Group #22560367 — subgroups

Subgroups — #22560367

Order 8576.

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

normal subgroups only

This group has 280 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×#14523977	C₂	1	1	yes	maximal, characteristic
2	C₂×#14523976	C₂	1	2	yes	maximal
3	C₂×C₈×Dic₆₇	C₂	1	1	yes	maximal, characteristic
4	#16863265	C₂	1	2	yes	maximal
5	C₂×#107×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₆₇	C₂²	1	2	yes	—
8	C₂×Dic₆₇⋊C₄	—	2	4	no	—
9	C₂×C₄×Dic₆₇	C₂²	1	1	yes	characteristic
10	#14523977	—	2	2	no	—
11	#14523976	—	2	4	no	—
12	C₈×Dic₆₇	C₄	1	4	yes	complemented
13	C₂²×C₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
15	#107×C₆₇	—	2	2	no	—
16	C₂²×Dic₆₇	—	2	2	no	—
17	C₂²×Dic₆₇	D₄	1	2	yes	—
18	C₂×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
19	C₂×C₆₇⋊C₈	D₄	1	2	yes	—
20	C₂×C₆₇⋊C₈	Q₈	1	2	yes	—
21	C₄⋊Dic₆₇	—	2	2	no	—
22	C₄⋊Dic₆₇	—	2	2	no	—
23	Dic₆₇⋊C₄	—	4	8	no	—
24	C₄×Dic₆₇	C₂×C₄	1	2	yes	—
25	C₄×Dic₆₇	Q₈	1	2	yes	—
26	C₂²×C₄×C₆₇	—	2	2	no	—
27	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
28	C₂×C₈×C₆₇	C₂×C₄	1	2	yes	—
29	C₂×C₈×C₆₇	D₄	1	2	yes	—
30	C₂×C₈×C₆₇	Q₈	1	2	yes	—
31	C₄⋊C₄×C₆₇	—	2	2	no	—
32	C₄⋊C₄×C₆₇	—	2	2	no	—
33	C₂³×C₆₇	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₆₇	—	4	2	no	—
35	C₂×C₄×C₆₇	—	4	4	no	—
36	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
42	C₂×Dic₆₇	—	4	2	no	—
43	C₂×Dic₆₇	—	4	4	no	—
44	C₂×Dic₆₇	C₄⋊C₄	1	4	yes	—
45	C₂×Dic₆₇	D₈	1	4	yes	—
46	C₂×Dic₆₇	Q₁₆	1	4	yes	—
47	C₈×C₆₇	C₄⋊C₄	1	4	yes	complemented
48	C₆₇⋊C₈	C₄⋊C₄	1	4	yes	complemented
49	C₂²×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₆₇	#127	1	1	yes	characteristic
51	C₂²×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
52	C₂²×C₆₇	C₂×D₈	1	2	yes	—
53	C₂²×C₆₇	C₂×Q₁₆	1	2	yes	—
54	C₄×C₆₇	—	8	2	no	—
55	C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
58	C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
59	Dic₆₇	—	8	2	no	—
60	Dic₆₇	#107	1	8	yes	complemented
61	C₂×C₆₇	#384	1	1	yes	characteristic
62	C₂×C₆₇	C₂×#107	1	2	yes	—
63	C₂×C₆₇	#493	1	1	yes	characteristic
64	C₂×C₆₇	#494	1	1	yes	characteristic
65	C₂×C₆₇	#500	1	2	yes	—
66	#1845	—	67	1	no	maximal, Sylow
67	C₆₇	#1845	1	1	yes	characteristic, complemented, Sylow
68	C₂×#107	—	67	1	no	—
69	C₂×#107	—	67	2	no	—
70	C₂×#107	D₆₇	1	1	yes	characteristic
71	C₂×C₄×C₈	—	67	1	no	—
72	#384	—	67	2	no	—
73	C₂²×C₈	—	67	1	no	—
74	C₂²×C₈	C₂×D₆₇	1	1	yes	characteristic
75	C₂×C₄⋊C₄	—	67	2	no	—
76	C₂×C₄⋊C₄	—	134	4	no	—
77	C₂×C₄⋊C₄	C₂×D₆₇	1	2	yes	—
78	C₂×C₄²	—	67	1	no	—
79	#107	—	2	2	no	—
80	#107	—	134	4	no	—
81	#107	—	134	2	no	—
82	C₄×C₈	—	67	4	no	—
83	C₂²×C₄	—	2	2	no	—
84	C₂²×C₄	—	67	2	no	—
85	C₂²×C₄	—	134	2	no	—
86	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
87	C₂×C₈	—	67	2	no	—
88	C₂×C₈	—	67	2	no	—
89	C₂×C₈	—	67	2	no	—
90	C₂×C₈	Dic₁₃₄	1	2	yes	—
91	C₂×C₈	C₄×D₆₇	1	2	yes	—
92	C₂×C₈	C₂²⋊D₆₇	1	2	yes	—
93	C₄⋊C₄	—	2	2	no	—
94	C₄⋊C₄	—	2	2	no	—
95	C₄⋊C₄	—	268	8	no	—
96	C₄⋊C₄	—	134	2	no	—
97	C₄⋊C₄	—	134	2	no	—
98	C₄²	—	67	2	no	—
99	C₄²	—	67	2	no	—
100	C₂³	D₄×D₆₇	1	1	yes	characteristic
101	C₂×C₄	—	4	2	no	—
102	C₂×C₄	—	4	4	no	—
103	C₂×C₄	—	268	2	no	—
104	C₂×C₄	—	268	4	no	—
105	C₂×C₄	—	67	4	no	—
106	C₂×C₄	—	67	4	no	—
107	C₂×C₄	—	67	4	no	—
108	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
109	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
110	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
111	C₂×C₄	Q₈×D₆₇	1	1	yes	characteristic
112	C₂×C₄	Q₈×D₆₇	1	1	yes	characteristic
113	C₂×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
114	C₈	—	67	4	no	—
115	C₈	Dic₆₇⋊C₄	1	4	yes	complemented
116	C₂²	C₄⋊C₄×D₆₇	1	1	yes	characteristic
117	C₂²	#14524039	1	1	yes	characteristic
118	C₂²	#14524042	1	1	yes	characteristic
119	C₂²	D₈×D₆₇	1	2	yes	—
120	C₂²	Q₁₆×D₆₇	1	2	yes	—
121	C₄	—	8	2	no	—
122	C₄	—	536	2	no	—
123	C₄	—	67	8	no	—
124	C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
125	C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
126	C₄	#14524041	1	1	yes	characteristic
127	C₄	#14524042	1	1	yes	characteristic
128	C₂	#16863265	1	1	yes	characteristic
129	C₂	#107×D₆₇	1	2	yes	—
130	C₂	#16863771	1	1	yes	characteristic
131	C₂	#16863774	1	1	yes	characteristic
132	C₂	#16863800	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14523977

C₂

yes

maximal, characteristic

C₂×#14523976

C₂

yes

maximal

C₂×C₈×Dic₆₇

C₂

yes

maximal, characteristic

#16863265

C₂

yes

maximal

C₂×#107×C₆₇

C₂

yes

maximal, characteristic

C₂²×C₆₇⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₆₇

C₂²

yes

—

C₂×Dic₆₇⋊C₄

—

C₂×C₄×Dic₆₇

C₂²

yes

characteristic

#14523977

—

#14523976

—

C₈×Dic₆₇

C₄

yes

complemented

C₂²×C₈×C₆₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₆₇

C₂²

yes

—

#107×C₆₇

—

C₂²×Dic₆₇

—

C₂²×Dic₆₇

D₄

yes

—

C₂×C₆₇⋊C₈

C₂×C₄

yes

—

C₂×C₆₇⋊C₈

D₄

yes

—

C₂×C₆₇⋊C₈

Q₈

yes

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

Dic₆₇⋊C₄

—

C₄×Dic₆₇

C₂×C₄

yes

—

C₄×Dic₆₇

Q₈

yes

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₂×C₈×C₆₇

C₂×C₄

yes

—

C₂×C₈×C₆₇

D₄

yes

—

C₂×C₈×C₆₇

Q₈

yes

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₂³×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₆₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₆₇

C₂×Q₈

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

C₄⋊C₄

yes

—

C₂×Dic₆₇

D₈

yes

—

C₂×Dic₆₇

Q₁₆

yes

—

C₈×C₆₇

C₄⋊C₄

yes

complemented

C₆₇⋊C₈

C₄⋊C₄

yes

complemented

C₂²×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₆₇

#127

yes

characteristic

C₂²×C₆₇

C₄⋊Q₈

yes

characteristic

C₂²×C₆₇

C₂×D₈

yes

—

C₂²×C₆₇

C₂×Q₁₆

yes

—

C₄×C₆₇

—

C₄×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₆₇

C₄⋊Q₈

yes

characteristic

C₄×C₆₇

C₄⋊Q₈

yes

characteristic

Dic₆₇

—

Dic₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

#1845

—

maximal, Sylow

C₆₇

#1845

yes

characteristic, complemented, Sylow

C₂×#107

—

C₂×#107

—

C₂×#107

D₆₇

yes

characteristic

C₂×C₄×C₈

—

#384

—

C₂²×C₈

—

C₂²×C₈

C₂×D₆₇

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

134

—

C₂×C₄⋊C₄

C₂×D₆₇

yes

—

C₂×C₄²

—

#107

—

#107

—

134

—

#107

—

134

—

C₄×C₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

134

—

C₂²×C₄

C₂²×D₆₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₁₃₄

yes

—

C₂×C₈

C₄×D₆₇

yes

—

C₂×C₈

C₂²⋊D₆₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

268

—

C₄⋊C₄

—

134

—

C₄⋊C₄

—

134

—

C₄²

—

C₄²

—

100

C₂³

D₄×D₆₇

yes

characteristic

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

268

—

104

C₂×C₄

—

268

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

C₂×C₄×D₆₇

yes

characteristic

109

C₂×C₄

D₄×D₆₇

yes

characteristic

110

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

111

C₂×C₄

Q₈×D₆₇

yes

characteristic

112

C₂×C₄

Q₈×D₆₇

yes

characteristic

113

C₂×C₄

C₂×Dic₁₃₄

yes

characteristic

114

C₈

—

115

C₈

Dic₆₇⋊C₄

yes

complemented

116

C₂²

C₄⋊C₄×D₆₇

yes

characteristic

117

C₂²

#14524039

yes

characteristic

118

C₂²

#14524042

yes

characteristic

119

C₂²

D₈×D₆₇

yes

—

120

C₂²

Q₁₆×D₆₇

yes

—

121

C₄

—

122

C₄

—

536

—

123

C₄

—

124

C₄

C₄⋊C₄×D₆₇

yes

characteristic

125

C₄

C₂×Dic₆₇⋊C₄

yes

characteristic

126

C₄

#14524041

yes

characteristic

127

C₄

#14524042

yes

characteristic

128

C₂

#16863265

yes

characteristic

129

C₂

#107×D₆₇

yes

—

130

C₂

#16863771

yes

characteristic

131

C₂

#16863774

yes

characteristic

132

C₂

#16863800

yes

—

	subgroup	G/H
center	C₂³	D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₆₇
Fitting	C₂×#107×C₆₇	C₂
socle	C₂³×C₆₇	C₂×D₄
hypercenter	C₂×#107	D₆₇
hypocenter	C₆₇	#1845
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

subgroup

G/H

center

C₂³

D₄×D₆₇

commutator

C₂×C₄×C₆₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₆₇

Fitting

C₂×#107×C₆₇

C₂

socle

C₂³×C₆₇

C₂×D₄

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

N	H ≅ G/N	type
12. C₈×Dic₆₇	C₄	semidirect
47. C₈×C₆₇	C₄⋊C₄	semidirect
48. C₆₇⋊C₈	C₄⋊C₄	semidirect
60. Dic₆₇	#107	semidirect
67. C₆₇	#1845	semidirect
115. C₈	Dic₆₇⋊C₄	semidirect

H ≅ G/N

type

12. C₈×Dic₆₇

C₄

semidirect

47. C₈×C₆₇

C₄⋊C₄

semidirect

48. C₆₇⋊C₈

C₄⋊C₄

semidirect

60. Dic₆₇

#107

semidirect

67. C₆₇

#1845

semidirect

115. C₈

Dic₆₇⋊C₄

semidirect

Subgroups — #22560367

Conjugacy classes of subgroups

Complements