Group #16854503 — subgroups

Subgroups — C₂×#14521160 #16854503

Order 3968.

608 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 608 conjugacy classes of subgroups — too many to draw. Only its 197 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	C₂²×D₂₄₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#11758470	C₂	1	2	yes	maximal, complemented
3	#14521160	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11758397	C₂	1	2	yes	maximal, complemented
5	C₂×#11758403	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃₁×#106	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₁₂₄	C₂²	1	2	yes	—
8	C₂²×C₄×D₃₁	—	2	2	no	—
9	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
11	C₂×(C₂×D₃₁)⋊C₄	—	2	2	no	—
12	C₂×C₄×Dic₃₁	C₂²	1	1	yes	characteristic
13	C₂²×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×D₂₄₈	C₄	1	8	yes	complemented
15	C₂×D₂₄₈	C₂²	1	4	yes	—
16	#11758470	—	2	4	no	—
17	#11758470	C₂²	1	8	yes	complemented
18	C₃₁×#106	C₂²	1	4	yes	complemented
19	#11758403	C₂²	1	4	yes	complemented
20	#11758397	C₂²	1	8	yes	complemented
21	C₂³×D₃₁	—	2	2	no	—
22	C₂²×C₄×C₃₁	—	2	2	no	—
23	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
24	C₂²×Dic₃₁	D₄	1	2	yes	—
25	C₂×D₁₂₄	—	2	2	no	—
26	C₂×D₁₂₄	—	2	4	no	—
27	C₂×D₁₂₄	C₂×C₄	1	8	yes	complemented
28	C₂×D₁₂₄	C₂³	1	4	yes	—
29	C₂×C₄×D₃₁	—	2	8	no	—
30	C₂×C₄×D₃₁	—	4	8	no	—
31	C₂×C₈×C₃₁	C₂×C₄	1	4	yes	complemented
32	C₂×C₈×C₃₁	C₂³	1	2	yes	—
33	C₄⋊C₄×C₃₁	—	2	2	no	—
34	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
35	(C₂×D₃₁)⋊C₄	—	2	8	no	—
36	C₄×Dic₃₁	D₄	1	2	yes	—
37	C₄×Dic₃₁	C₂³	1	2	yes	—
38	C₂×C₃₁⋊C₈	—	2	2	no	—
39	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
40	D₂₄₈	C₂×C₄	1	16	yes	complemented
41	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
42	C₂²×D₃₁	—	2	8	no	—
43	C₂²×D₃₁	—	2	4	no	—
44	C₂²×D₃₁	—	4	4	no	—
45	C₂²×D₃₁	—	4	4	no	—
46	C₂×C₄×C₃₁	—	2	4	no	—
47	C₂×C₄×C₃₁	—	4	4	no	—
48	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
49	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₃₁	#41	1	2	yes	—
51	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
52	C₂×Dic₃₁	—	2	4	no	—
53	C₂×Dic₃₁	C₂×D₄	1	4	yes	—
54	D₁₂₄	—	2	8	no	—
55	D₁₂₄	—	2	4	no	—
56	D₁₂₄	C₂²×C₄	1	8	yes	complemented
57	C₄×D₃₁	—	4	16	no	—
58	C₈×C₃₁	C₂²×C₄	1	4	yes	complemented
59	C₃₁⋊C₈	—	2	2	no	—
60	C₂×#442	—	31	1	no	maximal, Sylow
61	C₂²×C₃₁	C₄×D₄	1	2	yes	—
62	C₂²×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
63	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
64	C₂×D₃₁	—	2	8	no	—
65	C₂×D₃₁	—	4	4	no	—
66	C₂×D₃₁	—	4	4	no	—
67	C₂×D₃₁	—	4	16	no	—
68	C₄×C₃₁	—	4	4	no	—
69	C₄×C₃₁	C₄×D₄	1	2	yes	—
70	C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
71	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
72	Dic₃₁	—	2	4	no	—
73	C₂²×D₈	—	31	1	no	—
74	C₂×C₄×D₄	—	31	2	no	—
75	#442	—	31	8	no	—
76	C₂×#106	D₃₁	1	1	yes	characteristic, complemented
77	C₂×#102	—	31	2	no	—
78	C₂×#97	—	31	1	no	—
79	C₂×C₃₁	#442	1	4	yes	complemented
80	C₂×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
81	C₂×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
82	D₃₁	—	4	8	no	—
83	C₂²×D₄	—	31	2	no	—
84	C₂³×C₄	—	62	2	no	—
85	C₂×D₈	—	31	8	no	—
86	C₂×D₈	—	31	4	no	—
87	C₂²×C₈	—	31	1	no	—
88	C₂²×C₈	C₂×D₃₁	1	1	yes	characteristic
89	C₄×D₄	—	31	8	no	—
90	C₄×D₄	—	62	4	no	—
91	C₂×C₄⋊C₄	C₂×D₃₁	1	2	yes	—
92	C₂×C₂²⋊C₄	—	62	2	no	—
93	C₂×C₄²	—	31	1	no	—
94	#106	C₂×D₃₁	1	4	yes	complemented
95	#102	—	31	8	no	—
96	#97	—	31	4	no	—
97	C₃₁	C₂×#442	1	1	yes	characteristic, complemented, Sylow
98	C₂⁴	—	62	2	no	—
99	C₂×D₄	—	31	8	no	—
100	C₂×D₄	—	31	4	no	—
101	C₂×D₄	—	62	2	no	—
102	C₂×D₄	—	62	4	no	—
103	C₂²×C₄	—	2	2	no	—
104	C₂²×C₄	—	31	2	no	—
105	C₂²×C₄	—	62	8	no	—
106	C₂²×C₄	—	124	8	no	—
107	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
108	D₈	—	31	16	no	—
109	C₂×C₈	—	31	2	no	—
110	C₂×C₈	—	62	2	no	—
111	C₂×C₈	C₄×D₃₁	1	4	yes	complemented
112	C₂×C₈	C₂²×D₃₁	1	2	yes	—
113	C₄⋊C₄	—	2	2	no	—
114	C₄⋊C₄	C₂²×D₃₁	1	4	yes	—
115	C₂²⋊C₄	—	62	8	no	—
116	C₄²	—	31	2	no	—
117	C₄²	—	31	2	no	—
118	C₂³	—	62	8	no	—
119	C₂³	—	62	4	no	—
120	C₂³	—	124	4	no	—
121	C₂³	—	124	4	no	—
122	C₂³	D₄×D₃₁	1	1	yes	characteristic
123	D₄	—	31	8	no	—
124	D₄	—	62	8	no	—
125	D₄	—	62	4	no	—
126	C₂×C₄	—	2	4	no	—
127	C₂×C₄	—	4	4	no	—
128	C₂×C₄	—	31	4	no	—
129	C₂×C₄	—	62	4	no	—
130	C₂×C₄	—	124	16	no	—
131	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
132	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
133	C₂×C₄	#92499	1	2	yes	—
134	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
135	C₈	—	62	2	no	—
136	C₈	C₂×C₄×D₃₁	1	4	yes	complemented
137	C₂²	—	62	8	no	—
138	C₂²	—	124	16	no	—
139	C₂²	—	124	4	no	—
140	C₂²	—	124	4	no	—
141	C₂²	#11758470	1	2	yes	—
142	C₂²	#11758491	1	4	yes	—
143	C₂²	C₂×D₄×D₃₁	1	1	yes	characteristic
144	C₄	—	4	4	no	—
145	C₄	—	62	4	no	—
146	C₄	#11758470	1	2	yes	—
147	C₄	C₂²×C₄×D₃₁	1	1	yes	characteristic
148	C₄	C₂×#92499	1	1	yes	characteristic
149	C₂	—	124	8	no	—
150	C₂	#14521160	1	4	yes	direct product factor
151	C₂	C₂×#11758470	1	1	yes	characteristic
152	C₂	C₂×#11758491	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×D₂₄₈

C₂

yes

maximal, characteristic

C₂×#11758470

C₂

yes

maximal, complemented

#14521160

C₂

yes

maximal, direct product factor

C₂×#11758397

C₂

yes

maximal, complemented

C₂×#11758403

C₂

yes

maximal, characteristic, complemented

C₂×C₃₁×#106

C₂

yes

maximal, characteristic, complemented

C₂²×D₁₂₄

C₂²

yes

—

C₂²×C₄×D₃₁

—

C₂²×C₈×C₃₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₃₁

C₂²

yes

—

C₂×(C₂×D₃₁)⋊C₄

—

C₂×C₄×Dic₃₁

C₂²

yes

characteristic

C₂²×C₃₁⋊C₈

C₂²

yes

characteristic

C₂×D₂₄₈

C₄

yes

complemented

C₂×D₂₄₈

C₂²

yes

—

#11758470

—

#11758470

C₂²

yes

complemented

C₃₁×#106

C₂²

yes

complemented

#11758403

C₂²

yes

complemented

#11758397

C₂²

yes

complemented

C₂³×D₃₁

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×Dic₃₁

D₄

yes

—

C₂×D₁₂₄

—

C₂×D₁₂₄

—

C₂×D₁₂₄

C₂×C₄

yes

complemented

C₂×D₁₂₄

C₂³

yes

—

C₂×C₄×D₃₁

—

C₂×C₄×D₃₁

—

C₂×C₈×C₃₁

C₂×C₄

yes

complemented

C₂×C₈×C₃₁

C₂³

yes

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

C₂³

yes

—

(C₂×D₃₁)⋊C₄

—

C₄×Dic₃₁

D₄

yes

—

C₄×Dic₃₁

C₂³

yes

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

C₂³

yes

—

D₂₄₈

C₂×C₄

yes

complemented

C₂³×C₃₁

C₂×D₄

yes

characteristic

C₂²×D₃₁

—

C₂²×D₃₁

—

C₂²×D₃₁

—

C₂²×D₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂²×C₄

yes

—

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

#41

yes

—

C₂×C₄×C₃₁

C₂⁴

yes

characteristic

C₂×Dic₃₁

—

C₂×Dic₃₁

C₂×D₄

yes

—

D₁₂₄

—

D₁₂₄

—

D₁₂₄

C₂²×C₄

yes

complemented

C₄×D₃₁

—

C₈×C₃₁

C₂²×C₄

yes

complemented

C₃₁⋊C₈

—

C₂×#442

—

maximal, Sylow

C₂²×C₃₁

C₄×D₄

yes

—

C₂²×C₃₁

AGL(1,ℤ/8)

yes

—

C₂²×C₃₁

C₂²×D₄

yes

characteristic

C₂×D₃₁

—

C₂×D₃₁

—

C₂×D₃₁

—

C₂×D₃₁

—

C₄×C₃₁

—

C₄×C₃₁

C₄×D₄

yes

—

C₄×C₃₁

C₂³×C₄

yes

characteristic

C₄×C₃₁

C₂×#41

yes

characteristic

Dic₃₁

—

C₂²×D₈

—

C₂×C₄×D₄

—

#442

—

C₂×#106

D₃₁

yes

characteristic, complemented

C₂×#102

—

C₂×#97

—

C₂×C₃₁

#442

yes

complemented

C₂×C₃₁

C₂×C₄×D₄

yes

characteristic

C₂×C₃₁

C₂×AGL(1,ℤ/8)

yes

—

D₃₁

—

C₂²×D₄

—

C₂³×C₄

—

C₂×D₈

—

C₂×D₈

—

C₂²×C₈

—

C₂²×C₈

C₂×D₃₁

yes

characteristic

C₄×D₄

—

C₄×D₄

—

C₂×C₄⋊C₄

C₂×D₃₁

yes

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

#106

C₂×D₃₁

yes

complemented

#102

—

#97

—

C₃₁

C₂×#442

yes

characteristic, complemented, Sylow

C₂⁴

—

C₂×D₄

—

100

C₂×D₄

—

101

C₂×D₄

—

102

C₂×D₄

—

103

C₂²×C₄

—

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

—

124

—

107

C₂²×C₄

C₂²×D₃₁

yes

characteristic

108

D₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

C₄×D₃₁

yes

complemented

112

C₂×C₈

C₂²×D₃₁

yes

—

113

C₄⋊C₄

—

114

C₄⋊C₄

C₂²×D₃₁

yes

—

115

C₂²⋊C₄

—

116

C₄²

—

117

C₄²

—

118

C₂³

—

119

C₂³

—

120

C₂³

—

124

—

121

C₂³

—

124

—

122

C₂³

D₄×D₃₁

yes

characteristic

123

D₄

—

124

D₄

—

125

D₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

124

—

131

C₂×C₄

C₂×C₄×D₃₁

yes

—

132

C₂×C₄

D₄×D₃₁

yes

characteristic

133

C₂×C₄

#92499

yes

—

134

C₂×C₄

C₂³×D₃₁

yes

characteristic

135

C₈

—

136

C₈

C₂×C₄×D₃₁

yes

complemented

137

C₂²

—

138

C₂²

—

124

—

139

C₂²

—

124

—

140

C₂²

—

124

—

141

C₂²

#11758470

yes

—

142

C₂²

#11758491

yes

—

143

C₂²

C₂×D₄×D₃₁

yes

characteristic

144

C₄

—

145

C₄

—

146

C₄

#11758470

yes

—

147

C₄

C₂²×C₄×D₃₁

yes

characteristic

148

C₄

C₂×#92499

yes

characteristic

149

C₂

—

124

—

150

C₂

#14521160

yes

direct product factor

151

C₂

C₂×#11758470

yes

characteristic

152

C₂

C₂×#11758491

yes

—

	subgroup	G/H
center	C₂³	D₄×D₃₁
commutator	C₄×C₃₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₂×C₃₁×#106	C₂
socle	C₂³×C₃₁	C₂×D₄
prime radical	C₂²×D₂₄₈	C₂
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	C₂×#106	D₃₁
hypocenter	C₃₁	C₂×#442
LCS	G ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

subgroup

G/H

center

C₂³

D₄×D₃₁

commutator

C₄×C₃₁

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₃₁

Fitting

C₂×C₃₁×#106

C₂

socle

C₂³×C₃₁

C₂×D₄

prime radical

C₂²×D₂₄₈

C₂

cosocle

C₂×C₄×C₃₁

C₂⁴

D₃₁

C₃₁

G ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

N	H ≅ G/N	type
2. C₂×#11758470	C₂	semidirect
3. #14521160	C₂	direct (N × H)
4. C₂×#11758397	C₂	semidirect
5. C₂×#11758403	C₂	semidirect
6. C₂×C₃₁×#106	C₂	semidirect
14. C₂×D₂₄₈	C₄	semidirect
17. #11758470	C₂²	semidirect
18. C₃₁×#106	C₂²	semidirect
19. #11758403	C₂²	semidirect
20. #11758397	C₂²	semidirect
27. C₂×D₁₂₄	C₂×C₄	semidirect
31. C₂×C₈×C₃₁	C₂×C₄	semidirect
40. D₂₄₈	C₂×C₄	semidirect
56. D₁₂₄	C₂²×C₄	semidirect
58. C₈×C₃₁	C₂²×C₄	semidirect
76. C₂×#106	D₃₁	semidirect
79. C₂×C₃₁	#442	semidirect
94. #106	C₂×D₃₁	semidirect
97. C₃₁	C₂×#442	semidirect
111. C₂×C₈	C₄×D₃₁	semidirect
136. C₈	C₂×C₄×D₃₁	semidirect
150. C₂	#14521160	direct (N × H)

H ≅ G/N

type

2. C₂×#11758470

C₂

semidirect

3. #14521160

C₂

direct (N × H)

4. C₂×#11758397

C₂

semidirect

5. C₂×#11758403

C₂

semidirect

6. C₂×C₃₁×#106

C₂

semidirect

14. C₂×D₂₄₈

C₄

semidirect

17. #11758470

C₂²

semidirect

18. C₃₁×#106

C₂²

semidirect

19. #11758403

C₂²

semidirect

20. #11758397

C₂²

semidirect

27. C₂×D₁₂₄

C₂×C₄

semidirect

31. C₂×C₈×C₃₁

C₂×C₄

semidirect

40. D₂₄₈

C₂×C₄

semidirect

56. D₁₂₄

C₂²×C₄

semidirect

58. C₈×C₃₁

C₂²×C₄

semidirect

76. C₂×#106

D₃₁

semidirect

79. C₂×C₃₁

#442

semidirect

94. #106

C₂×D₃₁

semidirect

97. C₃₁

C₂×#442

semidirect

111. C₂×C₈

C₄×D₃₁

semidirect

136. C₈

C₂×C₄×D₃₁

semidirect

150. C₂

#14521160

direct (N × H)

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

subgroup	index
C₂²×D₂₄₈	2
C₂×#11758470	2
#14521160	2
C₂×#11758397	2
C₂×#11758403	2
C₂×C₃₁×#106	2
C₂×#442	31

subgroup

index

C₂²×D₂₄₈

C₂×C₃₁×#106

Subgroups — C₂×#14521160 #16854503

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#14521160 #16854503

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14521160 #16854503