Group #20251986 — subgroups

Subgroups — #20251986

Order 6016.

476 conjugacy classes of subgroups; only the 253 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 253 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	C₂×#13094031	C₂	1	1	yes	maximal, characteristic
2	#15689606	C₂	1	1	yes	maximal, characteristic
3	#15689166	C₂	1	3	yes	maximal, complemented
4	#15689163	C₂	1	3	yes	maximal, complemented
5	#15689159	C₂	1	3	yes	maximal, complemented
6	C₂×#13093957	C₂	1	3	yes	maximal, complemented
7	C₄₇×#520	C₂	1	1	yes	maximal, characteristic
8	#13094031	C₄	1	8	yes	complemented
9	C₂×#10625329	C₂²	1	3	yes	—
10	C₂²×C₄₇⋊C₈	C₂²	1	3	yes	—
11	Q₈×Dic₄₇	—	2	1	no	—
12	D₄×Dic₄₇	—	2	3	no	—
13	#13093983	C₂²	1	3	yes	—
14	C₂×C₄⋊Dic₄₇	C₂²	1	3	yes	—
15	#13093957	C₂²	1	3	yes	—
16	#13093957	C₂²	1	6	yes	—
17	#13093957	C₂²	1	1	yes	characteristic
18	#13093955	C₂²	1	3	yes	—
19	#13093955	C₂²	1	3	yes	—
20	C₂×#41×C₄₇	C₂²	1	1	yes	characteristic
21	C₄×Q₈×C₄₇	—	2	1	no	—
22	C₄×D₄×C₄₇	—	2	3	no	—
23	#117×C₄₇	C₂²	1	3	yes	—
24	C₂×C₄⋊C₄×C₄₇	C₂²	1	3	yes	—
25	#41×C₄₇	C₂×C₄	1	4	yes	—
26	#41×C₄₇	Q₈	1	4	yes	—
27	C₂×Q₈×C₄₇	D₄	1	1	yes	characteristic
28	C₂×D₄×C₄₇	D₄	1	3	yes	—
29	C₂²×C₄×C₄₇	—	2	3	no	—
30	C₂²×C₄×C₄₇	C₂³	1	3	yes	—
31	C₂²×Dic₄₇	—	2	3	no	—
32	C₄⋊C₄×C₄₇	—	2	3	no	—
33	C₄⋊C₄×C₄₇	C₂³	1	3	yes	—
34	C₄⋊C₄×C₄₇	C₂³	1	1	yes	characteristic
35	C₂²⋊C₄×C₄₇	—	2	3	no	—
36	C₄²×C₄₇	—	2	3	no	—
37	C₂²⋊Dic₄₇	—	2	3	no	—
38	C₄⋊Dic₄₇	—	2	3	no	—
39	C₄⋊Dic₄₇	C₂³	1	1	yes	characteristic
40	C₄⋊Dic₄₇	C₂³	1	3	yes	—
41	C₄×Dic₄₇	—	2	3	no	—
42	#10625329	C₂×C₄	1	12	yes	complemented
43	C₂×C₄₇⋊C₈	C₂×C₄	1	12	yes	complemented
44	C₂×C₄₇⋊C₈	C₂³	1	3	yes	—
45	C₂×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
46	C₂³×C₄₇	C₂×D₄	1	3	yes	—
47	Q₈×C₄₇	C₄⋊C₄	1	4	yes	—
48	D₄×C₄₇	C₄⋊C₄	1	12	yes	—
49	C₂×C₄×C₄₇	—	2	1	no	—
50	C₂×C₄×C₄₇	—	2	3	no	—
51	C₂×C₄×C₄₇	—	8	3	no	—
52	C₂×C₄×C₄₇	C₂²×C₄	1	6	yes	—
53	C₂×C₄×C₄₇	C₂×D₄	1	3	yes	—
54	C₂×C₄×C₄₇	C₂×Q₈	1	6	yes	—
55	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
56	C₂×Dic₄₇	—	2	1	no	—
57	C₂×Dic₄₇	—	2	3	no	—
58	C₂×Dic₄₇	—	8	3	no	—
59	C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
60	C₄₇⋊C₈	C₂²×C₄	1	6	yes	—
61	C₂²×C₄₇	—	2	3	no	—
62	C₂²×C₄₇	—	2	3	no	—
63	C₂²×C₄₇	C₂×C₄⋊C₄	1	6	yes	—
64	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
65	C₄×C₄₇	—	4	3	no	—
66	C₄×C₄₇	—	4	1	no	—
67	C₄×C₄₇	C₂×C₄⋊C₄	1	6	yes	—
68	C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
69	C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
70	Dic₄₇	—	4	3	no	—
71	Dic₄₇	—	4	1	no	—
72	#2913	—	47	1	no	maximal, Sylow
73	C₂×C₄₇	—	2	6	no	—
74	C₂×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₄₇	#576	1	1	yes	characteristic
76	C₂×C₄₇	#578	1	1	yes	characteristic
77	C₂×#131	—	47	1	no	—
78	#520	—	47	1	no	—
79	#520	D₄₇	1	1	yes	characteristic
80	#428	—	47	3	no	—
81	#428	—	47	3	no	—
82	#427	—	47	3	no	—
83	C₂×#107	—	47	3	no	—
84	C₄₇	#2913	1	1	yes	characteristic, complemented, Sylow
85	C₂×#41	C₂×D₄₇	1	1	yes	characteristic
86	#131	—	47	8	no	—
87	C₂×M₁₆	—	47	3	no	—
88	C₂²×C₈	—	47	3	no	—
89	C₄×Q₈	—	2	1	no	—
90	C₄×Q₈	—	94	1	no	—
91	C₄×D₄	—	2	3	no	—
92	C₄×D₄	—	94	3	no	—
93	#117	—	47	3	no	—
94	#117	C₂×D₄₇	1	3	yes	—
95	C₂×C₄⋊C₄	—	47	3	no	—
96	C₂×C₄⋊C₄	C₂×D₄₇	1	3	yes	—
97	#107	—	47	3	no	—
98	#107	—	47	6	no	—
99	#107	—	47	1	no	—
100	#106	—	47	3	no	—
101	#106	—	47	3	no	—
102	#41	Dic₉₄	1	4	yes	—
103	#41	C₄×D₄₇	1	4	yes	—
104	C₂×Q₈	C₂²⋊D₄₇	1	1	yes	characteristic
105	C₂×D₄	C₂²⋊D₄₇	1	3	yes	—
106	C₂²×C₄	—	2	3	no	—
107	C₂²×C₄	—	94	3	no	—
108	C₂²×C₄	C₂²×D₄₇	1	3	yes	—
109	M₁₆	—	47	12	no	—
110	C₂×C₈	—	47	12	no	—
111	C₂×C₈	—	47	3	no	—
112	C₂×C₈	—	47	1	no	—
113	C₄⋊C₄	—	2	3	no	—
114	C₄⋊C₄	—	47	1	no	—
115	C₄⋊C₄	—	47	3	no	—
116	C₄⋊C₄	—	94	3	no	—
117	C₄⋊C₄	C₂²×D₄₇	1	1	yes	characteristic
118	C₄⋊C₄	C₂²×D₄₇	1	3	yes	—
119	C₂²⋊C₄	—	2	3	no	—
120	C₂²⋊C₄	—	94	3	no	—
121	C₄²	—	2	3	no	—
122	C₄²	—	94	3	no	—
123	C₂³	C₂×C₂²⋊D₄₇	1	3	yes	—
124	Q₈	Dic₄₇⋊C₄	1	4	yes	—
125	D₄	Dic₄₇⋊C₄	1	12	yes	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	3	no	—
128	C₂×C₄	—	8	3	no	—
129	C₂×C₄	—	94	1	no	—
130	C₂×C₄	—	94	3	no	—
131	C₂×C₄	—	376	3	no	—
132	C₂×C₄	C₂×Dic₉₄	1	6	yes	—
133	C₂×C₄	C₂×C₄×D₄₇	1	6	yes	—
134	C₂×C₄	C₂×C₂²⋊D₄₇	1	3	yes	—
135	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
136	C₈	—	47	2	no	—
137	C₈	—	47	6	no	—
138	C₂²	—	2	3	no	—
139	C₂²	—	2	3	no	—
140	C₂²	C₂×Dic₄₇⋊C₄	1	6	yes	—
141	C₂²	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
142	C₄	—	4	3	no	—
143	C₄	—	4	1	no	—
144	C₄	—	188	3	no	—
145	C₄	—	188	1	no	—
146	C₄	C₂×Dic₄₇⋊C₄	1	6	yes	—
147	C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
148	C₄	C₂²×Dic₉₄	1	1	yes	characteristic
149	C₂	—	2	6	no	—
150	C₂	C₂²×Dic₄₇⋊C₄	1	1	yes	characteristic
151	C₂	#15689926	1	1	yes	characteristic
152	C₂	#15689938	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13094031

C₂

yes

maximal, characteristic

#15689606

C₂

yes

maximal, characteristic

#15689166

C₂

yes

maximal, complemented

#15689163

C₂

yes

maximal, complemented

#15689159

C₂

yes

maximal, complemented

C₂×#13093957

C₂

yes

maximal, complemented

C₄₇×#520

C₂

yes

maximal, characteristic

#13094031

C₄

yes

complemented

C₂×#10625329

C₂²

yes

—

C₂²×C₄₇⋊C₈

C₂²

yes

—

Q₈×Dic₄₇

—

D₄×Dic₄₇

—

#13093983

C₂²

yes

—

C₂×C₄⋊Dic₄₇

C₂²

yes

—

#13093957

C₂²

yes

—

#13093957

C₂²

yes

—

#13093957

C₂²

yes

characteristic

#13093955

C₂²

yes

—

#13093955

C₂²

yes

—

C₂×#41×C₄₇

C₂²

yes

characteristic

C₄×Q₈×C₄₇

—

C₄×D₄×C₄₇

—

#117×C₄₇

C₂²

yes

—

C₂×C₄⋊C₄×C₄₇

C₂²

yes

—

#41×C₄₇

C₂×C₄

yes

—

#41×C₄₇

Q₈

yes

—

C₂×Q₈×C₄₇

D₄

yes

characteristic

C₂×D₄×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×Dic₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

C₂³

yes

—

C₄⋊C₄×C₄₇

C₂³

yes

characteristic

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₂²⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

C₂³

yes

characteristic

C₄⋊Dic₄₇

C₂³

yes

—

C₄×Dic₄₇

—

#10625329

C₂×C₄

yes

complemented

C₂×C₄₇⋊C₈

C₂×C₄

yes

complemented

C₂×C₄₇⋊C₈

C₂³

yes

—

C₂×C₄₇⋊C₈

C₂³

yes

characteristic

C₂³×C₄₇

C₂×D₄

yes

—

Q₈×C₄₇

C₄⋊C₄

yes

—

D₄×C₄₇

C₄⋊C₄

yes

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×Q₈

yes

—

C₂×C₄×C₄₇

C₂⁴

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₄₇⋊C₈

C₂²×C₄

yes

—

C₄₇⋊C₈

C₂²×C₄

yes

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×C₄⋊C₄

yes

—

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×C₄⋊C₄

yes

—

C₄×C₄₇

C₂³×C₄

yes

characteristic

C₄×C₄₇

C₂²×Q₈

yes

characteristic

Dic₄₇

—

Dic₄₇

—

#2913

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

C₂²×C₄⋊C₄

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

—

D₄₇

yes

—

—

—

—

C₄₇

#2913

yes

characteristic, complemented, Sylow

C₂×#41

C₂×D₄₇

yes

characteristic

#131

—

C₂×M₁₆

—

C₂²×C₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×D₄

—

C₄×D₄

—

#117

—

#117

C₂×D₄₇

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₄₇

yes

—

#107

—

#107

—

#107

—

100

#106

—

101

#106

—

102

#41

Dic₉₄

yes

—

103

#41

C₄×D₄₇

yes

—

104

C₂×Q₈

C₂²⋊D₄₇

yes

characteristic

105

C₂×D₄

C₂²⋊D₄₇

yes

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

C₂²×D₄₇

yes

—

109

M₁₆

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

C₂²×D₄₇

yes

characteristic

118

C₄⋊C₄

C₂²×D₄₇

yes

—

119

C₂²⋊C₄

—

120

C₂²⋊C₄

—

121

C₄²

—

122

C₄²

—

123

C₂³

C₂×C₂²⋊D₄₇

yes

—

124

Q₈

Dic₄₇⋊C₄

yes

—

125

D₄

Dic₄₇⋊C₄

yes

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

376

—

132

C₂×C₄

C₂×Dic₉₄

yes

—

133

C₂×C₄

C₂×C₄×D₄₇

yes

—

134

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

135

C₂×C₄

C₂³×D₄₇

yes

characteristic

136

C₈

—

137

C₈

—

138

C₂²

—

139

C₂²

—

140

C₂²

C₂×Dic₄₇⋊C₄

yes

—

141

C₂²

C₂²×C₂²⋊D₄₇

yes

characteristic

142

C₄

—

143

C₄

—

144

C₄

—

188

—

145

C₄

—

188

—

146

C₄

C₂×Dic₄₇⋊C₄

yes

—

147

C₄

C₂²×C₄×D₄₇

yes

characteristic

148

C₄

C₂²×Dic₉₄

yes

characteristic

149

C₂

—

150

C₂

C₂²×Dic₄₇⋊C₄

yes

characteristic

151

C₂

#15689926

yes

characteristic

152

C₂

#15689938

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₄₇
commutator	C₄×C₄₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₄₇
Fitting	C₄₇×#520	C₂
socle	C₂²×C₄₇	C₂²×D₄
hypercenter	#520	D₄₇
hypocenter	C₄₇	#2913
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ #520

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₄₇

commutator

C₄×C₄₇

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₄₇

Fitting

C₄₇×#520

C₂

socle

C₂²×C₄₇

C₂²×D₄

D₄₇

C₄₇

G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂×#41 ◁ #520

N	H ≅ G/N	type
3. #15689166	C₂	semidirect
4. #15689163	C₂	semidirect
5. #15689159	C₂	semidirect
6. C₂×#13093957	C₂	semidirect
8. #13094031	C₄	semidirect
42. #10625329	C₂×C₄	semidirect
43. C₂×C₄₇⋊C₈	C₂×C₄	semidirect
84. C₄₇	#2913	semidirect

H ≅ G/N

type

3. #15689166

C₂

semidirect

4. #15689163

C₂

semidirect

5. #15689159

C₂

semidirect

6. C₂×#13093957

C₂

semidirect

8. #13094031

C₄

semidirect

42. #10625329

C₂×C₄

semidirect

43. C₂×C₄₇⋊C₈

C₂×C₄

semidirect

84. C₄₇

#2913

semidirect

Subgroups — #20251986

Conjugacy classes of subgroups

Complements