Group #19927 — subgroups

Subgroups — #19927

Order 256.

303 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 303 conjugacy classes of subgroups — too many to draw. Only its 109 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	#3384	C₂	1	2	yes	maximal
2	#3299	C₂	1	2	yes	maximal, complemented
3	#1707	C₂	1	1	yes	maximal, characteristic, complemented
4	#1692	C₂	1	2	yes	maximal, complemented
5	#1683	C₂	1	2	yes	maximal, complemented
6	#1649	C₂	1	2	yes	maximal, complemented
7	#1632	C₂	1	2	yes	maximal, complemented
8	#1591	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈×D₈	C₂	1	1	yes	maximal, characteristic
10	#554	C₂²	1	2	yes	—
11	D₄×Q₈	C₂²	1	2	yes	—
12	#501	—	2	2	no	—
13	#500	C₂²	1	1	yes	characteristic
14	#498	C₂²	1	2	yes	—
15	#494	C₂²	1	1	yes	characteristic
16	#487	C₂²	1	2	yes	—
17	#484	—	2	2	no	—
18	#478	—	2	2	no	—
19	#477	C₂²	1	2	yes	—
20	#476	—	2	2	no	—
21	#476	C₂²	1	2	yes	—
22	#474	C₂²	1	2	yes	—
23	#467	—	2	2	no	—
24	#462	C₂²	1	2	yes	—
25	#451	—	2	2	no	—
26	#446	C₂²	1	2	yes	—
27	#442	—	2	2	no	—
28	C₄×Q₁₆	C₂²	1	1	yes	characteristic
29	C₄×D₈	C₂²	1	1	yes	characteristic
30	C₈×D₄	C₂²	1	2	yes	—
31	C₂×#103	—	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	C₂²×Q₈	—	2	2	no	—
40	C₂×Q₁₆	—	2	2	no	—
41	C₂×Q₁₆	—	4	1	no	—
42	C₂×Q₁₆	D₄	1	1	yes	characteristic
43	C₂×D₈	D₄	1	1	yes	characteristic
44	C₂²×C₈	—	2	2	no	—
45	C₄⋊Q₈	—	2	2	no	—
46	C₄⋊Q₈	C₂³	1	2	yes	—
47	C₄⋊Q₈	C₂³	1	2	yes	—
48	#125	—	2	2	no	—
49	C₂²⋊Q₈	—	2	2	no	—
50	C₂²⋊Q₈	—	2	2	no	—
51	C₂²⋊Q₈	—	4	2	no	—
52	C₂²⋊Q₈	—	4	2	no	—
53	C₄×Q₈	C₂³	1	2	yes	—
54	C₄×D₄	—	2	2	no	—
55	C₄×D₄	—	2	2	no	—
56	C₄×D₄	C₂³	1	2	yes	—
57	C₂×C₄⋊C₄	—	2	2	no	—
58	#107	—	2	2	no	—
59	#107	—	2	2	no	—
60	#107	—	4	1	no	—
61	#107	D₄	1	1	yes	characteristic
62	#107	D₄	1	1	yes	characteristic
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	D₄	1	2	yes	—
74	#102	—	2	2	no	—
75	#102	—	2	2	no	—
76	#102	D₄	1	2	yes	—
77	C₂²⋊C₈	—	2	2	no	—
78	#97	—	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	—	4	2	no	—
85	C₂×Q₈	—	4	4	no	—
86	C₂×Q₈	—	8	2	no	—
87	C₂×Q₈	C₂×D₄	1	2	yes	—
88	C₂×D₄	C₂×D₄	1	2	yes	—
89	C₂²×C₄	—	2	2	no	—
90	C₂²×C₄	—	4	2	no	—
91	C₂²×C₄	—	4	2	no	—
92	Q₁₆	—	4	1	no	—
93	Q₁₆	—	8	2	no	—
94	Q₁₆	—	8	2	no	—
95	D₈	Q₁₆	1	4	yes	complemented
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	—	2	2	no	—
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	8	2	no	—
100	C₂×C₈	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈	C₂×D₄	1	1	yes	characteristic
104	C₄⋊C₄	—	2	2	no	—
105	C₄⋊C₄	—	2	2	no	—
106	C₄⋊C₄	—	2	2	no	—
107	C₄⋊C₄	—	2	2	no	—
108	C₄⋊C₄	—	2	2	no	—
109	C₄⋊C₄	—	4	2	no	—
110	C₄⋊C₄	—	4	2	no	—
111	C₄⋊C₄	—	4	2	no	—
112	C₄⋊C₄	C₂×D₄	1	2	yes	—
113	C₄⋊C₄	C₂×D₄	1	2	yes	—
114	C₂²⋊C₄	—	2	2	no	—
115	C₂²⋊C₄	—	4	2	no	—
116	C₂²⋊C₄	—	4	2	no	—
117	C₄²	—	2	2	no	—
118	C₄²	C₂⁴	1	1	yes	characteristic
119	C₂³	—	2	2	no	—
120	Q₈	—	2	2	no	—
121	Q₈	—	2	2	no	—
122	Q₈	—	4	2	no	—
123	Q₈	—	4	2	no	—
124	Q₈	—	8	2	no	—
125	Q₈	—	8	4	no	—
126	D₄	—	2	2	no	—
127	D₄	C₂×Q₁₆	1	4	yes	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	4	2	no	—
132	C₂×C₄	—	4	2	no	—
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	—	4	2	no	—
135	C₂×C₄	—	8	4	no	—
136	C₂×C₄	—	16	2	no	—
137	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
138	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
140	C₈	—	2	2	no	—
141	C₈	—	2	1	no	—
142	C₈	—	2	1	no	—
143	C₈	—	2	1	no	—
144	C₈	—	4	2	no	—
145	C₈	—	4	2	no	—
146	C₈	C₂×Q₁₆	1	2	yes	—
147	C₂²	—	2	4	no	—
148	C₂²	—	4	2	no	—
149	C₂²	—	4	2	no	—
150	C₂²	D₄²	1	1	yes	characteristic
151	C₄	—	2	1	no	—
152	C₄	—	4	2	no	—
153	C₄	—	4	2	no	—
154	C₄	—	4	2	no	—
155	C₄	—	8	2	no	—
156	C₄	—	8	2	no	—
157	C₄	C₂²×Q₁₆	1	1	yes	characteristic
158	C₄	#575	1	1	yes	characteristic
159	C₄	#576	1	1	yes	characteristic
160	C₄	#578	1	1	yes	characteristic
161	C₂	—	4	4	no	—
162	C₂	#3273	1	1	yes	characteristic
163	C₂	D₄×Q₁₆	1	1	yes	characteristic
164	C₂	#3291	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#3384

C₂

yes

maximal

#3299

C₂

yes

maximal, complemented

#1707

C₂

yes

maximal, characteristic, complemented

#1692

C₂

yes

maximal, complemented

#1683

C₂

yes

maximal, complemented

#1649

C₂

yes

maximal, complemented

#1632

C₂

yes

maximal, complemented

#1591

C₂

yes

maximal, characteristic, complemented

C₈×D₈

C₂

yes

maximal, characteristic

#554

C₂²

yes

—

D₄×Q₈

C₂²

yes

—

#501

—

#500

C₂²

yes

characteristic

#498

C₂²

yes

—

#494

C₂²

yes

characteristic

#487

C₂²

yes

—

#484

—

#478

—

#477

C₂²

yes

—

#476

—

#476

C₂²

yes

—

#474

C₂²

yes

—

#467

—

#462

C₂²

yes

—

#451

—

#446

C₂²

yes

—

#442

—

C₄×Q₁₆

C₂²

yes

characteristic

C₄×D₈

C₂²

yes

characteristic

C₈×D₄

C₂²

yes

—

C₂×#103

—

#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

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

D₄

yes

characteristic

C₂×D₈

D₄

yes

characteristic

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂³

yes

—

C₄⋊Q₈

C₂³

yes

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

C₂³

yes

—

C₄×D₄

—

C₄×D₄

—

C₄×D₄

C₂³

yes

—

C₂×C₄⋊C₄

—

—

—

—

D₄

yes

characteristic

#107

D₄

yes

—

—

—

C₄⋊C₈

—

C₄⋊C₈

C₂³

yes

—

C₄⋊C₈

C₂³

yes

—

—

—

—

—

D₄

yes

—

#102

—

#102

—

#102

D₄

yes

—

C₂²⋊C₈

—

#97

—

C₄×C₈

C₂³

yes

characteristic

C₄×C₈

C₂³

yes

characteristic

C₄×C₈

C₂³

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×D₄

yes

—

C₂×D₄

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

D₈

Q₁₆

yes

complemented

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

C₂×D₄

yes

characteristic

101

C₂×C₈

C₂×D₄

yes

characteristic

102

C₂×C₈

C₂×D₄

yes

characteristic

103

C₂×C₈

C₂×D₄

yes

characteristic

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

C₂×D₄

yes

—

113

C₄⋊C₄

C₂×D₄

yes

—

114

C₂²⋊C₄

—

115

C₂²⋊C₄

—

116

C₂²⋊C₄

—

117

C₄²

—

118

C₄²

C₂⁴

yes

characteristic

119

C₂³

—

120

Q₈

—

121

Q₈

—

122

Q₈

—

123

Q₈

—

124

Q₈

—

125

Q₈

—

126

D₄

—

127

D₄

C₂×Q₁₆

yes

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

C₂²×D₄

yes

characteristic

138

C₂×C₄

C₂²×D₄

yes

characteristic

139

C₂×C₄

H(5,𝔽₂)

yes

characteristic

140

C₈

—

141

C₈

—

142

C₈

—

143

C₈

—

144

C₈

—

145

C₈

—

146

C₈

C₂×Q₁₆

yes

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

150

C₂²

D₄²

yes

characteristic

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

C₂²×Q₁₆

yes

characteristic

158

C₄

#575

yes

characteristic

159

C₄

#576

yes

characteristic

160

C₄

#578

yes

characteristic

161

C₂

—

162

C₂

#3273

yes

characteristic

163

C₂

D₄×Q₁₆

yes

characteristic

164

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. #3299	C₂	semidirect
3. #1707	C₂	semidirect
4. #1692	C₂	semidirect
5. #1683	C₂	semidirect
6. #1649	C₂	semidirect
7. #1632	C₂	semidirect
8. #1591	C₂	semidirect
95. D₈	Q₁₆	semidirect

H ≅ G/N

type

2. #3299

C₂

semidirect

3. #1707

C₂

semidirect

4. #1692

C₂

semidirect

5. #1683

C₂

semidirect

6. #1649

C₂

semidirect

7. #1632

C₂

semidirect

8. #1591

C₂

semidirect

95. D₈

Q₁₆

semidirect

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

subgroup	index
#3384	2
#3299	2
#1707	2
#1692	2
#1683	2
#1649	2
#1632	2
#1591	2
C₈×D₈	2

subgroup

index

C₈×D₈

Subgroups — #19927

Conjugacy classes of subgroups

Complements

Maximal subgroups