Group #23776260 — subgroups

Subgroups — Q₈×#11929612 #23776260

Order 10112.

370 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 149 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×Dic₁₅₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#19098959	C₂	1	3	yes	maximal, complemented
3	#19098853	C₂	1	3	yes	maximal
4	#19098845	C₂	1	3	yes	maximal, complemented
5	Q₈×C₇₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#11929612	C₂	1	3	yes	maximal
7	D₄×Q₈×C₇₉	C₂	1	1	yes	maximal, characteristic
8	C₂×#11929612	C₂²	1	1	yes	characteristic
9	#14579161	C₂²	1	3	yes	—
10	#14579160	—	2	3	no	—
11	Q₈×Dic₇₉	—	2	1	no	—
12	C₄×Dic₁₅₈	C₂²	1	3	yes	—
13	#14579092	C₂²	1	3	yes	—
14	#14579092	C₂²	1	6	yes	—
15	C₄⋊(C₇₉⋊C₈)	C₂²	1	3	yes	—
16	#14579084	C₂²	1	3	yes	—
17	#14579082	C₂²	1	3	yes	—
18	C₄×C₇₉⋊C₈	C₂²	1	3	yes	—
19	C₂²×Q₈×C₇₉	—	2	1	no	—
20	C₄⋊Q₈×C₇₉	C₂²	1	3	yes	—
21	C₂²⋊Q₈×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	C₂×Dic₁₅₈	C₂³	1	1	yes	characteristic
26	#11929612	Q₈	1	4	yes	direct product factor
27	C₂×C₇₉⋊C₈	C₂³	1	3	yes	—
28	C₂×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
29	C₄⋊Dic₇₉	—	2	3	no	—
30	C₄⋊Dic₇₉	C₂³	1	3	yes	—
31	Dic₇₉⋊C₄	—	2	3	no	—
32	C₄×Dic₇₉	—	2	3	no	—
33	C₂×Q₈×C₇₉	—	2	3	no	—
34	C₂×Q₈×C₇₉	—	4	4	no	—
35	C₂×Q₈×C₇₉	D₄	1	1	yes	characteristic
36	C₂×D₄×C₇₉	C₂³	1	1	yes	characteristic
37	C₂²×C₄×C₇₉	—	2	3	no	—
38	C₄⋊C₄×C₇₉	—	2	3	no	—
39	C₄⋊C₄×C₇₉	D₄	1	3	yes	—
40	C₄⋊C₄×C₇₉	C₂³	1	3	yes	—
41	C₂²⋊C₄×C₇₉	—	2	3	no	—
42	C₄²×C₇₉	C₂³	1	3	yes	—
43	C₂³×C₇₉	—	2	1	no	—
44	Q₈×C₇₉	—	4	6	no	—
45	Q₈×C₇₉	SD₁₆	1	4	yes	complemented
46	D₄×C₇₉	—	2	1	no	—
47	D₄×C₇₉	C₂×Q₈	1	2	yes	—
48	C₂×C₄×C₇₉	—	2	3	no	—
49	C₂×C₄×C₇₉	—	4	6	no	—
50	C₂×C₄×C₇₉	C₂×D₄	1	3	yes	—
51	C₂×C₄×C₇₉	C₂×D₄	1	3	yes	—
52	C₂×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
53	C₂×Dic₇₉	—	2	1	no	—
54	C₂×Dic₇₉	—	2	3	no	—
55	Dic₁₅₈	—	2	1	no	—
56	Dic₁₅₈	—	8	3	no	—
57	Dic₁₅₈	C₂×Q₈	1	2	yes	complemented
58	C₇₉⋊C₈	—	2	3	no	—
59	C₇₉⋊C₈	C₂×Q₈	1	2	yes	complemented
60	C₂²×C₇₉	—	2	2	no	—
61	C₂²×C₇₉	—	4	1	no	—
62	C₂²×C₇₉	—	4	1	no	—
63	C₂²×C₇₉	C₂²×D₄	1	1	yes	characteristic
64	C₄×C₇₉	—	2	3	no	—
65	C₄×C₇₉	—	4	3	no	—
66	C₄×C₇₉	C₂×SD₁₆	1	6	yes	—
67	C₄×C₇₉	C₂²×Q₈	1	1	yes	characteristic
68	C₄×C₇₉	2₋¹⁺⁴	1	1	yes	characteristic
69	Dic₇₉	—	2	2	no	—
70	Dic₇₉	—	4	3	no	—
71	C₂×C₇₉	—	4	2	no	—
72	C₂×C₇₉	D₄×Q₈	1	1	yes	characteristic
73	C₂×C₇₉	C₂²×SD₁₆	1	1	yes	characteristic
74	C₂×C₇₉	#577	1	1	yes	characteristic
75	Q₈×SD₁₆	—	79	1	no	maximal, Sylow
76	C₇₉	Q₈×SD₁₆	1	1	yes	characteristic, complemented, Sylow
77	Q₈²	—	79	1	no	—
78	D₄×Q₈	D₇₉	1	1	yes	characteristic
79	#498	—	79	3	no	—
80	#476	—	79	3	no	—
81	#475	—	79	3	no	—
82	C₈×Q₈	—	79	1	no	—
83	C₄×SD₁₆	—	79	3	no	—
84	C₂²×Q₈	—	2	1	no	—
85	C₂×SD₁₆	—	79	1	no	—
86	C₄⋊Q₈	—	79	3	no	—
87	C₄⋊Q₈	—	158	3	no	—
88	C₄⋊Q₈	C₂×D₇₉	1	3	yes	—
89	C₂²⋊Q₈	—	2	3	no	—
90	C₄×Q₈	—	79	3	no	—
91	C₄×Q₈	—	158	1	no	—
92	C₄×Q₈	C₂×D₇₉	1	1	yes	characteristic
93	C₄×D₄	C₂×D₇₉	1	3	yes	—
94	#106	—	79	3	no	—
95	#106	—	79	6	no	—
96	C₄⋊C₈	—	79	3	no	—
97	#103	—	79	3	no	—
98	#102	—	79	3	no	—
99	C₄×C₈	—	79	3	no	—
100	C₂×Q₈	—	2	3	no	—
101	C₂×Q₈	—	4	4	no	—
102	C₂×Q₈	—	79	1	no	—
103	C₂×Q₈	—	158	3	no	—
104	C₂×Q₈	C₂²⋊D₇₉	1	1	yes	characteristic
105	C₂×D₄	C₂²×D₇₉	1	1	yes	characteristic
106	C₂²×C₄	—	2	3	no	—
107	SD₁₆	—	79	4	no	—
108	C₂×C₈	—	79	1	no	—
109	C₂×C₈	—	79	3	no	—
110	C₄⋊C₄	—	2	3	no	—
111	C₄⋊C₄	—	79	3	no	—
112	C₄⋊C₄	—	158	3	no	—
113	C₄⋊C₄	—	158	3	no	—
114	C₄⋊C₄	C₂²⋊D₇₉	1	3	yes	—
115	C₄⋊C₄	C₂²×D₇₉	1	3	yes	—
116	C₂²⋊C₄	—	2	3	no	—
117	C₄²	—	158	3	no	—
118	C₄²	C₂²×D₇₉	1	3	yes	—
119	C₂³	—	2	1	no	—
120	Q₈	—	4	6	no	—
121	Q₈	—	79	2	no	—
122	Q₈	—	632	3	no	—
123	Q₈	—	158	1	no	—
124	Q₈	#11929612	1	4	yes	direct product factor
125	D₄	—	2	1	no	—
126	D₄	Q₈×D₇₉	1	2	yes	—
127	C₂×C₄	—	2	3	no	—
128	C₂×C₄	—	4	6	no	—
129	C₂×C₄	—	158	3	no	—
130	C₂×C₄	—	158	1	no	—
131	C₂×C₄	C₂×C₂²⋊D₇₉	1	3	yes	—
132	C₂×C₄	C₂×C₂²⋊D₇₉	1	3	yes	—
133	C₂×C₄	C₂³×D₇₉	1	1	yes	characteristic
134	C₈	—	79	2	no	—
135	C₈	—	158	3	no	—
136	C₂²	—	2	2	no	—
137	C₂²	—	4	1	no	—
138	C₂²	—	4	1	no	—
139	C₂²	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
140	C₄	—	2	3	no	—
141	C₄	—	4	3	no	—
142	C₄	—	316	3	no	—
143	C₄	—	158	2	no	—
144	C₄	C₂×#11929612	1	6	yes	—
145	C₄	C₂×Q₈×D₇₉	1	1	yes	characteristic
146	C₄	#14579209	1	1	yes	characteristic
147	C₂	—	4	2	no	—
148	C₂	Q₈×C₂²⋊D₇₉	1	1	yes	characteristic
149	C₂	C₂²×#11929612	1	1	yes	characteristic
150	C₂	#19099371	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×Dic₁₅₈

C₂

yes

maximal, characteristic, complemented

#19098959

C₂

yes

maximal, complemented

#19098853

C₂

yes

maximal

#19098845

C₂

yes

maximal, complemented

Q₈×C₇₉⋊C₈

C₂

yes

maximal, characteristic, complemented

C₄×#11929612

C₂

yes

maximal

D₄×Q₈×C₇₉

C₂

yes

maximal, characteristic

C₂×#11929612

C₂²

yes

characteristic

#14579161

C₂²

yes

—

#14579160

—

Q₈×Dic₇₉

—

C₄×Dic₁₅₈

C₂²

yes

—

#14579092

C₂²

yes

—

#14579092

C₂²

yes

—

C₄⋊(C₇₉⋊C₈)

C₂²

yes

—

#14579084

C₂²

yes

—

#14579082

C₂²

yes

—

C₄×C₇₉⋊C₈

C₂²

yes

—

C₂²×Q₈×C₇₉

—

C₄⋊Q₈×C₇₉

C₂²

yes

—

C₂²⋊Q₈×C₇₉

—

C₄×Q₈×C₇₉

C₂²

yes

characteristic

C₄×D₄×C₇₉

C₂²

yes

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

C₂³

yes

characteristic

#11929612

Q₈

yes

direct product factor

C₂×C₇₉⋊C₈

C₂³

yes

—

C₂×C₇₉⋊C₈

C₂³

yes

characteristic

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

C₂³

yes

—

Dic₇₉⋊C₄

—

C₄×Dic₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

D₄

yes

characteristic

C₂×D₄×C₇₉

C₂³

yes

characteristic

C₂²×C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

D₄

yes

—

C₄⋊C₄×C₇₉

C₂³

yes

—

C₂²⋊C₄×C₇₉

—

C₄²×C₇₉

C₂³

yes

—

C₂³×C₇₉

—

Q₈×C₇₉

—

Q₈×C₇₉

SD₁₆

yes

complemented

D₄×C₇₉

—

D₄×C₇₉

C₂×Q₈

yes

—

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

—

C₂×Dic₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

Dic₁₅₈

C₂×Q₈

yes

complemented

C₇₉⋊C₈

—

C₇₉⋊C₈

C₂×Q₈

yes

complemented

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

C₂²×D₄

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

C₂×SD₁₆

yes

—

C₄×C₇₉

C₂²×Q₈

yes

characteristic

C₄×C₇₉

2₋¹⁺⁴

yes

characteristic

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

—

C₂×C₇₉

D₄×Q₈

yes

characteristic

C₂×C₇₉

C₂²×SD₁₆

yes

characteristic

C₂×C₇₉

#577

yes

characteristic

Q₈×SD₁₆

—

maximal, Sylow

C₇₉

Q₈×SD₁₆

yes

characteristic, complemented, Sylow

Q₈²

—

D₄×Q₈

D₇₉

yes

—

—

—

C₈×Q₈

—

C₄×SD₁₆

—

C₂²×Q₈

—

C₂×SD₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

158

—

C₄⋊Q₈

C₂×D₇₉

yes

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

158

—

C₄×Q₈

C₂×D₇₉

yes

characteristic

C₄×D₄

C₂×D₇₉

yes

—

#106

—

#106

—

C₄⋊C₈

—

#103

—

#102

—

C₄×C₈

—

100

C₂×Q₈

—

101

C₂×Q₈

—

102

C₂×Q₈

—

103

C₂×Q₈

—

158

—

104

C₂×Q₈

C₂²⋊D₇₉

yes

characteristic

105

C₂×D₄

C₂²×D₇₉

yes

characteristic

106

C₂²×C₄

—

107

SD₁₆

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

158

—

113

C₄⋊C₄

—

158

—

114

C₄⋊C₄

C₂²⋊D₇₉

yes

—

115

C₄⋊C₄

C₂²×D₇₉

yes

—

116

C₂²⋊C₄

—

117

C₄²

—

158

—

118

C₄²

C₂²×D₇₉

yes

—

119

C₂³

—

120

Q₈

—

121

Q₈

—

122

Q₈

—

632

—

123

Q₈

—

158

—

124

Q₈

#11929612

yes

direct product factor

125

D₄

—

126

D₄

Q₈×D₇₉

yes

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

158

—

130

C₂×C₄

—

158

—

131

C₂×C₄

C₂×C₂²⋊D₇₉

yes

—

132

C₂×C₄

C₂×C₂²⋊D₇₉

yes

—

133

C₂×C₄

C₂³×D₇₉

yes

characteristic

134

C₈

—

135

C₈

—

158

—

136

C₂²

—

137

C₂²

—

138

C₂²

—

139

C₂²

C₂²×C₂²⋊D₇₉

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

—

316

—

143

C₄

—

158

—

144

C₄

C₂×#11929612

yes

—

145

C₄

C₂×Q₈×D₇₉

yes

characteristic

146

C₄

#14579209

yes

characteristic

147

C₂

—

148

C₂

Q₈×C₂²⋊D₇₉

yes

characteristic

149

C₂

C₂²×#11929612

yes

characteristic

150

C₂

#19099371

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₇₉
commutator, cosocle	C₂×C₄×C₇₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₉
Fitting	D₄×Q₈×C₇₉	C₂
socle	C₂²×C₇₉	C₂²×D₄
prime radical	C₂×D₄×C₇₉	C₂³
hypercenter	D₄×Q₈	D₇₉
hypocenter	C₇₉	Q₈×SD₁₆
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ D₄×Q₈

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₇₉

commutator, cosocle

C₂×C₄×C₇₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₉

Fitting

D₄×Q₈×C₇₉

C₂

socle

C₂²×C₇₉

C₂²×D₄

prime radical

C₂×D₄×C₇₉

C₂³

hypercenter

D₄×Q₈

D₇₉

hypocenter

C₇₉

Q₈×SD₁₆

LCS

G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ D₄×Q₈

N	H ≅ G/N	type
1. Q₈×Dic₁₅₈	C₂	semidirect
2. #19098959	C₂	semidirect
4. #19098845	C₂	semidirect
5. Q₈×C₇₉⋊C₈	C₂	semidirect
26. #11929612	Q₈	direct (N × H)
45. Q₈×C₇₉	SD₁₆	semidirect
57. Dic₁₅₈	C₂×Q₈	semidirect
59. C₇₉⋊C₈	C₂×Q₈	semidirect
76. C₇₉	Q₈×SD₁₆	semidirect
124. Q₈	#11929612	direct (N × H)

H ≅ G/N

type

1. Q₈×Dic₁₅₈

C₂

semidirect

2. #19098959

C₂

semidirect

4. #19098845

C₂

semidirect

5. Q₈×C₇₉⋊C₈

C₂

semidirect

26. #11929612

Q₈

direct (N × H)

45. Q₈×C₇₉

SD₁₆

semidirect

57. Dic₁₅₈

C₂×Q₈

semidirect

59. C₇₉⋊C₈

C₂×Q₈

semidirect

76. C₇₉

Q₈×SD₁₆

semidirect

124. Q₈

#11929612

direct (N × H)

Subgroups — Q₈×#11929612 #23776260

Conjugacy classes of subgroups

Complements

Subgroups — Q8×#11929612 #23776260

Conjugacy classes of subgroups

Complements

Subgroups — Q₈×#11929612 #23776260