Group #72246480 — subgroups

Subgroups — C₄×D₂₁⋊D₇ #72246480

Order 2352.

264 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 63 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×D₂₁⋊D₇	C₂	1	1	yes	maximal, characteristic
2	C₄×C₇×D₂₁	C₂	1	2	yes	maximal, complemented
3	C₃×C₄×C₇⋊D₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#11924252	C₂	1	1	yes	maximal, characteristic, complemented
5	D₂₁⋊Dic₇	C₂	1	2	yes	maximal, complemented
6	C₄×D₇²	—	3	1	no	maximal, Hall
7	C₂×C₇×D₂₁	C₂²	1	2	yes	—
8	C₂×C₃×C₇⋊D₇	C₂²	1	1	yes	characteristic
9	D₂₁⋊D₇	C₄	1	4	yes	direct product factor
10	C₃×C₄×C₇²	C₂²	1	1	yes	characteristic, complemented
11	C₇×Dic₂₁	C₂²	1	2	yes	complemented
12	C₃×C₇⋊Dic₇	C₂²	1	1	yes	characteristic, complemented
13	C₂×D₇²	—	3	1	no	—
14	C₄×C₇⋊D₇	D₃	1	1	yes	characteristic, complemented
15	C₄×C₇×D₇	—	3	2	no	—
16	#87639	—	3	1	no	—
17	D₇×Dic₇	—	3	2	no	—
18	C₄×D₃×D₇	—	7	2	no	maximal
19	C₂×C₃×C₇²	C₂³	1	1	yes	characteristic
20	C₇×D₂₁	C₂×C₄	1	4	yes	complemented
21	C₃×C₇⋊D₇	C₂×C₄	1	2	yes	complemented
22	C₂×C₇⋊D₇	C₂×D₃	1	1	yes	characteristic
23	C₂×C₇×D₇	—	3	2	no	—
24	D₇²	—	3	4	no	—
25	C₄×C₇²	C₂×D₃	1	1	yes	characteristic, complemented
26	C₇⋊Dic₇	C₂×D₃	1	1	yes	characteristic, complemented
27	C₇×Dic₇	—	3	2	no	—
28	C₂×D₃×D₇	—	7	2	no	—
29	C₄×D₂₁	D₇	1	2	yes	complemented
30	C₄×D₃×C₇	—	7	2	no	—
31	C₃×C₄×D₇	—	7	2	no	—
32	C₃×C₄×D₇	—	14	3	no	—
33	D₂₁⋊C₄	—	7	2	no	—
34	D₃×Dic₇	—	7	2	no	—
35	Dic₃×D₇	—	7	2	no	—
36	C₃×C₇²	C₂²×C₄	1	1	yes	characteristic, complemented, Hall
37	C₂×C₄×D₇	—	21	2	no	—
38	C₂×C₇²	C₂²×D₃	1	1	yes	characteristic
39	C₇⋊D₇	C₄×D₃	1	2	yes	complemented
40	C₇×D₇	—	3	4	no	—
41	C₂×D₂₁	C₂×D₇	1	2	yes	—
42	C₂×D₃×C₇	—	7	2	no	—
43	C₂×C₃×D₇	—	7	2	no	—
44	C₂×C₃×D₇	—	14	3	no	—
45	D₃×D₇	—	7	8	no	—
46	C₃×C₄×C₇	—	2	3	no	—
47	C₃×C₄×C₇	C₂×D₇	1	2	yes	complemented
48	Dic₂₁	C₂×D₇	1	2	yes	complemented
49	C₃×Dic₇	—	7	2	no	—
50	C₃×Dic₇	—	14	3	no	—
51	C₇×Dic₃	—	7	2	no	—
52	C₂²×D₇	—	21	2	no	—
53	C₂×C₄×C₇	—	21	2	no	—
54	C₂×Dic₇	—	21	2	no	—
55	C₄×D₇	—	3	2	no	—
56	C₄×D₇	—	7	2	no	—
57	C₄×D₇	—	14	3	no	—
58	C₄×D₇	—	21	2	no	—
59	C₄×D₇	—	21	2	no	—
60	C₇²	C₂×C₄×D₃	1	1	yes	characteristic, complemented, Sylow
61	C₂×C₄×D₃	—	49	1	no	Hall
62	C₂×C₃×C₇	—	2	3	no	—
63	C₂×C₃×C₇	C₂²×D₇	1	2	yes	—
64	D₂₁	C₄×D₇	1	4	yes	complemented
65	C₃×D₇	—	7	4	no	—
66	C₃×D₇	—	14	6	no	—
67	D₃×C₇	—	7	4	no	—
68	C₂²×C₇	—	21	2	no	—
69	C₂×D₇	—	3	2	no	—
70	C₂×D₇	—	7	2	no	—
71	C₂×D₇	—	14	3	no	—
72	C₂×D₇	—	21	8	no	—
73	C₄×C₇	—	2	3	no	—
74	C₄×C₇	—	21	2	no	—
75	C₄×C₇	D₃×D₇	1	2	yes	complemented
76	Dic₇	—	3	2	no	—
77	Dic₇	—	7	2	no	—
78	Dic₇	—	14	3	no	—
79	C₂²×D₃	—	49	1	no	—
80	C₂×C₃×C₄	—	49	1	no	—
81	C₂×Dic₃	—	49	1	no	—
82	C₄×D₃	—	7	2	no	—
83	C₄×D₃	—	49	2	no	—
84	C₃×C₇	—	2	3	no	—
85	C₃×C₇	C₂×C₄×D₇	1	2	yes	complemented
86	C₂²×C₄	—	147	1	no	Sylow
87	C₂×C₇	—	2	3	no	—
88	C₂×C₇	—	21	4	no	—
89	C₂×C₇	C₂×D₃×D₇	1	2	yes	—
90	D₇	—	3	4	no	—
91	D₇	—	7	4	no	—
92	D₇	—	14	6	no	—
93	C₂²×C₃	—	49	1	no	—
94	C₂×D₃	—	7	2	no	—
95	C₂×D₃	—	49	4	no	—
96	C₃×C₄	—	49	1	no	—
97	C₃×C₄	D₇²	1	1	yes	characteristic, complemented
98	Dic₃	—	7	2	no	—
99	C₂³	—	147	1	no	—
100	C₂×C₄	—	21	2	no	—
101	C₂×C₄	—	49	1	no	—
102	C₂×C₄	—	147	1	no	—
103	C₂×C₄	—	147	2	no	—
104	C₇	—	2	3	no	—
105	C₇	C₄×D₃×D₇	1	2	yes	complemented
106	C₂×C₃	—	49	2	no	—
107	C₂×C₃	C₂×D₇²	1	1	yes	characteristic
108	D₃	—	7	4	no	—
109	C₂²	—	21	2	no	—
110	C₂²	—	49	1	no	—
111	C₂²	—	147	4	no	—
112	C₄	—	21	2	no	—
113	C₄	—	49	1	no	—
114	C₄	D₂₁⋊D₇	1	1	yes	characteristic, direct product factor
115	C₃	C₄×D₇²	1	1	yes	characteristic, complemented, Sylow
116	C₂	—	21	4	no	—
117	C₂	—	49	2	no	—
118	C₂	C₂×D₂₁⋊D₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×D₂₁⋊D₇

C₂

yes

maximal, characteristic

C₄×C₇×D₂₁

C₂

yes

maximal, complemented

C₃×C₄×C₇⋊D₇

C₂

yes

maximal, characteristic, complemented

#11924252

C₂

yes

maximal, characteristic, complemented

D₂₁⋊Dic₇

C₂

yes

maximal, complemented

C₄×D₇²

—

maximal, Hall

C₂×C₇×D₂₁

C₂²

yes

—

C₂×C₃×C₇⋊D₇

C₂²

yes

characteristic

D₂₁⋊D₇

C₄

yes

direct product factor

C₃×C₄×C₇²

C₂²

yes

characteristic, complemented

C₇×Dic₂₁

C₂²

yes

complemented

C₃×C₇⋊Dic₇

C₂²

yes

characteristic, complemented

C₂×D₇²

—

C₄×C₇⋊D₇

D₃

yes

characteristic, complemented

C₄×C₇×D₇

—

#87639

—

D₇×Dic₇

—

C₄×D₃×D₇

—

maximal

C₂×C₃×C₇²

C₂³

yes

characteristic

C₇×D₂₁

C₂×C₄

yes

complemented

C₃×C₇⋊D₇

C₂×C₄

yes

complemented

C₂×C₇⋊D₇

C₂×D₃

yes

characteristic

C₂×C₇×D₇

—

D₇²

—

C₄×C₇²

C₂×D₃

yes

characteristic, complemented

C₇⋊Dic₇

C₂×D₃

yes

characteristic, complemented

C₇×Dic₇

—

C₂×D₃×D₇

—

C₄×D₂₁

D₇

yes

complemented

C₄×D₃×C₇

—

C₃×C₄×D₇

—

C₃×C₄×D₇

—

D₂₁⋊C₄

—

D₃×Dic₇

—

Dic₃×D₇

—

C₃×C₇²

C₂²×C₄

yes

characteristic, complemented, Hall

C₂×C₄×D₇

—

C₂×C₇²

C₂²×D₃

yes

characteristic

C₇⋊D₇

C₄×D₃

yes

complemented

C₇×D₇

—

C₂×D₂₁

C₂×D₇

yes

—

C₂×D₃×C₇

—

C₂×C₃×D₇

—

C₂×C₃×D₇

—

D₃×D₇

—

C₃×C₄×C₇

—

C₃×C₄×C₇

C₂×D₇

yes

complemented

Dic₂₁

C₂×D₇

yes

complemented

C₃×Dic₇

—

C₃×Dic₇

—

C₇×Dic₃

—

C₂²×D₇

—

C₂×C₄×C₇

—

C₂×Dic₇

—

C₄×D₇

—

C₄×D₇

—

C₄×D₇

—

C₄×D₇

—

C₄×D₇

—

C₇²

C₂×C₄×D₃

yes

characteristic, complemented, Sylow

C₂×C₄×D₃

—

Hall

C₂×C₃×C₇

—

C₂×C₃×C₇

C₂²×D₇

yes

—

D₂₁

C₄×D₇

yes

complemented

C₃×D₇

—

C₃×D₇

—

D₃×C₇

—

C₂²×C₇

—

C₂×D₇

—

C₂×D₇

—

C₂×D₇

—

C₂×D₇

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

D₃×D₇

yes

complemented

Dic₇

—

Dic₇

—

Dic₇

—

C₂²×D₃

—

C₂×C₃×C₄

—

C₂×Dic₃

—

C₄×D₃

—

C₄×D₃

—

C₃×C₇

—

C₃×C₇

C₂×C₄×D₇

yes

complemented

C₂²×C₄

—

147

Sylow

C₂×C₇

—

C₂×C₇

—

C₂×C₇

C₂×D₃×D₇

yes

—

D₇

—

D₇

—

D₇

—

C₂²×C₃

—

C₂×D₃

—

C₂×D₃

—

C₃×C₄

—

C₃×C₄

D₇²

yes

characteristic, complemented

Dic₃

—

C₂³

—

147

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

147

—

103

C₂×C₄

—

147

—

104

C₇

—

105

C₇

C₄×D₃×D₇

yes

complemented

106

C₂×C₃

—

107

C₂×C₃

C₂×D₇²

yes

characteristic

108

D₃

—

109

C₂²

—

110

C₂²

—

111

C₂²

—

147

—

112

C₄

—

113

C₄

—

114

C₄

D₂₁⋊D₇

yes

characteristic, direct product factor

115

C₃

C₄×D₇²

yes

characteristic, complemented, Sylow

116

C₂

—

117

C₂

—

118

C₂

C₂×D₂₁⋊D₇

yes

characteristic

	subgroup	G/H
center, hypercenter	C₄	D₂₁⋊D₇
commutator, hypocenter	C₃×C₇²	C₂²×C₄
Frattini	C₂	C₂×D₂₁⋊D₇
Fitting	C₃×C₄×C₇²	C₂²
socle, cosocle	C₂×C₃×C₇²	C₂³
prime radical	C₂×D₂₁⋊D₇	C₂
LCS	G ▷ C₃×C₇²
UCS	C₁ ◁ C₄

subgroup

G/H

center, hypercenter

C₄

D₂₁⋊D₇

commutator, hypocenter

C₃×C₇²

C₂²×C₄

Frattini

C₂

C₂×D₂₁⋊D₇

Fitting

C₃×C₄×C₇²

C₂²

socle, cosocle

C₂×C₃×C₇²

C₂³

prime radical

C₂×D₂₁⋊D₇

C₂

LCS

G ▷ C₃×C₇²

UCS

C₁ ◁ C₄

N	H ≅ G/N	type
2. C₄×C₇×D₂₁	C₂	semidirect
3. C₃×C₄×C₇⋊D₇	C₂	semidirect
4. #11924252	C₂	semidirect
5. D₂₁⋊Dic₇	C₂	semidirect
9. D₂₁⋊D₇	C₄	direct (N × H)
10. C₃×C₄×C₇²	C₂²	semidirect
11. C₇×Dic₂₁	C₂²	semidirect
12. C₃×C₇⋊Dic₇	C₂²	semidirect
14. C₄×C₇⋊D₇	D₃	semidirect
20. C₇×D₂₁	C₂×C₄	semidirect
21. C₃×C₇⋊D₇	C₂×C₄	semidirect
25. C₄×C₇²	C₂×D₃	semidirect
26. C₇⋊Dic₇	C₂×D₃	semidirect
29. C₄×D₂₁	D₇	semidirect
36. C₃×C₇²	C₂²×C₄	semidirect
39. C₇⋊D₇	C₄×D₃	semidirect
47. C₃×C₄×C₇	C₂×D₇	semidirect
48. Dic₂₁	C₂×D₇	semidirect
60. C₇²	C₂×C₄×D₃	semidirect
64. D₂₁	C₄×D₇	semidirect
75. C₄×C₇	D₃×D₇	semidirect
85. C₃×C₇	C₂×C₄×D₇	semidirect
97. C₃×C₄	D₇²	semidirect
105. C₇	C₄×D₃×D₇	semidirect
114. C₄	D₂₁⋊D₇	direct (N × H)
115. C₃	C₄×D₇²	semidirect

H ≅ G/N

type

2. C₄×C₇×D₂₁

C₂

semidirect

3. C₃×C₄×C₇⋊D₇

C₂

semidirect

4. #11924252

C₂

semidirect

5. D₂₁⋊Dic₇

C₂

semidirect

9. D₂₁⋊D₇

C₄

direct (N × H)

10. C₃×C₄×C₇²

C₂²

semidirect

11. C₇×Dic₂₁

C₂²

semidirect

12. C₃×C₇⋊Dic₇

C₂²

semidirect

14. C₄×C₇⋊D₇

D₃

semidirect

20. C₇×D₂₁

C₂×C₄

semidirect

21. C₃×C₇⋊D₇

C₂×C₄

semidirect

25. C₄×C₇²

C₂×D₃

semidirect

26. C₇⋊Dic₇

C₂×D₃

semidirect

29. C₄×D₂₁

D₇

semidirect

36. C₃×C₇²

C₂²×C₄

semidirect

39. C₇⋊D₇

C₄×D₃

semidirect

47. C₃×C₄×C₇

C₂×D₇

semidirect

48. Dic₂₁

C₂×D₇

semidirect

60. C₇²

C₂×C₄×D₃

semidirect

64. D₂₁

C₄×D₇

semidirect

75. C₄×C₇

D₃×D₇

semidirect

85. C₃×C₇

C₂×C₄×D₇

semidirect

97. C₃×C₄

D₇²

semidirect

105. C₇

C₄×D₃×D₇

semidirect

114. C₄

D₂₁⋊D₇

direct (N × H)

115. C₃

C₄×D₇²

semidirect

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

subgroup	index
C₂×D₂₁⋊D₇	2
C₄×C₇×D₂₁	2
C₃×C₄×C₇⋊D₇	2
#11924252	2
D₂₁⋊Dic₇	2
C₄×D₇²	3
C₄×D₃×D₇	7

subgroup

index

C₂×D₂₁⋊D₇

C₄×C₇×D₂₁

C₃×C₄×C₇⋊D₇

#11924252

D₂₁⋊Dic₇

C₄×D₇²

C₄×D₃×D₇

Subgroups — C₄×D₂₁⋊D₇ #72246480

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C4×D21⋊D7 #72246480

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×D₂₁⋊D₇ #72246480