Group #30626705 — subgroups

Subgroups — C₂³×D₆₁⋊C₁₆ #30626705

Order 15616.

2590 conjugacy classes of subgroups; only the 1464 normal subgroups are shown.

normal subgroups only

This group has 2590 conjugacy classes of subgroups — too many to draw. Only its 1464 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×D₆₁⋊C₁₆	C₂	1	28	yes	maximal, direct product factor
2	C₂³×#11758187	C₂	1	2	yes	maximal, complemented
3	C₂³×C₈×D₆₁	C₂	1	1	yes	maximal, characteristic
4	C₂×D₆₁⋊C₁₆	C₂²	1	112	yes	direct product factor
5	C₂²×#11758187	C₂²	1	28	yes	complemented
6	C₂³×C₄×D₆₁	C₄	1	1	yes	characteristic
7	C₂³×#92418	C₂²	1	1	yes	characteristic
8	C₂²×C₈×D₆₁	C₄	1	14	yes	—
9	C₂²×C₈×D₆₁	C₂²	1	14	yes	—
10	C₂³×C₈×C₆₁	C₄	1	1	yes	characteristic
11	D₆₁⋊C₁₆	C₂³	1	64	yes	direct product factor
12	C₂×#11758187	C₂³	1	56	yes	complemented
13	C₂⁴×D₆₁	C₈	1	1	yes	characteristic
14	C₂³×Dic₆₁	C₈	1	1	yes	characteristic
15	C₂²×C₄×D₆₁	C₈	1	14	yes	—
16	C₂²×C₄×D₆₁	C₂×C₄	1	14	yes	—
17	C₂²×#92418	C₂×C₄	1	7	yes	—
18	C₂²×#92418	C₂³	1	7	yes	—
19	C₂×C₈×D₆₁	C₂×C₄	1	84	yes	—
20	C₂×C₈×D₆₁	C₂³	1	28	yes	—
21	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₆₁	C₂×C₄	1	14	yes	—
23	C₂⁴×C₆₁	C₂×C₈	1	1	yes	characteristic
24	C₂³×D₆₁	C₁₆	1	16	yes	complemented
25	C₂³×D₆₁	C₂×C₈	1	14	yes	—
26	C₂²×C₄×C₆₁	C₂×C₈	1	7	yes	—
27	C₂²×C₄×C₆₁	C₂²×C₄	1	7	yes	—
28	C₂²×Dic₆₁	C₂×C₈	1	14	yes	—
29	C₂×C₄×D₆₁	C₂×C₈	1	84	yes	—
30	C₂×C₄×D₆₁	C₂²×C₄	1	28	yes	—
31	C₂×C₈×C₆₁	C₂²×C₄	1	28	yes	—
32	C₂×#92418	C₂²×C₄	1	21	yes	—
33	C₂×#92418	C₂⁴	1	7	yes	—
34	C₈×D₆₁	C₂²×C₄	1	56	yes	—
35	C₈×D₆₁	C₂⁴	1	8	yes	—
36	#11758187	C₂⁴	1	16	yes	complemented
37	C₂³×C₆₁	C₂×C₁₆	1	8	yes	complemented
38	C₂³×C₆₁	C₂²×C₈	1	7	yes	—
39	C₂²×D₆₁	C₂×C₁₆	1	112	yes	complemented
40	C₂²×D₆₁	C₂²×C₈	1	28	yes	—
41	C₂×C₄×C₆₁	C₂²×C₈	1	21	yes	—
42	C₂×C₄×C₆₁	C₂³×C₄	1	7	yes	—
43	C₂×Dic₆₁	C₂²×C₈	1	28	yes	—
44	C₄×D₆₁	C₂²×C₈	1	56	yes	—
45	C₄×D₆₁	C₂³×C₄	1	8	yes	—
46	C₈×C₆₁	C₂³×C₄	1	8	yes	—
47	#92418	C₂³×C₄	1	7	yes	—
48	#92418	C₂⁵	1	1	yes	characteristic
49	C₂⁴×C₁₆	—	61	1	no	maximal, Sylow
50	C₂²×C₆₁	C₂²×C₁₆	1	28	yes	complemented
51	C₂²×C₆₁	C₂³×C₈	1	7	yes	—
52	C₂×D₆₁	C₂²×C₁₆	1	112	yes	complemented
53	C₂×D₆₁	C₂³×C₈	1	8	yes	—
54	C₄×C₆₁	C₂³×C₈	1	7	yes	—
55	C₄×C₆₁	C₂⁴×C₄	1	1	yes	characteristic
56	Dic₆₁	C₂³×C₈	1	8	yes	—
57	C₂⁴×C₈	—	61	1	no	—
58	C₂³×C₁₆	—	61	2	no	—
59	C₂³×C₁₆	—	61	28	no	—
60	C₂×C₆₁	C₂³×C₁₆	1	14	yes	complemented
61	C₂×C₆₁	C₂⁴×C₈	1	1	yes	characteristic
62	D₆₁	C₂³×C₁₆	1	16	yes	complemented
63	C₂⁴×C₄	—	61	1	no	—
64	C₂³×C₈	—	61	1	no	—
65	C₂³×C₈	—	61	14	no	—
66	C₂³×C₈	—	61	14	no	—
67	C₂³×C₈	#6923	1	1	yes	characteristic
68	C₂²×C₁₆	—	61	28	no	—
69	C₂²×C₁₆	—	61	112	no	—
70	C₆₁	C₂⁴×C₁₆	1	1	yes	characteristic, complemented, Sylow
71	C₂⁵	—	61	1	no	—
72	C₂³×C₄	—	61	1	no	—
73	C₂³×C₄	—	61	14	no	—
74	C₂³×C₄	—	61	14	no	—
75	C₂³×C₄	C₂×#6923	1	1	yes	characteristic
76	C₂²×C₈	—	61	7	no	—
77	C₂²×C₈	—	61	7	no	—
78	C₂²×C₈	—	61	84	no	—
79	C₂²×C₈	—	61	28	no	—
80	C₂²×C₈	C₂×#6923	1	14	yes	—
81	C₂×C₁₆	—	61	56	no	—
82	C₂×C₁₆	—	61	64	no	—
83	C₂⁴	—	61	16	no	—
84	C₂⁴	—	61	14	no	—
85	C₂⁴	D₆₁⋊C₈	1	1	yes	characteristic
86	C₂²×C₄	—	61	14	no	—
87	C₂²×C₄	—	61	84	no	—
88	C₂²×C₄	—	61	28	no	—
89	C₂²×C₄	D₆₁⋊C₈	1	7	yes	—
90	C₂²×C₄	C₂²×#6923	1	7	yes	—
91	C₂×C₈	—	61	7	no	—
92	C₂×C₈	—	61	21	no	—
93	C₂×C₈	—	61	56	no	—
94	C₂×C₈	—	61	8	no	—
95	C₂×C₈	C₂²×#6923	1	28	yes	—
96	C₁₆	—	61	16	no	—
97	C₂³	—	61	112	no	—
98	C₂³	—	61	28	no	—
99	C₂³	D₆₁⋊C₁₆	1	8	yes	direct product factor
100	C₂³	C₂×D₆₁⋊C₈	1	7	yes	—
101	C₂×C₄	—	61	28	no	—
102	C₂×C₄	—	61	56	no	—
103	C₂×C₄	—	61	8	no	—
104	C₂×C₄	C₂×D₆₁⋊C₈	1	21	yes	—
105	C₂×C₄	C₂³×#6923	1	7	yes	—
106	C₈	—	61	7	no	—
107	C₈	—	61	1	no	—
108	C₈	C₂³×#6923	1	8	yes	—
109	C₂²	—	61	8	no	—
110	C₂²	—	61	112	no	—
111	C₂²	C₂×D₆₁⋊C₁₆	1	28	yes	direct product factor
112	C₂²	C₂²×D₆₁⋊C₈	1	7	yes	—
113	C₄	—	61	8	no	—
114	C₄	C₂²×D₆₁⋊C₈	1	7	yes	—
115	C₄	C₂⁴×#6923	1	1	yes	characteristic
116	C₂	—	61	16	no	—
117	C₂	C₂²×D₆₁⋊C₁₆	1	14	yes	direct product factor
118	C₂	C₂³×D₆₁⋊C₈	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×D₆₁⋊C₁₆

C₂

yes

maximal, direct product factor

C₂³×#11758187

C₂

yes

maximal, complemented

C₂³×C₈×D₆₁

C₂

yes

maximal, characteristic

C₂×D₆₁⋊C₁₆

C₂²

112

yes

direct product factor

C₂²×#11758187

C₂²

yes

complemented

C₂³×C₄×D₆₁

C₄

yes

characteristic

C₂³×#92418

C₂²

yes

characteristic

C₂²×C₈×D₆₁

C₄

yes

—

C₂²×C₈×D₆₁

C₂²

yes

—

C₂³×C₈×C₆₁

C₄

yes

characteristic

D₆₁⋊C₁₆

C₂³

yes

direct product factor

C₂×#11758187

C₂³

yes

complemented

C₂⁴×D₆₁

C₈

yes

characteristic

C₂³×Dic₆₁

C₈

yes

characteristic

C₂²×C₄×D₆₁

C₈

yes

—

C₂²×C₄×D₆₁

C₂×C₄

yes

—

C₂²×#92418

C₂×C₄

yes

—

C₂²×#92418

C₂³

yes

—

C₂×C₈×D₆₁

C₂×C₄

yes

—

C₂×C₈×D₆₁

C₂³

yes

—

C₂³×C₄×C₆₁

C₂×C₄

yes

characteristic

C₂²×C₈×C₆₁

C₂×C₄

yes

—

C₂⁴×C₆₁

C₂×C₈

yes

characteristic

C₂³×D₆₁

C₁₆

yes

complemented

C₂³×D₆₁

C₂×C₈

yes

—

C₂²×C₄×C₆₁

C₂×C₈

yes

—

C₂²×C₄×C₆₁

C₂²×C₄

yes

—

C₂²×Dic₆₁

C₂×C₈

yes

—

C₂×C₄×D₆₁

C₂×C₈

yes

—

C₂×C₄×D₆₁

C₂²×C₄

yes

—

C₂×C₈×C₆₁

C₂²×C₄

yes

—

C₂×#92418

C₂²×C₄

yes

—

C₂×#92418

C₂⁴

yes

—

C₈×D₆₁

C₂²×C₄

yes

—

C₈×D₆₁

C₂⁴

yes

—

#11758187

C₂⁴

yes

complemented

C₂³×C₆₁

C₂×C₁₆

yes

complemented

C₂³×C₆₁

C₂²×C₈

yes

—

C₂²×D₆₁

C₂×C₁₆

112

yes

complemented

C₂²×D₆₁

C₂²×C₈

yes

—

C₂×C₄×C₆₁

C₂²×C₈

yes

—

C₂×C₄×C₆₁

C₂³×C₄

yes

—

C₂×Dic₆₁

C₂²×C₈

yes

—

C₄×D₆₁

C₂²×C₈

yes

—

C₄×D₆₁

C₂³×C₄

yes

—

C₈×C₆₁

C₂³×C₄

yes

—

#92418

C₂³×C₄

yes

—

#92418

C₂⁵

yes

characteristic

C₂⁴×C₁₆

—

maximal, Sylow

C₂²×C₆₁

C₂²×C₁₆

yes

complemented

C₂²×C₆₁

C₂³×C₈

yes

—

C₂×D₆₁

C₂²×C₁₆

112

yes

complemented

C₂×D₆₁

C₂³×C₈

yes

—

C₄×C₆₁

C₂³×C₈

yes

—

C₄×C₆₁

C₂⁴×C₄

yes

characteristic

Dic₆₁

C₂³×C₈

yes

—

C₂⁴×C₈

—

C₂³×C₁₆

—

C₂³×C₁₆

—

C₂×C₆₁

C₂³×C₁₆

yes

complemented

C₂×C₆₁

C₂⁴×C₈

yes

characteristic

D₆₁

C₂³×C₁₆

yes

complemented

C₂⁴×C₄

—

C₂³×C₈

—

C₂³×C₈

—

C₂³×C₈

—

C₂³×C₈

#6923

yes

characteristic

C₂²×C₁₆

—

C₂²×C₁₆

—

112

—

C₆₁

C₂⁴×C₁₆

yes

characteristic, complemented, Sylow

C₂⁵

—

C₂³×C₄

—

C₂³×C₄

—

C₂³×C₄

—

C₂³×C₄

C₂×#6923

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×#6923

yes

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂⁴

—

C₂⁴

—

C₂⁴

D₆₁⋊C₈

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₆₁⋊C₈

yes

—

C₂²×C₄

C₂²×#6923

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²×#6923

yes

—

C₁₆

—

C₂³

—

112

—

C₂³

—

C₂³

D₆₁⋊C₁₆

yes

direct product factor

100

C₂³

C₂×D₆₁⋊C₈

yes

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

C₂×D₆₁⋊C₈

yes

—

105

C₂×C₄

C₂³×#6923

yes

—

106

C₈

—

107

C₈

—

108

C₈

C₂³×#6923

yes

—

109

C₂²

—

110

C₂²

—

112

—

111

C₂²

C₂×D₆₁⋊C₁₆

yes

direct product factor

112

C₂²

C₂²×D₆₁⋊C₈

yes

—

113

C₄

—

114

C₄

C₂²×D₆₁⋊C₈

yes

—

115

C₄

C₂⁴×#6923

yes

characteristic

116

C₂

—

117

C₂

C₂²×D₆₁⋊C₁₆

yes

direct product factor

118

C₂

C₂³×D₆₁⋊C₈

yes

characteristic

	subgroup	G/H
center, hypercenter	C₂³×C₈	#6923
commutator, hypocenter	C₆₁	C₂⁴×C₁₆
Frattini	C₄	C₂⁴×#6923
Fitting	C₂³×C₈×C₆₁	C₄
socle	C₂⁴×C₆₁	C₂×C₈
prime radical	C₂⁴×D₆₁	C₈
cosocle	#92418	C₂⁵
LCS	G ▷ C₆₁
UCS	C₁ ◁ C₂³×C₈

subgroup

G/H

center, hypercenter

C₂³×C₈

#6923

commutator, hypocenter

C₆₁

C₂⁴×C₁₆

Frattini

C₄

C₂⁴×#6923

Fitting

C₂³×C₈×C₆₁

C₄

socle

C₂⁴×C₆₁

C₂×C₈

prime radical

C₂⁴×D₆₁

C₈

C₂⁵

G ▷ C₆₁

C₁ ◁ C₂³×C₈

N	H ≅ G/N	type
1. C₂²×D₆₁⋊C₁₆	C₂	direct (N × H)
2. C₂³×#11758187	C₂	semidirect
4. C₂×D₆₁⋊C₁₆	C₂²	direct (N × H)
5. C₂²×#11758187	C₂²	semidirect
11. D₆₁⋊C₁₆	C₂³	direct (N × H)
12. C₂×#11758187	C₂³	semidirect
24. C₂³×D₆₁	C₁₆	semidirect
36. #11758187	C₂⁴	semidirect
37. C₂³×C₆₁	C₂×C₁₆	semidirect
39. C₂²×D₆₁	C₂×C₁₆	semidirect
50. C₂²×C₆₁	C₂²×C₁₆	semidirect
52. C₂×D₆₁	C₂²×C₁₆	semidirect
60. C₂×C₆₁	C₂³×C₁₆	semidirect
62. D₆₁	C₂³×C₁₆	semidirect
70. C₆₁	C₂⁴×C₁₆	semidirect
99. C₂³	D₆₁⋊C₁₆	direct (N × H)
111. C₂²	C₂×D₆₁⋊C₁₆	direct (N × H)
117. C₂	C₂²×D₆₁⋊C₁₆	direct (N × H)

H ≅ G/N

type

1. C₂²×D₆₁⋊C₁₆

C₂

direct (N × H)

2. C₂³×#11758187

C₂

semidirect

4. C₂×D₆₁⋊C₁₆

C₂²

direct (N × H)

5. C₂²×#11758187

C₂²

semidirect

11. D₆₁⋊C₁₆

C₂³

direct (N × H)

12. C₂×#11758187

C₂³

semidirect

24. C₂³×D₆₁

C₁₆

semidirect

36. #11758187

C₂⁴

semidirect

37. C₂³×C₆₁

C₂×C₁₆

semidirect

39. C₂²×D₆₁

C₂×C₁₆

semidirect

50. C₂²×C₆₁

C₂²×C₁₆

semidirect

52. C₂×D₆₁

C₂²×C₁₆

semidirect

60. C₂×C₆₁

C₂³×C₁₆

semidirect

62. D₆₁

C₂³×C₁₆

semidirect

70. C₆₁

C₂⁴×C₁₆

semidirect

99. C₂³

D₆₁⋊C₁₆

direct (N × H)

111. C₂²

C₂×D₆₁⋊C₁₆

direct (N × H)

117. C₂

C₂²×D₆₁⋊C₁₆

direct (N × H)

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

subgroup	index
C₂²×D₆₁⋊C₁₆	2
C₂³×#11758187	2
C₂³×C₈×D₆₁	2
C₂⁴×C₁₆	61

subgroup

index

C₂²×D₆₁⋊C₁₆

C₂³×#11758187

C₂³×C₈×D₆₁

C₂⁴×C₁₆

Subgroups — C₂³×D₆₁⋊C₁₆ #30626705

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×D61⋊C16 #30626705

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×D₆₁⋊C₁₆ #30626705