	2	2	no	—
106	C₂×Q₈	—	109	2	no	—
107	C₂×D₄	—	109	2	no	—
108	C₂×D₄	—	218	2	no	—
109	C₂×D₄	—	218	2	no	—
110	C₂²×C₄	—	218	2	no	—
111	C₂²×C₄	C₂²⋊D₁₀₉	1	2	yes	—
112	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
113	C₂×C₈	—	109	4	no	—
114	C₂×C₈	—	218	4	no	—
115	C₄⋊C₄	—	2	4	no	—
116	C₄⋊C₄	—	2	2	no	—
117	C₄⋊C₄	—	218	2	no	—
118	C₄⋊C₄	C₂²×D₁₀₉	1	4	yes	—
119	C₂²⋊C₄	—	2	2	no	—
120	C₂²⋊C₄	—	218	4	no	—
121	C₄²	—	2	2	no	—
122	C₄²	C₂²⋊D₁₀₉	1	2	yes	—
123	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
124	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
125	C₂³	—	218	2	no	—
126	C₂³	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
127	Q₈	—	4	4	no	—
128	Q₈	—	218	4	no	—
129	D₄	—	436	4	no	—
130	D₄	—	436	4	no	—
131	D₄	—	218	4	no	—
132	C₂×C₄	—	2	4	no	—
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	—	436	4	no	—
135	C₂×C₄	—	218	2	no	—
136	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	2	yes	—
137	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	2	yes	—
138	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
139	C₂×C₄	#13182030	1	4	yes	—
140	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
141	C₈	—	218	4	no	—
142	C₂²	—	2	2	no	—
143	C₂²	—	436	2	no	—
144	C₂²	—	436	4	no	—
145	C₂²	#16812688	1	2	yes	—
146	C₂²	C₂²×C₂²⋊D₁₀₉	1	1	yes	characteristic
147	C₄	—	2	2	no	—
148	C₄	—	2	2	no	—
149	C₄	—	4	4	no	—
150	C₄	—	436	2	no	—
151	C₄	#16812688	1	2	yes	—
152	C₄	C₂×#13182030	1	2	yes	—
153	C₂	—	2	2	no	—
154	C₂	—	872	2	no	—
155	C₂	C₂×#16812688	1	1	yes	characteristic
156	C₂	#20295824	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20295576

C₂

yes

maximal, characteristic

#20295396

C₂

yes

maximal, complemented

#20295392

C₂

yes

maximal, complemented

#20295388

C₂

yes

maximal, complemented

#20295020

C₂

yes

maximal, complemented

C₂×#16812610

C₂

yes

maximal, characteristic, complemented

#533×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂×#13182029

C₂²

yes

—

C₂²×#11724784

C₂²

yes

—

#16812699

C₂²

yes

characteristic

#16812696

C₂²

yes

characteristic

#16812687

C₂²

yes

—

#16812673

—

C₄×D₄₃₆

—

#16812622

C₂²

yes

—

#16812619

C₂²

yes

—

#16812610

C₂²

yes

—

#16812610

C₂²

yes

—

C₄⋊Q₈×C₁₀₉

C₂²

yes

—

#125×C₁₀₉

C₂²

yes

—

C₂²⋊Q₈×C₁₀₉

—

C₄×Q₈×C₁₀₉

—

#117×C₁₀₉

C₂²

yes

—

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

#13182029

—

C₂×Dic₂₁₈

C₂³

yes

—

C₂×D₄₃₆

—

C₂×D₄₃₆

C₂³

yes

—

C₂×C₂²⋊D₁₀₉

—

C₂×C₄×D₁₀₉

—

C₂×#11724784

—

C₂×#11724784

C₂³

yes

—

C₄⋊Dic₁₀₉

—

#13182006

—

C₂×Q₈×C₁₀₉

—

C₂²×C₄×C₁₀₉

D₄

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

C₂³

yes

—

C₂²⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

—

C₄²×C₁₀₉

D₄

yes

—

C₄²×C₁₀₉

C₂³

yes

characteristic

C₄²×C₁₀₉

C₂³

yes

characteristic

C₂³×C₁₀₉

C₂×D₄

yes

characteristic

C₂²×D₁₀₉

—

Q₈×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

#41

yes

—

C₂×C₄×C₁₀₉

C₂⁴

yes

characteristic

C₂²⋊D₁₀₉

—

C₂×Dic₁₀₉

—

D₄₃₆

—

D₄₃₆

—

C₄×D₁₀₉

—

Dic₂₁₈

—

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

#124

yes

—

C₂²×C₁₀₉

C₂²×D₄

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#124

yes

—

C₄×C₁₀₉

C₂×#41

yes

—

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

C₂×#124

yes

characteristic

C₂×C₁₀₉

#575

yes

—

D₁₀₉

—

#3135

—

109

maximal, Sylow

C₁₀₉

#3135

yes

characteristic, complemented, Sylow

#533

D₁₀₉

yes

characteristic, complemented

#532

—

109

—

#491

—

109

—

#490

—

109

—

#489

—

109

—

#416

—

109

—

C₂×#97

—

109

—

C₂×#41

—

109

—

C₂²×C₈

—

109

—

C₄⋊Q₈

—

109

—

C₄⋊Q₈

C₂×D₁₀₉

yes

—

#127

—

109

—

#125

C₂×D₁₀₉

yes

—

#124

—

109

—

C₂²⋊Q₈

—

#121

—

218

—

C₄×Q₈

—

C₄×D₄

—

218

—

#117

C₂×D₁₀₉

yes

—

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

100

#103

—

109

—

101

#102

—

109

—

102

#97

—

109

—

103

#97

—

109

—

104

#41

—

218

—

105

C₂×Q₈

—

106

C₂×Q₈

—

109

—

107

C₂×D₄

—

109

—

108

C₂×D₄

—

218

—

109

C₂×D₄

—

218

—

110

C₂²×C₄

—

218

—

111

C₂²×C₄

C₂²⋊D₁₀₉

yes

—

112

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

113

C₂×C₈

—

109

—

114

C₂×C₈

—

218

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

218

—

118

C₄⋊C₄

C₂²×D₁₀₉

yes

—

119

C₂²⋊C₄

—

120

C₂²⋊C₄

—

218

—

121

C₄²

—

122

C₄²

C₂²⋊D₁₀₉

yes

—

123

C₄²

C₂²×D₁₀₉

yes

characteristic

124

C₄²

C₂²×D₁₀₉

yes

characteristic

125

C₂³

—

218

—

126

C₂³

C₂×C₂²⋊D₁₀₉

yes

characteristic

127

Q₈

—

128

Q₈

—

218

—

129

D₄

—

436

—

130

D₄

—

436

—

131

D₄

—

218

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

436

—

135

C₂×C₄

—

218

—

136

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

—

137

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

—

138

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

139

C₂×C₄

#13182030

yes

—

140

C₂×C₄

C₂³×D₁₀₉

yes

characteristic

141

C₈

—

218

—

142

C₂²

—

143

C₂²

—

436

—

144

C₂²

—

436

—

145

C₂²

#16812688

yes

—

146

C₂²

C₂²×C₂²⋊D₁₀₉

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

436

—

151

C₄

#16812688

yes

—

152

C₄

C₂×#13182030

yes

—

153

C₂

—

154

C₂

—

872

—

155

C₂

C₂×#16812688

yes

characteristic

156

C₂

#20295824

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#533×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	#20295576	C₂
hypercenter	#533	D₁₀₉
hypocenter	C₁₀₉	#3135
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₁₀₉

commutator, cosocle

C₂×C₄×C₁₀₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₀₉

Fitting

#533×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂²×D₄

C₂

D₁₀₉

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
2. #20295396	C₂	semidirect
3. #20295392	C₂	semidirect
4. #20295388	C₂	semidirect
5. #20295020	C₂	semidirect
6. C₂×#16812610	C₂	semidirect
7. #533×C₁₀₉	C₂	semidirect
79. C₁₀₉	#3135	semidirect
80. #533	D₁₀₉	semidirect

H ≅ G/N

type

2. #20295396

C₂

semidirect

3. #20295392

C₂

semidirect

4. #20295388

C₂

semidirect

5. #20295020

C₂

semidirect

6. C₂×#16812610

C₂

semidirect

7. #533×C₁₀₉

C₂

semidirect

79. C₁₀₉

#3135

semidirect

80. #533

D₁₀₉

semidirect

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

subgroup	index
#20295576	2
#20295396	2
#20295392	2
#20295388	2
#20295020	2
C₂×#16812610	2
#533×C₁₀₉	2
#3135	109

subgroup

index

#533×C₁₀₉

109

Subgroups — #28373380

Conjugacy classes of subgroups

Complements

Maximal subgroups