Group #33969 — subgroups

Subgroups — #33969

Order 256.

252 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 99 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	#3393	C₂	1	2	yes	maximal, complemented
2	#2251	C₂	1	2	yes	maximal, complemented
3	#2249	C₂	1	1	yes	maximal, characteristic
4	#2211	C₂	1	2	yes	maximal, complemented
5	#2210	C₂	1	2	yes	maximal
6	#2209	C₂	1	2	yes	maximal, complemented
7	#2183	C₂	1	1	yes	maximal, characteristic, complemented
8	#2175	C₂	1	2	yes	maximal, complemented
9	C₄×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
10	#556	—	2	2	no	—
11	C₂×Q₃₂	C₂²	1	1	yes	characteristic
12	C₂×Q₃₂	C₂²	1	1	yes	characteristic
13	C₂×Q₃₂	C₂²	1	2	yes	—
14	C₂×SD₃₂	C₂²	1	2	yes	—
15	C₂×SD₃₂	C₂²	1	2	yes	—
16	C₂×D₁₆	C₂²	1	1	yes	characteristic
17	#495	C₂²	1	2	yes	—
18	#495	C₂²	1	2	yes	—
19	#494	C₂²	1	2	yes	—
20	#464	—	2	2	no	—
21	#464	—	2	2	no	—
22	#461	—	2	2	no	—
23	C₈×Q₈	C₂²	1	1	yes	characteristic
24	C₄×Q₁₆	C₂²	1	2	yes	—
25	C₄×Q₁₆	C₂²	1	2	yes	—
26	C₄×D₈	C₂²	1	2	yes	—
27	#366	C₂²	1	1	yes	characteristic
28	C₄⋊C₁₆	C₂²	1	2	yes	—
29	C₄⋊C₁₆	C₂²	1	1	yes	characteristic
30	#358	C₂²	1	2	yes	—
31	#358	C₂²	1	2	yes	—
32	#357	C₂²	1	2	yes	—
33	#346	C₂²	1	2	yes	—
34	C₄×C₁₆	C₂²	1	1	yes	characteristic
35	C₂×Q₁₆	—	2	2	no	—
36	C₂×Q₁₆	C₂³	1	2	yes	—
37	C₂×Q₁₆	C₂³	1	2	yes	—
38	C₂×SD₁₆	—	2	2	no	—
39	C₂×SD₁₆	—	2	2	no	—
40	C₂×D₈	C₂³	1	2	yes	—
41	C₄⋊Q₈	—	2	2	no	—
42	#126	—	4	2	no	—
43	#126	—	4	2	no	—
44	#124	—	2	2	no	—
45	#124	—	2	2	no	—
46	C₂²⋊Q₈	—	4	2	no	—
47	C₄×Q₈	—	2	2	no	—
48	C₄×Q₈	—	2	2	no	—
49	C₄×Q₈	D₄	1	1	yes	characteristic
50	C₄×D₄	—	2	2	no	—
51	#117	—	4	2	no	—
52	Q₃₂	—	2	2	no	—
53	Q₃₂	—	4	1	no	—
54	Q₃₂	—	4	2	no	—
55	SD₃₂	—	2	4	no	—
56	SD₃₂	—	4	2	no	—
57	D₁₆	—	4	1	no	—
58	C₂×C₁₆	C₂³	1	1	yes	characteristic
59	C₂×C₁₆	C₂³	1	2	yes	—
60	C₂×C₁₆	C₂³	1	1	yes	characteristic
61	#107	C₂³	1	2	yes	—
62	C₄⋊C₈	D₄	1	2	yes	—
63	C₄⋊C₈	D₄	1	1	yes	characteristic
64	#103	—	2	2	no	—
65	#103	—	2	2	no	—
66	#102	—	2	2	no	—
67	C₄×C₈	C₂³	1	1	yes	characteristic
68	C₄×C₈	C₂³	1	2	yes	—
69	C₂×Q₈	—	2	2	no	—
70	C₂×Q₈	—	2	2	no	—
71	C₂×Q₈	D₈	1	1	yes	characteristic
72	C₂×D₄	—	2	2	no	—
73	C₂²×C₄	—	4	2	no	—
74	Q₁₆	—	2	2	no	—
75	Q₁₆	—	2	2	no	—
76	Q₁₆	—	2	2	no	—
77	Q₁₆	—	2	2	no	—
78	Q₁₆	—	8	2	no	—
79	SD₁₆	—	8	2	no	—
80	SD₁₆	—	8	2	no	—
81	D₈	—	2	2	no	—
82	D₈	—	2	2	no	—
83	C₂×C₈	C₂×D₄	1	2	yes	—
84	C₂×C₈	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈	C₂⁴	1	1	yes	characteristic
86	C₄⋊C₄	—	2	2	no	—
87	C₄⋊C₄	—	4	2	no	—
88	C₄⋊C₄	—	4	2	no	—
89	C₄⋊C₄	—	4	2	no	—
90	C₄⋊C₄	—	4	2	no	—
91	C₄⋊C₄	D₈	1	2	yes	—
92	C₄⋊C₄	D₈	1	1	yes	characteristic
93	C₂²⋊C₄	—	4	2	no	—
94	C₂²⋊C₄	—	4	2	no	—
95	C₂²⋊C₄	—	4	2	no	—
96	C₂²⋊C₄	—	4	2	no	—
97	C₂²⋊C₄	—	4	2	no	—
98	C₄²	—	4	2	no	—
99	C₄²	—	4	2	no	—
100	C₄²	C₂×D₄	1	2	yes	—
101	C₄²	C₂×D₄	1	1	yes	characteristic
102	C₁₆	—	2	1	no	—
103	C₁₆	—	2	2	no	—
104	C₁₆	#41	1	2	yes	—
105	C₂³	—	4	2	no	—
106	Q₈	—	4	1	no	—
107	Q₈	—	4	2	no	—
108	Q₈	—	4	2	no	—
109	Q₈	—	4	2	no	—
110	Q₈	—	4	2	no	—
111	D₄	—	4	2	no	—
112	D₄	—	4	2	no	—
113	C₂×C₄	—	4	2	no	—
114	C₂×C₄	—	4	2	no	—
115	C₂×C₄	—	4	2	no	—
116	C₂×C₄	—	8	4	no	—
117	C₂×C₄	C₂×D₈	1	1	yes	characteristic
118	C₂×C₄	C₂×D₈	1	2	yes	—
119	C₂×C₄	C₂×D₈	1	1	yes	characteristic
120	C₂×C₄	C₂×D₈	1	2	yes	—
121	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
122	C₈	—	2	1	no	—
123	C₈	—	2	2	no	—
124	C₈	C₂×#41	1	1	yes	characteristic
125	C₈	H(5,𝔽₂)	1	1	yes	characteristic
126	C₂²	—	8	2	no	—
127	C₂²	—	8	2	no	—
128	C₂²	—	8	2	no	—
129	C₂²	C₂²×D₈	1	1	yes	characteristic
130	C₄	—	2	2	no	—
131	C₄	—	2	1	no	—
132	C₄	—	2	1	no	—
133	C₄	—	2	2	no	—
134	C₄	—	8	2	no	—
135	C₄	—	8	2	no	—
136	C₄	—	8	2	no	—
137	C₄	#550	1	1	yes	characteristic
138	C₄	#576	1	1	yes	characteristic
139	C₂	—	16	2	no	—
140	C₂	#3391	1	1	yes	characteristic
141	C₂	#3414	1	1	yes	characteristic
142	C₂	#3417	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#3393

C₂

yes

maximal, complemented

#2251

C₂

yes

maximal, complemented

#2249

C₂

yes

maximal, characteristic

#2211

C₂

yes

maximal, complemented

#2210

C₂

yes

maximal

#2209

C₂

yes

maximal, complemented

#2183

C₂

yes

maximal, characteristic, complemented

#2175

C₂

yes

maximal, complemented

C₄×Q₃₂

C₂

yes

maximal, characteristic, complemented

#556

—

C₂×Q₃₂

C₂²

yes

characteristic

C₂×Q₃₂

C₂²

yes

characteristic

C₂×Q₃₂

C₂²

yes

—

C₂×SD₃₂

C₂²

yes

—

C₂×SD₃₂

C₂²

yes

—

C₂×D₁₆

C₂²

yes

characteristic

#495

C₂²

yes

—

#495

C₂²

yes

—

#494

C₂²

yes

—

#464

—

#464

—

#461

—

C₈×Q₈

C₂²

yes

characteristic

C₄×Q₁₆

C₂²

yes

—

C₄×Q₁₆

C₂²

yes

—

C₄×D₈

C₂²

yes

—

#366

C₂²

yes

characteristic

C₄⋊C₁₆

C₂²

yes

—

C₄⋊C₁₆

C₂²

yes

characteristic

#358

C₂²

yes

—

#358

C₂²

yes

—

#357

C₂²

yes

—

#346

C₂²

yes

—

C₄×C₁₆

C₂²

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

C₂³

yes

—

C₂×Q₁₆

C₂³

yes

—

C₂×SD₁₆

—

C₂×SD₁₆

—

C₂×D₈

C₂³

yes

—

C₄⋊Q₈

—

—

—

—

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

D₄

yes

characteristic

C₄×D₄

—

#117

—

Q₃₂

—

Q₃₂

—

Q₃₂

—

SD₃₂

—

SD₃₂

—

D₁₆

—

C₂×C₁₆

C₂³

yes

characteristic

C₂×C₁₆

C₂³

yes

—

C₂×C₁₆

C₂³

yes

characteristic

#107

C₂³

yes

—

C₄⋊C₈

D₄

yes

—

C₄⋊C₈

D₄

yes

—

—

—

C₄×C₈

C₂³

yes

characteristic

C₄×C₈

C₂³

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

D₈

yes

characteristic

C₂×D₄

—

C₂²×C₄

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

SD₁₆

—

SD₁₆

—

D₈

—

D₈

—

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂⁴

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

D₈

yes

—

C₄⋊C₄

D₈

yes

characteristic

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

100

C₄²

C₂×D₄

yes

—

101

C₄²

C₂×D₄

yes

characteristic

102

C₁₆

—

103

C₁₆

—

104

C₁₆

#41

yes

—

105

C₂³

—

106

Q₈

—

107

Q₈

—

108

Q₈

—

109

Q₈

—

110

Q₈

—

111

D₄

—

112

D₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

C₂×D₈

yes

characteristic

118

C₂×C₄

C₂×D₈

yes

—

119

C₂×C₄

C₂×D₈

yes

characteristic

120

C₂×C₄

C₂×D₈

yes

—

121

C₂×C₄

C₂²×D₄

yes

characteristic

122

C₈

—

123

C₈

—

124

C₈

C₂×#41

yes

characteristic

125

C₈

H(5,𝔽₂)

yes

characteristic

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

C₂²×D₈

yes

characteristic

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

#550

yes

characteristic

138

C₄

#576

yes

characteristic

139

C₂

—

140

C₂

#3391

yes

characteristic

141

C₂

#3414

yes

characteristic

142

C₂

#3417

yes

characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
prime radical	C₂×D₁₆	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

C₂²×D₈

commutator, Frattini, cosocle

C₂×C₈

C₂⁴

prime radical

C₂×D₁₆

C₂²

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ G

LECS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
1. #3393	C₂	semidirect
2. #2251	C₂	semidirect
4. #2211	C₂	semidirect
6. #2209	C₂	semidirect
7. #2183	C₂	semidirect
8. #2175	C₂	semidirect
9. C₄×Q₃₂	C₂	semidirect

H ≅ G/N

type

1. #3393

C₂

semidirect

2. #2251

C₂

semidirect

4. #2211

C₂

semidirect

6. #2209

C₂

semidirect

7. #2183

C₂

semidirect

8. #2175

C₂

semidirect

9. C₄×Q₃₂

C₂

semidirect

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

subgroup	index
#3393	2
#2251	2
#2249	2
#2211	2
#2210	2
#2209	2
#2183	2
#2175	2
C₄×Q₃₂	2

subgroup

index

C₄×Q₃₂

Subgroups — #33969

Conjugacy classes of subgroups

Complements

Maximal subgroups