Group #19958 — subgroups

Subgroups — #19958

Order 256.

285 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 285 conjugacy classes of subgroups — too many to draw. Only its 101 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	#3360	C₂	1	2	yes	maximal
2	#3356	C₂	1	2	yes	maximal, complemented
3	#1707	C₂	1	1	yes	maximal, characteristic, complemented
4	#1691	C₂	1	2	yes	maximal, complemented
5	#1684	C₂	1	2	yes	maximal, complemented
6	#1650	C₂	1	2	yes	maximal, complemented
7	#1632	C₂	1	2	yes	maximal, complemented
8	#1589	C₂	1	1	yes	maximal, characteristic
9	C₈×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
10	#556	C₂²	1	2	yes	—
11	#554	C₂²	1	2	yes	—
12	#500	C₂²	1	1	yes	characteristic
13	#498	C₂²	1	2	yes	—
14	#494	C₂²	1	1	yes	characteristic
15	#488	—	2	2	no	—
16	#487	C₂²	1	2	yes	—
17	#485	—	2	2	no	—
18	#483	—	2	2	no	—
19	#479	—	2	2	no	—
20	#477	C₂²	1	2	yes	—
21	#476	C₂²	1	2	yes	—
22	#475	C₂²	1	2	yes	—
23	#470	—	2	2	no	—
24	#463	—	2	2	no	—
25	#461	C₂²	1	2	yes	—
26	C₈×Q₈	C₂²	1	2	yes	—
27	C₄×Q₁₆	C₂²	1	1	yes	characteristic
28	C₄×SD₁₆	—	2	2	no	—
29	C₄×D₈	C₂²	1	1	yes	characteristic
30	#436	C₂²	1	2	yes	—
31	#428	—	2	2	no	—
32	#335	C₂²	1	1	yes	characteristic
33	#335	C₂²	1	1	yes	characteristic
34	#333	C₂²	1	2	yes	—
35	#332	C₂²	1	2	yes	—
36	Q₈⋊C₈	C₂²	1	2	yes	—
37	D₄⋊C₈	C₂²	1	2	yes	—
38	C₈²	C₂²	1	1	yes	characteristic
39	C₂×Q₁₆	—	4	1	no	—
40	C₂×Q₁₆	D₄	1	1	yes	characteristic
41	C₂×SD₁₆	—	2	2	no	—
42	C₂×D₈	D₄	1	1	yes	characteristic
43	C₂×M₁₆	—	2	2	no	—
44	C₄⋊Q₈	C₂³	1	2	yes	—
45	C₄⋊Q₈	C₂³	1	2	yes	—
46	#126	—	4	2	no	—
47	#125	—	2	2	no	—
48	#124	—	2	2	no	—
49	C₂²⋊Q₈	—	2	2	no	—
50	C₂²⋊Q₈	—	2	2	no	—
51	C₂²⋊Q₈	—	4	2	no	—
52	C₄×Q₈	—	2	2	no	—
53	C₄×Q₈	C₂³	1	2	yes	—
54	C₄×D₄	—	2	2	no	—
55	C₄×D₄	C₂³	1	2	yes	—
56	#117	—	2	2	no	—
57	C₂×C₄⋊C₄	—	2	2	no	—
58	#107	—	2	2	no	—
59	#107	—	4	1	no	—
60	#107	D₄	1	1	yes	characteristic
61	#107	D₄	1	1	yes	characteristic
62	#106	—	2	2	no	—
63	#106	—	2	2	no	—
64	#106	—	2	2	no	—
65	#106	—	2	2	no	—
66	C₄⋊C₈	—	2	2	no	—
67	C₄⋊C₈	C₂³	1	2	yes	—
68	C₄⋊C₈	C₂³	1	2	yes	—
69	#103	—	2	2	no	—
70	#103	—	2	2	no	—
71	#103	—	2	2	no	—
72	#103	—	2	2	no	—
73	#103	—	2	2	no	—
74	#103	D₄	1	2	yes	—
75	#102	—	2	2	no	—
76	#102	D₄	1	2	yes	—
77	C₂²⋊C₈	—	2	2	no	—
78	C₄×C₈	—	2	2	no	—
79	C₄×C₈	C₂³	1	1	yes	characteristic
80	C₄×C₈	C₂³	1	1	yes	characteristic
81	C₄×C₈	C₂³	1	1	yes	characteristic
82	C₂×Q₈	—	2	2	no	—
83	C₂×Q₈	—	2	2	no	—
84	C₂×Q₈	C₂×D₄	1	2	yes	—
85	C₂×D₄	C₂×D₄	1	2	yes	—
86	C₂²×C₄	—	2	2	no	—
87	C₂²×C₄	—	4	2	no	—
88	Q₁₆	—	2	2	no	—
89	Q₁₆	—	8	2	no	—
90	SD₁₆	—	4	4	no	—
91	D₈	—	4	1	no	—
92	M₁₆	—	4	4	no	—
93	C₂×C₈	—	2	1	no	—
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈	C₂×D₄	1	1	yes	characteristic
100	C₄⋊C₄	—	2	2	no	—
101	C₄⋊C₄	—	2	2	no	—
102	C₄⋊C₄	—	2	2	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	2	2	no	—
105	C₄⋊C₄	—	4	2	no	—
106	C₄⋊C₄	—	4	2	no	—
107	C₄⋊C₄	—	4	2	no	—
108	C₄⋊C₄	C₂×D₄	1	2	yes	—
109	C₄⋊C₄	C₂×D₄	1	2	yes	—
110	C₂²⋊C₄	—	2	2	no	—
111	C₂²⋊C₄	—	4	2	no	—
112	C₂²⋊C₄	—	4	2	no	—
113	C₂²⋊C₄	—	4	2	no	—
114	C₄²	—	2	2	no	—
115	C₄²	—	4	2	no	—
116	C₄²	C₂⁴	1	1	yes	characteristic
117	C₂³	—	2	2	no	—
118	Q₈	—	2	2	no	—
119	Q₈	—	2	2	no	—
120	Q₈	—	4	2	no	—
121	Q₈	—	4	2	no	—
122	Q₈	—	8	2	no	—
123	D₄	—	2	2	no	—
124	D₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	4	2	no	—
129	C₂×C₄	—	4	2	no	—
130	C₂×C₄	—	4	2	no	—
131	C₂×C₄	—	4	2	no	—
132	C₂×C₄	—	16	2	no	—
133	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
135	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
136	C₈	—	2	2	no	—
137	C₈	—	2	1	no	—
138	C₈	—	2	1	no	—
139	C₈	—	2	1	no	—
140	C₈	—	4	2	no	—
141	C₈	—	4	2	no	—
142	C₈	#135	1	2	yes	—
143	C₂²	—	4	2	no	—
144	C₂²	—	4	2	no	—
145	C₂²	—	4	2	no	—
146	C₂²	D₄²	1	1	yes	characteristic
147	C₄	—	2	1	no	—
148	C₄	—	4	2	no	—
149	C₄	—	4	2	no	—
150	C₄	—	4	2	no	—
151	C₄	—	8	2	no	—
152	C₄	—	8	2	no	—
153	C₄	C₂×#135	1	1	yes	characteristic
154	C₄	#575	1	1	yes	characteristic
155	C₄	#578	1	1	yes	characteristic
156	C₄	#578	1	1	yes	characteristic
157	C₂	—	8	2	no	—
158	C₂	#3277	1	1	yes	characteristic
159	C₂	#3283	1	1	yes	characteristic
160	C₂	#3291	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#3360

C₂

yes

maximal

#3356

C₂

yes

maximal, complemented

#1707

C₂

yes

maximal, characteristic, complemented

#1691

C₂

yes

maximal, complemented

#1684

C₂

yes

maximal, complemented

#1650

C₂

yes

maximal, complemented

#1632

C₂

yes

maximal, complemented

#1589

C₂

yes

maximal, characteristic

C₈×Q₁₆

C₂

yes

maximal, characteristic, complemented

#556

C₂²

yes

—

#554

C₂²

yes

—

#500

C₂²

yes

characteristic

#498

C₂²

yes

—

#494

C₂²

yes

characteristic

#488

—

#487

C₂²

yes

—

—

—

—

C₂²

yes

—

#476

C₂²

yes

—

#475

C₂²

yes

—

#470

—

#463

—

#461

C₂²

yes

—

C₈×Q₈

C₂²

yes

—

C₄×Q₁₆

C₂²

yes

characteristic

C₄×SD₁₆

—

C₄×D₈

C₂²

yes

characteristic

#436

C₂²

yes

—

#428

—

#335

C₂²

yes

characteristic

#335

C₂²

yes

characteristic

#333

C₂²

yes

—

#332

C₂²

yes

—

Q₈⋊C₈

C₂²

yes

—

D₄⋊C₈

C₂²

yes

—

C₈²

C₂²

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

D₄

yes

characteristic

C₂×SD₁₆

—

C₂×D₈

D₄

yes

characteristic

C₂×M₁₆

—

C₄⋊Q₈

C₂³

yes

—

C₄⋊Q₈

C₂³

yes

—

#126

—

#125

—

#124

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

C₂³

yes

—

C₄×D₄

—

C₄×D₄

C₂³

yes

—

#117

—

C₂×C₄⋊C₄

—

#107

—

#107

—

#107

D₄

yes

characteristic

#107

D₄

yes

—

—

—

—

C₄⋊C₈

—

C₄⋊C₈

C₂³

yes

—

C₄⋊C₈

C₂³

yes

—

—

—

—

—

—

D₄

yes

—

#102

—

#102

D₄

yes

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

C₂³

yes

characteristic

C₄×C₈

C₂³

yes

characteristic

C₄×C₈

C₂³

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×D₄

yes

—

C₂×D₄

yes

—

C₂²×C₄

—

C₂²×C₄

—

Q₁₆

—

Q₁₆

—

SD₁₆

—

D₈

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

characteristic

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

C₂×D₄

yes

—

109

C₄⋊C₄

C₂×D₄

yes

—

110

C₂²⋊C₄

—

111

C₂²⋊C₄

—

112

C₂²⋊C₄

—

113

C₂²⋊C₄

—

114

C₄²

—

115

C₄²

—

116

C₄²

C₂⁴

yes

characteristic

117

C₂³

—

118

Q₈

—

119

Q₈

—

120

Q₈

—

121

Q₈

—

122

Q₈

—

123

D₄

—

124

D₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

C₂²×D₄

yes

characteristic

134

C₂×C₄

C₂²×D₄

yes

characteristic

135

C₂×C₄

H(5,𝔽₂)

yes

characteristic

136

C₈

—

137

C₈

—

138

C₈

—

139

C₈

—

140

C₈

—

141

C₈

—

142

C₈

#135

yes

—

143

C₂²

—

144

C₂²

—

145

C₂²

—

146

C₂²

D₄²

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

C₂×#135

yes

characteristic

154

C₄

#575

yes

characteristic

155

C₄

#578

yes

characteristic

156

C₄

#578

yes

characteristic

157

C₂

—

158

C₂

#3277

yes

characteristic

159

C₂

#3283

yes

characteristic

160

C₂

#3291

yes

characteristic

	subgroup	G/H
center, socle	C₂²	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴
prime radical	C₂×D₈	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center, socle

C₂²

D₄²

commutator, Frattini, cosocle

C₄²

C₂⁴

prime radical

C₂×D₈

D₄

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ G

LECS

G ▷ C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
2. #3356	C₂	semidirect
3. #1707	C₂	semidirect
4. #1691	C₂	semidirect
5. #1684	C₂	semidirect
6. #1650	C₂	semidirect
7. #1632	C₂	semidirect
9. C₈×Q₁₆	C₂	semidirect

H ≅ G/N

type

2. #3356

C₂

semidirect

3. #1707

C₂

semidirect

4. #1691

C₂

semidirect

5. #1684

C₂

semidirect

6. #1650

C₂

semidirect

7. #1632

C₂

semidirect

9. C₈×Q₁₆

C₂

semidirect

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

subgroup	index
#3360	2
#3356	2
#1707	2
#1691	2
#1684	2
#1650	2
#1632	2
#1589	2
C₈×Q₁₆	2

subgroup

index

C₈×Q₁₆

Subgroups — #19958

Conjugacy classes of subgroups

Complements

Maximal subgroups