Group #22562003 — subgroups

Subgroups — #22562003

Order 8576.

140 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 140 conjugacy classes of subgroups — too many to draw. Only its 81 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×#14523983	C₂	1	1	yes	maximal, characteristic
2	C₂×#14523979	C₂	1	1	yes	maximal, characteristic
3	#16863225	C₂	1	4	yes	maximal, complemented
4	C₂×#108×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#11761612	—	2	1	no	—
6	#14523983	C₄	1	2	yes	—
7	#14523983	C₄	1	2	yes	—
8	C₂×C₆₇⋊C₁₆	C₂²	1	2	yes	—
9	#14523979	—	2	1	no	—
10	#14523979	C₂²	1	2	yes	—
11	C₂×M₁₆×C₆₇	—	2	1	no	—
12	C₂²×C₈×C₆₇	C₂²	1	1	yes	characteristic
13	#108×C₆₇	—	2	1	no	—
14	#108×C₆₇	C₂²	1	2	yes	—
15	#11761612	—	2	2	no	—
16	#11761612	—	4	1	no	—
17	C₂×C₆₇⋊C₈	—	4	1	no	—
18	C₆₇⋊C₁₆	C₂×C₄	1	4	yes	—
19	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
20	M₁₆×C₆₇	—	2	2	no	—
21	M₁₆×C₆₇	—	4	1	no	—
22	C₂×C₈×C₆₇	—	4	1	no	—
23	C₂×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
26	C₂×C₈×C₆₇	Q₈	1	1	yes	characteristic
27	C₂×C₈×C₆₇	Q₈	1	1	yes	characteristic
28	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
29	C₂³×C₆₇	Q₁₆	1	1	yes	characteristic
30	C₂×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₄×C₆₇	D₈	1	1	yes	characteristic
33	C₂×C₄×C₆₇	D₈	1	1	yes	characteristic
34	C₂×C₄×C₆₇	Q₁₆	1	1	yes	characteristic
35	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
36	C₈×C₆₇	—	4	2	no	—
37	C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
38	C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
39	C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
40	C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
41	C₆₇⋊C₈	—	4	2	no	—
42	C₂²×C₆₇	—	4	1	no	—
43	C₂²×C₆₇	#107	1	1	yes	characteristic
44	C₂²×C₆₇	#107	1	1	yes	characteristic
45	C₂²×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
46	C₄×C₆₇	#107	1	1	yes	characteristic
47	C₄×C₆₇	#107	1	1	yes	characteristic
48	C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
50	C₂×C₆₇	—	2	1	no	—
51	C₂×C₆₇	—	2	1	no	—
52	C₂×C₆₇	—	2	1	no	—
53	C₂×C₆₇	C₂×#107	1	1	yes	characteristic
54	#2161	—	67	1	no	maximal, Sylow
55	C₆₇	#2161	1	1	yes	characteristic, complemented, Sylow
56	C₂×M₃₂	—	67	1	no	—
57	C₂×#108	—	67	1	no	—
58	C₂×#108	D₆₇	1	1	yes	characteristic
59	#368	—	67	4	no	—
60	C₂×M₁₆	—	2	1	no	—
61	C₂×M₁₆	—	134	1	no	—
62	C₂²×C₈	C₂×D₆₇	1	1	yes	characteristic
63	M₃₂	—	67	2	no	—
64	M₃₂	—	67	2	no	—
65	C₂×C₁₆	—	67	2	no	—
66	#108	—	2	1	no	—
67	#108	—	67	2	no	—
68	#108	—	134	1	no	—
69	#108	C₂×D₆₇	1	2	yes	—
70	C₂²×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
71	M₁₆	—	2	2	no	—
72	M₁₆	—	4	1	no	—
73	M₁₆	—	268	1	no	—
74	M₁₆	—	134	2	no	—
75	C₂×C₈	—	4	1	no	—
76	C₂×C₈	—	268	1	no	—
77	C₂×C₈	Dic₁₃₄	1	1	yes	characteristic
78	C₂×C₈	Dic₁₃₄	1	1	yes	characteristic
79	C₂×C₈	C₄×D₆₇	1	1	yes	characteristic
80	C₂×C₈	C₄×D₆₇	1	1	yes	characteristic
81	C₂×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
82	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
83	C₁₆	—	67	4	no	—
84	C₂³	#11761618	1	1	yes	characteristic
85	C₂×C₄	Dic₆₇⋊C₄	1	1	yes	characteristic
86	C₂×C₄	Dic₆₇⋊C₄	1	1	yes	characteristic
87	C₂×C₄	#11761613	1	1	yes	characteristic
88	C₂×C₄	#11761613	1	1	yes	characteristic
89	C₂×C₄	#11761618	1	1	yes	characteristic
90	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
91	C₈	—	4	2	no	—
92	C₈	—	268	2	no	—
93	C₈	Dic₆₇⋊C₄	1	1	yes	characteristic
94	C₈	Dic₆₇⋊C₄	1	1	yes	characteristic
95	C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
96	C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
97	C₂²	—	4	1	no	—
98	C₂²	#14523976	1	1	yes	characteristic
99	C₂²	#14523976	1	1	yes	characteristic
100	C₂²	C₂×#11761618	1	1	yes	characteristic
101	C₄	#14523976	1	1	yes	characteristic
102	C₄	#14523976	1	1	yes	characteristic
103	C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
104	C₄	C₂×#11761613	1	1	yes	characteristic
105	C₂	—	2	1	no	—
106	C₂	—	2	1	no	—
107	C₂	—	2	1	no	—
108	C₂	C₂×#14523976	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14523983

C₂

yes

maximal, characteristic

C₂×#14523979

C₂

yes

maximal, characteristic

#16863225

C₂

yes

maximal, complemented

C₂×#108×C₆₇

C₂

yes

maximal, characteristic

C₂×#11761612

—

#14523983

C₄

yes

—

#14523983

C₄

yes

—

C₂×C₆₇⋊C₁₆

C₂²

yes

—

#14523979

—

#14523979

C₂²

yes

—

C₂×M₁₆×C₆₇

—

C₂²×C₈×C₆₇

C₂²

yes

characteristic

#108×C₆₇

—

#108×C₆₇

C₂²

yes

—

#11761612

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₆₇⋊C₁₆

C₂×C₄

yes

—

C₂²×C₄×C₆₇

D₄

yes

characteristic

M₁₆×C₆₇

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂×C₈×C₆₇

D₄

yes

characteristic

C₂×C₈×C₆₇

Q₈

yes

characteristic

C₂×C₈×C₆₇

Q₈

yes

characteristic

C₂×C₈×C₆₇

C₂³

yes

characteristic

C₂³×C₆₇

Q₁₆

yes

characteristic

C₂×C₄×C₆₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

D₈

yes

characteristic

C₂×C₄×C₆₇

D₈

yes

characteristic

C₂×C₄×C₆₇

Q₁₆

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₈×C₆₇

—

C₈×C₆₇

C₄⋊C₄

yes

characteristic

C₈×C₆₇

C₄⋊C₄

yes

characteristic

C₈×C₆₇

C₂²×C₄

yes

characteristic

C₈×C₆₇

C₂×Q₈

yes

characteristic

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

C₂×Q₁₆

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₆₇

C₂×D₈

yes

characteristic

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂×#108

—

C₂×#108

D₆₇

yes

characteristic

#368

—

C₂×M₁₆

—

C₂×M₁₆

—

134

—

C₂²×C₈

C₂×D₆₇

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

#108

—

#108

—

#108

—

134

—

#108

C₂×D₆₇

yes

—

C₂²×C₄

C₂²⋊D₆₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

268

—

M₁₆

—

134

—

C₂×C₈

—

C₂×C₈

—

268

—

C₂×C₈

Dic₁₃₄

yes

characteristic

C₂×C₈

Dic₁₃₄

yes

characteristic

C₂×C₈

C₄×D₆₇

yes

characteristic

C₂×C₈

C₄×D₆₇

yes

characteristic

C₂×C₈

C₂²⋊D₆₇

yes

characteristic

C₂×C₈

C₂²×D₆₇

yes

characteristic

C₁₆

—

C₂³

#11761618

yes

characteristic

C₂×C₄

Dic₆₇⋊C₄

yes

characteristic

C₂×C₄

Dic₆₇⋊C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

C₈

—

C₈

—

268

—

C₈

Dic₆₇⋊C₄

yes

characteristic

C₈

Dic₆₇⋊C₄

yes

characteristic

C₈

C₂×C₄×D₆₇

yes

characteristic

C₈

C₂×Dic₁₃₄

yes

characteristic

C₂²

—

C₂²

yes

C₂²

yes

100

C₂²

C₂×#11761618

yes

characteristic

101

C₄

#14523976

yes

characteristic

102

C₄

#14523976

yes

characteristic

103

C₄

C₂×Dic₆₇⋊C₄

yes

characteristic

104

C₄

C₂×#11761613

yes

characteristic

105

C₂

—

106

C₂

—

107

C₂

—

108

C₂

C₂×#14523976

yes

characteristic

	subgroup	G/H
center	C₄	C₂×#11761613
commutator	C₈×C₆₇	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₆₇
Fitting	C₂×#108×C₆₇	C₂
socle	C₂×C₆₇	C₂×#107
hypercenter	C₂×#108	D₆₇
hypocenter	C₆₇	#2161
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#108

subgroup

G/H

center

C₄

C₂×#11761613

commutator

C₈×C₆₇

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₆₇

Fitting

C₂×#108×C₆₇

C₂

C₂×C₆₇

D₆₇

C₆₇

G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#108

N	H ≅ G/N	type
3. #16863225	C₂	semidirect
55. C₆₇	#2161	semidirect

H ≅ G/N

type

3. #16863225

C₂

semidirect

55. C₆₇

#2161

semidirect

Subgroups — #22562003

Conjugacy classes of subgroups

Complements