Group #22561719 — subgroups

Subgroups — #22561719

Order 8576.

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

normal subgroups only

This group has 176 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	#16863439	C₂	1	2	yes	maximal
2	C₂×#14523975	C₂	1	1	yes	maximal, characteristic
3	#16863164	C₂	1	2	yes	maximal
4	#16863161	C₂	1	1	yes	maximal, characteristic
5	#424×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#11761612	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₆₇	C₂²	1	2	yes	—
8	#14523976	—	2	2	no	—
9	#14523975	—	2	2	no	—
10	#14523974	—	2	2	no	—
11	C₂×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
12	C₂²×C₈×C₆₇	C₂²	1	1	yes	characteristic
13	#117×C₆₇	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₆₇	—	2	2	no	—
15	C₂²×Dic₆₇	—	2	2	no	—
16	#11761612	—	2	2	no	—
17	#11761612	Q₈	1	4	yes	—
18	C₂×C₆₇⋊C₈	—	2	2	no	—
19	C₄⋊Dic₆₇	—	2	2	no	—
20	C₄⋊Dic₆₇	—	2	2	no	—
21	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
22	M₁₆×C₆₇	—	2	2	no	—
23	C₂×C₈×C₆₇	—	2	2	no	—
24	C₂×C₈×C₆₇	—	2	1	no	—
25	C₂×C₈×C₆₇	D₄	1	2	yes	—
26	C₄⋊C₄×C₆₇	D₄	1	2	yes	—
27	C₂²⋊C₄×C₆₇	—	2	1	no	—
28	C₄²×C₆₇	—	2	1	no	—
29	C₂³×C₆₇	#41	1	1	yes	characteristic
30	C₂×C₄×C₆₇	—	2	2	no	—
31	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₆₇	C₂×Q₈	1	2	yes	—
34	C₂×C₄×C₆₇	#41	1	2	yes	—
35	C₂×Dic₆₇	—	4	2	no	—
36	C₂×Dic₆₇	—	8	2	no	—
37	C₈×C₆₇	—	2	2	no	—
38	C₈×C₆₇	—	4	1	no	—
39	C₆₇⋊C₈	—	2	4	no	—
40	C₂²×C₆₇	—	2	2	no	—
41	C₂²×C₆₇	#121	1	1	yes	characteristic
42	C₂²×C₆₇	#125	1	2	yes	—
43	C₄×C₆₇	—	4	2	no	—
44	C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
45	C₄×C₆₇	#123	1	1	yes	characteristic
46	C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
47	Dic₆₇	—	8	2	no	—
48	C₂×C₆₇	—	2	2	no	—
49	C₂×C₆₇	#398	1	1	yes	characteristic
50	C₂×C₆₇	#471	1	2	yes	—
51	#2090	—	67	1	no	maximal, Sylow
52	C₆₇	#2090	1	1	yes	characteristic, complemented, Sylow
53	#428	—	67	2	no	—
54	C₂×#106	—	67	1	no	—
55	#424	D₆₇	1	1	yes	characteristic
56	#343	—	67	2	no	—
57	#341	—	67	1	no	—
58	C₂×M₁₆	—	67	2	no	—
59	C₂×M₁₆	C₂×D₆₇	1	1	yes	characteristic
60	C₂²×C₈	C₂×D₆₇	1	1	yes	characteristic
61	#117	C₂×D₆₇	1	1	yes	characteristic
62	C₂×C₄⋊C₄	—	67	2	no	—
63	#107	—	134	2	no	—
64	#106	—	134	2	no	—
65	#106	—	134	2	no	—
66	C₄⋊C₈	—	2	2	no	—
67	C₂²×C₄	—	134	2	no	—
68	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
69	M₁₆	—	2	2	no	—
70	M₁₆	—	67	4	no	—
71	M₁₆	—	134	2	no	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	—	2	1	no	—
74	C₂×C₈	—	134	2	no	—
75	C₂×C₈	C₂²⋊D₆₇	1	2	yes	—
76	C₄⋊C₄	—	134	2	no	—
77	C₄⋊C₄	—	134	2	no	—
78	C₄⋊C₄	D₂₆₈	1	2	yes	—
79	C₂²⋊C₄	—	2	1	no	—
80	C₄²	—	2	1	no	—
81	C₂³	#11761628	1	1	yes	characteristic
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	268	2	no	—
84	C₂×C₄	—	536	2	no	—
85	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
86	C₂×C₄	C₂×D₂₆₈	1	1	yes	characteristic
87	C₂×C₄	Q₈×D₆₇	1	2	yes	—
88	C₂×C₄	#11761627	1	2	yes	—
89	C₈	—	2	2	no	—
90	C₈	—	4	1	no	—
91	C₈	—	134	4	no	—
92	C₂²	—	2	2	no	—
93	C₂²	#14524017	1	1	yes	characteristic
94	C₂²	#14524033	1	2	yes	—
95	C₄	—	4	2	no	—
96	C₄	—	536	2	no	—
97	C₄	#14524023	1	2	yes	—
98	C₄	#14524024	1	1	yes	characteristic
99	C₄	#14524041	1	1	yes	characteristic
100	C₂	—	2	2	no	—
101	C₂	#16863326	1	1	yes	characteristic
102	C₂	#16863659	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16863439

C₂

yes

maximal

C₂×#14523975

C₂

yes

maximal, characteristic

#16863164

C₂

yes

maximal

#16863161

C₂

yes

maximal, characteristic

#424×C₆₇

C₂

yes

maximal, characteristic

C₂×#11761612

C₂²

yes

—

C₂×C₄⋊Dic₆₇

C₂²

yes

—

#14523976

—

#14523975

—

#14523974

—

C₂×M₁₆×C₆₇

C₂²

yes

characteristic

C₂²×C₈×C₆₇

C₂²

yes

characteristic

#117×C₆₇

C₂²

yes

characteristic

C₄⋊C₈×C₆₇

—

C₂²×Dic₆₇

—

#11761612

—

#11761612

Q₈

yes

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

D₄

yes

—

C₄⋊C₄×C₆₇

D₄

yes

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₂³×C₆₇

#41

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×Q₈

yes

—

C₂×C₄×C₆₇

#41

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₈×C₆₇

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

#121

yes

characteristic

C₂²×C₆₇

#125

yes

—

C₄×C₆₇

—

C₄×C₆₇

C₂²⋊Q₈

yes

—

C₄×C₆₇

#123

yes

characteristic

C₄×C₆₇

C₄⋊Q₈

yes

characteristic

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#398

yes

characteristic

C₂×C₆₇

#471

yes

—

#2090

—

maximal, Sylow

C₆₇

#2090

yes

characteristic, complemented, Sylow

#428

—

C₂×#106

—

#424

D₆₇

yes

characteristic

#343

—

#341

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₆₇

yes

characteristic

C₂²×C₈

C₂×D₆₇

yes

characteristic

#117

C₂×D₆₇

yes

characteristic

C₂×C₄⋊C₄

—

#107

—

134

—

#106

—

134

—

#106

—

134

—

C₄⋊C₈

—

C₂²×C₄

—

134

—

C₂²×C₄

C₂²×D₆₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

134

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

134

—

C₂×C₈

C₂²⋊D₆₇

yes

—

C₄⋊C₄

—

134

—

C₄⋊C₄

—

134

—

C₄⋊C₄

D₂₆₈

yes

—

C₂²⋊C₄

—

C₄²

—

C₂³

#11761628

yes

characteristic

C₂×C₄

—

C₂×C₄

—

268

—

C₂×C₄

—

536

—

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

C₂×C₄

C₂×D₂₆₈

yes

characteristic

C₂×C₄

Q₈×D₆₇

yes

—

C₂×C₄

#11761627

yes

—

C₈

—

C₈

—

C₈

—

134

—

C₂²

—

C₂²

#14524017

yes

characteristic

C₂²

#14524033

yes

—

C₄

—

C₄

—

536

—

C₄

#14524023

yes

—

C₄

yes

C₄

yes

100

C₂

—

101

C₂

#16863326

yes

characteristic

102

C₂

#16863659

yes

—

	subgroup	G/H
center	C₂²	#14524017
commutator	C₂²×C₄×C₆₇	C₂³
Frattini	C₂²×C₄	C₂²×D₆₇
Fitting	#424×C₆₇	C₂
socle	C₂²×C₆₇	#121
hypercenter	#424	D₆₇
hypocenter	C₆₇	#2090
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #424

subgroup

G/H

center

C₂²

#14524017

commutator

C₂²×C₄×C₆₇

C₂³

Frattini

C₂²×C₄

C₂²×D₆₇

Fitting

#424×C₆₇

C₂

C₂²×C₆₇

D₆₇

C₆₇

G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #424

N	H ≅ G/N	type
52. C₆₇	#2090	semidirect

H ≅ G/N

type

52. C₆₇

#2090

semidirect

Subgroups — #22561719

Conjugacy classes of subgroups

Complements