Group #22558321 — subgroups

Subgroups — #22558321

Order 8576.

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

normal subgroups only

This group has 124 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	C₂×#14523979	C₂	1	1	yes	maximal, characteristic
2	C₂×#14523979	C₂	1	1	yes	maximal, characteristic
3	C₂×M₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#11761612	—	2	1	no	—
5	C₂×#11761612	—	2	1	no	—
6	#14523979	—	2	1	no	—
7	#14523979	—	2	1	no	—
8	#14523979	C₄	1	2	yes	—
9	#14523979	C₄	1	2	yes	—
10	C₂²×C₈×C₆₇	C₂²	1	1	yes	characteristic
11	M₃₂×C₆₇	—	2	1	no	—
12	M₃₂×C₆₇	C₄	1	2	yes	—
13	C₂×C₁₆×C₆₇	—	2	1	no	—
14	#11761612	—	2	2	no	—
15	#11761612	—	2	2	no	—
16	#11761612	—	4	1	no	—
17	#11761612	—	4	1	no	—
18	C₂×C₆₇⋊C₈	—	4	1	no	—
19	C₂×C₆₇⋊C₈	—	4	1	no	—
20	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
21	C₂×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
25	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
26	C₂×C₈×C₆₇	Q₈	1	1	yes	characteristic
27	C₁₆×C₆₇	—	2	2	no	—
28	C₂³×C₆₇	Q₁₆	1	1	yes	characteristic
29	C₂×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
30	C₂×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₄×C₆₇	D₈	1	1	yes	characteristic
33	C₂×C₄×C₆₇	SD₁₆	1	1	yes	characteristic
34	C₂×C₄×C₆₇	SD₁₆	1	1	yes	characteristic
35	C₈×C₆₇	C₄²	1	1	yes	characteristic
36	C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
37	C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
38	C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
39	C₆₇⋊C₈	—	4	2	no	—
40	C₆₇⋊C₈	—	4	2	no	—
41	C₂²×C₆₇	—	4	1	no	—
42	C₂²×C₆₇	#103	1	1	yes	characteristic
43	C₂²×C₆₇	#103	1	1	yes	characteristic
44	C₂²×C₆₇	#107	1	1	yes	characteristic
45	C₄×C₆₇	#95	1	1	yes	characteristic
46	C₄×C₆₇	#102	1	1	yes	characteristic
47	C₄×C₆₇	#102	1	1	yes	characteristic
48	C₄×C₆₇	#106	1	1	yes	characteristic
49	C₂×C₆₇	—	2	1	no	—
50	C₂×C₆₇	—	2	1	no	—
51	C₂×C₆₇	—	2	1	no	—
52	C₂×C₆₇	#340	1	1	yes	characteristic
53	#1388	—	67	1	no	maximal, Sylow
54	C₆₇	#1388	1	1	yes	characteristic, complemented, Sylow
55	C₂×M₃₂	D₆₇	1	1	yes	characteristic
56	C₂×#108	—	67	1	no	—
57	C₂×#108	—	67	1	no	—
58	C₂×M₁₆	—	134	1	no	—
59	C₂×M₁₆	—	134	1	no	—
60	C₂²×C₈	C₂×D₆₇	1	1	yes	characteristic
61	M₃₂	—	2	1	no	—
62	M₃₂	Dic₆₇	1	2	yes	—
63	C₂×C₁₆	—	2	1	no	—
64	#108	—	67	2	no	—
65	#108	—	67	2	no	—
66	#108	—	134	1	no	—
67	#108	—	134	1	no	—
68	C₂²×C₄	D₂₆₈	1	1	yes	characteristic
69	M₁₆	—	268	1	no	—
70	M₁₆	—	268	1	no	—
71	M₁₆	—	134	2	no	—
72	M₁₆	—	134	2	no	—
73	C₂×C₈	—	268	1	no	—
74	C₂×C₈	—	268	1	no	—
75	C₂×C₈	Dic₁₃₄	1	1	yes	characteristic
76	C₂×C₈	C₄×D₆₇	1	1	yes	characteristic
77	C₂×C₈	C₄×D₆₇	1	1	yes	characteristic
78	C₂×C₈	C₂×Dic₆₇	1	1	yes	characteristic
79	C₂×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
80	C₂×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
81	C₁₆	—	2	2	no	—
82	C₂³	Dic₂₆₈	1	1	yes	characteristic
83	C₂×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
84	C₂×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
85	C₂×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
86	C₂×C₄	D₅₃₆	1	1	yes	characteristic
87	C₂×C₄	#11761617	1	1	yes	characteristic
88	C₂×C₄	#11761617	1	1	yes	characteristic
89	C₈	—	268	2	no	—
90	C₈	—	268	2	no	—
91	C₈	C₄×Dic₆₇	1	1	yes	characteristic
92	C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
93	C₈	Dic₆₇⋊C₄	1	1	yes	characteristic
94	C₈	Dic₆₇⋊C₄	1	1	yes	characteristic
95	C₂²	—	4	1	no	—
96	C₂²	#14523968	1	1	yes	characteristic
97	C₂²	#14523968	1	1	yes	characteristic
98	C₂²	#14523977	1	1	yes	characteristic
99	C₄	C₆₇⋊#95	1	1	yes	characteristic
100	C₄	#14523965	1	1	yes	characteristic
101	C₄	#14523965	1	1	yes	characteristic
102	C₄	#14523975	1	1	yes	characteristic
103	C₂	—	2	1	no	—
104	C₂	—	2	1	no	—
105	C₂	—	2	1	no	—
106	C₂	#16863159	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14523979

C₂

yes

maximal, characteristic

C₂×#14523979

C₂

yes

maximal, characteristic

C₂×M₃₂×C₆₇

C₂

yes

maximal, characteristic

—

—

—

—

C₄

yes

—

#14523979

C₄

yes

—

C₂²×C₈×C₆₇

C₂²

yes

characteristic

M₃₂×C₆₇

—

M₃₂×C₆₇

C₄

yes

—

C₂×C₁₆×C₆₇

—

—

—

—

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂²×C₄×C₆₇

D₄

yes

characteristic

C₂×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂×C₈×C₆₇

D₄

yes

characteristic

C₂×C₈×C₆₇

D₄

yes

characteristic

C₂×C₈×C₆₇

Q₈

yes

characteristic

C₁₆×C₆₇

—

C₂³×C₆₇

Q₁₆

yes

characteristic

C₂×C₄×C₆₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

D₈

yes

characteristic

C₂×C₄×C₆₇

SD₁₆

yes

characteristic

C₂×C₄×C₆₇

SD₁₆

yes

characteristic

C₈×C₆₇

C₄²

yes

characteristic

C₈×C₆₇

C₂²⋊C₄

yes

characteristic

C₈×C₆₇

C₄⋊C₄

yes

characteristic

C₈×C₆₇

C₄⋊C₄

yes

characteristic

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

C₂×M₃₂

D₆₇

yes

characteristic

C₂×#108

—

C₂×#108

—

C₂×M₁₆

—

134

—

C₂×M₁₆

—

134

—

C₂²×C₈

C₂×D₆₇

yes

characteristic

M₃₂

—

M₃₂

Dic₆₇

yes

—

C₂×C₁₆

—

#108

—

#108

—

#108

—

134

—

#108

—

134

—

C₂²×C₄

D₂₆₈

yes

characteristic

M₁₆

—

268

—

M₁₆

—

268

—

M₁₆

—

134

—

M₁₆

—

134

—

C₂×C₈

—

268

—

C₂×C₈

—

268

—

C₂×C₈

Dic₁₃₄

yes

characteristic

C₂×C₈

C₄×D₆₇

yes

characteristic

C₂×C₈

C₄×D₆₇

yes

characteristic

C₂×C₈

C₂×Dic₆₇

yes

characteristic

C₂×C₈

C₂²⋊D₆₇

yes

characteristic

C₂×C₈

C₂²⋊D₆₇

yes

characteristic

C₁₆

—

C₂³

Dic₂₆₈

yes

characteristic

C₂×C₄

(C₂×D₆₇)⋊C₄

yes

characteristic

C₂×C₄

(C₂×D₆₇)⋊C₄

yes

characteristic

C₂×C₄

C₄⋊Dic₆₇

yes

characteristic

C₂×C₄

D₅₃₆

yes

C₂×C₄

yes

C₂×C₄

yes

C₈

—

268

—

C₈

—

268

—

C₈

C₄×Dic₆₇

yes

characteristic

C₈

C₂²⋊Dic₆₇

yes

characteristic

C₈

Dic₆₇⋊C₄

yes

characteristic

C₈

Dic₆₇⋊C₄

yes

characteristic

C₂²

—

C₂²

yes

C₂²

yes

C₂²

yes

C₄

C₆₇⋊#95

yes

characteristic

100

C₄

#14523965

yes

characteristic

101

C₄

#14523965

yes

characteristic

102

C₄

#14523975

yes

characteristic

103

C₂

—

104

C₂

—

105

C₂

—

106

C₂

#16863159

yes

characteristic

	subgroup	G/H
center	C₄	#14523965
commutator	C₈×C₆₇	C₄²
Frattini	C₂²×C₈	C₂×D₆₇
Fitting	C₂×M₃₂×C₆₇	C₂
socle	C₂×C₆₇	#340
hypercenter	C₂×M₃₂	D₆₇
hypocenter	C₆₇	#1388
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

subgroup

G/H

center

C₄

#14523965

commutator

C₈×C₆₇

C₄²

Frattini

C₂²×C₈

C₂×D₆₇

Fitting

C₂×M₃₂×C₆₇

C₂

socle

C₂×C₆₇

#340

hypercenter

C₂×M₃₂

D₆₇

hypocenter

C₆₇

#1388

LCS

G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

N	H ≅ G/N	type
54. C₆₇	#1388	semidirect

H ≅ G/N

type

54. C₆₇

#1388

semidirect

Subgroups — #22558321

Conjugacy classes of subgroups

Complements