Group #20251956 — subgroups

Subgroups — #20251956

Order 6016.

484 conjugacy classes of subgroups; only the 297 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 297 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	2	yes	maximal
2	#15689150	C₂	1	6	yes	maximal, complemented
3	#15689147	C₂	1	3	yes	maximal, complemented
4	C₂×C₄⋊(C₄₇⋊C₈)	C₂	1	3	yes	maximal, complemented
5	C₄×#41×C₄₇	C₂	1	1	yes	maximal, characteristic
6	#13094031	—	2	8	no	—
7	C₂×#10625329	C₂²	1	6	yes	—
8	C₂²×C₄₇⋊C₈	C₂²	1	6	yes	—
9	C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
10	C₄⋊(C₄₇⋊C₈)	C₂²	1	3	yes	—
11	C₄⋊(C₄₇⋊C₈)	C₂²	1	6	yes	—
12	C₄⋊(C₄₇⋊C₈)	C₂²	1	6	yes	—
13	C₂×#41×C₄₇	C₂²	1	1	yes	characteristic
14	C₄×Q₈×C₄₇	C₄	1	1	yes	characteristic
15	C₄×Q₈×C₄₇	C₄	1	1	yes	characteristic
16	C₄×D₄×C₄₇	C₄	1	3	yes	—
17	C₄×D₄×C₄₇	C₄	1	3	yes	—
18	#117×C₄₇	C₂²	1	3	yes	—
19	C₂×C₄²×C₄₇	C₂²	1	3	yes	—
20	#41×C₄₇	D₄	1	4	yes	—
21	#41×C₄₇	Q₈	1	4	yes	—
22	C₂×Q₈×C₄₇	C₂×C₄	1	1	yes	characteristic
23	C₂×D₄×C₄₇	C₂×C₄	1	3	yes	—
24	C₂²×C₄×C₄₇	C₂×C₄	1	3	yes	—
25	C₂²×C₄×C₄₇	C₂×C₄	1	3	yes	—
26	C₂²×C₄×C₄₇	C₂³	1	3	yes	—
27	C₄⋊C₄×C₄₇	C₂×C₄	1	3	yes	—
28	C₄⋊C₄×C₄₇	C₂×C₄	1	3	yes	—
29	C₂²⋊C₄×C₄₇	C₂×C₄	1	6	yes	—
30	C₄²×C₄₇	C₂×C₄	1	3	yes	—
31	C₄²×C₄₇	C₂×C₄	1	3	yes	—
32	C₄²×C₄₇	C₂³	1	1	yes	characteristic
33	C₄²×C₄₇	C₂³	1	3	yes	—
34	#10625329	—	2	12	no	—
35	C₂×C₄₇⋊C₈	—	2	12	no	—
36	C₂×C₄₇⋊C₈	C₂³	1	2	yes	—
37	C₂×C₄₇⋊C₈	C₂³	1	6	yes	—
38	C₂³×C₄₇	C₂²×C₄	1	3	yes	—
39	Q₈×C₄₇	C₄⋊C₄	1	4	yes	—
40	D₄×C₄₇	C₄⋊C₄	1	12	yes	—
41	C₂×C₄×C₄₇	—	2	6	no	—
42	C₂×C₄×C₄₇	—	4	3	no	—
43	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₄₇	C₂²×C₄	1	3	yes	—
45	C₂×C₄×C₄₇	C₂²×C₄	1	3	yes	—
46	C₂×C₄×C₄₇	C₂²×C₄	1	3	yes	—
47	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₄₇	C₂×D₄	1	6	yes	—
49	C₂×C₄×C₄₇	C₂×Q₈	1	6	yes	—
50	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
51	C₄₇⋊C₈	—	2	2	no	—
52	C₄₇⋊C₈	—	2	6	no	—
53	C₂²×C₄₇	—	2	3	no	—
54	C₂²×C₄₇	—	2	3	no	—
55	C₂²×C₄₇	C₂×C₄⋊C₄	1	6	yes	—
56	C₂²×C₄₇	C₂³×C₄	1	1	yes	characteristic
57	C₄×C₄₇	—	2	3	no	—
58	C₄×C₄₇	—	2	1	no	—
59	C₄×C₄₇	—	2	3	no	—
60	C₄×C₄₇	C₂×C₄⋊C₄	1	6	yes	—
61	C₄×C₄₇	#131	1	2	yes	complemented
62	C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
63	C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
64	#2906	—	47	1	no	maximal, Sylow
65	C₂×C₄₇	—	2	6	no	—
66	C₂×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄₇	C₂×#131	1	1	yes	characteristic
68	C₂×C₄₇	#568	1	1	yes	characteristic
69	C₂×#131	—	47	2	no	—
70	C₄×#41	D₄₇	1	1	yes	characteristic
71	#424	—	47	6	no	—
72	#423	—	47	3	no	—
73	C₂×C₄⋊C₈	—	47	3	no	—
74	C₄₇	#2906	1	1	yes	characteristic, complemented, Sylow
75	C₂×#41	C₂×D₄₇	1	1	yes	characteristic
76	#131	—	94	8	no	—
77	C₂×M₁₆	—	47	6	no	—
78	C₂²×C₈	—	47	6	no	—
79	C₄×Q₈	Dic₄₇	1	1	yes	characteristic
80	C₄×Q₈	Dic₄₇	1	1	yes	characteristic
81	C₄×D₄	Dic₄₇	1	3	yes	—
82	C₄×D₄	Dic₄₇	1	3	yes	—
83	#117	C₂×D₄₇	1	3	yes	—
84	C₂×C₄²	C₂×D₄₇	1	3	yes	—
85	C₄⋊C₈	—	47	1	no	—
86	C₄⋊C₈	—	47	3	no	—
87	C₄⋊C₈	—	47	6	no	—
88	C₄⋊C₈	—	47	6	no	—
89	#41	Dic₉₄	1	4	yes	—
90	#41	D₁₈₈	1	4	yes	—
91	C₂×Q₈	C₂×Dic₄₇	1	1	yes	characteristic
92	C₂×D₄	C₂×Dic₄₇	1	3	yes	—
93	C₂²×C₄	C₂×Dic₄₇	1	3	yes	—
94	C₂²×C₄	C₂×Dic₄₇	1	3	yes	—
95	C₂²×C₄	C₂²×D₄₇	1	3	yes	—
96	M₁₆	—	94	12	no	—
97	C₂×C₈	—	47	2	no	—
98	C₂×C₈	—	47	6	no	—
99	C₂×C₈	—	94	12	no	—
100	C₄⋊C₄	C₂×Dic₄₇	1	3	yes	—
101	C₄⋊C₄	C₂×Dic₄₇	1	3	yes	—
102	C₂²⋊C₄	C₂×Dic₄₇	1	6	yes	—
103	C₄²	C₂×Dic₄₇	1	3	yes	—
104	C₄²	C₂×Dic₄₇	1	3	yes	—
105	C₄²	C₂²×D₄₇	1	1	yes	characteristic
106	C₄²	C₂²×D₄₇	1	3	yes	—
107	C₂³	C₂²×Dic₄₇	1	3	yes	—
108	Q₈	C₄⋊Dic₄₇	1	4	yes	—
109	D₄	C₄⋊Dic₄₇	1	12	yes	—
110	C₂×C₄	—	2	6	no	—
111	C₂×C₄	—	4	3	no	—
112	C₂×C₄	C₂×Dic₉₄	1	6	yes	—
113	C₂×C₄	C₂×D₁₈₈	1	6	yes	—
114	C₂×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
115	C₂×C₄	C₂²×Dic₄₇	1	3	yes	—
116	C₂×C₄	C₂²×Dic₄₇	1	3	yes	—
117	C₂×C₄	C₂²×Dic₄₇	1	3	yes	—
118	C₂×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
119	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
120	C₈	—	94	2	no	—
121	C₈	—	94	6	no	—
122	C₂²	—	2	3	no	—
123	C₂²	—	2	3	no	—
124	C₂²	C₂×C₄⋊Dic₄₇	1	6	yes	—
125	C₂²	C₂³×Dic₄₇	1	1	yes	characteristic
126	C₄	—	2	3	no	—
127	C₄	—	2	1	no	—
128	C₄	—	2	3	no	—
129	C₄	C₂×C₄⋊Dic₄₇	1	6	yes	—
130	C₄	#13094031	1	2	yes	complemented
131	C₄	C₂²×D₁₈₈	1	1	yes	characteristic
132	C₄	C₂²×Dic₉₄	1	1	yes	characteristic
133	C₂	—	2	6	no	—
134	C₂	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
135	C₂	C₂×#13094031	1	1	yes	characteristic
136	C₂	#15689890	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13094031

C₂

yes

maximal

#15689150

C₂

yes

maximal, complemented

#15689147

C₂

yes

maximal, complemented

C₂×C₄⋊(C₄₇⋊C₈)

C₂

yes

maximal, complemented

C₄×#41×C₄₇

C₂

yes

maximal, characteristic

#13094031

—

C₂×#10625329

C₂²

yes

—

C₂²×C₄₇⋊C₈

C₂²

yes

—

C₄⋊(C₄₇⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

C₂×#41×C₄₇

C₂²

yes

characteristic

C₄×Q₈×C₄₇

C₄

yes

characteristic

C₄×Q₈×C₄₇

C₄

yes

characteristic

C₄×D₄×C₄₇

C₄

yes

—

C₄×D₄×C₄₇

C₄

yes

—

#117×C₄₇

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

—

#41×C₄₇

D₄

yes

—

#41×C₄₇

Q₈

yes

—

C₂×Q₈×C₄₇

C₂×C₄

yes

characteristic

C₂×D₄×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

C₂³

yes

—

C₄⋊C₄×C₄₇

C₂×C₄

yes

—

C₄⋊C₄×C₄₇

C₂×C₄

yes

—

C₂²⋊C₄×C₄₇

C₂×C₄

yes

—

C₄²×C₄₇

C₂×C₄

yes

—

C₄²×C₄₇

C₂×C₄

yes

—

C₄²×C₄₇

C₂³

yes

characteristic

C₄²×C₄₇

C₂³

yes

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

C₂³

yes

—

C₂×C₄₇⋊C₈

C₂³

yes

—

C₂³×C₄₇

C₂²×C₄

yes

—

Q₈×C₄₇

C₄⋊C₄

yes

—

D₄×C₄₇

C₄⋊C₄

yes

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×Q₈

yes

—

C₂×C₄×C₄₇

C₂⁴

yes

characteristic

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×C₄⋊C₄

yes

—

C₂²×C₄₇

C₂³×C₄

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×C₄⋊C₄

yes

—

C₄×C₄₇

#131

yes

complemented

C₄×C₄₇

C₂²×D₄

yes

characteristic

C₄×C₄₇

C₂²×Q₈

yes

characteristic

#2906

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

C₂²×C₄⋊C₄

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

C₄×#41

D₄₇

yes

characteristic

#424

—

#423

—

C₂×C₄⋊C₈

—

C₄₇

#2906

yes

characteristic, complemented, Sylow

C₂×#41

C₂×D₄₇

yes

characteristic

#131

—

C₂×M₁₆

—

C₂²×C₈

—

C₄×Q₈

Dic₄₇

yes

characteristic

C₄×Q₈

Dic₄₇

yes

characteristic

C₄×D₄

Dic₄₇

yes

—

C₄×D₄

Dic₄₇

yes

—

#117

C₂×D₄₇

yes

—

C₂×C₄²

C₂×D₄₇

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#41

Dic₉₄

yes

—

#41

D₁₈₈

yes

—

C₂×Q₈

C₂×Dic₄₇

yes

characteristic

C₂×D₄

C₂×Dic₄₇

yes

—

C₂²×C₄

C₂×Dic₄₇

yes

—

C₂²×C₄

C₂×Dic₄₇

yes

—

C₂²×C₄

C₂²×D₄₇

yes

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₄⋊C₄

C₂×Dic₄₇

yes

—

101

C₄⋊C₄

C₂×Dic₄₇

yes

—

102

C₂²⋊C₄

C₂×Dic₄₇

yes

—

103

C₄²

C₂×Dic₄₇

yes

—

104

C₄²

C₂×Dic₄₇

yes

—

105

C₄²

C₂²×D₄₇

yes

characteristic

106

C₄²

C₂²×D₄₇

yes

—

107

C₂³

C₂²×Dic₄₇

yes

—

108

Q₈

C₄⋊Dic₄₇

yes

—

109

D₄

C₄⋊Dic₄₇

yes

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

C₂×Dic₉₄

yes

—

113

C₂×C₄

C₂×D₁₈₈

yes

—

114

C₂×C₄

C₂²×Dic₄₇

yes

characteristic

115

C₂×C₄

C₂²×Dic₄₇

yes

—

116

C₂×C₄

C₂²×Dic₄₇

yes

—

117

C₂×C₄

C₂²×Dic₄₇

yes

—

118

C₂×C₄

C₂²×Dic₄₇

yes

characteristic

119

C₂×C₄

C₂³×D₄₇

yes

characteristic

120

C₈

—

121

C₈

—

122

C₂²

—

123

C₂²

—

124

C₂²

C₂×C₄⋊Dic₄₇

yes

—

125

C₂²

C₂³×Dic₄₇

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

C₂×C₄⋊Dic₄₇

yes

—

130

C₄

#13094031

yes

complemented

131

C₄

C₂²×D₁₈₈

yes

characteristic

132

C₄

C₂²×Dic₉₄

yes

characteristic

133

C₂

—

134

C₂

C₂²×C₄⋊Dic₄₇

yes

characteristic

135

C₂

C₂×#13094031

yes

characteristic

136

C₂

#15689890

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₄₇
commutator, socle	C₂²×C₄₇	C₂³×C₄
Fitting	C₄×#41×C₄₇	C₂
hypercenter	C₄×#41	D₄₇
hypocenter	C₄₇	#2906
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₄₇

commutator, socle

C₂²×C₄₇

C₂³×C₄

Fitting

C₄×#41×C₄₇

C₂

hypercenter

C₄×#41

D₄₇

C₄₇

G ▷ C₂²×C₄₇ ▷ C₄₇

C₁ ◁ C₂×C₄ ◁ C₄×#41

N	H ≅ G/N	type
2. #15689150	C₂	semidirect
3. #15689147	C₂	semidirect
4. C₂×C₄⋊(C₄₇⋊C₈)	C₂	semidirect
61. C₄×C₄₇	#131	semidirect
74. C₄₇	#2906	semidirect
130. C₄	#13094031	semidirect

H ≅ G/N

type

2. #15689150

C₂

semidirect

3. #15689147

C₂

semidirect

4. C₂×C₄⋊(C₄₇⋊C₈)

C₂

semidirect

61. C₄×C₄₇

#131

semidirect

74. C₄₇

#2906

semidirect

130. C₄

#13094031

semidirect

Subgroups — #20251956

Conjugacy classes of subgroups

Complements