Group #19662 — subgroups

Subgroups — #19662

Order 256.

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

normal subgroups only

This group has 419 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	#3226	C₂	1	4	yes	maximal, complemented
2	#3147	C₂	1	1	yes	maximal, characteristic
3	C₂×#494	C₂	1	1	yes	maximal, characteristic, complemented
4	#1722	C₂	1	4	yes	maximal, complemented
5	#1713	C₂	1	2	yes	maximal, complemented
6	#1709	C₂	1	2	yes	maximal, complemented
7	#1450	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#135	—	2	2	no	—
9	C₂²×Q₁₆	—	2	2	no	—
10	#532	C₂²	1	2	yes	—
11	C₂×C₄⋊Q₈	C₂²	1	2	yes	—
12	#500	C₂²	1	8	yes	—
13	#495	—	2	2	no	—
14	#494	—	2	2	no	—
15	#494	C₂²	1	2	yes	—
16	#494	C₂²	1	2	yes	—
17	#494	C₂²	1	1	yes	characteristic
18	#493	C₂²	1	1	yes	characteristic
19	#492	C₂²	1	2	yes	—
20	#486	C₂²	1	8	yes	—
21	#480	—	2	8	no	—
22	#474	—	2	8	no	—
23	#431	C₂²	1	2	yes	—
24	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
25	#322	C₂²	1	2	yes	—
26	C₈²	C₂²	1	2	yes	—
27	C₂×#41	—	2	2	no	—
28	C₂²×Q₈	—	2	2	no	—
29	#135	—	4	8	no	—
30	C₂×Q₁₆	—	2	4	no	—
31	C₂×Q₁₆	—	2	4	no	—
32	C₂×Q₁₆	—	2	2	no	—
33	C₂×Q₁₆	—	4	8	no	—
34	C₂×SD₁₆	—	2	4	no	—
35	C₂×D₈	—	2	2	no	—
36	C₂²×C₈	D₄	1	2	yes	—
37	C₄⋊Q₈	—	2	4	no	—
38	C₄⋊Q₈	—	2	2	no	—
39	C₄⋊Q₈	C₂³	1	4	yes	—
40	C₄⋊Q₈	C₂³	1	2	yes	—
41	#127	C₂³	1	2	yes	—
42	#124	—	2	2	no	—
43	#121	—	2	4	no	—
44	C₄×D₄	—	2	4	no	—
45	C₂×C₄⋊C₄	—	2	4	no	—
46	C₂×C₄²	C₂³	1	1	yes	characteristic
47	#107	—	2	16	no	—
48	C₄⋊C₈	D₄	1	4	yes	—
49	#102	—	2	16	no	—
50	C₂²⋊C₈	D₄	1	4	yes	—
51	C₄×C₈	D₄	1	2	yes	—
52	C₄×C₈	C₂³	1	1	yes	characteristic
53	C₄×C₈	C₂³	1	4	yes	—
54	C₄×C₈	C₂³	1	1	yes	characteristic
55	#41	—	4	8	no	—
56	C₂×Q₈	—	2	2	no	—
57	C₂×Q₈	—	2	4	no	—
58	C₂×Q₈	—	4	2	no	—
59	C₂×Q₈	—	4	8	no	—
60	C₂×D₄	—	2	4	no	—
61	C₂×D₄	—	2	2	no	—
62	C₂×D₄	—	4	2	no	—
63	C₂²×C₄	—	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	8	no	—
68	Q₁₆	—	4	8	no	—
69	Q₁₆	—	4	4	no	—
70	SD₁₆	—	4	8	no	—
71	D₈	—	4	4	no	—
72	C₂×C₈	D₈	1	8	yes	—
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	8	no	—
77	C₄⋊C₄	—	2	4	no	—
78	C₄⋊C₄	—	4	4	no	—
79	C₂²⋊C₄	—	4	4	no	—
80	C₄²	C₂×D₄	1	1	yes	characteristic
81	C₄²	C₂×D₄	1	2	yes	—
82	C₄²	C₂⁴	1	1	yes	characteristic
83	C₂³	—	4	2	no	—
84	C₂³	#127	1	1	yes	characteristic
85	Q₈	—	4	4	no	—
86	Q₈	—	4	8	no	—
87	Q₈	—	8	4	no	—
88	D₄	—	4	4	no	—
89	D₄	—	4	8	no	—
90	D₄	—	8	4	no	—
91	C₂×C₄	—	2	4	no	—
92	C₂×C₄	—	4	2	no	—
93	C₂×C₄	—	4	4	no	—
94	C₂×C₄	—	8	4	no	—
95	C₂×C₄	—	8	4	no	—
96	C₂×C₄	#127	1	2	yes	—
97	C₂×C₄	#127	1	1	yes	characteristic
98	C₂×C₄	C₂×D₈	1	4	yes	—
99	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
100	C₂×C₄	C₂²×D₄	1	2	yes	—
101	C₈	—	2	8	no	—
102	C₈	C₂×D₈	1	4	yes	—
103	C₈	C₂×D₈	1	4	yes	—
104	C₂²	—	2	2	no	—
105	C₂²	—	8	2	no	—
106	C₂²	—	8	4	no	—
107	C₂²	#493	1	2	yes	—
108	C₂²	C₂×#127	1	1	yes	characteristic
109	C₄	—	2	2	no	—
110	C₄	—	8	2	no	—
111	C₄	—	8	4	no	—
112	C₄	#493	1	2	yes	—
113	C₄	C₂²×D₈	1	2	yes	—
114	C₄	#578	1	4	yes	—
115	C₂	—	2	2	no	—
116	C₂	—	16	2	no	—
117	C₂	C₂×#493	1	1	yes	characteristic
118	C₂	#3156	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#3226

C₂

yes

maximal, complemented

#3147

C₂

yes

maximal, characteristic

C₂×#494

C₂

yes

maximal, characteristic, complemented

#1722

C₂

yes

maximal, complemented

#1713

C₂

yes

maximal, complemented

#1709

C₂

yes

maximal, complemented

#1450

C₂

yes

maximal, characteristic, complemented

C₂×#135

—

C₂²×Q₁₆

—

#532

C₂²

yes

—

C₂×C₄⋊Q₈

C₂²

yes

—

#500

C₂²

yes

—

#495

—

#494

—

#494

C₂²

yes

—

#494

C₂²

yes

—

#494

C₂²

yes

characteristic

#493

C₂²

yes

characteristic

#492

C₂²

yes

—

#486

C₂²

yes

—

#480

—

#474

—

#431

C₂²

yes

—

C₂×C₄×C₈

C₂²

yes

characteristic

#322

C₂²

yes

—

C₈²

C₂²

yes

—

C₂×#41

—

C₂²×Q₈

—

#135

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×SD₁₆

—

C₂×D₈

—

C₂²×C₈

D₄

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂³

yes

—

C₄⋊Q₈

C₂³

yes

—

#127

C₂³

yes

—

#124

—

#121

—

C₄×D₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂³

yes

characteristic

#107

—

C₄⋊C₈

D₄

yes

—

#102

—

C₂²⋊C₈

D₄

yes

—

C₄×C₈

D₄

yes

—

C₄×C₈

C₂³

yes

characteristic

C₄×C₈

C₂³

yes

—

C₄×C₈

C₂³

yes

characteristic

#41

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

—

C₂²×C₄

C₂×D₄

yes

characteristic

Q₁₆

—

Q₁₆

—

Q₁₆

—

SD₁₆

—

D₈

—

C₂×C₈

D₈

yes

—

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×D₄

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×D₄

yes

—

C₄²

C₂⁴

yes

characteristic

C₂³

—

C₂³

#127

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#127

yes

—

C₂×C₄

#127

yes

characteristic

C₂×C₄

C₂×D₈

yes

—

C₂×C₄

C₂²×D₄

yes

characteristic

100

C₂×C₄

C₂²×D₄

yes

—

101

C₈

—

102

C₈

C₂×D₈

yes

—

103

C₈

C₂×D₈

yes

—

104

C₂²

—

105

C₂²

—

106

C₂²

—

107

C₂²

#493

yes

—

108

C₂²

C₂×#127

yes

characteristic

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

#493

yes

—

113

C₄

C₂²×D₈

yes

—

114

C₄

#578

yes

—

115

C₂

—

116

C₂

—

117

C₂

C₂×#493

yes

characteristic

118

C₂

#3156

yes

—

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

subgroup

G/H

center, socle

C₂²

C₂×#127

commutator, Frattini, cosocle

C₄²

C₂⁴

C₂

G ▷ C₄² ▷ C₂² ▷ C₁

C₁ ◁ C₂² ◁ C₂×C₄² ◁ G

LECS

G ▷ C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
1. #3226	C₂	semidirect
3. C₂×#494	C₂	semidirect
4. #1722	C₂	semidirect
5. #1713	C₂	semidirect
6. #1709	C₂	semidirect
7. #1450	C₂	semidirect

H ≅ G/N

type

1. #3226

C₂

semidirect

3. C₂×#494

C₂

semidirect

4. #1722

C₂

semidirect

5. #1713

C₂

semidirect

6. #1709

C₂

semidirect

7. #1450

C₂

semidirect

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

subgroup	index
#3226	2
#3147	2
C₂×#494	2
#1722	2
#1713	2
#1709	2
#1450	2

subgroup

index

Subgroups — #19662

Conjugacy classes of subgroups

Complements

Maximal subgroups