Group #499707 — subgroups

Subgroups — #499707

Order 512.

419 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 419 conjugacy classes of subgroups — too many to draw. Only its 160 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	#21736	C₂	1	2	yes	maximal
2	#13511	C₂	1	4	yes	maximal
3	#13459	C₂	1	4	yes	maximal
4	#12668	C₂	1	4	yes	maximal
5	#12594	C₂	1	1	yes	maximal, characteristic
6	#3162	C₂²	1	2	yes	—
7	#2961	—	2	2	no	—
8	#2155	C₂²	1	8	yes	—
9	C₄×M₃₂	C₂²	1	2	yes	—
10	#1994	—	2	2	no	—
11	#1946	C₂²	1	4	yes	—
12	#1945	—	2	2	no	—
13	#1849	C₂²	1	2	yes	—
14	#1773	—	2	2	no	—
15	#1772	C₂²	1	4	yes	—
16	C₄×#97	C₂²	1	1	yes	characteristic
17	#1385	C₂²	1	4	yes	—
18	#1384	C₂²	1	4	yes	—
19	#1375	C₂²	1	4	yes	—
20	#533	—	2	2	no	—
21	C₂×M₃₂	C₂³	1	4	yes	—
22	#501	—	2	8	no	—
23	#446	—	4	8	no	—
24	C₂×#108	—	2	2	no	—
25	C₂×#108	C₂³	1	4	yes	—
26	#427	—	2	2	no	—
27	#427	C₂³	1	4	yes	—
28	#424	—	2	4	no	—
29	#423	—	2	2	no	—
30	#405	—	2	4	no	—
31	C₂×#97	D₄	1	2	yes	—
32	C₂×#97	C₂³	1	2	yes	—
33	C₂×C₄×C₈	D₄	1	1	yes	characteristic
34	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
35	C₄³	D₄	1	1	yes	characteristic
36	#365	—	2	16	no	—
37	#346	Q₈	1	4	yes	complemented
38	C₄×C₁₆	Q₈	1	4	yes	complemented
39	#339	—	2	4	no	—
40	#339	—	2	4	no	—
41	C₂×M₁₆	—	2	4	no	—
42	C₂²×C₈	C₂×D₄	1	2	yes	—
43	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₈	C₂⁴	1	1	yes	characteristic
45	C₄⋊Q₈	—	2	4	no	—
46	#125	—	2	4	no	—
47	C₂²⋊Q₈	—	4	4	no	—
48	C₄×Q₈	—	4	4	no	—
49	#117	—	2	4	no	—
50	C₂×C₄²	—	4	1	no	—
51	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄²	C₂×D₄	1	2	yes	—
53	M₃₂	—	2	8	no	—
54	C₂×C₁₆	C₂×Q₈	1	8	yes	—
55	#108	—	2	8	no	—
56	#108	—	4	4	no	—
57	#107	—	2	4	no	—
58	#107	—	2	4	no	—
59	#106	—	2	4	no	—
60	#106	#41	1	8	yes	—
61	C₄⋊C₈	—	4	8	no	—
62	C₄⋊C₈	—	4	4	no	—
63	#97	—	2	4	no	—
64	#97	—	4	4	no	—
65	#97	SD₁₆	1	8	yes	—
66	C₄×C₈	—	4	1	no	—
67	C₄×C₈	—	4	4	no	—
68	C₄×C₈	C₂×Q₈	1	2	yes	—
69	C₄×C₈	C₂×Q₈	1	2	yes	—
70	C₂×Q₈	—	4	4	no	—
71	C₂²×C₄	—	2	1	no	—
72	C₂²×C₄	—	2	2	no	—
73	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
74	M₁₆	—	4	4	no	—
75	M₁₆	—	4	4	no	—
76	C₂×C₈	—	2	2	no	—
77	C₂×C₈	—	2	2	no	—
78	C₂×C₈	—	4	4	no	—
79	C₂×C₈	C₄⋊Q₈	1	2	yes	—
80	C₂×C₈	C₂×SD₁₆	1	4	yes	—
81	C₂×C₈	C₂×SD₁₆	1	4	yes	—
82	C₂×C₈	C₂²×Q₈	1	2	yes	—
83	C₂×C₈	C₂×#41	1	4	yes	—
84	C₄⋊C₄	—	2	8	no	—
85	C₄⋊C₄	—	4	8	no	—
86	C₄⋊C₄	—	4	4	no	—
87	C₂²⋊C₄	—	4	4	no	—
88	C₄²	—	2	2	no	—
89	C₄²	—	2	1	no	—
90	C₄²	—	4	4	no	—
91	C₄²	—	8	2	no	—
92	C₄²	C₄⋊Q₈	1	1	yes	characteristic
93	C₄²	C₄⋊Q₈	1	1	yes	characteristic
94	C₄²	C₂×SD₁₆	1	4	yes	—
95	C₁₆	—	2	8	no	—
96	C₂³	#575	1	1	yes	characteristic
97	Q₈	—	16	4	no	—
98	C₂×C₄	—	2	1	no	—
99	C₂×C₄	—	2	4	no	—
100	C₂×C₄	—	2	1	no	—
101	C₂×C₄	—	4	2	no	—
102	C₂×C₄	—	4	8	no	—
103	C₂×C₄	—	8	2	no	—
104	C₂×C₄	#525	1	2	yes	—
105	C₂×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
106	C₂×C₄	C₂²×SD₁₆	1	2	yes	—
107	C₂×C₄	#575	1	1	yes	characteristic
108	C₈	—	2	4	no	—
109	C₈	—	8	4	no	—
110	C₈	#498	1	4	yes	—
111	C₈	#533	1	4	yes	—
112	C₂²	—	4	1	no	—
113	C₂²	#3101	1	2	yes	—
114	C₂²	#3162	1	1	yes	characteristic
115	C₄	—	2	2	no	—
116	C₄	—	4	1	no	—
117	C₄	—	4	4	no	—
118	C₄	—	8	8	no	—
119	C₄	#2399	1	1	yes	characteristic
120	C₄	#3103	1	2	yes	—
121	C₄	C₂×#498	1	1	yes	characteristic
122	C₂	—	2	1	no	—
123	C₂	—	2	2	no	—
124	C₂	#23837	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21736

C₂

yes

maximal

#13511

C₂

yes

maximal

#13459

C₂

yes

maximal

#12668

C₂

yes

maximal

#12594

C₂

yes

maximal, characteristic

#3162

C₂²

yes

—

#2961

—

#2155

C₂²

yes

—

C₄×M₃₂

C₂²

yes

—

#1994

—

#1946

C₂²

yes

—

#1945

—

#1849

C₂²

yes

—

#1773

—

#1772

C₂²

yes

—

C₄×#97

C₂²

yes

characteristic

#1385

C₂²

yes

—

#1384

C₂²

yes

—

#1375

C₂²

yes

—

#533

—

C₂×M₃₂

C₂³

yes

—

—

—

—

C₂³

yes

—

#427

—

#427

C₂³

yes

—

#424

—

#423

—

#405

—

C₂×#97

D₄

yes

—

C₂×#97

C₂³

yes

—

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

characteristic

C₄³

D₄

yes

characteristic

#365

—

#346

Q₈

yes

complemented

C₄×C₁₆

Q₈

yes

complemented

#339

—

#339

—

C₂×M₁₆

—

C₂²×C₈

C₂×D₄

yes

—

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

C₂⁴

yes

characteristic

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

—

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₄²

C₂×D₄

yes

—

M₃₂

—

C₂×C₁₆

C₂×Q₈

yes

—

—

—

—

—

—

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

#97

—

#97

SD₁₆

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×Q₈

yes

—

C₄×C₈

C₂×Q₈

yes

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₄

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊Q₈

yes

—

C₂×C₈

C₂×SD₁₆

yes

—

C₂×C₈

C₂×SD₁₆

yes

—

C₂×C₈

C₂²×Q₈

yes

—

C₂×C₈

C₂×#41

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₄⋊Q₈

yes

characteristic

C₄²

C₄⋊Q₈

yes

characteristic

C₄²

C₂×SD₁₆

yes

—

C₁₆

—

C₂³

#575

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

#525

yes

—

105

C₂×C₄

C₂×C₄⋊Q₈

yes

characteristic

106

C₂×C₄

C₂²×SD₁₆

yes

—

107

C₂×C₄

#575

yes

characteristic

108

C₈

—

109

C₈

—

110

C₈

#498

yes

—

111

C₈

#533

yes

—

112

C₂²

—

113

C₂²

#3101

yes

—

114

C₂²

#3162

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

#2399

yes

characteristic

120

C₄

#3103

yes

—

121

C₄

C₂×#498

yes

characteristic

122

C₂

—

123

C₂

—

124

C₂

#23837

yes

characteristic

	subgroup	G/H
center	C₄	C₂×#498
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#23837
prime radical	C₂³	#575
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₄

C₂×#498

commutator, Frattini, cosocle

C₂²×C₈

C₂⁴

C₂

C₂³

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₄ ◁ C₂²×C₈ ◁ C₄×#97 ◁ G

LECS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
37. #346	Q₈	semidirect
38. C₄×C₁₆	Q₈	semidirect

H ≅ G/N

type

37. #346

Q₈

semidirect

38. C₄×C₁₆

Q₈

semidirect

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

subgroup	index
#21736	2
#13511	2
#13459	2
#12668	2
#12594	2

subgroup

index

Subgroups — #499707

Conjugacy classes of subgroups

Complements

Maximal subgroups