Group #22562349 — subgroups

Subgroups — #22562349

Order 8576.

130 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 130 conjugacy classes of subgroups — too many to draw. Only its 51 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	#16863800	C₂	1	1	yes	maximal, characteristic
2	Q₁₆×Dic₆₇	C₂	1	1	yes	maximal, characteristic
3	#16863224	C₂	1	1	yes	maximal, characteristic
4	#16863223	C₂	1	1	yes	maximal, characteristic
5	Dic₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	#16863200	C₂	1	1	yes	maximal, characteristic
7	#358×C₆₇	C₂	1	1	yes	maximal, characteristic
8	#14524042	—	2	1	no	—
9	Q₈×Dic₆₇	—	2	1	no	—
10	C₂×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
11	#14523977	C₂²	1	1	yes	characteristic
12	#14523977	C₂²	1	1	yes	characteristic
13	#14523976	—	2	1	no	—
14	#14523967	—	2	1	no	—
15	C₈×Dic₆₇	C₂²	1	1	yes	characteristic
16	C₂×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
18	#107×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×Dic₁₃₄	—	4	1	no	—
20	C₂×C₆₇⋊C₈	D₄	1	1	yes	characteristic
21	C₄⋊Dic₆₇	—	2	1	no	—
22	C₄⋊Dic₆₇	—	2	1	no	—
23	C₄⋊Dic₆₇	—	4	1	no	—
24	Dic₆₇⋊C₄	—	4	1	no	—
25	C₄×Dic₆₇	—	4	1	no	—
26	C₄×Dic₆₇	D₄	1	1	yes	characteristic
27	C₆₇⋊C₁₆	—	2	1	no	—
28	C₂×Q₈×C₆₇	—	2	1	no	—
29	Q₁₆×C₆₇	—	2	1	no	—
30	Q₁₆×C₆₇	Q₈	1	2	yes	complemented
31	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
32	C₄⋊C₄×C₆₇	—	2	1	no	—
33	C₁₆×C₆₇	—	2	1	no	—
34	Q₈×C₆₇	—	2	2	no	—
35	Q₈×C₆₇	—	4	1	no	—
36	C₂×C₄×C₆₇	—	4	1	no	—
37	C₂×C₄×C₆₇	—	4	1	no	—
38	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
39	C₂×Dic₆₇	—	4	1	no	—
40	C₂×Dic₆₇	—	4	1	no	—
41	C₂×Dic₆₇	D₈	1	1	yes	characteristic
42	C₂×Dic₆₇	D₈	1	1	yes	characteristic
43	Dic₁₃₄	—	8	2	no	—
44	C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
45	C₈×C₆₇	#41	1	1	yes	characteristic
46	C₆₇⋊C₈	—	2	1	no	—
47	C₂²×C₆₇	C₂×D₈	1	1	yes	characteristic
48	C₄×C₆₇	—	4	2	no	—
49	C₄×C₆₇	—	8	1	no	—
50	C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
51	C₄×C₆₇	#137	1	1	yes	characteristic
52	Dic₆₇	—	2	1	no	—
53	Dic₆₇	—	8	1	no	—
54	Dic₆₇	—	8	1	no	—
55	Dic₆₇	SD₃₂	1	2	yes	—
56	C₂×C₆₇	#474	1	1	yes	characteristic
57	C₂×C₆₇	C₂×SD₃₂	1	1	yes	characteristic
58	C₂×C₆₇	#510	1	1	yes	characteristic
59	#2225	—	67	1	no	maximal, Sylow
60	C₆₇	#2225	1	1	yes	characteristic, complemented, Sylow
61	#500	—	67	1	no	—
62	C₄×Q₁₆	—	67	1	no	—
63	#367	—	67	1	no	—
64	#367	—	67	1	no	—
65	C₄⋊C₁₆	—	67	1	no	—
66	#358	—	67	1	no	—
67	#358	D₆₇	1	1	yes	characteristic
68	C₂×Q₁₆	C₂×D₆₇	1	1	yes	characteristic
69	C₄⋊Q₈	—	134	1	no	—
70	C₄×Q₈	—	134	1	no	—
71	C₂×C₁₆	—	67	1	no	—
72	C₂×C₁₆	C₂×D₆₇	1	1	yes	characteristic
73	#107	—	67	1	no	—
74	#107	—	67	1	no	—
75	#107	—	134	1	no	—
76	#107	C₂×D₆₇	1	1	yes	characteristic
77	#103	—	134	1	no	—
78	C₄×C₈	—	67	1	no	—
79	C₂×Q₈	—	2	1	no	—
80	C₂×Q₈	—	268	1	no	—
81	Q₁₆	—	2	1	no	—
82	Q₁₆	Dic₁₃₄	1	2	yes	complemented
83	C₂×C₈	—	67	1	no	—
84	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
85	C₄⋊C₄	—	2	1	no	—
86	C₄⋊C₄	—	268	1	no	—
87	C₄⋊C₄	—	268	1	no	—
88	C₄⋊C₄	—	134	1	no	—
89	C₄⋊C₄	—	134	1	no	—
90	C₄²	—	268	1	no	—
91	C₄²	—	67	1	no	—
92	C₁₆	—	2	1	no	—
93	C₁₆	—	134	1	no	—
94	Q₈	—	2	2	no	—
95	Q₈	—	4	1	no	—
96	Q₈	—	536	2	no	—
97	C₂×C₄	—	4	1	no	—
98	C₂×C₄	—	4	1	no	—
99	C₂×C₄	—	268	1	no	—
100	C₂×C₄	—	268	1	no	—
101	C₂×C₄	—	67	1	no	—
102	C₂×C₄	—	67	1	no	—
103	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
104	C₈	—	134	1	no	—
105	C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
106	C₈	#11761627	1	1	yes	characteristic
107	C₂²	D₈×D₆₇	1	1	yes	characteristic
108	C₄	—	4	2	no	—
109	C₄	—	8	1	no	—
110	C₄	—	536	1	no	—
111	C₄	—	536	1	no	—
112	C₄	—	67	2	no	—
113	C₄	—	134	1	no	—
114	C₄	#14524020	1	1	yes	characteristic
115	C₄	#14524072	1	1	yes	characteristic
116	C₂	#16863674	1	1	yes	characteristic
117	C₂	SD₃₂×D₆₇	1	1	yes	characteristic
118	C₂	#16863838	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863800

C₂

yes

maximal, characteristic

Q₁₆×Dic₆₇

C₂

yes

maximal, characteristic

#16863224

C₂

yes

maximal, characteristic

#16863223

C₂

yes

maximal, characteristic

Dic₆₇⋊C₁₆

C₂

yes

maximal, characteristic

#16863200

C₂

yes

maximal, characteristic

#358×C₆₇

C₂

yes

maximal, characteristic

#14524042

—

Q₈×Dic₆₇

—

C₂×C₆₇⋊C₁₆

C₂²

yes

characteristic

#14523977

C₂²

yes

characteristic

#14523977

C₂²

yes

characteristic

#14523976

—

#14523967

—

C₈×Dic₆₇

C₂²

yes

characteristic

C₂×Q₁₆×C₆₇

C₂²

yes

characteristic

C₂×C₁₆×C₆₇

C₂²

yes

characteristic

#107×C₆₇

C₂²

yes

characteristic

C₂×Dic₁₃₄

—

C₂×C₆₇⋊C₈

D₄

yes

characteristic

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

Dic₆₇⋊C₄

—

C₄×Dic₆₇

—

C₄×Dic₆₇

D₄

yes

characteristic

C₆₇⋊C₁₆

—

C₂×Q₈×C₆₇

—

Q₁₆×C₆₇

—

Q₁₆×C₆₇

Q₈

yes

complemented

C₂×C₈×C₆₇

C₂³

yes

characteristic

C₄⋊C₄×C₆₇

—

C₁₆×C₆₇

—

Q₈×C₆₇

—

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

D₈

yes

characteristic

C₂×Dic₆₇

D₈

yes

characteristic

Dic₁₃₄

—

C₈×C₆₇

C₂×Q₈

yes

characteristic

C₈×C₆₇

#41

yes

characteristic

C₆₇⋊C₈

—

C₂²×C₆₇

C₂×D₈

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂²⋊Q₈

yes

characteristic

C₄×C₆₇

#137

yes

characteristic

Dic₆₇

—

Dic₆₇

—

Dic₆₇

—

Dic₆₇

SD₃₂

yes

—

C₂×C₆₇

#474

yes

characteristic

C₂×C₆₇

C₂×SD₃₂

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

#500

—

C₄×Q₁₆

—

#367

—

#367

—

C₄⋊C₁₆

—

#358

—

#358

D₆₇

yes

characteristic

C₂×Q₁₆

C₂×D₆₇

yes

characteristic

C₄⋊Q₈

—

134

—

C₄×Q₈

—

134

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₆₇

yes

—

—

—

134

—

#107

C₂×D₆₇

yes

characteristic

#103

—

134

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

268

—

Q₁₆

—

Q₁₆

Dic₁₃₄

yes

complemented

C₂×C₈

—

C₂×C₈

C₂²×D₆₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

268

—

C₄⋊C₄

—

268

—

C₄⋊C₄

—

134

—

C₄⋊C₄

—

134

—

C₄²

—

268

—

C₄²

—

C₁₆

—

C₁₆

—

134

—

Q₈

—

Q₈

—

Q₈

—

536

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

268

—

100

C₂×C₄

—

268

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

D₄×D₆₇

yes

characteristic

104

C₈

—

134

—

105

C₈

C₂×Dic₁₃₄

yes

characteristic

106

C₈

#11761627

yes

characteristic

107

C₂²

D₈×D₆₇

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

536

—

111

C₄

—

536

—

112

C₄

—

113

C₄

—

134

—

114

C₄

#14524020

yes

characteristic

115

C₄

#14524072

yes

characteristic

116

C₂

#16863674

yes

characteristic

117

C₂

SD₃₂×D₆₇

yes

characteristic

118

C₂

#16863838

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₆₇
commutator	C₂×C₈×C₆₇	C₂³
Frattini	C₂×C₈	C₂²×D₆₇
Fitting	#358×C₆₇	C₂
socle	C₂²×C₆₇	C₂×D₈
hypercenter	#358	D₆₇
hypocenter	C₆₇	#2225
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

subgroup

G/H

center

C₂²

D₈×D₆₇

commutator

C₂×C₈×C₆₇

C₂³

Frattini

C₂×C₈

C₂²×D₆₇

Fitting

#358×C₆₇

C₂

socle

C₂²×C₆₇

C₂×D₈

D₆₇

C₆₇

G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

N	H ≅ G/N	type
30. Q₁₆×C₆₇	Q₈	semidirect
60. C₆₇	#2225	semidirect
82. Q₁₆	Dic₁₃₄	semidirect

H ≅ G/N

type

30. Q₁₆×C₆₇

Q₈

semidirect

60. C₆₇

#2225

semidirect

82. Q₁₆

Dic₁₃₄

semidirect

Subgroups — #22562349

Conjugacy classes of subgroups

Complements