Group #33970 — subgroups

Subgroups — #33970

Order 256.

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

normal subgroups only

This group has 264 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	#3396	C₂	1	2	yes	maximal, complemented
2	#2250	C₂	1	2	yes	maximal, complemented
3	#2249	C₂	1	1	yes	maximal, characteristic, complemented
4	#2211	C₂	1	2	yes	maximal, complemented
5	#2210	C₂	1	2	yes	maximal, complemented
6	#2206	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×D₁₆	C₂	1	1	yes	maximal, characteristic, complemented
10	#559	—	2	2	no	—
11	C₂×Q₃₂	C₂²	1	1	yes	characteristic
12	C₂×SD₃₂	C₂²	1	2	yes	—
13	C₂×SD₃₂	C₂²	1	2	yes	—
14	C₂×D₁₆	C₂²	1	1	yes	characteristic
15	C₂×D₁₆	C₂²	1	1	yes	characteristic
16	C₂×D₁₆	C₂²	1	2	yes	—
17	#495	C₂²	1	2	yes	—
18	#495	C₂²	1	2	yes	—
19	#493	C₂²	1	2	yes	—
20	#463	—	2	2	no	—
21	#463	—	2	2	no	—
22	#460	—	2	2	no	—
23	C₈×Q₈	C₂²	1	1	yes	characteristic
24	C₄×Q₁₆	C₂²	1	2	yes	—
25	C₄×D₈	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	#357	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	C₂³	1	2	yes	—
36	C₂×SD₁₆	—	2	2	no	—
37	C₂×SD₁₆	—	2	2	no	—
38	C₂×D₈	—	2	2	no	—
39	C₂×D₈	C₂³	1	2	yes	—
40	C₂×D₈	C₂³	1	2	yes	—
41	#127	—	2	2	no	—
42	#124	—	2	2	no	—
43	#124	—	2	2	no	—
44	#124	—	4	2	no	—
45	#123	—	4	2	no	—
46	#123	—	4	2	no	—
47	C₄×Q₈	—	2	2	no	—
48	C₄×Q₈	D₄	1	1	yes	characteristic
49	C₄×D₄	—	2	2	no	—
50	C₄×D₄	—	2	2	no	—
51	C₄×D₄	—	4	2	no	—
52	Q₃₂	—	4	1	no	—
53	SD₃₂	—	2	4	no	—
54	SD₃₂	—	4	2	no	—
55	D₁₆	—	2	2	no	—
56	D₁₆	—	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	#102	—	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	D₈	1	1	yes	characteristic
71	C₂×D₄	—	2	2	no	—
72	C₂×D₄	—	2	2	no	—
73	C₂×D₄	—	4	2	no	—
74	C₂×D₄	—	4	2	no	—
75	C₂²×C₄	—	4	2	no	—
76	C₂²×C₄	—	4	2	no	—
77	Q₁₆	—	2	2	no	—
78	Q₁₆	—	2	2	no	—
79	SD₁₆	—	8	2	no	—
80	SD₁₆	—	8	2	no	—
81	D₈	—	2	2	no	—
82	D₈	—	2	2	no	—
83	D₈	—	2	2	no	—
84	D₈	—	2	2	no	—
85	D₈	—	8	2	no	—
86	C₂×C₈	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈	C₂×D₄	1	2	yes	—
88	C₂×C₈	C₂⁴	1	1	yes	characteristic
89	C₄⋊C₄	—	2	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	C₂²⋊C₄	—	4	2	no	—
99	C₄²	—	4	2	no	—
100	C₄²	C₂×D₄	1	1	yes	characteristic
101	C₄²	C₂×D₄	1	2	yes	—
102	C₁₆	—	2	2	no	—
103	C₁₆	—	2	1	no	—
104	C₁₆	#41	1	2	yes	—
105	C₂³	—	4	2	no	—
106	C₂³	—	4	2	no	—
107	Q₈	—	4	1	no	—
108	Q₈	—	4	2	no	—
109	Q₈	—	4	2	no	—
110	D₄	—	4	2	no	—
111	D₄	—	4	2	no	—
112	D₄	—	4	2	no	—
113	D₄	—	4	2	no	—
114	D₄	—	8	4	no	—
115	D₄	—	16	2	no	—
116	C₂×C₄	—	4	2	no	—
117	C₂×C₄	—	4	2	no	—
118	C₂×C₄	—	16	2	no	—
119	C₂×C₄	—	16	2	no	—
120	C₂×C₄	C₂×D₈	1	2	yes	—
121	C₂×C₄	C₂×D₈	1	1	yes	characteristic
122	C₂×C₄	C₂×D₈	1	1	yes	characteristic
123	C₂×C₄	C₂×D₈	1	2	yes	—
124	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
125	C₈	—	2	2	no	—
126	C₈	—	2	1	no	—
127	C₈	C₂×#41	1	1	yes	characteristic
128	C₈	H(5,𝔽₂)	1	1	yes	characteristic
129	C₂²	—	8	2	no	—
130	C₂²	—	8	2	no	—
131	C₂²	—	8	2	no	—
132	C₂²	—	8	2	no	—
133	C₂²	—	8	2	no	—
134	C₂²	—	8	2	no	—
135	C₂²	C₂²×D₈	1	1	yes	characteristic
136	C₄	—	2	2	no	—
137	C₄	—	2	1	no	—
138	C₄	—	2	1	no	—
139	C₄	—	2	2	no	—
140	C₄	—	8	2	no	—
141	C₄	—	8	2	no	—
142	C₄	#550	1	1	yes	characteristic
143	C₄	#576	1	1	yes	characteristic
144	C₂	—	16	2	no	—
145	C₂	—	16	2	no	—
146	C₂	#3391	1	1	yes	characteristic
147	C₂	#3414	1	1	yes	characteristic
148	C₂	#3415	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#3396

C₂

yes

maximal, complemented

#2250

C₂

yes

maximal, complemented

#2249

C₂

yes

maximal, characteristic, complemented

#2211

C₂

yes

maximal, complemented

#2210

C₂

yes

maximal, complemented

#2206

C₂

yes

maximal, complemented

#2183

C₂

yes

maximal, characteristic, complemented

#2175

C₂

yes

maximal, complemented

C₄×D₁₆

C₂

yes

maximal, characteristic, complemented

#559

—

C₂×Q₃₂

C₂²

yes

characteristic

C₂×SD₃₂

C₂²

yes

—

C₂×SD₃₂

C₂²

yes

—

C₂×D₁₆

C₂²

yes

characteristic

C₂×D₁₆

C₂²

yes

characteristic

C₂×D₁₆

C₂²

yes

—

#495

C₂²

yes

—

#495

C₂²

yes

—

#493

C₂²

yes

—

#463

—

#463

—

#460

—

C₈×Q₈

C₂²

yes

characteristic

C₄×Q₁₆

C₂²

yes

—

C₄×D₈

C₂²

yes

—

C₄×D₈

C₂²

yes

—

#366

C₂²

yes

characteristic

C₄⋊C₁₆

C₂²

yes

—

C₄⋊C₁₆

C₂²

yes

characteristic

#358

C₂²

yes

—

#357

C₂²

yes

—

#357

C₂²

yes

—

#346

C₂²

yes

—

C₄×C₁₆

C₂²

yes

characteristic

C₂×Q₁₆

C₂³

yes

—

C₂×SD₁₆

—

C₂×SD₁₆

—

C₂×D₈

—

C₂×D₈

C₂³

yes

—

C₂×D₈

C₂³

yes

—

—

—

—

—

—

—

C₄×Q₈

—

C₄×Q₈

D₄

yes

characteristic

C₄×D₄

—

C₄×D₄

—

C₄×D₄

—

Q₃₂

—

SD₃₂

—

SD₃₂

—

D₁₆

—

D₁₆

—

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

D₈

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

Q₁₆

—

Q₁₆

—

SD₁₆

—

SD₁₆

—

D₈

—

D₈

—

D₈

—

D₈

—

D₈

—

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂⁴

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

D₈

yes

—

C₄⋊C₄

D₈

yes

characteristic

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

100

C₄²

C₂×D₄

yes

characteristic

101

C₄²

C₂×D₄

yes

—

102

C₁₆

—

103

C₁₆

—

104

C₁₆

#41

yes

—

105

C₂³

—

106

C₂³

—

107

Q₈

—

108

Q₈

—

109

Q₈

—

110

D₄

—

111

D₄

—

112

D₄

—

113

D₄

—

114

D₄

—

115

D₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

C₂×D₈

yes

—

121

C₂×C₄

C₂×D₈

yes

characteristic

122

C₂×C₄

C₂×D₈

yes

characteristic

123

C₂×C₄

C₂×D₈

yes

—

124

C₂×C₄

C₂²×D₄

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

C₂×#41

yes

characteristic

128

C₈

H(5,𝔽₂)

yes

characteristic

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

C₂²×D₈

yes

characteristic

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

#550

yes

characteristic

143

C₄

#576

yes

characteristic

144

C₂

—

145

C₂

—

146

C₂

#3391

yes

characteristic

147

C₂

#3414

yes

characteristic

148

C₂

#3415

yes

characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
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

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. #3396	C₂	semidirect
2. #2250	C₂	semidirect
3. #2249	C₂	semidirect
4. #2211	C₂	semidirect
5. #2210	C₂	semidirect
6. #2206	C₂	semidirect
7. #2183	C₂	semidirect
8. #2175	C₂	semidirect
9. C₄×D₁₆	C₂	semidirect

H ≅ G/N

type

1. #3396

C₂

semidirect

2. #2250

C₂

semidirect

3. #2249

C₂

semidirect

4. #2211

C₂

semidirect

5. #2210

C₂

semidirect

6. #2206

C₂

semidirect

7. #2183

C₂

semidirect

8. #2175

C₂

semidirect

9. C₄×D₁₆

C₂

semidirect

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

subgroup	index
#3396	2
#2250	2
#2249	2
#2211	2
#2210	2
#2206	2
#2183	2
#2175	2
C₄×D₁₆	2

subgroup

index

C₄×D₁₆

Subgroups — #33970

Conjugacy classes of subgroups

Complements

Maximal subgroups