Group #22561747 — subgroups

Subgroups — #22561747

Order 8576.

254 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 93 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×#14523976	C₂	1	2	yes	maximal
2	C₂×#14523967	C₂	1	2	yes	maximal
3	#16863266	C₂	1	1	yes	maximal, characteristic
4	#16863151	C₂	1	1	yes	maximal, characteristic
5	#391×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₆₇⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₆₇	—	2	2	no	—
8	C₂×C₄⋊Dic₆₇	C₂²	1	2	yes	—
9	#14523976	—	2	4	no	—
10	#14523967	—	2	4	no	—
11	C₂²×Q₈×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
14	#95×C₆₇	—	2	2	no	—
15	C₂²×Dic₆₇	—	2	2	no	—
16	C₂×C₆₇⋊C₈	—	2	2	no	—
17	C₂×C₆₇⋊C₈	—	2	2	no	—
18	C₂×C₆₇⋊C₈	D₄	1	4	yes	—
19	C₄⋊Dic₆₇	—	2	2	no	—
20	C₄⋊Dic₆₇	—	2	2	no	—
21	C₄⋊Dic₆₇	—	4	4	no	—
22	C₂×Q₈×C₆₇	—	2	1	no	—
23	C₂×Q₈×C₆₇	—	2	2	no	—
24	C₂×Q₈×C₆₇	—	2	2	no	—
25	C₂×Q₈×C₆₇	Q₈	1	2	yes	—
26	C₂²×C₄×C₆₇	—	2	1	no	—
27	C₂²×C₄×C₆₇	—	2	1	no	—
28	C₂²×C₄×C₆₇	D₄	1	2	yes	—
29	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
30	C₄⋊C₄×C₆₇	—	2	2	no	—
31	C₄²×C₆₇	—	2	2	no	—
32	C₂³×C₆₇	C₂×D₄	1	1	yes	characteristic
33	Q₈×C₆₇	—	2	4	no	—
34	Q₈×C₆₇	—	4	2	no	—
35	C₂×C₄×C₆₇	—	2	2	no	—
36	C₂×C₄×C₆₇	—	2	2	no	—
37	C₂×C₄×C₆₇	—	2	2	no	—
38	C₂×C₄×C₆₇	—	4	1	no	—
39	C₂×C₄×C₆₇	—	4	1	no	—
40	C₂×C₄×C₆₇	—	4	1	no	—
41	C₂×C₄×C₆₇	—	4	2	no	—
42	C₂×C₄×C₆₇	Q₁₆	1	4	yes	—
43	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₄×C₆₇	#41	1	2	yes	—
46	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
47	C₂×Dic₆₇	—	4	2	no	—
48	C₂×Dic₆₇	—	4	4	no	—
49	C₆₇⋊C₈	—	2	4	no	—
50	C₂²×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
51	C₂²×C₆₇	#123	1	1	yes	characteristic
52	C₂²×C₆₇	#127	1	1	yes	characteristic
53	C₂²×C₆₇	C₂×Q₁₆	1	2	yes	—
54	C₂²×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
55	C₄×C₆₇	—	2	4	no	—
56	C₄×C₆₇	—	4	2	no	—
57	C₄×C₆₇	—	8	1	no	—
58	C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
59	C₄×C₆₇	#123	1	2	yes	—
60	Dic₆₇	—	8	2	no	—
61	C₂×C₆₇	#399	1	1	yes	characteristic
62	C₂×C₆₇	#477	1	2	yes	—
63	C₂×C₆₇	#484	1	2	yes	—
64	C₂×C₆₇	#494	1	1	yes	characteristic
65	C₂×C₆₇	#496	1	1	yes	characteristic
66	#2095	—	67	1	no	maximal, Sylow
67	C₆₇	#2095	1	1	yes	characteristic, complemented, Sylow
68	C₂×#107	—	67	2	no	—
69	C₂×#103	—	67	2	no	—
70	#391	D₆₇	1	1	yes	characteristic
71	#384	—	67	1	no	—
72	#336	—	67	1	no	—
73	C₂²×Q₈	C₂×D₆₇	1	1	yes	characteristic
74	C₂²×C₈	—	67	2	no	—
75	C₂×C₄⋊C₄	—	67	2	no	—
76	C₂×C₄⋊C₄	—	134	2	no	—
77	C₂×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
78	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
79	#107	—	134	4	no	—
80	#103	—	134	4	no	—
81	#95	—	2	2	no	—
82	C₂×Q₈	—	2	1	no	—
83	C₂×Q₈	—	2	2	no	—
84	C₂×Q₈	—	2	2	no	—
85	C₂×Q₈	Dic₁₃₄	1	2	yes	—
86	C₂²×C₄	—	2	1	no	—
87	C₂²×C₄	—	2	1	no	—
88	C₂²×C₄	—	134	2	no	—
89	C₂²×C₄	C₂²⋊D₆₇	1	2	yes	—
90	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
91	C₂×C₈	—	67	4	no	—
92	C₂×C₈	—	134	2	no	—
93	C₂×C₈	—	134	2	no	—
94	C₄⋊C₄	—	2	2	no	—
95	C₄⋊C₄	—	268	4	no	—
96	C₄⋊C₄	—	134	2	no	—
97	C₄⋊C₄	—	134	2	no	—
98	C₄²	—	2	2	no	—
99	C₂³	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
100	Q₈	—	2	4	no	—
101	Q₈	—	4	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	4	1	no	—
106	C₂×C₄	—	4	1	no	—
107	C₂×C₄	—	4	1	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	—	268	2	no	—
110	C₂×C₄	—	268	4	no	—
111	C₂×C₄	#11761618	1	4	yes	—
112	C₂×C₄	D₄×D₆₇	1	2	yes	—
113	C₂×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
114	C₂×C₄	#11761627	1	2	yes	—
115	C₂×C₄	#11761628	1	1	yes	characteristic
116	C₈	—	134	4	no	—
117	C₂²	#14524022	1	1	yes	characteristic
118	C₂²	#14524024	1	1	yes	characteristic
119	C₂²	#14524039	1	1	yes	characteristic
120	C₂²	C₂×#11761618	1	2	yes	—
121	C₂²	#14524068	1	2	yes	—
122	C₄	—	2	4	no	—
123	C₄	—	4	2	no	—
124	C₄	—	8	1	no	—
125	C₄	—	536	2	no	—
126	C₄	#14524020	1	2	yes	—
127	C₄	#14524025	1	2	yes	—
128	C₂	#16863332	1	1	yes	characteristic
129	C₂	#16863691	1	2	yes	—
130	C₂	#16863729	1	2	yes	—
131	C₂	#16863775	1	1	yes	characteristic
132	C₂	#16863786	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14523976

C₂

yes

maximal

C₂×#14523967

C₂

yes

maximal

#16863266

C₂

yes

maximal, characteristic

#16863151

C₂

yes

maximal, characteristic

#391×C₆₇

C₂

yes

maximal, characteristic

C₂²×C₆₇⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₆₇

—

C₂×C₄⋊Dic₆₇

C₂²

yes

—

#14523976

—

#14523967

—

C₂²×Q₈×C₆₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₆₇

C₂²

yes

characteristic

C₂×C₄²×C₆₇

C₂²

yes

characteristic

#95×C₆₇

—

C₂²×Dic₆₇

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

D₄

yes

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

Q₈

yes

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

D₄

yes

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₂³×C₆₇

C₂×D₄

yes

characteristic

Q₈×C₆₇

—

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

Q₁₆

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₆₇

#41

yes

—

C₂×C₄×C₆₇

#41

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₆₇⋊C₈

—

C₂²×C₆₇

C₂²⋊Q₈

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

C₂×Q₁₆

yes

—

C₂²×C₆₇

AGL(1,ℤ/8)

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂²⋊Q₈

yes

—

C₄×C₆₇

#123

yes

—

Dic₆₇

—

C₂×C₆₇

#399

yes

characteristic

C₂×C₆₇

#477

yes

—

C₂×C₆₇

#484

yes

—

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

C₂×#107

—

C₂×#103

—

#391

D₆₇

yes

characteristic

#384

—

#336

—

C₂²×Q₈

C₂×D₆₇

yes

characteristic

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

134

—

C₂×C₄⋊C₄

C₂×D₆₇

yes

characteristic

C₂×C₄²

C₂×D₆₇

yes

characteristic

#107

—

134

—

#103

—

134

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

Dic₁₃₄

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

134

—

C₂²×C₄

C₂²⋊D₆₇

yes

—

C₂²×C₄

C₂²×D₆₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

134

—

C₂×C₈

—

134

—

C₄⋊C₄

—

C₄⋊C₄

—

268

—

C₄⋊C₄

—

134

—

C₄⋊C₄

—

134

—

C₄²

—

C₂³

C₂×C₂²⋊D₆₇

yes

characteristic

100

Q₈

—

101

Q₈

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

268

—

110

C₂×C₄

—

268

—

111

C₂×C₄

#11761618

yes

—

112

C₂×C₄

D₄×D₆₇

yes

—

113

C₂×C₄

C₂×Dic₁₃₄

yes

characteristic

114

C₂×C₄

#11761627

yes

—

115

C₂×C₄

#11761628

yes

characteristic

116

C₈

—

134

—

117

C₂²

#14524022

yes

characteristic

118

C₂²

#14524024

yes

characteristic

119

C₂²

#14524039

yes

characteristic

120

C₂²

C₂×#11761618

yes

—

121

C₂²

#14524068

yes

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

536

—

126

C₄

#14524020

yes

—

127

C₄

#14524025

yes

—

128

C₂

#16863332

yes

characteristic

129

C₂

#16863691

yes

—

130

C₂

#16863729

yes

—

131

C₂

#16863775

yes

characteristic

132

C₂

#16863786

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₆₇
commutator	C₂²×C₄×C₆₇	C₂³
Frattini	C₂²×C₄	C₂²×D₆₇
Fitting	#391×C₆₇	C₂
socle	C₂³×C₆₇	C₂×D₄
hypercenter	#391	D₆₇
hypocenter	C₆₇	#2095
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #391

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₆₇

commutator

C₂²×C₄×C₆₇

C₂³

Frattini

C₂²×C₄

C₂²×D₆₇

Fitting

#391×C₆₇

C₂

socle

C₂³×C₆₇

C₂×D₄

D₆₇

C₆₇

G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #391

N	H ≅ G/N	type
67. C₆₇	#2095	semidirect

H ≅ G/N

type

67. C₆₇

#2095

semidirect

Subgroups — #22561747

Conjugacy classes of subgroups

Complements