Group #22562017 — subgroups

Subgroups — #22562017

Order 8576.

138 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 138 conjugacy classes of subgroups — too many to draw. Only its 69 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	#16863350	C₂	1	1	yes	maximal, characteristic
2	Dic₆₇⋊C₁₆	C₂	1	2	yes	maximal, complemented
3	(C₂×D₆₇)⋊C₁₆	C₂	1	2	yes	maximal
4	C₄×C₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	#346×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11761611	C₄	1	1	yes	characteristic
7	C₂×C₈×D₆₇	C₂²	1	1	yes	characteristic
8	#14524000	C₄	1	1	yes	characteristic
9	C₂×C₆₇⋊C₁₆	C₂²	1	2	yes	—
10	#14523954	C₄	1	1	yes	characteristic
11	#14523953	C₄	1	1	yes	characteristic
12	C₈×Dic₆₇	C₂²	1	1	yes	characteristic
13	C₂×C₁₆×C₆₇	C₂²	1	2	yes	—
14	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₆₇	C₂×C₄	1	1	yes	characteristic
16	#11761611	—	2	2	no	—
17	C₂×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
18	C₂×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
19	C₈×D₆₇	—	2	2	no	—
20	Dic₆₇⋊C₄	C₈	1	2	yes	—
21	(C₂×D₆₇)⋊C₄	C₈	1	2	yes	—
22	C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
23	C₆₇⋊C₁₆	—	2	2	no	—
24	C₂×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
26	C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
27	C₁₆×C₆₇	—	2	2	no	—
28	C₂²×D₆₇	C₂×C₈	1	1	yes	characteristic
29	C₂×C₄×C₆₇	C₂×C₈	1	2	yes	—
30	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
31	C₂×Dic₆₇	C₂×C₈	1	2	yes	—
32	C₂×Dic₆₇	C₂×C₈	1	1	yes	characteristic
33	C₄×D₆₇	—	2	1	no	—
34	C₄×D₆₇	—	2	1	no	—
35	C₈×C₆₇	M₁₆	1	2	yes	—
36	C₈×C₆₇	#41	1	2	yes	—
37	C₆₇⋊C₈	—	2	1	no	—
38	C₆₇⋊C₈	—	2	1	no	—
39	C₂²×C₆₇	C₂²×C₈	1	1	yes	characteristic
40	C₂×D₆₇	—	2	1	no	—
41	C₂×D₆₇	—	2	2	no	—
42	C₄×C₆₇	—	2	2	no	—
43	C₄×C₆₇	#117	1	1	yes	characteristic
44	C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
45	Dic₆₇	—	2	1	no	—
46	Dic₆₇	—	2	2	no	—
47	C₂×C₆₇	#431	1	1	yes	characteristic
48	C₂×C₆₇	#504	1	2	yes	—
49	D₆₇	—	4	1	no	—
50	#2164	—	67	1	no	maximal, Sylow
51	C₆₇	#2164	1	1	yes	characteristic, complemented, Sylow
52	#405	—	67	1	no	—
53	C₄⋊C₁₆	—	67	2	no	—
54	C₂²⋊C₁₆	—	67	2	no	—
55	#346	D₆₇	1	1	yes	characteristic, complemented
56	C₄×C₁₆	—	67	1	no	—
57	C₂×M₁₆	—	67	1	no	—
58	C₂²×C₈	—	67	1	no	—
59	#117	—	67	1	no	—
60	C₂×C₁₆	—	67	2	no	—
61	C₂×C₁₆	C₂×D₆₇	1	2	yes	—
62	#97	—	67	1	no	—
63	#97	—	67	1	no	—
64	C₄×C₈	—	67	1	no	—
65	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
66	C₂²×C₄	—	67	1	no	—
67	M₁₆	—	134	2	no	—
68	C₂×C₈	—	67	1	no	—
69	C₂×C₈	—	67	1	no	—
70	C₂×C₈	—	134	2	no	—
71	C₂×C₈	C₄×D₆₇	1	1	yes	characteristic
72	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
73	C₄⋊C₄	—	67	2	no	—
74	C₂²⋊C₄	—	67	2	no	—
75	C₄²	—	67	1	no	—
76	C₄²	C₄×D₆₇	1	1	yes	characteristic
77	C₁₆	—	2	2	no	—
78	C₁₆	—	134	2	no	—
79	C₂³	—	67	1	no	—
80	C₂×C₄	—	67	2	no	—
81	C₂×C₄	—	67	1	no	—
82	C₂×C₄	—	134	1	no	—
83	C₂×C₄	—	134	1	no	—
84	C₂×C₄	C₈×D₆₇	1	2	yes	—
85	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
86	C₈	—	134	1	no	—
87	C₈	—	134	1	no	—
88	C₈	#11761611	1	2	yes	—
89	C₈	#11761628	1	2	yes	—
90	C₂²	—	134	2	no	—
91	C₂²	—	134	1	no	—
92	C₂²	C₂×C₈×D₆₇	1	1	yes	characteristic
93	C₄	—	2	2	no	—
94	C₄	—	134	2	no	—
95	C₄	—	134	1	no	—
96	C₄	#14524000	1	1	yes	characteristic
97	C₄	C₂×#11761611	1	1	yes	characteristic
98	C₂	—	268	1	no	—
99	C₂	#16863447	1	1	yes	characteristic
100	C₂	#16863814	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16863350

C₂

yes

maximal, characteristic

Dic₆₇⋊C₁₆

C₂

yes

maximal, complemented

(C₂×D₆₇)⋊C₁₆

C₂

yes

maximal

C₄×C₆₇⋊C₁₆

C₂

yes

maximal, characteristic, complemented

#346×C₆₇

C₂

yes

maximal, characteristic, complemented

C₂×#11761611

C₄

yes

characteristic

C₂×C₈×D₆₇

C₂²

yes

characteristic

#14524000

C₄

yes

characteristic

C₂×C₆₇⋊C₁₆

C₂²

yes

—

#14523954

C₄

yes

characteristic

#14523953

C₄

yes

characteristic

C₈×Dic₆₇

C₂²

yes

characteristic

C₂×C₁₆×C₆₇

C₂²

yes

—

C₄×C₈×C₆₇

C₂²

yes

characteristic

C₂×C₄×D₆₇

C₂×C₄

yes

characteristic

#11761611

—

C₂×C₆₇⋊C₈

C₂×C₄

yes

characteristic

C₂×C₆₇⋊C₈

C₂×C₄

yes

characteristic

C₈×D₆₇

—

Dic₆₇⋊C₄

C₈

yes

—

(C₂×D₆₇)⋊C₄

C₈

yes

—

C₄×Dic₆₇

C₂×C₄

yes

characteristic

C₆₇⋊C₁₆

—

C₂×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂×C₈×C₆₇

C₂³

yes

characteristic

C₄²×C₆₇

C₂×C₄

yes

characteristic

C₁₆×C₆₇

—

C₂²×D₆₇

C₂×C₈

yes

characteristic

C₂×C₄×C₆₇

C₂×C₈

yes

—

C₂×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×Dic₆₇

C₂×C₈

yes

—

C₂×Dic₆₇

C₂×C₈

yes

characteristic

C₄×D₆₇

—

C₄×D₆₇

—

C₈×C₆₇

M₁₆

yes

—

C₈×C₆₇

#41

yes

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

C₂²×C₈

yes

characteristic

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#117

yes

characteristic

C₄×C₆₇

C₂×M₁₆

yes

characteristic

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

#431

yes

characteristic

C₂×C₆₇

#504

yes

—

D₆₇

—

#2164

—

maximal, Sylow

C₆₇

#2164

yes

characteristic, complemented, Sylow

#405

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

#346

D₆₇

yes

characteristic, complemented

C₄×C₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

#117

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₆₇

yes

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

C₂×D₆₇

yes

characteristic

C₂²×C₄

—

M₁₆

—

134

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

134

—

C₂×C₈

C₄×D₆₇

yes

characteristic

C₂×C₈

C₂²×D₆₇

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₄×D₆₇

yes

characteristic

C₁₆

—

C₁₆

—

134

—

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

134

—

C₂×C₄

—

134

—

C₂×C₄

C₈×D₆₇

yes

—

C₂×C₄

C₂×C₄×D₆₇

yes

characteristic

C₈

—

134

—

C₈

—

134

—

C₈

#11761611

yes

—

C₈

#11761628

yes

—

C₂²

—

134

—

C₂²

—

134

—

C₂²

C₂×C₈×D₆₇

yes

characteristic

C₄

—

C₄

—

134

—

C₄

—

134

—

C₄

yes

C₄

yes

C₂

—

268

—

C₂

#16863447

yes

characteristic

100

C₂

#16863814

yes

—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₆₇
commutator, socle	C₂²×C₆₇	C₂²×C₈
Fitting	#346×C₆₇	C₂
hypercenter	#346	D₆₇
hypocenter	C₆₇	#2164
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ #346

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₆₇

commutator, socle

C₂²×C₆₇

C₂²×C₈

Fitting

#346×C₆₇

C₂

D₆₇

C₆₇

G ▷ C₂²×C₆₇ ▷ C₆₇

C₁ ◁ C₂×C₈ ◁ #346

N	H ≅ G/N	type
2. Dic₆₇⋊C₁₆	C₂	semidirect
4. C₄×C₆₇⋊C₁₆	C₂	semidirect
5. #346×C₆₇	C₂	semidirect
51. C₆₇	#2164	semidirect
55. #346	D₆₇	semidirect

H ≅ G/N

type

2. Dic₆₇⋊C₁₆

C₂

semidirect

4. C₄×C₆₇⋊C₁₆

C₂

semidirect

5. #346×C₆₇

C₂

semidirect

51. C₆₇

#2164

semidirect

55. #346

D₆₇

semidirect

Subgroups — #22562017

Conjugacy classes of subgroups

Complements