Group #33607 — subgroups

Subgroups — #33607

Order 256.

357 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 357 conjugacy classes of subgroups — too many to draw. Only its 119 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	C₂²×D₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#504	C₂	1	1	yes	maximal, characteristic, complemented
3	#3058	C₂	1	2	yes	maximal, complemented
4	#2218	C₂	1	4	yes	maximal, complemented
5	#2215	C₂	1	4	yes	maximal, complemented
6	#2158	C₂	1	1	yes	maximal, characteristic, complemented
7	#2140	C₂	1	2	yes	maximal, complemented
8	C₂²×D₈	C₂²	1	2	yes	—
9	C₂×#131	C₂²	1	1	yes	characteristic
10	#535	—	2	2	no	—
11	C₂×D₁₆	—	2	4	no	—
12	C₂×D₁₆	C₂²	1	2	yes	—
13	C₂×D₁₆	C₂²	1	2	yes	—
14	#504	—	2	4	no	—
15	C₂×M₃₂	C₂²	1	1	yes	characteristic
16	C₂×M₃₂	C₂²	1	2	yes	complemented
17	C₂²×C₁₆	C₂²	1	1	yes	characteristic
18	C₂²×C₁₆	C₂²	1	2	yes	complemented
19	#469	—	2	4	no	—
20	#466	C₂²	1	8	yes	complemented
21	#427	C₂²	1	2	yes	—
22	#418	—	2	2	no	—
23	#367	C₂²	1	2	yes	complemented
24	#366	C₂²	1	2	yes	complemented
25	#357	C₂²	1	8	yes	complemented
26	C₂×#41	D₄	1	1	yes	characteristic
27	C₂²×D₄	—	2	2	no	—
28	C₂×D₈	—	2	4	no	—
29	C₂×D₈	—	2	2	no	—
30	C₂×D₈	—	2	4	no	—
31	C₂×D₈	C₂³	1	4	yes	—
32	#131	—	2	2	no	—
33	#131	—	2	2	no	—
34	C₂×M₁₆	D₄	1	1	yes	characteristic
35	C₂×M₁₆	D₄	1	2	yes	—
36	C₂²×C₈	C₂³	1	2	yes	—
37	C₂²×C₈	C₂³	1	1	yes	characteristic
38	#127	—	4	2	no	—
39	#124	—	4	2	no	—
40	#121	—	2	8	no	—
41	#120	—	4	4	no	—
42	#117	—	2	2	no	—
43	D₁₆	—	2	4	no	—
44	D₁₆	—	2	4	no	—
45	M₃₂	—	2	2	no	—
46	M₃₂	—	2	4	no	—
47	C₂×C₁₆	—	2	4	no	—
48	C₂×C₁₆	D₄	1	4	yes	complemented
49	C₂×C₁₆	C₂³	1	2	yes	—
50	C₂×C₁₆	C₂³	1	1	yes	characteristic
51	C₂×C₁₆	C₂³	1	1	yes	characteristic
52	#107	C₂³	1	4	yes	—
53	#106	—	2	2	no	—
54	#102	—	2	8	no	—
55	C₂⁴	—	4	2	no	—
56	#41	—	2	2	no	—
57	#41	—	2	2	no	—
58	C₂×Q₈	D₈	1	1	yes	characteristic
59	C₂×D₄	—	2	4	no	—
60	C₂×D₄	—	4	8	no	—
61	C₂×D₄	—	4	4	no	—
62	C₂×D₄	—	4	2	no	—
63	C₂×D₄	—	4	4	no	—
64	C₂×D₄	D₈	1	2	yes	—
65	C₂×D₄	D₈	1	1	yes	characteristic
66	C₂²×C₄	C₂×D₄	1	2	yes	—
67	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
68	D₈	—	2	4	no	—
69	D₈	—	2	4	no	—
70	D₈	—	4	4	no	—
71	M₁₆	—	2	2	no	—
72	M₁₆	—	2	4	no	—
73	C₂×C₈	—	2	2	no	—
74	C₂×C₈	—	2	2	no	—
75	C₂×C₈	C₂×D₄	1	2	yes	—
76	C₂×C₈	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈	C₂×D₄	1	2	yes	—
78	C₂×C₈	#41	1	2	yes	—
79	C₂×C₈	C₂⁴	1	1	yes	characteristic
80	C₄⋊C₄	—	2	4	no	—
81	C₂²⋊C₄	—	4	2	no	—
82	C₂²⋊C₄	—	4	8	no	—
83	C₄²	—	4	2	no	—
84	C₁₆	—	2	2	no	—
85	C₁₆	C₂×D₄	1	2	yes	—
86	C₁₆	C₂×D₄	1	2	yes	—
87	C₂³	—	4	4	no	—
88	C₂³	—	8	4	no	—
89	C₂³	—	16	4	no	—
90	C₂³	C₂×D₈	1	2	yes	—
91	C₂³	C₂×D₈	1	1	yes	characteristic
92	Q₈	—	2	2	no	—
93	D₄	—	2	4	no	—
94	D₄	—	2	2	no	—
95	D₄	—	4	4	no	—
96	D₄	—	4	4	no	—
97	D₄	—	8	4	no	—
98	D₄	—	16	8	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	4	4	no	—
102	C₂×C₄	#121	1	2	yes	—
103	C₂×C₄	C₂×D₈	1	2	yes	—
104	C₂×C₄	C₂×D₈	1	1	yes	characteristic
105	C₂×C₄	#135	1	2	yes	—
106	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
107	C₈	—	2	2	no	—
108	C₈	#121	1	2	yes	—
109	C₈	C₂²×D₄	1	1	yes	characteristic
110	C₈	C₂×#41	1	1	yes	characteristic
111	C₂²	—	2	2	no	—
112	C₂²	—	2	4	no	—
113	C₂²	—	2	2	no	—
114	C₂²	—	8	8	no	—
115	C₂²	—	8	4	no	—
116	C₂²	—	16	8	no	—
117	C₂²	#466	1	2	yes	—
118	C₂²	C₂²×D₈	1	1	yes	characteristic
119	C₄	—	2	2	no	—
120	C₄	—	8	4	no	—
121	C₄	#466	1	2	yes	—
122	C₄	C₂×#121	1	1	yes	characteristic
123	C₄	C₂×#135	1	1	yes	characteristic
124	C₂	—	2	2	no	—
125	C₂	—	4	2	no	—
126	C₂	—	16	4	no	—
127	C₂	C₂×#466	1	1	yes	characteristic
128	C₂	#3415	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×D₁₆

C₂

yes

maximal, characteristic, complemented

C₂×#504

C₂

yes

maximal, characteristic, complemented

#3058

C₂

yes

maximal, complemented

#2218

C₂

yes

maximal, complemented

#2215

C₂

yes

maximal, complemented

#2158

C₂

yes

maximal, characteristic, complemented

#2140

C₂

yes

maximal, complemented

C₂²×D₈

C₂²

yes

—

C₂×#131

C₂²

yes

characteristic

#535

—

C₂×D₁₆

—

C₂×D₁₆

C₂²

yes

—

C₂×D₁₆

C₂²

yes

—

#504

—

C₂×M₃₂

C₂²

yes

characteristic

C₂×M₃₂

C₂²

yes

complemented

C₂²×C₁₆

C₂²

yes

characteristic

C₂²×C₁₆

C₂²

yes

complemented

#469

—

#466

C₂²

yes

complemented

#427

C₂²

yes

—

#418

—

#367

C₂²

yes

complemented

#366

C₂²

yes

complemented

#357

C₂²

yes

complemented

C₂×#41

D₄

yes

characteristic

C₂²×D₄

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

C₂³

yes

—

#131

—

#131

—

C₂×M₁₆

D₄

yes

characteristic

C₂×M₁₆

D₄

yes

—

C₂²×C₈

C₂³

yes

—

C₂²×C₈

C₂³

yes

—

—

—

—

—

D₁₆

—

D₁₆

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

D₄

yes

complemented

C₂×C₁₆

C₂³

yes

—

C₂×C₁₆

C₂³

yes

characteristic

C₂×C₁₆

C₂³

yes

characteristic

#107

C₂³

yes

—

#106

—

#102

—

C₂⁴

—

#41

—

#41

—

C₂×Q₈

D₈

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

D₈

yes

—

C₂×D₄

D₈

yes

characteristic

C₂²×C₄

C₂×D₄

yes

—

C₂²×C₄

C₂×D₄

yes

characteristic

D₈

—

D₈

—

D₈

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

#41

yes

—

C₂×C₈

C₂⁴

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₁₆

—

C₁₆

C₂×D₄

yes

—

C₁₆

C₂×D₄

yes

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂×D₈

yes

—

C₂³

C₂×D₈

yes

characteristic

Q₈

—

D₄

—

D₄

—

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

#121

yes

—

103

C₂×C₄

C₂×D₈

yes

—

104

C₂×C₄

C₂×D₈

yes

characteristic

105

C₂×C₄

#135

yes

—

106

C₂×C₄

C₂²×D₄

yes

characteristic

107

C₈

—

108

C₈

#121

yes

—

109

C₈

C₂²×D₄

yes

characteristic

110

C₈

C₂×#41

yes

characteristic

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

—

115

C₂²

—

116

C₂²

—

117

C₂²

#466

yes

—

118

C₂²

C₂²×D₈

yes

characteristic

119

C₄

—

120

C₄

—

121

C₄

#466

yes

—

122

C₄

C₂×#121

yes

characteristic

123

C₄

C₂×#135

yes

characteristic

124

C₂

—

125

C₂

—

126

C₂

—

127

C₂

C₂×#466

yes

characteristic

128

C₂

#3415

yes

—

	subgroup	G/H
center, socle	C₂²	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131 ◁ 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 ◁ C₂×#41 ◁ C₂×#131 ◁ G

LECS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
1. C₂²×D₁₆	C₂	semidirect
2. C₂×#504	C₂	semidirect
3. #3058	C₂	semidirect
4. #2218	C₂	semidirect
5. #2215	C₂	semidirect
6. #2158	C₂	semidirect
7. #2140	C₂	semidirect
16. C₂×M₃₂	C₂²	semidirect
18. C₂²×C₁₆	C₂²	semidirect
20. #466	C₂²	semidirect
23. #367	C₂²	semidirect
24. #366	C₂²	semidirect
25. #357	C₂²	semidirect
48. C₂×C₁₆	D₄	semidirect

H ≅ G/N

type

1. C₂²×D₁₆

C₂

semidirect

2. C₂×#504

C₂

semidirect

3. #3058

C₂

semidirect

4. #2218

C₂

semidirect

5. #2215

C₂

semidirect

6. #2158

C₂

semidirect

7. #2140

C₂

semidirect

16. C₂×M₃₂

C₂²

semidirect

18. C₂²×C₁₆

C₂²

semidirect

20. #466

C₂²

semidirect

23. #367

C₂²

semidirect

24. #366

C₂²

semidirect

25. #357

C₂²

semidirect

48. C₂×C₁₆

D₄

semidirect

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

subgroup	index
C₂²×D₁₆	2
C₂×#504	2
#3058	2
#2218	2
#2215	2
#2158	2
#2140	2

subgroup

index

C₂²×D₁₆

Subgroups — #33607

Conjugacy classes of subgroups

Complements

Maximal subgroups