Group #19594 — subgroups

Subgroups — C₂×#1707 #19594

Order 256.

407 conjugacy classes of subgroups; only the 175 normal subgroups are shown.

normal subgroups only

This group has 407 conjugacy classes of subgroups — too many to draw. Only its 175 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×#500	C₂	1	1	yes	maximal, characteristic
2	C₂×#498	C₂	1	2	yes	maximal
3	C₂×#494	C₂	1	1	yes	maximal, characteristic
4	C₂×#487	C₂	1	2	yes	maximal
5	#1707	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₈²	C₂	1	1	yes	maximal, characteristic
7	C₂²×Q₁₆	—	2	2	no	—
8	C₂×C₄⋊Q₈	C₂²	1	2	yes	—
9	C₂×C₄⋊Q₈	C₂²	1	2	yes	—
10	#500	—	2	2	no	—
11	#500	C₂²	1	4	yes	—
12	#498	—	2	4	no	—
13	#498	C₂²	1	8	yes	—
14	#494	—	2	2	no	—
15	#494	C₂²	1	4	yes	—
16	#487	—	2	4	no	—
17	#487	C₂²	1	8	yes	—
18	C₂×#107	—	2	2	no	—
19	C₂×#106	—	2	4	no	—
20	C₂×#103	—	2	4	no	—
21	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
24	C₈²	C₂²	1	4	yes	—
25	C₂²×Q₈	—	2	2	no	—
26	C₂²×Q₈	—	2	2	no	—
27	C₂×Q₁₆	—	2	8	no	—
28	C₂×Q₁₆	—	4	8	no	—
29	C₂²×C₈	D₄	1	2	yes	—
30	C₂²×C₈	D₄	1	2	yes	—
31	C₂²×C₈	D₄	1	2	yes	—
32	C₄⋊Q₈	—	2	2	no	—
33	C₄⋊Q₈	—	2	2	no	—
34	C₄⋊Q₈	—	2	2	no	—
35	C₄⋊Q₈	—	2	2	no	—
36	C₄⋊Q₈	—	2	2	no	—
37	C₄⋊Q₈	—	2	2	no	—
38	C₄⋊Q₈	C₂³	1	4	yes	—
39	C₄⋊Q₈	C₂³	1	4	yes	—
40	C₂×C₄⋊C₄	—	2	2	no	—
41	C₂×C₄⋊C₄	—	2	2	no	—
42	C₂×C₄⋊C₄	—	2	2	no	—
43	C₂×C₄⋊C₄	—	2	2	no	—
44	C₂×C₄²	C₂³	1	1	yes	characteristic
45	#107	—	2	8	no	—
46	#106	—	2	16	no	—
47	#103	—	2	16	no	—
48	C₄×C₈	D₄	1	2	yes	—
49	C₄×C₈	D₄	1	2	yes	—
50	C₄×C₈	D₄	1	2	yes	—
51	C₄×C₈	C₂³	1	2	yes	—
52	C₄×C₈	C₂³	1	2	yes	—
53	C₄×C₈	C₂³	1	2	yes	—
54	C₂×Q₈	—	2	4	no	—
55	C₂×Q₈	—	2	4	no	—
56	C₂×Q₈	—	4	2	no	—
57	C₂×Q₈	—	4	2	no	—
58	C₂×Q₈	—	4	8	no	—
59	C₂×Q₈	—	4	8	no	—
60	C₂²×C₄	—	4	2	no	—
61	C₂²×C₄	—	4	2	no	—
62	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
65	Q₁₆	—	4	16	no	—
66	C₂×C₈	—	2	4	no	—
67	C₂×C₈	SD₁₆	1	8	yes	—
68	C₂×C₈	Q₁₆	1	8	yes	complemented
69	C₂×C₈	C₂×D₄	1	4	yes	—
70	C₂×C₈	C₂×D₄	1	4	yes	—
71	C₂×C₈	C₂×D₄	1	4	yes	—
72	C₄⋊C₄	—	2	4	no	—
73	C₄⋊C₄	—	2	4	no	—
74	C₄⋊C₄	—	2	4	no	—
75	C₄⋊C₄	—	2	4	no	—
76	C₄⋊C₄	—	4	2	no	—
77	C₄⋊C₄	—	4	2	no	—
78	C₄⋊C₄	—	4	2	no	—
79	C₄⋊C₄	—	4	2	no	—
80	C₄²	C₂×D₄	1	1	yes	characteristic
81	C₄²	C₂×D₄	1	1	yes	characteristic
82	C₄²	C₂×D₄	1	1	yes	characteristic
83	C₄²	C₂⁴	1	1	yes	characteristic
84	C₂³	#127	1	1	yes	characteristic
85	Q₈	—	4	8	no	—
86	Q₈	—	4	8	no	—
87	Q₈	—	8	4	no	—
88	Q₈	—	8	4	no	—
89	C₂×C₄	—	4	4	no	—
90	C₂×C₄	—	4	4	no	—
91	C₂×C₄	—	8	4	no	—
92	C₂×C₄	—	8	4	no	—
93	C₂×C₄	#127	1	1	yes	characteristic
94	C₂×C₄	#127	1	1	yes	characteristic
95	C₂×C₄	#127	1	1	yes	characteristic
96	C₂×C₄	C₂×SD₁₆	1	4	yes	—
97	C₂×C₄	C₂×Q₁₆	1	4	yes	—
98	C₂×C₄	#135	1	4	yes	—
99	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
100	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
101	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
102	C₈	—	2	4	no	—
103	C₈	C₂×SD₁₆	1	8	yes	—
104	C₈	C₂×Q₁₆	1	8	yes	complemented
105	C₂²	#492	1	2	yes	—
106	C₂²	#494	1	2	yes	—
107	C₂²	#495	1	2	yes	—
108	C₂²	C₂×#127	1	1	yes	characteristic
109	C₄	—	8	4	no	—
110	C₄	—	8	4	no	—
111	C₄	#492	1	2	yes	—
112	C₄	#494	1	2	yes	—
113	C₄	#495	1	2	yes	—
114	C₄	C₂²×SD₁₆	1	2	yes	—
115	C₄	C₂²×Q₁₆	1	2	yes	—
116	C₄	C₂×#135	1	2	yes	—
117	C₂	#1707	1	4	yes	direct product factor
118	C₂	C₂×#492	1	1	yes	characteristic
119	C₂	C₂×#494	1	1	yes	characteristic
120	C₂	C₂×#495	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#500

C₂

yes

maximal, characteristic

C₂×#498

C₂

yes

maximal

C₂×#494

C₂

yes

maximal, characteristic

C₂×#487

C₂

yes

maximal

#1707

C₂

yes

maximal, direct product factor

C₂×C₈²

C₂

yes

maximal, characteristic

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

C₂²

yes

—

C₂×C₄⋊Q₈

C₂²

yes

—

#500

—

#500

C₂²

yes

—

#498

—

#498

C₂²

yes

—

#494

—

#494

C₂²

yes

—

#487

—

#487

C₂²

yes

—

C₂×#107

—

C₂×#106

—

C₂×#103

—

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

C₈²

C₂²

yes

—

C₂²×Q₈

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂²×C₈

D₄

yes

—

C₂²×C₈

D₄

yes

—

C₂²×C₈

D₄

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂³

yes

—

C₄⋊Q₈

C₂³

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂³

yes

—

—

—

C₄×C₈

D₄

yes

—

C₄×C₈

D₄

yes

—

C₄×C₈

D₄

yes

—

C₄×C₈

C₂³

yes

—

C₄×C₈

C₂³

yes

—

C₄×C₈

C₂³

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

Q₁₆

—

C₂×C₈

—

C₂×C₈

SD₁₆

yes

—

C₂×C₈

Q₁₆

yes

complemented

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×D₄

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂⁴

yes

characteristic

C₂³

#127

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂×SD₁₆

yes

—

C₂×C₄

C₂×Q₁₆

yes

—

C₂×C₄

#135

yes

—

C₂×C₄

C₂²×D₄

yes

characteristic

100

C₂×C₄

C₂²×D₄

yes

characteristic

101

C₂×C₄

C₂²×D₄

yes

characteristic

102

C₈

—

103

C₈

C₂×SD₁₆

yes

—

104

C₈

C₂×Q₁₆

yes

complemented

105

C₂²

#492

yes

—

106

C₂²

#494

yes

—

107

C₂²

#495

yes

—

108

C₂²

C₂×#127

yes

characteristic

109

C₄

—

110

C₄

—

111

C₄

#492

yes

—

112

C₄

#494

yes

—

113

C₄

#495

yes

—

114

C₄

C₂²×SD₁₆

yes

—

115

C₄

C₂²×Q₁₆

yes

—

116

C₄

C₂×#135

yes

—

117

C₂

#1707

yes

direct product factor

118

C₂

C₂×#492

yes

characteristic

119

C₂

C₂×#494

yes

characteristic

120

C₂

C₂×#495

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	#127
commutator, Frattini, cosocle	C₄²	C₂⁴
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center, socle, prime radical

C₂³

#127

commutator, Frattini, cosocle

C₄²

C₂⁴

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G

LECS

G ▷ C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
5. #1707	C₂	direct (N × H)
68. C₂×C₈	Q₁₆	semidirect
104. C₈	C₂×Q₁₆	semidirect
117. C₂	#1707	direct (N × H)

H ≅ G/N

type

5. #1707

C₂

direct (N × H)

68. C₂×C₈

Q₁₆

semidirect

104. C₈

C₂×Q₁₆

semidirect

117. C₂

#1707

direct (N × H)

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

subgroup	index
C₂×#500	2
C₂×#498	2
C₂×#494	2
C₂×#487	2
#1707	2
C₂×C₈²	2

subgroup

index

C₂×C₈²

Subgroups — C₂×#1707 #19594

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#1707 #19594

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#1707 #19594