Group #33741 — subgroups

Subgroups — #33741

Order 256.

242 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 103 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	#3177	C₂	1	2	yes	maximal
2	#2253	C₂	1	1	yes	maximal, characteristic, complemented
3	#2247	C₂	1	1	yes	maximal, characteristic, complemented
4	#2245	C₂	1	2	yes	maximal, complemented
5	#2219	C₂	1	2	yes	maximal
6	#2216	C₂	1	2	yes	maximal
7	#2176	C₂	1	2	yes	maximal, complemented
8	C₄×Q₃₂	C₂	1	2	yes	maximal, complemented
9	#2108	C₂	1	1	yes	maximal, characteristic
10	#541	—	2	2	no	—
11	C₂×Q₃₂	C₂²	1	2	yes	—
12	C₂×Q₃₂	C₂²	1	2	yes	—
13	C₂×M₃₂	C₂²	1	1	yes	characteristic
14	C₂²×C₁₆	C₂²	1	1	yes	characteristic
15	#499	C₂²	1	2	yes	—
16	#494	C₂²	1	2	yes	—
17	#491	—	2	2	no	—
18	#470	—	2	2	no	—
19	#467	C₂²	1	4	yes	—
20	#441	—	2	2	no	—
21	C₄×Q₁₆	C₂²	1	4	yes	—
22	#405	C₂²	1	1	yes	characteristic
23	#367	C₂²	1	2	yes	—
24	#366	C₂²	1	2	yes	—
25	#358	C₂²	1	4	yes	—
26	#358	C₂²	1	4	yes	—
27	#346	C₂²	1	1	yes	characteristic
28	#346	C₂²	1	1	yes	characteristic
29	C₄×C₁₆	C₂²	1	1	yes	characteristic
30	C₄×C₁₆	C₂²	1	1	yes	characteristic
31	C₂×Q₁₆	—	2	2	no	—
32	C₂×Q₁₆	C₂³	1	4	yes	—
33	C₂×M₁₆	D₄	1	1	yes	characteristic
34	C₂²×C₈	C₂³	1	1	yes	characteristic
35	C₄⋊Q₈	—	2	2	no	—
36	#126	—	4	4	no	—
37	#125	—	2	2	no	—
38	#125	—	4	4	no	—
39	C₂²⋊Q₈	—	2	4	no	—
40	C₄×Q₈	—	2	4	no	—
41	#117	D₄	1	1	yes	characteristic
42	Q₃₂	—	2	4	no	—
43	Q₃₂	—	2	4	no	—
44	M₃₂	—	2	2	no	—
45	C₂×C₁₆	—	2	2	no	—
46	C₂×C₁₆	C₂³	1	1	yes	characteristic
47	C₂×C₁₆	C₂³	1	1	yes	characteristic
48	C₂×C₁₆	C₂³	1	1	yes	characteristic
49	C₂×C₁₆	C₂³	1	1	yes	characteristic
50	#107	C₂³	1	4	yes	—
51	#106	—	2	2	no	—
52	#103	—	2	4	no	—
53	#103	—	2	4	no	—
54	#97	D₄	1	1	yes	characteristic
55	#97	D₄	1	1	yes	characteristic
56	C₄×C₈	C₂³	1	1	yes	characteristic
57	C₄×C₈	C₂³	1	1	yes	characteristic
58	C₂×Q₈	—	2	4	no	—
59	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
60	Q₁₆	—	2	4	no	—
61	Q₁₆	—	2	4	no	—
62	Q₁₆	—	4	4	no	—
63	M₁₆	—	2	2	no	—
64	C₂×C₈	—	2	1	no	—
65	C₂×C₈	—	2	1	no	—
66	C₂×C₈	C₂×D₄	1	1	yes	characteristic
67	C₂×C₈	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈	C₂⁴	1	1	yes	characteristic
70	C₄⋊C₄	—	2	4	no	—
71	C₄⋊C₄	—	4	4	no	—
72	C₄⋊C₄	—	4	4	no	—
73	C₄⋊C₄	—	4	4	no	—
74	C₄⋊C₄	—	4	4	no	—
75	C₄⋊C₄	D₈	1	2	yes	—
76	C₂²⋊C₄	—	4	4	no	—
77	C₂²⋊C₄	D₈	1	2	yes	—
78	C₄²	—	4	4	no	—
79	C₄²	C₂×D₄	1	1	yes	characteristic
80	C₄²	C₂×D₄	1	1	yes	characteristic
81	C₁₆	—	2	1	no	—
82	C₁₆	—	2	1	no	—
83	C₁₆	#41	1	2	yes	—
84	C₁₆	#41	1	2	yes	—
85	C₂³	C₂×D₈	1	1	yes	characteristic
86	Q₈	—	4	4	no	—
87	Q₈	—	4	4	no	—
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	2	1	no	—
90	C₂×C₄	—	4	4	no	—
91	C₂×C₄	—	4	4	no	—
92	C₂×C₄	C₂×D₈	1	2	yes	—
93	C₂×C₄	C₂×D₈	1	2	yes	—
94	C₂×C₄	C₂×D₈	1	1	yes	characteristic
95	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
96	C₈	—	2	1	no	—
97	C₈	—	2	1	no	—
98	C₈	#135	1	2	yes	—
99	C₈	C₂×#41	1	1	yes	characteristic
100	C₈	C₂×#41	1	1	yes	characteristic
101	C₂²	—	2	2	no	—
102	C₂²	—	2	1	no	—
103	C₂²	C₂²×D₈	1	1	yes	characteristic
104	C₄	—	2	2	no	—
105	C₄	—	2	2	no	—
106	C₄	—	2	1	no	—
107	C₄	—	8	4	no	—
108	C₄	—	8	4	no	—
109	C₄	#532	1	1	yes	characteristic
110	C₄	C₂×#135	1	1	yes	characteristic
111	C₂	—	4	1	no	—
112	C₂	#3169	1	1	yes	characteristic
113	C₂	#3417	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#3177

C₂

yes

maximal

#2253

C₂

yes

maximal, characteristic, complemented

#2247

C₂

yes

maximal, characteristic, complemented

#2245

C₂

yes

maximal, complemented

#2219

C₂

yes

maximal

#2216

C₂

yes

maximal

#2176

C₂

yes

maximal, complemented

C₄×Q₃₂

C₂

yes

maximal, complemented

#2108

C₂

yes

maximal, characteristic

#541

—

C₂×Q₃₂

C₂²

yes

—

C₂×Q₃₂

C₂²

yes

—

C₂×M₃₂

C₂²

yes

characteristic

C₂²×C₁₆

C₂²

yes

characteristic

#499

C₂²

yes

—

#494

C₂²

yes

—

#491

—

#470

—

#467

C₂²

yes

—

#441

—

C₄×Q₁₆

C₂²

yes

—

#405

C₂²

yes

characteristic

#367

C₂²

yes

—

#366

C₂²

yes

—

#358

C₂²

yes

—

#358

C₂²

yes

—

#346

C₂²

yes

characteristic

#346

C₂²

yes

characteristic

C₄×C₁₆

C₂²

yes

characteristic

C₄×C₁₆

C₂²

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

C₂³

yes

—

C₂×M₁₆

D₄

yes

characteristic

C₂²×C₈

C₂³

yes

characteristic

C₄⋊Q₈

—

#126

—

#125

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

D₄

yes

characteristic

Q₃₂

—

Q₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₂³

yes

characteristic

C₂×C₁₆

C₂³

yes

characteristic

C₂×C₁₆

C₂³

yes

characteristic

C₂×C₁₆

C₂³

yes

characteristic

#107

C₂³

yes

—

—

—

—

D₄

yes

characteristic

#97

D₄

yes

characteristic

C₄×C₈

C₂³

yes

characteristic

C₄×C₈

C₂³

yes

characteristic

C₂×Q₈

—

C₂²×C₄

C₂×D₄

yes

characteristic

Q₁₆

—

Q₁₆

—

Q₁₆

—

M₁₆

—

C₂×C₈

—

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

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

D₈

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

D₈

yes

—

C₄²

—

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×D₄

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

#41

yes

—

C₁₆

#41

yes

—

C₂³

C₂×D₈

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×D₈

yes

—

C₂×C₄

C₂×D₈

yes

—

C₂×C₄

C₂×D₈

yes

characteristic

C₂×C₄

C₂²×D₄

yes

characteristic

C₈

—

C₈

—

C₈

#135

yes

—

C₈

C₂×#41

yes

characteristic

100

C₈

C₂×#41

yes

characteristic

101

C₂²

—

102

C₂²

—

103

C₂²

C₂²×D₈

yes

characteristic

104

C₄

—

105

C₄

—

106

C₄

—

107

C₄

—

108

C₄

—

109

C₄

#532

yes

characteristic

110

C₄

C₂×#135

yes

characteristic

111

C₂

—

112

C₂

#3169

yes

characteristic

113

C₂

#3417

yes

—

	subgroup	G/H
center, socle	C₂²	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
prime radical	C₂³	C₂×D₈
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

C₂³

C₂×D₈

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. #2253	C₂	semidirect
3. #2247	C₂	semidirect
4. #2245	C₂	semidirect
7. #2176	C₂	semidirect
8. C₄×Q₃₂	C₂	semidirect

H ≅ G/N

type

2. #2253

C₂

semidirect

3. #2247

C₂

semidirect

4. #2245

C₂

semidirect

7. #2176

C₂

semidirect

8. C₄×Q₃₂

C₂

semidirect

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

subgroup	index
#3177	2
#2253	2
#2247	2
#2245	2
#2219	2
#2216	2
#2176	2
C₄×Q₃₂	2
#2108	2

subgroup

index

C₄×Q₃₂

Subgroups — #33741

Conjugacy classes of subgroups

Complements

Maximal subgroups