Group #90418 — subgroups

Subgroups — #90418

Order 448.

280 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 115 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	C₂×Q₈×D₇	C₂	1	2	yes	maximal, complemented
2	C₇×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#6509	C₂	1	2	yes	maximal, complemented
4	#6459	C₂	1	4	yes	maximal, complemented
5	#6456	C₂	1	2	yes	maximal
6	#6451	C₂	1	2	yes	maximal, complemented
7	C₄×D₂₈	C₂	1	1	yes	maximal, characteristic
8	C₄×Dic₁₄	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₇×Q₈	C₂²	1	2	yes	—
10	Q₈×D₇	—	2	8	no	—
11	C₂×D₂₈	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₇	C₂²	1	2	yes	—
13	C₂×C₄×D₇	C₂²	1	4	yes	—
14	C₂×Dic₁₄	C₂²	1	4	yes	—
15	C₂×Dic₁₄	C₂²	1	1	yes	characteristic
16	C₇×C₄⋊C₄	C₂²	1	2	yes	—
17	C₇×C₄⋊C₄	C₂²	1	2	yes	—
18	C₄²×C₇	C₂²	1	1	yes	characteristic
19	(C₂×D₇)⋊C₄	C₂²	1	2	yes	—
20	(C₂×D₇)⋊C₄	C₂²	1	4	yes	—
21	C₄⋊Dic₇	C₂²	1	1	yes	characteristic
22	C₄⋊Dic₇	C₂²	1	1	yes	characteristic
23	Dic₇⋊C₄	C₂²	1	2	yes	—
24	Dic₇⋊C₄	C₂²	1	4	yes	—
25	C₄×Dic₇	C₂²	1	2	yes	—
26	D₄×Q₈	—	7	1	no	maximal, Sylow
27	C₂²×D₇	C₂³	1	2	yes	—
28	C₇×Q₈	—	2	4	no	—
29	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
30	C₂×C₄×C₇	C₂³	1	4	yes	—
31	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
32	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
33	C₂×Dic₇	C₂³	1	2	yes	—
34	C₂×Dic₇	C₂³	1	2	yes	—
35	C₂×Dic₇	C₂³	1	2	yes	—
36	D₂₈	Q₈	1	4	yes	complemented
37	C₄×D₇	—	2	4	no	—
38	C₄×D₇	—	2	8	no	—
39	Dic₁₄	—	2	8	no	—
40	Dic₁₄	D₄	1	4	yes	complemented
41	C₂²×Q₈	—	7	2	no	—
42	C₄⋊Q₈	—	7	2	no	—
43	C₄⋊Q₈	D₇	1	1	yes	characteristic, complemented
44	C₂²⋊Q₈	—	7	2	no	—
45	C₂²⋊Q₈	—	7	4	no	—
46	C₄×Q₈	—	7	1	no	—
47	C₄×D₄	—	7	2	no	—
48	C₄×D₄	—	7	1	no	—
49	C₂²×C₇	C₂⁴	1	1	yes	characteristic
50	C₂×D₇	—	2	2	no	—
51	C₂×D₇	—	2	2	no	—
52	C₂×D₇	C₂×Q₈	1	4	yes	complemented
53	C₄×C₇	—	2	4	no	—
54	C₄×C₇	—	2	1	no	—
55	C₄×C₇	C₂×D₄	1	2	yes	—
56	C₄×C₇	C₂×Q₈	1	2	yes	complemented
57	Dic₇	—	2	2	no	—
58	Dic₇	—	2	2	no	—
59	Dic₇	C₂×D₄	1	4	yes	—
60	C₂×Q₈	—	7	1	no	—
61	C₂×Q₈	—	7	4	no	—
62	C₂×Q₈	—	14	8	no	—
63	C₂×Q₈	C₂×D₇	1	2	yes	—
64	C₂×D₄	—	7	1	no	—
65	C₂²×C₄	—	7	4	no	—
66	C₂²×C₄	—	7	2	no	—
67	C₄⋊C₄	—	7	2	no	—
68	C₄⋊C₄	—	7	4	no	—
69	C₄⋊C₄	—	7	1	no	—
70	C₄⋊C₄	—	7	1	no	—
71	C₄⋊C₄	C₂×D₇	1	2	yes	—
72	C₄⋊C₄	C₂×D₇	1	2	yes	—
73	C₂²⋊C₄	—	7	4	no	—
74	C₂²⋊C₄	—	7	2	no	—
75	C₄²	—	7	2	no	—
76	C₄²	C₂×D₇	1	1	yes	characteristic
77	C₂×C₇	C₂²×D₄	1	1	yes	characteristic
78	C₂×C₇	C₂²×Q₈	1	1	yes	characteristic
79	C₂×C₇	2₋¹⁺⁴	1	1	yes	characteristic
80	D₇	—	2	4	no	—
81	C₂³	—	7	2	no	—
82	Q₈	—	2	4	no	—
83	Q₈	—	7	4	no	—
84	Q₈	—	14	8	no	—
85	D₄	—	7	4	no	—
86	C₂×C₄	—	7	2	no	—
87	C₂×C₄	—	7	2	no	—
88	C₂×C₄	—	7	2	no	—
89	C₂×C₄	—	14	8	no	—
90	C₂×C₄	—	14	4	no	—
91	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
92	C₂×C₄	C₂²×D₇	1	4	yes	—
93	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
94	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
95	C₇	D₄×Q₈	1	1	yes	characteristic, complemented, Sylow
96	C₂²	—	7	4	no	—
97	C₂²	—	14	2	no	—
98	C₂²	—	14	2	no	—
99	C₂²	C₂³×D₇	1	1	yes	characteristic
100	C₄	—	2	4	no	—
101	C₄	—	2	1	no	—
102	C₄	—	7	4	no	—
103	C₄	—	14	2	no	—
104	C₄	—	14	2	no	—
105	C₄	D₄×D₇	1	2	yes	—
106	C₄	Q₈×D₇	1	2	yes	complemented
107	C₂	—	14	4	no	—
108	C₂	C₂×D₄×D₇	1	1	yes	characteristic
109	C₂	C₂×Q₈×D₇	1	1	yes	characteristic
110	C₂	#6551	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×D₇

C₂

yes

maximal, complemented

C₇×C₄⋊Q₈

C₂

yes

maximal, characteristic, complemented

#6509

C₂

yes

maximal, complemented

#6459

C₂

yes

maximal, complemented

#6456

C₂

yes

maximal

#6451

C₂

yes

maximal, complemented

C₄×D₂₈

C₂

yes

maximal, characteristic

C₄×Dic₁₄

C₂

yes

maximal, characteristic, complemented

C₂×C₇×Q₈

C₂²

yes

—

Q₈×D₇

—

C₂×D₂₈

C₂²

yes

characteristic

C₂×C₄×D₇

C₂²

yes

—

C₂×C₄×D₇

C₂²

yes

—

C₂×Dic₁₄

C₂²

yes

—

C₂×Dic₁₄

C₂²

yes

characteristic

C₇×C₄⋊C₄

C₂²

yes

—

C₇×C₄⋊C₄

C₂²

yes

—

C₄²×C₇

C₂²

yes

characteristic

(C₂×D₇)⋊C₄

C₂²

yes

—

(C₂×D₇)⋊C₄

C₂²

yes

—

C₄⋊Dic₇

C₂²

yes

characteristic

C₄⋊Dic₇

C₂²

yes

characteristic

Dic₇⋊C₄

C₂²

yes

—

Dic₇⋊C₄

C₂²

yes

—

C₄×Dic₇

C₂²

yes

—

D₄×Q₈

—

maximal, Sylow

C₂²×D₇

C₂³

yes

—

C₇×Q₈

—

C₂×C₄×C₇

C₂³

yes

characteristic

C₂×C₄×C₇

C₂³

yes

—

C₂×C₄×C₇

C₂³

yes

characteristic

C₂×C₄×C₇

C₂³

yes

characteristic

C₂×Dic₇

C₂³

yes

—

C₂×Dic₇

C₂³

yes

—

C₂×Dic₇

C₂³

yes

—

D₂₈

Q₈

yes

complemented

C₄×D₇

—

C₄×D₇

—

Dic₁₄

—

Dic₁₄

D₄

yes

complemented

C₂²×Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

D₇

yes

characteristic, complemented

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×D₄

—

C₄×D₄

—

C₂²×C₇

C₂⁴

yes

characteristic

C₂×D₇

—

C₂×D₇

—

C₂×D₇

C₂×Q₈

yes

complemented

C₄×C₇

—

C₄×C₇

—

C₄×C₇

C₂×D₄

yes

—

C₄×C₇

C₂×Q₈

yes

complemented

Dic₇

—

Dic₇

—

Dic₇

C₂×D₄

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×D₇

yes

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₇

yes

—

C₄⋊C₄

C₂×D₇

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂×D₇

yes

characteristic

C₂×C₇

C₂²×D₄

yes

characteristic

C₂×C₇

C₂²×Q₈

yes

characteristic

C₂×C₇

2₋¹⁺⁴

yes

characteristic

D₇

—

C₂³

—

Q₈

—

Q₈

—

Q₈

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²×D₇

yes

characteristic

C₂×C₄

C₂²×D₇

yes

—

C₂×C₄

C₂²×D₇

yes

characteristic

C₂×C₄

C₂²×D₇

yes

characteristic

C₇

D₄×Q₈

yes

characteristic, complemented, Sylow

C₂²

—

C₂²

—

C₂²

—

C₂²

C₂³×D₇

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

D₄×D₇

yes

—

106

C₄

Q₈×D₇

yes

complemented

107

C₂

—

108

C₂

C₂×D₄×D₇

yes

characteristic

109

C₂

C₂×Q₈×D₇

yes

characteristic

110

C₂

#6551

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₇
commutator, socle, cosocle	C₂²×C₇	C₂⁴
Fitting	C₇×C₄⋊Q₈	C₂
prime radical	C₂×D₂₈	C₂²
hypercenter	C₄⋊Q₈	D₇
hypocenter	C₇	D₄×Q₈
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₇

commutator, socle, cosocle

C₂²×C₇

C₂⁴

Fitting

C₇×C₄⋊Q₈

C₂

prime radical

C₂×D₂₈

C₂²

hypercenter

C₄⋊Q₈

D₇

hypocenter

C₇

D₄×Q₈

LCS

G ▷ C₂²×C₇ ▷ C₇

UCS

C₁ ◁ C₂² ◁ C₄⋊Q₈

N	H ≅ G/N	type
1. C₂×Q₈×D₇	C₂	semidirect
2. C₇×C₄⋊Q₈	C₂	semidirect
3. #6509	C₂	semidirect
4. #6459	C₂	semidirect
6. #6451	C₂	semidirect
8. C₄×Dic₁₄	C₂	semidirect
36. D₂₈	Q₈	semidirect
40. Dic₁₄	D₄	semidirect
43. C₄⋊Q₈	D₇	semidirect
52. C₂×D₇	C₂×Q₈	semidirect
56. C₄×C₇	C₂×Q₈	semidirect
95. C₇	D₄×Q₈	semidirect
106. C₄	Q₈×D₇	semidirect

H ≅ G/N

type

1. C₂×Q₈×D₇

C₂

semidirect

2. C₇×C₄⋊Q₈

C₂

semidirect

3. #6509

C₂

semidirect

4. #6459

C₂

semidirect

6. #6451

C₂

semidirect

8. C₄×Dic₁₄

C₂

semidirect

36. D₂₈

Q₈

semidirect

40. Dic₁₄

D₄

semidirect

43. C₄⋊Q₈

D₇

semidirect

52. C₂×D₇

C₂×Q₈

semidirect

56. C₄×C₇

C₂×Q₈

semidirect

95. C₇

D₄×Q₈

semidirect

106. C₄

Q₈×D₇

semidirect

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

subgroup	index
C₂×Q₈×D₇	2
C₇×C₄⋊Q₈	2
#6509	2
#6459	2
#6456	2
#6451	2
C₄×D₂₈	2
C₄×Dic₁₄	2
D₄×Q₈	7

subgroup

index

C₂×Q₈×D₇

C₇×C₄⋊Q₈

C₄×D₂₈

C₄×Dic₁₄

D₄×Q₈

Subgroups — #90418

Conjugacy classes of subgroups

Complements

Maximal subgroups