Group #17847 — subgroups

Subgroups — C₂×#1579 #17847

Order 256.

359 conjugacy classes of subgroups; only the 215 normal subgroups are shown.

normal subgroups only

This group has 359 conjugacy classes of subgroups — too many to draw. Only its 215 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×C₈×Q₈	C₂	1	2	yes	maximal
2	C₂×C₄×Q₁₆	C₂	1	1	yes	maximal, characteristic
3	#1579	C₂	1	8	yes	maximal, direct product factor
4	C₂×#334	C₂	1	1	yes	maximal, characteristic
5	C₂×Q₈⋊C₈	C₂	1	2	yes	maximal
6	C₂×#322	C₂	1	1	yes	maximal, characteristic
7	C₂²×Q₁₆	C₄	1	1	yes	characteristic
8	C₂×C₄×Q₈	C₂²	1	2	yes	—
9	C₈×Q₈	—	2	4	no	—
10	C₈×Q₈	C₂²	1	8	yes	—
11	C₄×Q₁₆	C₄	1	4	yes	—
12	C₄×Q₁₆	C₂²	1	4	yes	—
13	C₂×#107	C₄	1	1	yes	characteristic
14	C₂×C₄⋊C₈	—	2	2	no	—
15	C₂×C₄⋊C₈	C₂²	1	2	yes	—
16	C₂×#103	C₄	1	2	yes	—
17	C₂×C₄×C₈	—	2	2	no	—
18	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
21	#334	C₂²	1	4	yes	—
22	Q₈⋊C₈	C₂²	1	8	yes	—
23	#322	C₂²	1	4	yes	—
24	C₂²×Q₈	C₂×C₄	1	2	yes	—
25	C₂×Q₁₆	C₈	1	8	yes	complemented
26	C₂×Q₁₆	C₂×C₄	1	4	yes	—
27	C₂²×C₈	—	2	2	no	—
28	C₂²×C₈	—	2	1	no	—
29	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈	D₄	1	2	yes	—
32	C₄×Q₈	—	2	2	no	—
33	C₄×Q₈	—	2	2	no	—
34	C₄×Q₈	C₂×C₄	1	4	yes	—
35	C₄×Q₈	C₂³	1	4	yes	—
36	C₂×C₄⋊C₄	—	2	2	no	—
37	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
38	C₂×C₄²	—	2	2	no	—
39	C₂×C₄²	C₂³	1	1	yes	characteristic
40	#107	C₂×C₄	1	4	yes	—
41	C₄⋊C₈	—	2	8	no	—
42	C₄⋊C₈	—	2	2	no	—
43	C₄⋊C₈	C₂³	1	4	yes	—
44	#103	C₂×C₄	1	8	yes	—
45	C₄×C₈	—	2	8	no	—
46	C₄×C₈	—	2	1	no	—
47	C₄×C₈	C₂×C₄	1	2	yes	—
48	C₄×C₈	D₄	1	2	yes	—
49	C₄×C₈	C₂³	1	2	yes	—
50	C₄×C₈	C₂³	1	2	yes	—
51	C₄×C₈	C₂³	1	2	yes	—
52	C₂×Q₈	—	2	2	no	—
53	C₂×Q₈	—	2	4	no	—
54	C₂×Q₈	C₂×C₈	1	8	yes	—
55	C₂×Q₈	C₂²×C₄	1	4	yes	—
56	C₂²×C₄	—	2	2	no	—
57	C₂²×C₄	—	2	2	no	—
58	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄	#41	1	1	yes	characteristic
61	Q₁₆	C₂×C₈	1	16	yes	complemented
62	C₂×C₈	—	2	2	no	—
63	C₂×C₈	—	2	4	no	—
64	C₂×C₈	—	2	4	no	—
65	C₂×C₈	—	2	2	no	—
66	C₂×C₈	—	4	2	no	—
67	C₂×C₈	—	4	4	no	—
68	C₂×C₈	C₂×C₈	1	4	yes	—
69	C₂×C₈	C₂²×C₄	1	2	yes	—
70	C₂×C₈	C₂²×C₄	1	2	yes	—
71	C₂×C₈	C₂×D₄	1	4	yes	—
72	C₄⋊C₄	—	2	4	no	—
73	C₄⋊C₄	—	2	2	no	—
74	C₄⋊C₄	—	2	4	no	—
75	C₄⋊C₄	C₂²×C₄	1	4	yes	—
76	C₄²	—	2	4	no	—
77	C₄²	—	2	4	no	—
78	C₄²	C₂²×C₄	1	1	yes	characteristic
79	C₄²	C₂×D₄	1	1	yes	characteristic
80	C₄²	#41	1	1	yes	characteristic
81	C₄²	C₂⁴	1	1	yes	characteristic
82	C₂³	C₄×D₄	1	1	yes	characteristic
83	Q₈	—	2	4	no	—
84	Q₈	—	2	8	no	—
85	Q₈	C₂²×C₈	1	8	yes	—
86	C₂×C₄	—	2	4	no	—
87	C₂×C₄	—	2	8	no	—
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	2	4	no	—
90	C₂×C₄	—	4	4	no	—
91	C₂×C₄	C₄×D₄	1	1	yes	characteristic
92	C₂×C₄	C₄×D₄	1	1	yes	characteristic
93	C₂×C₄	C₄×D₄	1	1	yes	characteristic
94	C₂×C₄	C₂²×C₈	1	2	yes	—
95	C₂×C₄	#131	1	2	yes	—
96	C₂×C₄	#137	1	4	yes	—
97	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
98	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
99	C₂×C₄	C₂×#41	1	1	yes	characteristic
100	C₈	—	2	2	no	—
101	C₈	—	2	4	no	—
102	C₈	—	4	2	no	—
103	C₈	—	4	4	no	—
104	C₈	C₂²×C₈	1	4	yes	—
105	C₂²	C₈×D₄	1	2	yes	—
106	C₂²	#441	1	2	yes	—
107	C₂²	#444	1	2	yes	—
108	C₂²	C₂×C₄×D₄	1	1	yes	characteristic
109	C₄	—	2	8	no	—
110	C₄	—	2	1	no	—
111	C₄	—	2	1	no	—
112	C₄	—	4	4	no	—
113	C₄	C₈×D₄	1	2	yes	—
114	C₄	#441	1	2	yes	—
115	C₄	C₂³×C₈	1	1	yes	characteristic
116	C₄	C₂×#131	1	1	yes	characteristic
117	C₄	C₂×#137	1	2	yes	—
118	C₂	#1579	1	4	yes	direct product factor
119	C₂	C₂×C₈×D₄	1	1	yes	characteristic
120	C₂	C₂×#441	1	1	yes	characteristic
121	C₂	C₂×#444	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₈×Q₈

C₂

yes

maximal

C₂×C₄×Q₁₆

C₂

yes

maximal, characteristic

#1579

C₂

yes

maximal, direct product factor

C₂×#334

C₂

yes

maximal, characteristic

C₂×Q₈⋊C₈

C₂

yes

maximal

C₂×#322

C₂

yes

maximal, characteristic

C₂²×Q₁₆

C₄

yes

characteristic

C₂×C₄×Q₈

C₂²

yes

—

C₈×Q₈

—

C₈×Q₈

C₂²

yes

—

C₄×Q₁₆

C₄

yes

—

C₄×Q₁₆

C₂²

yes

—

C₂×#107

C₄

yes

characteristic

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₂²

yes

—

C₂×#103

C₄

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

#334

C₂²

yes

—

Q₈⋊C₈

C₂²

yes

—

#322

C₂²

yes

—

C₂²×Q₈

C₂×C₄

yes

—

C₂×Q₁₆

C₈

yes

complemented

C₂×Q₁₆

C₂×C₄

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

D₄

yes

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

C₂×C₄

yes

—

C₄×Q₈

C₂³

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×C₄

yes

—

C₂×C₄²

—

C₂×C₄²

C₂³

yes

characteristic

#107

C₂×C₄

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂³

yes

—

#103

C₂×C₄

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×C₄

yes

—

C₄×C₈

D₄

yes

—

C₄×C₈

C₂³

yes

—

C₄×C₈

C₂³

yes

—

C₄×C₈

C₂³

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×C₈

yes

—

C₂×Q₈

C₂²×C₄

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

#41

yes

characteristic

Q₁₆

C₂×C₈

yes

complemented

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

yes

—

C₂×C₈

C₂²×C₄

yes

—

C₂×C₈

C₂²×C₄

yes

—

C₂×C₈

C₂×D₄

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²×C₄

yes

—

C₄²

—

C₄²

—

C₄²

C₂²×C₄

yes

characteristic

C₄²

C₂×D₄

yes

characteristic

C₄²

#41

yes

characteristic

C₄²

C₂⁴

yes

characteristic

C₂³

C₄×D₄

yes

characteristic

Q₈

—

Q₈

—

Q₈

C₂²×C₈

yes

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×D₄

yes

characteristic

C₂×C₄

C₄×D₄

yes

characteristic

C₂×C₄

C₄×D₄

yes

characteristic

C₂×C₄

C₂²×C₈

yes

—

C₂×C₄

#131

yes

—

C₂×C₄

#137

yes

—

C₂×C₄

C₂³×C₄

yes

characteristic

C₂×C₄

C₂²×D₄

yes

characteristic

C₂×C₄

C₂×#41

yes

characteristic

100

C₈

—

101

C₈

—

102

C₈

—

103

C₈

—

104

C₈

C₂²×C₈

yes

—

105

C₂²

C₈×D₄

yes

—

106

C₂²

#441

yes

—

107

C₂²

#444

yes

—

108

C₂²

C₂×C₄×D₄

yes

characteristic

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

C₈×D₄

yes

—

114

C₄

#441

yes

—

115

C₄

C₂³×C₈

yes

characteristic

116

C₄

C₂×#131

yes

characteristic

117

C₄

C₂×#137

yes

—

118

C₂

#1579

yes

direct product factor

119

C₂

C₂×C₈×D₄

yes

characteristic

120

C₂

C₂×#441

yes

characteristic

121

C₂

C₂×#444

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₄
commutator	C₄	C₂³×C₈
Frattini, cosocle	C₄²	C₂⁴
socle, prime radical	C₂³	C₄×D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center

C₂²×C₄

C₂×D₄

commutator

C₄

C₂³×C₈

Frattini, cosocle

C₄²

C₂⁴

socle, prime radical

C₂³

C₄×D₄

LCS

G ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G

LECS

G ▷ C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
3. #1579	C₂	direct (N × H)
25. C₂×Q₁₆	C₈	semidirect
61. Q₁₆	C₂×C₈	semidirect
118. C₂	#1579	direct (N × H)

H ≅ G/N

type

3. #1579

C₂

direct (N × H)

25. C₂×Q₁₆

C₈

semidirect

61. Q₁₆

C₂×C₈

semidirect

118. C₂

#1579

direct (N × H)

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

subgroup	index
C₂×C₈×Q₈	2
C₂×C₄×Q₁₆	2
#1579	2
C₂×#334	2
C₂×Q₈⋊C₈	2
C₂×#322	2

subgroup

index

C₂×C₈×Q₈

C₂×C₄×Q₁₆

#1579

C₂×#334

C₂×Q₈⋊C₈

C₂×#322

Subgroups — C₂×#1579 #17847

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#1579 #17847

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#1579 #17847