Group #16858850 — subgroups

Subgroups — #16858850

Order 3968.

370 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 129 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	#14521854	C₂	1	1	yes	maximal, characteristic, complemented
2	#14521485	C₂	1	3	yes	maximal, complemented
3	#14521484	C₂	1	3	yes	maximal, complemented
4	#14521479	C₂	1	3	yes	maximal, complemented
5	C₂×#11758414	C₂	1	3	yes	maximal, complemented
6	#14521466	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#576	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×H(5,𝔽₂)	—	2	1	no	—
9	#11758566	—	2	1	no	—
10	C₃₁×AGL(1,ℤ/8)	—	2	3	no	—
11	C₃₁×#135	C₂²	1	3	yes	complemented
12	C₂×D₈×C₃₁	C₂²	1	3	yes	complemented
13	C₃₁×#131	C₂²	1	1	yes	characteristic
14	#11758485	—	2	3	no	—
15	#11758481	C₂²	1	3	yes	complemented
16	C₂×D₂₄₈	C₂²	1	3	yes	complemented
17	#11758417	C₂²	1	1	yes	characteristic, complemented
18	#11758416	C₂²	1	3	yes	complemented
19	#11758415	C₂²	1	3	yes	complemented
20	#11758414	C₂²	1	3	yes	complemented
21	#11758414	C₂²	1	6	yes	complemented
22	#11758400	C₂²	1	3	yes	complemented
23	C₂×C₃₁⋊C₁₆	C₂²	1	3	yes	complemented
24	#41×C₃₁	—	2	3	no	—
25	#41×C₃₁	—	4	1	no	—
26	#41×C₃₁	D₄	1	1	yes	characteristic
27	C₂×D₄×C₃₁	—	2	3	no	—
28	C₂×D₄×C₃₁	—	4	3	no	—
29	#92499	—	4	1	no	—
30	D₄×D₃₁	—	4	3	no	—
31	#92495	—	2	3	no	—
32	C₂×D₁₂₄	—	2	3	no	—
33	Q₁₆×C₃₁	C₂³	1	1	yes	characteristic
34	SD₁₆×C₃₁	—	2	3	no	—
35	D₈×C₃₁	—	2	3	no	—
36	D₈×C₃₁	C₂³	1	3	yes	—
37	M₁₆×C₃₁	D₄	1	3	yes	—
38	C₂×C₈×C₃₁	C₂³	1	3	yes	—
39	Dic₁₂₄	C₂³	1	1	yes	characteristic
40	D₂₄₈	—	2	3	no	—
41	D₂₄₈	C₂³	1	3	yes	—
42	#92470	—	2	3	no	—
43	C₃₁⋊C₁₆	C₂³	1	1	yes	characteristic
44	C₃₁⋊C₁₆	C₂³	1	3	yes	—
45	C₂³×C₃₁	—	4	3	no	—
46	C₂²×D₃₁	—	4	3	no	—
47	Q₈×C₃₁	—	2	1	no	—
48	Q₈×C₃₁	D₈	1	1	yes	characteristic
49	D₄×C₃₁	—	2	3	no	—
50	D₄×C₃₁	—	4	3	no	—
51	D₄×C₃₁	—	4	3	no	—
52	D₄×C₃₁	D₈	1	3	yes	—
53	C₂×C₄×C₃₁	—	4	3	no	—
54	C₂×C₄×C₃₁	C₂×D₄	1	3	yes	—
55	C₂²⋊D₃₁	—	4	3	no	—
56	D₁₂₄	—	2	3	no	—
57	D₁₂₄	—	4	3	no	—
58	C₄×D₃₁	—	4	3	no	—
59	Dic₆₂	—	2	1	no	—
60	C₈×C₃₁	C₂×D₄	1	3	yes	—
61	C₈×C₃₁	C₂⁴	1	1	yes	characteristic
62	#3415	—	31	1	no	maximal, Sylow
63	C₂²×C₃₁	—	4	3	no	—
64	C₂²×C₃₁	—	16	3	no	—
65	C₂²×C₃₁	C₂×D₈	1	3	yes	—
66	C₂×D₃₁	—	4	3	no	—
67	C₂×D₃₁	—	16	3	no	—
68	C₄×C₃₁	—	4	1	no	—
69	C₄×C₃₁	C₂×D₈	1	3	yes	—
70	C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
71	Dic₃₁	—	4	1	no	—
72	#576	—	31	1	no	—
73	#576	D₃₁	1	1	yes	characteristic, complemented
74	#509	—	31	3	no	—
75	#509	—	31	3	no	—
76	#508	—	31	3	no	—
77	C₂×D₁₆	—	31	3	no	—
78	#504	—	31	1	no	—
79	C₂×C₃₁	—	2	3	no	—
80	C₂×C₃₁	—	8	3	no	—
81	C₂×C₃₁	C₂²×D₈	1	1	yes	characteristic
82	D₃₁	—	8	3	no	—
83	H(5,𝔽₂)	—	2	1	no	—
84	H(5,𝔽₂)	—	62	1	no	—
85	AGL(1,ℤ/8)	—	2	3	no	—
86	AGL(1,ℤ/8)	—	62	3	no	—
87	#135	—	31	3	no	—
88	#135	C₂×D₃₁	1	3	yes	complemented
89	C₂×D₈	—	31	3	no	—
90	C₂×D₈	C₂×D₃₁	1	3	yes	complemented
91	#131	C₂×D₃₁	1	1	yes	characteristic
92	Q₃₂	—	31	1	no	—
93	SD₃₂	—	31	3	no	—
94	SD₃₂	—	31	3	no	—
95	D₁₆	—	31	3	no	—
96	D₁₆	—	31	6	no	—
97	M₃₂	—	31	3	no	—
98	C₂×C₁₆	—	31	3	no	—
99	C₃₁	#3415	1	1	yes	characteristic, complemented, Sylow
100	#41	—	2	3	no	—
101	#41	—	4	1	no	—
102	#41	—	62	3	no	—
103	#41	—	124	1	no	—
104	#41	C₂²⋊D₃₁	1	1	yes	characteristic
105	C₂×D₄	—	2	3	no	—
106	C₂×D₄	—	4	3	no	—
107	C₂×D₄	—	62	3	no	—
108	C₂×D₄	—	124	3	no	—
109	Q₁₆	—	31	1	no	—
110	Q₁₆	C₂²×D₃₁	1	1	yes	characteristic
111	SD₁₆	—	2	3	no	—
112	SD₁₆	—	62	3	no	—
113	D₈	—	2	3	no	—
114	D₈	—	31	3	no	—
115	D₈	—	62	3	no	—
116	D₈	C₂²×D₃₁	1	3	yes	—
117	M₁₆	C₂²⋊D₃₁	1	3	yes	—
118	C₂×C₈	C₂²×D₃₁	1	3	yes	—
119	C₁₆	—	31	1	no	—
120	C₁₆	—	31	3	no	—
121	C₂³	—	4	3	no	—
122	C₂³	—	124	3	no	—
123	Q₈	—	2	1	no	—
124	Q₈	—	62	1	no	—
125	Q₈	#92479	1	1	yes	characteristic
126	D₄	—	2	3	no	—
127	D₄	—	4	3	no	—
128	D₄	—	4	3	no	—
129	D₄	—	62	3	no	—
130	D₄	—	124	3	no	—
131	D₄	—	124	3	no	—
132	D₄	#92479	1	3	yes	—
133	C₂×C₄	—	4	3	no	—
134	C₂×C₄	—	124	3	no	—
135	C₂×C₄	C₂×C₂²⋊D₃₁	1	3	yes	—
136	C₈	C₂×C₂²⋊D₃₁	1	3	yes	—
137	C₈	C₂³×D₃₁	1	1	yes	characteristic
138	C₂²	—	4	3	no	—
139	C₂²	—	16	3	no	—
140	C₂²	—	124	3	no	—
141	C₂²	—	496	3	no	—
142	C₂²	C₂×#92479	1	3	yes	—
143	C₄	—	4	1	no	—
144	C₄	—	124	1	no	—
145	C₄	C₂×#92479	1	3	yes	—
146	C₄	C₂²×C₂²⋊D₃₁	1	1	yes	characteristic
147	C₂	—	2	3	no	—
148	C₂	—	8	3	no	—
149	C₂	—	248	3	no	—
150	C₂	C₂²×#92479	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14521854

C₂

yes

maximal, characteristic, complemented

#14521485

C₂

yes

maximal, complemented

#14521484

C₂

yes

maximal, complemented

#14521479

C₂

yes

maximal, complemented

C₂×#11758414

C₂

yes

maximal, complemented

#14521466

C₂

yes

maximal, characteristic, complemented

C₃₁×#576

C₂

yes

maximal, characteristic, complemented

C₃₁×H(5,𝔽₂)

—

#11758566

—

C₃₁×AGL(1,ℤ/8)

—

C₃₁×#135

C₂²

yes

complemented

C₂×D₈×C₃₁

C₂²

yes

complemented

C₃₁×#131

C₂²

yes

characteristic

#11758485

—

#11758481

C₂²

yes

complemented

C₂×D₂₄₈

C₂²

yes

complemented

#11758417

C₂²

yes

characteristic, complemented

#11758416

C₂²

yes

complemented

#11758415

C₂²

yes

complemented

#11758414

C₂²

yes

complemented

#11758414

C₂²

yes

complemented

#11758400

C₂²

yes

complemented

C₂×C₃₁⋊C₁₆

C₂²

yes

complemented

#41×C₃₁

—

#41×C₃₁

—

#41×C₃₁

D₄

yes

characteristic

C₂×D₄×C₃₁

—

C₂×D₄×C₃₁

—

#92499

—

D₄×D₃₁

—

#92495

—

C₂×D₁₂₄

—

Q₁₆×C₃₁

C₂³

yes

characteristic

SD₁₆×C₃₁

—

D₈×C₃₁

—

D₈×C₃₁

C₂³

yes

—

M₁₆×C₃₁

D₄

yes

—

C₂×C₈×C₃₁

C₂³

yes

—

Dic₁₂₄

C₂³

yes

characteristic

D₂₄₈

—

D₂₄₈

C₂³

yes

—

#92470

—

C₃₁⋊C₁₆

C₂³

yes

characteristic

C₃₁⋊C₁₆

C₂³

yes

—

C₂³×C₃₁

—

C₂²×D₃₁

—

Q₈×C₃₁

—

Q₈×C₃₁

D₈

yes

characteristic

D₄×C₃₁

—

D₄×C₃₁

—

D₄×C₃₁

—

D₄×C₃₁

D₈

yes

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂×D₄

yes

—

C₂²⋊D₃₁

—

D₁₂₄

—

D₁₂₄

—

C₄×D₃₁

—

Dic₆₂

—

C₈×C₃₁

C₂×D₄

yes

—

C₈×C₃₁

C₂⁴

yes

characteristic

#3415

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

C₂×D₈

yes

—

C₂×D₃₁

—

C₂×D₃₁

—

C₄×C₃₁

—

C₄×C₃₁

C₂×D₈

yes

—

C₄×C₃₁

C₂²×D₄

yes

characteristic

Dic₃₁

—

#576

—

#576

D₃₁

yes

characteristic, complemented

#509

—

#509

—

#508

—

C₂×D₁₆

—

#504

—

C₂×C₃₁

—

C₂×C₃₁

—

C₂×C₃₁

C₂²×D₈

yes

characteristic

D₃₁

—

—

—

—

—

—

C₂×D₃₁

yes

complemented

C₂×D₈

—

C₂×D₈

C₂×D₃₁

yes

complemented

#131

C₂×D₃₁

yes

characteristic

Q₃₂

—

SD₃₂

—

SD₃₂

—

D₁₆

—

D₁₆

—

M₃₂

—

C₂×C₁₆

—

C₃₁

#3415

yes

characteristic, complemented, Sylow

100

#41

—

101

#41

—

102

#41

—

103

#41

—

124

—

104

#41

C₂²⋊D₃₁

yes

characteristic

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂×D₄

—

124

—

109

Q₁₆

—

110

Q₁₆

C₂²×D₃₁

yes

characteristic

111

SD₁₆

—

112

SD₁₆

—

113

D₈

—

114

D₈

—

115

D₈

—

116

D₈

C₂²×D₃₁

yes

—

117

M₁₆

C₂²⋊D₃₁

yes

—

118

C₂×C₈

C₂²×D₃₁

yes

—

119

C₁₆

—

120

C₁₆

—

121

C₂³

—

122

C₂³

—

124

—

123

Q₈

—

124

Q₈

—

125

Q₈

#92479

yes

characteristic

126

D₄

—

127

D₄

—

128

D₄

—

129

D₄

—

130

D₄

—

124

—

131

D₄

—

124

—

132

D₄

#92479

yes

—

133

C₂×C₄

—

134

C₂×C₄

—

124

—

135

C₂×C₄

C₂×C₂²⋊D₃₁

yes

—

136

C₈

C₂×C₂²⋊D₃₁

yes

—

137

C₈

C₂³×D₃₁

yes

characteristic

138

C₂²

—

139

C₂²

—

140

C₂²

—

124

—

141

C₂²

—

496

—

142

C₂²

C₂×#92479

yes

—

143

C₄

—

144

C₄

—

124

—

145

C₄

C₂×#92479

yes

—

146

C₄

C₂²×C₂²⋊D₃₁

yes

characteristic

147

C₂

—

148

C₂

—

149

C₂

—

248

—

150

C₂

C₂²×#92479

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×#92479
commutator, cosocle	C₈×C₃₁	C₂⁴
Frattini	C₈	C₂³×D₃₁
Fitting	C₃₁×#576	C₂
socle	C₂×C₃₁	C₂²×D₈
hypercenter	#576	D₃₁
hypocenter	C₃₁	#3415
LCS	G ▷ C₈×C₃₁ ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131 ◁ #576

subgroup

G/H

center

C₂

C₂²×#92479

commutator, cosocle

C₈×C₃₁

C₂⁴

Frattini

C₈

C₂³×D₃₁

Fitting

C₃₁×#576

C₂

socle

C₂×C₃₁

C₂²×D₈

D₃₁

C₃₁

G ▷ C₈×C₃₁ ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂ ◁ #41 ◁ #131 ◁ #576

N	H ≅ G/N	type
1. #14521854	C₂	semidirect
2. #14521485	C₂	semidirect
3. #14521484	C₂	semidirect
4. #14521479	C₂	semidirect
5. C₂×#11758414	C₂	semidirect
6. #14521466	C₂	semidirect
7. C₃₁×#576	C₂	semidirect
11. C₃₁×#135	C₂²	semidirect
12. C₂×D₈×C₃₁	C₂²	semidirect
15. #11758481	C₂²	semidirect
16. C₂×D₂₄₈	C₂²	semidirect
17. #11758417	C₂²	semidirect
18. #11758416	C₂²	semidirect
19. #11758415	C₂²	semidirect
20. #11758414	C₂²	semidirect
21. #11758414	C₂²	semidirect
22. #11758400	C₂²	semidirect
23. C₂×C₃₁⋊C₁₆	C₂²	semidirect
73. #576	D₃₁	semidirect
88. #135	C₂×D₃₁	semidirect
90. C₂×D₈	C₂×D₃₁	semidirect
99. C₃₁	#3415	semidirect

H ≅ G/N

type

1. #14521854

C₂

semidirect

2. #14521485

C₂

semidirect

3. #14521484

C₂

semidirect

4. #14521479

C₂

semidirect

5. C₂×#11758414

C₂

semidirect

6. #14521466

C₂

semidirect

7. C₃₁×#576

C₂

semidirect

11. C₃₁×#135

C₂²

semidirect

12. C₂×D₈×C₃₁

C₂²

semidirect

15. #11758481

C₂²

semidirect

16. C₂×D₂₄₈

C₂²

semidirect

17. #11758417

C₂²

semidirect

18. #11758416

C₂²

semidirect

19. #11758415

C₂²

semidirect

20. #11758414

C₂²

semidirect

21. #11758414

C₂²

semidirect

22. #11758400

C₂²

semidirect

23. C₂×C₃₁⋊C₁₆

C₂²

semidirect

73. #576

D₃₁

semidirect

88. #135

C₂×D₃₁

semidirect

90. C₂×D₈

C₂×D₃₁

semidirect

99. C₃₁

#3415

semidirect

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

subgroup	index
#14521854	2
#14521485	2
#14521484	2
#14521479	2
C₂×#11758414	2
#14521466	2
C₃₁×#576	2
#3415	31

subgroup

index

C₃₁×#576

Subgroups — #16858850

Conjugacy classes of subgroups

Complements

Maximal subgroups