Group #22561964 — subgroups

Subgroups — #22561964

Order 8576.

172 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 97 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×C₈×Dic₆₇	C₂	1	1	yes	maximal, characteristic
3	Dic₆₇⋊C₁₆	C₂	1	4	yes	maximal, complemented
4	C₂×M₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₆₇⋊C₈	C₄	1	1	yes	characteristic
6	C₂×C₄×Dic₆₇	C₄	1	1	yes	characteristic
7	#14523983	—	2	2	no	—
8	C₂×C₆₇⋊C₁₆	C₂²	1	2	yes	—
9	C₈×Dic₆₇	C₄	1	2	yes	—
10	C₈×Dic₆₇	C₂²	1	2	yes	—
11	C₂²×C₈×C₆₇	C₂²	1	1	yes	characteristic
12	M₃₂×C₆₇	—	2	2	no	—
13	C₂×C₁₆×C₆₇	C₂²	1	2	yes	—
14	C₂²×Dic₆₇	C₈	1	2	yes	—
15	C₂×C₆₇⋊C₈	—	2	2	no	—
16	C₂×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
17	C₄×Dic₆₇	C₈	1	1	yes	characteristic
18	C₄×Dic₆₇	C₈	1	1	yes	characteristic
19	C₄×Dic₆₇	C₂×C₄	1	2	yes	—
20	C₆₇⋊C₁₆	—	2	2	no	—
21	C₂²×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₈×C₆₇	D₄	1	2	yes	—
24	C₂×C₈×C₆₇	Q₈	1	2	yes	—
25	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
26	C₁₆×C₆₇	—	2	2	no	—
27	C₂³×C₆₇	C₂×C₈	1	1	yes	characteristic
28	C₂×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
29	C₂×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
30	C₂×C₄×C₆₇	M₁₆	1	2	yes	—
31	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
32	C₂×Dic₆₇	—	2	4	no	—
33	C₂×Dic₆₇	C₂×C₈	1	2	yes	—
34	C₂×Dic₆₇	C₂×C₈	1	2	yes	—
35	C₈×C₆₇	C₄⋊C₄	1	2	yes	—
36	C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
37	C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
38	C₆₇⋊C₈	—	2	2	no	—
39	C₂²×C₆₇	—	2	2	no	—
40	C₂²×C₆₇	C₄⋊C₈	1	2	yes	—
41	C₂²×C₆₇	C₂²×C₈	1	1	yes	characteristic
42	C₄×C₆₇	C₄⋊C₈	1	2	yes	—
43	C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
45	Dic₆₇	—	2	2	no	—
46	Dic₆₇	M₃₂	1	4	yes	complemented
47	C₂×C₆₇	—	2	2	no	—
48	C₂×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
49	C₂×C₆₇	C₂×M₃₂	1	2	yes	—
50	#2150	—	67	1	no	maximal, Sylow
51	C₆₇	#2150	1	1	yes	characteristic, complemented, Sylow
52	C₂×M₃₂	—	67	1	no	—
53	C₂×M₃₂	D₆₇	1	1	yes	characteristic
54	C₂×C₄×C₈	—	67	1	no	—
55	C₄⋊C₁₆	—	67	4	no	—
56	C₂²×C₈	—	67	1	no	—
57	C₂²×C₈	C₂×D₆₇	1	1	yes	characteristic
58	C₂×C₄²	—	67	1	no	—
59	M₃₂	—	2	2	no	—
60	M₃₂	—	134	2	no	—
61	C₂×C₁₆	—	67	2	no	—
62	C₂×C₁₆	C₂×D₆₇	1	2	yes	—
63	C₄×C₈	—	67	2	no	—
64	C₄×C₈	—	67	2	no	—
65	C₂²×C₄	—	67	2	no	—
66	C₂²×C₄	C₄×D₆₇	1	1	yes	characteristic
67	C₂×C₈	—	67	2	no	—
68	C₂×C₈	—	134	2	no	—
69	C₂×C₈	Dic₁₃₄	1	2	yes	—
70	C₂×C₈	C₄×D₆₇	1	1	yes	characteristic
71	C₂×C₈	C₂²⋊D₆₇	1	2	yes	—
72	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
73	C₄²	—	67	1	no	—
74	C₄²	—	67	1	no	—
75	C₄²	—	67	2	no	—
76	C₁₆	—	2	2	no	—
77	C₁₆	—	134	2	no	—
78	C₂³	C₈×D₆₇	1	1	yes	characteristic
79	C₂×C₄	—	67	2	no	—
80	C₂×C₄	—	67	2	no	—
81	C₂×C₄	—	134	4	no	—
82	C₂×C₄	Dic₆₇⋊C₄	1	2	yes	—
83	C₂×C₄	C₈×D₆₇	1	1	yes	characteristic
84	C₂×C₄	#11761611	1	2	yes	—
85	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
86	C₈	—	134	2	no	—
87	C₈	Dic₆₇⋊C₄	1	2	yes	—
88	C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
89	C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
90	C₂²	—	2	2	no	—
91	C₂²	Dic₆₇⋊C₈	1	2	yes	—
92	C₂²	C₂×C₈×D₆₇	1	1	yes	characteristic
93	C₄	—	67	4	no	—
94	C₄	—	134	2	no	—
95	C₄	Dic₆₇⋊C₈	1	2	yes	—
96	C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
97	C₄	C₂×#11761611	1	1	yes	characteristic
98	C₂	—	2	2	no	—
99	C₂	C₂×Dic₆₇⋊C₈	1	1	yes	characteristic
100	C₂	M₃₂×D₆₇	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14523983

C₂

yes

maximal, characteristic

C₂×C₈×Dic₆₇

C₂

yes

maximal, characteristic

Dic₆₇⋊C₁₆

C₂

yes

maximal, complemented

C₂×M₃₂×C₆₇

C₂

yes

maximal, characteristic

C₂²×C₆₇⋊C₈

C₄

yes

characteristic

C₂×C₄×Dic₆₇

C₄

yes

characteristic

#14523983

—

C₂×C₆₇⋊C₁₆

C₂²

yes

—

C₈×Dic₆₇

C₄

yes

—

C₈×Dic₆₇

C₂²

yes

—

C₂²×C₈×C₆₇

C₂²

yes

characteristic

M₃₂×C₆₇

—

C₂×C₁₆×C₆₇

C₂²

yes

—

C₂²×Dic₆₇

C₈

yes

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

C₂×C₄

yes

—

C₄×Dic₆₇

C₈

yes

characteristic

C₄×Dic₆₇

C₈

yes

characteristic

C₄×Dic₆₇

C₂×C₄

yes

—

C₆₇⋊C₁₆

—

C₂²×C₄×C₆₇

C₂×C₄

yes

characteristic

C₂×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂×C₈×C₆₇

D₄

yes

—

C₂×C₈×C₆₇

Q₈

yes

—

C₂×C₈×C₆₇

C₂³

yes

characteristic

C₁₆×C₆₇

—

C₂³×C₆₇

C₂×C₈

yes

characteristic

C₂×C₄×C₆₇

C₄⋊C₄

yes

—

C₂×C₄×C₆₇

C₂×C₈

yes

characteristic

C₂×C₄×C₆₇

M₁₆

yes

—

C₂×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

C₂×C₈

yes

—

C₂×Dic₆₇

C₂×C₈

yes

—

C₈×C₆₇

C₄⋊C₄

yes

—

C₈×C₆₇

C₂×D₄

yes

characteristic

C₈×C₆₇

C₂×Q₈

yes

characteristic

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

C₄⋊C₈

yes

—

C₂²×C₆₇

C₂²×C₈

yes

characteristic

C₄×C₆₇

C₄⋊C₈

yes

—

C₄×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₆₇

C₂×M₁₆

yes

characteristic

Dic₆₇

—

Dic₆₇

M₃₂

yes

complemented

C₂×C₆₇

—

C₂×C₆₇

C₂×C₄⋊C₈

yes

characteristic

C₂×C₆₇

C₂×M₃₂

yes

—

#2150

—

maximal, Sylow

C₆₇

#2150

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂×M₃₂

D₆₇

yes

characteristic

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂×D₆₇

yes

characteristic

C₂×C₄²

—

M₃₂

—

M₃₂

—

134

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₆₇

yes

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₄

—

C₂²×C₄

C₄×D₆₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

134

—

C₂×C₈

Dic₁₃₄

yes

—

C₂×C₈

C₄×D₆₇

yes

characteristic

C₂×C₈

C₂²⋊D₆₇

yes

—

C₂×C₈

C₂²×D₆₇

yes

characteristic

C₄²

—

C₄²

—

C₄²

—

C₁₆

—

C₁₆

—

134

—

C₂³

C₈×D₆₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

134

—

C₂×C₄

Dic₆₇⋊C₄

yes

—

C₂×C₄

C₈×D₆₇

yes

characteristic

C₂×C₄

#11761611

yes

—

C₂×C₄

C₂×C₄×D₆₇

yes

characteristic

C₈

—

134

—

C₈

Dic₆₇⋊C₄

yes

—

C₈

C₂×C₂²⋊D₆₇

yes

characteristic

C₈

C₂×Dic₁₃₄

yes

characteristic

C₂²

—

C₂²

Dic₆₇⋊C₈

yes

—

C₂²

C₂×C₈×D₆₇

yes

characteristic

C₄

—

C₄

—

134

—

C₄

Dic₆₇⋊C₈

yes

—

C₄

C₂×Dic₆₇⋊C₄

yes

characteristic

C₄

C₂×#11761611

yes

characteristic

C₂

—

C₂

C₂×Dic₆₇⋊C₈

yes

characteristic

100

C₂

M₃₂×D₆₇

yes

—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₆₇
commutator, socle	C₂²×C₆₇	C₂²×C₈
Fitting	C₂×M₃₂×C₆₇	C₂
hypercenter	C₂×M₃₂	D₆₇
hypocenter	C₆₇	#2150
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₆₇

commutator, socle

C₂²×C₆₇

C₂²×C₈

Fitting

C₂×M₃₂×C₆₇

C₂

hypercenter

C₂×M₃₂

D₆₇

C₆₇

G ▷ C₂²×C₆₇ ▷ C₆₇

C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

N	H ≅ G/N	type
3. Dic₆₇⋊C₁₆	C₂	semidirect
46. Dic₆₇	M₃₂	semidirect
51. C₆₇	#2150	semidirect

H ≅ G/N

type

3. Dic₆₇⋊C₁₆

C₂

semidirect

46. Dic₆₇

M₃₂

semidirect

51. C₆₇

#2150

semidirect

Subgroups — #22561964

Conjugacy classes of subgroups

Complements