Group #22559124 — subgroups

Subgroups — #22559124

Order 8576.

176 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 65 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	C₄×D₅₃₆	C₂	1	1	yes	maximal, characteristic
2	#16863475	C₂	1	2	yes	maximal, complemented
3	#16863147	C₂	1	1	yes	maximal, characteristic, complemented
4	#16863125	C₂	1	2	yes	maximal, complemented
5	#322×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₅₃₆	C₄	1	1	yes	characteristic
7	C₂×#11761611	—	2	2	no	—
8	C₄×D₂₆₈	C₂²	1	2	yes	—
9	#14523977	C₄	1	1	yes	characteristic
10	C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
11	#14523965	C₄	1	2	yes	—
12	(C₂×D₆₇)⋊C₈	—	2	2	no	—
13	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×D₂₆₈	C₂×C₄	1	2	yes	—
17	C₂×C₄×D₆₇	—	2	2	no	—
18	D₅₃₆	—	2	2	no	—
19	#11761611	—	4	4	no	—
20	C₂×C₆₇⋊C₈	—	2	2	no	—
21	C₄⋊Dic₆₇	C₂×C₄	1	2	yes	—
22	(C₂×D₆₇)⋊C₄	—	2	2	no	—
23	C₂×C₈×C₆₇	—	2	1	no	—
24	C₂×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
26	C₂×C₈×C₆₇	D₄	1	2	yes	—
27	C₄²×C₆₇	C₂³	1	1	yes	characteristic
28	C₂²×D₆₇	—	2	2	no	—
29	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
30	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
32	C₂×Dic₆₇	—	2	2	no	—
33	D₂₆₈	—	2	2	no	—
34	D₂₆₈	—	2	2	no	—
35	C₄×D₆₇	—	4	2	no	—
36	C₄×D₆₇	—	4	2	no	—
37	C₈×C₆₇	—	2	1	no	—
38	C₈×C₆₇	—	2	2	no	—
39	C₈×C₆₇	—	4	1	no	—
40	C₈×C₆₇	M₁₆	1	2	yes	complemented
41	C₆₇⋊C₈	—	4	2	no	—
42	C₂²×C₆₇	C₄×D₄	1	1	yes	characteristic
43	C₂×D₆₇	—	4	2	no	—
44	C₂×D₆₇	—	4	2	no	—
45	C₂×D₆₇	—	4	2	no	—
46	C₄×C₆₇	—	2	1	no	—
47	C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
48	C₄×C₆₇	#131	1	1	yes	characteristic
49	C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
50	Dic₆₇	—	4	2	no	—
51	C₂×C₆₇	#436	1	1	yes	characteristic
52	C₂×C₆₇	#442	1	1	yes	characteristic
53	C₂×C₆₇	#444	1	1	yes	characteristic
54	D₆₇	—	8	2	no	—
55	#1594	—	67	1	no	maximal, Sylow
56	C₆₇	#1594	1	1	yes	characteristic, complemented, Sylow
57	C₄×D₈	—	67	1	no	—
58	#436	—	67	2	no	—
59	#334	—	67	1	no	—
60	D₄⋊C₈	—	67	2	no	—
61	#322	D₆₇	1	1	yes	characteristic, complemented
62	C₂×D₈	—	67	1	no	—
63	C₂×M₁₆	—	134	2	no	—
64	C₄×D₄	—	67	2	no	—
65	#107	—	67	1	no	—
66	C₄⋊C₈	—	67	2	no	—
67	#102	—	67	2	no	—
68	C₂²⋊C₈	—	134	2	no	—
69	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
70	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
71	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
72	C₂×D₄	—	67	2	no	—
73	C₂²×C₄	—	134	2	no	—
74	D₈	—	134	2	no	—
75	M₁₆	—	268	4	no	—
76	C₂×C₈	—	2	1	no	—
77	C₂×C₈	—	134	2	no	—
78	C₂×C₈	C₄×D₆₇	1	1	yes	characteristic
79	C₂×C₈	C₄×D₆₇	1	1	yes	characteristic
80	C₂×C₈	D₂₆₈	1	2	yes	—
81	C₄⋊C₄	—	67	2	no	—
82	C₂²⋊C₄	—	134	2	no	—
83	C₄²	C₂²×D₆₇	1	1	yes	characteristic
84	C₂³	—	134	2	no	—
85	D₄	—	134	2	no	—
86	D₄	—	134	2	no	—
87	C₂×C₄	—	268	2	no	—
88	C₂×C₄	—	268	2	no	—
89	C₂×C₄	—	134	2	no	—
90	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
91	C₂×C₄	C₂×D₂₆₈	1	1	yes	characteristic
92	C₂×C₄	#11761628	1	1	yes	characteristic
93	C₈	—	2	1	no	—
94	C₈	—	2	2	no	—
95	C₈	—	4	1	no	—
96	C₈	—	268	2	no	—
97	C₈	#11761611	1	2	yes	complemented
98	C₂²	—	268	2	no	—
99	C₂²	—	268	2	no	—
100	C₂²	—	268	2	no	—
101	C₂²	C₄×D₂₆₈	1	1	yes	characteristic
102	C₄	—	2	1	no	—
103	C₄	—	268	2	no	—
104	C₄	C₂×#11761611	1	1	yes	characteristic
105	C₄	#14524049	1	1	yes	characteristic
106	C₄	#14524071	1	2	yes	—
107	C₂	—	536	2	no	—
108	C₂	#16863475	1	1	yes	characteristic
109	C₂	#16863509	1	1	yes	characteristic
110	C₂	#16863519	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×D₅₃₆

C₂

yes

maximal, characteristic

#16863475

C₂

yes

maximal, complemented

#16863147

C₂

yes

maximal, characteristic, complemented

#16863125

C₂

yes

maximal, complemented

#322×C₆₇

C₂

yes

maximal, characteristic, complemented

C₂×D₅₃₆

C₄

yes

characteristic

C₂×#11761611

—

C₄×D₂₆₈

C₂²

yes

—

#14523977

C₄

yes

characteristic

C₄⋊(C₆₇⋊C₈)

C₂²

yes

—

#14523965

C₄

yes

—

(C₂×D₆₇)⋊C₈

—

C₄×C₈×C₆₇

C₂²

yes

characteristic

C₄×C₈×C₆₇

C₂²

yes

characteristic

C₄×C₈×C₆₇

C₂²

yes

characteristic

C₂×D₂₆₈

C₂×C₄

yes

—

C₂×C₄×D₆₇

—

D₅₃₆

—

#11761611

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

C₂×C₄

yes

—

(C₂×D₆₇)⋊C₄

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂×C₈×C₆₇

D₄

yes

—

C₄²×C₆₇

C₂³

yes

characteristic

C₂²×D₆₇

—

C₂×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

#41

yes

characteristic

C₂×Dic₆₇

—

D₂₆₈

—

D₂₆₈

—

C₄×D₆₇

—

C₄×D₆₇

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

M₁₆

yes

complemented

C₆₇⋊C₈

—

C₂²×C₆₇

C₄×D₄

yes

characteristic

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂×M₁₆

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

—

Dic₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

D₆₇

—

#1594

—

maximal, Sylow

C₆₇

#1594

yes

characteristic, complemented, Sylow

C₄×D₈

—

#436

—

#334

—

D₄⋊C₈

—

#322

D₆₇

yes

characteristic, complemented

C₂×D₈

—

C₂×M₁₆

—

134

—

C₄×D₄

—

#107

—

C₄⋊C₈

—

#102

—

C₂²⋊C₈

—

134

—

C₄×C₈

C₂×D₆₇

yes

characteristic

C₄×C₈

C₂×D₆₇

yes

characteristic

C₄×C₈

C₂×D₆₇

yes

characteristic

C₂×D₄

—

C₂²×C₄

—

134

—

D₈

—

134

—

M₁₆

—

268

—

C₂×C₈

—

C₂×C₈

—

134

—

C₂×C₈

C₄×D₆₇

yes

characteristic

C₂×C₈

C₄×D₆₇

yes

characteristic

C₂×C₈

D₂₆₈

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

134

—

C₄²

C₂²×D₆₇

yes

characteristic

C₂³

—

134

—

D₄

—

134

—

D₄

—

134

—

C₂×C₄

—

268

—

C₂×C₄

—

268

—

C₂×C₄

—

134

—

C₂×C₄

C₂×C₄×D₆₇

yes

characteristic

C₂×C₄

C₂×D₂₆₈

yes

characteristic

C₂×C₄

#11761628

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

—

268

—

C₈

#11761611

yes

complemented

C₂²

—

268

—

C₂²

—

268

—

100

C₂²

—

268

—

101

C₂²

C₄×D₂₆₈

yes

characteristic

102

C₄

—

103

C₄

—

268

—

104

C₄

C₂×#11761611

yes

characteristic

105

C₄

#14524049

yes

characteristic

106

C₄

#14524071

yes

—

107

C₂

—

536

—

108

C₂

#16863475

yes

characteristic

109

C₂

#16863509

yes

characteristic

110

C₂

#16863519

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂²×C₄
Frattini	C₄²	C₂²×D₆₇
Fitting	#322×C₆₇	C₂
socle	C₂²×C₆₇	C₄×D₄
hypercenter	#322	D₆₇
hypocenter	C₆₇	#1594
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #322

subgroup

G/H

center

C₂×C₄

C₂×D₂₆₈

commutator

C₂×C₄×C₆₇

C₂²×C₄

Frattini

C₄²

C₂²×D₆₇

Fitting

#322×C₆₇

C₂

socle

C₂²×C₆₇

C₄×D₄

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #322

N	H ≅ G/N	type
2. #16863475	C₂	semidirect
3. #16863147	C₂	semidirect
4. #16863125	C₂	semidirect
5. #322×C₆₇	C₂	semidirect
40. C₈×C₆₇	M₁₆	semidirect
56. C₆₇	#1594	semidirect
61. #322	D₆₇	semidirect
97. C₈	#11761611	semidirect

H ≅ G/N

type

2. #16863475

C₂

semidirect

3. #16863147

C₂

semidirect

4. #16863125

C₂

semidirect

5. #322×C₆₇

C₂

semidirect

40. C₈×C₆₇

M₁₆

semidirect

56. C₆₇

#1594

semidirect

61. #322

D₆₇

semidirect

97. C₈

#11761611

semidirect

Subgroups — #22559124

Conjugacy classes of subgroups

Complements