Group #19655 — subgroups

Subgroups — #19655

Order 256.

399 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 399 conjugacy classes of subgroups — too many to draw. Only its 139 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	#3227	C₂	1	2	yes	maximal, complemented
2	#3206	C₂	1	2	yes	maximal, complemented
3	C₂×#500	C₂	1	1	yes	maximal, characteristic, complemented
4	#3147	C₂	1	1	yes	maximal, characteristic
5	#1714	C₂	1	4	yes	maximal, complemented
6	#1707	C₂	1	2	yes	maximal, complemented
7	#1706	C₂	1	2	yes	maximal, complemented
8	#1450	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#135	—	2	2	no	—
10	#532	C₂²	1	2	yes	—
11	C₂×C₄⋊Q₈	C₂²	1	2	yes	—
12	#500	—	2	2	no	—
13	#500	C₂²	1	2	yes	—
14	#500	C₂²	1	2	yes	—
15	#498	C₂²	1	4	yes	—
16	#495	—	2	2	no	—
17	#494	C₂²	1	1	yes	characteristic
18	#493	C₂²	1	1	yes	characteristic
19	#492	C₂²	1	2	yes	—
20	#492	C₂²	1	4	yes	—
21	#487	C₂²	1	4	yes	—
22	#486	C₂²	1	4	yes	—
23	#481	—	2	4	no	—
24	#476	—	2	4	no	—
25	#461	—	2	4	no	—
26	#448	—	2	4	no	—
27	#431	C₂²	1	1	yes	characteristic
28	#431	C₂²	1	1	yes	characteristic
29	C₂×#107	—	2	2	no	—
30	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
31	#322	C₂²	1	2	yes	—
32	C₈²	C₂²	1	2	yes	—
33	C₂×#41	—	2	2	no	—
34	C₂²×Q₈	—	2	2	no	—
35	#135	—	4	8	no	—
36	C₂×Q₁₆	—	2	2	no	—
37	C₂×SD₁₆	—	2	4	no	—
38	C₂×SD₁₆	—	2	8	no	—
39	C₂×D₈	—	2	2	no	—
40	C₂²×C₈	D₄	1	2	yes	—
41	C₄⋊Q₈	—	2	2	no	—
42	C₄⋊Q₈	—	2	2	no	—
43	C₄⋊Q₈	—	2	2	no	—
44	C₄⋊Q₈	C₂³	1	4	yes	—
45	C₄⋊Q₈	C₂³	1	2	yes	—
46	#127	C₂³	1	2	yes	—
47	#124	—	2	2	no	—
48	#121	—	2	2	no	—
49	#121	—	2	2	no	—
50	C₄×D₄	—	2	2	no	—
51	C₄×D₄	—	2	2	no	—
52	C₂×C₄⋊C₄	—	2	2	no	—
53	C₂×C₄⋊C₄	—	2	2	no	—
54	C₂×C₄²	C₂³	1	1	yes	characteristic
55	#107	—	2	4	no	—
56	#107	—	2	4	no	—
57	#106	—	2	8	no	—
58	C₄⋊C₈	D₄	1	2	yes	—
59	C₄⋊C₈	D₄	1	2	yes	—
60	#103	—	2	8	no	—
61	#102	—	2	8	no	—
62	C₂²⋊C₈	D₄	1	2	yes	—
63	C₂²⋊C₈	D₄	1	2	yes	—
64	C₄×C₈	D₄	1	2	yes	—
65	C₄×C₈	C₂³	1	1	yes	characteristic
66	C₄×C₈	C₂³	1	2	yes	—
67	C₄×C₈	C₂³	1	2	yes	—
68	C₄×C₈	C₂³	1	1	yes	characteristic
69	#41	—	4	8	no	—
70	C₂×Q₈	—	2	4	no	—
71	C₂×Q₈	—	2	2	no	—
72	C₂×Q₈	—	4	2	no	—
73	C₂×Q₈	—	8	4	no	—
74	C₂×D₄	—	2	2	no	—
75	C₂×D₄	—	2	2	no	—
76	C₂×D₄	—	2	2	no	—
77	C₂×D₄	—	4	2	no	—
78	C₂²×C₄	—	4	2	no	—
79	C₂²×C₄	—	4	2	no	—
80	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
83	Q₁₆	—	4	4	no	—
84	SD₁₆	—	4	8	no	—
85	SD₁₆	—	8	8	no	—
86	D₈	—	4	4	no	—
87	C₂×C₈	SD₁₆	1	8	yes	complemented
88	C₂×C₈	C₂×D₄	1	4	yes	—
89	C₂×C₈	C₂×D₄	1	4	yes	—
90	C₂×C₈	C₂×D₄	1	2	yes	—
91	C₂×C₈	C₂×D₄	1	2	yes	—
92	C₄⋊C₄	—	2	4	no	—
93	C₄⋊C₄	—	2	2	no	—
94	C₄⋊C₄	—	2	2	no	—
95	C₄⋊C₄	—	2	4	no	—
96	C₄⋊C₄	—	4	2	no	—
97	C₄⋊C₄	—	4	2	no	—
98	C₂²⋊C₄	—	4	2	no	—
99	C₂²⋊C₄	—	4	2	no	—
100	C₄²	C₂×D₄	1	1	yes	characteristic
101	C₄²	C₂×D₄	1	1	yes	characteristic
102	C₄²	C₂×D₄	1	1	yes	characteristic
103	C₄²	C₂⁴	1	1	yes	characteristic
104	C₂³	—	4	2	no	—
105	C₂³	#127	1	1	yes	characteristic
106	Q₈	—	4	4	no	—
107	Q₈	—	4	8	no	—
108	Q₈	—	8	4	no	—
109	D₄	—	4	4	no	—
110	D₄	—	4	4	no	—
111	D₄	—	4	4	no	—
112	D₄	—	8	4	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	4	2	no	—
116	C₂×C₄	—	4	4	no	—
117	C₂×C₄	—	8	4	no	—
118	C₂×C₄	—	8	4	no	—
119	C₂×C₄	#127	1	1	yes	characteristic
120	C₂×C₄	#127	1	1	yes	characteristic
121	C₂×C₄	#127	1	1	yes	characteristic
122	C₂×C₄	C₂×SD₁₆	1	4	yes	—
123	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
126	C₈	—	2	4	no	—
127	C₈	—	2	4	no	—
128	C₈	C₂×SD₁₆	1	4	yes	—
129	C₈	C₂×SD₁₆	1	4	yes	—
130	C₂²	—	2	1	no	—
131	C₂²	—	2	1	no	—
132	C₂²	—	8	2	no	—
133	C₂²	—	8	2	no	—
134	C₂²	—	8	2	no	—
135	C₂²	#492	1	2	yes	—
136	C₂²	C₂×#127	1	1	yes	characteristic
137	C₄	—	2	1	no	—
138	C₄	—	2	1	no	—
139	C₄	—	8	2	no	—
140	C₄	—	8	4	no	—
141	C₄	#492	1	2	yes	—
142	C₄	C₂²×SD₁₆	1	2	yes	—
143	C₄	#576	1	2	yes	—
144	C₄	#578	1	2	yes	—
145	C₂	—	2	2	no	—
146	C₂	—	16	2	no	—
147	C₂	C₂×#492	1	1	yes	characteristic
148	C₂	#3155	1	1	yes	characteristic
149	C₂	#3156	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#3227

C₂

yes

maximal, complemented

#3206

C₂

yes

maximal, complemented

C₂×#500

C₂

yes

maximal, characteristic, complemented

#3147

C₂

yes

maximal, characteristic

#1714

C₂

yes

maximal, complemented

#1707

C₂

yes

maximal, complemented

#1706

C₂

yes

maximal, complemented

#1450

C₂

yes

maximal, characteristic, complemented

C₂×#135

—

#532

C₂²

yes

—

C₂×C₄⋊Q₈

C₂²

yes

—

#500

—

#500

C₂²

yes

—

#500

C₂²

yes

—

#498

C₂²

yes

—

#495

—

#494

C₂²

yes

characteristic

#493

C₂²

yes

characteristic

#492

C₂²

yes

—

#492

C₂²

yes

—

#487

C₂²

yes

—

#486

C₂²

yes

—

—

—

—

—

C₂²

yes

characteristic

#431

C₂²

yes

characteristic

C₂×#107

—

C₂×C₄×C₈

C₂²

yes

characteristic

#322

C₂²

yes

—

C₈²

C₂²

yes

—

C₂×#41

—

C₂²×Q₈

—

#135

—

C₂×Q₁₆

—

C₂×SD₁₆

—

C₂×SD₁₆

—

C₂×D₈

—

C₂²×C₈

D₄

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂³

yes

—

C₄⋊Q₈

C₂³

yes

—

#127

C₂³

yes

—

#124

—

#121

—

#121

—

C₄×D₄

—

C₄×D₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂³

yes

—

—

—

C₄⋊C₈

D₄

yes

—

C₄⋊C₈

D₄

yes

—

#103

—

#102

—

C₂²⋊C₈

D₄

yes

—

C₂²⋊C₈

D₄

yes

—

C₄×C₈

D₄

yes

—

C₄×C₈

C₂³

yes

characteristic

C₄×C₈

C₂³

yes

—

C₄×C₈

C₂³

yes

—

C₄×C₈

C₂³

yes

characteristic

#41

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

Q₁₆

—

SD₁₆

—

SD₁₆

—

D₈

—

C₂×C₈

SD₁₆

yes

complemented

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×D₄

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

100

C₄²

C₂×D₄

yes

characteristic

101

C₄²

C₂×D₄

yes

characteristic

102

C₄²

C₂×D₄

yes

characteristic

103

C₄²

C₂⁴

yes

characteristic

104

C₂³

—

105

C₂³

#127

yes

characteristic

106

Q₈

—

107

Q₈

—

108

Q₈

—

109

D₄

—

110

D₄

—

111

D₄

—

112

D₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

#127

yes

characteristic

120

C₂×C₄

#127

yes

characteristic

121

C₂×C₄

#127

yes

characteristic

122

C₂×C₄

C₂×SD₁₆

yes

—

123

C₂×C₄

C₂²×D₄

yes

characteristic

124

C₂×C₄

C₂²×D₄

yes

characteristic

125

C₂×C₄

C₂²×D₄

yes

characteristic

126

C₈

—

127

C₈

—

128

C₈

C₂×SD₁₆

yes

—

129

C₈

C₂×SD₁₆

yes

—

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

#492

yes

—

136

C₂²

C₂×#127

yes

characteristic

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

#492

yes

—

142

C₄

C₂²×SD₁₆

yes

—

143

C₄

#576

yes

—

144

C₄

#578

yes

—

145

C₂

—

146

C₂

—

147

C₂

C₂×#492

yes

characteristic

148

C₂

#3155

yes

characteristic

149

C₂

#3156

yes

characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴
prime radical	#3147	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center, socle

C₂²

C₂×#127

commutator, Frattini, cosocle

C₄²

C₂⁴

C₂

G ▷ C₄² ▷ C₂² ▷ C₁

C₁ ◁ C₂² ◁ C₂×C₄² ◁ G

LECS

G ▷ C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
1. #3227	C₂	semidirect
2. #3206	C₂	semidirect
3. C₂×#500	C₂	semidirect
5. #1714	C₂	semidirect
6. #1707	C₂	semidirect
7. #1706	C₂	semidirect
8. #1450	C₂	semidirect
87. C₂×C₈	SD₁₆	semidirect

H ≅ G/N

type

1. #3227

C₂

semidirect

2. #3206

C₂

semidirect

3. C₂×#500

C₂

semidirect

5. #1714

C₂

semidirect

6. #1707

C₂

semidirect

7. #1706

C₂

semidirect

8. #1450

C₂

semidirect

87. C₂×C₈

SD₁₆

semidirect

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

subgroup	index
#3227	2
#3206	2
C₂×#500	2
#3147	2
#1714	2
#1707	2
#1706	2
#1450	2

subgroup

index

Subgroups — #19655

Conjugacy classes of subgroups

Complements

Maximal subgroups