Group #20251434 — subgroups

Subgroups — #20251434

Order 6016.

580 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 580 conjugacy classes of subgroups — too many to draw. Only its 133 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×#13094012	C₂	1	2	yes	maximal, complemented
2	#15689040	C₂	1	4	yes	maximal, complemented
3	#15689036	C₂	1	4	yes	maximal, complemented
4	#15689033	C₂	1	4	yes	maximal, complemented
5	C₄₇×#382	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×Dic₉₄	C₂²	1	2	yes	—
7	C₂²×D₁₈₈	C₂²	1	2	yes	—
8	C₂²×D₁₈₈	C₂²	1	4	yes	—
9	#13094012	—	2	8	no	—
10	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	4	yes	—
11	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	4	yes	—
12	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	8	yes	—
13	C₄₇⋊#95	C₂²	1	4	yes	—
14	C₂×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
15	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
16	#95×C₄₇	C₂²	1	4	yes	—
17	C₂³×D₄₇	C₂³	1	4	yes	—
18	C₂²×C₄×C₄₇	C₂³	1	4	yes	—
19	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
20	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
21	C₂²×Dic₄₇	C₂³	1	4	yes	—
22	C₂×D₁₈₈	—	2	4	no	—
23	C₂×D₁₈₈	—	4	4	no	—
24	C₂×D₁₈₈	—	4	8	no	—
25	C₂×D₁₈₈	—	4	4	no	—
26	C₂×Dic₉₄	—	2	4	no	—
27	C₂×Dic₉₄	—	4	4	no	—
28	C₄⋊C₄×C₄₇	—	4	2	no	—
29	C₄²×C₄₇	D₄	1	4	yes	complemented
30	(C₂×D₄₇)⋊C₄	—	2	8	no	—
31	(C₂×D₄₇)⋊C₄	—	2	8	no	—
32	(C₂×D₄₇)⋊C₄	—	4	8	no	—
33	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
34	C₂²×D₄₇	—	2	4	no	—
35	C₂²×D₄₇	—	2	8	no	—
36	C₂²×D₄₇	—	2	4	no	—
37	C₂²×D₄₇	—	2	8	no	—
38	C₂²×D₄₇	—	2	4	no	—
39	C₂×C₄×C₄₇	—	2	1	no	—
40	C₂×C₄×C₄₇	—	2	4	no	—
41	C₂×C₄×C₄₇	—	2	1	no	—
42	C₂×C₄×C₄₇	—	2	8	no	—
43	C₂×C₄×C₄₇	—	2	4	no	—
44	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₄₇	C₂×D₄	1	4	yes	—
46	C₂×Dic₄₇	—	2	4	no	—
47	C₂×Dic₄₇	—	2	8	no	—
48	D₁₈₈	—	4	8	no	—
49	D₁₈₈	—	4	16	no	—
50	Dic₉₄	—	4	8	no	—
51	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
52	C₂²×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
53	C₂²×C₄₇	H(5,𝔽₂)	1	4	yes	—
54	C₂²×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
55	C₂×D₄₇	—	4	4	no	—
56	C₂×D₄₇	—	4	4	no	—
57	C₂×D₄₇	—	4	4	no	—
58	C₂×D₄₇	—	4	8	no	—
59	C₂×D₄₇	—	4	8	no	—
60	C₄×C₄₇	—	2	4	no	—
61	C₄×C₄₇	—	2	2	no	—
62	C₄×C₄₇	—	4	4	no	—
63	Dic₄₇	—	4	4	no	—
64	#2819	—	47	1	no	maximal, Sylow
65	C₂×C₄₇	#535	1	2	yes	—
66	C₂×C₄₇	#537	1	1	yes	characteristic
67	C₂×C₄₇	H(3,𝔽₄)	1	2	yes	—
68	C₂×C₄₇	#562	1	2	yes	—
69	D₄₇	—	8	4	no	—
70	C₂×#124	—	47	2	no	—
71	#394	—	47	4	no	—
72	#393	—	47	4	no	—
73	#392	—	47	4	no	—
74	#382	D₄₇	1	1	yes	characteristic, complemented
75	C₄₇	#2819	1	1	yes	characteristic, complemented, Sylow
76	C₂²×Q₈	—	47	2	no	—
77	C₂²×D₄	—	47	2	no	—
78	C₂²×D₄	—	47	4	no	—
79	#124	—	94	8	no	—
80	C₂×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
81	C₂×C₂²⋊C₄	—	47	4	no	—
82	C₂×C₂²⋊C₄	—	47	4	no	—
83	C₂×C₂²⋊C₄	—	47	8	no	—
84	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
85	#95	—	47	4	no	—
86	#95	C₂×D₄₇	1	4	yes	—
87	C₂⁴	—	47	4	no	—
88	C₂×Q₈	—	94	4	no	—
89	C₂×Q₈	—	188	4	no	—
90	C₂×D₄	—	94	4	no	—
91	C₂×D₄	—	188	4	no	—
92	C₂×D₄	—	188	4	no	—
93	C₂×D₄	—	188	8	no	—
94	C₂²×C₄	—	47	4	no	—
95	C₂²×C₄	C₂²×D₄₇	1	4	yes	—
96	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
97	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
98	C₄⋊C₄	—	4	2	no	—
99	C₂²⋊C₄	—	94	8	no	—
100	C₂²⋊C₄	—	94	8	no	—
101	C₂²⋊C₄	—	188	8	no	—
102	C₄²	D₁₈₈	1	4	yes	complemented
103	C₂³	—	94	4	no	—
104	C₂³	—	94	4	no	—
105	C₂³	—	94	8	no	—
106	C₂³	—	94	8	no	—
107	C₂³	—	94	4	no	—
108	C₂³	C₂³×D₄₇	1	1	yes	characteristic
109	Q₈	—	188	8	no	—
110	D₄	—	188	8	no	—
111	D₄	—	188	16	no	—
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	2	8	no	—
117	C₂×C₄	—	94	4	no	—
118	C₂×C₄	—	94	8	no	—
119	C₂×C₄	C₂×D₁₈₈	1	2	yes	—
120	C₂×C₄	C₂×D₁₈₈	1	4	yes	—
121	C₂²	—	188	8	no	—
122	C₂²	—	188	8	no	—
123	C₂²	—	188	4	no	—
124	C₂²	—	188	4	no	—
125	C₂²	—	188	4	no	—
126	C₂²	C₂²×D₁₈₈	1	1	yes	characteristic
127	C₂²	#13094069	1	1	yes	characteristic
128	C₂²	#13094070	1	4	yes	—
129	C₂²	#13094072	1	1	yes	characteristic
130	C₄	—	2	4	no	—
131	C₄	—	2	2	no	—
132	C₄	—	4	4	no	—
133	C₄	—	188	4	no	—
134	C₂	—	376	4	no	—
135	C₂	#15689692	1	2	yes	—
136	C₂	#15689704	1	1	yes	characteristic
137	C₂	#15689865	1	2	yes	—
138	C₂	#15689871	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13094012

C₂

yes

maximal, complemented

#15689040

C₂

yes

maximal, complemented

#15689036

C₂

yes

maximal, complemented

#15689033

C₂

yes

maximal, complemented

C₄₇×#382

C₂

yes

maximal, characteristic, complemented

C₂²×Dic₉₄

C₂²

yes

—

C₂²×D₁₈₈

C₂²

yes

—

C₂²×D₁₈₈

C₂²

yes

—

#13094012

—

C₂×(C₂×D₄₇)⋊C₄

C₂²

yes

—

C₂×(C₂×D₄₇)⋊C₄

C₂²

yes

—

C₂×(C₂×D₄₇)⋊C₄

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₂×C₄⋊C₄×C₄₇

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

#95×C₄₇

C₂²

yes

—

C₂³×D₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

D₄

yes

complemented

(C₂×D₄₇)⋊C₄

—

(C₂×D₄₇)⋊C₄

—

(C₂×D₄₇)⋊C₄

—

C₂³×C₄₇

C₂⁴

yes

characteristic

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

D₁₈₈

—

D₁₈₈

—

Dic₉₄

—

C₂²×C₄₇

C₂²×D₄

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₂²×C₄₇

2₋¹⁺⁴

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

Dic₄₇

—

#2819

—

maximal, Sylow

C₂×C₄₇

#535

yes

—

C₂×C₄₇

#537

yes

characteristic

C₂×C₄₇

H(3,𝔽₄)

yes

—

C₂×C₄₇

#562

yes

—

D₄₇

—

—

—

—

—

D₄₇

yes

characteristic, complemented

C₄₇

#2819

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×D₄

—

C₂²×D₄

—

#124

—

C₂×C₄⋊C₄

C₂×D₄₇

yes

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₄₇

yes

characteristic

#95

—

#95

C₂×D₄₇

yes

—

C₂⁴

—

C₂×Q₈

—

C₂×Q₈

—

188

—

C₂×D₄

—

C₂×D₄

—

188

—

C₂×D₄

—

188

—

C₂×D₄

—

188

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₄₇

yes

—

C₂²×C₄

C₂²×D₄₇

yes

—

C₂²×C₄

C₂²×D₄₇

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

100

C₂²⋊C₄

—

101

C₂²⋊C₄

—

188

—

102

C₄²

D₁₈₈

yes

complemented

103

C₂³

—

104

C₂³

—

105

C₂³

—

106

C₂³

—

107

C₂³

—

108

C₂³

C₂³×D₄₇

yes

characteristic

109

Q₈

—

188

—

110

D₄

—

188

—

111

D₄

—

188

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

C₂×D₁₈₈

yes

—

120

C₂×C₄

C₂×D₁₈₈

yes

—

121

C₂²

—

188

—

122

C₂²

—

188

—

123

C₂²

—

188

—

124

C₂²

—

188

—

125

C₂²

—

188

—

126

C₂²

C₂²×D₁₈₈

yes

characteristic

127

C₂²

#13094069

yes

characteristic

128

C₂²

#13094070

yes

—

129

C₂²

#13094072

yes

characteristic

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

—

188

—

134

C₂

—

376

—

135

C₂

#15689692

yes

—

136

C₂

#15689704

yes

characteristic

137

C₂

#15689865

yes

—

138

C₂

#15689871

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂⁴
Fitting	C₄₇×#382	C₂
hypercenter	#382	D₄₇
hypocenter	C₄₇	#2819
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ #382

subgroup

G/H

center, Frattini

C₂³

C₂³×D₄₇

commutator, socle

C₂³×C₄₇

C₂⁴

Fitting

C₄₇×#382

C₂

D₄₇

C₄₇

G ▷ C₂³×C₄₇ ▷ C₄₇

C₁ ◁ C₂³ ◁ #382

N	H ≅ G/N	type
1. C₂×#13094012	C₂	semidirect
2. #15689040	C₂	semidirect
3. #15689036	C₂	semidirect
4. #15689033	C₂	semidirect
5. C₄₇×#382	C₂	semidirect
29. C₄²×C₄₇	D₄	semidirect
74. #382	D₄₇	semidirect
75. C₄₇	#2819	semidirect
102. C₄²	D₁₈₈	semidirect

H ≅ G/N

type

1. C₂×#13094012

C₂

semidirect

2. #15689040

C₂

semidirect

3. #15689036

C₂

semidirect

4. #15689033

C₂

semidirect

5. C₄₇×#382

C₂

semidirect

29. C₄²×C₄₇

D₄

semidirect

74. #382

D₄₇

semidirect

75. C₄₇

#2819

semidirect

102. C₄²

D₁₈₈

semidirect

Subgroups — #20251434

Conjugacy classes of subgroups

Complements