Group #28375699 — subgroups

Subgroups — #28375699

Order 13952.

392 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 139 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	Q₈×Dic₂₁₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#20295372	C₂	1	3	yes	maximal, complemented
3	#20295316	C₂	1	3	yes	maximal, complemented
4	#20295223	C₂	1	3	yes	maximal, complemented
5	Q₈×#11724784	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#13182017	C₂	1	3	yes	maximal, complemented
7	#550×C₁₀₉	C₂	1	1	yes	maximal, characteristic
8	C₂×#13182017	C₂²	1	3	yes	—
9	#16812699	C₂²	1	3	yes	—
10	#16812698	—	2	3	no	—
11	Q₈×Dic₁₀₉	—	2	1	no	—
12	C₄×Dic₂₁₈	C₂²	1	3	yes	—
13	#16812630	C₂²	1	3	yes	—
14	C₄⋊#11724784	C₂²	1	3	yes	—
15	#16812622	C₂²	1	1	yes	characteristic
16	#16812620	C₂²	1	6	yes	—
17	#16812620	C₂²	1	3	yes	—
18	C₄×#11724784	C₂²	1	3	yes	—
19	C₂×#41×C₁₀₉	—	2	1	no	—
20	#127×C₁₀₉	C₂²	1	3	yes	—
21	#121×C₁₀₉	—	2	3	no	—
22	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
23	C₄×D₄×C₁₀₉	C₂²	1	3	yes	—
24	C₂×Dic₂₁₈	—	2	3	no	—
25	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
26	#13182017	—	2	6	no	—
27	C₂×#11724784	C₂³	1	1	yes	characteristic
28	C₂×#11724784	C₂³	1	3	yes	—
29	C₄⋊Dic₁₀₉	—	2	3	no	—
30	C₄⋊Dic₁₀₉	C₂³	1	3	yes	—
31	#13182008	—	2	3	no	—
32	C₄×Dic₁₀₉	—	2	3	no	—
33	#41×C₁₀₉	—	4	4	no	—
34	C₂×Q₈×C₁₀₉	D₄	1	1	yes	characteristic
35	C₂×D₄×C₁₀₉	—	2	3	no	—
36	C₂×D₄×C₁₀₉	—	2	3	no	—
37	C₂×D₄×C₁₀₉	C₂³	1	3	yes	—
38	C₂²×C₄×C₁₀₉	—	2	3	no	—
39	C₄⋊C₄×C₁₀₉	D₄	1	3	yes	—
40	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
41	C₂²⋊C₄×C₁₀₉	—	2	3	no	—
42	C₄²×C₁₀₉	C₂³	1	3	yes	—
43	C₂³×C₁₀₉	—	2	3	no	—
44	Q₈×C₁₀₉	SD₁₆	1	4	yes	complemented
45	D₄×C₁₀₉	—	2	3	no	—
46	D₄×C₁₀₉	—	2	3	no	—
47	D₄×C₁₀₉	—	4	6	no	—
48	D₄×C₁₀₉	—	8	3	no	—
49	C₂×C₄×C₁₀₉	—	2	1	no	—
50	C₂×C₄×C₁₀₉	—	4	6	no	—
51	C₂×C₄×C₁₀₉	C₂×D₄	1	3	yes	—
52	C₂×C₄×C₁₀₉	C₂×D₄	1	3	yes	—
53	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
54	C₂×Dic₁₀₉	—	2	1	no	—
55	C₂×Dic₁₀₉	—	2	3	no	—
56	Dic₂₁₈	—	2	1	no	—
57	Dic₂₁₈	—	8	3	no	—
58	Dic₂₁₈	#41	1	2	yes	complemented
59	#11724784	—	2	3	no	—
60	#11724784	—	2	1	no	—
61	C₂²×C₁₀₉	—	4	3	no	—
62	C₂²×C₁₀₉	—	4	3	no	—
63	C₂²×C₁₀₉	—	4	3	no	—
64	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
65	C₄×C₁₀₉	—	2	3	no	—
66	C₄×C₁₀₉	—	4	1	no	—
67	C₄×C₁₀₉	C₂×SD₁₆	1	6	yes	—
68	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
69	C₄×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
70	Dic₁₀₉	—	2	2	no	—
71	Dic₁₀₉	—	4	3	no	—
72	C₂×C₁₀₉	—	8	3	no	—
73	C₂×C₁₀₉	#548	1	1	yes	characteristic
74	C₂×C₁₀₉	C₂²×SD₁₆	1	1	yes	characteristic
75	C₂×C₁₀₉	#576	1	1	yes	characteristic
76	#3362	—	109	1	no	maximal, Sylow
77	C₁₀₉	#3362	1	1	yes	characteristic, complemented, Sylow
78	Q₈²	—	109	1	no	—
79	#550	D₁₀₉	1	1	yes	characteristic
80	#486	—	109	3	no	—
81	#476	—	109	3	no	—
82	#460	—	109	3	no	—
83	C₈×Q₈	—	109	1	no	—
84	C₄×SD₁₆	—	109	3	no	—
85	C₂×#41	—	2	1	no	—
86	C₂×SD₁₆	—	109	3	no	—
87	C₄⋊Q₈	—	109	3	no	—
88	C₄⋊Q₈	—	218	3	no	—
89	#127	C₂×D₁₀₉	1	3	yes	—
90	#121	—	2	3	no	—
91	C₄×Q₈	—	109	3	no	—
92	C₄×Q₈	—	218	1	no	—
93	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
94	C₄×D₄	C₂×D₁₀₉	1	3	yes	—
95	#106	—	109	3	no	—
96	C₄⋊C₈	—	109	3	no	—
97	#103	—	109	1	no	—
98	#102	—	109	3	no	—
99	#102	—	109	6	no	—
100	C₄×C₈	—	109	3	no	—
101	#41	—	4	4	no	—
102	C₂×Q₈	—	109	1	no	—
103	C₂×Q₈	—	218	3	no	—
104	C₂×Q₈	C₂²⋊D₁₀₉	1	1	yes	characteristic
105	C₂×D₄	—	2	3	no	—
106	C₂×D₄	—	2	3	no	—
107	C₂×D₄	C₂²×D₁₀₉	1	3	yes	—
108	C₂²×C₄	—	2	3	no	—
109	SD₁₆	—	218	6	no	—
110	C₂×C₈	—	109	1	no	—
111	C₂×C₈	—	109	3	no	—
112	C₄⋊C₄	—	109	3	no	—
113	C₄⋊C₄	—	218	3	no	—
114	C₄⋊C₄	—	218	3	no	—
115	C₄⋊C₄	C₂²⋊D₁₀₉	1	3	yes	—
116	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
117	C₂²⋊C₄	—	2	3	no	—
118	C₄²	—	218	3	no	—
119	C₄²	C₂²×D₁₀₉	1	3	yes	—
120	C₂³	—	2	3	no	—
121	Q₈	—	872	3	no	—
122	Q₈	—	109	2	no	—
123	Q₈	—	218	1	no	—
124	Q₈	#13182017	1	4	yes	complemented
125	D₄	—	2	3	no	—
126	D₄	—	2	3	no	—
127	D₄	—	4	6	no	—
128	D₄	—	8	3	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	4	6	no	—
131	C₂×C₄	—	218	3	no	—
132	C₂×C₄	—	218	1	no	—
133	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	3	yes	—
134	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	3	yes	—
135	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
136	C₈	—	218	3	no	—
137	C₈	—	218	1	no	—
138	C₂²	—	4	3	no	—
139	C₂²	—	4	3	no	—
140	C₂²	—	4	3	no	—
141	C₂²	C₂²×C₂²⋊D₁₀₉	1	1	yes	characteristic
142	C₄	—	2	3	no	—
143	C₄	—	4	1	no	—
144	C₄	—	436	3	no	—
145	C₄	—	218	2	no	—
146	C₄	C₂×#13182017	1	6	yes	—
147	C₄	#41×D₁₀₉	1	1	yes	characteristic
148	C₄	#16812746	1	1	yes	characteristic
149	C₂	—	8	3	no	—
150	C₂	#20295683	1	1	yes	characteristic
151	C₂	C₂²×#13182017	1	1	yes	characteristic
152	C₂	#20295829	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×Dic₂₁₈

C₂

yes

maximal, characteristic, complemented

#20295372

C₂

yes

maximal, complemented

#20295316

C₂

yes

maximal, complemented

#20295223

C₂

yes

maximal, complemented

Q₈×#11724784

C₂

yes

maximal, characteristic, complemented

C₄×#13182017

C₂

yes

maximal, complemented

#550×C₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182017

C₂²

yes

—

#16812699

C₂²

yes

—

#16812698

—

Q₈×Dic₁₀₉

—

C₄×Dic₂₁₈

C₂²

yes

—

#16812630

C₂²

yes

—

C₄⋊#11724784

C₂²

yes

—

#16812622

C₂²

yes

characteristic

#16812620

C₂²

yes

—

#16812620

C₂²

yes

—

C₄×#11724784

C₂²

yes

—

C₂×#41×C₁₀₉

—

#127×C₁₀₉

C₂²

yes

—

#121×C₁₀₉

—

C₄×Q₈×C₁₀₉

C₂²

yes

characteristic

C₄×D₄×C₁₀₉

C₂²

yes

—

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

C₂³

yes

characteristic

#13182017

—

C₂×#11724784

C₂³

yes

characteristic

C₂×#11724784

C₂³

yes

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

C₂³

yes

—

#13182008

—

C₄×Dic₁₀₉

—

#41×C₁₀₉

—

C₂×Q₈×C₁₀₉

D₄

yes

characteristic

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

C₂³

yes

—

C₂²×C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

D₄

yes

—

C₄⋊C₄×C₁₀₉

C₂³

yes

characteristic

C₂²⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

C₂³

yes

—

C₂³×C₁₀₉

—

Q₈×C₁₀₉

SD₁₆

yes

complemented

D₄×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂⁴

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

Dic₂₁₈

—

Dic₂₁₈

—

Dic₂₁₈

yes

—

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂²×D₄

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×SD₁₆

yes

—

C₄×C₁₀₉

C₂×#41

yes

characteristic

C₄×C₁₀₉

2₋¹⁺⁴

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

#548

yes

characteristic

C₂×C₁₀₉

C₂²×SD₁₆

yes

C₂×C₁₀₉

yes

—

109

maximal, Sylow

C₁₀₉

#3362

yes

characteristic, complemented, Sylow

Q₈²

—

109

—

#550

D₁₀₉

yes

characteristic

#486

—

109

—

#476

—

109

—

#460

—

109

—

C₈×Q₈

—

109

—

C₄×SD₁₆

—

109

—

C₂×#41

—

C₂×SD₁₆

—

109

—

C₄⋊Q₈

—

109

—

C₄⋊Q₈

—

218

—

#127

C₂×D₁₀₉

yes

—

#121

—

C₄×Q₈

—

109

—

C₄×Q₈

—

218

—

C₄×Q₈

C₂×D₁₀₉

yes

characteristic

C₄×D₄

C₂×D₁₀₉

yes

—

#106

—

109

—

C₄⋊C₈

—

109

—

#103

—

109

—

#102

—

109

—

#102

—

109

—

100

C₄×C₈

—

109

—

101

#41

—

102

C₂×Q₈

—

109

—

103

C₂×Q₈

—

218

—

104

C₂×Q₈

C₂²⋊D₁₀₉

yes

characteristic

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂×D₄

C₂²×D₁₀₉

yes

—

108

C₂²×C₄

—

109

SD₁₆

—

218

—

110

C₂×C₈

—

109

—

111

C₂×C₈

—

109

—

112

C₄⋊C₄

—

109

—

113

C₄⋊C₄

—

218

—

114

C₄⋊C₄

—

218

—

115

C₄⋊C₄

C₂²⋊D₁₀₉

yes

—

116

C₄⋊C₄

C₂²×D₁₀₉

yes

characteristic

117

C₂²⋊C₄

—

118

C₄²

—

218

—

119

C₄²

C₂²×D₁₀₉

yes

—

120

C₂³

—

121

Q₈

—

872

—

122

Q₈

—

109

—

123

Q₈

—

218

—

124

Q₈

#13182017

yes

complemented

125

D₄

—

126

D₄

—

127

D₄

—

128

D₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

218

—

132

C₂×C₄

—

218

—

133

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

—

134

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

—

135

C₂×C₄

C₂³×D₁₀₉

yes

characteristic

136

C₈

—

218

—

137

C₈

—

218

—

138

C₂²

—

139

C₂²

—

140

C₂²

—

141

C₂²

C₂²×C₂²⋊D₁₀₉

yes

characteristic

142

C₄

—

143

C₄

—

144

C₄

—

436

—

145

C₄

—

218

—

146

C₄

C₂×#13182017

yes

—

147

C₄

#41×D₁₀₉

yes

characteristic

148

C₄

#16812746

yes

characteristic

149

C₂

—

150

C₂

#20295683

yes

characteristic

151

C₂

C₂²×#13182017

yes

characteristic

152

C₂

#20295829

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting, prime radical	#550×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
hypercenter	#550	D₁₀₉
hypocenter	C₁₀₉	#3362
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #550

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₁₀₉

commutator, cosocle

C₂×C₄×C₁₀₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₀₉

Fitting, prime radical

#550×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂²×D₄

D₁₀₉

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #550

N	H ≅ G/N	type
1. Q₈×Dic₂₁₈	C₂	semidirect
2. #20295372	C₂	semidirect
3. #20295316	C₂	semidirect
4. #20295223	C₂	semidirect
5. Q₈×#11724784	C₂	semidirect
6. C₄×#13182017	C₂	semidirect
44. Q₈×C₁₀₉	SD₁₆	semidirect
58. Dic₂₁₈	#41	semidirect
77. C₁₀₉	#3362	semidirect
124. Q₈	#13182017	semidirect

H ≅ G/N

type

1. Q₈×Dic₂₁₈

C₂

semidirect

2. #20295372

C₂

semidirect

3. #20295316

C₂

semidirect

4. #20295223

C₂

semidirect

5. Q₈×#11724784

C₂

semidirect

6. C₄×#13182017

C₂

semidirect

44. Q₈×C₁₀₉

SD₁₆

semidirect

58. Dic₂₁₈

#41

semidirect

77. C₁₀₉

#3362

semidirect

124. Q₈

#13182017

semidirect

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

subgroup	index
Q₈×Dic₂₁₈	2
#20295372	2
#20295316	2
#20295223	2
Q₈×#11724784	2
C₄×#13182017	2
#550×C₁₀₉	2
#3362	109

subgroup

index

Q₈×Dic₂₁₈

#550×C₁₀₉

109

Subgroups — #28375699

Conjugacy classes of subgroups

Complements

Maximal subgroups