Group #19615 — subgroups

Subgroups — C₂×#1722 #19615

Order 256.

457 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 457 conjugacy classes of subgroups — too many to draw. Only its 159 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	2	yes	maximal, complemented
2	C₂×#494	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#492	C₂	1	1	yes	maximal, characteristic
4	C₂×#486	C₂	1	2	yes	maximal
5	#1722	C₂	1	8	yes	maximal, direct product factor
6	C₂×#322	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×Q₁₆	—	2	2	no	—
8	C₂²×SD₁₆	—	2	2	no	—
9	C₂×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₂×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₂×#127	C₂²	1	1	yes	characteristic
12	#500	—	2	4	no	—
13	#500	C₂²	1	8	yes	complemented
14	#494	—	2	2	no	—
15	#494	C₂²	1	4	yes	complemented
16	#492	—	2	2	no	—
17	#492	C₂²	1	4	yes	—
18	#486	—	2	4	no	—
19	#486	C₂²	1	8	yes	—
20	C₂×#107	—	2	4	no	—
21	C₂×#102	—	2	4	no	—
22	C₂×C₄×C₈	C₂²	1	2	yes	—
23	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
24	#322	C₂²	1	4	yes	complemented
25	C₂²×Q₈	—	2	1	no	—
26	C₂²×Q₈	—	2	2	no	—
27	C₂²×D₄	—	2	1	no	—
28	C₂²×D₄	—	2	2	no	—
29	C₂×Q₁₆	—	2	8	no	—
30	C₂×Q₁₆	—	4	8	no	—
31	C₂×SD₁₆	—	2	8	no	—
32	C₂×SD₁₆	—	4	8	no	—
33	C₂²×C₈	D₄	1	4	yes	—
34	C₂²×C₈	D₄	1	2	yes	—
35	C₄⋊Q₈	—	2	2	no	—
36	C₄⋊Q₈	—	2	2	no	—
37	C₄⋊Q₈	—	2	2	no	—
38	C₄⋊Q₈	—	2	1	no	—
39	C₄⋊Q₈	—	2	2	no	—
40	C₄⋊Q₈	C₂³	1	4	yes	—
41	C₄⋊Q₈	C₂³	1	2	yes	—
42	#127	—	2	2	no	—
43	#127	—	2	1	no	—
44	#127	C₂³	1	2	yes	—
45	C₂×C₄⋊C₄	—	2	2	no	—
46	C₂×C₄⋊C₄	—	2	2	no	—
47	C₂×C₄⋊C₄	—	2	2	no	—
48	C₂×C₄²	C₂³	1	1	yes	characteristic
49	#107	—	2	16	no	—
50	#102	—	2	16	no	—
51	C₄×C₈	D₄	1	4	yes	—
52	C₄×C₈	D₄	1	2	yes	—
53	C₄×C₈	C₂³	1	4	yes	—
54	C₄×C₈	C₂³	1	2	yes	—
55	C₂⁴	—	4	1	no	—
56	C₂×Q₈	—	2	2	no	—
57	C₂×Q₈	—	2	4	no	—
58	C₂×Q₈	—	4	2	no	—
59	C₂×Q₈	—	4	1	no	—
60	C₂×Q₈	—	4	4	no	—
61	C₂×Q₈	—	4	8	no	—
62	C₂×D₄	—	2	4	no	—
63	C₂×D₄	—	2	2	no	—
64	C₂×D₄	—	4	1	no	—
65	C₂×D₄	—	4	2	no	—
66	C₂×D₄	—	4	4	no	—
67	C₂×D₄	—	4	8	no	—
68	C₂²×C₄	—	4	2	no	—
69	C₂²×C₄	—	4	1	no	—
70	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄	C₂×D₄	1	2	yes	—
72	Q₁₆	—	4	16	no	—
73	SD₁₆	—	4	16	no	—
74	C₂×C₈	—	2	8	no	—
75	C₂×C₈	D₈	1	8	yes	—
76	C₂×C₈	C₂×D₄	1	8	yes	—
77	C₂×C₈	C₂×D₄	1	4	yes	—
78	C₄⋊C₄	—	2	4	no	—
79	C₄⋊C₄	—	2	4	no	—
80	C₄⋊C₄	—	2	4	no	—
81	C₄⋊C₄	—	4	2	no	—
82	C₄⋊C₄	—	4	2	no	—
83	C₄⋊C₄	—	4	2	no	—
84	C₄²	C₂×D₄	1	2	yes	—
85	C₄²	C₂×D₄	1	1	yes	characteristic
86	C₄²	C₂⁴	1	1	yes	characteristic
87	C₂³	—	4	2	no	—
88	C₂³	—	8	2	no	—
89	C₂³	—	8	4	no	—
90	C₂³	#127	1	1	yes	characteristic
91	Q₈	—	4	4	no	—
92	Q₈	—	4	8	no	—
93	Q₈	—	8	2	no	—
94	Q₈	—	8	4	no	—
95	D₄	—	4	4	no	—
96	D₄	—	4	8	no	—
97	D₄	—	8	2	no	—
98	D₄	—	8	4	no	—
99	C₂×C₄	—	4	4	no	—
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	—	8	2	no	—
102	C₂×C₄	—	8	4	no	—
103	C₂×C₄	#127	1	1	yes	characteristic
104	C₂×C₄	#127	1	2	yes	—
105	C₂×C₄	C₂×D₈	1	4	yes	—
106	C₂×C₄	#137	1	8	yes	—
107	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
108	C₂×C₄	C₂²×D₄	1	2	yes	—
109	C₈	—	2	8	no	—
110	C₈	C₂×D₈	1	8	yes	—
111	C₂²	—	8	2	no	—
112	C₂²	—	8	4	no	—
113	C₂²	—	16	4	no	—
114	C₂²	#493	1	2	yes	—
115	C₂²	#497	1	4	yes	—
116	C₂²	C₂×#127	1	1	yes	characteristic
117	C₄	—	8	2	no	—
118	C₄	—	8	4	no	—
119	C₄	#493	1	2	yes	—
120	C₄	#497	1	4	yes	—
121	C₄	C₂²×D₈	1	2	yes	—
122	C₄	C₂×#137	1	4	yes	—
123	C₂	—	16	2	no	—
124	C₂	#1722	1	4	yes	direct product factor
125	C₂	C₂×#493	1	1	yes	characteristic
126	C₂	C₂×#497	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#500

C₂

yes

maximal, complemented

C₂×#494

C₂

yes

maximal, characteristic, complemented

C₂×#492

C₂

yes

maximal, characteristic

C₂×#486

C₂

yes

maximal

#1722

C₂

yes

maximal, direct product factor

C₂×#322

C₂

yes

maximal, characteristic, complemented

C₂²×Q₁₆

—

C₂²×SD₁₆

—

C₂×C₄⋊Q₈

C₂²

yes

—

C₂×C₄⋊Q₈

C₂²

yes

characteristic

C₂×#127

C₂²

yes

characteristic

#500

—

#500

C₂²

yes

complemented

#494

—

#494

C₂²

yes

complemented

#492

—

#492

C₂²

yes

—

#486

—

#486

C₂²

yes

—

C₂×#107

—

C₂×#102

—

C₂×C₄×C₈

C₂²

yes

—

C₂×C₄×C₈

C₂²

yes

characteristic

#322

C₂²

yes

complemented

C₂²×Q₈

—

C₂²×Q₈

—

C₂²×D₄

—

C₂²×D₄

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×SD₁₆

—

C₂×SD₁₆

—

C₂²×C₈

D₄

yes

—

C₂²×C₈

D₄

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂³

yes

—

C₄⋊Q₈

C₂³

yes

—

#127

—

#127

—

#127

C₂³

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂³

yes

characteristic

#107

—

#102

—

C₄×C₈

D₄

yes

—

C₄×C₈

D₄

yes

—

C₄×C₈

C₂³

yes

—

C₄×C₈

C₂³

yes

—

C₂⁴

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

—

Q₁₆

—

SD₁₆

—

C₂×C₈

—

C₂×C₈

D₈

yes

—

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×D₄

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×D₄

yes

—

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂⁴

yes

characteristic

C₂³

—

C₂³

—

C₂³

—

C₂³

#127

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

Q₈

—

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

#127

yes

characteristic

104

C₂×C₄

#127

yes

—

105

C₂×C₄

C₂×D₈

yes

—

106

C₂×C₄

#137

yes

—

107

C₂×C₄

C₂²×D₄

yes

characteristic

108

C₂×C₄

C₂²×D₄

yes

—

109

C₈

—

110

C₈

C₂×D₈

yes

—

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

#493

yes

—

115

C₂²

#497

yes

—

116

C₂²

C₂×#127

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

#493

yes

—

120

C₄

#497

yes

—

121

C₄

C₂²×D₈

yes

—

122

C₄

C₂×#137

yes

—

123

C₂

—

124

C₂

#1722

yes

direct product factor

125

C₂

C₂×#493

yes

characteristic

126

C₂

C₂×#497

yes

—

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

subgroup

G/H

center, socle

C₂³

#127

commutator, Frattini, cosocle

C₄²

C₂⁴

C₂²

G ▷ C₄² ▷ C₂² ▷ C₁

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G

LECS

G ▷ C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
1. C₂×#500	C₂	semidirect
2. C₂×#494	C₂	semidirect
5. #1722	C₂	direct (N × H)
6. C₂×#322	C₂	semidirect
13. #500	C₂²	semidirect
15. #494	C₂²	semidirect
24. #322	C₂²	semidirect
124. C₂	#1722	direct (N × H)

H ≅ G/N

type

1. C₂×#500

C₂

semidirect

2. C₂×#494

C₂

semidirect

5. #1722

C₂

direct (N × H)

6. C₂×#322

C₂

semidirect

13. #500

C₂²

semidirect

15. #494

C₂²

semidirect

24. #322

C₂²

semidirect

124. C₂

#1722

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×#494	2
C₂×#492	2
C₂×#486	2
#1722	2
C₂×#322	2

subgroup

index

Subgroups — C₂×#1722 #19615

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#1722 #19615

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#1722 #19615