Group #20243372 — subgroups

Subgroups — #20243372

Order 6016.

312 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 125 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×C₄⋊(C₄₇⋊C₈)	C₂	1	2	yes	maximal, complemented
2	C₂×C₂²⋊(C₄₇⋊C₈)	C₂	1	2	yes	maximal
3	#15688878	C₂	1	2	yes	maximal, complemented
4	C₂×#117×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄₇⋊C₈	C₂²	1	4	yes	—
6	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
7	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
8	C₂³×C₄×C₄₇	C₂²	1	1	yes	characteristic
9	#117×C₄₇	—	2	4	no	—
10	C₂×C₄⋊C₄×C₄₇	C₄	1	2	yes	—
11	C₂×C₂²⋊C₄×C₄₇	C₄	1	2	yes	—
12	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
13	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
14	C₂⁴×C₄₇	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄×C₄₇	—	2	2	no	—
16	C₂²×C₄×C₄₇	—	2	2	no	—
17	C₂²×C₄×C₄₇	C₂×C₄	1	2	yes	—
18	C₂²×C₄×C₄₇	C₂×C₄	1	2	yes	—
19	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₄₇	D₄	1	2	yes	—
21	C₂²×C₄×C₄₇	Q₈	1	2	yes	—
22	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
23	C₄⋊C₄×C₄₇	—	2	4	no	—
24	C₂²⋊C₄×C₄₇	—	2	4	no	—
25	C₄²×C₄₇	—	2	2	no	—
26	C₄²×C₄₇	D₄	1	4	yes	—
27	C₂×C₄₇⋊C₈	—	2	4	no	—
28	C₂×C₄₇⋊C₈	—	2	4	no	—
29	C₂×C₄₇⋊C₈	—	2	4	no	—
30	C₂³×C₄₇	—	2	1	no	—
31	C₂³×C₄₇	—	2	1	no	—
32	C₂³×C₄₇	—	2	4	no	—
33	C₂³×C₄₇	C₄⋊C₄	1	2	yes	—
34	C₂³×C₄₇	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₄₇	—	2	1	no	—
36	C₂×C₄×C₄₇	—	2	1	no	—
37	C₂×C₄×C₄₇	—	2	4	no	—
38	C₂×C₄×C₄₇	—	2	4	no	—
39	C₂×C₄×C₄₇	—	2	4	no	—
40	C₂×C₄×C₄₇	—	2	2	no	—
41	C₂×C₄×C₄₇	—	2	4	no	—
42	C₂×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
43	C₂×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
44	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₄×C₄₇	#41	1	2	yes	—
48	C₄₇⋊C₈	—	4	4	no	—
49	C₂²×C₄₇	—	2	2	no	—
50	C₂²×C₄₇	—	2	4	no	—
51	C₂²×C₄₇	—	4	2	no	—
52	C₂²×C₄₇	—	4	2	no	—
53	C₂²×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₄₇	#117	1	1	yes	characteristic
56	C₂²×C₄₇	#131	1	4	yes	—
57	C₄×C₄₇	—	2	2	no	—
58	C₄×C₄₇	—	2	4	no	—
59	C₄×C₄₇	—	4	2	no	—
60	C₄×C₄₇	#121	1	2	yes	—
61	C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
62	#1798	—	47	1	no	maximal, Sylow
63	C₂×C₄₇	—	4	2	no	—
64	C₂×C₄₇	#380	1	1	yes	characteristic
65	C₂×C₄₇	#408	1	2	yes	—
66	C₂×C₄₇	#424	1	2	yes	—
67	C₂×C₄₇	#433	1	2	yes	—
68	C₂×#117	D₄₇	1	1	yes	characteristic
69	C₂×C₄⋊C₈	—	47	2	no	—
70	C₂×C₂²⋊C₈	—	47	2	no	—
71	#336	—	47	2	no	—
72	C₄₇	#1798	1	1	yes	characteristic, complemented, Sylow
73	C₂³×C₄	C₂×D₄₇	1	1	yes	characteristic
74	C₂²×C₈	—	47	4	no	—
75	#117	—	2	4	no	—
76	C₂×C₄⋊C₄	Dic₄₇	1	2	yes	—
77	C₂×C₂²⋊C₄	Dic₄₇	1	2	yes	—
78	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
79	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
80	C₄⋊C₈	—	94	4	no	—
81	C₂²⋊C₈	—	94	4	no	—
82	C₂⁴	C₂×Dic₄₇	1	1	yes	characteristic
83	C₂²×C₄	—	2	2	no	—
84	C₂²×C₄	—	2	2	no	—
85	C₂²×C₄	Dic₉₄	1	2	yes	—
86	C₂²×C₄	D₁₈₈	1	2	yes	—
87	C₂²×C₄	C₂×Dic₄₇	1	2	yes	—
88	C₂²×C₄	C₂×Dic₄₇	1	1	yes	characteristic
89	C₂²×C₄	C₂×Dic₄₇	1	2	yes	—
90	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
91	C₂×C₈	—	94	4	no	—
92	C₂×C₈	—	94	4	no	—
93	C₂×C₈	—	94	4	no	—
94	C₄⋊C₄	—	2	4	no	—
95	C₂²⋊C₄	—	2	4	no	—
96	C₄²	—	2	2	no	—
97	C₄²	C₂²⋊D₄₇	1	4	yes	—
98	C₂³	—	2	1	no	—
99	C₂³	—	2	1	no	—
100	C₂³	—	2	4	no	—
101	C₂³	C₄⋊Dic₄₇	1	2	yes	—
102	C₂³	C₂²×Dic₄₇	1	1	yes	characteristic
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	2	1	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	2	4	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	4	no	—
110	C₂×C₄	C₄⋊Dic₄₇	1	2	yes	—
111	C₂×C₄	C₂²⋊Dic₄₇	1	4	yes	—
112	C₂×C₄	C₂×Dic₉₄	1	1	yes	characteristic
113	C₂×C₄	C₂×D₁₈₈	1	1	yes	characteristic
114	C₂×C₄	#10625350	1	2	yes	—
115	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
116	C₈	—	188	4	no	—
117	C₂²	—	2	2	no	—
118	C₂²	—	2	4	no	—
119	C₂²	—	4	2	no	—
120	C₂²	—	4	2	no	—
121	C₂²	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
122	C₂²	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
123	C₂²	#13093983	1	1	yes	characteristic
124	C₂²	#13094031	1	4	yes	—
125	C₄	—	2	2	no	—
126	C₄	—	2	4	no	—
127	C₄	—	4	2	no	—
128	C₄	#13093998	1	2	yes	—
129	C₄	#13094003	1	2	yes	—
130	C₂	—	4	2	no	—
131	C₂	#15688983	1	1	yes	characteristic
132	C₂	#15689090	1	2	yes	—
133	C₂	#15689150	1	2	yes	—
134	C₂	#15689184	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₄₇⋊C₈)

C₂

yes

maximal, complemented

C₂×C₂²⋊(C₄₇⋊C₈)

C₂

yes

maximal

#15688878

C₂

yes

maximal, complemented

C₂×#117×C₄₇

C₂

yes

maximal, characteristic

C₂²×C₄₇⋊C₈

C₂²

yes

—

C₄⋊(C₄₇⋊C₈)

—

C₂²⋊(C₄₇⋊C₈)

—

C₂³×C₄×C₄₇

C₂²

yes

characteristic

#117×C₄₇

—

C₂×C₄⋊C₄×C₄₇

C₄

yes

—

C₂×C₂²⋊C₄×C₄₇

C₄

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₂⁴×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

Q₈

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₄⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

D₄

yes

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

C₄⋊C₄

yes

—

C₂³×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂×C₄×C₄₇

C₄⋊C₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₄₇

#41

yes

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₄₇

C₂×C₄⋊C₄

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#121

yes

—

C₄×C₄₇

C₂²⋊Q₈

yes

—

#1798

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

#380

yes

characteristic

C₂×C₄₇

#408

yes

—

C₂×C₄₇

#424

yes

—

C₂×C₄₇

#433

yes

—

C₂×#117

D₄₇

yes

characteristic

C₂×C₄⋊C₈

—

C₂×C₂²⋊C₈

—

#336

—

C₄₇

#1798

yes

characteristic, complemented, Sylow

C₂³×C₄

C₂×D₄₇

yes

characteristic

C₂²×C₈

—

#117

—

C₂×C₄⋊C₄

Dic₄₇

yes

—

C₂×C₂²⋊C₄

Dic₄₇

yes

—

C₂×C₄²

C₂×D₄₇

yes

characteristic

C₂×C₄²

C₂×D₄₇

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

C₂⁴

C₂×Dic₄₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

Dic₉₄

yes

—

C₂²×C₄

D₁₈₈

yes

—

C₂²×C₄

C₂×Dic₄₇

yes

—

C₂²×C₄

C₂×Dic₄₇

yes

characteristic

C₂²×C₄

C₂×Dic₄₇

yes

—

C₂²×C₄

C₂²×D₄₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂²⋊D₄₇

yes

—

C₂³

—

C₂³

—

100

C₂³

—

101

C₂³

C₄⋊Dic₄₇

yes

—

102

C₂³

C₂²×Dic₄₇

yes

characteristic

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

C₄⋊Dic₄₇

yes

—

111

C₂×C₄

C₂²⋊Dic₄₇

yes

—

112

C₂×C₄

C₂×Dic₉₄

yes

characteristic

113

C₂×C₄

C₂×D₁₈₈

yes

characteristic

114

C₂×C₄

#10625350

yes

—

115

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

116

C₈

—

188

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

C₂×C₂²⋊Dic₄₇

yes

characteristic

122

C₂²

C₂×C₄⋊Dic₄₇

yes

characteristic

123

C₂²

#13093983

yes

characteristic

124

C₂²

#13094031

yes

—

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

#13093998

yes

—

129

C₄

#13094003

yes

—

130

C₂

—

131

C₂

#15688983

yes

characteristic

132

C₂

#15689090

yes

—

133

C₂

#15689150

yes

—

134

C₂

#15689184

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₄₇
commutator, socle	C₂³×C₄₇	C₂²×C₄
Fitting	C₂×#117×C₄₇	C₂
hypercenter	C₂×#117	D₄₇
hypocenter	C₄₇	#1798
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#117

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₄₇

commutator, socle

C₂³×C₄₇

C₂²×C₄

Fitting

C₂×#117×C₄₇

C₂

D₄₇

C₄₇

G ▷ C₂³×C₄₇ ▷ C₄₇

C₁ ◁ C₂²×C₄ ◁ C₂×#117

N	H ≅ G/N	type
1. C₂×C₄⋊(C₄₇⋊C₈)	C₂	semidirect
3. #15688878	C₂	semidirect
72. C₄₇	#1798	semidirect

H ≅ G/N

type

1. C₂×C₄⋊(C₄₇⋊C₈)

C₂

semidirect

3. #15688878

C₂

semidirect

72. C₄₇

#1798

semidirect

Subgroups — #20243372

Conjugacy classes of subgroups

Complements