Group #21423580 — subgroups

Subgroups — #21423580

Order 7552.

184 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 57 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	#16837153	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837145	C₂	1	2	yes	maximal, complemented
5	C₅₉×#446	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	#97×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	2	no	—
34	C₂×C₄×C₅₉	—	2	1	no	—
35	C₂×C₄×C₅₉	—	2	1	no	—
36	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
39	D₂₃₆	—	4	2	no	—
40	D₂₃₆	—	4	2	no	—
41	D₂₃₆	—	4	2	no	—
42	D₂₃₆	—	4	2	no	—
43	D₂₃₆	—	8	2	no	—
44	C₈×C₅₉	—	2	1	no	—
45	C₈×C₅₉	—	4	1	no	—
46	C₈×C₅₉	SD₁₆	1	2	yes	complemented
47	C₅₉⋊C₈	—	4	2	no	—
48	C₂²×C₅₉	#121	1	1	yes	characteristic
49	C₂×D₅₉	—	8	2	no	—
50	C₂×D₅₉	—	8	2	no	—
51	C₂×D₅₉	—	8	2	no	—
52	C₄×C₅₉	—	2	1	no	—
53	C₄×C₅₉	—	4	1	no	—
54	C₄×C₅₉	—	4	1	no	—
55	C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
56	C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
57	C₄×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
58	#1688	—	59	1	no	maximal, Sylow
59	C₂×C₅₉	#460	1	1	yes	characteristic
60	C₂×C₅₉	#469	1	1	yes	characteristic
61	C₂×C₅₉	#472	1	1	yes	characteristic
62	D₅₉	—	16	2	no	—
63	#493	—	59	1	no	—
64	#460	—	59	2	no	—
65	#446	D₅₉	1	1	yes	characteristic, complemented
66	#334	—	59	1	no	—
67	#331	—	59	2	no	—
68	C₅₉	#1688	1	1	yes	characteristic, complemented, Sylow
69	C₂×SD₁₆	—	118	2	no	—
70	C₂×D₈	—	118	1	no	—
71	C₂×D₈	—	118	1	no	—
72	#127	—	59	2	no	—
73	C₄×Q₈	C₂×D₅₉	1	1	yes	characteristic
74	C₄⋊C₈	—	2	1	no	—
75	C₄⋊C₈	—	59	2	no	—
76	C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
77	#102	—	118	2	no	—
78	#97	—	2	1	no	—
79	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
80	C₂×Q₈	D₂₃₆	1	1	yes	characteristic
81	C₂×D₄	—	118	2	no	—
82	C₂×D₄	—	118	2	no	—
83	C₂×D₄	—	118	2	no	—
84	SD₁₆	—	236	4	no	—
85	D₈	—	236	2	no	—
86	D₈	—	236	2	no	—
87	C₂×C₈	—	2	1	no	—
88	C₂×C₈	—	118	2	no	—
89	C₂×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
90	C₂×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
91	C₄⋊C₄	—	2	1	no	—
92	C₄⋊C₄	D₂₃₆	1	1	yes	characteristic
93	C₄²	—	2	1	no	—
94	C₄²	C₂²×D₅₉	1	1	yes	characteristic
95	C₂³	—	236	2	no	—
96	Q₈	—	2	2	no	—
97	D₄	—	472	2	no	—
98	D₄	—	236	2	no	—
99	D₄	—	236	2	no	—
100	D₄	—	236	2	no	—
101	D₄	—	236	2	no	—
102	C₂×C₄	—	2	1	no	—
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
105	C₂×C₄	#11745685	1	1	yes	characteristic
106	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
107	C₈	—	2	1	no	—
108	C₈	—	4	1	no	—
109	C₈	—	236	2	no	—
110	C₈	#11745671	1	2	yes	complemented
111	C₂²	—	472	2	no	—
112	C₂²	—	472	2	no	—
113	C₂²	—	472	2	no	—
114	C₂²	#14273701	1	1	yes	characteristic
115	C₄	—	2	1	no	—
116	C₄	—	4	1	no	—
117	C₄	—	4	1	no	—
118	C₄	C₂×#11745671	1	1	yes	characteristic
119	C₄	#14273752	1	1	yes	characteristic
120	C₄	#14273755	1	2	yes	—
121	C₂	—	944	2	no	—
122	C₂	#16837602	1	1	yes	characteristic
123	C₂	#16837655	1	1	yes	characteristic
124	C₂	#16837672	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837779

C₂

yes

maximal, characteristic

#16837602

C₂

yes

maximal, complemented

#16837153

C₂

yes

maximal, characteristic, complemented

#16837145

C₂

yes

maximal, complemented

C₅₉×#446

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

#97×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

—

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×SD₁₆

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

D₅₉

—

#493

—

#460

—

#446

D₅₉

yes

characteristic, complemented

#334

—

#331

—

C₅₉

#1688

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

—

#97

—

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

—

D₄

—

472

—

D₄

—

236

—

D₄

—

236

—

100

D₄

—

236

—

101

D₄

—

236

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

C₂×D₂₃₆

yes

characteristic

105

C₂×C₄

#11745685

yes

characteristic

106

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

107

C₈

—

108

C₈

—

109

C₈

—

236

—

110

C₈

#11745671

yes

complemented

111

C₂²

—

472

—

112

C₂²

—

472

—

113

C₂²

—

472

—

114

C₂²

#14273701

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

C₂×#11745671

yes

characteristic

119

C₄

#14273752

yes

characteristic

120

C₄

#14273755

yes

—

121

C₂

—

944

—

122

C₂

#16837602

yes

characteristic

123

C₂

#16837655

yes

characteristic

124

C₂

#16837672

yes

characteristic

	subgroup	G/H
center	C₂²	#14273701
commutator	C₄²×C₅₉	C₂³
Frattini	C₄²	C₂²×D₅₉
Fitting	C₅₉×#446	C₂
socle	C₂²×C₅₉	#121
hypercenter	#446	D₅₉
hypocenter	C₅₉	#1688
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #446

subgroup

G/H

center

C₂²

#14273701

commutator

C₄²×C₅₉

C₂³

Frattini

C₄²

C₂²×D₅₉

Fitting

C₅₉×#446

C₂

C₂²×C₅₉

D₅₉

C₅₉

G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #446

N	H ≅ G/N	type
2. #16837602	C₂	semidirect
3. #16837153	C₂	semidirect
4. #16837145	C₂	semidirect
5. C₅₉×#446	C₂	semidirect
46. C₈×C₅₉	SD₁₆	semidirect
65. #446	D₅₉	semidirect
68. C₅₉	#1688	semidirect
110. C₈	#11745671	semidirect

H ≅ G/N

type

2. #16837602

C₂

semidirect

3. #16837153

C₂

semidirect

4. #16837145

C₂

semidirect

5. C₅₉×#446

C₂

semidirect

46. C₈×C₅₉

SD₁₆

semidirect

65. #446

D₅₉

semidirect

68. C₅₉

#1688

semidirect

110. C₈

#11745671

semidirect

Subgroups — #21423580

Conjugacy classes of subgroups

Complements