Group #22562290 — subgroups

Subgroups — #22562290

Order 8576.

150 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 49 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	#16863836	C₂	1	2	yes	maximal
2	#16863776	C₂	1	1	yes	maximal, characteristic, complemented
3	#16863217	C₂	1	1	yes	maximal, characteristic, complemented
4	#16863211	C₂	1	2	yes	maximal, complemented
5	#443×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₂₆₈	—	2	1	no	—
7	C₂×Dic₂₆₈	C₂²	1	2	yes	—
8	#14524043	—	2	1	no	—
9	#14523989	—	2	2	no	—
10	#14523987	—	2	2	no	—
11	#14523983	C₂²	1	2	yes	—
12	#135×C₆₇	C₂²	1	1	yes	characteristic
13	#131×C₆₇	—	2	1	no	—
14	#108×C₆₇	C₂²	1	1	yes	characteristic
15	#104×C₆₇	—	2	1	no	—
16	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₃₄	—	2	2	no	—
18	Dic₂₆₈	—	2	2	no	—
19	Dic₂₆₈	—	2	2	no	—
20	Dic₂₆₈	—	4	1	no	—
21	Dic₂₆₈	—	4	1	no	—
22	C₄⋊Dic₆₇	—	4	2	no	—
23	C₆₇⋊C₁₆	—	2	2	no	—
24	#41×C₆₇	—	2	1	no	—
25	Q₁₆×C₆₇	D₄	1	1	yes	characteristic
26	SD₁₆×C₆₇	—	2	1	no	—
27	D₈×C₆₇	D₄	1	1	yes	characteristic
28	M₁₆×C₆₇	—	2	1	no	—
29	M₁₆×C₆₇	—	4	1	no	—
30	C₂×C₈×C₆₇	—	4	1	no	—
31	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
32	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
33	C₄²×C₆₇	D₄	1	1	yes	characteristic
34	Q₈×C₆₇	—	2	1	no	—
35	D₄×C₆₇	—	2	1	no	—
36	D₄×C₆₇	—	4	1	no	—
37	C₂×C₄×C₆₇	—	2	1	no	—
38	C₂×C₄×C₆₇	—	4	1	no	—
39	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
40	C₂×Dic₆₇	—	4	2	no	—
41	Dic₁₃₄	—	4	2	no	—
42	Dic₁₃₄	—	4	2	no	—
43	C₈×C₆₇	—	4	1	no	—
44	C₈×C₆₇	D₈	1	1	yes	characteristic
45	C₈×C₆₇	D₈	1	1	yes	characteristic
46	C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
47	C₈×C₆₇	#41	1	1	yes	characteristic
48	C₂²×C₆₇	—	4	1	no	—
49	C₂²×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
50	C₄×C₆₇	—	2	2	no	—
51	C₄×C₆₇	—	4	1	no	—
52	C₄×C₆₇	#121	1	1	yes	characteristic
53	C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
54	Dic₆₇	—	8	2	no	—
55	C₂×C₆₇	—	2	1	no	—
56	C₂×C₆₇	—	8	1	no	—
57	C₂×C₆₇	#459	1	1	yes	characteristic
58	#2214	—	67	1	no	maximal, Sylow
59	C₆₇	#2214	1	1	yes	characteristic, complemented, Sylow
60	#510	—	67	2	no	—
61	#494	—	67	1	no	—
62	#443	D₆₇	1	1	yes	characteristic
63	#364	—	67	1	no	—
64	#362	—	67	2	no	—
65	#135	C₂×D₆₇	1	1	yes	characteristic
66	C₂×Q₁₆	—	67	2	no	—
67	C₂×Q₁₆	—	134	1	no	—
68	#131	—	2	1	no	—
69	C₄⋊Q₈	—	134	1	no	—
70	Q₃₂	—	134	2	no	—
71	SD₃₂	—	134	2	no	—
72	M₃₂	—	67	2	no	—
73	#108	C₂×D₆₇	1	1	yes	characteristic
74	#104	—	2	1	no	—
75	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
76	#41	—	2	1	no	—
77	C₂×Q₈	—	134	2	no	—
78	Q₁₆	—	268	1	no	—
79	Q₁₆	—	268	1	no	—
80	Q₁₆	—	134	2	no	—
81	Q₁₆	—	134	2	no	—
82	Q₁₆	D₂₆₈	1	1	yes	characteristic
83	SD₁₆	—	2	1	no	—
84	D₈	D₂₆₈	1	1	yes	characteristic
85	M₁₆	—	2	1	no	—
86	M₁₆	—	4	1	no	—
87	C₂×C₈	—	4	1	no	—
88	C₂×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
89	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
90	C₄⋊C₄	—	268	2	no	—
91	C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
92	C₁₆	—	134	2	no	—
93	Q₈	—	2	1	no	—
94	Q₈	—	268	2	no	—
95	Q₈	—	268	2	no	—
96	D₄	—	2	1	no	—
97	D₄	—	4	1	no	—
98	C₂×C₄	—	2	1	no	—
99	C₂×C₄	—	4	1	no	—
100	C₂×C₄	—	268	2	no	—
101	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
102	C₈	—	4	1	no	—
103	C₈	#11761613	1	1	yes	characteristic
104	C₈	#11761613	1	1	yes	characteristic
105	C₈	C₂×D₂₆₈	1	1	yes	characteristic
106	C₈	#11761628	1	1	yes	characteristic
107	C₂²	—	4	1	no	—
108	C₂²	#14524068	1	1	yes	characteristic
109	C₄	—	2	2	no	—
110	C₄	—	4	1	no	—
111	C₄	—	536	2	no	—
112	C₄	#14524017	1	1	yes	characteristic
113	C₄	C₂×#11761613	1	1	yes	characteristic
114	C₂	—	2	1	no	—
115	C₂	—	8	1	no	—
116	C₂	#16863591	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863836

C₂

yes

maximal

#16863776

C₂

yes

maximal, characteristic, complemented

#16863217

C₂

yes

maximal, characteristic, complemented

#16863211

C₂

yes

maximal, complemented

#443×C₆₇

C₂

yes

maximal, characteristic

C₂×Dic₂₆₈

—

C₂×Dic₂₆₈

C₂²

yes

—

—

—

—

C₂²

yes

—

#135×C₆₇

C₂²

yes

characteristic

#131×C₆₇

—

#108×C₆₇

C₂²

yes

characteristic

#104×C₆₇

—

C₄×C₈×C₆₇

C₂²

yes

characteristic

C₂×Dic₁₃₄

—

Dic₂₆₈

—

Dic₂₆₈

—

Dic₂₆₈

—

Dic₂₆₈

—

C₄⋊Dic₆₇

—

C₆₇⋊C₁₆

—

#41×C₆₇

—

Q₁₆×C₆₇

D₄

yes

characteristic

SD₁₆×C₆₇

—

D₈×C₆₇

D₄

yes

characteristic

M₁₆×C₆₇

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

D₄

yes

characteristic

C₂×C₈×C₆₇

C₂³

yes

characteristic

C₄²×C₆₇

D₄

yes

characteristic

Q₈×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

D₈

yes

characteristic

C₈×C₆₇

D₈

yes

characteristic

C₈×C₆₇

C₂×D₄

yes

characteristic

C₈×C₆₇

#41

yes

characteristic

C₂²×C₆₇

—

C₂²×C₆₇

AGL(1,ℤ/8)

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#121

yes

characteristic

C₄×C₆₇

C₂×D₈

yes

characteristic

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

#510

—

#494

—

#443

D₆₇

yes

—

—

C₂×D₆₇

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

134

—

#131

—

C₄⋊Q₈

—

134

—

Q₃₂

—

134

—

SD₃₂

—

134

—

M₃₂

—

#108

C₂×D₆₇

yes

characteristic

#104

—

C₄×C₈

C₂×D₆₇

yes

characteristic

#41

—

C₂×Q₈

—

134

—

Q₁₆

—

268

—

Q₁₆

—

268

—

Q₁₆

—

134

—

Q₁₆

—

134

—

Q₁₆

D₂₆₈

yes

characteristic

SD₁₆

—

D₈

D₂₆₈

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₆₇

yes

characteristic

C₂×C₈

C₂²×D₆₇

yes

characteristic

C₄⋊C₄

—

268

—

C₄²

C₂²⋊D₆₇

yes

characteristic

C₁₆

—

134

—

Q₈

—

Q₈

—

268

—

Q₈

—

268

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

268

—

101

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

102

C₈

—

103

C₈

#11761613

yes

characteristic

104

C₈

#11761613

yes

characteristic

105

C₈

C₂×D₂₆₈

yes

characteristic

106

C₈

#11761628

yes

characteristic

107

C₂²

—

108

C₂²

#14524068

yes

characteristic

109

C₄

—

110

C₄

—

111

C₄

—

536

—

112

C₄

#14524017

yes

characteristic

113

C₄

C₂×#11761613

yes

characteristic

114

C₂

—

115

C₂

—

116

C₂

#16863591

yes

characteristic

	subgroup	G/H
center	C₂	#16863591
commutator	C₂×C₈×C₆₇	C₂³
Frattini	C₂×C₈	C₂²×D₆₇
Fitting	#443×C₆₇	C₂
socle	C₂×C₆₇	#459
hypercenter	#443	D₆₇
hypocenter	C₆₇	#2214
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #443

subgroup

G/H

center

C₂

#16863591

commutator

C₂×C₈×C₆₇

C₂³

Frattini

C₂×C₈

C₂²×D₆₇

Fitting

#443×C₆₇

C₂

C₂×C₆₇

D₆₇

C₆₇

G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #443

N	H ≅ G/N	type
2. #16863776	C₂	semidirect
3. #16863217	C₂	semidirect
4. #16863211	C₂	semidirect
59. C₆₇	#2214	semidirect

H ≅ G/N

type

2. #16863776

C₂

semidirect

3. #16863217

C₂

semidirect

4. #16863211

C₂

semidirect

59. C₆₇

#2214

semidirect

Subgroups — #22562290

Conjugacy classes of subgroups

Complements