Group #19651 — subgroups

Subgroups — #19651

Order 256.

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

normal subgroups only

This group has 403 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	#3229	C₂	1	2	yes	maximal, complemented
2	#3208	C₂	1	2	yes	maximal
3	#3160	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#492	C₂	1	1	yes	maximal, characteristic
5	#1714	C₂	1	4	yes	maximal, complemented
6	#1708	C₂	1	2	yes	maximal, complemented
7	#1705	C₂	1	2	yes	maximal, complemented
8	#1450	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×SD₁₆	—	2	2	no	—
10	#533	C₂²	1	2	yes	—
11	C₂×C₄⋊Q₈	C₂²	1	1	yes	characteristic
12	C₂×#127	C₂²	1	1	yes	characteristic
13	#500	C₂²	1	2	yes	complemented
14	#499	—	2	2	no	—
15	#498	C₂²	1	1	yes	characteristic, complemented
16	#498	C₂²	1	1	yes	characteristic, complemented
17	#498	C₂²	1	4	yes	complemented
18	#492	—	2	2	no	—
19	#492	C₂²	1	2	yes	—
20	#492	C₂²	1	2	yes	—
21	#492	C₂²	1	4	yes	—
22	#487	C₂²	1	4	yes	complemented
23	#486	C₂²	1	4	yes	—
24	#483	—	2	4	no	—
25	#475	—	2	4	no	—
26	#460	—	2	4	no	—
27	#450	—	2	4	no	—
28	#431	C₂²	1	2	yes	—
29	#427	—	2	2	no	—
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×Q₈	—	2	1	no	—
34	C₂²×D₄	—	2	1	no	—
35	C₂²×D₄	—	2	2	no	—
36	C₂×SD₁₆	—	2	4	no	—
37	C₂×SD₁₆	—	2	4	no	—
38	C₂×SD₁₆	—	2	8	no	—
39	C₂×SD₁₆	—	4	8	no	—
40	C₂²×C₈	D₄	1	2	yes	—
41	C₄⋊Q₈	—	2	2	no	—
42	C₄⋊Q₈	—	2	1	no	—
43	C₄⋊Q₈	C₂³	1	2	yes	—
44	C₄⋊Q₈	C₂³	1	2	yes	—
45	C₄⋊Q₈	C₂³	1	2	yes	—
46	#127	—	2	2	no	—
47	#127	—	2	1	no	—
48	#127	C₂³	1	2	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×C₄⋊C₄	—	2	2	no	—
56	C₂×C₄²	C₂³	1	1	yes	characteristic
57	#107	—	2	4	no	—
58	#106	—	2	2	no	—
59	#106	—	2	2	no	—
60	#106	—	2	8	no	—
61	C₄⋊C₈	D₄	1	4	yes	—
62	#103	—	2	8	no	—
63	#102	—	2	8	no	—
64	C₂²⋊C₈	D₄	1	4	yes	—
65	C₄×C₈	D₄	1	2	yes	—
66	C₄×C₈	C₂³	1	1	yes	characteristic
67	C₄×C₈	C₂³	1	4	yes	—
68	C₄×C₈	C₂³	1	1	yes	characteristic
69	C₂⁴	—	4	1	no	—
70	C₂×Q₈	—	2	2	no	—
71	C₂×Q₈	—	2	2	no	—
72	C₂×Q₈	—	2	2	no	—
73	C₂×Q₈	—	4	1	no	—
74	C₂×Q₈	—	4	4	no	—
75	C₂×D₄	—	2	4	no	—
76	C₂×D₄	—	2	2	no	—
77	C₂×D₄	—	4	2	no	—
78	C₂×D₄	—	4	1	no	—
79	C₂×D₄	—	4	4	no	—
80	C₂×D₄	—	8	4	no	—
81	C₂²×C₄	—	4	1	no	—
82	C₂²×C₄	C₂×D₄	1	2	yes	—
83	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
84	SD₁₆	—	4	8	no	—
85	SD₁₆	—	4	8	no	—
86	SD₁₆	—	8	8	no	—
87	C₂×C₈	SD₁₆	1	8	yes	complemented
88	C₂×C₈	C₂×D₄	1	8	yes	—
89	C₂×C₈	C₂×D₄	1	2	yes	—
90	C₂×C₈	C₂×D₄	1	2	yes	—
91	C₄⋊C₄	—	2	2	no	—
92	C₄⋊C₄	—	2	4	no	—
93	C₄⋊C₄	—	2	2	no	—
94	C₄⋊C₄	—	2	2	no	—
95	C₄⋊C₄	—	2	2	no	—
96	C₄⋊C₄	—	4	2	no	—
97	C₄⋊C₄	—	4	2	no	—
98	C₄⋊C₄	—	4	2	no	—
99	C₂²⋊C₄	—	4	2	no	—
100	C₄²	—	4	2	no	—
101	C₄²	C₂×D₄	1	1	yes	characteristic
102	C₄²	C₂×D₄	1	2	yes	—
103	C₄²	C₂⁴	1	1	yes	characteristic
104	C₂³	—	4	2	no	—
105	C₂³	—	8	4	no	—
106	C₂³	—	8	2	no	—
107	C₂³	#127	1	1	yes	characteristic
108	Q₈	—	4	4	no	—
109	Q₈	—	4	4	no	—
110	Q₈	—	4	4	no	—
111	Q₈	—	8	2	no	—
112	D₄	—	4	4	no	—
113	D₄	—	4	8	no	—
114	D₄	—	8	4	no	—
115	D₄	—	8	2	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	4	2	no	—
118	C₂×C₄	—	4	2	no	—
119	C₂×C₄	—	4	2	no	—
120	C₂×C₄	—	8	2	no	—
121	C₂×C₄	#127	1	2	yes	—
122	C₂×C₄	#127	1	1	yes	characteristic
123	C₂×C₄	C₂×SD₁₆	1	4	yes	—
124	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄	C₂²×D₄	1	2	yes	—
126	C₈	—	2	8	no	—
127	C₈	C₂×SD₁₆	1	4	yes	—
128	C₈	C₂×SD₁₆	1	4	yes	—
129	C₂²	—	2	2	no	—
130	C₂²	—	8	2	no	—
131	C₂²	—	8	4	no	—
132	C₂²	—	16	4	no	—
133	C₂²	#492	1	2	yes	—
134	C₂²	C₂×#127	1	1	yes	characteristic
135	C₄	—	2	2	no	—
136	C₄	—	8	2	no	—
137	C₄	—	8	2	no	—
138	C₄	—	8	2	no	—
139	C₄	#492	1	2	yes	—
140	C₄	C₂²×SD₁₆	1	2	yes	—
141	C₄	#577	1	4	yes	—
142	C₂	—	2	2	no	—
143	C₂	—	16	2	no	—
144	C₂	C₂×#492	1	1	yes	characteristic
145	C₂	#3154	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#3229

C₂

yes

maximal, complemented

#3208

C₂

yes

maximal

#3160

C₂

yes

maximal, characteristic, complemented

C₂×#492

C₂

yes

maximal, characteristic

#1714

C₂

yes

maximal, complemented

#1708

C₂

yes

maximal, complemented

#1705

C₂

yes

maximal, complemented

#1450

C₂

yes

maximal, characteristic, complemented

C₂²×SD₁₆

—

#533

C₂²

yes

—

C₂×C₄⋊Q₈

C₂²

yes

characteristic

C₂×#127

C₂²

yes

characteristic

#500

C₂²

yes

complemented

#499

—

#498

C₂²

yes

characteristic, complemented

#498

C₂²

yes

characteristic, complemented

#498

C₂²

yes

complemented

#492

—

#492

C₂²

yes

—

#492

C₂²

yes

—

#492

C₂²

yes

—

#487

C₂²

yes

complemented

#486

C₂²

yes

—

—

—

—

—

C₂²

yes

—

#427

—

C₂×C₄×C₈

C₂²

yes

characteristic

#322

C₂²

yes

complemented

C₈²

C₂²

yes

complemented

C₂²×Q₈

—

C₂²×D₄

—

C₂²×D₄

—

C₂×SD₁₆

—

C₂×SD₁₆

—

C₂×SD₁₆

—

C₂×SD₁₆

—

C₂²×C₈

D₄

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂³

yes

—

C₄⋊Q₈

C₂³

yes

—

C₄⋊Q₈

C₂³

yes

—

#127

—

#127

—

#127

C₂³

yes

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂³

yes

—

—

—

—

C₄⋊C₈

D₄

yes

—

#103

—

#102

—

C₂²⋊C₈

D₄

yes

—

C₄×C₈

D₄

yes

—

C₄×C₈

C₂³

yes

characteristic

C₄×C₈

C₂³

yes

—

C₄×C₈

C₂³

yes

characteristic

C₂⁴

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

—

C₂²×C₄

C₂×D₄

yes

characteristic

SD₁₆

—

SD₁₆

—

SD₁₆

—

C₂×C₈

SD₁₆

yes

complemented

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

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

100

C₄²

—

101

C₄²

C₂×D₄

yes

characteristic

102

C₄²

C₂×D₄

yes

—

103

C₄²

C₂⁴

yes

characteristic

104

C₂³

—

105

C₂³

—

106

C₂³

—

107

C₂³

#127

yes

characteristic

108

Q₈

—

109

Q₈

—

110

Q₈

—

111

Q₈

—

112

D₄

—

113

D₄

—

114

D₄

—

115

D₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

#127

yes

—

122

C₂×C₄

#127

yes

characteristic

123

C₂×C₄

C₂×SD₁₆

yes

—

124

C₂×C₄

C₂²×D₄

yes

characteristic

125

C₂×C₄

C₂²×D₄

yes

—

126

C₈

—

127

C₈

C₂×SD₁₆

yes

—

128

C₈

C₂×SD₁₆

yes

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

#492

yes

—

134

C₂²

C₂×#127

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

#492

yes

—

140

C₄

C₂²×SD₁₆

yes

—

141

C₄

#577

yes

—

142

C₂

—

143

C₂

—

144

C₂

C₂×#492

yes

characteristic

145

C₂

#3154

yes

—

	subgroup	G/H
center, socle	C₂²	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴
prime radical	C₂×#127	C₂²
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

C₂²

G ▷ C₄² ▷ C₂² ▷ C₁

C₁ ◁ C₂² ◁ C₂×C₄² ◁ G

LECS

G ▷ C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
1. #3229	C₂	semidirect
3. #3160	C₂	semidirect
5. #1714	C₂	semidirect
6. #1708	C₂	semidirect
7. #1705	C₂	semidirect
8. #1450	C₂	semidirect
13. #500	C₂²	semidirect
15. #498	C₂²	semidirect
16. #498	C₂²	semidirect
17. #498	C₂²	semidirect
22. #487	C₂²	semidirect
31. #322	C₂²	semidirect
32. C₈²	C₂²	semidirect
87. C₂×C₈	SD₁₆	semidirect

H ≅ G/N

type

1. #3229

C₂

semidirect

3. #3160

C₂

semidirect

5. #1714

C₂

semidirect

6. #1708

C₂

semidirect

7. #1705

C₂

semidirect

8. #1450

C₂

semidirect

13. #500

C₂²

semidirect

15. #498

C₂²

semidirect

16. #498

C₂²

semidirect

17. #498

C₂²

semidirect

22. #487

C₂²

semidirect

31. #322

C₂²

semidirect

32. C₈²

C₂²

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
#3229	2
#3208	2
#3160	2
C₂×#492	2
#1714	2
#1708	2
#1705	2
#1450	2

subgroup

index

Subgroups — #19651

Conjugacy classes of subgroups

Complements

Maximal subgroups