Group #22558357 — subgroups

Subgroups — #22558357

Order 8576.

112 conjugacy classes of subgroups; only the 25 normal subgroups are shown.

normal subgroups only

This group has 112 conjugacy classes of subgroups — too many to draw. Only its 25 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	#16863193	C₂	1	1	yes	maximal, characteristic, complemented
2	#16863136	C₂	1	1	yes	maximal, characteristic, complemented
3	#455×C₆₇	C₂	1	1	yes	maximal, characteristic
4	#14523962	—	2	1	no	—
5	#14523954	—	2	1	no	—
6	H(5,𝔽₂)×C₆₇	C₄	1	1	yes	characteristic
7	#137×C₆₇	—	2	1	no	—
8	#124×C₆₇	C₂²	1	1	yes	characteristic
9	#104×C₆₇	—	2	1	no	—
10	#100×C₆₇	C₄	1	1	yes	characteristic
11	#11761612	—	4	1	no	—
12	C₂×C₆₇⋊C₈	—	4	1	no	—
13	#41×C₆₇	—	2	1	no	—
14	#41×C₆₇	—	4	1	no	—
15	C₂×Q₈×C₆₇	D₄	1	1	yes	characteristic
16	C₂×D₄×C₆₇	—	4	1	no	—
17	C₂×D₄×C₆₇	—	4	1	no	—
18	C₂×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
19	Q₁₆×C₆₇	—	4	1	no	—
20	SD₁₆×C₆₇	—	4	1	no	—
21	M₁₆×C₆₇	—	2	1	no	—
22	C₂²⋊C₄×C₆₇	—	4	1	no	—
23	C₄²×C₆₇	D₄	1	1	yes	characteristic
24	C₂³×C₆₇	—	2	1	no	—
25	C₂³×C₆₇	—	4	1	no	—
26	Q₈×C₆₇	—	2	1	no	—
27	Q₈×C₆₇	—	4	1	no	—
28	D₄×C₆₇	—	2	1	no	—
29	D₄×C₆₇	—	4	1	no	—
30	D₄×C₆₇	—	4	2	no	—
31	D₄×C₆₇	—	4	1	no	—
32	C₂×C₄×C₆₇	—	2	1	no	—
33	C₂×C₄×C₆₇	—	2	1	no	—
34	C₂×C₄×C₆₇	—	4	1	no	—
35	C₂×C₄×C₆₇	—	4	1	no	—
36	C₂×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
37	C₈×C₆₇	—	4	1	no	—
38	C₆₇⋊C₈	—	4	2	no	—
39	C₆₇⋊C₈	—	8	1	no	—
40	C₂²×C₆₇	—	4	1	no	—
41	C₂²×C₆₇	—	4	1	no	—
42	C₂²×C₆₇	—	8	1	no	—
43	C₂²×C₆₇	—	16	1	no	—
44	C₂²×C₆₇	#99	1	1	yes	characteristic
45	C₄×C₆₇	—	2	1	no	—
46	C₄×C₆₇	—	4	1	no	—
47	C₄×C₆₇	—	4	1	no	—
48	C₄×C₆₇	—	4	1	no	—
49	C₂×C₆₇	—	2	1	no	—
50	C₂×C₆₇	—	8	1	no	—
51	C₂×C₆₇	—	8	1	no	—
52	C₂×C₆₇	AΣL(1,𝔽₁₆)	1	1	yes	characteristic
53	#1403	—	67	1	no	maximal, Sylow
54	C₆₇	#1403	1	1	yes	characteristic, complemented, Sylow
55	#455	D₆₇	1	1	yes	characteristic
56	#355	—	67	1	no	—
57	#329	—	67	1	no	—
58	H(5,𝔽₂)	Dic₆₇	1	1	yes	characteristic
59	#137	—	2	1	no	—
60	#124	C₂×D₆₇	1	1	yes	characteristic
61	#104	—	2	1	no	—
62	#101	—	134	1	no	—
63	#100	Dic₆₇	1	1	yes	characteristic
64	#97	—	134	1	no	—
65	#41	—	2	1	no	—
66	#41	—	4	1	no	—
67	C₂×Q₈	C₂²⋊D₆₇	1	1	yes	characteristic
68	C₂×D₄	—	4	1	no	—
69	C₂×D₄	—	4	1	no	—
70	C₂×D₄	C₂×Dic₆₇	1	1	yes	characteristic
71	Q₁₆	—	4	1	no	—
72	SD₁₆	—	4	1	no	—
73	M₁₆	—	2	1	no	—
74	M₁₆	—	268	1	no	—
75	C₂×C₈	—	268	1	no	—
76	C₂²⋊C₄	—	4	1	no	—
77	C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
78	C₂³	—	2	1	no	—
79	C₂³	—	4	1	no	—
80	Q₈	—	2	1	no	—
81	Q₈	—	4	1	no	—
82	D₄	—	2	1	no	—
83	D₄	—	4	1	no	—
84	D₄	—	4	2	no	—
85	D₄	—	4	1	no	—
86	C₂×C₄	—	2	1	no	—
87	C₂×C₄	—	2	1	no	—
88	C₂×C₄	—	4	1	no	—
89	C₂×C₄	—	4	1	no	—
90	C₂×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
91	C₈	—	4	1	no	—
92	C₈	—	268	2	no	—
93	C₈	—	536	1	no	—
94	C₂²	—	4	1	no	—
95	C₂²	—	4	1	no	—
96	C₂²	—	8	1	no	—
97	C₂²	—	16	1	no	—
98	C₂²	#14523958	1	1	yes	characteristic
99	C₄	—	2	1	no	—
100	C₄	—	4	1	no	—
101	C₄	—	4	1	no	—
102	C₄	—	4	1	no	—
103	C₂	—	2	1	no	—
104	C₂	—	8	1	no	—
105	C₂	—	8	1	no	—
106	C₂	#16863183	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863193

C₂

yes

maximal, characteristic, complemented

#16863136

C₂

yes

maximal, characteristic, complemented

#455×C₆₇

C₂

yes

maximal, characteristic

#14523962

—

#14523954

—

H(5,𝔽₂)×C₆₇

C₄

yes

characteristic

#137×C₆₇

—

#124×C₆₇

C₂²

yes

characteristic

#104×C₆₇

—

#100×C₆₇

C₄

yes

characteristic

#11761612

—

C₂×C₆₇⋊C₈

—

#41×C₆₇

—

#41×C₆₇

—

C₂×Q₈×C₆₇

D₄

yes

characteristic

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

C₂×C₄

yes

characteristic

Q₁₆×C₆₇

—

SD₁₆×C₆₇

—

M₁₆×C₆₇

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

D₄

yes

characteristic

C₂³×C₆₇

—

C₂³×C₆₇

—

Q₈×C₆₇

—

Q₈×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂²⋊C₄

yes

characteristic

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#99

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

AΣL(1,𝔽₁₆)

yes

—

C₆₇

yes

characteristic, complemented, Sylow

#455

D₆₇

yes

—

—

Dic₆₇

yes

characteristic

#137

—

#124

C₂×D₆₇

yes

characteristic

#104

—

#101

—

134

—

#100

Dic₆₇

yes

characteristic

#97

—

134

—

#41

—

#41

—

C₂×Q₈

C₂²⋊D₆₇

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×Dic₆₇

yes

characteristic

Q₁₆

—

SD₁₆

—

M₁₆

—

M₁₆

—

268

—

C₂×C₈

—

268

—

C₂²⋊C₄

—

C₄²

C₂²⋊D₆₇

yes

characteristic

C₂³

—

C₂³

—

Q₈

—

Q₈

—

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊Dic₆₇

yes

characteristic

C₈

—

C₈

—

268

—

C₈

—

536

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

#14523958

yes

characteristic

C₄

—

100

C₄

—

101

C₄

—

102

C₄

—

103

C₂

—

104

C₂

—

105

C₂

—

106

C₂

#16863183

yes

characteristic

	subgroup	G/H
center	C₂	#16863183
commutator	C₂×D₄×C₆₇	C₂×C₄
Frattini	#124	C₂×D₆₇
Fitting	#455×C₆₇	C₂
socle	C₂×C₆₇	AΣL(1,𝔽₁₆)
hypercenter	#455	D₆₇
hypocenter	C₆₇	#1403
LCS	G ▷ C₂×D₄×C₆₇ ▷ C₂×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ #124 ◁ #455

subgroup

G/H

center

C₂

#16863183

commutator

C₂×D₄×C₆₇

C₂×C₄

Frattini

#124

C₂×D₆₇

Fitting

#455×C₆₇

C₂

socle

C₂×C₆₇

AΣL(1,𝔽₁₆)

D₆₇

C₆₇

G ▷ C₂×D₄×C₆₇ ▷ C₂×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ #124 ◁ #455

N	H ≅ G/N	type
1. #16863193	C₂	semidirect
2. #16863136	C₂	semidirect
54. C₆₇	#1403	semidirect

H ≅ G/N

type

1. #16863193

C₂

semidirect

2. #16863136

C₂

semidirect

54. C₆₇

#1403

semidirect

Subgroups — #22558357

Conjugacy classes of subgroups

Complements