Group #2441 — subgroups

Subgroups — #2441

Order 128.

246 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 112 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	C₂×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#117	C₂	1	1	yes	maximal, characteristic
3	#399	C₂	1	2	yes	maximal
4	#396	C₂	1	2	yes	maximal
5	#389	C₂	1	2	yes	maximal, complemented
6	#389	C₂	1	1	yes	maximal, characteristic, complemented
7	#389	C₂	1	1	yes	maximal, characteristic, complemented
8	#387	C₂	1	2	yes	maximal, complemented
9	#384	C₂	1	1	yes	maximal, characteristic, complemented
10	#380	C₂	1	1	yes	maximal, characteristic
11	C₄×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
12	C₂³×C₄	C₂²	1	1	yes	characteristic
13	#125	—	2	4	no	—
14	#117	—	2	4	no	—
15	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₄	C₂²	1	2	yes	—
18	C₂×C₄⋊C₄	C₂²	1	2	yes	—
19	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄	C₂²	1	2	yes	—
21	C₂×C₄⋊C₄	C₂²	1	2	yes	—
22	C₂×C₄⋊C₄	C₂²	1	2	yes	—
23	C₂×C₄⋊C₄	C₂²	1	2	yes	—
24	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊C₄	C₂²	1	2	yes	—
26	C₂×C₂²⋊C₄	C₂²	1	1	yes	characteristic
27	C₂×C₂²⋊C₄	C₂²	1	1	yes	characteristic
28	C₂×C₂²⋊C₄	C₂²	1	2	yes	—
29	C₂×C₄²	C₂²	1	2	yes	—
30	C₂×C₄²	C₂²	1	1	yes	characteristic
31	C₂×C₄²	C₂²	1	1	yes	characteristic
32	#95	C₂²	1	1	yes	characteristic
33	#95	C₂²	1	1	yes	characteristic
34	#95	C₂²	1	2	yes	—
35	#95	C₂²	1	2	yes	—
36	#95	C₂²	1	2	yes	—
37	C₂⁴	C₂³	1	1	yes	characteristic
38	C₂²×C₄	—	2	2	no	—
39	C₂²×C₄	—	2	2	no	—
40	C₂²×C₄	Q₈	1	4	yes	—
41	C₂²×C₄	C₂³	1	1	yes	characteristic
42	C₂²×C₄	C₂³	1	2	yes	—
43	C₂²×C₄	C₂³	1	2	yes	—
44	C₂²×C₄	C₂³	1	1	yes	characteristic
45	C₂²×C₄	C₂³	1	2	yes	—
46	C₂²×C₄	C₂³	1	2	yes	—
47	C₂²×C₄	C₂³	1	1	yes	characteristic
48	C₂²×C₄	C₂³	1	1	yes	characteristic
49	C₂²×C₄	C₂³	1	1	yes	characteristic
50	C₂²×C₄	C₂³	1	1	yes	characteristic
51	C₄⋊C₄	—	2	2	no	—
52	C₄⋊C₄	—	2	4	no	—
53	C₄⋊C₄	—	2	4	no	—
54	C₄⋊C₄	—	2	2	no	—
55	C₄⋊C₄	—	2	4	no	—
56	C₄⋊C₄	—	2	4	no	—
57	C₄⋊C₄	—	2	2	no	—
58	C₄⋊C₄	—	4	2	no	—
59	C₄⋊C₄	—	4	2	no	—
60	C₄⋊C₄	D₄	1	4	yes	—
61	C₂²⋊C₄	—	2	4	no	—
62	C₂²⋊C₄	—	2	2	no	—
63	C₂²⋊C₄	—	2	2	no	—
64	C₂²⋊C₄	—	4	2	no	—
65	C₄²	—	2	4	no	—
66	C₄²	—	2	2	no	—
67	C₄²	—	2	2	no	—
68	C₂³	—	2	1	no	—
69	C₂³	—	2	1	no	—
70	C₂³	—	2	1	no	—
71	C₂³	—	2	1	no	—
72	C₂³	—	2	1	no	—
73	C₂³	—	2	1	no	—
74	C₂³	C₂×Q₈	1	2	yes	—
75	C₂³	C₂⁴	1	1	yes	characteristic
76	C₂×C₄	—	2	1	no	—
77	C₂×C₄	—	2	1	no	—
78	C₂×C₄	—	2	1	no	—
79	C₂×C₄	—	2	2	no	—
80	C₂×C₄	—	2	1	no	—
81	C₂×C₄	—	2	1	no	—
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	2	1	no	—
84	C₂×C₄	—	2	8	no	—
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	2	1	no	—
87	C₂×C₄	—	2	1	no	—
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	2	1	no	—
90	C₂×C₄	—	2	1	no	—
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	—	2	1	no	—
99	C₂×C₄	—	2	1	no	—
100	C₂×C₄	C₂×D₄	1	4	yes	—
101	C₂×C₄	C₂×D₄	1	2	yes	—
102	C₂×C₄	C₂×Q₈	1	2	yes	—
103	C₂×C₄	C₂×Q₈	1	2	yes	—
104	C₂×C₄	#41	1	2	yes	—
105	C₂×C₄	#41	1	2	yes	—
106	C₂×C₄	#41	1	2	yes	—
107	C₂²	—	2	2	no	—
108	C₂²	—	2	2	no	—
109	C₂²	—	2	2	no	—
110	C₂²	—	4	1	no	—
111	C₂²	—	4	1	no	—
112	C₂²	—	4	1	no	—
113	C₂²	—	4	1	no	—
114	C₂²	C₂²×D₄	1	1	yes	characteristic
115	C₂²	C₂²×Q₈	1	1	yes	characteristic
116	C₂²	C₂×#41	1	1	yes	characteristic
117	C₂²	C₂×#41	1	1	yes	characteristic
118	C₂²	C₂×#41	1	1	yes	characteristic
119	C₂²	H(5,𝔽₂)	1	1	yes	characteristic
120	C₂²	2₋¹⁺⁴	1	1	yes	characteristic
121	C₄	—	2	2	no	—
122	C₄	—	2	4	no	—
123	C₄	—	2	2	no	—
124	C₄	—	2	2	no	—
125	C₄	—	4	1	no	—
126	C₄	—	4	2	no	—
127	C₄	—	4	1	no	—
128	C₄	—	4	2	no	—
129	C₄	—	4	2	no	—
130	C₄	C₂²⋊Q₈	1	4	yes	—
131	C₂	—	4	2	no	—
132	C₂	C₂×C₂²⋊Q₈	1	1	yes	characteristic
133	C₂	#533	1	1	yes	characteristic
134	C₂	#544	1	1	yes	characteristic
135	C₂	#547	1	1	yes	characteristic
136	C₂	#550	1	1	yes	characteristic
137	C₂	#552	1	1	yes	characteristic
138	C₂	#553	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#125

C₂

yes

maximal, characteristic, complemented

C₂×#117

C₂

yes

maximal, characteristic

#399

C₂

yes

maximal

#396

C₂

yes

maximal

#389

C₂

yes

maximal, complemented

#389

C₂

yes

maximal, characteristic, complemented

#389

C₂

yes

maximal, characteristic, complemented

#387

C₂

yes

maximal, complemented

#384

C₂

yes

maximal, characteristic, complemented

#380

C₂

yes

maximal, characteristic

C₄×C₂²⋊C₄

C₂

yes

maximal, characteristic

C₂³×C₄

C₂²

yes

characteristic

#125

—

#117

—

C₂×C₄⋊C₄

C₂²

yes

characteristic

C₂×C₄⋊C₄

C₂²

yes

characteristic

C₂×C₄⋊C₄

C₂²

yes

—

C₂×C₄⋊C₄

C₂²

yes

—

C₂×C₄⋊C₄

C₂²

yes

characteristic

C₂×C₄⋊C₄

C₂²

yes

—

C₂×C₄⋊C₄

C₂²

yes

—

C₂×C₄⋊C₄

C₂²

yes

—

C₂×C₄⋊C₄

C₂²

yes

—

C₂×C₄⋊C₄

C₂²

yes

characteristic

C₂×C₂²⋊C₄

C₂²

yes

—

C₂×C₂²⋊C₄

C₂²

yes

characteristic

C₂×C₂²⋊C₄

C₂²

yes

characteristic

C₂×C₂²⋊C₄

C₂²

yes

—

C₂×C₄²

C₂²

yes

—

C₂×C₄²

C₂²

yes

characteristic

C₂×C₄²

C₂²

yes

characteristic

#95

C₂²

yes

characteristic

#95

C₂²

yes

characteristic

#95

C₂²

yes

—

#95

C₂²

yes

—

#95

C₂²

yes

—

C₂⁴

C₂³

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

Q₈

yes

—

C₂²×C₄

C₂³

yes

characteristic

C₂²×C₄

C₂³

yes

—

C₂²×C₄

C₂³

yes

—

C₂²×C₄

C₂³

yes

characteristic

C₂²×C₄

C₂³

yes

—

C₂²×C₄

C₂³

yes

—

C₂²×C₄

C₂³

yes

characteristic

C₂²×C₄

C₂³

yes

characteristic

C₂²×C₄

C₂³

yes

characteristic

C₂²×C₄

C₂³

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

D₄

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂×Q₈

yes

—

C₂³

C₂⁴

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

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

—

102

C₂×C₄

C₂×Q₈

yes

—

103

C₂×C₄

C₂×Q₈

yes

—

104

C₂×C₄

#41

yes

—

105

C₂×C₄

#41

yes

—

106

C₂×C₄

#41

yes

—

107

C₂²

—

108

C₂²

—

109

C₂²

—

110

C₂²

—

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

C₂²×D₄

yes

characteristic

115

C₂²

C₂²×Q₈

yes

characteristic

116

C₂²

C₂×#41

yes

characteristic

117

C₂²

C₂×#41

yes

characteristic

118

C₂²

C₂×#41

yes

characteristic

119

C₂²

H(5,𝔽₂)

yes

characteristic

120

C₂²

2₋¹⁺⁴

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

C₂²⋊Q₈

yes

—

131

C₂

—

132

C₂

C₂×C₂²⋊Q₈

yes

characteristic

133

C₂

#533

yes

characteristic

134

C₂

#544

yes

characteristic

135

C₂

#547

yes

characteristic

136

C₂

#550

yes

characteristic

137

C₂

#552

yes

characteristic

138

C₂

#553

yes

characteristic

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁴
prime radical	C₂⁴	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

subgroup

G/H

center, commutator, Frattini, socle, cosocle

C₂³

C₂⁴

prime radical

C₂⁴

C₂³

LCS

G ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂³ ◁ G

LECS

G ▷ C₂³ ▷ C₁

N	H ≅ G/N	type
1. C₂×#125	C₂	semidirect
5. #389	C₂	semidirect
6. #389	C₂	semidirect
7. #389	C₂	semidirect
8. #387	C₂	semidirect
9. #384	C₂	semidirect

H ≅ G/N

type

1. C₂×#125

C₂

semidirect

5. #389

C₂

semidirect

6. #389

C₂

semidirect

7. #389

C₂

semidirect

8. #387

C₂

semidirect

9. #384

C₂

semidirect

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

subgroup	index
C₂×#125	2
C₂×#117	2
#399	2
#396	2
#389	2
#387	2
#384	2
#380	2
C₄×C₂²⋊C₄	2

subgroup

index

C₄×C₂²⋊C₄

Subgroups — #2441

Conjugacy classes of subgroups

Complements

Maximal subgroups