Group #22559347 — subgroups

Subgroups — #22559347

Order 8576.

206 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 57 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	#16864126	C₂	1	1	yes	maximal, characteristic
2	#16863688	C₂	1	2	yes	maximal
3	#16863145	C₂	1	1	yes	maximal, characteristic
4	#16863128	C₂	1	2	yes	maximal
5	#494×C₆₇	C₂	1	1	yes	maximal, characteristic
6	#14524042	—	2	2	no	—
7	#14524041	—	2	2	no	—
8	C₄×Dic₁₃₄	C₂²	1	2	yes	—
9	#14524008	—	2	2	no	—
10	#14523976	—	2	2	no	—
11	C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
12	#14523966	—	2	2	no	—
13	C₂×Q₁₆×C₆₇	—	2	2	no	—
14	C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₃₄	—	2	2	no	—
18	C₂×Dic₁₃₄	D₄	1	2	yes	—
19	C₂×C₆₇⋊C₈	—	2	2	no	—
20	C₄⋊Dic₆₇	D₄	1	2	yes	—
21	Dic₆₇⋊C₄	—	2	4	no	—
22	Dic₆₇⋊C₄	—	2	2	no	—
23	C₄×Dic₆₇	—	2	2	no	—
24	C₄×Dic₆₇	—	2	2	no	—
25	C₂×Q₈×C₆₇	—	2	1	no	—
26	C₂×Q₈×C₆₇	D₄	1	2	yes	—
27	Q₁₆×C₆₇	—	4	4	no	—
28	C₂×C₈×C₆₇	—	2	1	no	—
29	C₄⋊C₄×C₆₇	—	2	2	no	—
30	C₄⋊C₄×C₆₇	—	2	2	no	—
31	C₄²×C₆₇	C₂³	1	1	yes	characteristic
32	Q₈×C₆₇	—	4	2	no	—
33	Q₈×C₆₇	—	4	2	no	—
34	C₂×C₄×C₆₇	—	2	2	no	—
35	C₂×C₄×C₆₇	—	4	1	no	—
36	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
37	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
38	C₂×Dic₆₇	—	2	2	no	—
39	C₂×Dic₆₇	—	2	2	no	—
40	Dic₁₃₄	—	2	2	no	—
41	Dic₁₃₄	—	8	2	no	—
42	Dic₁₃₄	Q₁₆	1	4	yes	complemented
43	C₈×C₆₇	—	2	2	no	—
44	C₆₇⋊C₈	—	4	2	no	—
45	C₂²×C₆₇	#120	1	1	yes	characteristic
46	C₄×C₆₇	—	2	1	no	—
47	C₄×C₆₇	—	4	2	no	—
48	C₄×C₆₇	—	8	1	no	—
49	C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
50	C₄×C₆₇	#137	1	2	yes	—
51	Dic₆₇	—	2	4	no	—
52	Dic₆₇	—	4	2	no	—
53	C₂×C₆₇	#451	1	2	yes	—
54	C₂×C₆₇	#453	1	1	yes	characteristic
55	#1633	—	67	1	no	maximal, Sylow
56	C₆₇	#1633	1	1	yes	characteristic, complemented, Sylow
57	Q₈²	—	67	1	no	—
58	#494	D₆₇	1	1	yes	characteristic
59	#477	—	67	2	no	—
60	#333	—	67	1	no	—
61	Q₈⋊C₈	—	67	2	no	—
62	C₂×Q₁₆	—	2	2	no	—
63	C₄⋊Q₈	—	134	2	no	—
64	C₄⋊Q₈	—	134	2	no	—
65	C₄⋊Q₈	C₂×D₆₇	1	1	yes	characteristic
66	C₄⋊Q₈	C₂×D₆₇	1	1	yes	characteristic
67	C₄×Q₈	—	67	2	no	—
68	C₄×Q₈	—	134	2	no	—
69	#107	—	134	2	no	—
70	C₄⋊C₈	—	67	2	no	—
71	#103	—	134	2	no	—
72	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
73	C₂×Q₈	—	2	1	no	—
74	C₂×Q₈	—	67	2	no	—
75	C₂×Q₈	—	134	2	no	—
76	C₂×Q₈	C₂²⋊D₆₇	1	2	yes	—
77	Q₁₆	—	4	4	no	—
78	C₂×C₈	—	2	1	no	—
79	C₂×C₈	—	134	2	no	—
80	C₄⋊C₄	—	2	2	no	—
81	C₄⋊C₄	—	2	2	no	—
82	C₄⋊C₄	—	67	2	no	—
83	C₄⋊C₄	—	134	4	no	—
84	C₄⋊C₄	—	134	2	no	—
85	C₄²	—	134	2	no	—
86	C₄²	—	134	2	no	—
87	C₄²	C₂²×D₆₇	1	1	yes	characteristic
88	Q₈	—	4	2	no	—
89	Q₈	—	4	2	no	—
90	Q₈	—	536	2	no	—
91	Q₈	—	67	4	no	—
92	Q₈	—	134	2	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	4	1	no	—
95	C₂×C₄	—	134	2	no	—
96	C₂×C₄	—	134	2	no	—
97	C₂×C₄	D₄×D₆₇	1	2	yes	—
98	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
99	C₈	—	2	2	no	—
100	C₈	—	268	2	no	—
101	C₂²	#14524011	1	1	yes	characteristic
102	C₄	—	2	1	no	—
103	C₄	—	4	2	no	—
104	C₄	—	8	1	no	—
105	C₄	—	268	2	no	—
106	C₄	—	134	4	no	—
107	C₄	Q₁₆×D₆₇	1	2	yes	—
108	C₄	#14524075	1	2	yes	—
109	C₂	#16863550	1	2	yes	—
110	C₂	#16863563	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16864126

C₂

yes

maximal, characteristic

#16863688

C₂

yes

maximal

#16863145

C₂

yes

maximal, characteristic

#16863128

C₂

yes

maximal

#494×C₆₇

C₂

yes

maximal, characteristic

#14524042

—

#14524041

—

C₄×Dic₁₃₄

C₂²

yes

—

#14524008

—

#14523976

—

C₄⋊(C₆₇⋊C₈)

C₂²

yes

—

#14523966

—

C₂×Q₁₆×C₆₇

—

C₄⋊Q₈×C₆₇

C₂²

yes

characteristic

C₄⋊Q₈×C₆₇

C₂²

yes

characteristic

C₄×C₈×C₆₇

C₂²

yes

characteristic

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

D₄

yes

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

D₄

yes

—

Dic₆₇⋊C₄

—

Dic₆₇⋊C₄

—

C₄×Dic₆₇

—

C₄×Dic₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

D₄

yes

—

Q₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

C₂³

yes

characteristic

Q₈×C₆₇

—

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

Q₁₆

yes

complemented

C₈×C₆₇

—

C₆₇⋊C₈

—

C₂²×C₆₇

#120

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂×Q₁₆

yes

—

C₄×C₆₇

#137

yes

—

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

#451

yes

—

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

Q₈²

—

#494

D₆₇

yes

characteristic

#477

—

#333

—

Q₈⋊C₈

—

C₂×Q₁₆

—

C₄⋊Q₈

—

134

—

C₄⋊Q₈

—

134

—

C₄⋊Q₈

C₂×D₆₇

yes

characteristic

C₄⋊Q₈

C₂×D₆₇

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

134

—

#107

—

134

—

C₄⋊C₈

—

#103

—

134

—

C₄×C₈

C₂×D₆₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

134

—

C₂×Q₈

C₂²⋊D₆₇

yes

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

134

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

134

—

C₄⋊C₄

—

134

—

C₄²

—

134

—

C₄²

—

134

—

C₄²

C₂²×D₆₇

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

536

—

Q₈

—

Q₈

—

134

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

134

—

C₂×C₄

—

134

—

C₂×C₄

D₄×D₆₇

yes

—

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

C₈

—

100

C₈

—

268

—

101

C₂²

#14524011

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

—

268

—

106

C₄

—

134

—

107

C₄

Q₁₆×D₆₇

yes

—

108

C₄

#14524075

yes

—

109

C₂

#16863550

yes

—

110

C₂

#16863563

yes

characteristic

	subgroup	G/H
center	C₂²	#14524011
commutator	C₄²×C₆₇	C₂³
Frattini	C₄²	C₂²×D₆₇
Fitting	#494×C₆₇	C₂
socle	C₂²×C₆₇	#120
hypercenter	#494	D₆₇
hypocenter	C₆₇	#1633
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

subgroup

G/H

center

C₂²

#14524011

commutator

C₄²×C₆₇

C₂³

Frattini

C₄²

C₂²×D₆₇

Fitting

#494×C₆₇

C₂

C₂²×C₆₇

D₆₇

C₆₇

G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #494

N	H ≅ G/N	type
42. Dic₁₃₄	Q₁₆	semidirect
56. C₆₇	#1633	semidirect

H ≅ G/N

type

42. Dic₁₃₄

Q₁₆

semidirect

56. C₆₇

#1633

semidirect

Subgroups — #22559347

Conjugacy classes of subgroups

Complements