Group #20242106 — subgroups

Subgroups — #20242106

Order 6016.

228 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 77 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×#13094024	C₂	1	1	yes	maximal, characteristic
2	#15689180	C₂	1	1	yes	maximal, characteristic
3	#15688870	C₂	1	4	yes	maximal, complemented
4	C₄₇×#432	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₉₄	—	2	1	no	—
6	#13094024	—	2	1	no	—
7	#13094024	—	2	1	no	—
8	#13094024	C₄	1	2	yes	—
9	#13094024	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₄₇	—	2	1	no	—
11	C₂×C₄⋊Dic₄₇	C₄	1	2	yes	—
12	C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
13	C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
14	#13093935	—	2	1	no	—
15	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
17	C₂²⋊C₈×C₄₇	—	2	1	no	—
18	#97×C₄₇	—	2	1	no	—
19	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
22	C₂²×Dic₄₇	—	2	2	no	—
23	C₂×Dic₉₄	—	2	2	no	—
24	C₂×Dic₉₄	—	2	2	no	—
25	C₂×Dic₉₄	—	4	4	no	—
26	C₂×C₈×C₄₇	—	2	2	no	—
27	C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
28	C₄²×C₄₇	D₄	1	1	yes	characteristic
29	C₄²×C₄₇	D₄	1	1	yes	characteristic
30	C₄²×C₄₇	C₂³	1	1	yes	characteristic
31	C₄⋊Dic₄₇	—	2	2	no	—
32	C₄⋊Dic₄₇	—	2	2	no	—
33	C₄⋊Dic₄₇	—	2	2	no	—
34	C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
35	C₂×C₄₇⋊C₈	—	2	2	no	—
36	C₂³×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₄×C₄₇	—	2	2	no	—
38	C₂×C₄×C₄₇	—	2	2	no	—
39	C₂×C₄×C₄₇	—	2	2	no	—
40	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
46	C₂×Dic₄₇	—	2	4	no	—
47	C₂×Dic₄₇	—	4	4	no	—
48	Dic₉₄	—	4	4	no	—
49	Dic₉₄	—	4	4	no	—
50	C₈×C₄₇	—	4	2	no	—
51	C₄₇⋊C₈	—	4	2	no	—
52	C₂²×C₄₇	—	2	1	no	—
53	C₂²×C₄₇	—	2	1	no	—
54	C₂²×C₄₇	#101	1	2	yes	—
55	C₂²×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
56	C₄×C₄₇	—	2	1	no	—
57	C₄×C₄₇	—	2	1	no	—
58	C₄×C₄₇	—	2	1	no	—
59	C₄×C₄₇	—	2	1	no	—
60	C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
61	C₄×C₄₇	#137	1	2	yes	—
62	Dic₄₇	—	4	4	no	—
63	#1556	—	47	1	no	maximal, Sylow
64	C₂×C₄₇	—	2	2	no	—
65	C₂×C₄₇	C₂×#101	1	1	yes	characteristic
66	C₂×C₄₇	#418	1	1	yes	characteristic
67	C₂×C₄₇	#419	1	1	yes	characteristic
68	C₂×C₄⋊Q₈	—	47	1	no	—
69	#432	—	47	1	no	—
70	#432	D₄₇	1	1	yes	characteristic
71	#332	—	47	4	no	—
72	C₄₇	#1556	1	1	yes	characteristic, complemented, Sylow
73	C₂²×Q₈	—	94	1	no	—
74	C₄⋊Q₈	—	47	2	no	—
75	C₄⋊Q₈	—	47	2	no	—
76	C₄⋊Q₈	—	94	1	no	—
77	C₄⋊Q₈	—	94	1	no	—
78	C₂×C₄⋊C₄	—	47	2	no	—
79	C₂×C₄⋊C₄	—	94	1	no	—
80	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
81	C₄⋊C₈	—	47	2	no	—
82	C₄⋊C₈	C₂×D₄₇	1	2	yes	—
83	C₂²⋊C₈	—	2	1	no	—
84	C₂²⋊C₈	—	94	1	no	—
85	#97	—	2	1	no	—
86	#97	—	94	1	no	—
87	C₂×Q₈	—	94	2	no	—
88	C₂×Q₈	—	94	2	no	—
89	C₂×Q₈	—	188	4	no	—
90	C₂²×C₄	—	94	2	no	—
91	C₂²×C₄	C₄×D₄₇	1	1	yes	characteristic
92	C₂²×C₄	D₁₈₈	1	1	yes	characteristic
93	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	—	94	2	no	—
96	C₄⋊C₄	—	47	4	no	—
97	C₄⋊C₄	—	94	2	no	—
98	C₄⋊C₄	—	94	2	no	—
99	C₄⋊C₄	—	94	2	no	—
100	C₄²	C₄×D₄₇	1	1	yes	characteristic
101	C₄²	D₁₈₈	1	1	yes	characteristic
102	C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
103	C₄²	C₂²×D₄₇	1	1	yes	characteristic
104	C₂³	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
105	Q₈	—	188	4	no	—
106	Q₈	—	188	4	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	94	4	no	—
111	C₂×C₄	—	188	4	no	—
112	C₂×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
113	C₂×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
114	C₂×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
115	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
116	C₂×C₄	C₂×D₁₈₈	1	1	yes	characteristic
117	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
118	C₈	—	4	2	no	—
119	C₈	—	188	2	no	—
120	C₂²	—	2	1	no	—
121	C₂²	—	2	1	no	—
122	C₂²	#13093942	1	2	yes	—
123	C₂²	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
124	C₄	—	2	1	no	—
125	C₄	—	2	1	no	—
126	C₄	—	2	1	no	—
127	C₄	—	2	1	no	—
128	C₄	—	188	4	no	—
129	C₄	#13094049	1	2	yes	—
130	C₄	#13094057	1	2	yes	—
131	C₂	—	2	2	no	—
132	C₂	C₂×#13093942	1	1	yes	characteristic
133	C₂	#15689127	1	1	yes	characteristic
134	C₂	#15689133	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13094024

C₂

yes

maximal, characteristic

#15689180

C₂

yes

maximal, characteristic

#15688870

C₂

yes

maximal, complemented

C₄₇×#432

C₂

yes

maximal, characteristic

C₂²×Dic₉₄

—

#13094024

—

#13094024

—

#13094024

C₄

yes

—

#13094024

C₂²

yes

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

C₄

yes

—

C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

C₂²⋊(C₄₇⋊C₈)

—

#13093935

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₄⋊C₈×C₄₇

C₂²

yes

—

C₂²⋊C₈×C₄₇

—

#97×C₄₇

—

C₂²×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×Dic₄₇

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×C₈×C₄₇

—

C₄²×C₄₇

C₂×C₄

yes

characteristic

C₄²×C₄₇

D₄

yes

characteristic

C₄²×C₄₇

D₄

yes

characteristic

C₄²×C₄₇

C₂³

yes

characteristic

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

C₂×C₄

yes

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#101

yes

—

C₂²×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

AGL(1,ℤ/8)

yes

—

C₄×C₄₇

#137

yes

—

Dic₄₇

—

#1556

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄⋊Q₈

—

#432

—

#432

D₄₇

yes

characteristic

#332

—

C₄₇

#1556

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₄₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

C₂×D₄₇

yes

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

#97

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

188

—

C₂²×C₄

—

C₂²×C₄

C₄×D₄₇

yes

characteristic

C₂²×C₄

D₁₈₈

yes

characteristic

C₂²×C₄

C₂²⋊D₄₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄²

C₄×D₄₇

yes

characteristic

101

C₄²

D₁₈₈

yes

characteristic

102

C₄²

C₂²⋊D₄₇

yes

characteristic

103

C₄²

C₂²×D₄₇

yes

characteristic

104

C₂³

(C₂×D₄₇)⋊C₄

yes

characteristic

105

Q₈

—

188

—

106

Q₈

—

188

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

188

—

112

C₂×C₄

(C₂×D₄₇)⋊C₄

yes

characteristic

113

C₂×C₄

(C₂×D₄₇)⋊C₄

yes

characteristic

114

C₂×C₄

(C₂×D₄₇)⋊C₄

yes

characteristic

115

C₂×C₄

C₂×C₄×D₄₇

yes

characteristic

116

C₂×C₄

C₂×D₁₈₈

yes

characteristic

117

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

118

C₈

—

119

C₈

—

188

—

120

C₂²

—

121

C₂²

—

122

C₂²

#13093942

yes

—

123

C₂²

C₂×(C₂×D₄₇)⋊C₄

yes

characteristic

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

—

188

—

129

C₄

#13094049

yes

—

130

C₄

#13094057

yes

—

131

C₂

—

132

C₂

C₂×#13093942

yes

characteristic

133

C₂

#15689127

yes

characteristic

134

C₂

#15689133

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×(C₂×D₄₇)⋊C₄
commutator	C₂×C₄×C₄₇	C₂²×C₄
Frattini	C₄²	C₂²×D₄₇
Fitting	C₄₇×#432	C₂
socle	C₂²×C₄₇	C₂×C₂²⋊C₄
hypercenter	#432	D₄₇
hypocenter	C₄₇	#1556
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #432

subgroup

G/H

center

C₂²

C₂×(C₂×D₄₇)⋊C₄

commutator

C₂×C₄×C₄₇

C₂²×C₄

Frattini

C₄²

C₂²×D₄₇

Fitting

C₄₇×#432

C₂

socle

C₂²×C₄₇

C₂×C₂²⋊C₄

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #432

N	H ≅ G/N	type
3. #15688870	C₂	semidirect
72. C₄₇	#1556	semidirect

H ≅ G/N

type

3. #15688870

C₂

semidirect

72. C₄₇

#1556

semidirect

Subgroups — #20242106

Conjugacy classes of subgroups

Complements