Group #89436 — subgroups

Subgroups — #89436

Order 448.

218 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 71 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	C₂²×C₄×D₇	C₂	1	1	yes	maximal, characteristic
2	C₂×(C₂×D₇)⋊C₄	C₂	1	2	yes	maximal
3	C₇×#95	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇⋊#95	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇⋊#95	C₂	1	2	yes	maximal, complemented
6	C₂³×D₇	C₂²	1	1	yes	characteristic
7	C₂²×C₄×C₇	C₂²	1	2	yes	—
8	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
9	C₂²×Dic₇	C₂²	1	2	yes	—
10	C₂²×Dic₇	C₂²	1	1	yes	characteristic
11	C₂×C₄×D₇	—	2	4	no	—
12	C₂×C₄×D₇	C₄	1	4	yes	complemented
13	(C₂×D₇)⋊C₄	—	2	4	no	—
14	#381	—	7	1	no	maximal, Sylow
15	C₂³×C₇	C₂³	1	1	yes	characteristic
16	C₂²×D₇	—	2	2	no	—
17	C₂²×D₇	—	2	1	no	—
18	C₂²×D₇	—	2	1	no	—
19	C₂²×D₇	C₂×C₄	1	2	yes	—
20	C₂²×D₇	D₄	1	4	yes	—
21	C₂×C₄×C₇	—	2	2	no	—
22	C₂×C₄×C₇	—	2	2	no	—
23	C₂×C₄×C₇	—	2	2	no	—
24	C₂×C₄×C₇	—	2	2	no	—
25	C₂×C₄×C₇	C₂×C₄	1	2	yes	—
26	C₂×Dic₇	—	2	2	no	—
27	C₂×Dic₇	—	2	2	no	—
28	C₂×Dic₇	—	2	2	no	—
29	C₂×Dic₇	—	2	2	no	—
30	C₂×Dic₇	C₂×C₄	1	2	yes	—
31	C₄×D₇	—	2	8	no	—
32	C₂³×C₄	—	7	1	no	—
33	C₂×C₂²⋊C₄	—	7	2	no	—
34	#95	—	7	2	no	—
35	#95	—	7	1	no	—
36	#95	D₇	1	1	yes	characteristic, complemented
37	C₂²×C₇	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₇	C₂×D₄	1	2	yes	—
39	C₂²×C₇	#41	1	2	yes	—
40	C₂²×C₇	#41	1	1	yes	characteristic
41	C₂²×C₇	#41	1	1	yes	characteristic
42	C₂×D₇	—	2	2	no	—
43	C₂×D₇	—	2	2	no	—
44	C₂×D₇	—	2	4	no	—
45	C₂×D₇	—	2	4	no	—
46	C₂×D₇	C₂²⋊C₄	1	4	yes	—
47	C₄×C₇	—	2	2	no	—
48	C₄×C₇	—	4	2	no	—
49	Dic₇	—	2	2	no	—
50	Dic₇	—	4	2	no	—
51	C₂⁴	—	7	1	no	—
52	C₂²×C₄	—	7	1	no	—
53	C₂²×C₄	—	7	4	no	—
54	C₂²×C₄	—	7	2	no	—
55	C₂²×C₄	—	14	4	no	—
56	C₂²×C₄	C₂×D₇	1	2	yes	—
57	C₂²×C₄	C₂×D₇	1	1	yes	characteristic
58	C₂²⋊C₄	—	14	4	no	—
59	C₂×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₂×C₇	#117	1	1	yes	characteristic
61	C₂×C₇	#117	1	1	yes	characteristic
62	C₂×C₇	#123	1	2	yes	—
63	C₂×C₇	#123	1	2	yes	—
64	D₇	—	2	4	no	—
65	C₂³	—	7	2	no	—
66	C₂³	—	7	4	no	—
67	C₂³	—	14	2	no	—
68	C₂³	—	14	1	no	—
69	C₂³	—	14	1	no	—
70	C₂³	C₂²×D₇	1	1	yes	characteristic
71	C₂×C₄	—	2	2	no	—
72	C₂×C₄	—	2	2	no	—
73	C₂×C₄	—	2	2	no	—
74	C₂×C₄	—	2	2	no	—
75	C₂×C₄	—	7	2	no	—
76	C₂×C₄	—	14	8	no	—
77	C₂×C₄	—	14	2	no	—
78	C₂×C₄	—	14	2	no	—
79	C₂×C₄	—	14	2	no	—
80	C₂×C₄	—	14	2	no	—
81	C₂×C₄	C₄×D₇	1	2	yes	—
82	C₇	#381	1	1	yes	characteristic, complemented, Sylow
83	C₂²	—	7	4	no	—
84	C₂²	—	14	2	no	—
85	C₂²	—	14	4	no	—
86	C₂²	—	14	2	no	—
87	C₂²	—	14	4	no	—
88	C₂²	C₂×C₄×D₇	1	1	yes	characteristic
89	C₂²	#1157	1	2	yes	—
90	C₂²	D₄×D₇	1	2	yes	—
91	C₂²	#1159	1	1	yes	characteristic
92	C₂²	#1161	1	1	yes	characteristic
93	C₄	—	2	2	no	—
94	C₄	—	4	2	no	—
95	C₄	—	14	2	no	—
96	C₄	—	28	2	no	—
97	C₂	—	14	4	no	—
98	C₂	#6435	1	1	yes	characteristic
99	C₂	D₇×C₂²⋊C₄	1	1	yes	characteristic
100	C₂	#6446	1	2	yes	—
101	C₂	#6455	1	1	yes	characteristic
102	C₂	#6457	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₄×D₇

C₂

yes

maximal, characteristic

C₂×(C₂×D₇)⋊C₄

C₂

yes

maximal

C₇×#95

C₂

yes

maximal, characteristic, complemented

C₇⋊#95

C₂

yes

maximal, characteristic, complemented

C₇⋊#95

C₂

yes

maximal, complemented

C₂³×D₇

C₂²

yes

characteristic

C₂²×C₄×C₇

C₂²

yes

—

C₂²×C₄×C₇

C₂²

yes

characteristic

C₂²×Dic₇

C₂²

yes

—

C₂²×Dic₇

C₂²

yes

characteristic

C₂×C₄×D₇

—

C₂×C₄×D₇

C₄

yes

complemented

(C₂×D₇)⋊C₄

—

#381

—

maximal, Sylow

C₂³×C₇

C₂³

yes

characteristic

C₂²×D₇

—

C₂²×D₇

—

C₂²×D₇

—

C₂²×D₇

C₂×C₄

yes

—

C₂²×D₇

D₄

yes

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂×C₄

yes

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

C₂×C₄

yes

—

C₄×D₇

—

C₂³×C₄

—

C₂×C₂²⋊C₄

—

#95

—

#95

—

#95

D₇

yes

characteristic, complemented

C₂²×C₇

C₂²×C₄

yes

characteristic

C₂²×C₇

C₂×D₄

yes

—

C₂²×C₇

#41

yes

—

C₂²×C₇

yes

C₂²×C₇

yes

C₂×D₇

—

C₂×D₇

—

C₂×D₇

—

C₂×D₇

—

C₂×D₇

C₂²⋊C₄

yes

—

C₄×C₇

—

C₄×C₇

—

Dic₇

—

Dic₇

—

C₂⁴

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₇

yes

—

C₂²×C₄

C₂×D₇

yes

characteristic

C₂²⋊C₄

—

C₂×C₇

C₂×C₂²⋊C₄

yes

C₂×C₇

yes

C₂×C₇

yes

C₂×C₇

yes

—

C₂×C₇

#123

yes

—

D₇

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂²×D₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×D₇

yes

—

C₇

#381

yes

characteristic, complemented, Sylow

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

C₂×C₄×D₇

yes

characteristic

C₂²

#1157

yes

—

C₂²

D₄×D₇

yes

—

C₂²

yes

C₂²

yes

C₄

—

C₄

—

C₄

—

C₄

—

C₂

—

C₂

#6435

yes

characteristic

C₂

D₇×C₂²⋊C₄

yes

characteristic

100

C₂

#6446

yes

—

101

C₂

#6455

yes

characteristic

102

C₂

#6457

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₇
commutator	C₂²×C₇	C₂²×C₄
Fitting	C₇×#95	C₂
socle, cosocle	C₂³×C₇	C₂³
prime radical	C₂³×D₇	C₂²
hypercenter	#95	D₇
hypocenter	C₇	#381
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ #95

subgroup

G/H

center, Frattini

C₂³

C₂²×D₇

commutator

C₂²×C₇

C₂²×C₄

Fitting

C₇×#95

C₂

socle, cosocle

C₂³×C₇

C₂³

prime radical

C₂³×D₇

C₂²

D₇

C₇

G ▷ C₂²×C₇ ▷ C₇

C₁ ◁ C₂³ ◁ #95

N	H ≅ G/N	type
3. C₇×#95	C₂	semidirect
4. C₇⋊#95	C₂	semidirect
5. C₇⋊#95	C₂	semidirect
12. C₂×C₄×D₇	C₄	semidirect
36. #95	D₇	semidirect
82. C₇	#381	semidirect

H ≅ G/N

type

3. C₇×#95

C₂

semidirect

4. C₇⋊#95

C₂

semidirect

5. C₇⋊#95

C₂

semidirect

12. C₂×C₄×D₇

C₄

semidirect

36. #95

D₇

semidirect

82. C₇

#381

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₄×D₇	2
C₂×(C₂×D₇)⋊C₄	2
C₇×#95	2
C₇⋊#95	2
#381	7

subgroup

index

C₂²×C₄×D₇

C₂×(C₂×D₇)⋊C₄

C₇×#95

C₇⋊#95

#381

Subgroups — #89436

Conjugacy classes of subgroups

Complements

Maximal subgroups