Group #22561997 — subgroups

Subgroups — #22561997

Order 8576.

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

normal subgroups only

This group has 156 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	#16863434	C₂	1	1	yes	maximal, characteristic
2	C₂×#14523977	C₂	1	1	yes	maximal, characteristic
3	#16863224	C₂	1	2	yes	maximal, complemented
4	#16863222	C₂	1	2	yes	maximal, complemented
5	C₂×M₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic
6	#14524002	—	2	1	no	—
7	C₂×C₄⋊Dic₆₇	—	2	1	no	—
8	#14523977	—	2	1	no	—
9	#14523977	C₂²	1	2	yes	—
10	#14523977	C₂²	1	1	yes	characteristic
11	#14523977	C₂²	1	1	yes	characteristic
12	#14523975	—	2	1	no	—
13	C₂²×C₈×C₆₇	C₂²	1	1	yes	characteristic
14	M₃₂×C₆₇	C₄	1	4	yes	complemented
15	C₂×C₁₆×C₆₇	C₂²	1	2	yes	—
16	C₂²×Dic₆₇	—	4	1	no	—
17	C₄⋊Dic₆₇	—	2	1	no	—
18	C₄⋊Dic₆₇	—	2	1	no	—
19	C₄⋊Dic₆₇	—	2	2	no	—
20	C₄⋊Dic₆₇	—	4	1	no	—
21	C₂²⋊Dic₆₇	—	4	1	no	—
22	C₄×Dic₆₇	—	4	1	no	—
23	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
24	C₂×C₈×C₆₇	C₂×C₄	1	2	yes	—
25	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
26	C₂×C₈×C₆₇	Q₈	1	2	yes	—
27	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
28	C₁₆×C₆₇	C₂×C₄	1	4	yes	complemented
29	C₂³×C₆₇	D₈	1	1	yes	characteristic
30	C₂×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
31	C₂×C₄×C₆₇	D₈	1	1	yes	characteristic
32	C₂×C₄×C₆₇	Q₁₆	1	2	yes	—
33	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
34	C₂×Dic₆₇	—	4	1	no	—
35	C₂×Dic₆₇	—	4	1	no	—
36	C₂×Dic₆₇	—	4	2	no	—
37	C₂×Dic₆₇	—	8	2	no	—
38	C₈×C₆₇	C₄⋊C₄	1	2	yes	—
39	C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
40	C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
41	C₂²×C₆₇	—	2	1	no	—
42	C₂²×C₆₇	—	2	1	no	—
43	C₂²×C₆₇	#107	1	2	yes	—
44	C₂²×C₆₇	C₂×D₈	1	1	yes	characteristic
45	C₄×C₆₇	#107	1	2	yes	—
46	C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
48	Dic₆₇	—	8	1	no	—
49	Dic₆₇	—	8	2	no	—
50	Dic₆₇	—	8	1	no	—
51	C₂×C₆₇	—	2	2	no	—
52	C₂×C₆₇	C₂×#107	1	1	yes	characteristic
53	C₂×C₆₇	#509	1	1	yes	characteristic
54	C₂×C₆₇	#510	1	1	yes	characteristic
55	#2159	—	67	1	no	maximal, Sylow
56	C₆₇	#2159	1	1	yes	characteristic, complemented, Sylow
57	C₂×M₃₂	D₆₇	1	1	yes	characteristic
58	#427	—	67	1	no	—
59	C₂×#107	—	67	1	no	—
60	#367	—	67	2	no	—
61	#366	—	67	2	no	—
62	C₂²×C₈	C₂×D₆₇	1	1	yes	characteristic
63	#117	—	134	1	no	—
64	C₂×C₄⋊C₄	—	134	1	no	—
65	M₃₂	Dic₆₇	1	4	yes	complemented
66	C₂×C₁₆	C₂×D₆₇	1	2	yes	—
67	#107	—	67	1	no	—
68	#107	—	67	2	no	—
69	#107	—	67	1	no	—
70	#107	—	134	1	no	—
71	#106	—	134	1	no	—
72	C₂²×C₄	—	268	1	no	—
73	C₂²×C₄	D₂₆₈	1	1	yes	characteristic
74	C₂×C₈	Dic₁₃₄	1	2	yes	—
75	C₂×C₈	D₂₆₈	1	1	yes	characteristic
76	C₂×C₈	C₂×Dic₆₇	1	2	yes	—
77	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
78	C₄⋊C₄	—	268	1	no	—
79	C₄⋊C₄	—	134	1	no	—
80	C₄⋊C₄	—	134	1	no	—
81	C₄⋊C₄	—	134	2	no	—
82	C₂²⋊C₄	—	268	1	no	—
83	C₄²	—	268	1	no	—
84	C₁₆	C₂×Dic₆₇	1	4	yes	complemented
85	C₂³	D₅₃₆	1	1	yes	characteristic
86	C₂×C₄	—	268	1	no	—
87	C₂×C₄	—	268	1	no	—
88	C₂×C₄	—	268	2	no	—
89	C₂×C₄	—	536	2	no	—
90	C₂×C₄	C₄⋊Dic₆₇	1	2	yes	—
91	C₂×C₄	D₅₃₆	1	1	yes	characteristic
92	C₂×C₄	Dic₂₆₈	1	2	yes	—
93	C₂×C₄	C₂×D₂₆₈	1	1	yes	characteristic
94	C₈	C₄⋊Dic₆₇	1	2	yes	—
95	C₈	C₂²×Dic₆₇	1	1	yes	characteristic
96	C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
97	C₂²	—	2	1	no	—
98	C₂²	—	2	1	no	—
99	C₂²	#14523977	1	2	yes	—
100	C₂²	C₂×D₅₃₆	1	1	yes	characteristic
101	C₄	—	536	2	no	—
102	C₄	—	536	1	no	—
103	C₄	—	536	1	no	—
104	C₄	#14523977	1	2	yes	—
105	C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
106	C₄	C₂×Dic₂₆₈	1	1	yes	characteristic
107	C₂	—	2	2	no	—
108	C₂	C₂×#14523977	1	1	yes	characteristic
109	C₂	#16863834	1	1	yes	characteristic
110	C₂	#16863839	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863434

C₂

yes

maximal, characteristic

C₂×#14523977

C₂

yes

maximal, characteristic

#16863224

C₂

yes

maximal, complemented

#16863222

C₂

yes

maximal, complemented

C₂×M₃₂×C₆₇

C₂

yes

maximal, characteristic

#14524002

—

C₂×C₄⋊Dic₆₇

—

#14523977

—

#14523977

C₂²

yes

—

#14523977

C₂²

yes

characteristic

#14523977

C₂²

yes

characteristic

#14523975

—

C₂²×C₈×C₆₇

C₂²

yes

characteristic

M₃₂×C₆₇

C₄

yes

complemented

C₂×C₁₆×C₆₇

C₂²

yes

—

C₂²×Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

—

C₂²×C₄×C₆₇

D₄

yes

characteristic

C₂×C₈×C₆₇

C₂×C₄

yes

—

C₂×C₈×C₆₇

D₄

yes

characteristic

C₂×C₈×C₆₇

Q₈

yes

—

C₂×C₈×C₆₇

C₂³

yes

characteristic

C₁₆×C₆₇

C₂×C₄

yes

complemented

C₂³×C₆₇

D₈

yes

characteristic

C₂×C₄×C₆₇

C₄⋊C₄

yes

—

C₂×C₄×C₆₇

D₈

yes

characteristic

C₂×C₄×C₆₇

Q₁₆

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₈×C₆₇

C₄⋊C₄

yes

—

C₈×C₆₇

C₂²×C₄

yes

characteristic

C₈×C₆₇

C₂×Q₈

yes

characteristic

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#107

yes

—

C₂²×C₆₇

C₂×D₈

yes

characteristic

C₄×C₆₇

#107

yes

—

C₄×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₆₇

C₂×Q₁₆

yes

characteristic

Dic₆₇

—

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

C₂×M₃₂

D₆₇

yes

—

—

—

—

C₂²×C₈

C₂×D₆₇

yes

characteristic

#117

—

134

—

C₂×C₄⋊C₄

—

134

—

M₃₂

Dic₆₇

yes

complemented

C₂×C₁₆

C₂×D₆₇

yes

—

—

—

—

—

134

—

#106

—

134

—

C₂²×C₄

—

268

—

C₂²×C₄

D₂₆₈

yes

characteristic

C₂×C₈

Dic₁₃₄

yes

—

C₂×C₈

D₂₆₈

yes

characteristic

C₂×C₈

C₂×Dic₆₇

yes

—

C₂×C₈

C₂²×D₆₇

yes

characteristic

C₄⋊C₄

—

268

—

C₄⋊C₄

—

134

—

C₄⋊C₄

—

134

—

C₄⋊C₄

—

134

—

C₂²⋊C₄

—

268

—

C₄²

—

268

—

C₁₆

C₂×Dic₆₇

yes

complemented

C₂³

D₅₃₆

yes

characteristic

C₂×C₄

—

268

—

C₂×C₄

—

268

—

C₂×C₄

—

268

—

C₂×C₄

—

536

—

C₂×C₄

C₄⋊Dic₆₇

yes

—

C₂×C₄

D₅₃₆

yes

characteristic

C₂×C₄

Dic₂₆₈

yes

—

C₂×C₄

C₂×D₂₆₈

yes

characteristic

C₈

C₄⋊Dic₆₇

yes

—

C₈

C₂²×Dic₆₇

yes

characteristic

C₈

C₂×Dic₁₃₄

yes

characteristic

C₂²

—

C₂²

—

C₂²

#14523977

yes

—

100

C₂²

C₂×D₅₃₆

yes

characteristic

101

C₄

—

536

—

102

C₄

—

536

—

103

C₄

—

536

—

104

C₄

#14523977

yes

—

105

C₄

C₂×C₄⋊Dic₆₇

yes

characteristic

106

C₄

C₂×Dic₂₆₈

yes

characteristic

107

C₂

—

108

C₂

C₂×#14523977

yes

characteristic

109

C₂

#16863834

yes

characteristic

110

C₂

#16863839

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₅₃₆
commutator	C₈×C₆₇	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₆₇
Fitting	C₂×M₃₂×C₆₇	C₂
socle	C₂²×C₆₇	C₂×D₈
hypercenter	C₂×M₃₂	D₆₇
hypocenter	C₆₇	#2159
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

subgroup

G/H

center

C₂²

C₂×D₅₃₆

commutator

C₈×C₆₇

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₆₇

Fitting

C₂×M₃₂×C₆₇

C₂

socle

C₂²×C₆₇

C₂×D₈

hypercenter

C₂×M₃₂

D₆₇

hypocenter

C₆₇

#2159

LCS

G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

N	H ≅ G/N	type
3. #16863224	C₂	semidirect
4. #16863222	C₂	semidirect
14. M₃₂×C₆₇	C₄	semidirect
28. C₁₆×C₆₇	C₂×C₄	semidirect
56. C₆₇	#2159	semidirect
65. M₃₂	Dic₆₇	semidirect
84. C₁₆	C₂×Dic₆₇	semidirect

H ≅ G/N

type

3. #16863224

C₂

semidirect

4. #16863222

C₂

semidirect

14. M₃₂×C₆₇

C₄

semidirect

28. C₁₆×C₆₇

C₂×C₄

semidirect

56. C₆₇

#2159

semidirect

65. M₃₂

Dic₆₇

semidirect

84. C₁₆

C₂×Dic₆₇

semidirect

Subgroups — #22561997

Conjugacy classes of subgroups

Complements