Group #22561982 — subgroups

Subgroups — #22561982

Order 8576.

144 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 89 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×#14523979	C₂	1	1	yes	maximal, characteristic
2	#16863434	C₂	1	1	yes	maximal, characteristic
3	#16863219	C₂	1	4	yes	maximal, complemented
4	C₂×M₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#11761612	—	2	1	no	—
6	#14524002	—	2	1	no	—
7	#14523979	—	2	1	no	—
8	#14523979	C₂²	1	2	yes	—
9	#14523977	—	2	1	no	—
10	#14523975	C₂²	1	2	yes	—
11	C₂²×C₈×C₆₇	C₂²	1	1	yes	characteristic
12	M₃₂×C₆₇	C₄	1	2	yes	complemented
13	M₃₂×C₆₇	C₂²	1	2	yes	—
14	C₂×C₁₆×C₆₇	C₄	1	2	yes	complemented
15	#11761612	—	2	2	no	—
16	#11761612	—	4	1	no	—
17	C₂×C₆₇⋊C₈	—	4	1	no	—
18	C₄⋊Dic₆₇	—	2	2	no	—
19	C₂²⋊Dic₆₇	—	4	1	no	—
20	C₄×Dic₆₇	—	4	1	no	—
21	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
22	C₂×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
25	C₂×C₈×C₆₇	Q₈	1	1	yes	characteristic
26	C₂×C₈×C₆₇	Q₈	1	1	yes	characteristic
27	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
28	C₁₆×C₆₇	C₂×C₄	1	4	yes	—
29	C₂³×C₆₇	SD₁₆	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	SD₁₆	1	1	yes	characteristic
33	C₂×C₄×C₆₇	SD₁₆	1	1	yes	characteristic
34	C₂×C₄×C₆₇	SD₁₆	1	1	yes	characteristic
35	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
36	C₂×Dic₆₇	—	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	#106	1	1	yes	characteristic
44	C₂²×C₆₇	#106	1	1	yes	characteristic
45	C₂²×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
46	C₄×C₆₇	#106	1	1	yes	characteristic
47	C₄×C₆₇	#106	1	1	yes	characteristic
48	C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
50	Dic₆₇	—	8	2	no	—
51	C₂×C₆₇	—	2	1	no	—
52	C₂×C₆₇	—	2	1	no	—
53	C₂×C₆₇	—	2	1	no	—
54	C₂×C₆₇	C₂×#106	1	1	yes	characteristic
55	#2155	—	67	1	no	maximal, Sylow
56	C₆₇	#2155	1	1	yes	characteristic, complemented, Sylow
57	C₂×M₃₂	D₆₇	1	1	yes	characteristic
58	C₂×#108	—	67	1	no	—
59	#427	—	67	1	no	—
60	#365	—	67	4	no	—
61	C₂×M₁₆	—	134	1	no	—
62	C₂²×C₈	C₂×D₆₇	1	1	yes	characteristic
63	#117	—	134	1	no	—
64	M₃₂	Dic₆₇	1	2	yes	complemented
65	M₃₂	C₂×D₆₇	1	2	yes	—
66	C₂×C₁₆	Dic₆₇	1	2	yes	complemented
67	#108	—	67	2	no	—
68	#108	—	134	1	no	—
69	#107	—	134	1	no	—
70	#106	—	67	2	no	—
71	C₂²×C₄	D₂₆₈	1	1	yes	characteristic
72	M₁₆	—	268	1	no	—
73	M₁₆	—	134	2	no	—
74	C₂×C₈	—	268	1	no	—
75	C₂×C₈	Dic₁₃₄	1	1	yes	characteristic
76	C₂×C₈	Dic₁₃₄	1	1	yes	characteristic
77	C₂×C₈	D₂₆₈	1	1	yes	characteristic
78	C₂×C₈	C₂×Dic₆₇	1	1	yes	characteristic
79	C₂×C₈	C₂×Dic₆₇	1	1	yes	characteristic
80	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
81	C₄⋊C₄	—	134	2	no	—
82	C₂²⋊C₄	—	268	1	no	—
83	C₄²	—	268	1	no	—
84	C₁₆	C₂×Dic₆₇	1	4	yes	—
85	C₂³	#11761617	1	1	yes	characteristic
86	C₂×C₄	—	268	2	no	—
87	C₂×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
88	C₂×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
89	C₂×C₄	#11761617	1	1	yes	characteristic
90	C₂×C₄	#11761617	1	1	yes	characteristic
91	C₂×C₄	#11761617	1	1	yes	characteristic
92	C₂×C₄	C₂×D₂₆₈	1	1	yes	characteristic
93	C₈	—	268	2	no	—
94	C₈	C₄⋊Dic₆₇	1	1	yes	characteristic
95	C₈	C₄⋊Dic₆₇	1	1	yes	characteristic
96	C₈	C₂²×Dic₆₇	1	1	yes	characteristic
97	C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
98	C₂²	—	4	1	no	—
99	C₂²	#14523975	1	1	yes	characteristic
100	C₂²	#14523975	1	1	yes	characteristic
101	C₂²	C₂×#11761617	1	1	yes	characteristic
102	C₄	—	536	2	no	—
103	C₄	#14523975	1	1	yes	characteristic
104	C₄	#14523975	1	1	yes	characteristic
105	C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
106	C₄	C₂×#11761617	1	1	yes	characteristic
107	C₂	—	2	1	no	—
108	C₂	—	2	1	no	—
109	C₂	—	2	1	no	—
110	C₂	C₂×#14523975	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14523979

C₂

yes

maximal, characteristic

#16863434

C₂

yes

maximal, characteristic

#16863219

C₂

yes

maximal, complemented

C₂×M₃₂×C₆₇

C₂

yes

maximal, characteristic

—

—

—

C₂²

yes

—

#14523977

—

#14523975

C₂²

yes

—

C₂²×C₈×C₆₇

C₂²

yes

characteristic

M₃₂×C₆₇

C₄

yes

complemented

M₃₂×C₆₇

C₂²

yes

—

C₂×C₁₆×C₆₇

C₄

yes

complemented

#11761612

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

—

C₂²×C₄×C₆₇

D₄

yes

characteristic

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

C₁₆×C₆₇

C₂×C₄

yes

—

C₂³×C₆₇

SD₁₆

yes

characteristic

C₂×C₄×C₆₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

SD₁₆

yes

characteristic

C₂×C₄×C₆₇

SD₁₆

yes

characteristic

C₂×C₄×C₆₇

SD₁₆

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×Dic₆₇

—

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×SD₁₆

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₆₇

C₂×SD₁₆

yes

characteristic

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

C₂×M₃₂

D₆₇

yes

—

—

—

C₂×M₁₆

—

134

—

C₂²×C₈

C₂×D₆₇

yes

characteristic

#117

—

134

—

M₃₂

Dic₆₇

yes

complemented

M₃₂

C₂×D₆₇

yes

—

C₂×C₁₆

Dic₆₇

yes

complemented

#108

—

#108

—

134

—

#107

—

134

—

#106

—

C₂²×C₄

D₂₆₈

yes

characteristic

M₁₆

—

268

—

M₁₆

—

134

—

C₂×C₈

—

268

—

C₂×C₈

Dic₁₃₄

yes

characteristic

C₂×C₈

Dic₁₃₄

yes

characteristic

C₂×C₈

D₂₆₈

yes

characteristic

C₂×C₈

C₂×Dic₆₇

yes

characteristic

C₂×C₈

C₂×Dic₆₇

yes

characteristic

C₂×C₈

C₂²×D₆₇

yes

characteristic

C₄⋊C₄

—

134

—

C₂²⋊C₄

—

268

—

C₄²

—

268

—

C₁₆

C₂×Dic₆₇

yes

—

C₂³

#11761617

yes

characteristic

C₂×C₄

—

268

—

C₂×C₄

C₄⋊Dic₆₇

yes

characteristic

C₂×C₄

C₄⋊Dic₆₇

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂×D₂₆₈

yes

characteristic

C₈

—

268

—

C₈

C₄⋊Dic₆₇

yes

characteristic

C₈

C₄⋊Dic₆₇

yes

characteristic

C₈

C₂²×Dic₆₇

yes

characteristic

C₈

C₂×Dic₁₃₄

yes

characteristic

C₂²

—

C₂²

#14523975

yes

characteristic

100

C₂²

#14523975

yes

characteristic

101

C₂²

C₂×#11761617

yes

characteristic

102

C₄

—

536

—

103

C₄

#14523975

yes

characteristic

104

C₄

#14523975

yes

characteristic

105

C₄

C₂×C₄⋊Dic₆₇

yes

characteristic

106

C₄

C₂×#11761617

yes

characteristic

107

C₂

—

108

C₂

—

109

C₂

—

110

C₂

C₂×#14523975

yes

characteristic

	subgroup	G/H
center	C₄	#14523975
commutator	C₈×C₆₇	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₆₇
Fitting	C₂×M₃₂×C₆₇	C₂
socle	C₂×C₆₇	C₂×#106
hypercenter	C₂×M₃₂	D₆₇
hypocenter	C₆₇	#2155
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

subgroup

G/H

center

C₄

#14523975

commutator

C₈×C₆₇

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₆₇

Fitting

C₂×M₃₂×C₆₇

C₂

socle

C₂×C₆₇

C₂×#106

hypercenter

C₂×M₃₂

D₆₇

hypocenter

C₆₇

#2155

LCS

G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

N	H ≅ G/N	type
3. #16863219	C₂	semidirect
12. M₃₂×C₆₇	C₄	semidirect
14. C₂×C₁₆×C₆₇	C₄	semidirect
56. C₆₇	#2155	semidirect
64. M₃₂	Dic₆₇	semidirect
66. C₂×C₁₆	Dic₆₇	semidirect

H ≅ G/N

type

3. #16863219

C₂

semidirect

12. M₃₂×C₆₇

C₄

semidirect

14. C₂×C₁₆×C₆₇

C₄

semidirect

56. C₆₇

#2155

semidirect

64. M₃₂

Dic₆₇

semidirect

66. C₂×C₁₆

Dic₆₇

semidirect

Subgroups — #22561982

Conjugacy classes of subgroups

Complements