Group #21423456 — subgroups

Subgroups — #21423456

Order 7552.

188 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 61 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	#16837779	C₂	1	1	yes	maximal, characteristic
2	#16837602	C₂	1	2	yes	maximal, complemented
3	#16837155	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837145	C₂	1	2	yes	maximal, complemented
5	C₈×Q₈×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11745671	—	2	2	no	—
7	C₂×D₄₇₂	—	2	1	no	—
8	C₂×D₄₇₂	—	2	1	no	—
9	#14273724	C₂²	1	2	yes	—
10	C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
11	#14273647	—	2	2	no	—
12	C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
13	C₄⋊C₈×C₅₉	—	2	1	no	—
14	C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₅₉	—	2	1	no	—
16	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₅₉	D₄	1	1	yes	characteristic
18	C₂×D₂₃₆	—	2	2	no	—
19	C₂×D₂₃₆	—	2	2	no	—
20	C₂×D₂₃₆	—	2	2	no	—
21	C₂×C₈×C₅₉	—	2	1	no	—
22	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
23	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
24	C₄⋊C₄×C₅₉	—	2	1	no	—
25	C₄⋊C₄×C₅₉	D₄	1	1	yes	characteristic
26	C₄²×C₅₉	—	2	1	no	—
27	C₄²×C₅₉	C₂³	1	1	yes	characteristic
28	#11745671	—	4	4	no	—
29	C₂×C₅₉⋊C₈	—	2	2	no	—
30	D₄₇₂	—	4	2	no	—
31	D₄₇₂	—	4	2	no	—
32	C₂²×D₅₉	—	4	2	no	—
33	Q₈×C₅₉	—	2	1	no	—
34	Q₈×C₅₉	D₈	1	2	yes	complemented
35	C₂×C₄×C₅₉	—	2	1	no	—
36	C₂×C₄×C₅₉	—	2	1	no	—
37	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
40	D₂₃₆	—	4	2	no	—
41	D₂₃₆	—	4	2	no	—
42	D₂₃₆	—	4	2	no	—
43	D₂₃₆	—	4	2	no	—
44	D₂₃₆	—	8	2	no	—
45	C₈×C₅₉	—	2	1	no	—
46	C₈×C₅₉	—	4	1	no	—
47	C₈×C₅₉	SD₁₆	1	2	yes	complemented
48	C₅₉⋊C₈	—	4	2	no	—
49	C₂²×C₅₉	#121	1	1	yes	characteristic
50	C₂×D₅₉	—	8	2	no	—
51	C₂×D₅₉	—	8	2	no	—
52	C₂×D₅₉	—	8	2	no	—
53	C₄×C₅₉	—	2	2	no	—
54	C₄×C₅₉	—	2	1	no	—
55	C₄×C₅₉	—	4	1	no	—
56	C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
57	C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
58	C₄×C₅₉	#135	1	1	yes	characteristic
59	C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
60	#1668	—	59	1	no	maximal, Sylow
61	C₂×C₅₉	#460	1	1	yes	characteristic
62	C₂×C₅₉	#466	1	1	yes	characteristic
63	C₂×C₅₉	#472	1	1	yes	characteristic
64	D₅₉	—	16	2	no	—
65	#493	—	59	1	no	—
66	#460	—	59	2	no	—
67	C₈×Q₈	D₅₉	1	1	yes	characteristic, complemented
68	#335	—	59	1	no	—
69	#331	—	59	2	no	—
70	C₅₉	#1668	1	1	yes	characteristic, complemented, Sylow
71	C₂×SD₁₆	—	118	2	no	—
72	C₂×D₈	—	118	1	no	—
73	C₂×D₈	—	118	1	no	—
74	#127	—	59	2	no	—
75	C₄×Q₈	C₂×D₅₉	1	1	yes	characteristic
76	C₄⋊C₈	—	2	1	no	—
77	C₄⋊C₈	—	59	2	no	—
78	C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
79	#102	—	118	2	no	—
80	C₄×C₈	—	2	1	no	—
81	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
82	C₂×Q₈	D₂₃₆	1	1	yes	characteristic
83	C₂×D₄	—	118	2	no	—
84	C₂×D₄	—	118	2	no	—
85	C₂×D₄	—	118	2	no	—
86	SD₁₆	—	236	4	no	—
87	D₈	—	236	2	no	—
88	D₈	—	236	2	no	—
89	C₂×C₈	—	2	1	no	—
90	C₂×C₈	—	118	2	no	—
91	C₂×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
92	C₂×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
93	C₄⋊C₄	—	2	1	no	—
94	C₄⋊C₄	D₂₃₆	1	1	yes	characteristic
95	C₄²	—	2	1	no	—
96	C₄²	C₂²×D₅₉	1	1	yes	characteristic
97	C₂³	—	236	2	no	—
98	Q₈	—	2	1	no	—
99	Q₈	D₄₇₂	1	2	yes	complemented
100	D₄	—	472	2	no	—
101	D₄	—	236	2	no	—
102	D₄	—	236	2	no	—
103	D₄	—	236	2	no	—
104	D₄	—	236	2	no	—
105	C₂×C₄	—	2	1	no	—
106	C₂×C₄	—	2	1	no	—
107	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
108	C₂×C₄	#11745685	1	1	yes	characteristic
109	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
110	C₈	—	2	1	no	—
111	C₈	—	4	1	no	—
112	C₈	—	236	2	no	—
113	C₈	#11745671	1	2	yes	complemented
114	C₂²	—	472	2	no	—
115	C₂²	—	472	2	no	—
116	C₂²	—	472	2	no	—
117	C₂²	#14273701	1	1	yes	characteristic
118	C₄	—	2	2	no	—
119	C₄	—	2	1	no	—
120	C₄	—	4	1	no	—
121	C₄	C₂×D₄₇₂	1	1	yes	characteristic
122	C₄	C₂×#11745671	1	1	yes	characteristic
123	C₄	#14273749	1	1	yes	characteristic
124	C₄	#14273752	1	1	yes	characteristic
125	C₂	—	944	2	no	—
126	C₂	#16837602	1	1	yes	characteristic
127	C₂	#16837638	1	1	yes	characteristic
128	C₂	#16837672	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837779

C₂

yes

maximal, characteristic

#16837602

C₂

yes

maximal, complemented

#16837155

C₂

yes

maximal, characteristic, complemented

#16837145

C₂

yes

maximal, complemented

C₈×Q₈×C₅₉

C₂

yes

maximal, characteristic, complemented

C₂×#11745671

—

C₂×D₄₇₂

—

C₂×D₄₇₂

—

#14273724

C₂²

yes

—

C₄⋊(C₅₉⋊C₈)

C₂²

yes

—

#14273647

—

C₄×Q₈×C₅₉

C₂²

yes

characteristic

C₄⋊C₈×C₅₉

—

C₄⋊C₈×C₅₉

C₂²

yes

characteristic

C₄×C₈×C₅₉

—

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂×Q₈×C₅₉

D₄

yes

characteristic

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

D₄

yes

characteristic

C₂×C₈×C₅₉

D₄

yes

characteristic

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

D₄

yes

characteristic

C₄²×C₅₉

—

C₄²×C₅₉

C₂³

yes

characteristic

#11745671

—

C₂×C₅₉⋊C₈

—

D₄₇₂

—

D₄₇₂

—

C₂²×D₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

D₈

yes

complemented

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

#41

yes

characteristic

D₂₃₆

—

D₂₃₆

—

D₂₃₆

—

D₂₃₆

—

D₂₃₆

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

SD₁₆

yes

complemented

C₅₉⋊C₈

—

C₂²×C₅₉

#121

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂×D₈

yes

characteristic

C₄×C₅₉

C₂×SD₁₆

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

D₅₉

—

#493

—

#460

—

C₈×Q₈

D₅₉

yes

characteristic, complemented

#335

—

#331

—

C₅₉

#1668

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

118

—

C₂×D₈

—

118

—

C₂×D₈

—

118

—

#127

—

C₄×Q₈

C₂×D₅₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₅₉

yes

characteristic

#102

—

118

—

C₄×C₈

—

C₄×C₈

C₂×D₅₉

yes

characteristic

C₂×Q₈

D₂₃₆

yes

characteristic

C₂×D₄

—

118

—

C₂×D₄

—

118

—

C₂×D₄

—

118

—

SD₁₆

—

236

—

D₈

—

236

—

D₈

—

236

—

C₂×C₈

—

C₂×C₈

—

118

—

C₂×C₈

C₂²⋊D₅₉

yes

characteristic

C₂×C₈

C₂²⋊D₅₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

D₂₃₆

yes

characteristic

C₄²

—

C₄²

C₂²×D₅₉

yes

characteristic

C₂³

—

236

—

Q₈

—

Q₈

D₄₇₂

yes

complemented

100

D₄

—

472

—

101

D₄

—

236

—

102

D₄

—

236

—

103

D₄

—

236

—

104

D₄

—

236

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

C₂×D₂₃₆

yes

characteristic

108

C₂×C₄

#11745685

yes

characteristic

109

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

110

C₈

—

111

C₈

—

112

C₈

—

236

—

113

C₈

#11745671

yes

complemented

114

C₂²

—

472

—

115

C₂²

—

472

—

116

C₂²

—

472

—

117

C₂²

#14273701

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

C₂×D₄₇₂

yes

characteristic

122

C₄

C₂×#11745671

yes

characteristic

123

C₄

#14273749

yes

characteristic

124

C₄

#14273752

yes

characteristic

125

C₂

—

944

—

126

C₂

#16837602

yes

characteristic

127

C₂

#16837638

yes

characteristic

128

C₂

#16837672

yes

characteristic

	subgroup	G/H
center	C₂²	#14273701
commutator	C₄²×C₅₉	C₂³
Frattini	C₄²	C₂²×D₅₉
Fitting	C₈×Q₈×C₅₉	C₂
socle	C₂²×C₅₉	#121
hypercenter	C₈×Q₈	D₅₉
hypocenter	C₅₉	#1668
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈

subgroup

G/H

center

C₂²

#14273701

commutator

C₄²×C₅₉

C₂³

Frattini

C₄²

C₂²×D₅₉

Fitting

C₈×Q₈×C₅₉

C₂

socle

C₂²×C₅₉

#121

hypercenter

C₈×Q₈

D₅₉

hypocenter

C₅₉

#1668

LCS

G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈

N	H ≅ G/N	type
2. #16837602	C₂	semidirect
3. #16837155	C₂	semidirect
4. #16837145	C₂	semidirect
5. C₈×Q₈×C₅₉	C₂	semidirect
34. Q₈×C₅₉	D₈	semidirect
47. C₈×C₅₉	SD₁₆	semidirect
67. C₈×Q₈	D₅₉	semidirect
70. C₅₉	#1668	semidirect
99. Q₈	D₄₇₂	semidirect
113. C₈	#11745671	semidirect

H ≅ G/N

type

2. #16837602

C₂

semidirect

3. #16837155

C₂

semidirect

4. #16837145

C₂

semidirect

5. C₈×Q₈×C₅₉

C₂

semidirect

34. Q₈×C₅₉

D₈

semidirect

47. C₈×C₅₉

SD₁₆

semidirect

67. C₈×Q₈

D₅₉

semidirect

70. C₅₉

#1668

semidirect

99. Q₈

D₄₇₂

semidirect

113. C₈

#11745671

semidirect

Subgroups — #21423456

Conjugacy classes of subgroups

Complements