Group #19608 — subgroups

Subgroups — #19608

Order 256.

399 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 399 conjugacy classes of subgroups — too many to draw. Only its 139 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	#3167	C₂	1	4	yes	maximal, complemented
2	#3160	C₂	1	1	yes	maximal, characteristic, complemented
3	#3147	C₂	1	1	yes	maximal, characteristic
4	#1716	C₂	1	2	yes	maximal, complemented
5	#1715	C₂	1	2	yes	maximal, complemented
6	#1714	C₂	1	4	yes	maximal, complemented
7	C₂×#322	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#135	—	2	2	no	—
9	#533	C₂²	1	2	yes	—
10	#532	C₂²	1	2	yes	—
11	#500	C₂²	1	2	yes	—
12	#499	—	2	2	no	—
13	#498	C₂²	1	2	yes	—
14	#498	C₂²	1	4	yes	—
15	#495	—	2	2	no	—
16	#494	C₂²	1	1	yes	characteristic
17	#493	C₂²	1	1	yes	characteristic
18	#492	C₂²	1	2	yes	—
19	#492	C₂²	1	4	yes	—
20	#487	C₂²	1	4	yes	—
21	#486	C₂²	1	4	yes	—
22	#465	—	2	8	no	—
23	C₄×SD₁₆	—	2	8	no	—
24	#427	—	2	2	no	—
25	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈	C₂²	1	2	yes	—
27	#322	C₂²	1	2	yes	—
28	#322	C₂²	1	2	yes	—
29	C₂×#41	—	2	2	no	—
30	#135	—	4	8	no	—
31	C₂×Q₁₆	—	2	2	no	—
32	C₂×SD₁₆	—	2	4	no	—
33	C₂×SD₁₆	—	2	8	no	—
34	C₂×D₈	—	2	2	no	—
35	C₂²×C₈	D₄	1	4	yes	—
36	C₂²×C₈	D₄	1	2	yes	—
37	C₄⋊Q₈	C₂³	1	2	yes	—
38	C₄⋊Q₈	C₂³	1	4	yes	—
39	#127	C₂³	1	2	yes	—
40	#125	—	2	2	no	—
41	#124	—	2	2	no	—
42	C₂²⋊Q₈	—	2	4	no	—
43	#121	—	2	4	no	—
44	C₄×Q₈	—	2	4	no	—
45	C₄×D₄	—	2	4	no	—
46	#117	—	2	2	no	—
47	C₂×C₄²	C₂³	1	1	yes	characteristic
48	#107	—	2	4	no	—
49	#106	—	2	4	no	—
50	#106	—	2	8	no	—
51	#103	—	2	8	no	—
52	#102	—	2	8	no	—
53	C₄×C₈	D₄	1	4	yes	—
54	C₄×C₈	D₄	1	2	yes	—
55	C₄×C₈	C₂³	1	4	yes	—
56	C₄×C₈	C₂³	1	1	yes	characteristic
57	C₄×C₈	C₂³	1	1	yes	characteristic
58	#41	—	4	8	no	—
59	C₂×Q₈	—	2	4	no	—
60	C₂×Q₈	—	2	2	no	—
61	C₂×D₄	—	2	4	no	—
62	C₂×D₄	—	2	2	no	—
63	C₂×D₄	—	4	2	no	—
64	C₂²×C₄	—	4	2	no	—
65	C₂²×C₄	C₂×D₄	1	2	yes	—
66	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
67	Q₁₆	—	4	4	no	—
68	SD₁₆	—	4	16	no	—
69	SD₁₆	—	4	8	no	—
70	D₈	—	4	4	no	—
71	C₂×C₈	—	4	4	no	—
72	C₂×C₈	SD₁₆	1	8	yes	complemented
73	C₂×C₈	C₂×D₄	1	8	yes	—
74	C₂×C₈	C₂×D₄	1	2	yes	—
75	C₂×C₈	C₂×D₄	1	2	yes	—
76	C₄⋊C₄	—	2	4	no	—
77	C₄⋊C₄	—	2	4	no	—
78	C₄⋊C₄	—	2	4	no	—
79	C₄⋊C₄	—	4	4	no	—
80	C₂²⋊C₄	—	4	2	no	—
81	C₂²⋊C₄	—	4	4	no	—
82	C₄²	—	4	2	no	—
83	C₄²	C₂×D₄	1	1	yes	characteristic
84	C₄²	C₂×D₄	1	2	yes	—
85	C₄²	C₂⁴	1	1	yes	characteristic
86	C₂³	—	4	2	no	—
87	C₂³	#127	1	1	yes	characteristic
88	Q₈	—	4	4	no	—
89	Q₈	—	4	8	no	—
90	D₄	—	4	4	no	—
91	D₄	—	4	8	no	—
92	D₄	—	8	4	no	—
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	—	4	4	no	—
95	C₂×C₄	—	4	2	no	—
96	C₂×C₄	—	8	4	no	—
97	C₂×C₄	#127	1	2	yes	—
98	C₂×C₄	#127	1	1	yes	characteristic
99	C₂×C₄	C₂×SD₁₆	1	4	yes	—
100	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
101	C₂×C₄	C₂²×D₄	1	2	yes	—
102	C₈	—	2	8	no	—
103	C₈	C₂×SD₁₆	1	4	yes	—
104	C₈	C₂×SD₁₆	1	4	yes	—
105	C₂²	—	2	2	no	—
106	C₂²	—	8	2	no	—
107	C₂²	—	8	4	no	—
108	C₂²	#492	1	2	yes	—
109	C₂²	C₂×#127	1	1	yes	characteristic
110	C₄	—	2	2	no	—
111	C₄	—	8	2	no	—
112	C₄	—	8	4	no	—
113	C₄	#492	1	2	yes	—
114	C₄	C₂²×SD₁₆	1	2	yes	—
115	C₄	#575	1	4	yes	—
116	C₂	—	2	2	no	—
117	C₂	—	16	2	no	—
118	C₂	C₂×#492	1	1	yes	characteristic
119	C₂	#3150	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#3167

C₂

yes

maximal, complemented

#3160

C₂

yes

maximal, characteristic, complemented

#3147

C₂

yes

maximal, characteristic

#1716

C₂

yes

maximal, complemented

#1715

C₂

yes

maximal, complemented

#1714

C₂

yes

maximal, complemented

C₂×#322

C₂

yes

maximal, characteristic, complemented

C₂×#135

—

#533

C₂²

yes

—

#532

C₂²

yes

—

#500

C₂²

yes

—

#499

—

#498

C₂²

yes

—

#498

C₂²

yes

—

#495

—

#494

C₂²

yes

characteristic

#493

C₂²

yes

characteristic

#492

C₂²

yes

—

#492

C₂²

yes

—

#487

C₂²

yes

—

#486

C₂²

yes

—

#465

—

C₄×SD₁₆

—

#427

—

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

—

#322

C₂²

yes

—

#322

C₂²

yes

—

C₂×#41

—

#135

—

C₂×Q₁₆

—

C₂×SD₁₆

—

C₂×SD₁₆

—

C₂×D₈

—

C₂²×C₈

D₄

yes

—

C₂²×C₈

D₄

yes

—

C₄⋊Q₈

C₂³

yes

—

C₄⋊Q₈

C₂³

yes

—

#127

C₂³

yes

—

#125

—

#124

—

C₂²⋊Q₈

—

#121

—

C₄×Q₈

—

C₄×D₄

—

#117

—

C₂×C₄²

C₂³

yes

—

—

—

—

—

C₄×C₈

D₄

yes

—

C₄×C₈

D₄

yes

—

C₄×C₈

C₂³

yes

—

C₄×C₈

C₂³

yes

characteristic

C₄×C₈

C₂³

yes

characteristic

#41

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

—

C₂²×C₄

C₂×D₄

yes

characteristic

Q₁₆

—

SD₁₆

—

SD₁₆

—

D₈

—

C₂×C₈

—

C₂×C₈

SD₁₆

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

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×D₄

yes

—

C₄²

C₂⁴

yes

characteristic

C₂³

—

C₂³

#127

yes

characteristic

Q₈

—

Q₈

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#127

yes

—

C₂×C₄

#127

yes

characteristic

C₂×C₄

C₂×SD₁₆

yes

—

100

C₂×C₄

C₂²×D₄

yes

characteristic

101

C₂×C₄

C₂²×D₄

yes

—

102

C₈

—

103

C₈

C₂×SD₁₆

yes

—

104

C₈

C₂×SD₁₆

yes

—

105

C₂²

—

106

C₂²

—

107

C₂²

—

108

C₂²

#492

yes

—

109

C₂²

C₂×#127

yes

characteristic

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

#492

yes

—

114

C₄

C₂²×SD₁₆

yes

—

115

C₄

#575

yes

—

116

C₂

—

117

C₂

—

118

C₂

C₂×#492

yes

characteristic

119

C₂

#3150

yes

—

	subgroup	G/H
center	C₂×C₄	#127
commutator, Frattini, cosocle	C₄²	C₂⁴
socle	C₂²	C₂×#127
prime radical	#3147	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center

C₂×C₄

#127

commutator, Frattini, cosocle

C₄²

C₂⁴

C₂²

C₂

G ▷ C₄² ▷ C₂² ▷ C₁

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G

LECS

G ▷ C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
1. #3167	C₂	semidirect
2. #3160	C₂	semidirect
4. #1716	C₂	semidirect
5. #1715	C₂	semidirect
6. #1714	C₂	semidirect
7. C₂×#322	C₂	semidirect
72. C₂×C₈	SD₁₆	semidirect

H ≅ G/N

type

1. #3167

C₂

semidirect

2. #3160

C₂

semidirect

4. #1716

C₂

semidirect

5. #1715

C₂

semidirect

6. #1714

C₂

semidirect

7. C₂×#322

C₂

semidirect

72. C₂×C₈

SD₁₆

semidirect

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

subgroup	index
#3167	2
#3160	2
#3147	2
#1716	2
#1715	2
#1714	2
C₂×#322	2

subgroup

index

Subgroups — #19608

Conjugacy classes of subgroups

Complements

Maximal subgroups