Group #33686 — subgroups

Subgroups — #33686

Order 256.

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

normal subgroups only

This group has 272 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	#3176	C₂	1	1	yes	maximal, characteristic
2	#2950	C₂	1	1	yes	maximal, characteristic, complemented
3	#2234	C₂	1	2	yes	maximal, complemented
4	#2229	C₂	1	2	yes	maximal, complemented
5	#2207	C₂	1	2	yes	maximal, complemented
6	#2185	C₂	1	2	yes	maximal, complemented
7	#2175	C₂	1	2	yes	maximal, complemented
8	C₄×SD₃₂	C₂	1	2	yes	maximal, complemented
9	#2164	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×Q₁₆	C₂²	1	1	yes	characteristic
11	#540	—	2	1	no	—
12	#519	—	2	1	no	—
13	C₂×SD₃₂	C₂²	1	2	yes	—
14	C₂×SD₃₂	C₂²	1	2	yes	—
15	#499	C₂²	1	1	yes	characteristic
16	#493	C₂²	1	1	yes	characteristic
17	#490	—	2	1	no	—
18	#469	—	2	1	no	—
19	#466	C₂²	1	2	yes	—
20	#442	—	2	1	no	—
21	#441	—	2	1	no	—
22	#441	—	2	1	no	—
23	C₄×Q₁₆	C₂²	1	2	yes	—
24	C₄×Q₁₆	C₂²	1	2	yes	—
25	C₄×D₈	C₂²	1	2	yes	—
26	#427	C₂²	1	1	yes	characteristic
27	#419	—	2	1	no	—
28	#405	C₂²	1	1	yes	characteristic
29	#367	C₂²	1	2	yes	—
30	#366	C₂²	1	2	yes	—
31	C₄⋊C₁₆	C₂²	1	2	yes	—
32	#358	C₂²	1	2	yes	—
33	#358	C₂²	1	2	yes	—
34	#357	C₂²	1	2	yes	—
35	#357	C₂²	1	2	yes	—
36	C₂²⋊C₁₆	C₂²	1	2	yes	—
37	#346	C₂²	1	1	yes	characteristic
38	C₄×C₁₆	C₂²	1	1	yes	characteristic
39	C₂²×Q₈	—	2	1	no	—
40	C₂×Q₁₆	—	2	4	no	—
41	C₂×Q₁₆	—	2	1	no	—
42	C₂×Q₁₆	C₂³	1	2	yes	—
43	C₂×D₈	—	2	1	no	—
44	C₂×D₈	C₂³	1	2	yes	—
45	C₂×M₁₆	D₄	1	1	yes	characteristic
46	C₂²×C₈	C₂³	1	1	yes	characteristic
47	#127	—	2	1	no	—
48	#125	—	2	1	no	—
49	#123	—	4	2	no	—
50	#121	—	2	2	no	—
51	#121	—	4	2	no	—
52	C₄×Q₈	—	2	2	no	—
53	C₄×Q₈	—	2	2	no	—
54	C₄×Q₈	—	4	2	no	—
55	C₄×D₄	—	2	2	no	—
56	#117	—	2	1	no	—
57	#117	—	4	2	no	—
58	#117	D₄	1	1	yes	characteristic
59	SD₃₂	—	2	4	no	—
60	SD₃₂	—	2	4	no	—
61	C₂×C₁₆	C₂³	1	2	yes	—
62	C₂×C₁₆	C₂³	1	2	yes	—
63	#107	C₂³	1	2	yes	—
64	#107	C₂³	1	2	yes	—
65	#106	—	2	1	no	—
66	#106	—	2	1	no	—
67	#103	—	2	2	no	—
68	#103	—	2	2	no	—
69	#102	—	2	2	no	—
70	#102	—	2	2	no	—
71	#97	D₄	1	1	yes	characteristic
72	#97	D₄	1	1	yes	characteristic
73	C₄×C₈	C₂³	1	1	yes	characteristic
74	C₄×C₈	C₂³	1	1	yes	characteristic
75	C₂×Q₈	—	2	4	no	—
76	C₂×Q₈	—	2	2	no	—
77	C₂×Q₈	—	4	2	no	—
78	C₂×Q₈	—	4	1	no	—
79	C₂×D₄	—	2	2	no	—
80	C₂×D₄	—	4	2	no	—
81	C₂×D₄	—	4	2	no	—
82	C₂²×C₄	—	4	1	no	—
83	C₂²×C₄	—	4	2	no	—
84	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
85	Q₁₆	—	2	4	no	—
86	Q₁₆	—	2	2	no	—
87	Q₁₆	#41	1	4	yes	—
88	D₈	—	2	2	no	—
89	D₈	—	2	2	no	—
90	D₈	—	4	2	no	—
91	M₁₆	—	4	1	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈	C₂⁴	1	1	yes	characteristic
97	C₄⋊C₄	—	2	2	no	—
98	C₄⋊C₄	—	2	2	no	—
99	C₄⋊C₄	—	4	2	no	—
100	C₄⋊C₄	—	4	2	no	—
101	C₄⋊C₄	—	4	2	no	—
102	C₄⋊C₄	D₈	1	2	yes	—
103	C₂²⋊C₄	—	4	1	no	—
104	C₂²⋊C₄	—	4	2	no	—
105	C₂²⋊C₄	—	4	2	no	—
106	C₂²⋊C₄	D₈	1	2	yes	—
107	C₄²	—	4	1	no	—
108	C₄²	—	4	2	no	—
109	C₄²	—	4	2	no	—
110	C₄²	C₂×D₄	1	1	yes	characteristic
111	C₄²	C₂×D₄	1	1	yes	characteristic
112	C₁₆	—	2	2	no	—
113	C₁₆	—	2	2	no	—
114	C₂³	—	4	2	no	—
115	C₂³	C₂×D₈	1	1	yes	characteristic
116	Q₈	—	2	4	no	—
117	Q₈	—	4	4	no	—
118	Q₈	—	4	2	no	—
119	D₄	—	4	2	no	—
120	D₄	—	4	2	no	—
121	D₄	—	16	2	no	—
122	D₄	—	16	2	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	4	4	no	—
126	C₂×C₄	—	4	2	no	—
127	C₂×C₄	—	4	2	no	—
128	C₂×C₄	—	4	2	no	—
129	C₂×C₄	—	16	2	no	—
130	C₂×C₄	C₂×D₈	1	2	yes	—
131	C₂×C₄	C₂×D₈	1	2	yes	—
132	C₂×C₄	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
134	C₈	—	2	1	no	—
135	C₈	—	2	1	no	—
136	C₈	—	2	1	no	—
137	C₈	C₂×#41	1	2	yes	—
138	C₂²	—	2	1	no	—
139	C₂²	—	2	2	no	—
140	C₂²	—	8	2	no	—
141	C₂²	—	8	2	no	—
142	C₂²	—	8	2	no	—
143	C₂²	C₂²×D₈	1	1	yes	characteristic
144	C₄	—	2	1	no	—
145	C₄	—	2	2	no	—
146	C₄	—	2	2	no	—
147	C₄	—	4	4	no	—
148	C₄	—	8	2	no	—
149	C₄	—	8	2	no	—
150	C₄	#525	1	1	yes	characteristic
151	C₄	#575	1	1	yes	characteristic
152	C₂	—	4	1	no	—
153	C₂	—	16	2	no	—
154	C₂	#3100	1	1	yes	characteristic
155	C₂	#3416	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#3176

C₂

yes

maximal, characteristic

#2950

C₂

yes

maximal, characteristic, complemented

#2234

C₂

yes

maximal, complemented

#2229

C₂

yes

maximal, complemented

#2207

C₂

yes

maximal, complemented

#2185

C₂

yes

maximal, complemented

#2175

C₂

yes

maximal, complemented

C₄×SD₃₂

C₂

yes

maximal, complemented

#2164

C₂

yes

maximal, characteristic, complemented

C₂²×Q₁₆

C₂²

yes

characteristic

#540

—

#519

—

C₂×SD₃₂

C₂²

yes

—

C₂×SD₃₂

C₂²

yes

—

#499

C₂²

yes

characteristic

#493

C₂²

yes

—

—

C₂²

yes

—

#442

—

#441

—

#441

—

C₄×Q₁₆

C₂²

yes

—

C₄×Q₁₆

C₂²

yes

—

C₄×D₈

C₂²

yes

—

#427

C₂²

yes

characteristic

#419

—

#405

C₂²

yes

characteristic

#367

C₂²

yes

—

#366

C₂²

yes

—

C₄⋊C₁₆

C₂²

yes

—

#358

C₂²

yes

—

#358

C₂²

yes

—

#357

C₂²

yes

—

#357

C₂²

yes

—

C₂²⋊C₁₆

C₂²

yes

—

#346

C₂²

yes

characteristic

C₄×C₁₆

C₂²

yes

characteristic

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

C₂³

yes

—

C₂×D₈

—

C₂×D₈

C₂³

yes

—

C₂×M₁₆

D₄

yes

characteristic

C₂²×C₈

C₂³

yes

—

—

—

—

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×D₄

—

#117

—

#117

—

#117

D₄

yes

characteristic

SD₃₂

—

SD₃₂

—

C₂×C₁₆

C₂³

yes

—

C₂×C₁₆

C₂³

yes

—

#107

C₂³

yes

—

#107

C₂³

yes

—

—

—

—

—

—

—

D₄

yes

characteristic

#97

D₄

yes

characteristic

C₄×C₈

C₂³

yes

characteristic

C₄×C₈

C₂³

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

characteristic

Q₁₆

—

Q₁₆

—

Q₁₆

#41

yes

—

D₈

—

D₈

—

D₈

—

M₁₆

—

C₂×C₈

—

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂⁴

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

D₈

yes

—

103

C₂²⋊C₄

—

104

C₂²⋊C₄

—

105

C₂²⋊C₄

—

106

C₂²⋊C₄

D₈

yes

—

107

C₄²

—

108

C₄²

—

109

C₄²

—

110

C₄²

C₂×D₄

yes

characteristic

111

C₄²

C₂×D₄

yes

characteristic

112

C₁₆

—

113

C₁₆

—

114

C₂³

—

115

C₂³

C₂×D₈

yes

characteristic

116

Q₈

—

117

Q₈

—

118

Q₈

—

119

D₄

—

120

D₄

—

121

D₄

—

122

D₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

C₂×D₈

yes

—

131

C₂×C₄

C₂×D₈

yes

—

132

C₂×C₄

C₂×D₈

yes

characteristic

133

C₂×C₄

C₂²×D₄

yes

characteristic

134

C₈

—

135

C₈

—

136

C₈

—

137

C₈

C₂×#41

yes

—

138

C₂²

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

C₂²×D₈

yes

characteristic

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

#525

yes

characteristic

151

C₄

#575

yes

characteristic

152

C₂

—

153

C₂

—

154

C₂

#3100

yes

characteristic

155

C₂

#3416

yes

—

	subgroup	G/H
center, socle	C₂²	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
prime radical	#3176	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

C₂²×D₈

commutator, Frattini, cosocle

C₂×C₈

C₂⁴

prime radical

#3176

C₂

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ G

LECS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
2. #2950	C₂	semidirect
3. #2234	C₂	semidirect
4. #2229	C₂	semidirect
5. #2207	C₂	semidirect
6. #2185	C₂	semidirect
7. #2175	C₂	semidirect
8. C₄×SD₃₂	C₂	semidirect
9. #2164	C₂	semidirect

H ≅ G/N

type

2. #2950

C₂

semidirect

3. #2234

C₂

semidirect

4. #2229

C₂

semidirect

5. #2207

C₂

semidirect

6. #2185

C₂

semidirect

7. #2175

C₂

semidirect

8. C₄×SD₃₂

C₂

semidirect

9. #2164

C₂

semidirect

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

subgroup	index
#3176	2
#2950	2
#2234	2
#2229	2
#2207	2
#2185	2
#2175	2
C₄×SD₃₂	2
#2164	2

subgroup

index

C₄×SD₃₂

Subgroups — #33686

Conjugacy classes of subgroups

Complements

Maximal subgroups