Group #22561201 — subgroups

Subgroups — #22561201

Order 8576.

228 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 89 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	C₄×#11761612	C₂	1	1	yes	maximal, characteristic
3	#16863164	C₂	1	4	yes	maximal
4	#423×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#11761612	C₂²	1	2	yes	—
6	#14524035	—	2	1	no	—
7	#14524035	—	2	1	no	—
8	#14524035	—	2	2	no	—
9	C₂×C₄⋊Dic₆₇	C₂²	1	2	yes	—
10	C₂×Dic₆₇⋊C₄	C₄	1	4	yes	—
11	#14523954	—	2	1	no	—
12	C₄×C₆₇⋊C₈	—	2	1	no	—
13	C₂×M₁₆×C₆₇	C₂²	1	2	yes	—
14	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
15	C₄⋊C₈×C₆₇	—	2	2	no	—
16	C₂²×Dic₆₇	C₂×C₄	1	4	yes	—
17	#11761612	—	2	2	no	—
18	#11761612	Q₈	1	4	yes	—
19	C₂×C₆₇⋊C₈	—	2	2	no	—
20	C₄⋊Dic₆₇	—	2	2	no	—
21	C₄⋊Dic₆₇	—	2	2	no	—
22	Dic₆₇⋊C₄	—	2	4	no	—
23	Dic₆₇⋊C₄	—	2	4	no	—
24	C₂²×C₄×C₆₇	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
26	M₁₆×C₆₇	—	2	4	no	—
27	C₂×C₈×C₆₇	—	2	2	no	—
28	C₄²×C₆₇	D₄	1	1	yes	characteristic
29	C₄²×C₆₇	D₄	1	1	yes	characteristic
30	C₄²×C₆₇	D₄	1	2	yes	—
31	C₂³×C₆₇	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₄×C₆₇	—	4	2	no	—
33	C₂×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
34	C₂×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
35	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₆₇	C₂×Q₈	1	2	yes	—
38	C₂×C₄×C₆₇	#41	1	2	yes	—
39	C₂×Dic₆₇	—	2	4	no	—
40	C₂×Dic₆₇	—	4	4	no	—
41	C₈×C₆₇	—	4	2	no	—
42	C₆₇⋊C₈	—	2	4	no	—
43	C₂²×C₆₇	—	2	2	no	—
44	C₂²×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
45	C₂²×C₆₇	C₄×Q₈	1	2	yes	—
46	C₄×C₆₇	—	2	2	no	—
47	C₄×C₆₇	—	2	2	no	—
48	C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
49	C₄×C₆₇	#124	1	1	yes	characteristic
50	C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
51	Dic₆₇	—	4	4	no	—
52	C₂×C₆₇	—	2	2	no	—
53	C₂×C₆₇	#391	1	1	yes	characteristic
54	C₂×C₆₇	#413	1	2	yes	—
55	#1998	—	67	1	no	maximal, Sylow
56	C₆₇	#1998	1	1	yes	characteristic, complemented, Sylow
57	C₂×#125	—	67	1	no	—
58	#423	D₆₇	1	1	yes	characteristic
59	C₄×M₁₆	—	67	1	no	—
60	#343	—	67	4	no	—
61	C₂×M₁₆	—	67	2	no	—
62	C₂×M₁₆	C₂×D₆₇	1	2	yes	—
63	#125	—	134	1	no	—
64	#125	—	134	1	no	—
65	#125	—	134	2	no	—
66	C₂×C₄⋊C₄	—	67	4	no	—
67	C₂×C₄⋊C₄	—	67	2	no	—
68	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
69	C₄⋊C₈	—	2	2	no	—
70	#97	—	134	1	no	—
71	C₄×C₈	—	134	1	no	—
72	C₂²×C₄	—	67	4	no	—
73	C₂²×C₄	C₄×D₆₇	1	2	yes	—
74	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
75	M₁₆	—	2	4	no	—
76	M₁₆	—	67	4	no	—
77	M₁₆	—	134	2	no	—
78	C₂×C₈	—	2	2	no	—
79	C₂×C₈	—	134	2	no	—
80	C₄⋊C₄	—	134	4	no	—
81	C₄⋊C₄	—	134	4	no	—
82	C₄⋊C₄	—	134	2	no	—
83	C₄⋊C₄	—	134	2	no	—
84	C₄²	D₂₆₈	1	1	yes	characteristic
85	C₄²	D₂₆₈	1	1	yes	characteristic
86	C₄²	C₂²⋊D₆₇	1	2	yes	—
87	C₂³	C₂×C₄×D₆₇	1	1	yes	characteristic
88	C₂×C₄	—	4	2	no	—
89	C₂×C₄	—	268	4	no	—
90	C₂×C₄	—	134	4	no	—
91	C₂×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
92	C₂×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
93	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
94	C₂×C₄	C₂×D₂₆₈	1	1	yes	characteristic
95	C₂×C₄	Q₈×D₆₇	1	2	yes	—
96	C₂×C₄	#11761627	1	2	yes	—
97	C₈	—	4	2	no	—
98	C₈	—	134	4	no	—
99	C₂²	—	2	2	no	—
100	C₂²	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
101	C₂²	#14524008	1	2	yes	—
102	C₄	—	2	2	no	—
103	C₄	—	2	2	no	—
104	C₄	—	268	4	no	—
105	C₄	#14524023	1	2	yes	—
106	C₄	#14524030	1	1	yes	characteristic
107	C₄	#14524041	1	1	yes	characteristic
108	C₂	—	2	2	no	—
109	C₂	#16863301	1	1	yes	characteristic
110	C₂	#16863378	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14524035

C₂

yes

maximal, characteristic

C₄×#11761612

C₂

yes

maximal, characteristic

#16863164

C₂

yes

maximal

#423×C₆₇

C₂

yes

maximal, characteristic

C₂×#11761612

C₂²

yes

—

#14524035

—

#14524035

—

#14524035

—

C₂×C₄⋊Dic₆₇

C₂²

yes

—

C₂×Dic₆₇⋊C₄

C₄

yes

—

#14523954

—

C₄×C₆₇⋊C₈

—

C₂×M₁₆×C₆₇

C₂²

yes

—

C₂×C₄²×C₆₇

C₂²

yes

characteristic

C₄⋊C₈×C₆₇

—

C₂²×Dic₆₇

C₂×C₄

yes

—

#11761612

—

#11761612

Q₈

yes

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

Dic₆₇⋊C₄

—

Dic₆₇⋊C₄

—

C₂²×C₄×C₆₇

C₂×C₄

yes

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₄²×C₆₇

D₄

yes

characteristic

C₄²×C₆₇

D₄

yes

characteristic

C₄²×C₆₇

D₄

yes

—

C₂³×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂²⋊C₄

yes

—

C₂×C₄×C₆₇

C₂²⋊C₄

yes

—

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

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₆₇

C₄×Q₈

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂²⋊Q₈

yes

—

C₄×C₆₇

#124

yes

characteristic

C₄×C₆₇

C₄⋊Q₈

yes

characteristic

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#391

yes

characteristic

C₂×C₆₇

#413

yes

—

#1998

—

maximal, Sylow

C₆₇

#1998

yes

characteristic, complemented, Sylow

C₂×#125

—

#423

D₆₇

yes

characteristic

C₄×M₁₆

—

#343

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₆₇

yes

—

#125

—

134

—

#125

—

134

—

#125

—

134

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₆₇

yes

characteristic

C₄⋊C₈

—

#97

—

134

—

C₄×C₈

—

134

—

C₂²×C₄

—

C₂²×C₄

C₄×D₆₇

yes

—

C₂²×C₄

C₂²×D₆₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

134

—

C₂×C₈

—

C₂×C₈

—

134

—

C₄⋊C₄

—

134

—

C₄⋊C₄

—

134

—

C₄⋊C₄

—

134

—

C₄⋊C₄

—

134

—

C₄²

D₂₆₈

yes

characteristic

C₄²

D₂₆₈

yes

characteristic

C₄²

C₂²⋊D₆₇

yes

—

C₂³

C₂×C₄×D₆₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

268

—

C₂×C₄

—

134

—

C₂×C₄

(C₂×D₆₇)⋊C₄

yes

—

C₂×C₄

(C₂×D₆₇)⋊C₄

yes

—

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

C₂×C₄

C₂×D₂₆₈

yes

characteristic

C₂×C₄

Q₈×D₆₇

yes

—

C₂×C₄

#11761627

yes

—

C₈

—

C₈

—

134

—

C₂²

—

100

C₂²

C₂×(C₂×D₆₇)⋊C₄

yes

characteristic

101

C₂²

#14524008

yes

—

102

C₄

—

103

C₄

—

104

C₄

—

268

—

105

C₄

#14524023

yes

—

106

C₄

#14524030

yes

characteristic

107

C₄

#14524041

yes

characteristic

108

C₂

—

109

C₂

#16863301

yes

characteristic

110

C₂

#16863378

yes

—

	subgroup	G/H
center	C₂²	C₂×(C₂×D₆₇)⋊C₄
commutator	C₂³×C₆₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₆₇
Fitting	#423×C₆₇	C₂
socle	C₂²×C₆₇	C₂×C₂²⋊C₄
hypercenter	#423	D₆₇
hypocenter	C₆₇	#1998
LCS	G ▷ C₂³×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂²

C₂×(C₂×D₆₇)⋊C₄

commutator

C₂³×C₆₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₆₇

Fitting

#423×C₆₇

C₂

socle

C₂²×C₆₇

C₂×C₂²⋊C₄

D₆₇

C₆₇

G ▷ C₂³×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
56. C₆₇	#1998	semidirect

H ≅ G/N

type

56. C₆₇

#1998

semidirect

Subgroups — #22561201

Conjugacy classes of subgroups

Complements