Group #44529620 — subgroups

Subgroups — #44529620

Order 31168.

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×#21441260	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×D₄₈₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#29510215	C₂	1	1	yes	maximal, characteristic, complemented
4	#29510213	C₂	1	1	yes	maximal, characteristic, complemented
5	#29510202	C₂	1	1	yes	maximal, characteristic, complemented
6	#29510197	C₂	1	4	yes	maximal, complemented
7	#29510192	C₂	1	4	yes	maximal, complemented
8	#29510172	C₂	1	1	yes	maximal, characteristic, complemented
9	#124×C₄₈₇	C₂	1	1	yes	maximal, characteristic, complemented
10	#21441260	—	2	4	no	—
11	C₂×Dic₉₇₄	C₂²	1	2	yes	—
12	C₂×Dic₉₇₄	C₂²	1	1	yes	characteristic
13	C₂×Dic₉₇₄	C₂²	1	1	yes	characteristic
14	Q₈×D₄₈₇	—	2	4	no	—
15	C₂×C₂²⋊D₄₈₇	C₂²	1	2	yes	—
16	C₂²×Dic₄₈₇	C₂²	1	2	yes	—
17	C₂×C₄×D₄₈₇	C₂²	1	2	yes	—
18	C₂×C₄×D₄₈₇	C₂²	1	1	yes	characteristic
19	C₄⋊Dic₄₈₇	C₂²	1	4	yes	—
20	Dic₄₈₇⋊C₄	C₂²	1	2	yes	—
21	Dic₄₈₇⋊C₄	C₂²	1	4	yes	—
22	C₂²⋊Dic₄₈₇	C₂²	1	4	yes	—
23	(C₂×D₄₈₇)⋊C₄	C₂²	1	2	yes	—
24	C₄×Dic₄₈₇	C₂²	1	1	yes	characteristic
25	C₂×Q₈×C₄₈₇	C₂²	1	1	yes	characteristic
26	C₂×D₄×C₄₈₇	C₂²	1	1	yes	characteristic
27	C₂²⋊C₄×C₄₈₇	C₂²	1	4	yes	—
28	C₄²×C₄₈₇	C₂²	1	1	yes	characteristic
29	C₂²×D₄₈₇	C₂³	1	1	yes	characteristic
30	Dic₉₇₄	—	2	2	no	—
31	Dic₉₇₄	—	2	2	no	—
32	Dic₉₇₄	—	2	4	no	—
33	C₂²⋊D₄₈₇	—	2	4	no	—
34	C₂×Dic₄₈₇	—	2	4	no	—
35	C₂×Dic₄₈₇	C₂³	1	2	yes	—
36	C₂×Dic₄₈₇	C₂³	1	2	yes	—
37	C₂×Dic₄₈₇	C₂³	1	2	yes	—
38	C₂×Dic₄₈₇	C₂³	1	1	yes	characteristic
39	C₄×D₄₈₇	—	2	4	no	—
40	C₄×D₄₈₇	D₄	1	4	yes	—
41	C₂³×C₄₈₇	C₂³	1	2	yes	—
42	Q₈×C₄₈₇	—	2	2	no	—
43	D₄×C₄₈₇	—	2	2	no	—
44	C₂×C₄×C₄₈₇	C₂³	1	2	yes	—
45	C₂×C₄×C₄₈₇	C₂³	1	2	yes	—
46	C₂×C₄×C₄₈₇	C₂³	1	1	yes	characteristic
47	C₂²×C₄₈₇	—	2	2	no	—
48	C₂²×C₄₈₇	—	2	4	no	—
49	C₂²×C₄₈₇	C₂⁴	1	1	yes	characteristic
50	C₂×D₄₈₇	—	2	2	no	—
51	C₂×D₄₈₇	C₂×D₄	1	2	yes	—
52	C₄×C₄₈₇	—	2	2	no	—
53	C₄×C₄₈₇	—	2	2	no	—
54	C₄×C₄₈₇	C₂×D₄	1	2	yes	—
55	Dic₄₈₇	—	2	2	no	—
56	Dic₄₈₇	—	2	2	no	—
57	Dic₄₈₇	—	2	2	no	—
58	Dic₄₈₇	C₂×D₄	1	2	yes	—
59	C₂×C₄₈₇	—	4	2	no	—
60	C₂×C₄₈₇	C₂²×D₄	1	1	yes	characteristic
61	C₂×C₄₈₇	2₋¹⁺⁴	1	2	yes	—
62	D₄₈₇	—	2	2	no	—
63	C₄₈₇	#536	1	1	yes	characteristic, complemented, Sylow
64	#536	—	487	1	no	maximal, Sylow
65	C₂×#41	—	487	1	no	—
66	C₂²×Q₈	—	487	1	no	—
67	C₄⋊Q₈	—	487	1	no	—
68	C₄⋊Q₈	—	487	1	no	—
69	#124	—	487	1	no	—
70	#124	D₄₈₇	1	1	yes	characteristic, complemented
71	#123	—	487	4	no	—
72	C₂²⋊Q₈	—	487	4	no	—
73	#117	—	487	1	no	—
74	#41	—	974	4	no	—
75	C₂×Q₈	—	974	4	no	—
76	C₂×Q₈	—	487	2	no	—
77	C₂×Q₈	—	487	1	no	—
78	C₂×Q₈	—	487	1	no	—
79	C₂×Q₈	C₂×D₄₈₇	1	1	yes	characteristic
80	C₂×D₄	—	487	2	no	—
81	C₂×D₄	C₂×D₄₈₇	1	1	yes	characteristic
82	C₂²×C₄	—	487	2	no	—
83	C₂²×C₄	—	487	2	no	—
84	C₂²×C₄	—	487	1	no	—
85	C₄⋊C₄	—	487	2	no	—
86	C₄⋊C₄	—	487	4	no	—
87	C₄⋊C₄	—	487	4	no	—
88	C₂²⋊C₄	—	487	4	no	—
89	C₂²⋊C₄	—	487	2	no	—
90	C₂²⋊C₄	C₂×D₄₈₇	1	4	yes	—
91	C₄²	—	487	1	no	—
92	C₄²	C₂×D₄₈₇	1	1	yes	characteristic
93	C₂³	—	487	1	no	—
94	C₂³	C₂²×D₄₈₇	1	2	yes	—
95	Q₈	—	2	2	no	—
96	Q₈	—	974	4	no	—
97	Q₈	—	974	2	no	—
98	Q₈	—	974	2	no	—
99	D₄	—	2	2	no	—
100	D₄	—	974	4	no	—
101	C₂×C₄	—	974	4	no	—
102	C₂×C₄	—	974	4	no	—
103	C₂×C₄	—	487	2	no	—
104	C₂×C₄	—	487	2	no	—
105	C₂×C₄	—	487	2	no	—
106	C₂×C₄	—	487	1	no	—
107	C₂×C₄	—	487	4	no	—
108	C₂×C₄	C₂²×D₄₈₇	1	2	yes	—
109	C₂×C₄	C₂²×D₄₈₇	1	2	yes	—
110	C₂×C₄	C₂²×D₄₈₇	1	1	yes	characteristic
111	C₂²	—	2	2	no	—
112	C₂²	—	2	4	no	—
113	C₂²	—	974	2	no	—
114	C₂²	—	487	2	no	—
115	C₂²	C₂³×D₄₈₇	1	1	yes	characteristic
116	C₄	—	2	2	no	—
117	C₄	—	2	2	no	—
118	C₄	—	974	2	no	—
119	C₄	—	974	2	no	—
120	C₄	—	974	2	no	—
121	C₄	—	487	2	no	—
122	C₄	D₄×D₄₈₇	1	2	yes	—
123	C₂	—	4	2	no	—
124	C₂	—	974	2	no	—
125	C₂	C₂×D₄×D₄₈₇	1	1	yes	characteristic
126	C₂	#29510262	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#21441260

C₂

yes

maximal, characteristic

C₂×Q₈×D₄₈₇

C₂

yes

maximal, characteristic, complemented

#29510215

C₂

yes

maximal, characteristic, complemented

#29510213

C₂

yes

maximal, characteristic, complemented

#29510202

C₂

yes

maximal, characteristic, complemented

#29510197

C₂

yes

maximal, complemented

#29510192

C₂

yes

maximal, complemented

#29510172

C₂

yes

maximal, characteristic, complemented

#124×C₄₈₇

C₂

yes

maximal, characteristic, complemented

#21441260

—

C₂×Dic₉₇₄

C₂²

yes

—

C₂×Dic₉₇₄

C₂²

yes

characteristic

C₂×Dic₉₇₄

C₂²

yes

characteristic

Q₈×D₄₈₇

—

C₂×C₂²⋊D₄₈₇

C₂²

yes

—

C₂²×Dic₄₈₇

C₂²

yes

—

C₂×C₄×D₄₈₇

C₂²

yes

—

C₂×C₄×D₄₈₇

C₂²

yes

characteristic

C₄⋊Dic₄₈₇

C₂²

yes

—

Dic₄₈₇⋊C₄

C₂²

yes

—

Dic₄₈₇⋊C₄

C₂²

yes

—

C₂²⋊Dic₄₈₇

C₂²

yes

—

(C₂×D₄₈₇)⋊C₄

C₂²

yes

—

C₄×Dic₄₈₇

C₂²

yes

characteristic

C₂×Q₈×C₄₈₇

C₂²

yes

characteristic

C₂×D₄×C₄₈₇

C₂²

yes

characteristic

C₂²⋊C₄×C₄₈₇

C₂²

yes

—

C₄²×C₄₈₇

C₂²

yes

characteristic

C₂²×D₄₈₇

C₂³

yes

characteristic

Dic₉₇₄

—

Dic₉₇₄

—

Dic₉₇₄

—

C₂²⋊D₄₈₇

—

C₂×Dic₄₈₇

—

C₂×Dic₄₈₇

C₂³

yes

—

C₂×Dic₄₈₇

C₂³

yes

—

C₂×Dic₄₈₇

C₂³

yes

—

C₂×Dic₄₈₇

C₂³

yes

characteristic

C₄×D₄₈₇

—

C₄×D₄₈₇

D₄

yes

—

C₂³×C₄₈₇

C₂³

yes

—

Q₈×C₄₈₇

—

D₄×C₄₈₇

—

C₂×C₄×C₄₈₇

C₂³

yes

—

C₂×C₄×C₄₈₇

C₂³

yes

—

C₂×C₄×C₄₈₇

C₂³

yes

characteristic

C₂²×C₄₈₇

—

C₂²×C₄₈₇

—

C₂²×C₄₈₇

C₂⁴

yes

characteristic

C₂×D₄₈₇

—

C₂×D₄₈₇

C₂×D₄

yes

—

C₄×C₄₈₇

—

C₄×C₄₈₇

—

C₄×C₄₈₇

C₂×D₄

yes

—

Dic₄₈₇

—

Dic₄₈₇

—

Dic₄₈₇

—

Dic₄₈₇

C₂×D₄

yes

—

C₂×C₄₈₇

—

C₂×C₄₈₇

C₂²×D₄

yes

characteristic

C₂×C₄₈₇

2₋¹⁺⁴

yes

—

D₄₈₇

—

C₄₈₇

#536

yes

characteristic, complemented, Sylow

#536

—

487

maximal, Sylow

C₂×#41

—

487

—

C₂²×Q₈

—

487

—

C₄⋊Q₈

—

487

—

C₄⋊Q₈

—

487

—

#124

—

487

—

#124

D₄₈₇

yes

characteristic, complemented

#123

—

487

—

C₂²⋊Q₈

—

487

—

#117

—

487

—

#41

—

974

—

C₂×Q₈

—

974

—

C₂×Q₈

—

487

—

C₂×Q₈

—

487

—

C₂×Q₈

—

487

—

C₂×Q₈

C₂×D₄₈₇

yes

characteristic

C₂×D₄

—

487

—

C₂×D₄

C₂×D₄₈₇

yes

characteristic

C₂²×C₄

—

487

—

C₂²×C₄

—

487

—

C₂²×C₄

—

487

—

C₄⋊C₄

—

487

—

C₄⋊C₄

—

487

—

C₄⋊C₄

—

487

—

C₂²⋊C₄

—

487

—

C₂²⋊C₄

—

487

—

C₂²⋊C₄

C₂×D₄₈₇

yes

—

C₄²

—

487

—

C₄²

C₂×D₄₈₇

yes

characteristic

C₂³

—

487

—

C₂³

C₂²×D₄₈₇

yes

—

Q₈

—

Q₈

—

974

—

Q₈

—

974

—

Q₈

—

974

—

D₄

—

100

D₄

—

974

—

101

C₂×C₄

—

974

—

102

C₂×C₄

—

974

—

103

C₂×C₄

—

487

—

104

C₂×C₄

—

487

—

105

C₂×C₄

—

487

—

106

C₂×C₄

—

487

—

107

C₂×C₄

—

487

—

108

C₂×C₄

C₂²×D₄₈₇

yes

—

109

C₂×C₄

C₂²×D₄₈₇

yes

—

110

C₂×C₄

C₂²×D₄₈₇

yes

characteristic

111

C₂²

—

112

C₂²

—

113

C₂²

—

974

—

114

C₂²

—

487

—

115

C₂²

C₂³×D₄₈₇

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

974

—

119

C₄

—

974

—

120

C₄

—

974

—

121

C₄

—

487

—

122

C₄

D₄×D₄₈₇

yes

—

123

C₂

—

124

C₂

—

974

—

125

C₂

C₂×D₄×D₄₈₇

yes

characteristic

126

C₂

#29510262

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₄₈₇
commutator, socle, cosocle	C₂²×C₄₈₇	C₂⁴
Fitting	#124×C₄₈₇	C₂
prime radical	C₂×#21441260	C₂
hypercenter	#124	D₄₈₇
hypocenter	C₄₈₇	#536
LCS	G ▷ C₂²×C₄₈₇ ▷ C₄₈₇
UCS	C₁ ◁ C₂² ◁ #124

subgroup

G/H

center, Frattini

C₂²

C₂³×D₄₈₇

commutator, socle, cosocle

C₂²×C₄₈₇

C₂⁴

Fitting

#124×C₄₈₇

C₂

C₂

D₄₈₇

C₄₈₇

G ▷ C₂²×C₄₈₇ ▷ C₄₈₇

C₁ ◁ C₂² ◁ #124

N	H ≅ G/N	type
2. C₂×Q₈×D₄₈₇	C₂	semidirect
3. #29510215	C₂	semidirect
4. #29510213	C₂	semidirect
5. #29510202	C₂	semidirect
6. #29510197	C₂	semidirect
7. #29510192	C₂	semidirect
8. #29510172	C₂	semidirect
9. #124×C₄₈₇	C₂	semidirect
63. C₄₈₇	#536	semidirect
70. #124	D₄₈₇	semidirect

H ≅ G/N

type

2. C₂×Q₈×D₄₈₇

C₂

semidirect

3. #29510215

C₂

semidirect

4. #29510213

C₂

semidirect

5. #29510202

C₂

semidirect

6. #29510197

C₂

semidirect

7. #29510192

C₂

semidirect

8. #29510172

C₂

semidirect

9. #124×C₄₈₇

C₂

semidirect

63. C₄₈₇

#536

semidirect

70. #124

D₄₈₇

semidirect

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

subgroup	index
C₂×#21441260	2
C₂×Q₈×D₄₈₇	2
#29510215	2
#29510213	2
#29510202	2
#29510197	2
#29510192	2
#29510172	2
#124×C₄₈₇	2
#536	487

subgroup

index

C₂×Q₈×D₄₈₇

#124×C₄₈₇

487

Subgroups — #44529620

Conjugacy classes of subgroups

Complements

Maximal subgroups