Group #27692 — subgroups

Subgroups — #27692

Order 256.

313 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 313 conjugacy classes of subgroups — too many to draw. Only its 117 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	#2930	C₂	1	1	yes	maximal, characteristic
2	#2916	C₂	1	1	yes	maximal, characteristic, complemented
3	#2103	C₂	1	2	yes	maximal, complemented
4	#2091	C₂	1	2	yes	maximal, complemented
5	#1953	C₂	1	2	yes	maximal, complemented
6	#1891	C₂	1	2	yes	maximal
7	#1858	C₂	1	2	yes	maximal, complemented
8	#1813	C₂	1	1	yes	maximal, characteristic, complemented
9	#1811	C₂	1	2	yes	maximal, complemented
10	#568	C₂²	1	2	yes	—
11	C₂×#131	C₂²	1	1	yes	characteristic
12	C₂²×M₁₆	C₂²	1	1	yes	characteristic
13	#525	C₂²	1	1	yes	characteristic
14	#436	—	2	2	no	—
15	#435	—	2	2	no	—
16	#430	—	2	2	no	—
17	#430	C₂²	1	2	yes	—
18	#430	C₂²	1	4	yes	—
19	C₂×#108	—	2	2	no	—
20	C₂×#108	C₂²	1	2	yes	—
21	C₂×#108	C₂²	1	2	yes	—
22	#424	C₂²	1	1	yes	characteristic
23	#413	C₂²	1	2	yes	—
24	#408	C₂²	1	1	yes	characteristic
25	#407	C₂²	1	1	yes	characteristic
26	#407	C₂²	1	4	yes	—
27	#405	C₂²	1	1	yes	characteristic
28	#344	C₂²	1	4	yes	—
29	#341	C₂²	1	4	yes	—
30	#338	C₂²	1	2	yes	—
31	C₂×#41	C₂³	1	1	yes	characteristic
32	C₂³×C₄	C₂³	1	1	yes	characteristic
33	#131	—	2	4	no	—
34	#131	—	4	4	no	—
35	C₂×M₁₆	—	2	1	no	—
36	C₂×M₁₆	—	2	1	no	—
37	C₂×M₁₆	—	2	4	no	—
38	C₂×M₁₆	—	2	4	no	—
39	C₂×M₁₆	D₄	1	2	yes	—
40	C₂×M₁₆	C₂³	1	4	yes	—
41	C₂×M₁₆	C₂³	1	4	yes	—
42	C₂×M₁₆	C₂³	1	1	yes	characteristic
43	C₂×M₁₆	C₂³	1	1	yes	characteristic
44	C₂²×C₈	D₄	1	2	yes	—
45	C₂²×C₈	C₂³	1	1	yes	characteristic
46	C₂²×C₈	C₂³	1	1	yes	characteristic
47	#123	Q₈	1	2	yes	—
48	C₂²⋊Q₈	—	2	1	no	—
49	#121	—	2	1	no	—
50	#120	Q₈	1	2	yes	—
51	C₄×D₄	—	2	2	no	—
52	#117	C₂³	1	1	yes	characteristic
53	#108	—	2	2	no	—
54	#108	—	2	4	no	—
55	#108	—	2	2	no	—
56	#108	—	2	4	no	—
57	#108	—	4	4	no	—
58	C₄⋊C₈	—	2	2	no	—
59	#101	—	2	2	no	—
60	#100	—	2	2	no	—
61	C₂²⋊C₈	—	2	2	no	—
62	C₂²⋊C₈	—	2	2	no	—
63	C₂²⋊C₈	—	4	4	no	—
64	#97	—	2	1	no	—
65	C₄×C₈	—	2	1	no	—
66	C₂⁴	C₂×Q₈	1	1	yes	characteristic
67	#41	—	2	2	no	—
68	#41	—	2	2	no	—
69	C₂×Q₈	#41	1	1	yes	characteristic
70	C₂×D₄	—	2	2	no	—
71	C₂×D₄	C₂×Q₈	1	2	yes	—
72	C₂×D₄	#41	1	1	yes	characteristic
73	C₂²×C₄	—	2	1	no	—
74	C₂²×C₄	—	2	1	no	—
75	C₂²×C₄	—	2	2	no	—
76	C₂²×C₄	—	2	2	no	—
77	C₂²×C₄	C₂×D₄	1	2	yes	—
78	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
79	C₂²×C₄	C₂⁴	1	1	yes	characteristic
80	M₁₆	—	2	2	no	—
81	M₁₆	—	2	4	no	—
82	M₁₆	—	2	4	no	—
83	M₁₆	—	2	4	no	—
84	M₁₆	—	2	4	no	—
85	M₁₆	—	2	2	no	—
86	M₁₆	—	2	4	no	—
87	M₁₆	—	4	4	no	—
88	M₁₆	—	4	2	no	—
89	M₁₆	—	4	2	no	—
90	C₂×C₈	—	2	4	no	—
91	C₂×C₈	—	2	1	no	—
92	C₂×C₈	—	2	1	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	2	4	no	—
95	C₂×C₈	—	2	4	no	—
96	C₂×C₈	—	4	4	no	—
97	C₂×C₈	C₂×D₄	1	2	yes	—
98	C₂×C₈	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈	#41	1	4	yes	—
101	C₄⋊C₄	—	2	2	no	—
102	C₄⋊C₄	—	2	1	no	—
103	C₂²⋊C₄	—	2	2	no	—
104	C₂²⋊C₄	—	2	2	no	—
105	C₂²⋊C₄	C₂×Q₈	1	2	yes	—
106	C₄²	—	2	1	no	—
107	C₂³	—	2	1	no	—
108	C₂³	—	2	2	no	—
109	C₂³	—	8	1	no	—
110	C₂³	C₂²⋊Q₈	1	2	yes	—
111	C₂³	C₂²×Q₈	1	1	yes	characteristic
112	Q₈	—	2	2	no	—
113	D₄	—	2	4	no	—
114	D₄	—	2	2	no	—
115	D₄	—	8	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	4	1	no	—
122	C₂×C₄	—	4	1	no	—
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	C₂²⋊Q₈	1	2	yes	—
125	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄	C₂×#41	1	2	yes	—
127	C₂×C₄	C₂×#41	1	1	yes	characteristic
128	C₂×C₄	H(5,𝔽₂)	1	2	yes	—
129	C₈	—	2	2	no	—
130	C₈	—	4	1	no	—
131	C₈	—	4	4	no	—
132	C₈	—	4	1	no	—
133	C₈	—	4	4	no	—
134	C₈	#123	1	2	yes	—
135	C₈	#123	1	2	yes	—
136	C₂²	—	2	2	no	—
137	C₂²	—	4	1	no	—
138	C₂²	—	4	2	no	—
139	C₂²	—	4	1	no	—
140	C₂²	—	4	1	no	—
141	C₂²	—	4	4	no	—
142	C₂²	—	4	1	no	—
143	C₂²	C₂×C₂²⋊Q₈	1	1	yes	characteristic
144	C₂²	#554	1	2	yes	—
145	C₄	—	2	2	no	—
146	C₄	—	4	2	no	—
147	C₄	—	4	1	no	—
148	C₄	C₂×#123	1	1	yes	characteristic
149	C₄	#535	1	1	yes	characteristic
150	C₄	#553	1	2	yes	—
151	C₂	—	2	1	no	—
152	C₂	—	2	2	no	—
153	C₂	—	4	2	no	—
154	C₂	—	8	1	no	—
155	C₂	#2487	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#2930

C₂

yes

maximal, characteristic

#2916

C₂

yes

maximal, characteristic, complemented

#2103

C₂

yes

maximal, complemented

#2091

C₂

yes

maximal, complemented

#1953

C₂

yes

maximal, complemented

#1891

C₂

yes

maximal

#1858

C₂

yes

maximal, complemented

#1813

C₂

yes

maximal, characteristic, complemented

#1811

C₂

yes

maximal, complemented

#568

C₂²

yes

—

C₂×#131

C₂²

yes

characteristic

C₂²×M₁₆

C₂²

yes

characteristic

#525

C₂²

yes

—

—

—

C₂²

yes

—

#430

C₂²

yes

—

C₂×#108

—

C₂×#108

C₂²

yes

—

C₂×#108

C₂²

yes

—

#424

C₂²

yes

characteristic

#413

C₂²

yes

—

#408

C₂²

yes

characteristic

#407

C₂²

yes

characteristic

#407

C₂²

yes

—

#405

C₂²

yes

characteristic

#344

C₂²

yes

—

#341

C₂²

yes

—

#338

C₂²

yes

—

C₂×#41

C₂³

yes

characteristic

C₂³×C₄

C₂³

yes

characteristic

#131

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

D₄

yes

—

C₂×M₁₆

C₂³

yes

—

C₂×M₁₆

C₂³

yes

—

C₂×M₁₆

C₂³

yes

characteristic

C₂×M₁₆

C₂³

yes

characteristic

C₂²×C₈

D₄

yes

—

C₂²×C₈

C₂³

yes

characteristic

C₂²×C₈

C₂³

yes

characteristic

#123

Q₈

yes

—

C₂²⋊Q₈

—

#121

—

#120

Q₈

yes

—

C₄×D₄

—

#117

C₂³

yes

—

—

—

—

—

C₄⋊C₈

—

#101

—

#100

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₂⁴

C₂×Q₈

yes

characteristic

#41

—

#41

—

C₂×Q₈

#41

yes

characteristic

C₂×D₄

—

C₂×D₄

C₂×Q₈

yes

—

C₂×D₄

#41

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

—

C₂²×C₄

C₂×Q₈

yes

characteristic

C₂²×C₄

C₂⁴

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

characteristic

100

C₂×C₈

#41

yes

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₂²⋊C₄

—

104

C₂²⋊C₄

—

105

C₂²⋊C₄

C₂×Q₈

yes

—

106

C₄²

—

107

C₂³

—

108

C₂³

—

109

C₂³

—

110

C₂³

C₂²⋊Q₈

yes

—

111

C₂³

C₂²×Q₈

yes

characteristic

112

Q₈

—

113

D₄

—

114

D₄

—

115

D₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

C₂²⋊Q₈

yes

—

125

C₂×C₄

C₂²×D₄

yes

characteristic

126

C₂×C₄

C₂×#41

yes

—

127

C₂×C₄

C₂×#41

yes

characteristic

128

C₂×C₄

H(5,𝔽₂)

yes

—

129

C₈

—

130

C₈

—

131

C₈

—

132

C₈

—

133

C₈

—

134

C₈

#123

yes

—

135

C₈

#123

yes

—

136

C₂²

—

137

C₂²

—

138

C₂²

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

C₂×C₂²⋊Q₈

yes

characteristic

144

C₂²

#554

yes

—

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

C₂×#123

yes

characteristic

149

C₄

#535

yes

characteristic

150

C₄

#553

yes

—

151

C₂

—

152

C₂

—

153

C₂

—

154

C₂

—

155

C₂

#2487

yes

characteristic

	subgroup	G/H
center	C₄	#535
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴
socle	C₂	#2487
prime radical	#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₄

#535

commutator, Frattini, cosocle

C₂²×C₄

C₂⁴

C₂

C₂²

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ G

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
2. #2916	C₂	semidirect
3. #2103	C₂	semidirect
4. #2091	C₂	semidirect
5. #1953	C₂	semidirect
7. #1858	C₂	semidirect
8. #1813	C₂	semidirect
9. #1811	C₂	semidirect

H ≅ G/N

type

2. #2916

C₂

semidirect

3. #2103

C₂

semidirect

4. #2091

C₂

semidirect

5. #1953

C₂

semidirect

7. #1858

C₂

semidirect

8. #1813

C₂

semidirect

9. #1811

C₂

semidirect

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

subgroup	index
#2930	2
#2916	2
#2103	2
#2091	2
#1953	2
#1891	2
#1858	2
#1813	2
#1811	2

subgroup

index

Subgroups — #27692

Conjugacy classes of subgroups

Complements

Maximal subgroups