Group #20276396 — subgroups

Subgroups — C₂×#16810422 #20276396

Order 6784.

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

normal subgroups only

This group has 240 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×#13133136	C₂	1	2	yes	maximal
2	#16810422	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₄⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#11719917	—	2	2	no	—
5	#13133136	—	2	2	no	—
6	#13133136	C₂²	1	4	yes	—
7	C₂²×Q₈×C₅₃	C₄	1	1	yes	characteristic
8	C₂²×Q₈×C₅₃	C₂²	1	1	yes	characteristic
9	C₄⋊Q₈×C₅₃	—	2	2	no	—
10	C₄⋊Q₈×C₅₃	C₄	1	2	yes	—
11	C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
12	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
13	C₂×C₄²×C₅₃	C₄	1	1	yes	characteristic
14	C₂×Q₈×C₅₃	—	2	1	no	—
15	C₂×Q₈×C₅₃	—	4	1	no	—
16	C₂×Q₈×C₅₃	—	4	1	no	—
17	C₂×Q₈×C₅₃	—	4	1	no	—
18	C₂×Q₈×C₅₃	—	4	1	no	—
19	C₂×Q₈×C₅₃	C₂×C₄	1	2	yes	—
20	C₂×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂×Q₈×C₅₃	D₄	1	2	yes	—
22	C₂×Q₈×C₅₃	C₂³	1	1	yes	characteristic
23	C₂²×C₄×C₅₃	—	2	1	no	—
24	C₂²×C₄×C₅₃	—	2	1	no	—
25	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₅₃	D₄	1	2	yes	—
27	C₄⋊C₄×C₅₃	—	2	4	no	—
28	C₄⋊C₄×C₅₃	—	2	4	no	—
29	C₄²×C₅₃	—	2	1	no	—
30	C₄²×C₅₃	C₂×C₄	1	2	yes	—
31	#11719917	—	2	4	no	—
32	#11719917	—	4	2	no	—
33	C₂×#88322	—	4	2	no	—
34	C₂³×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
35	Q₈×C₅₃	—	4	1	no	—
36	Q₈×C₅₃	—	4	2	no	—
37	Q₈×C₅₃	—	4	1	no	—
38	Q₈×C₅₃	—	4	2	no	—
39	Q₈×C₅₃	—	4	2	no	—
40	C₂×C₄×C₅₃	—	2	4	no	—
41	C₂×C₄×C₅₃	—	2	1	no	—
42	C₂×C₄×C₅₃	—	2	1	no	—
43	C₂×C₄×C₅₃	—	2	2	no	—
44	C₂×C₄×C₅₃	—	2	2	no	—
45	C₂×C₄×C₅₃	—	2	2	no	—
46	C₂×C₄×C₅₃	—	2	2	no	—
47	C₂×C₄×C₅₃	—	4	1	no	—
48	C₂×C₄×C₅₃	—	4	1	no	—
49	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
51	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
53	#88322	—	4	4	no	—
54	C₂²×C₅₃	—	2	2	no	—
55	C₂²×C₅₃	#99	1	1	yes	characteristic
56	C₂²×C₅₃	#99	1	1	yes	characteristic
57	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₄×C₅₃	—	2	4	no	—
59	C₄×C₅₃	—	2	2	no	—
60	C₄×C₅₃	—	2	2	no	—
61	C₄×C₅₃	—	2	1	no	—
62	C₄×C₅₃	—	2	1	no	—
63	C₄×C₅₃	—	4	2	no	—
64	C₂×#356	—	53	1	no	maximal, Sylow
65	C₂×C₅₃	—	2	1	no	—
66	C₂×C₅₃	—	2	1	no	—
67	C₂×C₅₃	#356	1	2	yes	complemented
68	C₂×C₅₃	C₂×#99	1	1	yes	characteristic
69	C₂×C₄⋊Q₈	D₅₃	1	1	yes	characteristic
70	C₂×#101	—	53	2	no	—
71	#356	—	53	4	no	—
72	C₅₃	C₂×#356	1	1	yes	characteristic, complemented, Sylow
73	C₂²×Q₈	Dic₅₃	1	1	yes	characteristic
74	C₂²×Q₈	C₂×D₅₃	1	1	yes	characteristic
75	C₂×M₁₆	—	106	2	no	—
76	C₄⋊Q₈	—	2	2	no	—
77	C₄⋊Q₈	Dic₅₃	1	2	yes	—
78	C₄⋊Q₈	C₂×D₅₃	1	2	yes	—
79	C₂×C₄⋊C₄	—	2	2	no	—
80	C₂×C₄²	Dic₅₃	1	1	yes	characteristic
81	#101	—	53	4	no	—
82	#101	—	106	2	no	—
83	C₂×Q₈	—	2	1	no	—
84	C₂×Q₈	—	4	1	no	—
85	C₂×Q₈	—	4	1	no	—
86	C₂×Q₈	—	4	1	no	—
87	C₂×Q₈	—	4	1	no	—
88	C₂×Q₈	C₂×Dic₅₃	1	2	yes	—
89	C₂×Q₈	C₂×Dic₅₃	1	1	yes	characteristic
90	C₂×Q₈	C₂²⋊D₅₃	1	2	yes	—
91	C₂×Q₈	C₂²×D₅₃	1	1	yes	characteristic
92	C₂²×C₄	—	2	1	no	—
93	C₂²×C₄	—	2	1	no	—
94	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
95	C₂²×C₄	C₂²⋊D₅₃	1	2	yes	—
96	M₁₆	—	106	4	no	—
97	M₁₆	—	212	2	no	—
98	C₂×C₈	—	212	2	no	—
99	C₄⋊C₄	—	2	4	no	—
100	C₄⋊C₄	—	2	4	no	—
101	C₄²	—	2	1	no	—
102	C₄²	C₂×Dic₅₃	1	2	yes	—
103	C₂³	C₂²⋊Dic₅₃	1	1	yes	characteristic
104	Q₈	—	4	1	no	—
105	Q₈	—	4	2	no	—
106	Q₈	—	4	1	no	—
107	Q₈	—	4	2	no	—
108	Q₈	—	4	2	no	—
109	C₂×C₄	—	2	4	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	4	1	no	—
117	C₂×C₄	—	4	1	no	—
118	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
119	C₂×C₄	C₂²⋊Dic₅₃	1	2	yes	—
120	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
121	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
122	C₈	—	212	4	no	—
123	C₂²	—	2	2	no	—
124	C₂²	#13133132	1	1	yes	characteristic
125	C₂²	#13133132	1	1	yes	characteristic
126	C₂²	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
127	C₄	—	2	4	no	—
128	C₄	—	2	2	no	—
129	C₄	—	2	1	no	—
130	C₄	—	2	2	no	—
131	C₄	—	2	1	no	—
132	C₄	—	4	2	no	—
133	C₂	—	2	1	no	—
134	C₂	—	2	1	no	—
135	C₂	#16810422	1	2	yes	direct product factor
136	C₂	C₂×#13133132	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133136

C₂

yes

maximal

#16810422

C₂

yes

maximal, direct product factor

C₂×C₄⋊Q₈×C₅₃

C₂

yes

maximal, characteristic

C₂×#11719917

—

#13133136

—

#13133136

C₂²

yes

—

C₂²×Q₈×C₅₃

C₄

yes

characteristic

C₂²×Q₈×C₅₃

C₂²

yes

characteristic

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

C₄

yes

—

C₄⋊Q₈×C₅₃

C₂²

yes

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄²×C₅₃

C₄

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

C₂×C₄

yes

—

C₂×Q₈×C₅₃

C₂×C₄

yes

characteristic

C₂×Q₈×C₅₃

D₄

yes

—

C₂×Q₈×C₅₃

C₂³

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

—

#11719917

—

#11719917

—

C₂×#88322

—

C₂³×C₅₃

C₂²⋊C₄

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₂×#356

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#356

yes

complemented

C₂×C₅₃

C₂×#99

yes

characteristic

C₂×C₄⋊Q₈

D₅₃

yes

characteristic

C₂×#101

—

#356

—

C₅₃

C₂×#356

yes

characteristic, complemented, Sylow

C₂²×Q₈

Dic₅₃

yes

characteristic

C₂²×Q₈

C₂×D₅₃

yes

characteristic

C₂×M₁₆

—

106

—

C₄⋊Q₈

—

C₄⋊Q₈

Dic₅₃

yes

—

C₄⋊Q₈

C₂×D₅₃

yes

—

C₂×C₄⋊C₄

—

C₂×C₄²

Dic₅₃

yes

characteristic

#101

—

#101

—

106

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₅₃

yes

—

C₂×Q₈

C₂×Dic₅₃

yes

characteristic

C₂×Q₈

C₂²⋊D₅₃

yes

—

C₂×Q₈

C₂²×D₅₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₅₃

yes

characteristic

C₂²×C₄

C₂²⋊D₅₃

yes

—

M₁₆

—

106

—

M₁₆

—

212

—

C₂×C₈

—

212

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄²

—

102

C₄²

C₂×Dic₅₃

yes

—

103

C₂³

C₂²⋊Dic₅₃

yes

characteristic

104

Q₈

—

105

Q₈

—

106

Q₈

—

107

Q₈

—

108

Q₈

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

C₂²⋊Dic₅₃

yes

characteristic

119

C₂×C₄

C₂²⋊Dic₅₃

yes

—

120

C₂×C₄

C₂²×Dic₅₃

yes

characteristic

121

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

122

C₈

—

212

—

123

C₂²

—

124

C₂²

#13133132

yes

characteristic

125

C₂²

#13133132

yes

characteristic

126

C₂²

C₂×C₂²⋊Dic₅₃

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

—

133

C₂

—

134

C₂

—

135

C₂

#16810422

yes

direct product factor

136

C₂

C₂×#13133132

yes

characteristic

	subgroup	G/H
center	C₂²	#13133132
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂×Q₈	C₂²×D₅₃
Fitting	C₂×C₄⋊Q₈×C₅₃	C₂
socle	C₂²×C₅₃	#99
hypercenter	C₂×C₄⋊Q₈	D₅₃
hypocenter	C₅₃	C₂×#356
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×Q₈ ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂²

#13133132

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂×Q₈

C₂²×D₅₃

Fitting

C₂×C₄⋊Q₈×C₅₃

C₂

socle

C₂²×C₅₃

#99

hypercenter

C₂×C₄⋊Q₈

D₅₃

hypocenter

C₅₃

C₂×#356

LCS

G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×Q₈ ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
2. #16810422	C₂	direct (N × H)
67. C₂×C₅₃	#356	semidirect
72. C₅₃	C₂×#356	semidirect
135. C₂	#16810422	direct (N × H)

H ≅ G/N

type

2. #16810422

C₂

direct (N × H)

67. C₂×C₅₃

#356

semidirect

72. C₅₃

C₂×#356

semidirect

135. C₂

#16810422

direct (N × H)

Subgroups — C₂×#16810422 #20276396

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16810422 #20276396

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16810422 #20276396