Group #3098 — subgroups

Subgroups — #3098

Order 128.

199 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 199 conjugacy classes of subgroups — too many to draw. Only its 92 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	#537	C₂	1	1	yes	maximal, characteristic
2	#520	C₂	1	1	yes	maximal, characteristic, complemented
3	#485	C₂	1	2	yes	maximal, complemented
4	#484	C₂	1	2	yes	maximal, complemented
5	#482	C₂	1	2	yes	maximal, complemented
6	#480	C₂	1	2	yes	maximal, complemented
7	#428	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#107	C₂	1	1	yes	maximal, characteristic, complemented
9	#417	C₂	1	1	yes	maximal, characteristic, complemented
10	#416	C₂	1	1	yes	maximal, characteristic, complemented
11	#408	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41	C₂²	1	1	yes	characteristic
13	C₂×#41	C₂²	1	1	yes	characteristic
14	C₂×M₁₆	C₂²	1	1	yes	characteristic
15	C₂²×C₈	C₂²	1	1	yes	characteristic
16	C₂²⋊Q₈	—	2	1	no	—
17	C₂²⋊Q₈	C₂²	1	2	yes	—
18	#121	—	2	1	no	—
19	#121	—	2	2	no	—
20	#121	C₂²	1	2	yes	—
21	C₄×Q₈	—	2	1	no	—
22	C₄×D₄	—	2	2	no	—
23	C₄×D₄	—	2	1	no	—
24	#117	C₂²	1	1	yes	characteristic
25	#117	C₂²	1	2	yes	—
26	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊C₄	C₂²	1	2	yes	—
28	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
29	#107	C₂²	1	2	yes	—
30	#107	C₂²	1	2	yes	—
31	#107	C₂²	1	2	yes	—
32	#106	C₂²	1	2	yes	—
33	#103	C₂²	1	1	yes	characteristic
34	#103	C₂²	1	1	yes	characteristic
35	#103	C₂²	1	2	yes	—
36	#102	C₂²	1	1	yes	characteristic
37	#102	C₂²	1	1	yes	characteristic
38	#102	C₂²	1	2	yes	—
39	C₂²⋊C₈	C₂²	1	2	yes	—
40	C₂²⋊C₈	C₂²	1	2	yes	—
41	#41	—	2	2	no	—
42	#41	—	2	2	no	—
43	#41	—	2	2	no	—
44	#41	—	2	2	no	—
45	C₂×Q₈	D₄	1	1	yes	characteristic
46	C₂×Q₈	C₂³	1	1	yes	characteristic
47	C₂×D₄	—	2	1	no	—
48	C₂×D₄	—	2	2	no	—
49	C₂×D₄	D₄	1	2	yes	—
50	C₂×D₄	D₄	1	1	yes	characteristic
51	C₂×D₄	C₂³	1	1	yes	characteristic
52	C₂²×C₄	—	2	2	no	—
53	C₂²×C₄	—	2	1	no	—
54	C₂²×C₄	—	2	1	no	—
55	C₂²×C₄	—	2	1	no	—
56	C₂²×C₄	C₂³	1	2	yes	—
57	C₂²×C₄	C₂³	1	1	yes	characteristic
58	M₁₆	—	2	2	no	—
59	C₂×C₈	—	2	2	no	—
60	C₂×C₈	C₂³	1	2	yes	—
61	C₂×C₈	C₂³	1	1	yes	characteristic
62	C₂×C₈	C₂³	1	1	yes	characteristic
63	C₄⋊C₄	—	2	2	no	—
64	C₄⋊C₄	—	2	1	no	—
65	C₄⋊C₄	—	2	1	no	—
66	C₄⋊C₄	—	2	2	no	—
67	C₄⋊C₄	C₂³	1	1	yes	characteristic
68	C₄⋊C₄	C₂³	1	1	yes	characteristic
69	C₄⋊C₄	C₂³	1	2	yes	—
70	C₄⋊C₄	C₂³	1	2	yes	—
71	C₂²⋊C₄	—	2	2	no	—
72	C₂²⋊C₄	—	2	1	no	—
73	C₂²⋊C₄	—	2	2	no	—
74	C₂²⋊C₄	—	2	2	no	—
75	C₄²	—	2	2	no	—
76	C₄²	—	2	1	no	—
77	C₂³	—	2	1	no	—
78	C₂³	C₂×D₄	1	2	yes	—
79	C₂³	C₂×D₄	1	1	yes	characteristic
80	Q₈	—	2	2	no	—
81	Q₈	—	2	1	no	—
82	Q₈	—	2	1	no	—
83	D₄	—	2	4	no	—
84	D₄	—	2	2	no	—
85	D₄	—	2	1	no	—
86	D₄	—	2	1	no	—
87	D₄	—	4	2	no	—
88	D₄	—	8	2	no	—
89	C₂×C₄	—	2	2	no	—
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	—	2	1	no	—
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	2	1	no	—
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	2	1	no	—
96	C₂×C₄	—	4	2	no	—
97	C₂×C₄	—	8	1	no	—
98	C₂×C₄	—	8	2	no	—
99	C₂×C₄	—	8	1	no	—
100	C₂×C₄	C₂×D₄	1	2	yes	—
101	C₂×C₄	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄	#41	1	2	yes	—
103	C₂×C₄	#41	1	2	yes	—
104	C₂×C₄	C₂⁴	1	1	yes	characteristic
105	C₈	—	2	1	no	—
106	C₈	—	2	1	no	—
107	C₈	—	2	2	no	—
108	C₂²	—	2	1	no	—
109	C₂²	—	2	1	no	—
110	C₂²	—	2	2	no	—
111	C₂²	—	2	2	no	—
112	C₂²	—	2	2	no	—
113	C₂²	—	4	1	no	—
114	C₂²	—	4	1	no	—
115	C₂²	—	4	1	no	—
116	C₂²	#123	1	2	yes	—
117	C₂²	C₂²×D₄	1	1	yes	characteristic
118	C₄	—	2	2	no	—
119	C₄	—	4	1	no	—
120	C₄	—	4	2	no	—
121	C₄	—	4	1	no	—
122	C₄	—	4	2	no	—
123	C₄	—	4	1	no	—
124	C₄	#123	1	2	yes	—
125	C₄	C₂×#41	1	1	yes	characteristic
126	C₄	C₂×#41	1	1	yes	characteristic
127	C₂	—	2	2	no	—
128	C₂	—	4	2	no	—
129	C₂	—	8	1	no	—
130	C₂	C₂×#123	1	1	yes	characteristic
131	C₂	#576	1	1	yes	characteristic
132	C₂	#578	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#537

C₂

yes

maximal, characteristic

#520

C₂

yes

maximal, characteristic, complemented

#485

C₂

yes

maximal, complemented

#484

C₂

yes

maximal, complemented

#482

C₂

yes

maximal, complemented

#480

C₂

yes

maximal, complemented

#428

C₂

yes

maximal, characteristic, complemented

C₂×#107

C₂

yes

maximal, characteristic, complemented

#417

C₂

yes

maximal, characteristic, complemented

#416

C₂

yes

maximal, characteristic, complemented

#408

C₂

yes

maximal, characteristic, complemented

C₂×#41

C₂²

yes

characteristic

C₂×#41

C₂²

yes

characteristic

C₂×M₁₆

C₂²

yes

characteristic

C₂²×C₈

C₂²

yes

characteristic

C₂²⋊Q₈

—

C₂²⋊Q₈

C₂²

yes

—

#121

—

#121

—

#121

C₂²

yes

—

C₄×Q₈

—

C₄×D₄

—

C₄×D₄

—

#117

C₂²

yes

characteristic

#117

C₂²

yes

—

C₂×C₄⋊C₄

C₂²

yes

characteristic

C₂×C₄⋊C₄

C₂²

yes

—

C₂×C₄⋊C₄

C₂²

yes

characteristic

#107

C₂²

yes

—

#107

C₂²

yes

—

#107

C₂²

yes

—

#106

C₂²

yes

—

#103

C₂²

yes

characteristic

#103

C₂²

yes

characteristic

#103

C₂²

yes

—

#102

C₂²

yes

characteristic

#102

C₂²

yes

characteristic

#102

C₂²

yes

—

C₂²⋊C₈

C₂²

yes

—

C₂²⋊C₈

C₂²

yes

—

—

—

—

—

C₂×Q₈

D₄

yes

characteristic

C₂×Q₈

C₂³

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂×D₄

D₄

yes

—

C₂×D₄

D₄

yes

characteristic

C₂×D₄

C₂³

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂³

yes

—

C₂²×C₄

C₂³

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

C₂³

yes

—

C₂×C₈

C₂³

yes

characteristic

C₂×C₈

C₂³

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂³

yes

characteristic

C₄⋊C₄

C₂³

yes

characteristic

C₄⋊C₄

C₂³

yes

—

C₄⋊C₄

C₂³

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₂³

—

C₂³

C₂×D₄

yes

—

C₂³

C₂×D₄

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

D₄

—

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

—

100

C₂×C₄

C₂×D₄

yes

—

101

C₂×C₄

C₂×D₄

yes

characteristic

102

C₂×C₄

#41

yes

—

103

C₂×C₄

#41

yes

—

104

C₂×C₄

C₂⁴

yes

characteristic

105

C₈

—

106

C₈

—

107

C₈

—

108

C₂²

—

109

C₂²

—

110

C₂²

—

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

—

115

C₂²

—

116

C₂²

#123

yes

—

117

C₂²

C₂²×D₄

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

#123

yes

—

125

C₄

C₂×#41

yes

characteristic

126

C₄

C₂×#41

yes

characteristic

127

C₂

—

128

C₂

—

129

C₂

—

130

C₂

C₂×#123

yes

characteristic

131

C₂

#576

yes

characteristic

132

C₂

#578

yes

characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴
prime radical	#537	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ 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 ◁ C₂×#41 ◁ G

LECS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
2. #520	C₂	semidirect
3. #485	C₂	semidirect
4. #484	C₂	semidirect
5. #482	C₂	semidirect
6. #480	C₂	semidirect
7. #428	C₂	semidirect
8. C₂×#107	C₂	semidirect
9. #417	C₂	semidirect
10. #416	C₂	semidirect
11. #408	C₂	semidirect

H ≅ G/N

type

2. #520

C₂

semidirect

3. #485

C₂

semidirect

4. #484

C₂

semidirect

5. #482

C₂

semidirect

6. #480

C₂

semidirect

7. #428

C₂

semidirect

8. C₂×#107

C₂

semidirect

9. #417

C₂

semidirect

10. #416

C₂

semidirect

11. #408

C₂

semidirect

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

subgroup	index
#537	2
#520	2
#485	2
#484	2
#482	2
#480	2
#428	2
C₂×#107	2
#417	2
#416	2
#408	2

subgroup

index

Subgroups — #3098

Conjugacy classes of subgroups

Complements

Maximal subgroups