Group #16858378 — subgroups

Subgroups — #16858378

Order 3968.

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

normal subgroups only

This group has 392 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	Q₈×Dic₆₂	C₂	1	1	yes	maximal, characteristic, complemented
2	#14521395	C₂	1	3	yes	maximal, complemented
3	#14521339	C₂	1	3	yes	maximal, complemented
4	#14521246	C₂	1	3	yes	maximal, complemented
5	Q₈×C₃₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#92480	C₂	1	3	yes	maximal, complemented
7	C₃₁×#550	C₂	1	1	yes	maximal, characteristic
8	C₂×#41×C₃₁	—	2	1	no	—
9	C₃₁×#127	C₂²	1	3	yes	—
10	C₃₁×#121	—	2	3	no	—
11	C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
12	C₄×D₄×C₃₁	C₂²	1	3	yes	—
13	Q₈×Dic₃₁	—	2	1	no	—
14	C₂×#92480	C₂²	1	3	yes	—
15	#11758465	—	2	3	no	—
16	#11758446	C₂²	1	3	yes	—
17	C₄×Dic₆₂	C₂²	1	3	yes	—
18	#11758420	C₂²	1	3	yes	—
19	#11758420	C₂²	1	6	yes	—
20	#11758398	C₂²	1	1	yes	characteristic
21	#11758396	C₂²	1	3	yes	—
22	C₄⋊(C₃₁⋊C₈)	C₂²	1	3	yes	—
23	C₄×C₃₁⋊C₈	C₂²	1	3	yes	—
24	#41×C₃₁	—	4	4	no	—
25	C₂×Q₈×C₃₁	D₄	1	1	yes	characteristic
26	C₂×D₄×C₃₁	—	2	3	no	—
27	C₂×D₄×C₃₁	—	2	3	no	—
28	C₂×D₄×C₃₁	C₂³	1	3	yes	—
29	C₂²×C₄×C₃₁	—	2	3	no	—
30	C₂×Dic₆₂	—	2	3	no	—
31	C₂×Dic₆₂	C₂³	1	1	yes	characteristic
32	C₄⋊C₄×C₃₁	D₄	1	3	yes	—
33	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
34	C₂²⋊C₄×C₃₁	—	2	3	no	—
35	C₄²×C₃₁	C₂³	1	3	yes	—
36	#92480	—	2	6	no	—
37	C₄⋊Dic₃₁	—	2	3	no	—
38	C₄⋊Dic₃₁	C₂³	1	3	yes	—
39	Dic₃₁⋊C₄	—	2	3	no	—
40	C₄×Dic₃₁	—	2	3	no	—
41	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
42	C₂×C₃₁⋊C₈	C₂³	1	3	yes	—
43	C₂³×C₃₁	—	2	3	no	—
44	Q₈×C₃₁	SD₁₆	1	4	yes	complemented
45	D₄×C₃₁	—	2	3	no	—
46	D₄×C₃₁	—	2	3	no	—
47	D₄×C₃₁	—	4	6	no	—
48	D₄×C₃₁	—	8	3	no	—
49	C₂×C₄×C₃₁	—	2	1	no	—
50	C₂×C₄×C₃₁	—	4	6	no	—
51	C₂×C₄×C₃₁	C₂×D₄	1	3	yes	—
52	C₂×C₄×C₃₁	C₂×D₄	1	3	yes	—
53	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
54	C₂×Dic₃₁	—	2	1	no	—
55	C₂×Dic₃₁	—	2	3	no	—
56	Dic₆₂	—	2	1	no	—
57	Dic₆₂	—	8	3	no	—
58	Dic₆₂	#41	1	2	yes	complemented
59	C₃₁⋊C₈	—	2	3	no	—
60	C₃₁⋊C₈	—	2	1	no	—
61	#3362	—	31	1	no	maximal, Sylow
62	C₂²×C₃₁	—	4	3	no	—
63	C₂²×C₃₁	—	4	3	no	—
64	C₂²×C₃₁	—	4	3	no	—
65	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
66	C₄×C₃₁	—	2	3	no	—
67	C₄×C₃₁	—	4	1	no	—
68	C₄×C₃₁	C₂×SD₁₆	1	6	yes	—
69	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
70	C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
71	Dic₃₁	—	2	2	no	—
72	Dic₃₁	—	4	3	no	—
73	Q₈²	—	31	1	no	—
74	#550	D₃₁	1	1	yes	characteristic
75	#486	—	31	3	no	—
76	#476	—	31	3	no	—
77	#460	—	31	3	no	—
78	C₈×Q₈	—	31	1	no	—
79	C₄×SD₁₆	—	31	3	no	—
80	C₂×C₃₁	—	8	3	no	—
81	C₂×C₃₁	#548	1	1	yes	characteristic
82	C₂×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
83	C₂×C₃₁	#576	1	1	yes	characteristic
84	C₂×#41	—	2	1	no	—
85	C₂×SD₁₆	—	31	3	no	—
86	C₄⋊Q₈	—	31	3	no	—
87	C₄⋊Q₈	—	62	3	no	—
88	#127	C₂×D₃₁	1	3	yes	—
89	#121	—	2	3	no	—
90	C₄×Q₈	—	31	3	no	—
91	C₄×Q₈	—	62	1	no	—
92	C₄×Q₈	C₂×D₃₁	1	1	yes	characteristic
93	C₄×D₄	C₂×D₃₁	1	3	yes	—
94	#106	—	31	3	no	—
95	C₄⋊C₈	—	31	3	no	—
96	#103	—	31	1	no	—
97	#102	—	31	3	no	—
98	#102	—	31	6	no	—
99	C₄×C₈	—	31	3	no	—
100	C₃₁	#3362	1	1	yes	characteristic, complemented, Sylow
101	#41	—	4	4	no	—
102	C₂×Q₈	—	31	1	no	—
103	C₂×Q₈	—	62	3	no	—
104	C₂×Q₈	C₂²⋊D₃₁	1	1	yes	characteristic
105	C₂×D₄	—	2	3	no	—
106	C₂×D₄	—	2	3	no	—
107	C₂×D₄	C₂²×D₃₁	1	3	yes	—
108	C₂²×C₄	—	2	3	no	—
109	SD₁₆	—	62	6	no	—
110	C₂×C₈	—	31	1	no	—
111	C₂×C₈	—	31	3	no	—
112	C₄⋊C₄	—	31	3	no	—
113	C₄⋊C₄	—	62	3	no	—
114	C₄⋊C₄	—	62	3	no	—
115	C₄⋊C₄	C₂²⋊D₃₁	1	3	yes	—
116	C₄⋊C₄	C₂²×D₃₁	1	1	yes	characteristic
117	C₂²⋊C₄	—	2	3	no	—
118	C₄²	—	62	3	no	—
119	C₄²	C₂²×D₃₁	1	3	yes	—
120	C₂³	—	2	3	no	—
121	Q₈	—	31	2	no	—
122	Q₈	—	62	1	no	—
123	Q₈	—	248	3	no	—
124	Q₈	#92480	1	4	yes	complemented
125	D₄	—	2	3	no	—
126	D₄	—	2	3	no	—
127	D₄	—	4	6	no	—
128	D₄	—	8	3	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	4	6	no	—
131	C₂×C₄	—	62	1	no	—
132	C₂×C₄	—	62	3	no	—
133	C₂×C₄	C₂×C₂²⋊D₃₁	1	3	yes	—
134	C₂×C₄	C₂×C₂²⋊D₃₁	1	3	yes	—
135	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
136	C₈	—	62	3	no	—
137	C₈	—	62	1	no	—
138	C₂²	—	4	3	no	—
139	C₂²	—	4	3	no	—
140	C₂²	—	4	3	no	—
141	C₂²	C₂²×C₂²⋊D₃₁	1	1	yes	characteristic
142	C₄	—	2	3	no	—
143	C₄	—	4	1	no	—
144	C₄	—	62	2	no	—
145	C₄	—	124	3	no	—
146	C₄	C₂×#92480	1	6	yes	—
147	C₄	#41×D₃₁	1	1	yes	characteristic
148	C₄	#11758567	1	1	yes	characteristic
149	C₂	—	8	3	no	—
150	C₂	#14521706	1	1	yes	characteristic
151	C₂	C₂²×#92480	1	1	yes	characteristic
152	C₂	#14521852	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×Dic₆₂

C₂

yes

maximal, characteristic, complemented

#14521395

C₂

yes

maximal, complemented

#14521339

C₂

yes

maximal, complemented

#14521246

C₂

yes

maximal, complemented

Q₈×C₃₁⋊C₈

C₂

yes

maximal, characteristic, complemented

C₄×#92480

C₂

yes

maximal, complemented

C₃₁×#550

C₂

yes

maximal, characteristic

C₂×#41×C₃₁

—

C₃₁×#127

C₂²

yes

—

C₃₁×#121

—

C₄×Q₈×C₃₁

C₂²

yes

characteristic

C₄×D₄×C₃₁

C₂²

yes

—

Q₈×Dic₃₁

—

C₂×#92480

C₂²

yes

—

#11758465

—

#11758446

C₂²

yes

—

C₄×Dic₆₂

C₂²

yes

—

#11758420

C₂²

yes

—

#11758420

C₂²

yes

—

#11758398

C₂²

yes

characteristic

#11758396

C₂²

yes

—

C₄⋊(C₃₁⋊C₈)

C₂²

yes

—

C₄×C₃₁⋊C₈

C₂²

yes

—

#41×C₃₁

—

C₂×Q₈×C₃₁

D₄

yes

characteristic

C₂×D₄×C₃₁

—

C₂×D₄×C₃₁

—

C₂×D₄×C₃₁

C₂³

yes

—

C₂²×C₄×C₃₁

—

C₂×Dic₆₂

—

C₂×Dic₆₂

C₂³

yes

characteristic

C₄⋊C₄×C₃₁

D₄

yes

—

C₄⋊C₄×C₃₁

C₂³

yes

characteristic

C₂²⋊C₄×C₃₁

—

C₄²×C₃₁

C₂³

yes

—

#92480

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

C₂³

yes

—

Dic₃₁⋊C₄

—

C₄×Dic₃₁

—

C₂×C₃₁⋊C₈

C₂³

yes

characteristic

C₂×C₃₁⋊C₈

C₂³

yes

—

C₂³×C₃₁

—

Q₈×C₃₁

SD₁₆

yes

complemented

D₄×C₃₁

—

D₄×C₃₁

—

D₄×C₃₁

—

D₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂×D₄

yes

—

C₂×C₄×C₃₁

C₂×D₄

yes

—

C₂×C₄×C₃₁

C₂⁴

yes

characteristic

C₂×Dic₃₁

—

C₂×Dic₃₁

—

Dic₆₂

—

Dic₆₂

—

Dic₆₂

#41

yes

complemented

C₃₁⋊C₈

—

C₃₁⋊C₈

—

#3362

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

C₂²×D₄

yes

characteristic

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

C₂×SD₁₆

yes

—

C₄×C₃₁

C₂×#41

yes

characteristic

C₄×C₃₁

2₋¹⁺⁴

yes

characteristic

Dic₃₁

—

Dic₃₁

—

Q₈²

—

#550

D₃₁

yes

—

—

—

C₈×Q₈

—

C₄×SD₁₆

—

C₂×C₃₁

—

C₂×C₃₁

#548

yes

characteristic

C₂×C₃₁

C₂²×SD₁₆

yes

characteristic

C₂×C₃₁

#576

yes

characteristic

C₂×#41

—

C₂×SD₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#127

C₂×D₃₁

yes

—

#121

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

C₂×D₃₁

yes

characteristic

C₄×D₄

C₂×D₃₁

yes

—

#106

—

C₄⋊C₈

—

#103

—

#102

—

#102

—

C₄×C₈

—

100

C₃₁

#3362

yes

characteristic, complemented, Sylow

101

#41

—

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂×Q₈

C₂²⋊D₃₁

yes

characteristic

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂×D₄

C₂²×D₃₁

yes

—

108

C₂²×C₄

—

109

SD₁₆

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

C₂²⋊D₃₁

yes

—

116

C₄⋊C₄

C₂²×D₃₁

yes

characteristic

117

C₂²⋊C₄

—

118

C₄²

—

119

C₄²

C₂²×D₃₁

yes

—

120

C₂³

—

121

Q₈

—

122

Q₈

—

123

Q₈

—

248

—

124

Q₈

#92480

yes

complemented

125

D₄

—

126

D₄

—

127

D₄

—

128

D₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

C₂×C₂²⋊D₃₁

yes

—

134

C₂×C₄

C₂×C₂²⋊D₃₁

yes

—

135

C₂×C₄

C₂³×D₃₁

yes

characteristic

136

C₈

—

137

C₈

—

138

C₂²

—

139

C₂²

—

140

C₂²

—

141

C₂²

C₂²×C₂²⋊D₃₁

yes

characteristic

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

—

124

—

146

C₄

C₂×#92480

yes

—

147

C₄

#41×D₃₁

yes

characteristic

148

C₄

#11758567

yes

characteristic

149

C₂

—

150

C₂

#14521706

yes

characteristic

151

C₂

C₂²×#92480

yes

characteristic

152

C₂

#14521852

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₃₁
commutator, cosocle	C₂×C₄×C₃₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₁
Fitting, prime radical	C₃₁×#550	C₂
socle	C₂²×C₃₁	C₂²×D₄
hypercenter	#550	D₃₁
hypocenter	C₃₁	#3362
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #550

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₃₁

commutator, cosocle

C₂×C₄×C₃₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₃₁

Fitting, prime radical

C₃₁×#550

C₂

socle

C₂²×C₃₁

C₂²×D₄

D₃₁

C₃₁

G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #550

N	H ≅ G/N	type
1. Q₈×Dic₆₂	C₂	semidirect
2. #14521395	C₂	semidirect
3. #14521339	C₂	semidirect
4. #14521246	C₂	semidirect
5. Q₈×C₃₁⋊C₈	C₂	semidirect
6. C₄×#92480	C₂	semidirect
44. Q₈×C₃₁	SD₁₆	semidirect
58. Dic₆₂	#41	semidirect
100. C₃₁	#3362	semidirect
124. Q₈	#92480	semidirect

H ≅ G/N

type

1. Q₈×Dic₆₂

C₂

semidirect

2. #14521395

C₂

semidirect

3. #14521339

C₂

semidirect

4. #14521246

C₂

semidirect

5. Q₈×C₃₁⋊C₈

C₂

semidirect

6. C₄×#92480

C₂

semidirect

44. Q₈×C₃₁

SD₁₆

semidirect

58. Dic₆₂

#41

semidirect

100. C₃₁

#3362

semidirect

124. Q₈

#92480

semidirect

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

subgroup	index
Q₈×Dic₆₂	2
#14521395	2
#14521339	2
#14521246	2
Q₈×C₃₁⋊C₈	2
C₄×#92480	2
C₃₁×#550	2
#3362	31

subgroup

index

Q₈×Dic₆₂

Q₈×C₃₁⋊C₈

C₃₁×#550

Subgroups — #16858378

Conjugacy classes of subgroups

Complements

Maximal subgroups