Group #3197 — subgroups

Subgroups — #3197

Order 128.

209 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 209 conjugacy classes of subgroups — too many to draw. Only its 84 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	#525	C₂	1	2	yes	maximal, complemented
2	#485	C₂	1	2	yes	maximal, complemented
3	#482	C₂	1	2	yes	maximal, complemented
4	#470	C₂	1	1	yes	maximal, characteristic, complemented
5	#469	C₂	1	1	yes	maximal, characteristic, complemented
6	#468	C₂	1	2	yes	maximal, complemented
7	#452	C₂	1	2	yes	maximal, complemented
8	#449	C₂	1	2	yes	maximal, complemented
9	#407	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#41	C₂²	1	2	yes	—
11	C₂³×C₄	C₂²	1	1	yes	characteristic
12	C₂×Q₁₆	C₂²	1	1	yes	characteristic, complemented
13	C₂×SD₁₆	C₂²	1	2	yes	complemented
14	C₂×D₈	C₂²	1	1	yes	characteristic, complemented
15	C₂×M₁₆	C₂²	1	1	yes	characteristic
16	C₂×M₁₆	C₂²	1	1	yes	characteristic
17	#123	—	2	2	no	—
18	C₂²⋊Q₈	C₂²	1	2	yes	—
19	C₂²⋊Q₈	C₂²	1	2	yes	—
20	#121	C₂²	1	2	yes	—
21	#121	C₂²	1	2	yes	—
22	#120	—	2	2	no	—
23	C₄×D₄	—	2	2	no	—
24	C₄×D₄	—	2	2	no	—
25	#117	C₂²	1	2	yes	—
26	#107	C₂²	1	2	yes	complemented
27	#106	C₂²	1	1	yes	characteristic, complemented
28	#106	C₂²	1	1	yes	characteristic, complemented
29	#103	C₂²	1	2	yes	complemented
30	#103	C₂²	1	2	yes	complemented
31	#102	C₂²	1	2	yes	complemented
32	#102	C₂²	1	2	yes	complemented
33	C₂²⋊C₈	C₂²	1	2	yes	—
34	C₂²⋊C₈	C₂²	1	2	yes	—
35	C₂⁴	D₄	1	1	yes	characteristic
36	#41	—	4	2	no	—
37	#41	—	4	2	no	—
38	C₂×Q₈	C₂³	1	2	yes	—
39	C₂×D₄	—	2	2	no	—
40	C₂×D₄	—	2	2	no	—
41	C₂×D₄	—	2	2	no	—
42	C₂×D₄	C₂³	1	2	yes	—
43	C₂²×C₄	—	2	2	no	—
44	C₂²×C₄	—	4	1	no	—
45	C₂²×C₄	—	4	1	no	—
46	C₂²×C₄	D₄	1	1	yes	characteristic
47	C₂²×C₄	D₄	1	2	yes	—
48	C₂²×C₄	C₂³	1	2	yes	—
49	C₂²×C₄	C₂³	1	1	yes	characteristic
50	Q₁₆	—	4	1	no	—
51	SD₁₆	—	4	2	no	—
52	D₈	—	4	1	no	—
53	M₁₆	—	4	1	no	—
54	M₁₆	—	4	1	no	—
55	C₂×C₈	C₂³	1	1	yes	characteristic
56	C₂×C₈	C₂³	1	2	yes	—
57	C₂×C₈	C₂³	1	1	yes	characteristic
58	C₄⋊C₄	—	2	2	no	—
59	C₄⋊C₄	—	2	2	no	—
60	C₄⋊C₄	C₂³	1	2	yes	—
61	C₄⋊C₄	C₂³	1	2	yes	—
62	C₂²⋊C₄	—	2	2	no	—
63	C₂²⋊C₄	—	2	2	no	—
64	C₂²⋊C₄	—	2	2	no	—
65	C₂²⋊C₄	—	2	2	no	—
66	C₂²⋊C₄	—	2	2	no	—
67	C₄²	—	2	2	no	—
68	C₂³	—	2	2	no	—
69	C₂³	—	4	1	no	—
70	C₂³	—	4	2	no	—
71	C₂³	C₂×D₄	1	2	yes	—
72	C₂³	C₂×D₄	1	1	yes	characteristic
73	Q₈	—	2	2	no	—
74	Q₈	—	2	2	no	—
75	D₄	—	2	2	no	—
76	D₄	—	2	2	no	—
77	D₄	—	4	4	no	—
78	D₄	—	4	4	no	—
79	D₄	—	4	4	no	—
80	C₂×C₄	—	2	1	no	—
81	C₂×C₄	—	2	1	no	—
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	2	2	no	—
84	C₂×C₄	—	2	2	no	—
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	2	2	no	—
87	C₂×C₄	—	4	1	no	—
88	C₂×C₄	—	4	2	no	—
89	C₂×C₄	—	4	4	no	—
90	C₂×C₄	C₂×D₄	1	2	yes	—
91	C₂×C₄	C₂×D₄	1	1	yes	characteristic
92	C₂×C₄	C₂⁴	1	1	yes	characteristic
93	C₈	—	2	1	no	—
94	C₈	—	2	2	no	—
95	C₈	—	2	1	no	—
96	C₂²	—	2	2	no	—
97	C₂²	—	2	1	no	—
98	C₂²	—	2	2	no	—
99	C₂²	—	2	2	no	—
100	C₂²	—	2	2	no	—
101	C₂²	—	4	4	no	—
102	C₂²	—	4	2	no	—
103	C₂²	—	4	2	no	—
104	C₂²	—	4	2	no	—
105	C₂²	C₂²×D₄	1	1	yes	characteristic
106	C₄	—	2	2	no	—
107	C₄	—	2	1	no	—
108	C₄	—	4	2	no	—
109	C₄	—	4	2	no	—
110	C₄	—	4	2	no	—
111	C₄	H(5,𝔽₂)	1	1	yes	characteristic
112	C₄	H(5,𝔽₂)	1	1	yes	characteristic
113	C₂	—	4	2	no	—
114	C₂	—	4	1	no	—
115	C₂	—	8	2	no	—
116	C₂	#534	1	1	yes	characteristic
117	C₂	#575	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#525

C₂

yes

maximal, complemented

#485

C₂

yes

maximal, complemented

#482

C₂

yes

maximal, complemented

#470

C₂

yes

maximal, characteristic, complemented

#469

C₂

yes

maximal, characteristic, complemented

#468

C₂

yes

maximal, complemented

#452

C₂

yes

maximal, complemented

#449

C₂

yes

maximal, complemented

#407

C₂

yes

maximal, characteristic, complemented

C₂×#41

C₂²

yes

—

C₂³×C₄

C₂²

yes

characteristic

C₂×Q₁₆

C₂²

yes

characteristic, complemented

C₂×SD₁₆

C₂²

yes

complemented

C₂×D₈

C₂²

yes

characteristic, complemented

C₂×M₁₆

C₂²

yes

characteristic

C₂×M₁₆

C₂²

yes

characteristic

#123

—

C₂²⋊Q₈

C₂²

yes

—

C₂²⋊Q₈

C₂²

yes

—

#121

C₂²

yes

—

#121

C₂²

yes

—

#120

—

C₄×D₄

—

C₄×D₄

—

#117

C₂²

yes

—

#107

C₂²

yes

complemented

#106

C₂²

yes

characteristic, complemented

#106

C₂²

yes

characteristic, complemented

#103

C₂²

yes

complemented

#103

C₂²

yes

complemented

#102

C₂²

yes

complemented

#102

C₂²

yes

complemented

C₂²⋊C₈

C₂²

yes

—

C₂²⋊C₈

C₂²

yes

—

C₂⁴

D₄

yes

characteristic

#41

—

#41

—

C₂×Q₈

C₂³

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂³

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₄

yes

characteristic

C₂²×C₄

D₄

yes

—

C₂²×C₄

C₂³

yes

—

C₂²×C₄

C₂³

yes

characteristic

Q₁₆

—

SD₁₆

—

D₈

—

M₁₆

—

M₁₆

—

C₂×C₈

C₂³

yes

characteristic

C₂×C₈

C₂³

yes

—

C₂×C₈

C₂³

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂³

yes

—

C₄⋊C₄

C₂³

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂×D₄

yes

—

C₂³

C₂×D₄

yes

characteristic

Q₈

—

Q₈

—

D₄

—

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×D₄

yes

—

C₂×C₄

C₂×D₄

yes

characteristic

C₂×C₄

C₂⁴

yes

characteristic

C₈

—

C₈

—

C₈

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

100

C₂²

—

101

C₂²

—

102

C₂²

—

103

C₂²

—

104

C₂²

—

105

C₂²

C₂²×D₄

yes

characteristic

106

C₄

—

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

H(5,𝔽₂)

yes

characteristic

112

C₄

H(5,𝔽₂)

yes

characteristic

113

C₂

—

114

C₂

—

115

C₂

—

116

C₂

#534

yes

characteristic

117

C₂

#575

yes

—

	subgroup	G/H
center, socle	C₂²	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

C₂²×D₄

commutator, Frattini, cosocle

C₂×C₄

C₂⁴

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G

LECS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
1. #525	C₂	semidirect
2. #485	C₂	semidirect
3. #482	C₂	semidirect
4. #470	C₂	semidirect
5. #469	C₂	semidirect
6. #468	C₂	semidirect
7. #452	C₂	semidirect
8. #449	C₂	semidirect
9. #407	C₂	semidirect
12. C₂×Q₁₆	C₂²	semidirect
13. C₂×SD₁₆	C₂²	semidirect
14. C₂×D₈	C₂²	semidirect
26. #107	C₂²	semidirect
27. #106	C₂²	semidirect
28. #106	C₂²	semidirect
29. #103	C₂²	semidirect
30. #103	C₂²	semidirect
31. #102	C₂²	semidirect
32. #102	C₂²	semidirect

H ≅ G/N

type

1. #525

C₂

semidirect

2. #485

C₂

semidirect

3. #482

C₂

semidirect

4. #470

C₂

semidirect

5. #469

C₂

semidirect

6. #468

C₂

semidirect

7. #452

C₂

semidirect

8. #449

C₂

semidirect

9. #407

C₂

semidirect

12. C₂×Q₁₆

C₂²

semidirect

13. C₂×SD₁₆

C₂²

semidirect

14. C₂×D₈

C₂²

semidirect

26. #107

C₂²

semidirect

27. #106

C₂²

semidirect

28. #106

C₂²

semidirect

29. #103

C₂²

semidirect

30. #103

C₂²

semidirect

31. #102

C₂²

semidirect

32. #102

C₂²

semidirect

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

subgroup	index
#525	2
#485	2
#482	2
#470	2
#469	2
#468	2
#452	2
#449	2
#407	2

subgroup

index

Subgroups — #3197

Conjugacy classes of subgroups

Complements

Maximal subgroups