Group #22561173 — subgroups

Subgroups — #22561173

Order 8576.

252 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 105 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×#14524035	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#16863158	C₂	1	4	yes	maximal
4	#382×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₆₇⋊C₈	C₂²	1	2	yes	—
6	#14524035	—	2	2	no	—
7	#14524035	C₄	1	4	yes	complemented
8	C₂×C₄⋊Dic₆₇	C₂²	1	2	yes	—
9	C₂×Dic₆₇⋊C₄	—	2	2	no	—
10	C₄×C₆₇⋊C₈	—	2	2	no	—
11	C₂×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
12	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
13	#95×C₆₇	—	2	2	no	—
14	C₂²×Dic₆₇	—	2	2	no	—
15	C₂×C₆₇⋊C₈	—	2	2	no	—
16	C₂×C₆₇⋊C₈	—	2	2	no	—
17	C₂×C₆₇⋊C₈	Q₈	1	4	yes	—
18	C₄⋊Dic₆₇	—	2	2	no	—
19	C₄⋊Dic₆₇	C₂×C₄	1	4	yes	—
20	Dic₆₇⋊C₄	—	2	8	no	—
21	C₂²×C₄×C₆₇	—	2	2	no	—
22	C₂²×C₄×C₆₇	D₄	1	2	yes	—
23	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
24	C₄⋊C₄×C₆₇	—	2	4	no	—
25	C₄²×C₆₇	C₂×C₄	1	2	yes	—
26	C₄²×C₆₇	D₄	1	2	yes	—
27	C₂³×C₆₇	C₂×D₄	1	1	yes	characteristic
28	C₂×C₄×C₆₇	—	2	4	no	—
29	C₂×C₄×C₆₇	—	4	2	no	—
30	C₂×C₄×C₆₇	—	4	4	no	—
31	C₂×C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
32	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₆₇	C₂×Q₈	1	2	yes	—
35	C₂×C₄×C₆₇	#41	1	2	yes	—
36	C₂×Dic₆₇	—	2	4	no	—
37	C₂×Dic₆₇	—	4	4	no	—
38	C₆₇⋊C₈	—	2	4	no	—
39	C₂²×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
40	C₂²×C₆₇	#124	1	1	yes	characteristic
41	C₂²×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
42	C₂²×C₆₇	#135	1	4	yes	—
43	C₄×C₆₇	—	2	4	no	—
44	C₄×C₆₇	—	8	2	no	—
45	C₄×C₆₇	C₄×Q₈	1	2	yes	—
46	C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
47	Dic₆₇	—	4	4	no	—
48	C₂×C₆₇	#391	1	1	yes	characteristic
49	C₂×C₆₇	#416	1	2	yes	—
50	C₂×C₆₇	#488	1	2	yes	—
51	C₂×C₆₇	#499	1	2	yes	—
52	#1991	—	67	1	no	maximal, Sylow
53	C₆₇	#1991	1	1	yes	characteristic, complemented, Sylow
54	C₂×#125	—	67	1	no	—
55	C₂×C₄×C₈	—	67	1	no	—
56	#382	D₆₇	1	1	yes	characteristic
57	#340	—	67	4	no	—
58	C₂²×C₈	—	67	2	no	—
59	#125	—	67	4	no	—
60	#125	—	134	2	no	—
61	C₂×C₄⋊C₄	—	67	2	no	—
62	C₂×C₄⋊C₄	—	134	2	no	—
63	C₂×C₄⋊C₄	C₂×D₆₇	1	2	yes	—
64	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
65	C₄×C₈	—	134	2	no	—
66	#95	—	2	2	no	—
67	C₂²×C₄	—	2	2	no	—
68	C₂²×C₄	—	134	2	no	—
69	C₂²×C₄	C₂²⋊D₆₇	1	2	yes	—
70	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
71	C₂×C₈	—	67	4	no	—
72	C₂×C₈	—	134	2	no	—
73	C₂×C₈	—	134	2	no	—
74	C₄⋊C₄	—	2	4	no	—
75	C₄⋊C₄	—	67	4	no	—
76	C₄⋊C₄	—	134	8	no	—
77	C₄⋊C₄	—	134	2	no	—
78	C₄²	C₄×D₆₇	1	2	yes	—
79	C₄²	D₂₆₈	1	2	yes	—
80	C₂³	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
81	C₂×C₄	—	2	4	no	—
82	C₂×C₄	—	4	2	no	—
83	C₂×C₄	—	4	4	no	—
84	C₂×C₄	—	268	4	no	—
85	C₂×C₄	—	134	4	no	—
86	C₂×C₄	(C₂×D₆₇)⋊C₄	1	4	yes	—
87	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
88	C₂×C₄	C₂×D₂₆₈	1	1	yes	characteristic
89	C₂×C₄	Q₈×D₆₇	1	2	yes	—
90	C₂×C₄	#11761627	1	2	yes	—
91	C₈	—	134	4	no	—
92	C₂²	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
93	C₂²	#14524030	1	1	yes	characteristic
94	C₂²	#14524041	1	1	yes	characteristic
95	C₂²	#14524063	1	4	yes	—
96	C₄	—	2	4	no	—
97	C₄	—	8	2	no	—
98	C₄	—	268	4	no	—
99	C₄	#14524008	1	2	yes	—
100	C₄	#14524023	1	2	yes	—
101	C₂	#16863301	1	1	yes	characteristic
102	C₂	#16863390	1	2	yes	—
103	C₂	#16863750	1	2	yes	—
104	C₂	#16863795	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14524035

C₂

yes

maximal, characteristic

C₂×C₄×C₆₇⋊C₈

C₂

yes

maximal, characteristic

#16863158

C₂

yes

maximal

#382×C₆₇

C₂

yes

maximal, characteristic

C₂²×C₆₇⋊C₈

C₂²

yes

—

#14524035

—

#14524035

C₄

yes

complemented

C₂×C₄⋊Dic₆₇

C₂²

yes

—

C₂×Dic₆₇⋊C₄

—

C₄×C₆₇⋊C₈

—

C₂×C₄⋊C₄×C₆₇

C₂²

yes

—

C₂×C₄²×C₆₇

C₂²

yes

characteristic

#95×C₆₇

—

C₂²×Dic₆₇

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

Q₈

yes

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

C₂×C₄

yes

—

Dic₆₇⋊C₄

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

D₄

yes

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

C₂×C₄

yes

—

C₄²×C₆₇

D₄

yes

—

C₂³×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂²⋊C₄

yes

—

C₂×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×Q₈

yes

—

C₂×C₄×C₆₇

#41

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₆₇⋊C₈

—

C₂²×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₆₇

#124

yes

characteristic

C₂²×C₆₇

C₄⋊Q₈

yes

characteristic

C₂²×C₆₇

#135

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₄×Q₈

yes

—

C₄×C₆₇

C₂²⋊Q₈

yes

—

Dic₆₇

—

C₂×C₆₇

#391

yes

characteristic

C₂×C₆₇

#416

yes

—

C₂×C₆₇

#488

yes

—

C₂×C₆₇

#499

yes

—

#1991

—

maximal, Sylow

C₆₇

#1991

yes

characteristic, complemented, Sylow

C₂×#125

—

C₂×C₄×C₈

—

#382

D₆₇

yes

characteristic

#340

—

C₂²×C₈

—

#125

—

#125

—

134

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

134

—

C₂×C₄⋊C₄

C₂×D₆₇

yes

—

C₂×C₄²

C₂×D₆₇

yes

characteristic

C₄×C₈

—

134

—

#95

—

C₂²×C₄

—

C₂²×C₄

—

134

—

C₂²×C₄

C₂²⋊D₆₇

yes

—

C₂²×C₄

C₂²×D₆₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

134

—

C₂×C₈

—

134

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

134

—

C₄⋊C₄

—

134

—

C₄²

C₄×D₆₇

yes

—

C₄²

D₂₆₈

yes

—

C₂³

C₂×C₂²⋊D₆₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

268

—

C₂×C₄

—

134

—

C₂×C₄

(C₂×D₆₇)⋊C₄

yes

—

C₂×C₄

C₂×C₄×D₆₇

yes

characteristic

C₂×C₄

C₂×D₂₆₈

yes

characteristic

C₂×C₄

Q₈×D₆₇

yes

—

C₂×C₄

#11761627

yes

—

C₈

—

134

—

C₂²

C₂×(C₂×D₆₇)⋊C₄

yes

C₂²

yes

C₂²

yes

C₂²

yes

—

C₄

—

C₄

—

C₄

—

268

—

C₄

#14524008

yes

—

100

C₄

#14524023

yes

—

101

C₂

#16863301

yes

characteristic

102

C₂

#16863390

yes

—

103

C₂

#16863750

yes

—

104

C₂

#16863795

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₆₇
commutator	C₂×C₄×C₆₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₆₇
Fitting	#382×C₆₇	C₂
socle	C₂³×C₆₇	C₂×D₄
hypercenter	#382	D₆₇
hypocenter	C₆₇	#1991
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₆₇

commutator

C₂×C₄×C₆₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₆₇

Fitting

#382×C₆₇

C₂

socle

C₂³×C₆₇

C₂×D₄

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

N	H ≅ G/N	type
7. #14524035	C₄	semidirect
53. C₆₇	#1991	semidirect

H ≅ G/N

type

7. #14524035

C₄

semidirect

53. C₆₇

#1991

semidirect

Subgroups — #22561173

Conjugacy classes of subgroups

Complements