Group #13098779 — subgroups

Subgroups — (C₂×#1035)⋊Q₈ #13098779

Order 1600.

279 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 279 conjugacy classes of subgroups — too many to draw. Only its 75 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	C₂×Q₈×D₂₅	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#1035	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×#1035	C₂	1	1	yes	maximal, characteristic
4	(C₂×#1035)⋊C₄	C₂	1	4	yes	maximal
5	C₂²×#1035	C₂²	1	2	yes	—
6	C₂²×#1035	C₂²	1	2	yes	—
7	C₂×Q₈×C₂₅	C₄	1	1	yes	characteristic, complemented
8	Q₈×D₂₅	—	2	2	no	—
9	Q₈×D₂₅	—	2	2	no	—
10	C₂×C₄×D₂₅	C₂²	1	2	yes	—
11	C₂×C₄×D₂₅	C₂²	1	1	yes	characteristic
12	C₂×Dic₅₀	C₄	1	1	yes	characteristic, complemented
13	C₂×Dic₅₀	C₄	1	2	yes	complemented
14	C₄⋊#1035	—	2	2	no	—
15	C₄×#1035	—	2	2	no	—
16	(C₂×AGL(1,𝔽₅))⋊Q₈	—	5	1	no	maximal
17	C₂²×D₂₅	C₂³	1	1	yes	characteristic
18	C₂×#1035	—	2	2	no	—
19	C₂×#1035	—	2	2	no	—
20	C₂×#1035	—	2	2	no	—
21	C₂×#1035	—	2	4	no	—
22	C₂×#1035	Q₈	1	4	yes	complemented
23	Q₈×C₂₅	—	2	2	no	—
24	C₂×C₄×C₂₅	C₂×C₄	1	2	yes	—
25	C₂×C₄×C₂₅	C₂×C₄	1	1	yes	characteristic
26	C₂×Dic₂₅	C₂×C₄	1	2	yes	—
27	C₂×Dic₂₅	C₂×C₄	1	1	yes	characteristic
28	C₄×D₂₅	—	2	2	no	—
29	C₄×D₂₅	—	2	2	no	—
30	C₄×D₂₅	D₄	1	2	yes	—
31	C₄×D₂₅	D₄	1	2	yes	—
32	Dic₅₀	—	2	2	no	—
33	Dic₅₀	—	2	4	no	—
34	C₂×Q₈×D₅	—	5	1	no	—
35	C₂×C₄⋊AGL(1,𝔽₅)	—	5	1	no	—
36	C₂×C₄×AGL(1,𝔽₅)	—	5	1	no	—
37	(C₂×AGL(1,𝔽₅))⋊C₄	—	5	4	no	—
38	C₂²×C₂₅	C₂²×C₄	1	1	yes	characteristic
39	C₂×D₂₅	C₂×D₄	1	1	yes	characteristic
40	C₂×D₂₅	C₂×D₄	1	1	yes	characteristic
41	C₂×D₂₅	C₂×Q₈	1	2	yes	—
42	C₂×D₂₅	#41	1	2	yes	—
43	#1035	—	2	4	no	—
44	#1035	—	4	2	no	—
45	C₄×C₂₅	—	2	2	no	—
46	C₄×C₂₅	C₂²⋊C₄	1	2	yes	—
47	Dic₂₅	—	2	2	no	—
48	Dic₂₅	C₂²⋊C₄	1	2	yes	—
49	C₂²×AGL(1,𝔽₅)	—	5	2	no	—
50	C₂²×AGL(1,𝔽₅)	—	5	2	no	—
51	C₂×C₅×Q₈	AGL(1,𝔽₅)	1	1	yes	characteristic
52	Q₈×D₅	—	10	2	no	—
53	Q₈×D₅	—	10	2	no	—
54	C₂×C₄×D₅	—	5	1	no	—
55	C₂×C₄×D₅	—	5	2	no	—
56	C₂×Dic₁₀	—	5	2	no	—
57	C₂×Dic₁₀	—	5	1	no	—
58	C₄⋊AGL(1,𝔽₅)	—	10	2	no	—
59	C₄×AGL(1,𝔽₅)	—	10	2	no	—
60	#391	—	25	1	no	Sylow
61	C₂×C₂₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₂₅	C₄×Q₈	1	2	yes	—
63	D₂₅	C₂²⋊Q₈	1	2	yes	—
64	D₂₅	#124	1	1	yes	characteristic
65	D₂₅	C₄⋊Q₈	1	1	yes	characteristic
66	C₂²×D₅	—	5	1	no	—
67	C₂×AGL(1,𝔽₅)	—	5	4	no	—
68	C₂×AGL(1,𝔽₅)	—	10	2	no	—
69	C₂×AGL(1,𝔽₅)	—	10	2	no	—
70	C₂×AGL(1,𝔽₅)	—	10	2	no	—
71	C₂×AGL(1,𝔽₅)	—	10	4	no	—
72	C₅×Q₈	—	2	2	no	—
73	C₂×C₄×C₅	C₂×AGL(1,𝔽₅)	1	2	yes	—
74	C₂×C₄×C₅	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
75	C₂×Dic₅	—	5	1	no	—
76	C₂×Dic₅	—	5	2	no	—
77	C₄×D₅	—	5	2	no	—
78	C₄×D₅	—	5	2	no	—
79	C₄×D₅	—	10	2	no	—
80	C₄×D₅	—	10	2	no	—
81	Dic₁₀	—	10	4	no	—
82	Dic₁₀	—	10	2	no	—
83	C₂²×Q₈	—	25	1	no	—
84	C₂×C₄⋊C₄	—	25	1	no	—
85	C₂×C₄²	—	25	1	no	—
86	#95	—	25	4	no	—
87	C₂₅	#391	1	1	yes	characteristic, complemented, Sylow
88	C₂²×C₅	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
89	C₂×D₅	—	5	2	no	—
90	C₂×D₅	—	5	2	no	—
91	C₂×D₅	—	5	1	no	—
92	C₂×D₅	—	5	1	no	—
93	AGL(1,𝔽₅)	—	10	4	no	—
94	AGL(1,𝔽₅)	—	20	2	no	—
95	C₄×C₅	—	2	2	no	—
96	C₄×C₅	C₂²⋊AGL(1,𝔽₅)	1	2	yes	—
97	Dic₅	—	5	2	no	—
98	Dic₅	—	10	2	no	—
99	C₂×Q₈	—	25	2	no	—
100	C₂×Q₈	—	25	1	no	—
101	C₂×Q₈	—	50	2	no	—
102	C₂×Q₈	—	50	2	no	—
103	C₂×Q₈	#1035	1	1	yes	characteristic, complemented
104	C₂²×C₄	—	25	1	no	—
105	C₂²×C₄	—	25	2	no	—
106	C₂²×C₄	—	25	2	no	—
107	C₂²×C₄	—	25	2	no	—
108	C₄⋊C₄	—	50	2	no	—
109	C₄²	—	50	2	no	—
110	C₂×C₅	Q₈×AGL(1,𝔽₅)	1	2	yes	—
111	C₂×C₅	C₂×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
112	D₅	—	5	2	no	—
113	D₅	—	5	1	no	—
114	D₅	—	5	1	no	—
115	C₂³	—	25	1	no	—
116	Q₈	—	2	2	no	—
117	Q₈	—	50	4	no	—
118	Q₈	—	50	2	no	—
119	C₂×C₄	—	25	1	no	—
120	C₂×C₄	—	25	2	no	—
121	C₂×C₄	—	25	2	no	—
122	C₂×C₄	—	25	4	no	—
123	C₂×C₄	—	25	2	no	—
124	C₂×C₄	—	50	2	no	—
125	C₂×C₄	—	50	2	no	—
126	C₂×C₄	—	50	2	no	—
127	C₂×C₄	—	50	2	no	—
128	C₂×C₄	—	50	2	no	—
129	C₂×C₄	—	50	4	no	—
130	C₂×C₄	C₂×#1035	1	2	yes	—
131	C₂×C₄	C₂×#1035	1	1	yes	characteristic
132	C₅	(C₂×AGL(1,𝔽₅))⋊Q₈	1	1	yes	characteristic
133	C₂²	—	25	2	no	—
134	C₂²	—	25	2	no	—
135	C₂²	—	25	1	no	—
136	C₂²	—	25	1	no	—
137	C₂²	C₂²×#1035	1	1	yes	characteristic
138	C₄	—	2	2	no	—
139	C₄	—	25	2	no	—
140	C₄	—	50	4	no	—
141	C₄	—	50	2	no	—
142	C₄	—	100	2	no	—
143	C₄	C₂²⋊#1035	1	2	yes	—
144	C₂	—	25	2	no	—
145	C₂	—	25	1	no	—
146	C₂	—	25	1	no	—
147	C₂	Q₈×#1035	1	2	yes	—
148	C₂	C₂×C₂²⋊#1035	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×D₂₅

C₂

yes

maximal, characteristic

C₂×C₄⋊#1035

C₂

yes

maximal, characteristic

C₂×C₄×#1035

C₂

yes

maximal, characteristic

(C₂×#1035)⋊C₄

C₂

yes

maximal

C₂²×#1035

C₂²

yes

—

C₂²×#1035

C₂²

yes

—

C₂×Q₈×C₂₅

C₄

yes

characteristic, complemented

Q₈×D₂₅

—

Q₈×D₂₅

—

C₂×C₄×D₂₅

C₂²

yes

—

C₂×C₄×D₂₅

C₂²

yes

characteristic

C₂×Dic₅₀

C₄

yes

characteristic, complemented

C₂×Dic₅₀

C₄

yes

complemented

C₄⋊#1035

—

C₄×#1035

—

(C₂×AGL(1,𝔽₅))⋊Q₈

—

maximal

C₂²×D₂₅

C₂³

yes

—

—

—

—

Q₈

yes

complemented

Q₈×C₂₅

—

C₂×C₄×C₂₅

C₂×C₄

yes

—

C₂×C₄×C₂₅

C₂×C₄

yes

characteristic

C₂×Dic₂₅

C₂×C₄

yes

—

C₂×Dic₂₅

C₂×C₄

yes

characteristic

C₄×D₂₅

—

C₄×D₂₅

—

C₄×D₂₅

D₄

yes

—

C₄×D₂₅

D₄

yes

—

Dic₅₀

—

Dic₅₀

—

C₂×Q₈×D₅

—

C₂×C₄⋊AGL(1,𝔽₅)

—

C₂×C₄×AGL(1,𝔽₅)

—

(C₂×AGL(1,𝔽₅))⋊C₄

—

C₂²×C₂₅

C₂²×C₄

yes

characteristic

C₂×D₂₅

C₂×D₄

yes

characteristic

C₂×D₂₅

C₂×D₄

yes

characteristic

C₂×D₂₅

C₂×Q₈

yes

—

C₂×D₂₅

#41

yes

—

#1035

—

#1035

—

C₄×C₂₅

—

C₄×C₂₅

C₂²⋊C₄

yes

—

Dic₂₅

—

Dic₂₅

C₂²⋊C₄

yes

—

C₂²×AGL(1,𝔽₅)

—

C₂²×AGL(1,𝔽₅)

—

C₂×C₅×Q₈

AGL(1,𝔽₅)

yes

characteristic

Q₈×D₅

—

Q₈×D₅

—

C₂×C₄×D₅

—

C₂×C₄×D₅

—

C₂×Dic₁₀

—

C₂×Dic₁₀

—

—

—

—

C₂×C₂₅

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₂₅

C₄×Q₈

yes

—

D₂₅

C₂²⋊Q₈

yes

—

D₂₅

#124

yes

characteristic

D₂₅

C₄⋊Q₈

yes

characteristic

C₂²×D₅

—

—

—

—

—

—

C₅×Q₈

—

C₂×C₄×C₅

C₂×AGL(1,𝔽₅)

yes

—

C₂×C₄×C₅

C₂×AGL(1,𝔽₅)

yes

characteristic

C₂×Dic₅

—

C₂×Dic₅

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

Dic₁₀

—

Dic₁₀

—

C₂²×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

#95

—

C₂₅

#391

yes

characteristic, complemented, Sylow

C₂²×C₅

C₂²×AGL(1,𝔽₅)

yes

characteristic

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

AGL(1,𝔽₅)

—

AGL(1,𝔽₅)

—

C₄×C₅

—

C₄×C₅

C₂²⋊AGL(1,𝔽₅)

yes

—

Dic₅

—

Dic₅

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×Q₈

—

102

C₂×Q₈

—

103

C₂×Q₈

#1035

yes

characteristic, complemented

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₄⋊C₄

—

109

C₄²

—

110

C₂×C₅

Q₈×AGL(1,𝔽₅)

yes

—

111

C₂×C₅

C₂×C₂²⋊AGL(1,𝔽₅)

yes

characteristic

112

D₅

—

113

D₅

—

114

D₅

—

115

C₂³

—

116

Q₈

—

117

Q₈

—

118

Q₈

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

C₂×#1035

yes

—

131

C₂×C₄

C₂×#1035

yes

characteristic

132

C₅

(C₂×AGL(1,𝔽₅))⋊Q₈

yes

characteristic

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

C₂²×#1035

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

100

—

143

C₄

C₂²⋊#1035

yes

—

144

C₂

—

145

C₂

—

146

C₂

—

147

C₂

Q₈×#1035

yes

—

148

C₂

C₂×C₂²⋊#1035

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×#1035
commutator	C₂²×C₂₅	C₂²×C₄
Frattini, socle	C₂²×C₅	C₂²×AGL(1,𝔽₅)
Fitting	C₂×Q₈×C₂₅	C₄
prime radical, cosocle	C₂²×D₂₅	C₂³
hypercenter	C₂×Q₈	#1035
hypocenter	C₂₅	#391
LCS	G ▷ C₂²×C₂₅ ▷ C₂₅
UCS	C₁ ◁ C₂² ◁ C₂×Q₈

subgroup

G/H

center

C₂²

C₂²×#1035

commutator

C₂²×C₂₅

C₂²×C₄

Frattini, socle

C₂²×C₅

C₂²×AGL(1,𝔽₅)

Fitting

C₂×Q₈×C₂₅

C₄

prime radical, cosocle

C₂²×D₂₅

C₂³

C₂×Q₈

C₂₅

G ▷ C₂²×C₂₅ ▷ C₂₅

C₁ ◁ C₂² ◁ C₂×Q₈

N	H ≅ G/N	type
7. C₂×Q₈×C₂₅	C₄	semidirect
12. C₂×Dic₅₀	C₄	semidirect
13. C₂×Dic₅₀	C₄	semidirect
22. C₂×#1035	Q₈	semidirect
87. C₂₅	#391	semidirect
103. C₂×Q₈	#1035	semidirect

H ≅ G/N

type

7. C₂×Q₈×C₂₅

C₄

semidirect

12. C₂×Dic₅₀

C₄

semidirect

13. C₂×Dic₅₀

C₄

semidirect

22. C₂×#1035

Q₈

semidirect

87. C₂₅

#391

semidirect

103. C₂×Q₈

#1035

semidirect

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

subgroup	index
C₂×Q₈×D₂₅	2
C₂×C₄⋊#1035	2
C₂×C₄×#1035	2
(C₂×#1035)⋊C₄	2
(C₂×AGL(1,𝔽₅))⋊Q₈	5

subgroup

index

C₂×Q₈×D₂₅

C₂×C₄⋊#1035

C₂×C₄×#1035

(C₂×#1035)⋊C₄

(C₂×AGL(1,𝔽₅))⋊Q₈

Subgroups — (C₂×#1035)⋊Q₈ #13098779

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — (C2×#1035)⋊Q8 #13098779

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — (C₂×#1035)⋊Q₈ #13098779