Group #21438817 — subgroups

Subgroups — C₂²×Q₁₆×D₅₉ #21438817

Order 7552.

1464 conjugacy classes of subgroups; only the 565 normal subgroups are shown.

normal subgroups only

This group has 1464 conjugacy classes of subgroups — too many to draw. Only its 565 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×Q₈×D₅₉	C₂	1	2	yes	maximal
2	C₂²×Dic₂₃₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#11745672	C₂	1	2	yes	maximal, complemented
4	C₂×Q₁₆×D₅₉	C₂	1	24	yes	maximal, direct product factor
5	C₂²×C₈×D₅₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×Dic₁₁₈	—	2	2	no	—
8	C₂²×Dic₁₁₈	C₂²	1	2	yes	—
9	C₂×Q₈×D₅₉	—	2	12	no	—
10	C₂×Q₈×D₅₉	C₂²	1	24	yes	—
11	C₂²×C₄×D₅₉	—	2	2	no	—
12	C₂²×C₄×D₅₉	C₂²	1	1	yes	characteristic
13	C₂×Dic₂₃₆	C₂²	1	12	yes	complemented
14	C₂×#11745672	C₂²	1	24	yes	complemented
15	Q₁₆×D₅₉	C₂²	1	64	yes	direct product factor
16	C₂²×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₈×D₅₉	C₂²	1	12	yes	—
18	C₂²×Q₈×C₅₉	C₂²	1	2	yes	—
19	C₂×Q₁₆×C₅₉	C₂²	1	12	yes	complemented
20	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
21	C₂³×D₅₉	D₄	1	1	yes	characteristic
22	C₂×Q₈×C₅₉	—	2	6	no	—
23	C₂×Q₈×C₅₉	C₂³	1	12	yes	—
24	C₂²×C₄×C₅₉	—	2	2	no	—
25	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
26	C₂²×Dic₅₉	—	2	2	no	—
27	C₂²×Dic₅₉	D₄	1	1	yes	characteristic
28	Q₈×D₅₉	—	2	48	no	—
29	Q₈×D₅₉	C₂³	1	32	yes	—
30	C₂×C₄×D₅₉	—	2	24	no	—
31	C₂×C₄×D₅₉	D₄	1	6	yes	—
32	C₂×C₄×D₅₉	C₂³	1	6	yes	—
33	C₂×Dic₁₁₈	—	2	6	no	—
34	C₂×Dic₁₁₈	—	2	24	no	—
35	C₂×Dic₁₁₈	C₂³	1	12	yes	—
36	Q₁₆×C₅₉	C₂³	1	16	yes	complemented
37	C₂×C₈×C₅₉	C₂³	1	6	yes	—
38	#11745672	C₂³	1	32	yes	complemented
39	C₂×C₅₉⋊C₈	C₂³	1	6	yes	—
40	Dic₂₃₆	C₂³	1	16	yes	complemented
41	C₈×D₅₉	C₂³	1	16	yes	—
42	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
43	C₂²×D₅₉	Q₁₆	1	8	yes	direct product factor
44	C₂²×D₅₉	C₂×D₄	1	6	yes	—
45	Q₈×C₅₉	—	2	12	no	—
46	Q₈×C₅₉	C₂⁴	1	8	yes	—
47	C₂×C₄×C₅₉	—	2	12	no	—
48	C₂×C₄×C₅₉	C₂×D₄	1	3	yes	—
49	C₂×C₄×C₅₉	C₂⁴	1	3	yes	—
50	C₂×Dic₅₉	—	2	12	no	—
51	C₂×Dic₅₉	C₂×D₄	1	6	yes	—
52	C₄×D₅₉	—	2	32	no	—
53	C₄×D₅₉	C₂×D₄	1	12	yes	—
54	C₄×D₅₉	C₂⁴	1	4	yes	—
55	Dic₁₁₈	—	2	12	no	—
56	Dic₁₁₈	—	2	32	no	—
57	Dic₁₁₈	C₂⁴	1	8	yes	—
58	C₈×C₅₉	C₂⁴	1	4	yes	—
59	C₅₉⋊C₈	C₂⁴	1	4	yes	—
60	C₂²×C₅₉	C₂×Q₁₆	1	4	yes	complemented
61	C₂²×C₅₉	C₂²×D₄	1	3	yes	—
62	C₂×D₅₉	C₂×Q₁₆	1	24	yes	direct product factor
63	C₂×D₅₉	C₂²×D₄	1	4	yes	—
64	C₄×C₅₉	—	2	8	no	—
65	C₄×C₅₉	C₂²×D₄	1	3	yes	—
66	C₄×C₅₉	C₂⁵	1	1	yes	characteristic
67	Dic₅₉	—	2	8	no	—
68	Dic₅₉	C₂²×D₄	1	4	yes	—
69	C₂³×Q₁₆	—	59	1	no	maximal, Sylow
70	C₂×C₅₉	C₂²×Q₁₆	1	6	yes	complemented
71	C₂×C₅₉	C₂³×D₄	1	1	yes	characteristic
72	D₅₉	C₂²×Q₁₆	1	8	yes	direct product factor
73	C₂³×Q₈	—	59	2	no	—
74	C₂²×Q₁₆	—	59	2	no	—
75	C₂²×Q₁₆	—	59	24	no	—
76	C₂²×Q₁₆	—	59	1	no	—
77	C₂²×Q₁₆	D₅₉	1	1	yes	characteristic, direct product factor
78	C₂³×C₈	—	59	1	no	—
79	C₅₉	C₂³×Q₁₆	1	1	yes	characteristic, complemented, Sylow
80	C₂²×Q₈	—	59	24	no	—
81	C₂²×Q₈	—	59	2	no	—
82	C₂²×Q₈	—	118	2	no	—
83	C₂²×Q₈	—	118	12	no	—
84	C₂²×Q₈	C₂×D₅₉	1	2	yes	—
85	C₂³×C₄	—	59	1	no	—
86	C₂³×C₄	—	118	2	no	—
87	C₂×Q₁₆	—	59	24	no	—
88	C₂×Q₁₆	—	59	64	no	—
89	C₂×Q₁₆	—	59	12	no	—
90	C₂×Q₁₆	C₂×D₅₉	1	12	yes	direct product factor
91	C₂²×C₈	—	59	1	no	—
92	C₂²×C₈	—	59	12	no	—
93	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
94	C₂⁴	—	59	1	no	—
95	C₂×Q₈	—	2	6	no	—
96	C₂×Q₈	—	59	32	no	—
97	C₂×Q₈	—	59	12	no	—
98	C₂×Q₈	—	118	24	no	—
99	C₂×Q₈	—	118	48	no	—
100	C₂×Q₈	—	118	6	no	—
101	C₂×Q₈	C₂²×D₅₉	1	12	yes	—
102	C₂²×C₄	—	2	2	no	—
103	C₂²×C₄	—	59	1	no	—
104	C₂²×C₄	—	59	6	no	—
105	C₂²×C₄	—	59	6	no	—
106	C₂²×C₄	—	118	24	no	—
107	C₂²×C₄	—	118	2	no	—
108	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
109	Q₁₆	—	59	32	no	—
110	Q₁₆	—	59	16	no	—
111	Q₁₆	C₂²×D₅₉	1	16	yes	direct product factor
112	C₂×C₈	—	59	6	no	—
113	C₂×C₈	—	59	16	no	—
114	C₂×C₈	C₂²×D₅₉	1	6	yes	—
115	C₂³	—	59	8	no	—
116	C₂³	—	59	6	no	—
117	C₂³	D₄×D₅₉	1	1	yes	characteristic
118	Q₈	—	2	12	no	—
119	Q₈	—	59	8	no	—
120	Q₈	—	118	32	no	—
121	Q₈	—	118	12	no	—
122	Q₈	C₂³×D₅₉	1	8	yes	—
123	C₂×C₄	—	2	12	no	—
124	C₂×C₄	—	59	6	no	—
125	C₂×C₄	—	59	12	no	—
126	C₂×C₄	—	59	4	no	—
127	C₂×C₄	—	118	32	no	—
128	C₂×C₄	—	118	12	no	—
129	C₂×C₄	D₄×D₅₉	1	3	yes	—
130	C₂×C₄	C₂³×D₅₉	1	3	yes	—
131	C₈	—	59	4	no	—
132	C₈	C₂³×D₅₉	1	4	yes	—
133	C₂²	—	59	24	no	—
134	C₂²	—	59	4	no	—
135	C₂²	Q₁₆×D₅₉	1	4	yes	direct product factor
136	C₂²	C₂×D₄×D₅₉	1	3	yes	—
137	C₄	—	2	8	no	—
138	C₄	—	59	4	no	—
139	C₄	—	118	8	no	—
140	C₄	C₂×D₄×D₅₉	1	3	yes	—
141	C₄	C₂⁴×D₅₉	1	1	yes	characteristic
142	C₂	—	59	8	no	—
143	C₂	C₂×Q₁₆×D₅₉	1	6	yes	direct product factor
144	C₂	C₂²×D₄×D₅₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Q₈×D₅₉

C₂

yes

maximal

C₂²×Dic₂₃₆

C₂

yes

maximal, characteristic, complemented

C₂²×#11745672

C₂

yes

maximal, complemented

C₂×Q₁₆×D₅₉

C₂

yes

maximal, direct product factor

C₂²×C₈×D₅₉

C₂

yes

maximal, characteristic

C₂²×Q₁₆×C₅₉

C₂

yes

maximal, characteristic, complemented

C₂²×Dic₁₁₈

—

C₂²×Dic₁₁₈

C₂²

yes

—

C₂×Q₈×D₅₉

—

C₂×Q₈×D₅₉

C₂²

yes

—

C₂²×C₄×D₅₉

—

C₂²×C₄×D₅₉

C₂²

yes

characteristic

C₂×Dic₂₃₆

C₂²

yes

complemented

C₂×#11745672

C₂²

yes

complemented

Q₁₆×D₅₉

C₂²

yes

direct product factor

C₂²×C₅₉⋊C₈

C₂²

yes

characteristic

C₂×C₈×D₅₉

C₂²

yes

—

C₂²×Q₈×C₅₉

C₂²

yes

—

C₂×Q₁₆×C₅₉

C₂²

yes

complemented

C₂²×C₈×C₅₉

C₂²

yes

characteristic

C₂³×D₅₉

D₄

yes

characteristic

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

C₂³

yes

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

—

C₂²×Dic₅₉

D₄

yes

characteristic

Q₈×D₅₉

—

Q₈×D₅₉

C₂³

yes

—

C₂×C₄×D₅₉

—

C₂×C₄×D₅₉

D₄

yes

—

C₂×C₄×D₅₉

C₂³

yes

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

C₂³

yes

—

Q₁₆×C₅₉

C₂³

yes

complemented

C₂×C₈×C₅₉

C₂³

yes

—

#11745672

C₂³

yes

complemented

C₂×C₅₉⋊C₈

C₂³

yes

—

Dic₂₃₆

C₂³

yes

complemented

C₈×D₅₉

C₂³

yes

—

C₂³×C₅₉

C₂×D₄

yes

characteristic

C₂²×D₅₉

Q₁₆

yes

direct product factor

C₂²×D₅₉

C₂×D₄

yes

—

Q₈×C₅₉

—

Q₈×C₅₉

C₂⁴

yes

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

C₂⁴

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

C₂×D₄

yes

—

C₄×D₅₉

—

C₄×D₅₉

C₂×D₄

yes

—

C₄×D₅₉

C₂⁴

yes

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

C₂⁴

yes

—

C₈×C₅₉

C₂⁴

yes

—

C₅₉⋊C₈

C₂⁴

yes

—

C₂²×C₅₉

C₂×Q₁₆

yes

complemented

C₂²×C₅₉

C₂²×D₄

yes

—

C₂×D₅₉

C₂×Q₁₆

yes

direct product factor

C₂×D₅₉

C₂²×D₄

yes

—

C₄×C₅₉

—

C₄×C₅₉

C₂²×D₄

yes

—

C₄×C₅₉

C₂⁵

yes

characteristic

Dic₅₉

—

Dic₅₉

C₂²×D₄

yes

—

C₂³×Q₁₆

—

maximal, Sylow

C₂×C₅₉

C₂²×Q₁₆

yes

complemented

C₂×C₅₉

C₂³×D₄

yes

characteristic

D₅₉

C₂²×Q₁₆

yes

direct product factor

C₂³×Q₈

—

C₂²×Q₁₆

—

C₂²×Q₁₆

—

C₂²×Q₁₆

—

C₂²×Q₁₆

D₅₉

yes

characteristic, direct product factor

C₂³×C₈

—

C₅₉

C₂³×Q₁₆

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×Q₈

—

C₂²×Q₈

—

118

—

C₂²×Q₈

—

118

—

C₂²×Q₈

C₂×D₅₉

yes

—

C₂³×C₄

—

C₂³×C₄

—

118

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

C₂×D₅₉

yes

direct product factor

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×D₅₉

yes

characteristic

C₂⁴

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

118

—

C₂×Q₈

—

118

—

100

C₂×Q₈

—

118

—

101

C₂×Q₈

C₂²×D₅₉

yes

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

—

118

—

107

C₂²×C₄

—

118

—

108

C₂²×C₄

C₂²×D₅₉

yes

characteristic

109

Q₁₆

—

110

Q₁₆

—

111

Q₁₆

C₂²×D₅₉

yes

direct product factor

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

C₂²×D₅₉

yes

—

115

C₂³

—

116

C₂³

—

117

C₂³

D₄×D₅₉

yes

characteristic

118

Q₈

—

119

Q₈

—

120

Q₈

—

118

—

121

Q₈

—

118

—

122

Q₈

C₂³×D₅₉

yes

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

118

—

128

C₂×C₄

—

118

—

129

C₂×C₄

D₄×D₅₉

yes

—

130

C₂×C₄

C₂³×D₅₉

yes

—

131

C₈

—

132

C₈

C₂³×D₅₉

yes

—

133

C₂²

—

134

C₂²

—

135

C₂²

Q₁₆×D₅₉

yes

direct product factor

136

C₂²

C₂×D₄×D₅₉

yes

—

137

C₄

—

138

C₄

—

139

C₄

—

118

—

140

C₄

C₂×D₄×D₅₉

yes

—

141

C₄

C₂⁴×D₅₉

yes

characteristic

142

C₂

—

143

C₂

C₂×Q₁₆×D₅₉

yes

direct product factor

144

C₂

C₂²×D₄×D₅₉

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₅₉
commutator	C₄×C₅₉	C₂⁵
Frattini	C₄	C₂⁴×D₅₉
Fitting	C₂²×Q₁₆×C₅₉	C₂
socle	C₂³×C₅₉	C₂×D₄
hypercenter	C₂²×Q₁₆	D₅₉
hypocenter	C₅₉	C₂³×Q₁₆
LCS	G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×Q₁₆

subgroup

G/H

center

C₂³

D₄×D₅₉

commutator

C₄×C₅₉

C₂⁵

Frattini

C₄

C₂⁴×D₅₉

Fitting

C₂²×Q₁₆×C₅₉

C₂

socle

C₂³×C₅₉

C₂×D₄

hypercenter

C₂²×Q₁₆

D₅₉

hypocenter

C₅₉

C₂³×Q₁₆

LCS

G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×Q₁₆

N	H ≅ G/N	type
2. C₂²×Dic₂₃₆	C₂	semidirect
3. C₂²×#11745672	C₂	semidirect
4. C₂×Q₁₆×D₅₉	C₂	direct (N × H)
6. C₂²×Q₁₆×C₅₉	C₂	semidirect
13. C₂×Dic₂₃₆	C₂²	semidirect
14. C₂×#11745672	C₂²	semidirect
15. Q₁₆×D₅₉	C₂²	direct (N × H)
19. C₂×Q₁₆×C₅₉	C₂²	semidirect
36. Q₁₆×C₅₉	C₂³	semidirect
38. #11745672	C₂³	semidirect
40. Dic₂₃₆	C₂³	semidirect
43. C₂²×D₅₉	Q₁₆	direct (N × H)
60. C₂²×C₅₉	C₂×Q₁₆	semidirect
62. C₂×D₅₉	C₂×Q₁₆	direct (N × H)
70. C₂×C₅₉	C₂²×Q₁₆	semidirect
72. D₅₉	C₂²×Q₁₆	direct (N × H)
77. C₂²×Q₁₆	D₅₉	direct (N × H)
79. C₅₉	C₂³×Q₁₆	semidirect
90. C₂×Q₁₆	C₂×D₅₉	direct (N × H)
111. Q₁₆	C₂²×D₅₉	direct (N × H)
135. C₂²	Q₁₆×D₅₉	direct (N × H)
143. C₂	C₂×Q₁₆×D₅₉	direct (N × H)

H ≅ G/N

type

2. C₂²×Dic₂₃₆

C₂

semidirect

3. C₂²×#11745672

C₂

semidirect

4. C₂×Q₁₆×D₅₉

C₂

direct (N × H)

6. C₂²×Q₁₆×C₅₉

C₂

semidirect

13. C₂×Dic₂₃₆

C₂²

semidirect

14. C₂×#11745672

C₂²

semidirect

15. Q₁₆×D₅₉

C₂²

direct (N × H)

19. C₂×Q₁₆×C₅₉

C₂²

semidirect

36. Q₁₆×C₅₉

C₂³

semidirect

38. #11745672

C₂³

semidirect

40. Dic₂₃₆

C₂³

semidirect

43. C₂²×D₅₉

Q₁₆

direct (N × H)

60. C₂²×C₅₉

C₂×Q₁₆

semidirect

62. C₂×D₅₉

C₂×Q₁₆

direct (N × H)

70. C₂×C₅₉

C₂²×Q₁₆

semidirect

72. D₅₉

C₂²×Q₁₆

direct (N × H)

77. C₂²×Q₁₆

D₅₉

direct (N × H)

79. C₅₉

C₂³×Q₁₆

semidirect

90. C₂×Q₁₆

C₂×D₅₉

direct (N × H)

111. Q₁₆

C₂²×D₅₉

direct (N × H)

135. C₂²

Q₁₆×D₅₉

direct (N × H)

143. C₂

C₂×Q₁₆×D₅₉

direct (N × H)

Subgroups — C₂²×Q₁₆×D₅₉ #21438817

Conjugacy classes of subgroups

Complements

Subgroups — C22×Q16×D59 #21438817

Conjugacy classes of subgroups

Complements

Subgroups — C₂²×Q₁₆×D₅₉ #21438817