Group #20256663 — subgroups

Subgroups — C₂²×#13094023 #20256663

Order 6016.

1624 conjugacy classes of subgroups; only the 725 normal subgroups are shown.

normal subgroups only

This group has 1624 conjugacy classes of subgroups — too many to draw. Only its 725 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×Dic₉₄	C₂	1	2	yes	maximal
2	C₂×#13094023	C₂	1	24	yes	maximal, direct product factor
3	C₂²×C₄⋊Dic₄₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×Dic₄₇⋊C₄	C₂	1	2	yes	maximal
5	C₂²×C₄×Dic₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄⋊C₄×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₉₄	—	2	16	no	—
8	C₂²×Dic₉₄	C₂²	1	24	yes	—
9	C₂³×Dic₄₇	C₂²	1	2	yes	—
10	C₂³×Dic₄₇	C₂²	1	1	yes	characteristic
11	C₂³×Dic₄₇	C₂²	1	1	yes	characteristic
12	#13094023	C₂²	1	64	yes	direct product factor
13	C₂×C₄⋊Dic₄₇	C₂²	1	12	yes	—
14	C₂×Dic₄₇⋊C₄	C₂²	1	24	yes	—
15	C₂×C₄×Dic₄₇	C₂²	1	12	yes	—
16	C₂³×C₄×C₄₇	C₂²	1	2	yes	—
17	C₂³×C₄×C₄₇	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₄₇	C₂²	1	12	yes	—
19	C₂⁴×C₄₇	C₂³	1	1	yes	characteristic
20	C₂²×C₄×C₄₇	—	2	8	no	—
21	C₂²×C₄×C₄₇	Q₈	1	8	yes	complemented
22	C₂²×C₄×C₄₇	C₂³	1	12	yes	—
23	C₂²×C₄×C₄₇	C₂³	1	6	yes	—
24	C₂²×Dic₄₇	—	2	8	no	—
25	C₂²×Dic₄₇	D₄	1	8	yes	—
26	C₂²×Dic₄₇	Q₈	1	8	yes	complemented
27	C₂²×Dic₄₇	C₂³	1	12	yes	—
28	C₂²×Dic₄₇	C₂³	1	6	yes	—
29	C₂²×Dic₄₇	C₂³	1	6	yes	—
30	C₂×Dic₉₄	—	2	96	no	—
31	C₂×Dic₉₄	C₂³	1	32	yes	—
32	C₄⋊C₄×C₄₇	C₂³	1	16	yes	—
33	C₄⋊Dic₄₇	C₂³	1	16	yes	—
34	Dic₄₇⋊C₄	C₂³	1	32	yes	—
35	C₄×Dic₄₇	C₂³	1	16	yes	—
36	C₂³×C₄₇	C₂×D₄	1	4	yes	—
37	C₂³×C₄₇	C₂×Q₈	1	4	yes	—
38	C₂³×C₄₇	C₂×Q₈	1	4	yes	—
39	C₂³×C₄₇	C₂⁴	1	3	yes	—
40	C₂×C₄×C₄₇	—	2	24	no	—
41	C₂×C₄×C₄₇	C₂×Q₈	1	24	yes	complemented
42	C₂×C₄×C₄₇	C₂⁴	1	8	yes	—
43	C₂×C₄×C₄₇	C₂⁴	1	4	yes	—
44	C₂×Dic₄₇	—	2	24	no	—
45	C₂×Dic₄₇	C₂×D₄	1	24	yes	—
46	C₂×Dic₄₇	C₂×Q₈	1	24	yes	complemented
47	C₂×Dic₄₇	C₂⁴	1	8	yes	—
48	C₂×Dic₄₇	C₂⁴	1	4	yes	—
49	C₂×Dic₄₇	C₂⁴	1	4	yes	—
50	Dic₉₄	—	2	64	no	—
51	C₂²×C₄₇	C₄⋊Q₈	1	16	yes	complemented
52	C₂²×C₄₇	C₂²×D₄	1	6	yes	—
53	C₂²×C₄₇	C₂²×Q₈	1	6	yes	—
54	C₂²×C₄₇	C₂²×Q₈	1	6	yes	—
55	C₂²×C₄₇	C₂⁵	1	1	yes	characteristic
56	C₄×C₄₇	—	2	8	no	—
57	C₄×C₄₇	C₂²×Q₈	1	8	yes	complemented
58	Dic₄₇	—	2	8	no	—
59	Dic₄₇	C₂²×D₄	1	8	yes	—
60	Dic₄₇	C₂²×Q₈	1	8	yes	complemented
61	C₂²×C₄⋊Q₈	—	47	1	no	maximal, Sylow
62	C₂×C₄₇	C₂×C₄⋊Q₈	1	12	yes	complemented
63	C₂×C₄₇	C₂³×D₄	1	1	yes	characteristic
64	C₂×C₄₇	C₂³×Q₈	1	1	yes	characteristic
65	C₂×C₄₇	C₂³×Q₈	1	1	yes	characteristic
66	C₂³×Q₈	—	47	2	no	—
67	C₂×C₄⋊Q₈	—	47	24	no	—
68	C₂²×C₄⋊C₄	—	47	2	no	—
69	C₂²×C₄⋊C₄	—	47	1	no	—
70	C₂²×C₄⋊C₄	D₄₇	1	1	yes	characteristic
71	C₂²×C₄²	—	47	1	no	—
72	C₄₇	C₂²×C₄⋊Q₈	1	1	yes	characteristic, complemented, Sylow
73	C₂²×Q₈	—	47	24	no	—
74	C₂²×Q₈	—	94	16	no	—
75	C₂³×C₄	—	47	1	no	—
76	C₂³×C₄	—	47	1	no	—
77	C₂³×C₄	—	47	2	no	—
78	C₂³×C₄	C₂×D₄₇	1	2	yes	—
79	C₂³×C₄	C₂×D₄₇	1	1	yes	characteristic
80	C₄⋊Q₈	—	47	64	no	—
81	C₂×C₄⋊C₄	—	47	24	no	—
82	C₂×C₄⋊C₄	—	47	12	no	—
83	C₂×C₄⋊C₄	C₂×D₄₇	1	12	yes	—
84	C₂×C₄²	—	47	12	no	—
85	C₂⁴	C₂²×D₄₇	1	1	yes	characteristic
86	C₂×Q₈	—	47	32	no	—
87	C₂×Q₈	—	94	96	no	—
88	C₂²×C₄	—	2	8	no	—
89	C₂²×C₄	—	47	8	no	—
90	C₂²×C₄	—	47	8	no	—
91	C₂²×C₄	—	47	6	no	—
92	C₂²×C₄	—	47	6	no	—
93	C₂²×C₄	—	47	12	no	—
94	C₂²×C₄	—	94	8	no	—
95	C₂²×C₄	Dic₉₄	1	8	yes	complemented
96	C₂²×C₄	C₂²×D₄₇	1	12	yes	—
97	C₂²×C₄	C₂²×D₄₇	1	6	yes	—
98	C₄⋊C₄	—	47	32	no	—
99	C₄⋊C₄	—	47	16	no	—
100	C₄⋊C₄	C₂²×D₄₇	1	16	yes	—
101	C₄²	—	47	16	no	—
102	C₂³	C₂×Dic₉₄	1	4	yes	—
103	C₂³	D₄×D₄₇	1	4	yes	—
104	C₂³	Q₈×D₄₇	1	4	yes	—
105	C₂³	C₂³×D₄₇	1	3	yes	—
106	Q₈	—	94	64	no	—
107	C₂×C₄	—	2	24	no	—
108	C₂×C₄	—	47	24	no	—
109	C₂×C₄	—	47	24	no	—
110	C₂×C₄	—	47	4	no	—
111	C₂×C₄	—	47	4	no	—
112	C₂×C₄	—	47	8	no	—
113	C₂×C₄	—	94	24	no	—
114	C₂×C₄	C₂×Dic₉₄	1	24	yes	complemented
115	C₂×C₄	C₂³×D₄₇	1	8	yes	—
116	C₂×C₄	C₂³×D₄₇	1	4	yes	—
117	C₂²	#13094023	1	16	yes	direct product factor
118	C₂²	C₂×D₄×D₄₇	1	6	yes	—
119	C₂²	C₂×Q₈×D₄₇	1	6	yes	—
120	C₂²	C₂²×Dic₉₄	1	6	yes	—
121	C₂²	C₂⁴×D₄₇	1	1	yes	characteristic
122	C₄	—	2	8	no	—
123	C₄	—	47	8	no	—
124	C₄	—	47	8	no	—
125	C₄	—	94	8	no	—
126	C₄	C₂²×Dic₉₄	1	8	yes	complemented
127	C₂	C₂×#13094023	1	12	yes	direct product factor
128	C₂	C₂²×D₄×D₄₇	1	1	yes	characteristic
129	C₂	C₂²×Q₈×D₄₇	1	1	yes	characteristic
130	C₂	C₂³×Dic₉₄	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×Dic₉₄

C₂

yes

maximal

C₂×#13094023

C₂

yes

maximal, direct product factor

C₂²×C₄⋊Dic₄₇

C₂

yes

maximal, characteristic

C₂²×Dic₄₇⋊C₄

C₂

yes

maximal

C₂²×C₄×Dic₄₇

C₂

yes

maximal, characteristic

C₂²×C₄⋊C₄×C₄₇

C₂

yes

maximal, characteristic

C₂²×Dic₉₄

—

C₂²×Dic₉₄

C₂²

yes

—

C₂³×Dic₄₇

C₂²

yes

—

C₂³×Dic₄₇

C₂²

yes

characteristic

C₂³×Dic₄₇

C₂²

yes

characteristic

#13094023

C₂²

yes

direct product factor

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×Dic₄₇⋊C₄

C₂²

yes

—

C₂×C₄×Dic₄₇

C₂²

yes

—

C₂³×C₄×C₄₇

C₂²

yes

—

C₂³×C₄×C₄₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₄₇

C₂²

yes

—

C₂⁴×C₄₇

C₂³

yes

characteristic

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

Q₈

yes

complemented

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

D₄

yes

—

C₂²×Dic₄₇

Q₈

yes

complemented

C₂²×Dic₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

—

C₂×Dic₉₄

—

C₂×Dic₉₄

C₂³

yes

—

C₄⋊C₄×C₄₇

C₂³

yes

—

C₄⋊Dic₄₇

C₂³

yes

—

Dic₄₇⋊C₄

C₂³

yes

—

C₄×Dic₄₇

C₂³

yes

—

C₂³×C₄₇

C₂×D₄

yes

—

C₂³×C₄₇

C₂×Q₈

yes

—

C₂³×C₄₇

C₂×Q₈

yes

—

C₂³×C₄₇

C₂⁴

yes

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×Q₈

yes

complemented

C₂×C₄×C₄₇

C₂⁴

yes

—

C₂×C₄×C₄₇

C₂⁴

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

C₂×D₄

yes

—

C₂×Dic₄₇

C₂×Q₈

yes

complemented

C₂×Dic₄₇

C₂⁴

yes

—

C₂×Dic₄₇

C₂⁴

yes

—

C₂×Dic₄₇

C₂⁴

yes

—

Dic₉₄

—

C₂²×C₄₇

C₄⋊Q₈

yes

complemented

C₂²×C₄₇

C₂²×D₄

yes

—

C₂²×C₄₇

C₂²×Q₈

yes

—

C₂²×C₄₇

C₂²×Q₈

yes

—

C₂²×C₄₇

C₂⁵

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

C₂²×Q₈

yes

complemented

Dic₄₇

—

Dic₄₇

C₂²×D₄

yes

—

Dic₄₇

C₂²×Q₈

yes

complemented

C₂²×C₄⋊Q₈

—

maximal, Sylow

C₂×C₄₇

C₂×C₄⋊Q₈

yes

complemented

C₂×C₄₇

C₂³×D₄

yes

characteristic

C₂×C₄₇

C₂³×Q₈

yes

characteristic

C₂×C₄₇

C₂³×Q₈

yes

characteristic

C₂³×Q₈

—

C₂×C₄⋊Q₈

—

C₂²×C₄⋊C₄

—

C₂²×C₄⋊C₄

—

C₂²×C₄⋊C₄

D₄₇

yes

characteristic

C₂²×C₄²

—

C₄₇

C₂²×C₄⋊Q₈

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×Q₈

—

C₂³×C₄

—

C₂³×C₄

—

C₂³×C₄

—

C₂³×C₄

C₂×D₄₇

yes

—

C₂³×C₄

C₂×D₄₇

yes

characteristic

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₄₇

yes

—

C₂×C₄²

—

C₂⁴

C₂²×D₄₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

Dic₉₄

yes

complemented

C₂²×C₄

C₂²×D₄₇

yes

—

C₂²×C₄

C₂²×D₄₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

C₂²×D₄₇

yes

—

101

C₄²

—

102

C₂³

C₂×Dic₉₄

yes

—

103

C₂³

D₄×D₄₇

yes

—

104

C₂³

Q₈×D₄₇

yes

—

105

C₂³

C₂³×D₄₇

yes

—

106

Q₈

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

C₂×Dic₉₄

yes

complemented

115

C₂×C₄

C₂³×D₄₇

yes

—

116

C₂×C₄

C₂³×D₄₇

yes

—

117

C₂²

#13094023

yes

direct product factor

118

C₂²

C₂×D₄×D₄₇

yes

—

119

C₂²

C₂×Q₈×D₄₇

yes

—

120

C₂²

C₂²×Dic₉₄

yes

—

121

C₂²

C₂⁴×D₄₇

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

C₂²×Dic₉₄

yes

complemented

127

C₂

C₂×#13094023

yes

direct product factor

128

C₂

C₂²×D₄×D₄₇

yes

characteristic

129

C₂

C₂²×Q₈×D₄₇

yes

characteristic

130

C₂

C₂³×Dic₉₄

yes

characteristic

	subgroup	G/H
center	C₂⁴	C₂²×D₄₇
commutator	C₂²×C₄₇	C₂⁵
Frattini	C₂²	C₂⁴×D₄₇
Fitting	C₂²×C₄⋊C₄×C₄₇	C₂
socle	C₂⁴×C₄₇	C₂³
hypercenter	C₂²×C₄⋊C₄	D₄₇
hypocenter	C₄₇	C₂²×C₄⋊Q₈
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄⋊C₄

subgroup

G/H

center

C₂⁴

C₂²×D₄₇

commutator

C₂²×C₄₇

C₂⁵

Frattini

C₂²

C₂⁴×D₄₇

Fitting

C₂²×C₄⋊C₄×C₄₇

C₂

socle

C₂⁴×C₄₇

C₂³

hypercenter

C₂²×C₄⋊C₄

D₄₇

hypocenter

C₄₇

C₂²×C₄⋊Q₈

LCS

G ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂⁴ ◁ C₂²×C₄⋊C₄

N	H ≅ G/N	type
2. C₂×#13094023	C₂	direct (N × H)
12. #13094023	C₂²	direct (N × H)
21. C₂²×C₄×C₄₇	Q₈	semidirect
26. C₂²×Dic₄₇	Q₈	semidirect
41. C₂×C₄×C₄₇	C₂×Q₈	semidirect
46. C₂×Dic₄₇	C₂×Q₈	semidirect
51. C₂²×C₄₇	C₄⋊Q₈	semidirect
57. C₄×C₄₇	C₂²×Q₈	semidirect
60. Dic₄₇	C₂²×Q₈	semidirect
62. C₂×C₄₇	C₂×C₄⋊Q₈	semidirect
72. C₄₇	C₂²×C₄⋊Q₈	semidirect
95. C₂²×C₄	Dic₉₄	semidirect
114. C₂×C₄	C₂×Dic₉₄	semidirect
117. C₂²	#13094023	direct (N × H)
126. C₄	C₂²×Dic₉₄	semidirect
127. C₂	C₂×#13094023	direct (N × H)

H ≅ G/N

type

2. C₂×#13094023

C₂

direct (N × H)

12. #13094023

C₂²

direct (N × H)

21. C₂²×C₄×C₄₇

Q₈

semidirect

26. C₂²×Dic₄₇

Q₈

semidirect

41. C₂×C₄×C₄₇

C₂×Q₈

semidirect

46. C₂×Dic₄₇

C₂×Q₈

semidirect

51. C₂²×C₄₇

C₄⋊Q₈

semidirect

57. C₄×C₄₇

C₂²×Q₈

semidirect

60. Dic₄₇

C₂²×Q₈

semidirect

62. C₂×C₄₇

C₂×C₄⋊Q₈

semidirect

72. C₄₇

C₂²×C₄⋊Q₈

semidirect

95. C₂²×C₄

Dic₉₄

semidirect

114. C₂×C₄

C₂×Dic₉₄

semidirect

117. C₂²

#13094023

direct (N × H)

126. C₄

C₂²×Dic₉₄

semidirect

127. C₂

C₂×#13094023

direct (N × H)

Subgroups — C₂²×#13094023 #20256663

Conjugacy classes of subgroups

Complements

Subgroups — C22×#13094023 #20256663

Conjugacy classes of subgroups

Complements

Subgroups — C₂²×#13094023 #20256663