Group #33919 — subgroups

Subgroups — #33919

Order 256.

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

normal subgroups only

This group has 232 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	#3336	C₂	1	2	yes	maximal
2	#2253	C₂	1	1	yes	maximal, characteristic, complemented
3	#2237	C₂	1	2	yes	maximal, complemented
4	#2236	C₂	1	2	yes	maximal, complemented
5	#2229	C₂	1	2	yes	maximal, complemented
6	#2219	C₂	1	2	yes	maximal, complemented
7	C₄×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
8	#2169	C₂	1	1	yes	maximal, characteristic
9	#2159	C₂	1	2	yes	maximal, complemented
10	#552	—	2	2	no	—
11	C₂×Q₃₂	C₂²	1	1	yes	characteristic
12	C₂×M₃₂	C₂²	1	2	yes	—
13	#499	C₂²	1	2	yes	—
14	#485	—	2	2	no	—
15	#483	—	2	2	no	—
16	#479	—	2	2	no	—
17	#467	C₂²	1	2	yes	—
18	#467	C₂²	1	2	yes	—
19	C₄×Q₁₆	C₂²	1	2	yes	—
20	C₈×D₄	C₂²	1	1	yes	characteristic
21	#427	C₂²	1	2	yes	—
22	C₂×#107	C₂²	1	2	yes	—
23	#367	C₂²	1	2	yes	—
24	#367	C₂²	1	2	yes	—
25	#366	C₂²	1	1	yes	characteristic
26	#366	C₂²	1	2	yes	—
27	#366	C₂²	1	1	yes	characteristic
28	#366	C₂²	1	1	yes	characteristic
29	C₄⋊C₁₆	C₂²	1	1	yes	characteristic
30	#358	C₂²	1	2	yes	—
31	#358	C₂²	1	2	yes	—
32	#358	C₂²	1	2	yes	—
33	C₂²⋊C₁₆	C₂²	1	2	yes	—
34	C₄×C₁₆	C₂²	1	1	yes	characteristic
35	C₂×Q₁₆	C₂³	1	2	yes	—
36	C₂²×C₈	C₂³	1	2	yes	—
37	#126	—	4	2	no	—
38	#125	—	2	2	no	—
39	#125	—	4	2	no	—
40	#123	—	4	2	no	—
41	C₂²⋊Q₈	—	2	2	no	—
42	C₂²⋊Q₈	—	2	2	no	—
43	C₄×Q₈	—	2	2	no	—
44	C₄×D₄	D₄	1	1	yes	characteristic
45	#117	—	2	2	no	—
46	#117	—	4	2	no	—
47	C₂×C₄⋊C₄	—	2	2	no	—
48	Q₃₂	—	2	2	no	—
49	M₃₂	—	2	4	no	—
50	C₂×C₁₆	C₂³	1	1	yes	characteristic
51	C₂×C₁₆	C₂³	1	2	yes	—
52	C₂×C₁₆	C₂³	1	1	yes	characteristic
53	#107	—	2	2	no	—
54	#107	C₂³	1	2	yes	—
55	#107	C₂³	1	2	yes	—
56	#107	C₂³	1	2	yes	—
57	#106	—	2	2	no	—
58	#106	—	2	2	no	—
59	C₄⋊C₈	D₄	1	1	yes	characteristic
60	#103	—	2	2	no	—
61	#103	—	2	2	no	—
62	#103	—	2	2	no	—
63	C₂²⋊C₈	D₄	1	2	yes	—
64	C₄×C₈	C₂³	1	1	yes	characteristic
65	C₂×Q₈	—	2	2	no	—
66	C₂×D₄	D₈	1	1	yes	characteristic
67	C₂²×C₄	—	4	2	no	—
68	C₂²×C₄	C₂×D₄	1	2	yes	—
69	Q₁₆	—	2	2	no	—
70	Q₁₆	—	2	2	no	—
71	C₂×C₈	—	2	2	no	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈	C₂×D₄	1	2	yes	—
75	C₂×C₈	C₂⁴	1	1	yes	characteristic
76	C₄⋊C₄	—	2	2	no	—
77	C₄⋊C₄	—	2	2	no	—
78	C₄⋊C₄	—	2	2	no	—
79	C₄⋊C₄	—	4	2	no	—
80	C₄⋊C₄	—	4	2	no	—
81	C₄⋊C₄	—	4	2	no	—
82	C₄⋊C₄	—	4	2	no	—
83	C₄⋊C₄	—	4	2	no	—
84	C₄⋊C₄	—	4	2	no	—
85	C₄⋊C₄	D₈	1	1	yes	characteristic
86	C₂²⋊C₄	—	4	2	no	—
87	C₂²⋊C₄	—	4	2	no	—
88	C₂²⋊C₄	—	4	2	no	—
89	C₂²⋊C₄	D₈	1	2	yes	—
90	C₄²	—	4	2	no	—
91	C₄²	—	4	2	no	—
92	C₄²	C₂×D₄	1	1	yes	characteristic
93	C₁₆	—	2	2	no	—
94	C₁₆	—	2	1	no	—
95	C₁₆	#41	1	2	yes	—
96	C₂³	C₂×D₈	1	2	yes	—
97	Q₈	—	4	2	no	—
98	Q₈	—	4	2	no	—
99	D₄	—	4	1	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	2	2	no	—
102	C₂×C₄	—	4	2	no	—
103	C₂×C₄	—	4	2	no	—
104	C₂×C₄	—	4	2	no	—
105	C₂×C₄	—	4	2	no	—
106	C₂×C₄	—	8	4	no	—
107	C₂×C₄	C₂×D₈	1	1	yes	characteristic
108	C₂×C₄	C₂×D₈	1	2	yes	—
109	C₂×C₄	C₂×D₈	1	1	yes	characteristic
110	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
111	C₈	—	2	2	no	—
112	C₈	—	2	1	no	—
113	C₈	C₂×#41	1	1	yes	characteristic
114	C₈	2₋¹⁺⁴	1	1	yes	characteristic
115	C₂²	—	2	2	no	—
116	C₂²	—	2	2	no	—
117	C₂²	—	2	2	no	—
118	C₂²	C₂²×D₈	1	1	yes	characteristic
119	C₄	—	2	2	no	—
120	C₄	—	2	1	no	—
121	C₄	—	2	1	no	—
122	C₄	—	8	2	no	—
123	C₄	—	8	2	no	—
124	C₄	—	8	2	no	—
125	C₄	—	8	2	no	—
126	C₄	#547	1	1	yes	characteristic
127	C₄	#578	1	1	yes	characteristic
128	C₂	—	4	2	no	—
129	C₂	#3334	1	1	yes	characteristic
130	C₂	#3414	1	1	yes	characteristic
131	C₂	#3417	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#3336

C₂

yes

maximal

#2253

C₂

yes

maximal, characteristic, complemented

#2237

C₂

yes

maximal, complemented

#2236

C₂

yes

maximal, complemented

#2229

C₂

yes

maximal, complemented

#2219

C₂

yes

maximal, complemented

C₄×Q₃₂

C₂

yes

maximal, characteristic, complemented

#2169

C₂

yes

maximal, characteristic

#2159

C₂

yes

maximal, complemented

#552

—

C₂×Q₃₂

C₂²

yes

characteristic

C₂×M₃₂

C₂²

yes

—

#499

C₂²

yes

—

—

—

—

C₂²

yes

—

#467

C₂²

yes

—

C₄×Q₁₆

C₂²

yes

—

C₈×D₄

C₂²

yes

characteristic

#427

C₂²

yes

—

C₂×#107

C₂²

yes

—

#367

C₂²

yes

—

#367

C₂²

yes

—

#366

C₂²

yes

characteristic

#366

C₂²

yes

—

#366

C₂²

yes

characteristic

#366

C₂²

yes

characteristic

C₄⋊C₁₆

C₂²

yes

characteristic

#358

C₂²

yes

—

#358

C₂²

yes

—

#358

C₂²

yes

—

C₂²⋊C₁₆

C₂²

yes

—

C₄×C₁₆

C₂²

yes

characteristic

C₂×Q₁₆

C₂³

yes

—

C₂²×C₈

C₂³

yes

—

—

—

—

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×D₄

D₄

yes

characteristic

#117

—

#117

—

C₂×C₄⋊C₄

—

Q₃₂

—

M₃₂

—

C₂×C₁₆

C₂³

yes

characteristic

C₂×C₁₆

C₂³

yes

—

C₂×C₁₆

C₂³

yes

characteristic

#107

—

#107

C₂³

yes

—

#107

C₂³

yes

—

#107

C₂³

yes

—

#106

—

#106

—

C₄⋊C₈

D₄

yes

—

—

—

C₂²⋊C₈

D₄

yes

—

C₄×C₈

C₂³

yes

characteristic

C₂×Q₈

—

C₂×D₄

D₈

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

—

Q₁₆

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂⁴

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

D₈

yes

characteristic

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

D₈

yes

—

C₄²

—

C₄²

—

C₄²

C₂×D₄

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

#41

yes

—

C₂³

C₂×D₈

yes

—

Q₈

—

Q₈

—

D₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

C₂×D₈

yes

characteristic

108

C₂×C₄

C₂×D₈

yes

—

109

C₂×C₄

C₂×D₈

yes

characteristic

110

C₂×C₄

C₂²×D₄

yes

characteristic

111

C₈

—

112

C₈

—

113

C₈

C₂×#41

yes

characteristic

114

C₈

2₋¹⁺⁴

yes

characteristic

115

C₂²

—

116

C₂²

—

117

C₂²

—

118

C₂²

C₂²×D₈

yes

characteristic

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

#547

yes

characteristic

127

C₄

#578

yes

characteristic

128

C₂

—

129

C₂

#3334

yes

characteristic

130

C₂

#3414

yes

characteristic

131

C₂

#3417

yes

characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
prime radical	C₂×D₄	D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ 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×D₄

D₈

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ G

LECS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
2. #2253	C₂	semidirect
3. #2237	C₂	semidirect
4. #2236	C₂	semidirect
5. #2229	C₂	semidirect
6. #2219	C₂	semidirect
7. C₄×Q₃₂	C₂	semidirect
9. #2159	C₂	semidirect

H ≅ G/N

type

2. #2253

C₂

semidirect

3. #2237

C₂

semidirect

4. #2236

C₂

semidirect

5. #2229

C₂

semidirect

6. #2219

C₂

semidirect

7. C₄×Q₃₂

C₂

semidirect

9. #2159

C₂

semidirect

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

subgroup	index
#3336	2
#2253	2
#2237	2
#2236	2
#2229	2
#2219	2
C₄×Q₃₂	2
#2169	2
#2159	2

subgroup

index

C₄×Q₃₂

Subgroups — #33919

Conjugacy classes of subgroups

Complements

Maximal subgroups