Group #22562016 — subgroups

Subgroups — #22562016

Order 8576.

138 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 138 conjugacy classes of subgroups — too many to draw. Only its 85 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	C₄⋊(C₆₇⋊C₁₆)	C₂	1	2	yes	maximal, complemented
2	C₂²⋊(C₆₇⋊C₁₆)	C₂	1	2	yes	maximal
3	#16863172	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×C₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	#405×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₆₇⋊C₁₆	C₂²	1	2	yes	—
7	C₂×C₆₇⋊C₁₆	C₂²	1	2	yes	—
8	C₂×M₁₆×C₆₇	C₄	1	1	yes	characteristic
9	C₂²×C₈×C₆₇	C₂²	1	1	yes	characteristic
10	#117×C₆₇	C₄	1	1	yes	characteristic
11	#97×C₆₇	C₄	1	1	yes	characteristic
12	#97×C₆₇	C₄	1	1	yes	characteristic
13	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₆₇⋊C₁₆	—	2	2	no	—
16	C₆₇⋊C₁₆	—	2	2	no	—
17	C₂²×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
18	M₁₆×C₆₇	—	2	2	no	—
19	C₂×C₈×C₆₇	—	2	2	no	—
20	C₂×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
24	C₄⋊C₄×C₆₇	C₈	1	2	yes	—
25	C₂²⋊C₄×C₆₇	C₈	1	2	yes	—
26	C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
27	C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₆₇	C₂×C₈	1	1	yes	characteristic
29	C₂×C₄×C₆₇	—	2	1	no	—
30	C₂×C₄×C₆₇	—	2	1	no	—
31	C₂×C₄×C₆₇	C₂×C₈	1	2	yes	—
32	C₂×C₄×C₆₇	C₂×C₈	1	2	yes	—
33	C₂×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
34	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
35	C₈×C₆₇	—	2	1	no	—
36	C₈×C₆₇	—	2	1	no	—
37	C₈×C₆₇	M₁₆	1	2	yes	—
38	C₈×C₆₇	#41	1	2	yes	—
39	C₂²×C₆₇	—	2	1	no	—
40	C₂²×C₆₇	—	2	2	no	—
41	C₂²×C₆₇	C₂²×C₈	1	1	yes	characteristic
42	C₄×C₆₇	—	2	1	no	—
43	C₄×C₆₇	—	2	2	no	—
44	C₄×C₆₇	—	2	2	no	—
45	C₄×C₆₇	#117	1	1	yes	characteristic
46	C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
47	C₂×C₆₇	—	4	1	no	—
48	C₂×C₆₇	#431	1	1	yes	characteristic
49	C₂×C₆₇	#504	1	2	yes	—
50	#2164	—	67	1	no	maximal, Sylow
51	C₆₇	#2164	1	1	yes	characteristic, complemented, Sylow
52	#405	D₆₇	1	1	yes	characteristic
53	C₄⋊C₁₆	—	67	2	no	—
54	C₂²⋊C₁₆	—	67	2	no	—
55	#346	—	67	1	no	—
56	C₄×C₁₆	—	67	1	no	—
57	C₂×M₁₆	Dic₆₇	1	1	yes	characteristic
58	C₂²×C₈	C₂×D₆₇	1	1	yes	characteristic
59	#117	Dic₆₇	1	1	yes	characteristic
60	C₂×C₁₆	—	67	2	no	—
61	C₂×C₁₆	—	67	2	no	—
62	#97	Dic₆₇	1	1	yes	characteristic
63	#97	Dic₆₇	1	1	yes	characteristic
64	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
65	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
66	C₂²×C₄	C₂×Dic₆₇	1	1	yes	characteristic
67	M₁₆	—	2	2	no	—
68	C₂×C₈	—	2	2	no	—
69	C₂×C₈	C₂×Dic₆₇	1	1	yes	characteristic
70	C₂×C₈	C₂×Dic₆₇	1	1	yes	characteristic
71	C₂×C₈	C₂×Dic₆₇	1	1	yes	characteristic
72	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
73	C₄⋊C₄	C₆₇⋊C₈	1	2	yes	—
74	C₂²⋊C₄	C₆₇⋊C₈	1	2	yes	—
75	C₄²	C₂×Dic₆₇	1	1	yes	characteristic
76	C₄²	C₂×Dic₆₇	1	1	yes	characteristic
77	C₁₆	—	134	2	no	—
78	C₁₆	—	134	2	no	—
79	C₂³	C₂×C₆₇⋊C₈	1	1	yes	characteristic
80	C₂×C₄	—	2	1	no	—
81	C₂×C₄	—	2	1	no	—
82	C₂×C₄	C₂×C₆₇⋊C₈	1	2	yes	—
83	C₂×C₄	C₂×C₆₇⋊C₈	1	2	yes	—
84	C₂×C₄	C₂×C₆₇⋊C₈	1	1	yes	characteristic
85	C₂×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
86	C₈	—	2	1	no	—
87	C₈	—	2	1	no	—
88	C₈	#11761612	1	2	yes	—
89	C₈	#11761628	1	2	yes	—
90	C₂²	—	2	2	no	—
91	C₂²	—	2	1	no	—
92	C₂²	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
93	C₄	—	2	2	no	—
94	C₄	—	2	2	no	—
95	C₄	—	2	1	no	—
96	C₄	#14524002	1	1	yes	characteristic
97	C₄	C₂×#11761612	1	1	yes	characteristic
98	C₂	—	4	1	no	—
99	C₂	#16863449	1	1	yes	characteristic
100	C₂	#16863813	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄⋊(C₆₇⋊C₁₆)

C₂

yes

maximal, complemented

C₂²⋊(C₆₇⋊C₁₆)

C₂

yes

maximal

#16863172

C₂

yes

maximal, characteristic, complemented

C₄×C₆₇⋊C₁₆

C₂

yes

maximal, characteristic, complemented

#405×C₆₇

C₂

yes

maximal, characteristic

C₂×C₆₇⋊C₁₆

C₂²

yes

—

C₂×C₆₇⋊C₁₆

C₂²

yes

—

C₂×M₁₆×C₆₇

C₄

yes

characteristic

C₂²×C₈×C₆₇

C₂²

yes

characteristic

#117×C₆₇

C₄

yes

characteristic

#97×C₆₇

C₄

yes

characteristic

#97×C₆₇

C₄

yes

characteristic

C₄×C₈×C₆₇

C₂²

yes

characteristic

C₄×C₈×C₆₇

C₂²

yes

characteristic

C₆₇⋊C₁₆

—

C₆₇⋊C₁₆

—

C₂²×C₄×C₆₇

C₂×C₄

yes

characteristic

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂×C₈×C₆₇

C₂³

yes

characteristic

C₄⋊C₄×C₆₇

C₈

yes

—

C₂²⋊C₄×C₆₇

C₈

yes

—

C₄²×C₆₇

C₂×C₄

yes

characteristic

C₄²×C₆₇

C₂×C₄

yes

characteristic

C₂³×C₆₇

C₂×C₈

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×C₈

yes

—

C₂×C₄×C₆₇

C₂×C₈

yes

—

C₂×C₄×C₆₇

C₂×C₈

yes

characteristic

C₂×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

M₁₆

yes

—

C₈×C₆₇

#41

yes

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂²×C₈

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#117

yes

characteristic

C₄×C₆₇

C₂×M₁₆

yes

characteristic

C₂×C₆₇

—

C₂×C₆₇

#431

yes

characteristic

C₂×C₆₇

#504

yes

—

#2164

—

maximal, Sylow

C₆₇

#2164

yes

characteristic, complemented, Sylow

#405

D₆₇

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

#346

—

C₄×C₁₆

—

C₂×M₁₆

Dic₆₇

yes

characteristic

C₂²×C₈

C₂×D₆₇

yes

characteristic

#117

Dic₆₇

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

#97

Dic₆₇

yes

characteristic

#97

Dic₆₇

yes

characteristic

C₄×C₈

C₂×D₆₇

yes

characteristic

C₄×C₈

C₂×D₆₇

yes

characteristic

C₂²×C₄

C₂×Dic₆₇

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

C₂×Dic₆₇

yes

characteristic

C₂×C₈

C₂×Dic₆₇

yes

characteristic

C₂×C₈

C₂×Dic₆₇

yes

characteristic

C₂×C₈

C₂²×D₆₇

yes

characteristic

C₄⋊C₄

C₆₇⋊C₈

yes

—

C₂²⋊C₄

C₆₇⋊C₈

yes

—

C₄²

C₂×Dic₆₇

yes

characteristic

C₄²

C₂×Dic₆₇

yes

characteristic

C₁₆

—

134

—

C₁₆

—

134

—

C₂³

C₂×C₆₇⋊C₈

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₆₇⋊C₈

yes

—

C₂×C₄

C₂×C₆₇⋊C₈

yes

—

C₂×C₄

C₂×C₆₇⋊C₈

yes

characteristic

C₂×C₄

C₂²×Dic₆₇

yes

characteristic

C₈

—

C₈

—

C₈

#11761612

yes

—

C₈

#11761628

yes

—

C₂²

—

C₂²

—

C₂²

C₂²×C₆₇⋊C₈

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

yes

C₄

yes

C₂

—

C₂

#16863449

yes

characteristic

100

C₂

#16863813

yes

—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₆₇
commutator, socle	C₂²×C₆₇	C₂²×C₈
Fitting	#405×C₆₇	C₂
hypercenter	#405	D₆₇
hypocenter	C₆₇	#2164
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ #405

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₆₇

commutator, socle

C₂²×C₆₇

C₂²×C₈

Fitting

#405×C₆₇

C₂

D₆₇

C₆₇

G ▷ C₂²×C₆₇ ▷ C₆₇

C₁ ◁ C₂×C₈ ◁ #405

N	H ≅ G/N	type
1. C₄⋊(C₆₇⋊C₁₆)	C₂	semidirect
3. #16863172	C₂	semidirect
4. C₄×C₆₇⋊C₁₆	C₂	semidirect
51. C₆₇	#2164	semidirect

H ≅ G/N

type

1. C₄⋊(C₆₇⋊C₁₆)

C₂

semidirect

3. #16863172

C₂

semidirect

4. C₄×C₆₇⋊C₁₆

C₂

semidirect

51. C₆₇

#2164

semidirect

Subgroups — #22562016

Conjugacy classes of subgroups

Complements