Group #22566169 — subgroups

Subgroups — #22566169

Order 8576.

420 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 133 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	#16863325	C₂	1	6	yes	maximal, complemented
2	#16863320	C₂	1	2	yes	maximal, complemented
3	#16863298	C₂	1	3	yes	maximal
4	#16863294	C₂	1	3	yes	maximal, complemented
5	#376×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₂₆₈	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₆₇	C₂²	1	3	yes	—
8	C₂×Dic₆₇⋊C₄	C₂²	1	6	yes	—
9	C₂×(C₂×D₆₇)⋊C₄	C₂²	1	6	yes	—
10	C₂×(C₂×D₆₇)⋊C₄	C₂²	1	6	yes	—
11	C₆₇⋊#95	C₂²	1	6	yes	—
12	C₂×C₄²×C₆₇	C₂²	1	3	yes	—
13	#95×C₆₇	C₂²	1	1	yes	characteristic
14	#95×C₆₇	C₂²	1	3	yes	—
15	C₂³×D₆₇	C₂³	1	2	yes	—
16	C₂×D₂₆₈	—	2	6	no	—
17	C₂²×Dic₆₇	C₂³	1	6	yes	—
18	C₄⋊Dic₆₇	—	2	6	no	—
19	Dic₆₇⋊C₄	—	4	6	no	—
20	(C₂×D₆₇)⋊C₄	—	4	6	no	—
21	(C₂×D₆₇)⋊C₄	—	4	6	no	—
22	C₂²×C₄×C₆₇	C₂³	1	3	yes	—
23	C₂²×C₄×C₆₇	C₂³	1	3	yes	—
24	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
25	C₄²×C₆₇	—	2	6	no	—
26	C₂³×C₆₇	C₂⁴	1	1	yes	characteristic
27	C₂²×D₆₇	—	2	6	no	—
28	C₂²×D₆₇	—	2	6	no	—
29	C₂²×D₆₇	—	2	2	no	—
30	C₂×C₄×C₆₇	—	2	3	no	—
31	C₂×C₄×C₆₇	—	2	3	no	—
32	C₂×C₄×C₆₇	—	2	3	no	—
33	C₂×C₄×C₆₇	—	2	3	no	—
34	C₂×C₄×C₆₇	—	2	3	no	—
35	C₂×C₄×C₆₇	—	2	3	no	—
36	C₂×C₄×C₆₇	#41	1	6	yes	—
37	C₂×Dic₆₇	—	2	6	no	—
38	C₂×Dic₆₇	—	2	6	no	—
39	C₂×Dic₆₇	—	2	6	no	—
40	D₂₆₈	—	4	4	no	—
41	C₂²×C₆₇	C₂×#41	1	3	yes	—
42	C₂²×C₆₇	H(5,𝔽₂)	1	3	yes	—
43	C₂²×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
44	C₂×D₆₇	—	4	2	no	—
45	C₂×D₆₇	—	4	6	no	—
46	C₂×D₆₇	—	4	6	no	—
47	C₄×C₆₇	—	4	3	no	—
48	C₄×C₆₇	—	4	3	no	—
49	C₄×C₆₇	#126	1	4	yes	—
50	Dic₆₇	—	4	6	no	—
51	C₂×C₆₇	C₂×#126	1	1	yes	characteristic
52	C₂×C₆₇	#540	1	3	yes	—
53	C₂×C₆₇	#542	1	3	yes	—
54	D₆₇	—	8	2	no	—
55	#2644	—	67	1	no	maximal, Sylow
56	C₆₇	#2644	1	1	yes	characteristic, complemented, Sylow
57	#397	—	67	6	no	—
58	#396	—	67	2	no	—
59	#390	—	67	3	no	—
60	#389	—	67	3	no	—
61	#376	D₆₇	1	1	yes	characteristic, complemented
62	C₂²×D₄	—	67	1	no	—
63	C₂×C₄⋊C₄	—	67	6	no	—
64	C₂×C₄⋊C₄	—	67	3	no	—
65	C₂×C₂²⋊C₄	—	67	6	no	—
66	C₂×C₂²⋊C₄	—	67	6	no	—
67	C₂×C₄²	C₂×D₆₇	1	3	yes	—
68	#95	—	67	6	no	—
69	#95	C₂×D₆₇	1	1	yes	characteristic
70	#95	C₂×D₆₇	1	3	yes	—
71	C₂⁴	—	67	2	no	—
72	C₂×D₄	—	134	6	no	—
73	C₂²×C₄	—	67	6	no	—
74	C₂²×C₄	C₂²×D₆₇	1	3	yes	—
75	C₂²×C₄	C₂²×D₆₇	1	3	yes	—
76	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
77	C₄⋊C₄	—	268	6	no	—
78	C₄⋊C₄	—	134	6	no	—
79	C₂²⋊C₄	—	268	6	no	—
80	C₂²⋊C₄	—	268	6	no	—
81	C₄²	—	2	6	no	—
82	C₂³	—	134	6	no	—
83	C₂³	—	134	6	no	—
84	C₂³	—	134	2	no	—
85	C₂³	C₂³×D₆₇	1	1	yes	characteristic
86	D₄	—	268	4	no	—
87	C₂×C₄	—	2	3	no	—
88	C₂×C₄	—	2	3	no	—
89	C₂×C₄	—	2	3	no	—
90	C₂×C₄	—	2	3	no	—
91	C₂×C₄	—	2	3	no	—
92	C₂×C₄	—	2	3	no	—
93	C₂×C₄	—	134	6	no	—
94	C₂×C₄	—	134	6	no	—
95	C₂×C₄	—	134	6	no	—
96	C₂×C₄	#11761628	1	6	yes	—
97	C₂²	—	268	6	no	—
98	C₂²	—	268	6	no	—
99	C₂²	—	268	2	no	—
100	C₂²	C₂×#11761628	1	3	yes	—
101	C₂²	#14524089	1	3	yes	—
102	C₂²	#14524090	1	1	yes	characteristic
103	C₄	—	4	3	no	—
104	C₄	—	4	3	no	—
105	C₄	—	268	6	no	—
106	C₄	#14524037	1	4	yes	—
107	C₂	—	536	2	no	—
108	C₂	C₂×#14524037	1	1	yes	characteristic
109	C₂	#16864001	1	3	yes	—
110	C₂	#16864017	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16863325

C₂

yes

maximal, complemented

#16863320

C₂

yes

maximal, complemented

#16863298

C₂

yes

maximal

#16863294

C₂

yes

maximal, complemented

#376×C₆₇

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₆₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₆₇

C₂²

yes

—

C₂×Dic₆₇⋊C₄

C₂²

yes

—

C₂×(C₂×D₆₇)⋊C₄

C₂²

yes

—

C₂×(C₂×D₆₇)⋊C₄

C₂²

yes

—

C₆₇⋊#95

C₂²

yes

—

C₂×C₄²×C₆₇

C₂²

yes

—

#95×C₆₇

C₂²

yes

characteristic

#95×C₆₇

C₂²

yes

—

C₂³×D₆₇

C₂³

yes

—

C₂×D₂₆₈

—

C₂²×Dic₆₇

C₂³

yes

—

C₄⋊Dic₆₇

—

Dic₆₇⋊C₄

—

(C₂×D₆₇)⋊C₄

—

(C₂×D₆₇)⋊C₄

—

C₂²×C₄×C₆₇

C₂³

yes

—

C₂²×C₄×C₆₇

C₂³

yes

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₄²×C₆₇

—

C₂³×C₆₇

C₂⁴

yes

characteristic

C₂²×D₆₇

—

C₂²×D₆₇

—

C₂²×D₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

#41

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

D₂₆₈

—

C₂²×C₆₇

C₂×#41

yes

—

C₂²×C₆₇

H(5,𝔽₂)

yes

—

C₂²×C₆₇

2₋¹⁺⁴

yes

characteristic

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#126

yes

—

Dic₆₇

—

C₂×C₆₇

C₂×#126

yes

characteristic

C₂×C₆₇

#540

yes

—

C₂×C₆₇

#542

yes

—

D₆₇

—

#2644

—

maximal, Sylow

C₆₇

#2644

yes

characteristic, complemented, Sylow

—

—

—

—

D₆₇

yes

characteristic, complemented

C₂²×D₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₆₇

yes

—

#95

—

#95

C₂×D₆₇

yes

characteristic

#95

C₂×D₆₇

yes

—

C₂⁴

—

C₂×D₄

—

134

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₆₇

yes

—

C₂²×C₄

C₂²×D₆₇

yes

—

C₂²×C₄

C₂²×D₆₇

yes

characteristic

C₄⋊C₄

—

268

—

C₄⋊C₄

—

134

—

C₂²⋊C₄

—

268

—

C₂²⋊C₄

—

268

—

C₄²

—

C₂³

—

134

—

C₂³

—

134

—

C₂³

—

134

—

C₂³

C₂³×D₆₇

yes

characteristic

D₄

—

268

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

134

—

C₂×C₄

—

134

—

C₂×C₄

—

134

—

C₂×C₄

#11761628

yes

—

C₂²

—

268

—

C₂²

—

268

—

C₂²

—

268

—

100

C₂²

C₂×#11761628

yes

—

101

C₂²

#14524089

yes

—

102

C₂²

#14524090

yes

characteristic

103

C₄

—

104

C₄

—

105

C₄

—

268

—

106

C₄

#14524037

yes

—

107

C₂

—

536

—

108

C₂

C₂×#14524037

yes

characteristic

109

C₂

#16864001

yes

—

110

C₂

#16864017

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₆₇
commutator, socle	C₂³×C₆₇	C₂⁴
Fitting	#376×C₆₇	C₂
hypercenter	#376	D₆₇
hypocenter	C₆₇	#2644
LCS	G ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ #376

subgroup

G/H

center, Frattini

C₂³

C₂³×D₆₇

commutator, socle

C₂³×C₆₇

C₂⁴

Fitting

#376×C₆₇

C₂

D₆₇

C₆₇

G ▷ C₂³×C₆₇ ▷ C₆₇

C₁ ◁ C₂³ ◁ #376

N	H ≅ G/N	type
1. #16863325	C₂	semidirect
2. #16863320	C₂	semidirect
4. #16863294	C₂	semidirect
5. #376×C₆₇	C₂	semidirect
56. C₆₇	#2644	semidirect
61. #376	D₆₇	semidirect

H ≅ G/N

type

1. #16863325

C₂

semidirect

2. #16863320

C₂

semidirect

4. #16863294

C₂

semidirect

5. #376×C₆₇

C₂

semidirect

56. C₆₇

#2644

semidirect

61. #376

D₆₇

semidirect

Subgroups — #22566169

Conjugacy classes of subgroups

Complements