Group #20295718 — subgroups

Subgroups — #20295718

Order 6976.

214 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 99 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	#16812694	C₂	1	2	yes	maximal, complemented
2	#16812690	C₂	1	1	yes	maximal, characteristic, complemented
3	#16812690	C₂	1	2	yes	maximal, complemented
4	#16812683	C₂	1	2	yes	maximal
5	#16812675	C₂	1	1	yes	maximal, characteristic, complemented
6	Q₈×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×C₂²⋊D₁₀₉	C₂	1	1	yes	maximal, characteristic
8	#16812657	C₂	1	1	yes	maximal, characteristic, complemented
9	#16812655	C₂	1	2	yes	maximal, complemented
10	C₂×C₄⋊Dic₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²⋊Q₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₂²⋊D₁₀₉	C₂²	1	1	yes	characteristic
13	C₂²×Dic₁₀₉	C₂²	1	2	yes	—
14	C₂×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
15	C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
16	C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
17	C₄⋊Dic₁₀₉	C₂²	1	2	yes	—
18	C₄⋊Dic₁₀₉	C₂²	1	2	yes	—
19	C₄⋊Dic₁₀₉	C₂²	1	2	yes	—
20	#13182008	C₂²	1	2	yes	—
21	#13182008	C₂²	1	2	yes	—
22	#13182008	C₂²	1	1	yes	characteristic
23	C₂²⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
24	#13182006	C₂²	1	2	yes	—
25	#13182006	C₂²	1	1	yes	characteristic
26	#13182006	C₂²	1	2	yes	—
27	C₄×Dic₁₀₉	C₂²	1	2	yes	—
28	C₄×Dic₁₀₉	C₂²	1	2	yes	—
29	C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
30	C₂×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
31	C₂²×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
32	C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
33	C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
34	C₂²⋊C₄×C₁₀₉	C₂²	1	2	yes	—
35	C₂³×C₁₀₉	C₂³	1	1	yes	characteristic
36	C₂²×D₁₀₉	C₂³	1	1	yes	characteristic
37	Q₈×C₁₀₉	—	2	2	no	—
38	C₂×C₄×C₁₀₉	—	2	2	no	—
39	C₂×C₄×C₁₀₉	C₂³	1	2	yes	—
40	C₂×C₄×C₁₀₉	C₂³	1	2	yes	—
41	C₂×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
42	C₂×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
43	C₂²⋊D₁₀₉	—	2	2	no	—
44	C₂×Dic₁₀₉	—	2	4	no	—
45	C₂×Dic₁₀₉	C₂³	1	1	yes	characteristic
46	C₂×Dic₁₀₉	C₂³	1	2	yes	—
47	C₂×Dic₁₀₉	C₂³	1	1	yes	characteristic
48	C₂×Dic₁₀₉	C₂³	1	1	yes	characteristic
49	C₂×Dic₁₀₉	C₂³	1	2	yes	—
50	C₄×D₁₀₉	—	2	2	no	—
51	C₂²×C₁₀₉	—	2	1	no	—
52	C₂²×C₁₀₉	—	2	1	no	—
53	C₂²×C₁₀₉	#41	1	2	yes	—
54	C₂²×C₁₀₉	C₂⁴	1	1	yes	characteristic
55	C₂×D₁₀₉	—	2	1	no	—
56	C₂×D₁₀₉	—	2	1	no	—
57	C₂×D₁₀₉	—	2	1	no	—
58	C₄×C₁₀₉	—	2	1	no	—
59	C₄×C₁₀₉	—	2	2	no	—
60	C₄×C₁₀₉	—	2	2	no	—
61	C₄×C₁₀₉	#41	1	2	yes	—
62	Dic₁₀₉	—	2	1	no	—
63	Dic₁₀₉	—	2	2	no	—
64	Dic₁₀₉	—	2	2	no	—
65	Dic₁₀₉	—	2	1	no	—
66	Dic₁₀₉	—	2	1	no	—
67	C₂×C₁₀₉	—	2	2	no	—
68	C₂×C₁₀₉	C₂×#41	1	1	yes	characteristic
69	C₂×C₁₀₉	C₂×#41	1	1	yes	characteristic
70	C₂×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
71	D₁₀₉	—	4	1	no	—
72	C₁₀₉	#552	1	1	yes	characteristic, complemented, Sylow
73	#552	—	109	1	no	maximal, Sylow
74	#126	—	109	2	no	—
75	#125	—	109	2	no	—
76	#125	—	109	1	no	—
77	#123	—	109	2	no	—
78	C₂²⋊Q₈	—	109	1	no	—
79	C₂²⋊Q₈	D₁₀₉	1	1	yes	characteristic, complemented
80	C₄×Q₈	—	109	1	no	—
81	C₄×D₄	—	109	1	no	—
82	#117	—	109	1	no	—
83	#117	—	109	2	no	—
84	C₂×C₄⋊C₄	—	109	1	no	—
85	C₂×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
86	C₂×D₄	—	109	1	no	—
87	C₂²×C₄	—	109	2	no	—
88	C₂²×C₄	—	109	1	no	—
89	C₂²×C₄	C₂×D₁₀₉	1	1	yes	characteristic
90	C₄⋊C₄	—	109	2	no	—
91	C₄⋊C₄	—	109	2	no	—
92	C₄⋊C₄	—	109	1	no	—
93	C₄⋊C₄	—	109	1	no	—
94	C₄⋊C₄	—	109	1	no	—
95	C₄⋊C₄	—	109	2	no	—
96	C₄⋊C₄	—	109	2	no	—
97	C₄⋊C₄	—	109	2	no	—
98	C₄⋊C₄	C₂×D₁₀₉	1	1	yes	characteristic
99	C₄⋊C₄	C₂×D₁₀₉	1	2	yes	—
100	C₂²⋊C₄	—	109	1	no	—
101	C₂²⋊C₄	—	109	2	no	—
102	C₂²⋊C₄	—	109	1	no	—
103	C₂²⋊C₄	—	109	2	no	—
104	C₂²⋊C₄	C₂×D₁₀₉	1	2	yes	—
105	C₄²	—	109	2	no	—
106	C₄²	—	109	2	no	—
107	C₄²	—	109	1	no	—
108	C₂³	—	109	1	no	—
109	C₂³	C₂²×D₁₀₉	1	1	yes	characteristic
110	Q₈	—	2	2	no	—
111	D₄	—	218	2	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	109	1	no	—
114	C₂×C₄	—	109	1	no	—
115	C₂×C₄	—	109	2	no	—
116	C₂×C₄	—	109	1	no	—
117	C₂×C₄	—	109	2	no	—
118	C₂×C₄	—	218	4	no	—
119	C₂×C₄	—	218	2	no	—
120	C₂×C₄	C₂²×D₁₀₉	1	2	yes	—
121	C₂×C₄	C₂²×D₁₀₉	1	2	yes	—
122	C₂×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
123	C₂×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
124	C₂²	—	2	1	no	—
125	C₂²	—	2	1	no	—
126	C₂²	—	218	1	no	—
127	C₂²	—	218	1	no	—
128	C₂²	—	218	1	no	—
129	C₂²	#13182030	1	2	yes	—
130	C₂²	C₂³×D₁₀₉	1	1	yes	characteristic
131	C₄	—	2	1	no	—
132	C₄	—	2	2	no	—
133	C₄	—	2	2	no	—
134	C₄	—	218	1	no	—
135	C₄	—	218	2	no	—
136	C₄	—	218	2	no	—
137	C₄	—	218	1	no	—
138	C₄	—	218	1	no	—
139	C₄	#13182028	1	2	yes	—
140	C₂	—	2	2	no	—
141	C₂	—	436	1	no	—
142	C₂	C₂×#13182028	1	1	yes	characteristic
143	C₂	C₂×#13182030	1	1	yes	characteristic
144	C₂	#16812746	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16812694

C₂

yes

maximal, complemented

#16812690

C₂

yes

maximal, characteristic, complemented

#16812690

C₂

yes

maximal, complemented

#16812683

C₂

yes

maximal

#16812675

C₂

yes

maximal, characteristic, complemented

Q₈×Dic₁₀₉

C₂

yes

maximal, characteristic, complemented

C₄×C₂²⋊D₁₀₉

C₂

yes

maximal, characteristic

#16812657

C₂

yes

maximal, characteristic, complemented

#16812655

C₂

yes

maximal, complemented

C₂×C₄⋊Dic₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂²⋊Q₈×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂×C₂²⋊D₁₀₉

C₂²

yes

characteristic

C₂²×Dic₁₀₉

C₂²

yes

—

C₂×C₄×D₁₀₉

C₂²

yes

characteristic

C₄⋊Dic₁₀₉

C₂²

yes

characteristic

C₄⋊Dic₁₀₉

C₂²

yes

characteristic

C₄⋊Dic₁₀₉

C₂²

yes

—

C₄⋊Dic₁₀₉

C₂²

yes

—

C₄⋊Dic₁₀₉

C₂²

yes

—

#13182008

C₂²

yes

—

#13182008

C₂²

yes

—

#13182008

C₂²

yes

characteristic

C₂²⋊Dic₁₀₉

C₂²

yes

characteristic

#13182006

C₂²

yes

—

#13182006

C₂²

yes

characteristic

#13182006

C₂²

yes

—

C₄×Dic₁₀₉

C₂²

yes

—

C₄×Dic₁₀₉

C₂²

yes

—

C₄×Dic₁₀₉

C₂²

yes

characteristic

C₂×Q₈×C₁₀₉

C₂²

yes

characteristic

C₂²×C₄×C₁₀₉

C₂²

yes

characteristic

C₄⋊C₄×C₁₀₉

C₂²

yes

characteristic

C₄⋊C₄×C₁₀₉

C₂²

yes

—

C₂²⋊C₄×C₁₀₉

C₂²

yes

—

C₂³×C₁₀₉

C₂³

yes

characteristic

C₂²×D₁₀₉

C₂³

yes

characteristic

Q₈×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂³

yes

—

C₂×C₄×C₁₀₉

C₂³

yes

—

C₂×C₄×C₁₀₉

C₂³

yes

characteristic

C₂×C₄×C₁₀₉

C₂³

yes

characteristic

C₂²⋊D₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₂³

yes

characteristic

C₂×Dic₁₀₉

C₂³

yes

—

C₂×Dic₁₀₉

C₂³

yes

characteristic

C₂×Dic₁₀₉

C₂³

yes

characteristic

C₂×Dic₁₀₉

C₂³

yes

—

C₄×D₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

#41

yes

—

C₂²×C₁₀₉

C₂⁴

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#41

yes

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

C₂×#41

yes

characteristic

C₂×C₁₀₉

C₂×#41

yes

characteristic

C₂×C₁₀₉

2₋¹⁺⁴

yes

characteristic

D₁₀₉

—

C₁₀₉

#552

yes

characteristic, complemented, Sylow

#552

—

109

maximal, Sylow

#126

—

109

—

#125

—

109

—

#125

—

109

—

#123

—

109

—

C₂²⋊Q₈

—

109

—

C₂²⋊Q₈

D₁₀₉

yes

characteristic, complemented

C₄×Q₈

—

109

—

C₄×D₄

—

109

—

#117

—

109

—

#117

—

109

—

C₂×C₄⋊C₄

—

109

—

C₂×Q₈

C₂×D₁₀₉

yes

characteristic

C₂×D₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

C₂×D₁₀₉

yes

characteristic

C₄⋊C₄

—

109

—

C₄⋊C₄

—

109

—

C₄⋊C₄

—

109

—

C₄⋊C₄

—

109

—

C₄⋊C₄

—

109

—

C₄⋊C₄

—

109

—

C₄⋊C₄

—

109

—

C₄⋊C₄

—

109

—

C₄⋊C₄

C₂×D₁₀₉

yes

characteristic

C₄⋊C₄

C₂×D₁₀₉

yes

—

100

C₂²⋊C₄

—

109

—

101

C₂²⋊C₄

—

109

—

102

C₂²⋊C₄

—

109

—

103

C₂²⋊C₄

—

109

—

104

C₂²⋊C₄

C₂×D₁₀₉

yes

—

105

C₄²

—

109

—

106

C₄²

—

109

—

107

C₄²

—

109

—

108

C₂³

—

109

—

109

C₂³

C₂²×D₁₀₉

yes

characteristic

110

Q₈

—

111

D₄

—

218

—

112

C₂×C₄

—

113

C₂×C₄

—

109

—

114

C₂×C₄

—

109

—

115

C₂×C₄

—

109

—

116

C₂×C₄

—

109

—

117

C₂×C₄

—

109

—

118

C₂×C₄

—

218

—

119

C₂×C₄

—

218

—

120

C₂×C₄

C₂²×D₁₀₉

yes

—

121

C₂×C₄

C₂²×D₁₀₉

yes

—

122

C₂×C₄

C₂²×D₁₀₉

yes

characteristic

123

C₂×C₄

C₂²×D₁₀₉

yes

characteristic

124

C₂²

—

125

C₂²

—

126

C₂²

—

218

—

127

C₂²

—

218

—

128

C₂²

—

218

—

129

C₂²

#13182030

yes

—

130

C₂²

C₂³×D₁₀₉

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

218

—

135

C₄

—

218

—

136

C₄

—

218

—

137

C₄

—

218

—

138

C₄

—

218

—

139

C₄

#13182028

yes

—

140

C₂

—

141

C₂

—

436

—

142

C₂

C₂×#13182028

yes

characteristic

143

C₂

C₂×#13182030

yes

characteristic

144

C₂

#16812746

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₀₉
commutator, socle	C₂²×C₁₀₉	C₂⁴
Fitting	C₂²⋊Q₈×C₁₀₉	C₂
hypercenter	C₂²⋊Q₈	D₁₀₉
hypocenter	C₁₀₉	#552
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂²⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₀₉

commutator, socle

C₂²×C₁₀₉

C₂⁴

Fitting

C₂²⋊Q₈×C₁₀₉

C₂

hypercenter

C₂²⋊Q₈

D₁₀₉

C₁₀₉

G ▷ C₂²×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂² ◁ C₂²⋊Q₈

N	H ≅ G/N	type
1. #16812694	C₂	semidirect
2. #16812690	C₂	semidirect
3. #16812690	C₂	semidirect
5. #16812675	C₂	semidirect
6. Q₈×Dic₁₀₉	C₂	semidirect
8. #16812657	C₂	semidirect
9. #16812655	C₂	semidirect
10. C₂×C₄⋊Dic₁₀₉	C₂	semidirect
11. C₂²⋊Q₈×C₁₀₉	C₂	semidirect
72. C₁₀₉	#552	semidirect
79. C₂²⋊Q₈	D₁₀₉	semidirect

H ≅ G/N

type

1. #16812694

C₂

semidirect

2. #16812690

C₂

semidirect

3. #16812690

C₂

semidirect

5. #16812675

C₂

semidirect

6. Q₈×Dic₁₀₉

C₂

semidirect

8. #16812657

C₂

semidirect

9. #16812655

C₂

semidirect

10. C₂×C₄⋊Dic₁₀₉

C₂

semidirect

11. C₂²⋊Q₈×C₁₀₉

C₂

semidirect

72. C₁₀₉

#552

semidirect

79. C₂²⋊Q₈

D₁₀₉

semidirect

Subgroups — #20295718

Conjugacy classes of subgroups

Complements