Group #23776031 — subgroups

Subgroups — #23776031

Order 10112.

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

normal subgroups only

This group has 426 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×D₃₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	#19098909	C₂	1	3	yes	maximal, complemented
3	#19098843	C₂	1	3	yes	maximal, complemented
4	#19098755	C₂	1	3	yes	maximal, complemented
5	Q₈×C₇₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#11929609	C₂	1	3	yes	maximal, complemented
7	#550×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Q₈×D₇₉	—	2	1	no	—
9	C₂×#11929609	C₂²	1	3	yes	—
10	#14579161	C₂²	1	3	yes	—
11	#14579141	—	2	3	no	—
12	C₄×D₃₁₆	C₂²	1	3	yes	—
13	#14579094	C₂²	1	3	yes	—
14	C₄⋊(C₇₉⋊C₈)	C₂²	1	3	yes	—
15	#14579082	C₂²	1	3	yes	—
16	#14579082	C₂²	1	6	yes	—
17	#14579081	C₂²	1	1	yes	characteristic
18	C₄×C₇₉⋊C₈	C₂²	1	3	yes	—
19	C₂×#41×C₇₉	—	2	1	no	—
20	#127×C₇₉	C₂²	1	3	yes	—
21	#121×C₇₉	—	2	3	no	—
22	C₄×Q₈×C₇₉	C₂²	1	1	yes	characteristic
23	C₄×D₄×C₇₉	C₂²	1	3	yes	—
24	C₂×Dic₁₅₈	—	2	3	no	—
25	Q₈×D₇₉	—	4	4	no	—
26	C₂×D₃₁₆	C₂³	1	1	yes	characteristic
27	C₂×C₄×D₇₉	—	2	3	no	—
28	#11929609	—	2	6	no	—
29	C₂×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
30	C₂×C₇₉⋊C₈	C₂³	1	3	yes	—
31	C₄⋊Dic₇₉	—	2	3	no	—
32	C₄⋊Dic₇₉	C₂³	1	3	yes	—
33	(C₂×D₇₉)⋊C₄	—	2	3	no	—
34	#41×C₇₉	—	4	4	no	—
35	C₂×Q₈×C₇₉	D₄	1	1	yes	characteristic
36	C₂×D₄×C₇₉	—	2	3	no	—
37	C₂×D₄×C₇₉	—	2	3	no	—
38	C₂×D₄×C₇₉	C₂³	1	3	yes	—
39	C₂²×C₄×C₇₉	—	2	3	no	—
40	C₄⋊C₄×C₇₉	D₄	1	3	yes	—
41	C₄⋊C₄×C₇₉	C₂³	1	1	yes	characteristic
42	C₂²⋊C₄×C₇₉	—	2	3	no	—
43	C₄²×C₇₉	C₂³	1	3	yes	—
44	C₂³×C₇₉	—	2	3	no	—
45	C₂²×D₇₉	—	2	1	no	—
46	Q₈×C₇₉	D₈	1	4	yes	complemented
47	D₄×C₇₉	—	2	3	no	—
48	D₄×C₇₉	—	2	3	no	—
49	D₄×C₇₉	—	4	6	no	—
50	D₄×C₇₉	—	8	3	no	—
51	C₂×C₄×C₇₉	—	2	1	no	—
52	C₂×C₄×C₇₉	—	4	6	no	—
53	C₂×C₄×C₇₉	C₂×D₄	1	3	yes	—
54	C₂×C₄×C₇₉	C₂×D₄	1	3	yes	—
55	C₂×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
56	C₂×Dic₇₉	—	2	3	no	—
57	D₃₁₆	—	2	1	no	—
58	D₃₁₆	#41	1	2	yes	complemented
59	C₄×D₇₉	—	4	6	no	—
60	Dic₁₅₈	—	4	6	no	—
61	C₇₉⋊C₈	—	2	3	no	—
62	C₇₉⋊C₈	—	2	1	no	—
63	C₂²×C₇₉	—	4	3	no	—
64	C₂²×C₇₉	—	4	3	no	—
65	C₂²×C₇₉	—	4	3	no	—
66	C₂²×C₇₉	C₂²×D₄	1	1	yes	characteristic
67	C₂×D₇₉	—	2	2	no	—
68	C₂×D₇₉	—	4	1	no	—
69	C₂×D₇₉	—	4	1	no	—
70	C₄×C₇₉	—	2	3	no	—
71	C₄×C₇₉	—	4	1	no	—
72	C₄×C₇₉	C₂×D₈	1	6	yes	—
73	C₄×C₇₉	C₂×#41	1	1	yes	characteristic
74	C₄×C₇₉	2₋¹⁺⁴	1	1	yes	characteristic
75	Dic₇₉	—	4	3	no	—
76	C₂×C₇₉	—	8	3	no	—
77	C₂×C₇₉	#548	1	1	yes	characteristic
78	C₂×C₇₉	C₂²×D₈	1	1	yes	characteristic
79	C₂×C₇₉	#577	1	1	yes	characteristic
80	D₇₉	—	4	2	no	—
81	#3358	—	79	1	no	maximal, Sylow
82	C₇₉	#3358	1	1	yes	characteristic, complemented, Sylow
83	#550	D₇₉	1	1	yes	characteristic, complemented
84	D₄×Q₈	—	79	1	no	—
85	#486	—	79	3	no	—
86	#474	—	79	3	no	—
87	#459	—	79	3	no	—
88	C₈×Q₈	—	79	1	no	—
89	C₄×D₈	—	79	3	no	—
90	C₂×#41	—	2	1	no	—
91	C₂²×Q₈	—	158	1	no	—
92	C₂×D₈	—	79	3	no	—
93	C₄⋊Q₈	—	79	3	no	—
94	#127	C₂×D₇₉	1	3	yes	—
95	C₂²⋊Q₈	—	158	3	no	—
96	#121	—	2	3	no	—
97	C₄×Q₈	C₂×D₇₉	1	1	yes	characteristic
98	C₄×D₄	—	79	3	no	—
99	C₄×D₄	C₂×D₇₉	1	3	yes	—
100	#107	—	79	3	no	—
101	C₄⋊C₈	—	79	3	no	—
102	#102	—	79	3	no	—
103	#102	—	79	6	no	—
104	#102	—	79	1	no	—
105	C₄×C₈	—	79	3	no	—
106	#41	—	4	4	no	—
107	C₂×Q₈	—	316	4	no	—
108	C₂×Q₈	—	158	3	no	—
109	C₂×Q₈	C₂²⋊D₇₉	1	1	yes	characteristic
110	C₂×D₄	—	2	3	no	—
111	C₂×D₄	—	2	3	no	—
112	C₂×D₄	—	79	1	no	—
113	C₂×D₄	C₂²×D₇₉	1	3	yes	—
114	C₂²×C₄	—	2	3	no	—
115	C₂²×C₄	—	158	3	no	—
116	D₈	—	158	6	no	—
117	C₂×C₈	—	79	1	no	—
118	C₂×C₈	—	79	3	no	—
119	C₄⋊C₄	—	79	3	no	—
120	C₄⋊C₄	—	158	3	no	—
121	C₄⋊C₄	C₂²⋊D₇₉	1	3	yes	—
122	C₄⋊C₄	C₂²×D₇₉	1	1	yes	characteristic
123	C₂²⋊C₄	—	2	3	no	—
124	C₂²⋊C₄	—	158	3	no	—
125	C₄²	C₂²×D₇₉	1	3	yes	—
126	C₂³	—	2	3	no	—
127	C₂³	—	158	1	no	—
128	Q₈	—	316	6	no	—
129	Q₈	#11929609	1	4	yes	complemented
130	D₄	—	2	3	no	—
131	D₄	—	2	3	no	—
132	D₄	—	4	6	no	—
133	D₄	—	8	3	no	—
134	D₄	—	79	2	no	—
135	D₄	—	158	1	no	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	4	6	no	—
138	C₂×C₄	—	316	6	no	—
139	C₂×C₄	—	158	3	no	—
140	C₂×C₄	C₂×C₂²⋊D₇₉	1	3	yes	—
141	C₂×C₄	C₂×C₂²⋊D₇₉	1	3	yes	—
142	C₂×C₄	C₂³×D₇₉	1	1	yes	characteristic
143	C₈	—	158	3	no	—
144	C₈	—	158	1	no	—
145	C₂²	—	4	3	no	—
146	C₂²	—	4	3	no	—
147	C₂²	—	4	3	no	—
148	C₂²	—	316	1	no	—
149	C₂²	—	316	1	no	—
150	C₂²	—	158	2	no	—
151	C₂²	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
152	C₄	—	2	3	no	—
153	C₄	—	4	1	no	—
154	C₄	—	316	3	no	—
155	C₄	C₂×#11929609	1	6	yes	—
156	C₄	#41×D₇₉	1	1	yes	characteristic
157	C₄	#14579208	1	1	yes	characteristic
158	C₂	—	8	3	no	—
159	C₂	—	316	2	no	—
160	C₂	#19099220	1	1	yes	characteristic
161	C₂	C₂²×#11929609	1	1	yes	characteristic
162	C₂	#19099374	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×D₃₁₆

C₂

yes

maximal, characteristic, complemented

#19098909

C₂

yes

maximal, complemented

#19098843

C₂

yes

maximal, complemented

#19098755

C₂

yes

maximal, complemented

Q₈×C₇₉⋊C₈

C₂

yes

maximal, characteristic, complemented

C₄×#11929609

C₂

yes

maximal, complemented

#550×C₇₉

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₇₉

—

C₂×#11929609

C₂²

yes

—

#14579161

C₂²

yes

—

#14579141

—

C₄×D₃₁₆

C₂²

yes

—

#14579094

C₂²

yes

—

C₄⋊(C₇₉⋊C₈)

C₂²

yes

—

#14579082

C₂²

yes

—

#14579082

C₂²

yes

—

#14579081

C₂²

yes

characteristic

C₄×C₇₉⋊C₈

C₂²

yes

—

C₂×#41×C₇₉

—

#127×C₇₉

C₂²

yes

—

#121×C₇₉

—

C₄×Q₈×C₇₉

C₂²

yes

characteristic

C₄×D₄×C₇₉

C₂²

yes

—

C₂×Dic₁₅₈

—

Q₈×D₇₉

—

C₂×D₃₁₆

C₂³

yes

characteristic

C₂×C₄×D₇₉

—

#11929609

—

C₂×C₇₉⋊C₈

C₂³

yes

characteristic

C₂×C₇₉⋊C₈

C₂³

yes

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

C₂³

yes

—

(C₂×D₇₉)⋊C₄

—

#41×C₇₉

—

C₂×Q₈×C₇₉

D₄

yes

characteristic

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

C₂³

yes

—

C₂²×C₄×C₇₉

—

C₄⋊C₄×C₇₉

D₄

yes

—

C₄⋊C₄×C₇₉

C₂³

yes

characteristic

C₂²⋊C₄×C₇₉

—

C₄²×C₇₉

C₂³

yes

—

C₂³×C₇₉

—

C₂²×D₇₉

—

Q₈×C₇₉

D₈

yes

complemented

D₄×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₄×C₇₉

C₂⁴

yes

characteristic

C₂×Dic₇₉

—

D₃₁₆

—

D₃₁₆

#41

yes

complemented

C₄×D₇₉

—

Dic₁₅₈

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

C₂²×D₄

yes

characteristic

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

C₂×D₈

yes

—

C₄×C₇₉

C₂×#41

yes

characteristic

C₄×C₇₉

2₋¹⁺⁴

yes

characteristic

Dic₇₉

—

C₂×C₇₉

—

C₂×C₇₉

#548

yes

characteristic

C₂×C₇₉

C₂²×D₈

yes

characteristic

C₂×C₇₉

#577

yes

characteristic

D₇₉

—

#3358

—

maximal, Sylow

C₇₉

#3358

yes

characteristic, complemented, Sylow

#550

D₇₉

yes

characteristic, complemented

D₄×Q₈

—

#486

—

#474

—

#459

—

C₈×Q₈

—

C₄×D₈

—

C₂×#41

—

C₂²×Q₈

—

158

—

C₂×D₈

—

C₄⋊Q₈

—

#127

C₂×D₇₉

yes

—

C₂²⋊Q₈

—

158

—

#121

—

C₄×Q₈

C₂×D₇₉

yes

characteristic

C₄×D₄

—

C₄×D₄

C₂×D₇₉

yes

—

100

#107

—

101

C₄⋊C₈

—

102

#102

—

103

#102

—

104

#102

—

105

C₄×C₈

—

106

#41

—

107

C₂×Q₈

—

316

—

108

C₂×Q₈

—

158

—

109

C₂×Q₈

C₂²⋊D₇₉

yes

characteristic

110

C₂×D₄

—

111

C₂×D₄

—

112

C₂×D₄

—

113

C₂×D₄

C₂²×D₇₉

yes

—

114

C₂²×C₄

—

115

C₂²×C₄

—

158

—

116

D₈

—

158

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

158

—

121

C₄⋊C₄

C₂²⋊D₇₉

yes

—

122

C₄⋊C₄

C₂²×D₇₉

yes

characteristic

123

C₂²⋊C₄

—

124

C₂²⋊C₄

—

158

—

125

C₄²

C₂²×D₇₉

yes

—

126

C₂³

—

127

C₂³

—

158

—

128

Q₈

—

316

—

129

Q₈

#11929609

yes

complemented

130

D₄

—

131

D₄

—

132

D₄

—

133

D₄

—

134

D₄

—

135

D₄

—

158

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

316

—

139

C₂×C₄

—

158

—

140

C₂×C₄

C₂×C₂²⋊D₇₉

yes

—

141

C₂×C₄

C₂×C₂²⋊D₇₉

yes

—

142

C₂×C₄

C₂³×D₇₉

yes

characteristic

143

C₈

—

158

—

144

C₈

—

158

—

145

C₂²

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

316

—

149

C₂²

—

316

—

150

C₂²

—

158

—

151

C₂²

C₂²×C₂²⋊D₇₉

yes

characteristic

152

C₄

—

153

C₄

—

154

C₄

—

316

—

155

C₄

C₂×#11929609

yes

—

156

C₄

#41×D₇₉

yes

characteristic

157

C₄

#14579208

yes

characteristic

158

C₂

—

159

C₂

—

316

—

160

C₂

#19099220

yes

characteristic

161

C₂

C₂²×#11929609

yes

characteristic

162

C₂

#19099374

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₇₉
commutator, cosocle	C₂×C₄×C₇₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₉
Fitting	#550×C₇₉	C₂
socle	C₂²×C₇₉	C₂²×D₄
hypercenter	#550	D₇₉
hypocenter	C₇₉	#3358
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #550

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₇₉

commutator, cosocle

C₂×C₄×C₇₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₉

Fitting

#550×C₇₉

C₂

socle

C₂²×C₇₉

C₂²×D₄

D₇₉

C₇₉

G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #550

N	H ≅ G/N	type
1. Q₈×D₃₁₆	C₂	semidirect
2. #19098909	C₂	semidirect
3. #19098843	C₂	semidirect
4. #19098755	C₂	semidirect
5. Q₈×C₇₉⋊C₈	C₂	semidirect
6. C₄×#11929609	C₂	semidirect
7. #550×C₇₉	C₂	semidirect
46. Q₈×C₇₉	D₈	semidirect
58. D₃₁₆	#41	semidirect
82. C₇₉	#3358	semidirect
83. #550	D₇₉	semidirect
129. Q₈	#11929609	semidirect

H ≅ G/N

type

1. Q₈×D₃₁₆

C₂

semidirect

2. #19098909

C₂

semidirect

3. #19098843

C₂

semidirect

4. #19098755

C₂

semidirect

5. Q₈×C₇₉⋊C₈

C₂

semidirect

6. C₄×#11929609

C₂

semidirect

7. #550×C₇₉

C₂

semidirect

46. Q₈×C₇₉

D₈

semidirect

58. D₃₁₆

#41

semidirect

82. C₇₉

#3358

semidirect

83. #550

D₇₉

semidirect

129. Q₈

#11929609

semidirect

Subgroups — #23776031

Conjugacy classes of subgroups

Complements