Group #72246604 — subgroups

Subgroups — #72246604

Order 2352.

200 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 50 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	#11924252	C₂	1	1	yes	maximal, characteristic
2	#11924250	C₂	1	2	yes	maximal, complemented
3	Dic₃×C₇⋊D₇	C₂	1	1	yes	maximal, characteristic
4	C₇×#4248	C₂	1	2	yes	maximal, complemented
5	C₃×#87639	C₂	1	1	yes	maximal, characteristic
6	#11716101	—	3	1	no	maximal, Hall
7	C₂×C₃×C₇⋊D₇	C₂²	1	1	yes	characteristic
8	C₃⋊(C₇⋊Dic₇)	C₂²	1	1	yes	characteristic
9	C₇×Dic₂₁	C₂²	1	2	yes	—
10	C₇²×Dic₃	C₂²	1	1	yes	characteristic
11	C₃×C₇×Dic₇	C₂²	1	2	yes	—
12	C₄×C₇⋊D₇	—	3	1	no	—
13	C₇×Dic₁₄	—	3	2	no	—
14	#87642	—	3	2	no	—
15	#87639	—	3	1	no	—
16	#87639	D₃	1	1	yes	characteristic
17	D₇×Dic₆	—	7	2	no	maximal
18	C₂×C₃×C₇²	C₂³	1	1	yes	characteristic
19	C₃×C₇⋊D₇	Q₈	1	2	yes	complemented
20	C₂×C₇⋊D₇	C₂×D₃	1	1	yes	characteristic
21	C₄×C₇²	—	3	1	no	—
22	C₇⋊Dic₇	—	3	1	no	—
23	C₇×Dic₇	—	3	2	no	—
24	C₇×Dic₇	C₂×D₃	1	2	yes	—
25	Dic₄₂	—	7	2	no	—
26	C₇×Dic₆	—	7	2	no	—
27	C₃×C₄×D₇	—	7	2	no	—
28	#4248	—	7	2	no	—
29	#4248	D₇	1	2	yes	complemented
30	Dic₃×D₇	—	7	2	no	—
31	Dic₃×D₇	—	7	2	no	—
32	Dic₃×D₇	—	14	3	no	—
33	C₃×C₇²	C₂×Q₈	1	1	yes	characteristic, complemented, Hall
34	Q₈×D₇	—	21	2	no	—
35	C₂×C₇²	C₂²×D₃	1	1	yes	characteristic
36	C₇⋊D₇	Dic₆	1	2	yes	complemented
37	C₂×C₃×D₇	—	7	2	no	—
38	C₂×C₃×D₇	—	14	3	no	—
39	C₃×C₄×C₇	—	7	2	no	—
40	Dic₂₁	—	7	2	no	—
41	Dic₂₁	—	14	3	no	—
42	Dic₂₁	C₂×D₇	1	2	yes	—
43	C₃×Dic₇	C₂×D₇	1	2	yes	—
44	C₇×Dic₃	—	2	3	no	—
45	C₇×Dic₃	—	7	2	no	—
46	C₇×Dic₃	C₂×D₇	1	2	yes	—
47	C₇×Q₈	—	21	2	no	—
48	C₄×D₇	—	7	2	no	—
49	C₄×D₇	—	21	2	no	—
50	C₄×D₇	—	21	2	no	—
51	C₄×D₇	—	42	3	no	—
52	Dic₁₄	—	3	2	no	—
53	Dic₁₄	—	21	2	no	—
54	Dic₁₄	—	21	2	no	—
55	C₇²	C₂×Dic₆	1	1	yes	characteristic, complemented, Sylow
56	C₂×Dic₆	—	49	1	no	Hall
57	C₂×C₃×C₇	—	2	3	no	—
58	C₂×C₃×C₇	C₂²×D₇	1	2	yes	—
59	C₃×D₇	—	7	4	no	—
60	C₃×D₇	—	14	6	no	—
61	C₂×D₇	—	7	2	no	—
62	C₂×D₇	—	14	3	no	—
63	C₄×C₇	—	3	2	no	—
64	C₄×C₇	—	6	3	no	—
65	C₄×C₇	—	7	2	no	—
66	C₄×C₇	—	21	2	no	—
67	Dic₇	—	3	2	no	—
68	Dic₇	—	21	2	no	—
69	Dic₇	—	42	3	no	—
70	Dic₇	D₃×D₇	1	2	yes	—
71	C₂×C₃×C₄	—	49	1	no	—
72	C₂×Dic₃	—	49	1	no	—
73	C₂×Dic₃	—	49	1	no	—
74	Dic₆	—	7	2	no	—
75	Dic₆	—	49	2	no	—
76	C₃×C₇	—	2	3	no	—
77	C₃×C₇	Q₈×D₇	1	2	yes	complemented
78	C₂×Q₈	—	147	1	no	Sylow
79	C₂×C₇	—	2	3	no	—
80	C₂×C₇	C₂×D₃×D₇	1	2	yes	—
81	D₇	—	7	4	no	—
82	D₇	—	14	6	no	—
83	C₂²×C₃	—	49	1	no	—
84	C₃×C₄	—	7	2	no	—
85	Dic₃	—	7	2	no	—
86	Dic₃	—	49	1	no	—
87	Dic₃	D₇²	1	1	yes	characteristic
88	Q₈	—	21	2	no	—
89	Q₈	—	147	2	no	—
90	C₂×C₄	—	49	1	no	—
91	C₂×C₄	—	147	1	no	—
92	C₂×C₄	—	147	1	no	—
93	C₇	—	2	3	no	—
94	C₇	D₇×Dic₆	1	2	yes	complemented
95	C₂×C₃	—	49	2	no	—
96	C₂×C₃	C₂×D₇²	1	1	yes	characteristic
97	C₂²	—	49	1	no	—
98	C₄	—	3	1	no	—
99	C₄	—	7	2	no	—
100	C₄	—	21	2	no	—
101	C₄	—	147	1	no	—
102	C₃	#11716101	1	1	yes	characteristic, complemented, Sylow
103	C₂	—	49	2	no	—
104	C₂	D₃×D₇²	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11924252

C₂

yes

maximal, characteristic

#11924250

C₂

yes

maximal, complemented

Dic₃×C₇⋊D₇

C₂

yes

maximal, characteristic

C₇×#4248

C₂

yes

maximal, complemented

C₃×#87639

C₂

yes

maximal, characteristic

#11716101

—

maximal, Hall

C₂×C₃×C₇⋊D₇

C₂²

yes

characteristic

C₃⋊(C₇⋊Dic₇)

C₂²

yes

characteristic

C₇×Dic₂₁

C₂²

yes

—

C₇²×Dic₃

C₂²

yes

characteristic

C₃×C₇×Dic₇

C₂²

yes

—

C₄×C₇⋊D₇

—

C₇×Dic₁₄

—

#87642

—

#87639

—

#87639

D₃

yes

characteristic

D₇×Dic₆

—

maximal

C₂×C₃×C₇²

C₂³

yes

characteristic

C₃×C₇⋊D₇

Q₈

yes

complemented

C₂×C₇⋊D₇

C₂×D₃

yes

characteristic

C₄×C₇²

—

C₇⋊Dic₇

—

C₇×Dic₇

—

C₇×Dic₇

C₂×D₃

yes

—

Dic₄₂

—

C₇×Dic₆

—

C₃×C₄×D₇

—

#4248

—

#4248

D₇

yes

complemented

Dic₃×D₇

—

Dic₃×D₇

—

Dic₃×D₇

—

C₃×C₇²

C₂×Q₈

yes

characteristic, complemented, Hall

Q₈×D₇

—

C₂×C₇²

C₂²×D₃

yes

characteristic

C₇⋊D₇

Dic₆

yes

complemented

C₂×C₃×D₇

—

C₂×C₃×D₇

—

C₃×C₄×C₇

—

Dic₂₁

—

Dic₂₁

—

Dic₂₁

C₂×D₇

yes

—

C₃×Dic₇

C₂×D₇

yes

—

C₇×Dic₃

—

C₇×Dic₃

—

C₇×Dic₃

C₂×D₇

yes

—

C₇×Q₈

—

C₄×D₇

—

C₄×D₇

—

C₄×D₇

—

C₄×D₇

—

Dic₁₄

—

Dic₁₄

—

Dic₁₄

—

C₇²

C₂×Dic₆

yes

characteristic, complemented, Sylow

C₂×Dic₆

—

Hall

C₂×C₃×C₇

—

C₂×C₃×C₇

C₂²×D₇

yes

—

C₃×D₇

—

C₃×D₇

—

C₂×D₇

—

C₂×D₇

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

—

Dic₇

—

Dic₇

—

Dic₇

—

Dic₇

D₃×D₇

yes

—

C₂×C₃×C₄

—

C₂×Dic₃

—

C₂×Dic₃

—

Dic₆

—

Dic₆

—

C₃×C₇

—

C₃×C₇

Q₈×D₇

yes

complemented

C₂×Q₈

—

147

Sylow

C₂×C₇

—

C₂×C₇

C₂×D₃×D₇

yes

—

D₇

—

D₇

—

C₂²×C₃

—

C₃×C₄

—

Dic₃

—

Dic₃

—

Dic₃

D₇²

yes

characteristic

Q₈

—

Q₈

—

147

—

C₂×C₄

—

C₂×C₄

—

147

—

C₂×C₄

—

147

—

C₇

—

C₇

D₇×Dic₆

yes

complemented

C₂×C₃

—

C₂×C₃

C₂×D₇²

yes

characteristic

C₂²

—

C₄

—

C₄

—

100

C₄

—

101

C₄

—

147

—

102

C₃

#11716101

yes

characteristic, complemented, Sylow

103

C₂

—

104

C₂

D₃×D₇²

yes

characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₃×D₇²
commutator, Fitting, socle, cosocle	C₂×C₃×C₇²	C₂³
prime radical	C₂×C₃×C₇⋊D₇	C₂²
hypocenter	C₃×C₇²	C₂×Q₈
LCS	G ▷ C₂×C₃×C₇² ▷ C₃×C₇²
UCS	C₁ ◁ C₂

subgroup

G/H

center, Frattini, hypercenter

C₂

D₃×D₇²

commutator, Fitting, socle, cosocle

C₂×C₃×C₇²

C₂³

prime radical

C₂×C₃×C₇⋊D₇

C₂²

hypocenter

C₃×C₇²

C₂×Q₈

LCS

G ▷ C₂×C₃×C₇² ▷ C₃×C₇²

UCS

C₁ ◁ C₂

N	H ≅ G/N	type
2. #11924250	C₂	semidirect
4. C₇×#4248	C₂	semidirect
19. C₃×C₇⋊D₇	Q₈	semidirect
29. #4248	D₇	semidirect
33. C₃×C₇²	C₂×Q₈	semidirect
36. C₇⋊D₇	Dic₆	semidirect
55. C₇²	C₂×Dic₆	semidirect
77. C₃×C₇	Q₈×D₇	semidirect
94. C₇	D₇×Dic₆	semidirect
102. C₃	#11716101	semidirect

H ≅ G/N

type

2. #11924250

C₂

semidirect

4. C₇×#4248

C₂

semidirect

19. C₃×C₇⋊D₇

Q₈

semidirect

29. #4248

D₇

semidirect

33. C₃×C₇²

C₂×Q₈

semidirect

36. C₇⋊D₇

Dic₆

semidirect

55. C₇²

C₂×Dic₆

semidirect

77. C₃×C₇

Q₈×D₇

semidirect

94. C₇

D₇×Dic₆

semidirect

102. C₃

#11716101

semidirect

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

subgroup	index
#11924252	2
#11924250	2
Dic₃×C₇⋊D₇	2
C₇×#4248	2
C₃×#87639	2
#11716101	3
D₇×Dic₆	7

subgroup

index

Dic₃×C₇⋊D₇

D₇×Dic₆

Subgroups — #72246604

Conjugacy classes of subgroups

Complements

Maximal subgroups