Group #47983049 — subgroups

Subgroups — #47983049

Order 35456.

232 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 117 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	#31800040	C₂	1	1	yes	maximal, characteristic
2	#31799960	C₂	1	2	yes	maximal
3	C₂×C₈×Dic₂₇₇	C₂	1	1	yes	maximal, characteristic
4	#31799763	C₂	1	2	yes	maximal
5	C₂×#108×C₂₇₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#14555660	C₂²	1	1	yes	characteristic
7	#22579839	C₂²	1	2	yes	—
8	C₂×C₄×Dic₂₇₇	C₂²	1	1	yes	characteristic
9	#22579814	C₄	1	2	yes	complemented
10	#22579812	C₄	1	2	yes	complemented
11	#22579790	—	2	2	no	—
12	C₈×Dic₂₇₇	—	2	2	no	—
13	C₈×Dic₂₇₇	C₄	1	4	yes	complemented
14	C₂×M₁₆×C₂₇₇	C₂²	1	2	yes	—
15	C₂²×C₈×C₂₇₇	C₂²	1	1	yes	characteristic
16	#108×C₂₇₇	C₄	1	4	yes	complemented
17	C₂²×Dic₂₇₇	—	2	1	no	—
18	C₂×#14555660	C₂×C₄	1	2	yes	—
19	C₂×#14555660	D₄	1	1	yes	characteristic
20	C₂×#14555660	D₄	1	1	yes	characteristic
21	C₂×#14555660	Q₈	1	2	yes	—
22	C₄⋊Dic₂₇₇	C₂×C₄	1	4	yes	—
23	C₂²⋊Dic₂₇₇	—	2	2	no	—
24	C₄×Dic₂₇₇	—	2	2	no	—
25	C₄×Dic₂₇₇	—	2	1	no	—
26	C₄×Dic₂₇₇	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₂₇₇	C₂³	1	1	yes	characteristic
28	M₁₆×C₂₇₇	—	2	2	no	—
29	M₁₆×C₂₇₇	C₂×C₄	1	4	yes	—
30	C₂×C₈×C₂₇₇	—	2	2	no	—
31	C₂×C₈×C₂₇₇	C₂×C₄	1	2	yes	—
32	C₂×C₈×C₂₇₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₈×C₂₇₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₈×C₂₇₇	C₂×C₄	1	2	yes	—
35	C₂×Dic₂₇₇	—	2	4	no	—
36	C₂×Dic₂₇₇	—	2	2	no	—
37	C₂×Dic₂₇₇	—	2	4	no	—
38	#14555660	C₄⋊C₄	1	2	yes	—
39	#14555660	C₄⋊C₄	1	2	yes	—
40	C₂³×C₂₇₇	#41	1	1	yes	characteristic
41	C₂×C₄×C₂₇₇	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₂₇₇	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₂₇₇	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₂₇₇	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₂₇₇	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×C₂₇₇	#41	1	1	yes	characteristic
47	C₈×C₂₇₇	—	2	4	no	—
48	C₈×C₂₇₇	C₄²	1	2	yes	—
49	C₈×C₂₇₇	C₄²	1	2	yes	—
50	C₂²×C₂₇₇	—	2	2	no	—
51	C₂²×C₂₇₇	#117	1	1	yes	characteristic
52	C₂²×C₂₇₇	C₄×D₄	1	1	yes	characteristic
53	C₂²×C₂₇₇	C₄×Q₈	1	1	yes	characteristic
54	C₄×C₂₇₇	C₂×C₄²	1	1	yes	characteristic
55	C₄×C₂₇₇	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₄×C₂₇₇	C₄×D₄	1	1	yes	characteristic
57	C₄×C₂₇₇	C₄×Q₈	1	1	yes	characteristic
58	Dic₂₇₇	—	2	4	no	—
59	Dic₂₇₇	—	4	4	no	—
60	C₂×C₂₇₇	—	2	1	no	—
61	C₂×C₂₇₇	—	2	1	no	—
62	C₂×C₂₇₇	C₄×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₂₇₇	#443	1	2	yes	—
64	C₂₇₇	#1772	1	1	yes	characteristic, complemented, Sylow
65	#1772	—	277	1	no	maximal, Sylow
66	C₂×#108	D₂₇₇	1	1	yes	characteristic
67	#427	—	277	1	no	—
68	#405	—	277	2	no	—
69	C₂×C₄×C₈	—	277	1	no	—
70	#339	—	277	2	no	—
71	C₂×M₁₆	C₂×D₂₇₇	1	2	yes	—
72	C₂²×C₈	—	277	1	no	—
73	C₂²×C₈	C₂×D₂₇₇	1	1	yes	characteristic
74	#117	—	277	2	no	—
75	C₂×C₄²	—	277	1	no	—
76	#108	Dic₂₇₇	1	4	yes	complemented
77	#107	—	277	2	no	—
78	#106	—	277	2	no	—
79	#97	—	554	2	no	—
80	C₄×C₈	—	277	4	no	—
81	C₄×C₈	—	554	2	no	—
82	C₂²×C₄	—	554	1	no	—
83	C₂²×C₄	C₂²×D₂₇₇	1	1	yes	characteristic
84	M₁₆	—	2	2	no	—
85	M₁₆	C₂×Dic₂₇₇	1	4	yes	—
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	—	277	2	no	—
88	C₂×C₈	—	277	1	no	—
89	C₂×C₈	—	277	1	no	—
90	C₂×C₈	—	277	2	no	—
91	C₂×C₈	C₄×D₂₇₇	1	2	yes	—
92	C₂×C₈	C₄×D₂₇₇	1	1	yes	characteristic
93	C₂×C₈	C₄×D₂₇₇	1	1	yes	characteristic
94	C₂×C₈	C₂×Dic₂₇₇	1	2	yes	—
95	C₄⋊C₄	—	277	4	no	—
96	C₂²⋊C₄	—	554	2	no	—
97	C₄²	—	277	2	no	—
98	C₄²	—	554	2	no	—
99	C₄²	—	554	1	no	—
100	C₂³	#17984564	1	1	yes	characteristic
101	C₂×C₄	—	554	4	no	—
102	C₂×C₄	—	554	2	no	—
103	C₂×C₄	—	554	4	no	—
104	C₂×C₄	C₂×C₄×D₂₇₇	1	1	yes	characteristic
105	C₂×C₄	C₂×C₄×D₂₇₇	1	1	yes	characteristic
106	C₂×C₄	C₂²×Dic₂₇₇	1	1	yes	characteristic
107	C₂×C₄	D₄×D₂₇₇	1	1	yes	characteristic
108	C₂×C₄	Q₈×D₂₇₇	1	1	yes	characteristic
109	C₂×C₄	#17984562	1	1	yes	characteristic
110	C₈	—	2	4	no	—
111	C₈	—	277	2	no	—
112	C₈	—	277	2	no	—
113	C₈	C₄×Dic₂₇₇	1	2	yes	—
114	C₈	C₄×Dic₂₇₇	1	2	yes	—
115	C₂²	—	2	2	no	—
116	C₂²	#22579838	1	1	yes	characteristic
117	C₂²	#22579842	1	1	yes	characteristic
118	C₂²	Q₈×Dic₂₇₇	1	1	yes	characteristic
119	C₄	—	554	4	no	—
120	C₄	—	1108	4	no	—
121	C₄	C₂×C₄×Dic₂₇₇	1	1	yes	characteristic
122	C₄	C₄⋊C₄×D₂₇₇	1	1	yes	characteristic
123	C₄	D₄×Dic₂₇₇	1	1	yes	characteristic
124	C₄	#22579845	1	1	yes	characteristic
125	C₂	—	2	1	no	—
126	C₂	—	2	1	no	—
127	C₂	C₄⋊C₄×Dic₂₇₇	1	1	yes	characteristic
128	C₂	#31800118	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#31800040

C₂

yes

maximal, characteristic

#31799960

C₂

yes

maximal

C₂×C₈×Dic₂₇₇

C₂

yes

maximal, characteristic

#31799763

C₂

yes

maximal

C₂×#108×C₂₇₇

C₂

yes

maximal, characteristic

C₂²×#14555660

C₂²

yes

characteristic

#22579839

C₂²

yes

—

C₂×C₄×Dic₂₇₇

C₂²

yes

characteristic

#22579814

C₄

yes

complemented

#22579812

C₄

yes

complemented

#22579790

—

C₈×Dic₂₇₇

—

C₈×Dic₂₇₇

C₄

yes

complemented

C₂×M₁₆×C₂₇₇

C₂²

yes

—

C₂²×C₈×C₂₇₇

C₂²

yes

characteristic

#108×C₂₇₇

C₄

yes

complemented

C₂²×Dic₂₇₇

—

C₂×#14555660

C₂×C₄

yes

—

C₂×#14555660

D₄

yes

characteristic

C₂×#14555660

D₄

yes

characteristic

C₂×#14555660

Q₈

yes

—

C₄⋊Dic₂₇₇

C₂×C₄

yes

—

C₂²⋊Dic₂₇₇

—

C₄×Dic₂₇₇

—

C₄×Dic₂₇₇

—

C₄×Dic₂₇₇

C₂×C₄

yes

—

C₂²×C₄×C₂₇₇

C₂³

yes

characteristic

M₁₆×C₂₇₇

—

M₁₆×C₂₇₇

C₂×C₄

yes

—

C₂×C₈×C₂₇₇

—

C₂×C₈×C₂₇₇

C₂×C₄

yes

—

C₂×C₈×C₂₇₇

C₂×C₄

yes

characteristic

C₂×C₈×C₂₇₇

C₂×C₄

yes

characteristic

C₂×C₈×C₂₇₇

C₂×C₄

yes

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

—

#14555660

C₄⋊C₄

yes

—

#14555660

C₄⋊C₄

yes

—

C₂³×C₂₇₇

#41

yes

characteristic

C₂×C₄×C₂₇₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₇₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₇₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₇₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₇₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₇₇

#41

yes

characteristic

C₈×C₂₇₇

—

C₈×C₂₇₇

C₄²

yes

—

C₈×C₂₇₇

C₄²

yes

—

C₂²×C₂₇₇

—

C₂²×C₂₇₇

#117

yes

characteristic

C₂²×C₂₇₇

C₄×D₄

yes

characteristic

C₂²×C₂₇₇

C₄×Q₈

yes

characteristic

C₄×C₂₇₇

C₂×C₄²

yes

characteristic

C₄×C₂₇₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₂₇₇

C₄×D₄

yes

characteristic

C₄×C₂₇₇

C₄×Q₈

yes

characteristic

Dic₂₇₇

—

Dic₂₇₇

—

C₂×C₂₇₇

—

C₂×C₂₇₇

—

C₂×C₂₇₇

C₄×C₄⋊C₄

yes

characteristic

C₂×C₂₇₇

#443

yes

—

C₂₇₇

#1772

yes

characteristic, complemented, Sylow

#1772

—

277

maximal, Sylow

C₂×#108

D₂₇₇

yes

characteristic

#427

—

277

—

#405

—

277

—

C₂×C₄×C₈

—

277

—

#339

—

277

—

C₂×M₁₆

C₂×D₂₇₇

yes

—

C₂²×C₈

—

277

—

C₂²×C₈

C₂×D₂₇₇

yes

characteristic

#117

—

277

—

C₂×C₄²

—

277

—

#108

Dic₂₇₇

yes

complemented

#107

—

277

—

#106

—

277

—

#97

—

554

—

C₄×C₈

—

277

—

C₄×C₈

—

554

—

C₂²×C₄

—

554

—

C₂²×C₄

C₂²×D₂₇₇

yes

characteristic

M₁₆

—

M₁₆

C₂×Dic₂₇₇

yes

—

C₂×C₈

—

C₂×C₈

—

277

—

C₂×C₈

—

277

—

C₂×C₈

—

277

—

C₂×C₈

—

277

—

C₂×C₈

C₄×D₂₇₇

yes

—

C₂×C₈

C₄×D₂₇₇

yes

characteristic

C₂×C₈

C₄×D₂₇₇

yes

characteristic

C₂×C₈

C₂×Dic₂₇₇

yes

—

C₄⋊C₄

—

277

—

C₂²⋊C₄

—

554

—

C₄²

—

277

—

C₄²

—

554

—

C₄²

—

554

—

100

C₂³

#17984564

yes

characteristic

101

C₂×C₄

—

554

—

102

C₂×C₄

—

554

—

103

C₂×C₄

—

554

—

104

C₂×C₄

C₂×C₄×D₂₇₇

yes

characteristic

105

C₂×C₄

C₂×C₄×D₂₇₇

yes

characteristic

106

C₂×C₄

C₂²×Dic₂₇₇

yes

characteristic

107

C₂×C₄

D₄×D₂₇₇

yes

characteristic

108

C₂×C₄

Q₈×D₂₇₇

yes

characteristic

109

C₂×C₄

#17984562

yes

characteristic

110

C₈

—

111

C₈

—

277

—

112

C₈

—

277

—

113

C₈

C₄×Dic₂₇₇

yes

—

114

C₈

C₄×Dic₂₇₇

yes

—

115

C₂²

—

116

C₂²

#22579838

yes

characteristic

117

C₂²

#22579842

yes

characteristic

118

C₂²

Q₈×Dic₂₇₇

yes

characteristic

119

C₄

—

554

—

120

C₄

—

1108

—

121

C₄

C₂×C₄×Dic₂₇₇

yes

characteristic

122

C₄

C₄⋊C₄×D₂₇₇

yes

characteristic

123

C₄

D₄×Dic₂₇₇

yes

characteristic

124

C₄

#22579845

yes

characteristic

125

C₂

—

126

C₂

—

127

C₂

C₄⋊C₄×Dic₂₇₇

yes

characteristic

128

C₂

#31800118

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₂₇₇
commutator	C₄×C₂₇₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₂₇₇
Fitting	C₂×#108×C₂₇₇	C₂
socle	C₂²×C₂₇₇	C₄×D₄
prime radical	C₂³×C₂₇₇	#41
cosocle	C₂²×C₄×C₂₇₇	C₂³
hypercenter	C₂×#108	D₂₇₇
hypocenter	C₂₇₇	#1772
LCS	G ▷ C₄×C₂₇₇ ▷ C₂×C₂₇₇ ▷ C₂₇₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₂×C₄

D₄×D₂₇₇

commutator

C₄×C₂₇₇

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₂₇₇

Fitting

C₂×#108×C₂₇₇

C₂

socle

C₂²×C₂₇₇

C₄×D₄

prime radical

C₂³×C₂₇₇

#41

cosocle

C₂²×C₄×C₂₇₇

C₂³

D₂₇₇

C₂₇₇

G ▷ C₄×C₂₇₇ ▷ C₂×C₂₇₇ ▷ C₂₇₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
9. #22579814	C₄	semidirect
10. #22579812	C₄	semidirect
13. C₈×Dic₂₇₇	C₄	semidirect
16. #108×C₂₇₇	C₄	semidirect
64. C₂₇₇	#1772	semidirect
76. #108	Dic₂₇₇	semidirect

H ≅ G/N

type

9. #22579814

C₄

semidirect

10. #22579812

C₄

semidirect

13. C₈×Dic₂₇₇

C₄

semidirect

16. #108×C₂₇₇

C₄

semidirect

64. C₂₇₇

#1772

semidirect

76. #108

Dic₂₇₇

semidirect

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

subgroup	index
#31800040	2
#31799960	2
C₂×C₈×Dic₂₇₇	2
#31799763	2
C₂×#108×C₂₇₇	2
#1772	277

subgroup

index

C₂×C₈×Dic₂₇₇

C₂×#108×C₂₇₇

277

Subgroups — #47983049

Conjugacy classes of subgroups

Complements

Maximal subgroups