Group #19690 — subgroups

Subgroups — #19690

Order 256.

328 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 328 conjugacy classes of subgroups — too many to draw. Only its 119 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	#3231	C₂	1	2	yes	maximal
2	#3185	C₂	1	2	yes	maximal
3	#3170	C₂	1	1	yes	maximal, characteristic
4	#3160	C₂	1	1	yes	maximal, characteristic
5	#1723	C₂	1	2	yes	maximal, complemented
6	#1723	C₂	1	2	yes	maximal, complemented
7	#1718	C₂	1	2	yes	maximal, complemented
8	#1716	C₂	1	2	yes	maximal, complemented
9	#1451	C₂	1	1	yes	maximal, characteristic
10	#533	C₂²	1	2	yes	—
11	#533	C₂²	1	2	yes	—
12	#500	C₂²	1	1	yes	characteristic
13	#500	C₂²	1	1	yes	characteristic
14	#500	C₂²	1	1	yes	characteristic
15	#500	C₂²	1	2	yes	—
16	#499	—	2	2	no	—
17	#498	C₂²	1	2	yes	—
18	#498	C₂²	1	2	yes	—
19	#498	C₂²	1	2	yes	—
20	#494	C₂²	1	1	yes	characteristic
21	#494	C₂²	1	2	yes	—
22	#487	C₂²	1	2	yes	—
23	#487	C₂²	1	2	yes	—
24	#487	C₂²	1	2	yes	—
25	#487	C₂²	1	2	yes	—
26	#487	C₂²	1	2	yes	—
27	#485	—	2	4	no	—
28	#479	—	2	4	no	—
29	#470	—	2	4	no	—
30	#467	—	2	2	no	—
31	#441	—	2	4	no	—
32	C₄×Q₁₆	—	2	2	no	—
33	#431	C₂²	1	1	yes	characteristic
34	#427	—	2	2	no	—
35	C₄×M₁₆	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
37	#322	C₂²	1	2	yes	—
38	#322	C₂²	1	2	yes	—
39	C₂×Q₁₆	—	2	2	no	—
40	C₂×Q₁₆	—	2	4	no	—
41	C₂×M₁₆	D₄	1	2	yes	—
42	C₂²×C₈	D₄	1	2	yes	—
43	C₄⋊Q₈	C₂³	1	2	yes	—
44	C₄⋊Q₈	C₂³	1	2	yes	—
45	C₄⋊Q₈	C₂³	1	2	yes	—
46	C₄⋊Q₈	C₂³	1	2	yes	—
47	#125	—	2	2	no	—
48	#125	—	2	2	no	—
49	C₂²⋊Q₈	—	2	2	no	—
50	C₂²⋊Q₈	—	2	2	no	—
51	C₂²⋊Q₈	—	2	2	no	—
52	C₂²⋊Q₈	—	2	2	no	—
53	C₄×Q₈	—	2	2	no	—
54	C₄×Q₈	—	2	2	no	—
55	C₄×Q₈	—	2	2	no	—
56	C₄×Q₈	—	2	2	no	—
57	#117	—	2	2	no	—
58	#117	—	2	2	no	—
59	C₂×C₄²	C₂³	1	1	yes	characteristic
60	#107	—	2	2	no	—
61	#107	—	2	2	no	—
62	#107	—	2	2	no	—
63	#107	—	2	4	no	—
64	#106	—	2	4	no	—
65	#106	—	2	4	no	—
66	#106	—	2	4	no	—
67	C₄⋊C₈	D₄	1	2	yes	—
68	#103	—	2	4	no	—
69	#103	—	2	4	no	—
70	#103	—	2	4	no	—
71	#103	—	2	4	no	—
72	#103	—	2	4	no	—
73	C₂²⋊C₈	D₄	1	2	yes	—
74	#97	D₄	1	2	yes	—
75	C₄×C₈	D₄	1	2	yes	—
76	C₄×C₈	C₂³	1	1	yes	characteristic
77	C₄×C₈	C₂³	1	1	yes	characteristic
78	C₄×C₈	C₂³	1	2	yes	—
79	C₄×C₈	C₂³	1	2	yes	—
80	C₂×Q₈	—	2	2	no	—
81	C₂×Q₈	—	2	2	no	—
82	C₂×Q₈	—	2	2	no	—
83	C₂×Q₈	—	2	2	no	—
84	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
87	Q₁₆	—	4	8	no	—
88	Q₁₆	—	4	4	no	—
89	M₁₆	—	4	2	no	—
90	C₂×C₈	—	2	4	no	—
91	C₂×C₈	C₂×D₄	1	2	yes	—
92	C₂×C₈	C₂×D₄	1	4	yes	—
93	C₂×C₈	C₂×D₄	1	4	yes	—
94	C₂×C₈	C₂×D₄	1	2	yes	—
95	C₄⋊C₄	—	2	2	no	—
96	C₄⋊C₄	—	2	2	no	—
97	C₄⋊C₄	—	2	2	no	—
98	C₄⋊C₄	—	2	2	no	—
99	C₄⋊C₄	—	2	2	no	—
100	C₄⋊C₄	—	2	2	no	—
101	C₄⋊C₄	—	2	2	no	—
102	C₄⋊C₄	—	2	2	no	—
103	C₄⋊C₄	—	4	2	no	—
104	C₄⋊C₄	—	4	2	no	—
105	C₄⋊C₄	—	4	2	no	—
106	C₄⋊C₄	—	4	2	no	—
107	C₂²⋊C₄	—	4	2	no	—
108	C₂²⋊C₄	—	4	2	no	—
109	C₄²	—	4	2	no	—
110	C₄²	—	4	2	no	—
111	C₄²	C₂×D₄	1	1	yes	characteristic
112	C₄²	C₂×D₄	1	1	yes	characteristic
113	C₄²	C₂×D₄	1	1	yes	characteristic
114	C₄²	C₂⁴	1	1	yes	characteristic
115	C₂³	#127	1	1	yes	characteristic
116	Q₈	—	4	4	no	—
117	Q₈	—	4	4	no	—
118	Q₈	—	4	4	no	—
119	Q₈	—	4	4	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	—	4	2	no	—
125	C₂×C₄	—	4	2	no	—
126	C₂×C₄	—	4	2	no	—
127	C₂×C₄	#127	1	1	yes	characteristic
128	C₂×C₄	#127	1	1	yes	characteristic
129	C₂×C₄	#127	1	1	yes	characteristic
130	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
133	C₈	—	2	2	no	—
134	C₈	—	2	4	no	—
135	C₈	—	2	4	no	—
136	C₈	#137	1	4	yes	—
137	C₂²	—	2	1	no	—
138	C₂²	—	2	1	no	—
139	C₂²	—	2	1	no	—
140	C₂²	C₂×#127	1	1	yes	characteristic
141	C₄	—	2	1	no	—
142	C₄	—	2	1	no	—
143	C₄	—	2	1	no	—
144	C₄	—	8	2	no	—
145	C₄	—	8	2	no	—
146	C₄	—	8	2	no	—
147	C₄	—	8	2	no	—
148	C₄	C₂×#137	1	2	yes	—
149	C₄	#575	1	2	yes	—
150	C₄	#578	1	2	yes	—
151	C₂	—	4	1	no	—
152	C₂	#3150	1	1	yes	characteristic
153	C₂	#3152	1	1	yes	characteristic
154	C₂	#3156	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#3231

C₂

yes

maximal

#3185

C₂

yes

maximal

#3170

C₂

yes

maximal, characteristic

#3160

C₂

yes

maximal, characteristic

#1723

C₂

yes

maximal, complemented

#1723

C₂

yes

maximal, complemented

#1718

C₂

yes

maximal, complemented

#1716

C₂

yes

maximal, complemented

#1451

C₂

yes

maximal, characteristic

#533

C₂²

yes

—

#533

C₂²

yes

—

#500

C₂²

yes

characteristic

#500

C₂²

yes

characteristic

#500

C₂²

yes

characteristic

#500

C₂²

yes

—

#499

—

#498

C₂²

yes

—

#498

C₂²

yes

—

#498

C₂²

yes

—

#494

C₂²

yes

characteristic

#494

C₂²

yes

—

#487

C₂²

yes

—

#487

C₂²

yes

—

#487

C₂²

yes

—

#487

C₂²

yes

—

#487

C₂²

yes

—

—

—

—

—

—

C₄×Q₁₆

—

#431

C₂²

yes

characteristic

#427

—

C₄×M₁₆

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

#322

C₂²

yes

—

#322

C₂²

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×M₁₆

D₄

yes

—

C₂²×C₈

D₄

yes

—

C₄⋊Q₈

C₂³

yes

—

C₄⋊Q₈

C₂³

yes

—

C₄⋊Q₈

C₂³

yes

—

C₄⋊Q₈

C₂³

yes

—

#125

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

#117

—

C₂×C₄²

C₂³

yes

—

—

—

—

—

—

—

C₄⋊C₈

D₄

yes

—

—

—

—

—

—

C₂²⋊C₈

D₄

yes

—

#97

D₄

yes

—

C₄×C₈

D₄

yes

—

C₄×C₈

C₂³

yes

characteristic

C₄×C₈

C₂³

yes

characteristic

C₄×C₈

C₂³

yes

—

C₄×C₈

C₂³

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

Q₁₆

—

Q₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×D₄

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₂²⋊C₄

—

108

C₂²⋊C₄

—

109

C₄²

—

110

C₄²

—

111

C₄²

C₂×D₄

yes

characteristic

112

C₄²

C₂×D₄

yes

characteristic

113

C₄²

C₂×D₄

yes

characteristic

114

C₄²

C₂⁴

yes

characteristic

115

C₂³

#127

yes

characteristic

116

Q₈

—

117

Q₈

—

118

Q₈

—

119

Q₈

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

#127

yes

characteristic

128

C₂×C₄

#127

yes

characteristic

129

C₂×C₄

#127

yes

characteristic

130

C₂×C₄

C₂²×D₄

yes

characteristic

131

C₂×C₄

C₂²×D₄

yes

characteristic

132

C₂×C₄

C₂²×D₄

yes

characteristic

133

C₈

—

134

C₈

—

135

C₈

—

136

C₈

#137

yes

—

137

C₂²

—

138

C₂²

—

139

C₂²

—

140

C₂²

C₂×#127

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

C₂×#137

yes

—

149

C₄

#575

yes

—

150

C₄

#578

yes

—

151

C₂

—

152

C₂

#3150

yes

characteristic

153

C₂

#3152

yes

characteristic

154

C₂

#3156

yes

characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴
prime radical	C₂³	#127
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center, socle

C₂²

C₂×#127

commutator, Frattini, cosocle

C₄²

C₂⁴

C₂³

G ▷ C₄² ▷ C₂² ▷ C₁

C₁ ◁ C₂² ◁ C₂×C₄² ◁ G

LECS

G ▷ C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
5. #1723	C₂	semidirect
6. #1723	C₂	semidirect
7. #1718	C₂	semidirect
8. #1716	C₂	semidirect

H ≅ G/N

type

5. #1723

C₂

semidirect

6. #1723

C₂

semidirect

7. #1718

C₂

semidirect

8. #1716

C₂

semidirect

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

subgroup	index
#3231	2
#3185	2
#3170	2
#3160	2
#1723	2
#1718	2
#1716	2
#1451	2

subgroup

index

Subgroups — #19690

Conjugacy classes of subgroups

Complements

Maximal subgroups