Group #16837977 — subgroups

Subgroups — #16837977

Order 3776.

270 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 103 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	C₂×#11745689	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×D₅₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#14273725	C₂	1	1	yes	maximal, characteristic, complemented
4	#14273716	C₂	1	1	yes	maximal, characteristic, complemented
5	#14273715	C₂	1	1	yes	maximal, characteristic, complemented
6	#14273712	C₂	1	4	yes	maximal, complemented
7	#14273705	C₂	1	4	yes	maximal, complemented
8	#14273684	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄⋊Q₈×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Q₈×C₅₉	C₂²	1	1	yes	characteristic
11	C₂×Q₈×C₅₉	C₂²	1	1	yes	characteristic
12	#11745689	—	2	4	no	—
13	Q₈×D₅₉	—	2	4	no	—
14	C₂×D₂₃₆	C₂²	1	2	yes	—
15	C₂×D₂₃₆	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₅₉	C₂²	1	2	yes	—
17	C₂×C₄×D₅₉	C₂²	1	2	yes	—
18	C₂×C₄×D₅₉	C₂²	1	1	yes	characteristic
19	C₂×Dic₁₁₈	C₂²	1	2	yes	—
20	C₂×Dic₁₁₈	C₂²	1	1	yes	characteristic
21	C₄⋊C₄×C₅₉	C₂²	1	4	yes	—
22	C₄²×C₅₉	C₂²	1	1	yes	characteristic
23	(C₂×D₅₉)⋊C₄	C₂²	1	2	yes	—
24	(C₂×D₅₉)⋊C₄	C₂²	1	4	yes	—
25	(C₂×D₅₉)⋊C₄	C₂²	1	4	yes	—
26	Dic₅₉⋊C₄	C₂²	1	2	yes	—
27	Dic₅₉⋊C₄	C₂²	1	4	yes	—
28	C₄×Dic₅₉	C₂²	1	1	yes	characteristic
29	C₂²×D₅₉	C₂³	1	2	yes	—
30	C₂²×D₅₉	C₂³	1	1	yes	characteristic
31	Q₈×C₅₉	—	2	2	no	—
32	Q₈×C₅₉	—	2	2	no	—
33	C₂×C₄×C₅₉	C₂³	1	2	yes	—
34	C₂×C₄×C₅₉	C₂³	1	2	yes	—
35	C₂×C₄×C₅₉	C₂³	1	2	yes	—
36	C₂×C₄×C₅₉	C₂³	1	1	yes	characteristic
37	C₂×Dic₅₉	C₂³	1	2	yes	—
38	C₂×Dic₅₉	C₂³	1	2	yes	—
39	C₂×Dic₅₉	C₂³	1	1	yes	characteristic
40	D₂₃₆	—	2	4	no	—
41	D₂₃₆	—	2	2	no	—
42	C₄×D₅₉	—	2	4	no	—
43	C₄×D₅₉	—	2	4	no	—
44	C₄×D₅₉	D₄	1	4	yes	—
45	Dic₁₁₈	—	2	2	no	—
46	Dic₁₁₈	—	2	4	no	—
47	C₂²×C₅₉	C₂⁴	1	1	yes	characteristic
48	C₂×D₅₉	—	2	2	no	—
49	C₂×D₅₉	—	2	2	no	—
50	C₂×D₅₉	—	2	4	no	—
51	C₂×D₅₉	C₂×D₄	1	2	yes	—
52	C₄×C₅₉	—	2	2	no	—
53	C₄×C₅₉	—	2	2	no	—
54	C₄×C₅₉	—	2	2	no	—
55	C₄×C₅₉	C₂×D₄	1	2	yes	—
56	Dic₅₉	—	2	2	no	—
57	Dic₅₉	—	2	2	no	—
58	Dic₅₉	C₂×D₄	1	2	yes	—
59	C₂×C₅₉	C₂²×D₄	1	1	yes	characteristic
60	C₂×C₅₉	2₋¹⁺⁴	1	2	yes	—
61	D₅₉	—	2	2	no	—
62	D₅₉	—	4	2	no	—
63	#536	—	59	1	no	maximal, Sylow
64	C₅₉	#536	1	1	yes	characteristic, complemented, Sylow
65	C₂×#41	—	59	1	no	—
66	C₂²×Q₈	—	59	1	no	—
67	C₄⋊Q₈	—	59	1	no	—
68	C₄⋊Q₈	D₅₉	1	1	yes	characteristic, complemented
69	#124	—	59	1	no	—
70	#124	—	59	1	no	—
71	#123	—	59	4	no	—
72	C₂²⋊Q₈	—	59	4	no	—
73	#117	—	59	1	no	—
74	#41	—	118	4	no	—
75	C₂×Q₈	—	59	2	no	—
76	C₂×Q₈	—	59	1	no	—
77	C₂×Q₈	—	118	4	no	—
78	C₂×Q₈	C₂×D₅₉	1	1	yes	characteristic
79	C₂×Q₈	C₂×D₅₉	1	1	yes	characteristic
80	C₂×D₄	—	59	2	no	—
81	C₂×D₄	—	59	1	no	—
82	C₂²×C₄	—	59	1	no	—
83	C₂²×C₄	—	59	2	no	—
84	C₂²×C₄	—	59	2	no	—
85	C₄⋊C₄	—	59	2	no	—
86	C₄⋊C₄	—	59	4	no	—
87	C₄⋊C₄	C₂×D₅₉	1	4	yes	—
88	C₂²⋊C₄	—	59	2	no	—
89	C₂²⋊C₄	—	59	4	no	—
90	C₂²⋊C₄	—	59	4	no	—
91	C₄²	—	59	1	no	—
92	C₄²	C₂×D₅₉	1	1	yes	characteristic
93	C₂³	—	59	1	no	—
94	C₂³	—	59	2	no	—
95	Q₈	—	2	2	no	—
96	Q₈	—	2	2	no	—
97	Q₈	—	118	4	no	—
98	Q₈	—	118	2	no	—
99	D₄	—	118	4	no	—
100	D₄	—	118	2	no	—
101	C₂×C₄	—	59	1	no	—
102	C₂×C₄	—	59	4	no	—
103	C₂×C₄	—	59	2	no	—
104	C₂×C₄	—	59	2	no	—
105	C₂×C₄	—	118	4	no	—
106	C₂×C₄	—	118	4	no	—
107	C₂×C₄	C₂²×D₅₉	1	2	yes	—
108	C₂×C₄	C₂²×D₅₉	1	2	yes	—
109	C₂×C₄	C₂²×D₅₉	1	2	yes	—
110	C₂×C₄	C₂²×D₅₉	1	1	yes	characteristic
111	C₂²	—	59	2	no	—
112	C₂²	—	118	2	no	—
113	C₂²	—	118	2	no	—
114	C₂²	—	118	4	no	—
115	C₂²	C₂³×D₅₉	1	1	yes	characteristic
116	C₄	—	2	2	no	—
117	C₄	—	2	2	no	—
118	C₄	—	2	2	no	—
119	C₄	—	59	2	no	—
120	C₄	—	118	2	no	—
121	C₄	—	118	2	no	—
122	C₄	D₄×D₅₉	1	2	yes	—
123	C₂	—	118	2	no	—
124	C₂	—	236	2	no	—
125	C₂	C₂×D₄×D₅₉	1	1	yes	characteristic
126	C₂	#14273775	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11745689

C₂

yes

maximal, characteristic

C₂×Q₈×D₅₉

C₂

yes

maximal, characteristic, complemented

#14273725

C₂

yes

maximal, characteristic, complemented

#14273716

C₂

yes

maximal, characteristic, complemented

#14273715

C₂

yes

maximal, characteristic, complemented

#14273712

C₂

yes

maximal, complemented

#14273705

C₂

yes

maximal, complemented

#14273684

C₂

yes

maximal, characteristic, complemented

C₄⋊Q₈×C₅₉

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×C₅₉

C₂²

yes

characteristic

C₂×Q₈×C₅₉

C₂²

yes

characteristic

#11745689

—

Q₈×D₅₉

—

C₂×D₂₃₆

C₂²

yes

—

C₂×D₂₃₆

C₂²

yes

characteristic

C₂×C₄×D₅₉

C₂²

yes

—

C₂×C₄×D₅₉

C₂²

yes

—

C₂×C₄×D₅₉

C₂²

yes

characteristic

C₂×Dic₁₁₈

C₂²

yes

—

C₂×Dic₁₁₈

C₂²

yes

characteristic

C₄⋊C₄×C₅₉

C₂²

yes

—

C₄²×C₅₉

C₂²

yes

characteristic

(C₂×D₅₉)⋊C₄

C₂²

yes

—

(C₂×D₅₉)⋊C₄

C₂²

yes

—

(C₂×D₅₉)⋊C₄

C₂²

yes

—

Dic₅₉⋊C₄

C₂²

yes

—

Dic₅₉⋊C₄

C₂²

yes

—

C₄×Dic₅₉

C₂²

yes

characteristic

C₂²×D₅₉

C₂³

yes

—

C₂²×D₅₉

C₂³

yes

characteristic

Q₈×C₅₉

—

Q₈×C₅₉

—

C₂×C₄×C₅₉

C₂³

yes

—

C₂×C₄×C₅₉

C₂³

yes

—

C₂×C₄×C₅₉

C₂³

yes

—

C₂×C₄×C₅₉

C₂³

yes

characteristic

C₂×Dic₅₉

C₂³

yes

—

C₂×Dic₅₉

C₂³

yes

—

C₂×Dic₅₉

C₂³

yes

characteristic

D₂₃₆

—

D₂₃₆

—

C₄×D₅₉

—

C₄×D₅₉

—

C₄×D₅₉

D₄

yes

—

Dic₁₁₈

—

Dic₁₁₈

—

C₂²×C₅₉

C₂⁴

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

C₂×D₄

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂×D₄

yes

—

Dic₅₉

—

Dic₅₉

—

Dic₅₉

C₂×D₄

yes

—

C₂×C₅₉

C₂²×D₄

yes

characteristic

C₂×C₅₉

2₋¹⁺⁴

yes

—

D₅₉

—

D₅₉

—

#536

—

maximal, Sylow

C₅₉

#536

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂²×Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

D₅₉

yes

characteristic, complemented

#124

—

#124

—

#123

—

C₂²⋊Q₈

—

#117

—

#41

—

118

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

118

—

C₂×Q₈

C₂×D₅₉

yes

characteristic

C₂×Q₈

C₂×D₅₉

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₅₉

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂×D₅₉

yes

characteristic

C₂³

—

C₂³

—

Q₈

—

Q₈

—

Q₈

—

118

—

Q₈

—

118

—

D₄

—

118

—

100

D₄

—

118

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

118

—

106

C₂×C₄

—

118

—

107

C₂×C₄

C₂²×D₅₉

yes

—

108

C₂×C₄

C₂²×D₅₉

yes

—

109

C₂×C₄

C₂²×D₅₉

yes

—

110

C₂×C₄

C₂²×D₅₉

yes

characteristic

111

C₂²

—

112

C₂²

—

118

—

113

C₂²

—

118

—

114

C₂²

—

118

—

115

C₂²

C₂³×D₅₉

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

118

—

121

C₄

—

118

—

122

C₄

D₄×D₅₉

yes

—

123

C₂

—

118

—

124

C₂

—

236

—

125

C₂

C₂×D₄×D₅₉

yes

characteristic

126

C₂

#14273775

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₅₉
commutator, socle, cosocle	C₂²×C₅₉	C₂⁴
Fitting	C₄⋊Q₈×C₅₉	C₂
prime radical	C₂×#11745689	C₂
hypercenter	C₄⋊Q₈	D₅₉
hypocenter	C₅₉	#536
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₅₉

commutator, socle, cosocle

C₂²×C₅₉

C₂⁴

Fitting

C₄⋊Q₈×C₅₉

C₂

prime radical

C₂×#11745689

C₂

hypercenter

C₄⋊Q₈

D₅₉

C₅₉

G ▷ C₂²×C₅₉ ▷ C₅₉

C₁ ◁ C₂² ◁ C₄⋊Q₈

N	H ≅ G/N	type
2. C₂×Q₈×D₅₉	C₂	semidirect
3. #14273725	C₂	semidirect
4. #14273716	C₂	semidirect
5. #14273715	C₂	semidirect
6. #14273712	C₂	semidirect
7. #14273705	C₂	semidirect
8. #14273684	C₂	semidirect
9. C₄⋊Q₈×C₅₉	C₂	semidirect
64. C₅₉	#536	semidirect
68. C₄⋊Q₈	D₅₉	semidirect

H ≅ G/N

type

2. C₂×Q₈×D₅₉

C₂

semidirect

3. #14273725

C₂

semidirect

4. #14273716

C₂

semidirect

5. #14273715

C₂

semidirect

6. #14273712

C₂

semidirect

7. #14273705

C₂

semidirect

8. #14273684

C₂

semidirect

9. C₄⋊Q₈×C₅₉

C₂

semidirect

64. C₅₉

#536

semidirect

68. C₄⋊Q₈

D₅₉

semidirect

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

subgroup	index
C₂×#11745689	2
C₂×Q₈×D₅₉	2
#14273725	2
#14273716	2
#14273715	2
#14273712	2
#14273705	2
#14273684	2
C₄⋊Q₈×C₅₉	2
#536	59

subgroup

index

C₂×Q₈×D₅₉

C₄⋊Q₈×C₅₉

Subgroups — #16837977

Conjugacy classes of subgroups

Complements

Maximal subgroups