Group #19595 — subgroups

Subgroups — C₂×#1708 #19595

Order 256.

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

normal subgroups only

This group has 433 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, complemented
2	C₂×#492	C₂	1	1	yes	maximal, characteristic
3	C₂×#487	C₂	1	2	yes	maximal, complemented
4	C₂×#486	C₂	1	2	yes	maximal
5	#1708	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₈²	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×SD₁₆	—	2	2	no	—
8	C₂×C₄⋊Q₈	C₂²	1	2	yes	—
9	C₂×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₂×#127	C₂²	1	1	yes	characteristic
11	#498	—	2	2	no	—
12	#498	C₂²	1	4	yes	complemented
13	#492	—	2	2	no	—
14	#492	C₂²	1	4	yes	—
15	#487	—	2	4	no	—
16	#487	C₂²	1	8	yes	complemented
17	#486	—	2	4	no	—
18	#486	C₂²	1	8	yes	—
19	C₂×#106	—	2	2	no	—
20	C₂×#103	—	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	C₈²	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×SD₁₆	—	2	8	no	—
30	C₂×SD₁₆	—	4	8	no	—
31	C₂²×C₈	D₄	1	4	yes	—
32	C₂²×C₈	D₄	1	2	yes	—
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	C₂³	1	4	yes	—
39	C₄⋊Q₈	C₂³	1	2	yes	—
40	#127	—	2	2	no	—
41	#127	—	2	1	no	—
42	#127	C₂³	1	2	yes	—
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	8	no	—
48	#103	—	2	16	no	—
49	#102	—	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	C₂⁴	—	4	1	no	—
55	C₂×Q₈	—	2	4	no	—
56	C₂×Q₈	—	2	2	no	—
57	C₂×Q₈	—	4	1	no	—
58	C₂×Q₈	—	4	2	no	—
59	C₂×Q₈	—	4	4	no	—
60	C₂×Q₈	—	4	8	no	—
61	C₂×D₄	—	2	4	no	—
62	C₂×D₄	—	2	2	no	—
63	C₂×D₄	—	4	1	no	—
64	C₂×D₄	—	4	2	no	—
65	C₂×D₄	—	4	4	no	—
66	C₂×D₄	—	4	8	no	—
67	C₂²×C₄	—	4	1	no	—
68	C₂²×C₄	—	4	2	no	—
69	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄	C₂×D₄	1	2	yes	—
71	SD₁₆	—	4	16	no	—
72	C₂×C₈	—	2	8	no	—
73	C₂×C₈	SD₁₆	1	8	yes	—
74	C₂×C₈	C₂×D₄	1	8	yes	—
75	C₂×C₈	C₂×D₄	1	4	yes	—
76	C₄⋊C₄	—	2	4	no	—
77	C₄⋊C₄	—	2	4	no	—
78	C₄⋊C₄	—	2	4	no	—
79	C₄⋊C₄	—	4	2	no	—
80	C₄⋊C₄	—	4	2	no	—
81	C₄⋊C₄	—	4	2	no	—
82	C₄²	C₂×D₄	1	2	yes	—
83	C₄²	C₂×D₄	1	1	yes	characteristic
84	C₄²	C₂⁴	1	1	yes	characteristic
85	C₂³	—	4	2	no	—
86	C₂³	—	8	2	no	—
87	C₂³	—	8	4	no	—
88	C₂³	#127	1	1	yes	characteristic
89	Q₈	—	4	4	no	—
90	Q₈	—	4	8	no	—
91	Q₈	—	8	2	no	—
92	Q₈	—	8	4	no	—
93	D₄	—	4	4	no	—
94	D₄	—	4	8	no	—
95	D₄	—	8	2	no	—
96	D₄	—	8	4	no	—
97	C₂×C₄	—	4	2	no	—
98	C₂×C₄	—	4	4	no	—
99	C₂×C₄	—	8	2	no	—
100	C₂×C₄	—	8	4	no	—
101	C₂×C₄	#127	1	1	yes	characteristic
102	C₂×C₄	#127	1	2	yes	—
103	C₂×C₄	C₂×SD₁₆	1	4	yes	—
104	C₂×C₄	#135	1	8	yes	—
105	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
106	C₂×C₄	C₂²×D₄	1	2	yes	—
107	C₈	—	2	8	no	—
108	C₈	C₂×SD₁₆	1	8	yes	—
109	C₂²	—	8	2	no	—
110	C₂²	—	8	4	no	—
111	C₂²	—	16	4	no	—
112	C₂²	#492	1	2	yes	—
113	C₂²	#495	1	4	yes	—
114	C₂²	C₂×#127	1	1	yes	characteristic
115	C₄	—	8	2	no	—
116	C₄	—	8	4	no	—
117	C₄	#492	1	2	yes	—
118	C₄	#495	1	4	yes	—
119	C₄	C₂²×SD₁₆	1	2	yes	—
120	C₄	C₂×#135	1	4	yes	—
121	C₂	—	16	2	no	—
122	C₂	#1708	1	4	yes	direct product factor
123	C₂	C₂×#492	1	1	yes	characteristic
124	C₂	C₂×#495	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#498

C₂

yes

maximal, characteristic, complemented

C₂×#492

C₂

yes

maximal, characteristic

C₂×#487

C₂

yes

maximal, complemented

C₂×#486

C₂

yes

maximal

#1708

C₂

yes

maximal, direct product factor

C₂×C₈²

C₂

yes

maximal, characteristic, complemented

C₂²×SD₁₆

—

C₂×C₄⋊Q₈

C₂²

yes

—

C₂×C₄⋊Q₈

C₂²

yes

characteristic

C₂×#127

C₂²

yes

characteristic

#498

—

#498

C₂²

yes

complemented

#492

—

#492

C₂²

yes

—

#487

—

#487

C₂²

yes

complemented

#486

—

#486

C₂²

yes

—

C₂×#106

—

C₂×#103

—

C₂×#102

—

C₂×C₄×C₈

C₂²

yes

—

C₂×C₄×C₈

C₂²

yes

characteristic

C₈²

C₂²

yes

complemented

C₂²×Q₈

—

C₂²×Q₈

—

C₂²×D₄

—

C₂²×D₄

—

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

—

—

—

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

—

SD₁₆

—

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

—

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

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

#127

yes

characteristic

102

C₂×C₄

#127

yes

—

103

C₂×C₄

C₂×SD₁₆

yes

—

104

C₂×C₄

#135

yes

—

105

C₂×C₄

C₂²×D₄

yes

characteristic

106

C₂×C₄

C₂²×D₄

yes

—

107

C₈

—

108

C₈

C₂×SD₁₆

yes

—

109

C₂²

—

110

C₂²

—

111

C₂²

—

112

C₂²

#492

yes

—

113

C₂²

#495

yes

—

114

C₂²

C₂×#127

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

#492

yes

—

118

C₄

#495

yes

—

119

C₄

C₂²×SD₁₆

yes

—

120

C₄

C₂×#135

yes

—

121

C₂

—

122

C₂

#1708

yes

direct product factor

123

C₂

C₂×#492

yes

characteristic

124

C₂

C₂×#495

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×#498	C₂	semidirect
3. C₂×#487	C₂	semidirect
5. #1708	C₂	direct (N × H)
6. C₂×C₈²	C₂	semidirect
12. #498	C₂²	semidirect
16. #487	C₂²	semidirect
24. C₈²	C₂²	semidirect
122. C₂	#1708	direct (N × H)

H ≅ G/N

type

1. C₂×#498

C₂

semidirect

3. C₂×#487

C₂

semidirect

5. #1708

C₂

direct (N × H)

6. C₂×C₈²

C₂

semidirect

12. #498

C₂²

semidirect

16. #487

C₂²

semidirect

24. C₈²

C₂²

semidirect

122. C₂

#1708

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×#492	2
C₂×#487	2
C₂×#486	2
#1708	2
C₂×C₈²	2

subgroup

index

C₂×C₈²

Subgroups — C₂×#1708 #19595

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#1708 #19595

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#1708 #19595