Group #3173 — subgroups

Subgroups — #3173

Order 128.

181 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 181 conjugacy classes of subgroups — too many to draw. Only its 88 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	#542	C₂	1	2	yes	maximal, complemented
2	#498	C₂	1	1	yes	maximal, characteristic, complemented
3	#495	C₂	1	1	yes	maximal, characteristic, complemented
4	#489	C₂	1	2	yes	maximal, complemented
5	#470	C₂	1	1	yes	maximal, characteristic, complemented
6	#469	C₂	1	1	yes	maximal, characteristic
7	#465	C₂	1	2	yes	maximal, complemented
8	#442	C₂	1	1	yes	maximal, characteristic, complemented
9	#441	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×SD₁₆	C₂	1	2	yes	maximal, complemented
11	#405	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆	C₂²	1	1	yes	characteristic
13	C₂×SD₁₆	C₂²	1	2	yes	—
14	C₂×D₈	C₂²	1	1	yes	characteristic
15	C₂×M₁₆	C₂²	1	1	yes	characteristic
16	C₂²×C₈	C₂²	1	1	yes	characteristic
17	C₄⋊Q₈	C₂²	1	2	yes	—
18	#126	—	2	2	no	—
19	#124	—	2	2	no	—
20	#124	C₂²	1	2	yes	—
21	#123	—	2	2	no	—
22	C₂²⋊Q₈	—	2	2	no	—
23	C₂²⋊Q₈	C₂²	1	2	yes	—
24	#121	C₂²	1	2	yes	—
25	C₄×Q₈	C₂²	1	2	yes	—
26	C₄×D₄	C₂²	1	2	yes	—
27	#117	C₂²	1	1	yes	characteristic
28	#106	C₂²	1	2	yes	—
29	#106	C₂²	1	1	yes	characteristic
30	#106	C₂²	1	1	yes	characteristic
31	#103	C₂²	1	2	yes	—
32	#103	C₂²	1	2	yes	—
33	#102	C₂²	1	2	yes	—
34	#102	C₂²	1	2	yes	—
35	#97	C₂²	1	1	yes	characteristic
36	#97	C₂²	1	1	yes	characteristic
37	C₄×C₈	C₂²	1	1	yes	characteristic
38	C₄×C₈	C₂²	1	1	yes	characteristic
39	C₂×Q₈	—	2	2	no	—
40	C₂×Q₈	C₂³	1	2	yes	—
41	C₂×D₄	—	2	2	no	—
42	C₂×D₄	C₂³	1	2	yes	—
43	C₂²×C₄	—	2	2	no	—
44	C₂²×C₄	C₂³	1	1	yes	characteristic
45	Q₁₆	—	2	2	no	—
46	SD₁₆	—	2	4	no	—
47	D₈	—	2	2	no	—
48	M₁₆	—	2	2	no	—
49	C₂×C₈	—	2	2	no	—
50	C₂×C₈	C₂³	1	1	yes	characteristic
51	C₂×C₈	C₂³	1	1	yes	characteristic
52	C₂×C₈	C₂³	1	1	yes	characteristic
53	C₂×C₈	C₂³	1	1	yes	characteristic
54	C₄⋊C₄	—	2	2	no	—
55	C₄⋊C₄	—	2	2	no	—
56	C₄⋊C₄	—	2	2	no	—
57	C₄⋊C₄	D₄	1	2	yes	—
58	C₄⋊C₄	C₂³	1	2	yes	—
59	C₄⋊C₄	C₂³	1	2	yes	—
60	C₂²⋊C₄	—	2	2	no	—
61	C₂²⋊C₄	—	2	2	no	—
62	C₂²⋊C₄	—	2	2	no	—
63	C₂²⋊C₄	—	2	2	no	—
64	C₂²⋊C₄	—	2	2	no	—
65	C₂²⋊C₄	D₄	1	2	yes	—
66	C₄²	—	2	2	no	—
67	C₄²	C₂³	1	1	yes	characteristic
68	C₄²	C₂³	1	1	yes	characteristic
69	C₂³	—	2	2	no	—
70	C₂³	C₂×D₄	1	1	yes	characteristic
71	Q₈	—	2	2	no	—
72	Q₈	—	2	2	no	—
73	Q₈	—	8	2	no	—
74	D₄	—	2	2	no	—
75	D₄	—	2	2	no	—
76	D₄	—	8	2	no	—
77	C₂×C₄	—	2	1	no	—
78	C₂×C₄	—	2	1	no	—
79	C₂×C₄	—	2	2	no	—
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	—	2	2	no	—
82	C₂×C₄	—	8	2	no	—
83	C₂×C₄	C₂×D₄	1	2	yes	—
84	C₂×C₄	C₂×D₄	1	2	yes	—
85	C₂×C₄	C₂×D₄	1	1	yes	characteristic
86	C₂×C₄	C₂⁴	1	1	yes	characteristic
87	C₈	—	2	1	no	—
88	C₈	—	2	1	no	—
89	C₈	#41	1	2	yes	—
90	C₈	#41	1	2	yes	—
91	C₂²	—	2	1	no	—
92	C₂²	—	2	2	no	—
93	C₂²	—	4	2	no	—
94	C₂²	—	4	2	no	—
95	C₂²	—	4	2	no	—
96	C₂²	C₂²×D₄	1	1	yes	characteristic
97	C₄	—	2	1	no	—
98	C₄	—	2	2	no	—
99	C₄	—	2	2	no	—
100	C₄	—	4	2	no	—
101	C₄	—	4	2	no	—
102	C₄	—	4	2	no	—
103	C₄	C₂×#41	1	1	yes	characteristic
104	C₄	C₂×#41	1	1	yes	characteristic
105	C₂	—	4	1	no	—
106	C₂	—	8	2	no	—
107	C₂	#532	1	1	yes	characteristic
108	C₂	#577	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#542

C₂

yes

maximal, complemented

#498

C₂

yes

maximal, characteristic, complemented

#495

C₂

yes

maximal, characteristic, complemented

#489

C₂

yes

maximal, complemented

#470

C₂

yes

maximal, characteristic, complemented

#469

C₂

yes

maximal, characteristic

#465

C₂

yes

maximal, complemented

#442

C₂

yes

maximal, characteristic, complemented

#441

C₂

yes

maximal, characteristic, complemented

C₄×SD₁₆

C₂

yes

maximal, complemented

#405

C₂

yes

maximal, characteristic, complemented

C₂×Q₁₆

C₂²

yes

characteristic

C₂×SD₁₆

C₂²

yes

—

C₂×D₈

C₂²

yes

characteristic

C₂×M₁₆

C₂²

yes

characteristic

C₂²×C₈

C₂²

yes

characteristic

C₄⋊Q₈

C₂²

yes

—

#126

—

#124

—

#124

C₂²

yes

—

#123

—

C₂²⋊Q₈

—

C₂²⋊Q₈

C₂²

yes

—

#121

C₂²

yes

—

C₄×Q₈

C₂²

yes

—

C₄×D₄

C₂²

yes

—

#117

C₂²

yes

characteristic

#106

C₂²

yes

—

#106

C₂²

yes

characteristic

#106

C₂²

yes

characteristic

#103

C₂²

yes

—

#103

C₂²

yes

—

#102

C₂²

yes

—

#102

C₂²

yes

—

#97

C₂²

yes

characteristic

#97

C₂²

yes

characteristic

C₄×C₈

C₂²

yes

characteristic

C₄×C₈

C₂²

yes

characteristic

C₂×Q₈

—

C₂×Q₈

C₂³

yes

—

C₂×D₄

—

C₂×D₄

C₂³

yes

—

C₂²×C₄

—

C₂²×C₄

C₂³

yes

characteristic

Q₁₆

—

SD₁₆

—

D₈

—

M₁₆

—

C₂×C₈

—

C₂×C₈

C₂³

yes

characteristic

C₂×C₈

C₂³

yes

characteristic

C₂×C₈

C₂³

yes

characteristic

C₂×C₈

C₂³

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

D₄

yes

—

C₄⋊C₄

C₂³

yes

—

C₄⋊C₄

C₂³

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

D₄

yes

—

C₄²

—

C₄²

C₂³

yes

characteristic

C₄²

C₂³

yes

characteristic

C₂³

—

C₂³

C₂×D₄

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×D₄

yes

—

C₂×C₄

C₂×D₄

yes

—

C₂×C₄

C₂×D₄

yes

characteristic

C₂×C₄

C₂⁴

yes

characteristic

C₈

—

C₈

—

C₈

#41

yes

—

C₈

#41

yes

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

C₂²×D₄

yes

characteristic

C₄

—

C₄

—

C₄

—

100

C₄

—

101

C₄

—

102

C₄

—

103

C₄

C₂×#41

yes

characteristic

104

C₄

C₂×#41

yes

characteristic

105

C₂

—

106

C₂

—

107

C₂

#532

yes

characteristic

108

C₂

#577

yes

—

	subgroup	G/H
center, socle	C₂²	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴
prime radical	#469	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #117 ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

C₂²×D₄

commutator, Frattini, cosocle

C₂×C₄

C₂⁴

C₂

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

C₁ ◁ C₂² ◁ #117 ◁ G

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
1. #542	C₂	semidirect
2. #498	C₂	semidirect
3. #495	C₂	semidirect
4. #489	C₂	semidirect
5. #470	C₂	semidirect
7. #465	C₂	semidirect
8. #442	C₂	semidirect
9. #441	C₂	semidirect
10. C₄×SD₁₆	C₂	semidirect
11. #405	C₂	semidirect

H ≅ G/N

type

1. #542

C₂

semidirect

2. #498

C₂

semidirect

3. #495

C₂

semidirect

4. #489

C₂

semidirect

5. #470

C₂

semidirect

7. #465

C₂

semidirect

8. #442

C₂

semidirect

9. #441

C₂

semidirect

10. C₄×SD₁₆

C₂

semidirect

11. #405

C₂

semidirect

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

subgroup	index
#542	2
#498	2
#495	2
#489	2
#470	2
#469	2
#465	2
#442	2
#441	2
C₄×SD₁₆	2
#405	2

subgroup

index

C₄×SD₁₆

Subgroups — #3173

Conjugacy classes of subgroups

Complements

Maximal subgroups