Group #19607 — subgroups

Subgroups — C₂×#1716 #19607

Order 256.

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

normal subgroups only

This group has 399 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×#498	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	#1716	C₂	1	8	yes	maximal, direct product factor
6	C₂×#322	C₂	1	1	yes	maximal, characteristic
7	C₂²×Q₁₆	—	2	2	no	—
8	C₂×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₂×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	#498	—	2	4	no	—
12	#498	—	2	2	no	—
13	#498	C₂²	1	8	yes	—
14	#498	C₂²	1	4	yes	—
15	#494	—	2	2	no	—
16	#494	C₂²	1	4	yes	—
17	#487	—	2	4	no	—
18	#487	C₂²	1	8	yes	—
19	C₂×#106	—	2	4	no	—
20	C₂×#106	—	2	2	no	—
21	C₂×#103	—	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	—
25	C₂²×Q₈	—	2	1	no	—
26	C₂²×Q₈	—	2	1	no	—
27	C₂²×Q₈	—	2	2	no	—
28	C₂×Q₁₆	—	2	8	no	—
29	C₂×Q₁₆	—	4	8	no	—
30	C₂²×C₈	D₄	1	4	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	1	no	—
38	C₄⋊Q₈	—	2	1	no	—
39	C₄⋊Q₈	C₂³	1	2	yes	—
40	C₄⋊Q₈	C₂³	1	4	yes	—
41	C₄⋊Q₈	C₂³	1	2	yes	—
42	C₂×C₄⋊C₄	—	2	2	no	—
43	C₂×C₄⋊C₄	—	2	2	no	—
44	C₂×C₄⋊C₄	—	2	2	no	—
45	C₂×C₄⋊C₄	—	2	2	no	—
46	C₂×C₄²	C₂³	1	1	yes	characteristic
47	#106	—	2	16	no	—
48	#106	—	2	8	no	—
49	#103	—	2	16	no	—
50	C₄×C₈	D₄	1	4	yes	—
51	C₄×C₈	D₄	1	2	yes	—
52	C₄×C₈	C₂³	1	4	yes	—
53	C₄×C₈	C₂³	1	2	yes	—
54	C₂×Q₈	—	2	4	no	—
55	C₂×Q₈	—	2	2	no	—
56	C₂×Q₈	—	2	2	no	—
57	C₂×Q₈	—	4	1	no	—
58	C₂×Q₈	—	4	1	no	—
59	C₂×Q₈	—	4	2	no	—
60	C₂×Q₈	—	4	4	no	—
61	C₂×Q₈	—	4	4	no	—
62	C₂×Q₈	—	4	8	no	—
63	C₂²×C₄	—	4	1	no	—
64	C₂²×C₄	—	4	1	no	—
65	C₂²×C₄	—	4	2	no	—
66	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄	C₂×D₄	1	2	yes	—
68	Q₁₆	—	4	16	no	—
69	C₂×C₈	—	2	8	no	—
70	C₂×C₈	SD₁₆	1	8	yes	—
71	C₂×C₈	C₂×D₄	1	8	yes	—
72	C₂×C₈	C₂×D₄	1	4	yes	—
73	C₄⋊C₄	—	2	4	no	—
74	C₄⋊C₄	—	2	4	no	—
75	C₄⋊C₄	—	2	4	no	—
76	C₄⋊C₄	—	2	4	no	—
77	C₄⋊C₄	—	4	2	no	—
78	C₄⋊C₄	—	4	2	no	—
79	C₄⋊C₄	—	4	2	no	—
80	C₄⋊C₄	—	4	2	no	—
81	C₄²	C₂×D₄	1	2	yes	—
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	4	no	—
86	Q₈	—	4	4	no	—
87	Q₈	—	4	8	no	—
88	Q₈	—	8	2	no	—
89	Q₈	—	8	2	no	—
90	Q₈	—	8	4	no	—
91	C₂×C₄	—	4	2	no	—
92	C₂×C₄	—	4	2	no	—
93	C₂×C₄	—	4	4	no	—
94	C₂×C₄	—	8	2	no	—
95	C₂×C₄	—	8	2	no	—
96	C₂×C₄	—	8	4	no	—
97	C₂×C₄	#127	1	1	yes	characteristic
98	C₂×C₄	#127	1	2	yes	—
99	C₂×C₄	C₂×SD₁₆	1	4	yes	—
100	C₂×C₄	#137	1	8	yes	—
101	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
102	C₂×C₄	C₂²×D₄	1	2	yes	—
103	C₈	—	2	8	no	—
104	C₈	C₂×SD₁₆	1	8	yes	—
105	C₂²	#492	1	2	yes	—
106	C₂²	#497	1	4	yes	—
107	C₂²	C₂×#127	1	1	yes	characteristic
108	C₄	—	8	2	no	—
109	C₄	—	8	2	no	—
110	C₄	—	8	4	no	—
111	C₄	#492	1	2	yes	—
112	C₄	#497	1	4	yes	—
113	C₄	C₂²×SD₁₆	1	2	yes	—
114	C₄	C₂×#137	1	4	yes	—
115	C₂	#1716	1	4	yes	direct product factor
116	C₂	C₂×#492	1	1	yes	characteristic
117	C₂	C₂×#497	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#498

C₂

yes

maximal, characteristic

C₂×#498

C₂

yes

maximal

C₂×#494

C₂

yes

maximal, characteristic

C₂×#487

C₂

yes

maximal

#1716

C₂

yes

maximal, direct product factor

C₂×#322

C₂

yes

maximal, characteristic

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

C₂²

yes

characteristic

C₂×C₄⋊Q₈

C₂²

yes

—

C₂×C₄⋊Q₈

C₂²

yes

—

—

C₂²

yes

—

#498

C₂²

yes

—

#494

—

#494

C₂²

yes

—

#487

—

#487

C₂²

yes

—

C₂×#106

—

C₂×#106

—

C₂×#103

—

C₂×C₄×C₈

C₂²

yes

—

C₂×C₄×C₈

C₂²

yes

characteristic

#322

C₂²

yes

—

C₂²×Q₈

—

C₂²×Q₈

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂²×C₈

D₄

yes

—

C₂²×C₈

D₄

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂³

yes

—

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

C₂³

yes

—

C₄×C₈

C₂³

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

SD₁₆

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

—

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂⁴

yes

characteristic

C₂³

#127

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

Q₈

—

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#127

yes

characteristic

C₂×C₄

#127

yes

—

C₂×C₄

C₂×SD₁₆

yes

—

100

C₂×C₄

#137

yes

—

101

C₂×C₄

C₂²×D₄

yes

characteristic

102

C₂×C₄

C₂²×D₄

yes

—

103

C₈

—

104

C₈

C₂×SD₁₆

yes

—

105

C₂²

#492

yes

—

106

C₂²

#497

yes

—

107

C₂²

C₂×#127

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

#492

yes

—

112

C₄

#497

yes

—

113

C₄

C₂²×SD₁₆

yes

—

114

C₄

C₂×#137

yes

—

115

C₂

#1716

yes

direct product factor

116

C₂

C₂×#492

yes

characteristic

117

C₂

C₂×#497

yes

—

	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. #1716	C₂	direct (N × H)
115. C₂	#1716	direct (N × H)

H ≅ G/N

type

5. #1716

C₂

direct (N × H)

115. C₂

#1716

direct (N × H)

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

subgroup	index
C₂×#498	2
C₂×#494	2
C₂×#487	2
#1716	2
C₂×#322	2

subgroup

index

Subgroups — C₂×#1716 #19607

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#1716 #19607

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#1716 #19607