Group #26374443 — subgroups

Subgroups — C₂²×C₄²×Dic₄₇ #26374443

Order 12032.

8094 conjugacy classes of subgroups; only the 5547 normal subgroups are shown.

normal subgroups only

This group has 8094 conjugacy classes of subgroups — too many to draw. Only its 5547 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	C₂³×C₄×Dic₄₇	C₂	1	6	yes	maximal
2	C₂×C₄²×Dic₄₇	C₂	1	24	yes	maximal, direct product factor
3	C₂³×C₄²×C₄₇	C₂	1	1	yes	maximal, characteristic
4	C₂⁴×Dic₄₇	C₂²	1	4	yes	—
5	C₂²×C₄×Dic₄₇	C₄	1	96	yes	direct product factor
6	C₂²×C₄×Dic₄₇	C₂²	1	72	yes	—
7	C₄²×Dic₄₇	C₂²	1	64	yes	direct product factor
8	C₂⁴×C₄×C₄₇	C₂²	1	3	yes	—
9	C₂²×C₄²×C₄₇	C₄	1	16	yes	complemented
10	C₂²×C₄²×C₄₇	C₂²	1	12	yes	—
11	C₂³×Dic₄₇	C₂×C₄	1	96	yes	—
12	C₂³×Dic₄₇	C₂³	1	24	yes	—
13	C₂×C₄×Dic₄₇	C₂×C₄	1	576	yes	direct product factor
14	C₂×C₄×Dic₄₇	C₂³	1	96	yes	—
15	C₂⁵×C₄₇	C₂³	1	1	yes	characteristic
16	C₂³×C₄×C₄₇	C₂×C₄	1	48	yes	—
17	C₂³×C₄×C₄₇	C₂×C₄	1	24	yes	—
18	C₂³×C₄×C₄₇	C₂³	1	18	yes	—
19	C₂×C₄²×C₄₇	C₂×C₄	1	96	yes	complemented
20	C₂×C₄²×C₄₇	C₂³	1	16	yes	—
21	C₂⁴×C₄₇	C₂²×C₄	1	24	yes	—
22	C₂⁴×C₄₇	C₂²×C₄	1	4	yes	—
23	C₂⁴×C₄₇	C₂⁴	1	3	yes	—
24	C₂²×C₄×C₄₇	C₄²	1	192	yes	complemented
25	C₂²×C₄×C₄₇	C₂²×C₄	1	144	yes	—
26	C₂²×C₄×C₄₇	C₂²×C₄	1	72	yes	—
27	C₂²×C₄×C₄₇	C₂⁴	1	12	yes	—
28	C₂²×Dic₄₇	C₄²	1	256	yes	direct product factor
29	C₂²×Dic₄₇	C₂²×C₄	1	288	yes	—
30	C₂²×Dic₄₇	C₂⁴	1	16	yes	—
31	C₄²×C₄₇	C₂²×C₄	1	64	yes	complemented
32	C₄×Dic₄₇	C₂²×C₄	1	384	yes	direct product factor
33	C₂³×C₄₇	C₂×C₄²	1	64	yes	—
34	C₂³×C₄₇	C₂×C₄²	1	48	yes	—
35	C₂³×C₄₇	C₂³×C₄	1	36	yes	—
36	C₂³×C₄₇	C₂³×C₄	1	6	yes	—
37	C₂³×C₄₇	C₂⁵	1	1	yes	characteristic
38	C₂×C₄×C₄₇	C₂×C₄²	1	288	yes	complemented
39	C₂×C₄×C₄₇	C₂³×C₄	1	48	yes	—
40	C₂×C₄×C₄₇	C₂³×C₄	1	24	yes	—
41	C₂×Dic₄₇	C₂×C₄²	1	384	yes	direct product factor
42	C₂×Dic₄₇	C₂³×C₄	1	96	yes	—
43	C₂²×C₄³	—	47	1	no	maximal, Sylow
44	C₂²×C₄₇	C₄³	1	64	yes	complemented
45	C₂²×C₄₇	C₂²×C₄²	1	48	yes	—
46	C₂²×C₄₇	C₂²×C₄²	1	36	yes	—
47	C₂²×C₄₇	C₂⁴×C₄	1	6	yes	—
48	C₂²×C₄₇	C₂⁴×C₄	1	1	yes	characteristic
49	C₄×C₄₇	C₂²×C₄²	1	48	yes	complemented
50	Dic₄₇	C₂²×C₄²	1	64	yes	direct product factor
51	C₂³×C₄²	—	47	6	no	—
52	C₂³×C₄²	D₄₇	1	1	yes	characteristic
53	C₂×C₄³	—	47	24	no	—
54	C₂×C₄₇	C₂×C₄³	1	24	yes	complemented
55	C₂×C₄₇	C₂³×C₄²	1	4	yes	—
56	C₂×C₄₇	C₂³×C₄²	1	3	yes	—
57	C₂⁴×C₄	—	47	4	no	—
58	C₂⁴×C₄	C₂×D₄₇	1	3	yes	—
59	C₂²×C₄²	—	47	96	no	—
60	C₂²×C₄²	—	47	72	no	—
61	C₂²×C₄²	Dic₄₇	1	16	yes	direct product factor
62	C₂²×C₄²	C₂×D₄₇	1	12	yes	—
63	C₄³	—	47	64	no	—
64	C₄₇	C₂²×C₄³	1	1	yes	characteristic, complemented, Sylow
65	C₂⁵	C₂²×D₄₇	1	1	yes	characteristic
66	C₂³×C₄	—	47	96	no	—
67	C₂³×C₄	—	47	24	no	—
68	C₂³×C₄	C₄×D₄₇	1	48	yes	—
69	C₂³×C₄	C₂×Dic₄₇	1	24	yes	—
70	C₂³×C₄	C₂²×D₄₇	1	18	yes	—
71	C₂×C₄²	—	47	576	no	—
72	C₂×C₄²	—	47	96	no	—
73	C₂×C₄²	C₂×Dic₄₇	1	96	yes	direct product factor
74	C₂×C₄²	C₂²×D₄₇	1	16	yes	—
75	C₂⁴	C₂×C₄×D₄₇	1	24	yes	—
76	C₂⁴	C₂²×Dic₄₇	1	4	yes	—
77	C₂⁴	C₂³×D₄₇	1	3	yes	—
78	C₂²×C₄	—	47	256	no	—
79	C₂²×C₄	—	47	288	no	—
80	C₂²×C₄	—	47	16	no	—
81	C₂²×C₄	C₄×Dic₄₇	1	192	yes	direct product factor
82	C₂²×C₄	C₂×C₄×D₄₇	1	144	yes	—
83	C₂²×C₄	C₂²×Dic₄₇	1	72	yes	—
84	C₂²×C₄	C₂³×D₄₇	1	12	yes	—
85	C₄²	—	47	384	no	—
86	C₄²	C₂²×Dic₄₇	1	64	yes	direct product factor
87	C₂³	C₄²×D₄₇	1	64	yes	—
88	C₂³	C₂×C₄×Dic₄₇	1	48	yes	—
89	C₂³	C₂²×C₄×D₄₇	1	36	yes	—
90	C₂³	C₂³×Dic₄₇	1	6	yes	—
91	C₂³	C₂⁴×D₄₇	1	1	yes	characteristic
92	C₂×C₄	—	47	384	no	—
93	C₂×C₄	—	47	96	no	—
94	C₂×C₄	C₂×C₄×Dic₄₇	1	288	yes	direct product factor
95	C₂×C₄	C₂²×C₄×D₄₇	1	48	yes	—
96	C₂×C₄	C₂³×Dic₄₇	1	24	yes	—
97	C₂²	C₄²×Dic₄₇	1	64	yes	direct product factor
98	C₂²	C₂×C₄²×D₄₇	1	48	yes	—
99	C₂²	C₂²×C₄×Dic₄₇	1	36	yes	—
100	C₂²	C₂³×C₄×D₄₇	1	6	yes	—
101	C₂²	C₂⁴×Dic₄₇	1	1	yes	characteristic
102	C₄	—	47	64	no	—
103	C₄	C₂²×C₄×Dic₄₇	1	48	yes	direct product factor
104	C₂	C₂×C₄²×Dic₄₇	1	24	yes	direct product factor
105	C₂	C₂²×C₄²×D₄₇	1	4	yes	—
106	C₂	C₂³×C₄×Dic₄₇	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×C₄×Dic₄₇

C₂

yes

maximal

C₂×C₄²×Dic₄₇

C₂

yes

maximal, direct product factor

C₂³×C₄²×C₄₇

C₂

yes

maximal, characteristic

C₂⁴×Dic₄₇

C₂²

yes

—

C₂²×C₄×Dic₄₇

C₄

yes

direct product factor

C₂²×C₄×Dic₄₇

C₂²

yes

—

C₄²×Dic₄₇

C₂²

yes

direct product factor

C₂⁴×C₄×C₄₇

C₂²

yes

—

C₂²×C₄²×C₄₇

C₄

yes

complemented

C₂²×C₄²×C₄₇

C₂²

yes

—

C₂³×Dic₄₇

C₂×C₄

yes

—

C₂³×Dic₄₇

C₂³

yes

—

C₂×C₄×Dic₄₇

C₂×C₄

576

yes

direct product factor

C₂×C₄×Dic₄₇

C₂³

yes

—

C₂⁵×C₄₇

C₂³

yes

characteristic

C₂³×C₄×C₄₇

C₂×C₄

yes

—

C₂³×C₄×C₄₇

C₂×C₄

yes

—

C₂³×C₄×C₄₇

C₂³

yes

—

C₂×C₄²×C₄₇

C₂×C₄

yes

complemented

C₂×C₄²×C₄₇

C₂³

yes

—

C₂⁴×C₄₇

C₂²×C₄

yes

—

C₂⁴×C₄₇

C₂²×C₄

yes

—

C₂⁴×C₄₇

C₂⁴

yes

—

C₂²×C₄×C₄₇

C₄²

192

yes

complemented

C₂²×C₄×C₄₇

C₂²×C₄

144

yes

—

C₂²×C₄×C₄₇

C₂²×C₄

yes

—

C₂²×C₄×C₄₇

C₂⁴

yes

—

C₂²×Dic₄₇

C₄²

256

yes

direct product factor

C₂²×Dic₄₇

C₂²×C₄

288

yes

—

C₂²×Dic₄₇

C₂⁴

yes

—

C₄²×C₄₇

C₂²×C₄

yes

complemented

C₄×Dic₄₇

C₂²×C₄

384

yes

direct product factor

C₂³×C₄₇

C₂×C₄²

yes

—

C₂³×C₄₇

C₂×C₄²

yes

—

C₂³×C₄₇

C₂³×C₄

yes

—

C₂³×C₄₇

C₂³×C₄

yes

—

C₂³×C₄₇

C₂⁵

yes

characteristic

C₂×C₄×C₄₇

C₂×C₄²

288

yes

complemented

C₂×C₄×C₄₇

C₂³×C₄

yes

—

C₂×C₄×C₄₇

C₂³×C₄

yes

—

C₂×Dic₄₇

C₂×C₄²

384

yes

direct product factor

C₂×Dic₄₇

C₂³×C₄

yes

—

C₂²×C₄³

—

maximal, Sylow

C₂²×C₄₇

C₄³

yes

complemented

C₂²×C₄₇

C₂²×C₄²

yes

—

C₂²×C₄₇

C₂²×C₄²

yes

—

C₂²×C₄₇

C₂⁴×C₄

yes

—

C₂²×C₄₇

C₂⁴×C₄

yes

characteristic

C₄×C₄₇

C₂²×C₄²

yes

complemented

Dic₄₇

C₂²×C₄²

yes

direct product factor

C₂³×C₄²

—

C₂³×C₄²

D₄₇

yes

characteristic

C₂×C₄³

—

C₂×C₄₇

C₂×C₄³

yes

complemented

C₂×C₄₇

C₂³×C₄²

yes

—

C₂×C₄₇

C₂³×C₄²

yes

—

C₂⁴×C₄

—

C₂⁴×C₄

C₂×D₄₇

yes

—

C₂²×C₄²

—

C₂²×C₄²

—

C₂²×C₄²

Dic₄₇

yes

direct product factor

C₂²×C₄²

C₂×D₄₇

yes

—

C₄³

—

C₄₇

C₂²×C₄³

yes

characteristic, complemented, Sylow

C₂⁵

C₂²×D₄₇

yes

characteristic

C₂³×C₄

—

C₂³×C₄

—

C₂³×C₄

C₄×D₄₇

yes

—

C₂³×C₄

C₂×Dic₄₇

yes

—

C₂³×C₄

C₂²×D₄₇

yes

—

C₂×C₄²

—

576

—

C₂×C₄²

—

C₂×C₄²

C₂×Dic₄₇

yes

direct product factor

C₂×C₄²

C₂²×D₄₇

yes

—

C₂⁴

C₂×C₄×D₄₇

yes

—

C₂⁴

C₂²×Dic₄₇

yes

—

C₂⁴

C₂³×D₄₇

yes

—

C₂²×C₄

—

256

—

C₂²×C₄

—

288

—

C₂²×C₄

—

C₂²×C₄

C₄×Dic₄₇

192

yes

direct product factor

C₂²×C₄

C₂×C₄×D₄₇

144

yes

—

C₂²×C₄

C₂²×Dic₄₇

yes

—

C₂²×C₄

C₂³×D₄₇

yes

—

C₄²

—

384

—

C₄²

C₂²×Dic₄₇

yes

direct product factor

C₂³

C₄²×D₄₇

yes

—

C₂³

C₂×C₄×Dic₄₇

yes

—

C₂³

C₂²×C₄×D₄₇

yes

—

C₂³

C₂³×Dic₄₇

yes

—

C₂³

C₂⁴×D₄₇

yes

characteristic

C₂×C₄

—

384

—

C₂×C₄

—

C₂×C₄

C₂×C₄×Dic₄₇

288

yes

direct product factor

C₂×C₄

C₂²×C₄×D₄₇

yes

—

C₂×C₄

C₂³×Dic₄₇

yes

—

C₂²

C₄²×Dic₄₇

yes

direct product factor

C₂²

C₂×C₄²×D₄₇

yes

—

C₂²

C₂²×C₄×Dic₄₇

yes

—

100

C₂²

C₂³×C₄×D₄₇

yes

—

101

C₂²

C₂⁴×Dic₄₇

yes

characteristic

102

C₄

—

103

C₄

C₂²×C₄×Dic₄₇

yes

direct product factor

104

C₂

C₂×C₄²×Dic₄₇

yes

direct product factor

105

C₂

C₂²×C₄²×D₄₇

yes

—

106

C₂

C₂³×C₄×Dic₄₇

yes

—

	subgroup	G/H
center, hypercenter	C₂³×C₄²	D₄₇
commutator, hypocenter	C₄₇	C₂²×C₄³
Frattini	C₂³	C₂⁴×D₄₇
Fitting	C₂³×C₄²×C₄₇	C₂
socle, prime radical	C₂⁵×C₄₇	C₂³
cosocle	C₂³×C₄₇	C₂⁵
LCS	G ▷ C₄₇
UCS	C₁ ◁ C₂³×C₄²

subgroup

G/H

center, hypercenter

C₂³×C₄²

D₄₇

commutator, hypocenter

C₄₇

C₂²×C₄³

Frattini

C₂³

C₂⁴×D₄₇

Fitting

C₂³×C₄²×C₄₇

C₂

socle, prime radical

C₂⁵×C₄₇

C₂³

cosocle

C₂³×C₄₇

C₂⁵

LCS

G ▷ C₄₇

UCS

C₁ ◁ C₂³×C₄²

N	H ≅ G/N	type
2. C₂×C₄²×Dic₄₇	C₂	direct (N × H)
5. C₂²×C₄×Dic₄₇	C₄	direct (N × H)
7. C₄²×Dic₄₇	C₂²	direct (N × H)
9. C₂²×C₄²×C₄₇	C₄	semidirect
13. C₂×C₄×Dic₄₇	C₂×C₄	direct (N × H)
19. C₂×C₄²×C₄₇	C₂×C₄	semidirect
24. C₂²×C₄×C₄₇	C₄²	semidirect
28. C₂²×Dic₄₇	C₄²	direct (N × H)
31. C₄²×C₄₇	C₂²×C₄	semidirect
32. C₄×Dic₄₇	C₂²×C₄	direct (N × H)
38. C₂×C₄×C₄₇	C₂×C₄²	semidirect
41. C₂×Dic₄₇	C₂×C₄²	direct (N × H)
44. C₂²×C₄₇	C₄³	semidirect
49. C₄×C₄₇	C₂²×C₄²	semidirect
50. Dic₄₇	C₂²×C₄²	direct (N × H)
54. C₂×C₄₇	C₂×C₄³	semidirect
61. C₂²×C₄²	Dic₄₇	direct (N × H)
64. C₄₇	C₂²×C₄³	semidirect
73. C₂×C₄²	C₂×Dic₄₇	direct (N × H)
81. C₂²×C₄	C₄×Dic₄₇	direct (N × H)
86. C₄²	C₂²×Dic₄₇	direct (N × H)
94. C₂×C₄	C₂×C₄×Dic₄₇	direct (N × H)
97. C₂²	C₄²×Dic₄₇	direct (N × H)
103. C₄	C₂²×C₄×Dic₄₇	direct (N × H)
104. C₂	C₂×C₄²×Dic₄₇	direct (N × H)

H ≅ G/N

type

2. C₂×C₄²×Dic₄₇

C₂

direct (N × H)

5. C₂²×C₄×Dic₄₇

C₄

direct (N × H)

7. C₄²×Dic₄₇

C₂²

direct (N × H)

9. C₂²×C₄²×C₄₇

C₄

semidirect

13. C₂×C₄×Dic₄₇

C₂×C₄

direct (N × H)

19. C₂×C₄²×C₄₇

C₂×C₄

semidirect

24. C₂²×C₄×C₄₇

C₄²

semidirect

28. C₂²×Dic₄₇

C₄²

direct (N × H)

31. C₄²×C₄₇

C₂²×C₄

semidirect

32. C₄×Dic₄₇

C₂²×C₄

direct (N × H)

38. C₂×C₄×C₄₇

C₂×C₄²

semidirect

41. C₂×Dic₄₇

C₂×C₄²

direct (N × H)

44. C₂²×C₄₇

C₄³

semidirect

49. C₄×C₄₇

C₂²×C₄²

semidirect

50. Dic₄₇

C₂²×C₄²

direct (N × H)

54. C₂×C₄₇

C₂×C₄³

semidirect

61. C₂²×C₄²

Dic₄₇

direct (N × H)

64. C₄₇

C₂²×C₄³

semidirect

73. C₂×C₄²

C₂×Dic₄₇

direct (N × H)

81. C₂²×C₄

C₄×Dic₄₇

direct (N × H)

86. C₄²

C₂²×Dic₄₇

direct (N × H)

94. C₂×C₄

C₂×C₄×Dic₄₇

direct (N × H)

97. C₂²

C₄²×Dic₄₇

direct (N × H)

103. C₄

C₂²×C₄×Dic₄₇

direct (N × H)

104. C₂

C₂×C₄²×Dic₄₇

direct (N × H)

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

subgroup	index
C₂³×C₄×Dic₄₇	2
C₂×C₄²×Dic₄₇	2
C₂³×C₄²×C₄₇	2
C₂²×C₄³	47

subgroup

index

C₂³×C₄×Dic₄₇

C₂×C₄²×Dic₄₇

C₂³×C₄²×C₄₇

C₂²×C₄³

Subgroups — C₂²×C₄²×Dic₄₇ #26374443

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C22×C42×Dic47 #26374443

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄²×Dic₄₇ #26374443