Group #19635 — subgroups

Subgroups — #19635

Order 256.

439 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 439 conjugacy classes of subgroups — too many to draw. Only its 127 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	#3208	C₂	1	2	yes	maximal, complemented
2	#3170	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#493	C₂	1	1	yes	maximal, characteristic
4	#3138	C₂	1	2	yes	maximal, complemented
5	#1717	C₂	1	4	yes	maximal, complemented
6	#1709	C₂	1	2	yes	maximal, complemented
7	#1706	C₂	1	2	yes	maximal, complemented
8	C₈×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₈	—	2	2	no	—
10	#533	C₂²	1	2	yes	—
11	C₂×#127	C₂²	1	2	yes	—
12	#500	C₂²	1	1	yes	characteristic, complemented
13	#494	C₂²	1	1	yes	characteristic, complemented
14	#493	—	2	2	no	—
15	#493	C₂²	1	2	yes	—
16	#493	C₂²	1	2	yes	—
17	#492	C₂²	1	4	yes	complemented
18	#490	—	2	4	no	—
19	#487	C₂²	1	2	yes	complemented
20	#486	C₂²	1	4	yes	complemented
21	#486	C₂²	1	4	yes	complemented
22	#486	C₂²	1	4	yes	complemented
23	#467	—	2	2	no	—
24	#460	—	2	4	no	—
25	#450	—	2	4	no	—
26	C₄×Q₁₆	—	2	2	no	—
27	#431	C₂²	1	1	yes	characteristic
28	#418	—	2	4	no	—
29	C₄×M₁₆	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
31	#322	C₂²	1	2	yes	complemented
32	C₈²	C₂²	1	2	yes	complemented
33	C₂²×D₄	—	2	2	no	—
34	C₂²×D₄	—	2	2	no	—
35	C₂²×D₄	—	2	2	no	—
36	C₂×Q₁₆	—	2	2	no	—
37	C₂×SD₁₆	—	2	8	no	—
38	C₂×D₈	—	2	4	no	—
39	C₂×D₈	—	2	4	no	—
40	C₂×D₈	—	4	8	no	—
41	C₂×M₁₆	D₄	1	2	yes	—
42	C₂²×C₈	D₄	1	2	yes	—
43	C₄⋊Q₈	C₂³	1	2	yes	—
44	C₄⋊Q₈	C₂³	1	2	yes	—
45	#127	—	2	2	no	—
46	#127	—	2	2	no	—
47	#127	—	2	2	no	—
48	#127	C₂³	1	4	yes	—
49	#125	—	2	2	no	—
50	C₂²⋊Q₈	—	2	2	no	—
51	C₂²⋊Q₈	—	2	2	no	—
52	C₄×Q₈	—	2	2	no	—
53	C₄×Q₈	—	2	2	no	—
54	#117	—	2	2	no	—
55	C₂×C₄²	C₂³	1	1	yes	characteristic
56	#107	—	2	2	no	—
57	C₄⋊C₈	D₄	1	2	yes	—
58	#103	—	2	4	no	—
59	#102	—	2	8	no	—
60	#102	—	2	8	no	—
61	#102	—	2	8	no	—
62	C₂²⋊C₈	D₄	1	2	yes	—
63	#97	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	C₂⁴	—	4	2	no	—
70	C₂×Q₈	—	2	2	no	—
71	C₂×Q₈	—	2	2	no	—
72	C₂×D₄	—	2	4	no	—
73	C₂×D₄	—	2	4	no	—
74	C₂×D₄	—	2	4	no	—
75	C₂×D₄	—	4	8	no	—
76	C₂×D₄	—	4	2	no	—
77	C₂×D₄	—	4	2	no	—
78	C₂×D₄	—	4	2	no	—
79	C₂×D₄	—	4	8	no	—
80	C₂×D₄	—	8	4	no	—
81	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
84	Q₁₆	—	4	4	no	—
85	SD₁₆	—	8	8	no	—
86	D₈	—	4	8	no	—
87	D₈	—	4	8	no	—
88	M₁₆	—	2	4	no	—
89	C₂×C₈	—	2	4	no	—
90	C₂×C₈	C₂×D₄	1	2	yes	—
91	C₂×C₈	C₂×D₄	1	4	yes	—
92	C₂×C₈	C₂×D₄	1	4	yes	—
93	C₂×C₈	C₂×D₄	1	2	yes	—
94	C₄⋊C₄	—	2	2	no	—
95	C₄⋊C₄	—	2	2	no	—
96	C₄⋊C₄	—	2	2	no	—
97	C₄⋊C₄	—	2	2	no	—
98	C₄⋊C₄	—	4	2	no	—
99	C₄⋊C₄	—	4	2	no	—
100	C₂²⋊C₄	—	4	2	no	—
101	C₄²	—	4	2	no	—
102	C₄²	C₂×D₄	1	1	yes	characteristic
103	C₄²	C₂×D₄	1	1	yes	characteristic
104	C₄²	C₂×D₄	1	1	yes	characteristic
105	C₄²	C₂⁴	1	1	yes	characteristic
106	C₂³	—	4	4	no	—
107	C₂³	—	8	4	no	—
108	C₂³	—	8	4	no	—
109	C₂³	—	8	4	no	—
110	C₂³	#127	1	1	yes	characteristic
111	Q₈	—	4	4	no	—
112	Q₈	—	4	4	no	—
113	D₄	—	4	8	no	—
114	D₄	—	4	8	no	—
115	D₄	—	4	8	no	—
116	D₄	—	8	4	no	—
117	D₄	—	8	4	no	—
118	D₄	—	8	4	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	4	2	no	—
122	C₂×C₄	—	4	2	no	—
123	C₂×C₄	#127	1	1	yes	characteristic
124	C₂×C₄	#127	1	1	yes	characteristic
125	C₂×C₄	#127	1	1	yes	characteristic
126	C₂×C₄	#135	1	4	yes	—
127	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
129	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
130	C₈	—	2	4	no	—
131	C₈	—	2	2	no	—
132	C₈	—	2	4	no	—
133	C₈	AGL(1,ℤ/8)	1	4	yes	—
134	C₂²	—	2	1	no	—
135	C₂²	—	2	1	no	—
136	C₂²	—	8	4	no	—
137	C₂²	—	8	4	no	—
138	C₂²	—	8	4	no	—
139	C₂²	—	16	8	no	—
140	C₂²	#495	1	2	yes	—
141	C₂²	C₂×#127	1	1	yes	characteristic
142	C₄	—	2	1	no	—
143	C₄	—	2	1	no	—
144	C₄	—	8	2	no	—
145	C₄	—	8	2	no	—
146	C₄	#495	1	2	yes	—
147	C₄	C₂×#135	1	2	yes	—
148	C₄	C₂×AGL(1,ℤ/8)	1	2	yes	—
149	C₄	#576	1	2	yes	—
150	C₂	—	2	2	no	—
151	C₂	—	16	4	no	—
152	C₂	C₂×#495	1	1	yes	characteristic
153	C₂	#3151	1	1	yes	characteristic
154	C₂	#3155	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#3208

C₂

yes

maximal, complemented

#3170

C₂

yes

maximal, characteristic, complemented

C₂×#493

C₂

yes

maximal, characteristic

#3138

C₂

yes

maximal, complemented

#1717

C₂

yes

maximal, complemented

#1709

C₂

yes

maximal, complemented

#1706

C₂

yes

maximal, complemented

C₈×M₁₆

C₂

yes

maximal, characteristic, complemented

C₂²×D₈

—

#533

C₂²

yes

—

C₂×#127

C₂²

yes

—

#500

C₂²

yes

characteristic, complemented

#494

C₂²

yes

characteristic, complemented

#493

—

#493

C₂²

yes

—

#493

C₂²

yes

—

#492

C₂²

yes

complemented

#490

—

#487

C₂²

yes

complemented

#486

C₂²

yes

complemented

#486

C₂²

yes

complemented

#486

C₂²

yes

—

—

—

C₄×Q₁₆

—

#431

C₂²

yes

characteristic

#418

—

C₄×M₁₆

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

#322

C₂²

yes

complemented

C₈²

C₂²

yes

complemented

C₂²×D₄

—

C₂²×D₄

—

C₂²×D₄

—

C₂×Q₁₆

—

C₂×SD₁₆

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×M₁₆

D₄

yes

—

C₂²×C₈

D₄

yes

—

C₄⋊Q₈

C₂³

yes

—

C₄⋊Q₈

C₂³

yes

—

—

—

—

C₂³

yes

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂³

yes

characteristic

#107

—

C₄⋊C₈

D₄

yes

—

—

—

—

—

C₂²⋊C₈

D₄

yes

—

#97

D₄

yes

—

C₄×C₈

D₄

yes

—

C₄×C₈

C₂³

yes

characteristic

C₄×C₈

C₂³

yes

—

C₄×C₈

C₂³

yes

—

C₄×C₈

C₂³

yes

characteristic

C₂⁴

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

Q₁₆

—

SD₁₆

—

D₈

—

D₈

—

M₁₆

—

C₂×C₈

—

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

—

100

C₂²⋊C₄

—

101

C₄²

—

102

C₄²

C₂×D₄

yes

characteristic

103

C₄²

C₂×D₄

yes

characteristic

104

C₄²

C₂×D₄

yes

characteristic

105

C₄²

C₂⁴

yes

characteristic

106

C₂³

—

107

C₂³

—

108

C₂³

—

109

C₂³

—

110

C₂³

#127

yes

characteristic

111

Q₈

—

112

Q₈

—

113

D₄

—

114

D₄

—

115

D₄

—

116

D₄

—

117

D₄

—

118

D₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

#127

yes

characteristic

124

C₂×C₄

#127

yes

characteristic

125

C₂×C₄

#127

yes

characteristic

126

C₂×C₄

#135

yes

—

127

C₂×C₄

C₂²×D₄

yes

characteristic

128

C₂×C₄

C₂²×D₄

yes

characteristic

129

C₂×C₄

C₂²×D₄

yes

characteristic

130

C₈

—

131

C₈

—

132

C₈

—

133

C₈

AGL(1,ℤ/8)

yes

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

—

139

C₂²

—

140

C₂²

#495

yes

—

141

C₂²

C₂×#127

yes

characteristic

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

#495

yes

—

147

C₄

C₂×#135

yes

—

148

C₄

C₂×AGL(1,ℤ/8)

yes

—

149

C₄

#576

yes

—

150

C₂

—

151

C₂

—

152

C₂

C₂×#495

yes

characteristic

153

C₂

#3151

yes

characteristic

154

C₂

#3155

yes

characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴
prime radical	C₂×#493	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. #3208	C₂	semidirect
2. #3170	C₂	semidirect
4. #3138	C₂	semidirect
5. #1717	C₂	semidirect
6. #1709	C₂	semidirect
7. #1706	C₂	semidirect
8. C₈×M₁₆	C₂	semidirect
12. #500	C₂²	semidirect
13. #494	C₂²	semidirect
17. #492	C₂²	semidirect
19. #487	C₂²	semidirect
20. #486	C₂²	semidirect
21. #486	C₂²	semidirect
22. #486	C₂²	semidirect
31. #322	C₂²	semidirect
32. C₈²	C₂²	semidirect

H ≅ G/N

type

1. #3208

C₂

semidirect

2. #3170

C₂

semidirect

4. #3138

C₂

semidirect

5. #1717

C₂

semidirect

6. #1709

C₂

semidirect

7. #1706

C₂

semidirect

8. C₈×M₁₆

C₂

semidirect

12. #500

C₂²

semidirect

13. #494

C₂²

semidirect

17. #492

C₂²

semidirect

19. #487

C₂²

semidirect

20. #486

C₂²

semidirect

21. #486

C₂²

semidirect

22. #486

C₂²

semidirect

31. #322

C₂²

semidirect

32. C₈²

C₂²

semidirect

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

subgroup	index
#3208	2
#3170	2
C₂×#493	2
#3138	2
#1717	2
#1709	2
#1706	2
C₈×M₁₆	2

subgroup

index

C₈×M₁₆

Subgroups — #19635

Conjugacy classes of subgroups

Complements

Maximal subgroups