Group #47983947 — subgroups

Subgroups — #47983947

Order 35456.

196 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 93 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×#22579816	C₂	1	1	yes	maximal, characteristic
2	#31800045	C₂	1	2	yes	maximal
3	C₂×#22579812	C₂	1	1	yes	maximal, characteristic
4	#31799770	C₂	1	2	yes	maximal
5	#405×C₂₇₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#17984547	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₂₇₇	C₂²	1	2	yes	—
8	#22579816	C₄	1	4	yes	complemented
9	#22579813	—	2	2	no	—
10	#22579812	—	2	1	no	—
11	#22579812	—	2	1	no	—
12	#22579811	—	2	2	no	—
13	C₂×M₁₆×C₂₇₇	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₂₇₇	C₂²	1	1	yes	characteristic
15	#117×C₂₇₇	C₂²	1	1	yes	characteristic
16	#97×C₂₇₇	—	2	1	no	—
17	C₄×C₈×C₂₇₇	—	2	1	no	—
18	C₂²×Dic₂₇₇	—	2	2	no	—
19	#17984547	—	2	2	no	—
20	#17984547	C₂×C₄	1	4	yes	—
21	C₂×#14555660	—	2	2	no	—
22	C₄⋊Dic₂₇₇	—	2	2	no	—
23	C₄⋊Dic₂₇₇	—	2	2	no	—
24	C₂²×C₄×C₂₇₇	C₂³	1	1	yes	characteristic
25	M₁₆×C₂₇₇	—	2	1	no	—
26	M₁₆×C₂₇₇	Q₈	1	2	yes	—
27	C₂×C₈×C₂₇₇	—	2	1	no	—
28	C₂×C₈×C₂₇₇	C₂×C₄	1	2	yes	—
29	C₂×C₈×C₂₇₇	D₄	1	1	yes	characteristic
30	C₂×C₈×C₂₇₇	D₄	1	1	yes	characteristic
31	C₂×C₈×C₂₇₇	Q₈	1	2	yes	—
32	C₄⋊C₄×C₂₇₇	D₄	1	2	yes	—
33	C₂²⋊C₄×C₂₇₇	—	2	1	no	—
34	C₄²×C₂₇₇	—	2	1	no	—
35	C₂×Dic₂₇₇	—	4	2	no	—
36	C₂×Dic₂₇₇	—	8	2	no	—
37	#14555660	—	2	4	no	—
38	C₂³×C₂₇₇	#41	1	1	yes	characteristic
39	C₂×C₄×C₂₇₇	—	2	2	no	—
40	C₂×C₄×C₂₇₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₂₇₇	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₂₇₇	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₂₇₇	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₄×C₂₇₇	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₄×C₂₇₇	#41	1	1	yes	characteristic
46	C₈×C₂₇₇	—	2	2	no	—
47	C₈×C₂₇₇	C₄⋊C₄	1	2	yes	—
48	C₈×C₂₇₇	C₄⋊C₄	1	2	yes	—
49	C₂²×C₂₇₇	—	2	2	no	—
50	C₂²×C₂₇₇	C₄×Q₈	1	1	yes	characteristic
51	C₂²×C₂₇₇	#121	1	1	yes	characteristic
52	C₂²×C₂₇₇	#125	1	1	yes	characteristic
53	C₄×C₂₇₇	—	4	2	no	—
54	C₄×C₂₇₇	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₄×C₂₇₇	C₄×D₄	1	1	yes	characteristic
56	C₄×C₂₇₇	C₂²⋊Q₈	1	1	yes	characteristic
57	C₄×C₂₇₇	C₄⋊Q₈	1	1	yes	characteristic
58	Dic₂₇₇	—	8	2	no	—
59	C₂×C₂₇₇	—	2	1	no	—
60	C₂×C₂₇₇	—	2	1	no	—
61	C₂×C₂₇₇	#389	1	1	yes	characteristic
62	C₂×C₂₇₇	#471	1	2	yes	—
63	C₂₇₇	#1951	1	1	yes	characteristic, complemented, Sylow
64	#1951	—	277	1	no	maximal, Sylow
65	C₂×#108	—	277	1	no	—
66	#428	—	277	2	no	—
67	C₂×#106	—	277	1	no	—
68	#405	D₂₇₇	1	1	yes	characteristic
69	#343	—	277	2	no	—
70	C₂×M₁₆	—	277	2	no	—
71	C₂×M₁₆	C₂×D₂₇₇	1	1	yes	characteristic
72	C₂²×C₈	C₂×D₂₇₇	1	1	yes	characteristic
73	#117	C₂×D₂₇₇	1	1	yes	characteristic
74	C₂×C₄⋊C₄	—	277	2	no	—
75	#108	—	277	4	no	—
76	#107	—	554	2	no	—
77	#106	—	554	1	no	—
78	#106	—	554	1	no	—
79	#106	—	554	2	no	—
80	#97	—	2	1	no	—
81	C₄×C₈	—	2	1	no	—
82	C₂²×C₄	—	554	2	no	—
83	C₂²×C₄	C₂²×D₂₇₇	1	1	yes	characteristic
84	M₁₆	—	2	1	no	—
85	M₁₆	—	277	4	no	—
86	M₁₆	—	554	2	no	—
87	M₁₆	Dic₅₅₄	1	2	yes	—
88	C₂×C₈	—	2	1	no	—
89	C₂×C₈	—	554	2	no	—
90	C₂×C₈	C₄×D₂₇₇	1	2	yes	—
91	C₂×C₈	C₂²⋊D₂₇₇	1	1	yes	characteristic
92	C₂×C₈	C₂²⋊D₂₇₇	1	1	yes	characteristic
93	C₂×C₈	Dic₅₅₄	1	2	yes	—
94	C₄⋊C₄	—	554	2	no	—
95	C₄⋊C₄	—	554	2	no	—
96	C₄⋊C₄	D₁₁₀₈	1	2	yes	—
97	C₂²⋊C₄	—	2	1	no	—
98	C₄²	—	2	1	no	—
99	C₂³	#17984563	1	1	yes	characteristic
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	1108	2	no	—
102	C₂×C₄	—	2216	2	no	—
103	C₂×C₄	C₂×C₄×D₂₇₇	1	1	yes	characteristic
104	C₂×C₄	C₂×C₂²⋊D₂₇₇	1	1	yes	characteristic
105	C₂×C₄	C₂×D₁₁₀₈	1	1	yes	characteristic
106	C₂×C₄	C₂×Dic₅₅₄	1	1	yes	characteristic
107	C₂×C₄	C₂×Dic₅₅₄	1	1	yes	characteristic
108	C₂×C₄	#17984563	1	1	yes	characteristic
109	C₈	—	2	2	no	—
110	C₈	—	554	4	no	—
111	C₈	#17984542	1	2	yes	—
112	C₈	#17984542	1	2	yes	—
113	C₂²	—	2	2	no	—
114	C₂²	C₄×Dic₅₅₄	1	1	yes	characteristic
115	C₂²	#22579854	1	1	yes	characteristic
116	C₂²	#22579872	1	1	yes	characteristic
117	C₄	—	4	2	no	—
118	C₄	—	2216	2	no	—
119	C₄	C₂×#17984542	1	1	yes	characteristic
120	C₄	C₄×D₁₁₀₈	1	1	yes	characteristic
121	C₄	#22579859	1	1	yes	characteristic
122	C₄	#22579880	1	1	yes	characteristic
123	C₂	—	2	1	no	—
124	C₂	—	2	1	no	—
125	C₂	#31799900	1	1	yes	characteristic
126	C₂	#31800265	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22579816

C₂

yes

maximal, characteristic

#31800045

C₂

yes

maximal

C₂×#22579812

C₂

yes

maximal, characteristic

#31799770

C₂

yes

maximal

#405×C₂₇₇

C₂

yes

maximal, characteristic

C₂×#17984547

C₂²

yes

—

C₂×C₄⋊Dic₂₇₇

C₂²

yes

—

#22579816

C₄

yes

—

—

—

—

C₂×M₁₆×C₂₇₇

C₂²

yes

characteristic

C₂²×C₈×C₂₇₇

C₂²

yes

characteristic

#117×C₂₇₇

C₂²

yes

characteristic

#97×C₂₇₇

—

C₄×C₈×C₂₇₇

—

C₂²×Dic₂₇₇

—

#17984547

—

#17984547

C₂×C₄

yes

—

C₂×#14555660

—

C₄⋊Dic₂₇₇

—

C₄⋊Dic₂₇₇

—

C₂²×C₄×C₂₇₇

C₂³

yes

characteristic

M₁₆×C₂₇₇

—

M₁₆×C₂₇₇

Q₈

yes

—

C₂×C₈×C₂₇₇

—

C₂×C₈×C₂₇₇

C₂×C₄

yes

—

C₂×C₈×C₂₇₇

D₄

yes

characteristic

C₂×C₈×C₂₇₇

D₄

yes

characteristic

C₂×C₈×C₂₇₇

Q₈

yes

—

C₄⋊C₄×C₂₇₇

D₄

yes

—

C₂²⋊C₄×C₂₇₇

—

C₄²×C₂₇₇

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

—

#14555660

—

C₂³×C₂₇₇

#41

yes

characteristic

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₇₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₇₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₇₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₇₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₇₇

#41

yes

characteristic

C₈×C₂₇₇

—

C₈×C₂₇₇

C₄⋊C₄

yes

—

C₈×C₂₇₇

C₄⋊C₄

yes

—

C₂²×C₂₇₇

—

C₂²×C₂₇₇

C₄×Q₈

yes

C₂²×C₂₇₇

yes

C₂²×C₂₇₇

yes

C₄×C₂₇₇

—

C₄×C₂₇₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₂₇₇

C₄×D₄

yes

characteristic

C₄×C₂₇₇

C₂²⋊Q₈

yes

characteristic

C₄×C₂₇₇

C₄⋊Q₈

yes

characteristic

Dic₂₇₇

—

C₂×C₂₇₇

—

C₂×C₂₇₇

—

C₂×C₂₇₇

#389

yes

characteristic

C₂×C₂₇₇

#471

yes

—

C₂₇₇

#1951

yes

characteristic, complemented, Sylow

#1951

—

277

maximal, Sylow

C₂×#108

—

277

—

#428

—

277

—

C₂×#106

—

277

—

#405

D₂₇₇

yes

characteristic

#343

—

277

—

C₂×M₁₆

—

277

—

C₂×M₁₆

C₂×D₂₇₇

yes

characteristic

C₂²×C₈

C₂×D₂₇₇

yes

characteristic

#117

C₂×D₂₇₇

yes

characteristic

C₂×C₄⋊C₄

—

277

—

#108

—

277

—

#107

—

554

—

#106

—

554

—

#106

—

554

—

#106

—

554

—

#97

—

C₄×C₈

—

C₂²×C₄

—

554

—

C₂²×C₄

C₂²×D₂₇₇

yes

characteristic

M₁₆

—

M₁₆

—

277

—

M₁₆

—

554

—

M₁₆

Dic₅₅₄

yes

—

C₂×C₈

—

C₂×C₈

—

554

—

C₂×C₈

C₄×D₂₇₇

yes

—

C₂×C₈

C₂²⋊D₂₇₇

yes

characteristic

C₂×C₈

C₂²⋊D₂₇₇

yes

characteristic

C₂×C₈

Dic₅₅₄

yes

—

C₄⋊C₄

—

554

—

C₄⋊C₄

—

554

—

C₄⋊C₄

D₁₁₀₈

yes

—

C₂²⋊C₄

—

C₄²

—

C₂³

#17984563

yes

characteristic

100

C₂×C₄

—

101

C₂×C₄

—

1108

—

102

C₂×C₄

—

2216

—

103

C₂×C₄

C₂×C₄×D₂₇₇

yes

characteristic

104

C₂×C₄

C₂×C₂²⋊D₂₇₇

yes

characteristic

105

C₂×C₄

C₂×D₁₁₀₈

yes

characteristic

106

C₂×C₄

C₂×Dic₅₅₄

yes

characteristic

107

C₂×C₄

C₂×Dic₅₅₄

yes

characteristic

108

C₂×C₄

#17984563

yes

characteristic

109

C₈

—

110

C₈

—

554

—

111

C₈

#17984542

yes

—

112

C₈

#17984542

yes

—

113

C₂²

—

114

C₂²

C₄×Dic₅₅₄

yes

characteristic

115

C₂²

#22579854

yes

characteristic

116

C₂²

#22579872

yes

characteristic

117

C₄

—

118

C₄

—

2216

—

119

C₄

C₂×#17984542

yes

characteristic

120

C₄

C₄×D₁₁₀₈

yes

characteristic

121

C₄

#22579859

yes

characteristic

122

C₄

#22579880

yes

characteristic

123

C₂

—

124

C₂

—

125

C₂

#31799900

yes

characteristic

126

C₂

#31800265

yes

—

	subgroup	G/H
center	C₂²	#22579854
commutator	C₂×C₄×C₂₇₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₇₇
Fitting	#405×C₂₇₇	C₂
socle	C₂²×C₂₇₇	#121
prime radical	C₂³×C₂₇₇	#41
cosocle	C₂²×C₄×C₂₇₇	C₂³
hypercenter	#405	D₂₇₇
hypocenter	C₂₇₇	#1951
LCS	G ▷ C₂×C₄×C₂₇₇ ▷ C₂×C₂₇₇ ▷ C₂₇₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405

subgroup

G/H

center

C₂²

#22579854

commutator

C₂×C₄×C₂₇₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₇₇

Fitting

#405×C₂₇₇

C₂

C₂²×C₂₇₇

C₂³×C₂₇₇

C₂²×C₄×C₂₇₇

C₂³

D₂₇₇

C₂₇₇

G ▷ C₂×C₄×C₂₇₇ ▷ C₂×C₂₇₇ ▷ C₂₇₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405

N	H ≅ G/N	type
8. #22579816	C₄	semidirect
63. C₂₇₇	#1951	semidirect

H ≅ G/N

type

8. #22579816

C₄

semidirect

63. C₂₇₇

#1951

semidirect

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

subgroup	index
C₂×#22579816	2
#31800045	2
C₂×#22579812	2
#31799770	2
#405×C₂₇₇	2
#1951	277

subgroup

index

#405×C₂₇₇

277

Subgroups — #47983947

Conjugacy classes of subgroups

Complements

Maximal subgroups