Group #20242122 — subgroups

Subgroups — #20242122

Order 6016.

254 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 67 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	#15689674	C₂	1	1	yes	maximal, characteristic
2	#15689145	C₂	1	2	yes	maximal, complemented
3	#15688872	C₂	1	1	yes	maximal, characteristic, complemented
4	#15688866	C₂	1	2	yes	maximal, complemented
5	C₄₇×#333	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#10625354	C₄	1	2	yes	—
7	C₂×#10625324	—	2	2	no	—
8	#13094021	C₂²	1	1	yes	characteristic
9	#13094013	C₄	1	2	yes	—
10	#13093999	—	2	2	no	—
11	#13093986	—	2	2	no	—
12	C₄²×D₄₇	C₂²	1	1	yes	characteristic
13	C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
14	Dic₄₇⋊C₈	—	2	2	no	—
15	C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
17	C₂×Q₈×C₄₇	C₂×C₄	1	2	yes	—
18	#10625354	—	4	4	no	—
19	C₂×D₁₈₈	—	2	2	no	—
20	C₂×D₁₈₈	—	2	2	no	—
21	C₂×D₁₈₈	C₂×C₄	1	2	yes	—
22	C₂×C₄×D₄₇	—	2	2	no	—
23	C₂×C₄×D₄₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×D₄₇	D₄	1	2	yes	—
25	C₂×C₈×C₄₇	—	2	2	no	—
26	C₄⋊C₄×C₄₇	—	2	2	no	—
27	C₄²×C₄₇	C₂³	1	1	yes	characteristic
28	(C₂×D₄₇)⋊C₄	—	2	4	no	—
29	C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
30	C₄×Dic₄₇	D₄	1	2	yes	—
31	C₂×C₄₇⋊C₈	—	2	2	no	—
32	#10625324	—	4	4	no	—
33	C₂²×D₄₇	—	2	2	no	—
34	C₂²×D₄₇	C₂²⋊C₄	1	1	yes	characteristic
35	Q₈×C₄₇	—	4	2	no	—
36	C₂×C₄×C₄₇	—	2	2	no	—
37	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
39	C₂×Dic₄₇	C₂²⋊C₄	1	2	yes	—
40	C₂×Dic₄₇	C₂²⋊C₄	1	1	yes	characteristic
41	D₁₈₈	—	4	2	no	—
42	D₁₈₈	—	4	4	no	—
43	D₁₈₈	—	4	2	no	—
44	D₁₈₈	—	4	2	no	—
45	C₄×D₄₇	—	2	2	no	—
46	C₄×D₄₇	—	2	2	no	—
47	C₄×D₄₇	—	2	2	no	—
48	C₄×D₄₇	—	4	4	no	—
49	C₈×C₄₇	—	4	2	no	—
50	C₄₇⋊C₈	—	4	2	no	—
51	C₂²×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
52	C₂×D₄₇	—	2	1	no	—
53	C₂×D₄₇	—	2	2	no	—
54	C₂×D₄₇	—	4	2	no	—
55	C₂×D₄₇	—	4	4	no	—
56	C₄×C₄₇	—	2	1	no	—
57	C₄×C₄₇	—	4	2	no	—
58	C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
59	C₄×C₄₇	#137	1	2	yes	—
60	Dic₄₇	—	2	2	no	—
61	Dic₄₇	#100	1	2	yes	—
62	#1560	—	47	1	no	maximal, Sylow
63	C₂×C₄₇	C₂×#100	1	1	yes	characteristic
64	C₂×C₄₇	#417	1	2	yes	—
65	D₄₇	—	4	1	no	—
66	D₄₇	—	8	2	no	—
67	#532	—	47	1	no	—
68	#423	—	47	2	no	—
69	#333	—	47	1	no	—
70	#333	D₄₇	1	1	yes	characteristic, complemented
71	#331	—	47	2	no	—
72	C₄₇	#1560	1	1	yes	characteristic, complemented, Sylow
73	C₂×#41	—	47	2	no	—
74	C₂×M₁₆	—	94	2	no	—
75	C₄⋊Q₈	C₂×D₄₇	1	1	yes	characteristic
76	#127	—	47	1	no	—
77	#124	—	47	2	no	—
78	#121	—	94	2	no	—
79	C₄×D₄	—	94	2	no	—
80	C₂×C₄²	—	47	1	no	—
81	C₄⋊C₈	—	47	2	no	—
82	C₄⋊C₈	—	94	2	no	—
83	C₄⋊C₈	C₂×D₄₇	1	2	yes	—
84	#41	—	188	4	no	—
85	C₂×Q₈	C₄×D₄₇	1	2	yes	—
86	C₂×D₄	—	47	2	no	—
87	C₂×D₄	—	94	2	no	—
88	C₂×D₄	—	94	2	no	—
89	C₂²×C₄	—	47	1	no	—
90	C₂²×C₄	—	47	2	no	—
91	C₂²×C₄	—	94	2	no	—
92	M₁₆	—	188	4	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	94	2	no	—
95	C₄⋊C₄	—	2	2	no	—
96	C₂²⋊C₄	—	94	4	no	—
97	C₄²	—	47	1	no	—
98	C₄²	—	47	2	no	—
99	C₄²	C₂²×D₄₇	1	1	yes	characteristic
100	C₂³	—	47	1	no	—
101	C₂³	—	94	2	no	—
102	Q₈	—	4	2	no	—
103	D₄	—	188	2	no	—
104	D₄	—	188	4	no	—
105	D₄	—	188	2	no	—
106	D₄	—	188	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	47	1	no	—
109	C₂×C₄	—	47	2	no	—
110	C₂×C₄	—	94	2	no	—
111	C₂×C₄	—	94	2	no	—
112	C₂×C₄	—	94	2	no	—
113	C₂×C₄	—	188	4	no	—
114	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
115	C₂×C₄	D₄×D₄₇	1	2	yes	—
116	C₈	—	4	2	no	—
117	C₈	—	188	2	no	—
118	C₂²	—	94	1	no	—
119	C₂²	—	94	2	no	—
120	C₂²	—	188	2	no	—
121	C₂²	—	188	4	no	—
122	C₂²	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
123	C₄	—	2	1	no	—
124	C₄	—	4	2	no	—
125	C₄	—	47	2	no	—
126	C₄	—	94	2	no	—
127	C₄	#13094051	1	2	yes	—
128	C₄	#13094056	1	2	yes	—
129	C₂	—	376	2	no	—
130	C₂	—	188	1	no	—
131	C₂	#100×D₄₇	1	1	yes	characteristic
132	C₂	#15689123	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#15689674

C₂

yes

maximal, characteristic

#15689145

C₂

yes

maximal, complemented

#15688872

C₂

yes

maximal, characteristic, complemented

#15688866

C₂

yes

maximal, complemented

C₄₇×#333

C₂

yes

maximal, characteristic, complemented

C₂×#10625354

C₄

yes

—

C₂×#10625324

—

#13094021

C₂²

yes

characteristic

#13094013

C₄

yes

—

#13093999

—

#13093986

—

C₄²×D₄₇

C₂²

yes

characteristic

C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

Dic₄₇⋊C₈

—

C₄⋊Q₈×C₄₇

C₂²

yes

characteristic

C₄⋊C₈×C₄₇

C₂²

yes

—

C₂×Q₈×C₄₇

C₂×C₄

yes

—

#10625354

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

C₂×C₄

yes

—

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

C₂×C₄

yes

characteristic

C₂×C₄×D₄₇

D₄

yes

—

C₂×C₈×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

C₂³

yes

characteristic

(C₂×D₄₇)⋊C₄

—

C₄×Dic₄₇

C₂×C₄

yes

characteristic

C₄×Dic₄₇

D₄

yes

—

C₂×C₄₇⋊C₈

—

#10625324

—

C₂²×D₄₇

—

C₂²×D₄₇

C₂²⋊C₄

yes

characteristic

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×Dic₄₇

C₂²⋊C₄

yes

—

C₂×Dic₄₇

C₂²⋊C₄

yes

characteristic

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

C₄×D₄₇

—

C₄×D₄₇

—

C₄×D₄₇

—

C₄×D₄₇

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₂²×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

AGL(1,ℤ/8)

yes

—

C₄×C₄₇

#137

yes

—

Dic₄₇

—

Dic₄₇

#100

yes

—

—

C₂×C₄₇

yes

C₂×C₄₇

yes

—

D₄₇

—

D₄₇

—

—

—

—

D₄₇

yes

characteristic, complemented

#331

—

C₄₇

#1560

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂×M₁₆

—

C₄⋊Q₈

C₂×D₄₇

yes

—

—

—

C₄×D₄

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₄₇

yes

—

#41

—

188

—

C₂×Q₈

C₄×D₄₇

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

188

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂²×D₄₇

yes

characteristic

100

C₂³

—

101

C₂³

—

102

Q₈

—

103

D₄

—

188

—

104

D₄

—

188

—

105

D₄

—

188

—

106

D₄

—

188

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

188

—

114

C₂×C₄

C₂×C₄×D₄₇

yes

characteristic

115

C₂×C₄

D₄×D₄₇

yes

—

116

C₈

—

117

C₈

—

188

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

188

—

121

C₂²

—

188

—

122

C₂²

C₂²⋊C₄×D₄₇

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

#13094051

yes

—

128

C₄

#13094056

yes

—

129

C₂

—

376

—

130

C₂

—

188

—

131

C₂

#100×D₄₇

yes

characteristic

132

C₂

#15689123

yes

—

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂²×C₄
Frattini	C₄²	C₂²×D₄₇
Fitting	C₄₇×#333	C₂
socle	C₂²×C₄₇	C₂×C₂²⋊C₄
hypercenter	#333	D₄₇
hypocenter	C₄₇	#1560
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #333

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₄₇

commutator

C₂×C₄×C₄₇

C₂²×C₄

Frattini

C₄²

C₂²×D₄₇

Fitting

C₄₇×#333

C₂

socle

C₂²×C₄₇

C₂×C₂²⋊C₄

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #333

N	H ≅ G/N	type
2. #15689145	C₂	semidirect
3. #15688872	C₂	semidirect
4. #15688866	C₂	semidirect
5. C₄₇×#333	C₂	semidirect
70. #333	D₄₇	semidirect
72. C₄₇	#1560	semidirect

H ≅ G/N

type

2. #15689145

C₂

semidirect

3. #15688872

C₂

semidirect

4. #15688866

C₂

semidirect

5. C₄₇×#333

C₂

semidirect

70. #333

D₄₇

semidirect

72. C₄₇

#1560

semidirect

Subgroups — #20242122

Conjugacy classes of subgroups

Complements