Group #19667 — subgroups

Subgroups — #19667

Order 256.

390 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 119 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	#3207	C₂	1	4	yes	maximal, complemented
2	#3169	C₂	1	2	yes	maximal, complemented
3	#1717	C₂	1	4	yes	maximal, complemented
4	#1712	C₂	1	1	yes	maximal, characteristic, complemented
5	#1710	C₂	1	1	yes	maximal, characteristic, complemented
6	#1706	C₂	1	2	yes	maximal, complemented
7	#1450	C₂	1	1	yes	maximal, characteristic, complemented
8	#532	C₂²	1	4	yes	—
9	#500	C₂²	1	2	yes	—
10	#493	C₂²	1	2	yes	—
11	#493	C₂²	1	4	yes	—
12	#492	C₂²	1	4	yes	—
13	#487	C₂²	1	4	yes	—
14	#486	C₂²	1	4	yes	—
15	#486	C₂²	1	4	yes	—
16	#466	—	2	4	no	—
17	#463	—	2	8	no	—
18	#449	—	2	8	no	—
19	C₄×D₈	—	2	4	no	—
20	#431	C₂²	1	2	yes	—
21	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
22	#322	C₂²	1	2	yes	—
23	C₈²	C₂²	1	1	yes	characteristic
24	C₈²	C₂²	1	1	yes	characteristic
25	C₂×#41	—	2	4	no	—
26	C₂×SD₁₆	—	2	8	no	—
27	C₂×D₈	—	2	4	no	—
28	C₂×D₈	—	2	8	no	—
29	C₂²×C₈	D₄	1	2	yes	—
30	C₄⋊Q₈	C₂³	1	4	yes	—
31	#127	C₂³	1	4	yes	—
32	#124	—	2	4	no	—
33	#121	—	2	4	no	—
34	#121	—	2	4	no	—
35	C₄×D₄	—	2	4	no	—
36	C₄×D₄	—	2	4	no	—
37	C₂×C₄²	C₂³	1	1	yes	characteristic
38	#107	—	2	4	no	—
39	C₄⋊C₈	D₄	1	4	yes	—
40	#103	—	2	8	no	—
41	#102	—	2	8	no	—
42	#102	—	2	8	no	—
43	C₂²⋊C₈	D₄	1	4	yes	—
44	C₄×C₈	D₄	1	2	yes	—
45	C₄×C₈	C₂³	1	1	yes	characteristic
46	C₄×C₈	C₂³	1	2	yes	—
47	C₄×C₈	C₂³	1	2	yes	—
48	C₄×C₈	C₂³	1	1	yes	characteristic
49	#41	—	8	8	no	—
50	C₂×Q₈	—	2	4	no	—
51	C₂×D₄	—	2	4	no	—
52	C₂×D₄	—	2	4	no	—
53	C₂×D₄	—	2	4	no	—
54	C₂×D₄	—	4	4	no	—
55	C₂²×C₄	—	4	4	no	—
56	C₂²×C₄	C₂×D₄	1	2	yes	—
57	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
58	SD₁₆	—	8	8	no	—
59	D₈	—	4	8	no	—
60	D₈	—	8	8	no	—
61	C₂×C₈	—	2	4	no	—
62	C₂×C₈	C₂×D₄	1	2	yes	—
63	C₂×C₈	C₂×D₄	1	4	yes	—
64	C₂×C₈	C₂×D₄	1	4	yes	—
65	C₂×C₈	C₂×D₄	1	2	yes	—
66	C₄⋊C₄	—	2	4	no	—
67	C₄⋊C₄	—	2	4	no	—
68	C₂²⋊C₄	—	4	4	no	—
69	C₂²⋊C₄	—	4	4	no	—
70	C₄²	C₂×D₄	1	1	yes	characteristic
71	C₄²	C₂×D₄	1	2	yes	—
72	C₄²	C₂⁴	1	1	yes	characteristic
73	C₂³	—	4	4	no	—
74	C₂³	#127	1	1	yes	characteristic
75	Q₈	—	4	8	no	—
76	D₄	—	4	8	no	—
77	D₄	—	4	8	no	—
78	D₄	—	4	8	no	—
79	D₄	—	8	8	no	—
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	—	2	4	no	—
82	C₂×C₄	—	4	4	no	—
83	C₂×C₄	—	8	8	no	—
84	C₂×C₄	#127	1	1	yes	characteristic
85	C₂×C₄	#127	1	2	yes	—
86	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
87	C₂×C₄	C₂²×D₄	1	2	yes	—
88	C₈	—	2	2	no	—
89	C₈	—	2	4	no	—
90	C₈	—	2	4	no	—
91	C₈	#135	1	4	yes	—
92	C₂²	—	2	1	no	—
93	C₂²	—	2	2	no	—
94	C₂²	—	8	4	no	—
95	C₂²	—	8	4	no	—
96	C₂²	—	8	4	no	—
97	C₂²	C₂×#127	1	1	yes	characteristic
98	C₄	—	2	1	no	—
99	C₄	—	2	2	no	—
100	C₄	—	8	4	no	—
101	C₄	C₂×#135	1	2	yes	—
102	C₄	#576	1	4	yes	—
103	C₂	—	4	1	no	—
104	C₂	—	16	4	no	—
105	C₂	#3147	1	1	yes	characteristic
106	C₂	#3155	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#3207

C₂

yes

maximal, complemented

#3169

C₂

yes

maximal, complemented

#1717

C₂

yes

maximal, complemented

#1712

C₂

yes

maximal, characteristic, complemented

#1710

C₂

yes

maximal, characteristic, complemented

#1706

C₂

yes

maximal, complemented

#1450

C₂

yes

maximal, characteristic, complemented

#532

C₂²

yes

—

#500

C₂²

yes

—

#493

C₂²

yes

—

#493

C₂²

yes

—

#492

C₂²

yes

—

#487

C₂²

yes

—

#486

C₂²

yes

—

#486

C₂²

yes

—

#466

—

#463

—

#449

—

C₄×D₈

—

#431

C₂²

yes

—

C₂×C₄×C₈

C₂²

yes

characteristic

#322

C₂²

yes

—

C₈²

C₂²

yes

characteristic

C₈²

C₂²

yes

characteristic

C₂×#41

—

C₂×SD₁₆

—

C₂×D₈

—

C₂×D₈

—

C₂²×C₈

D₄

yes

—

C₄⋊Q₈

C₂³

yes

—

#127

C₂³

yes

—

#124

—

#121

—

#121

—

C₄×D₄

—

C₄×D₄

—

C₂×C₄²

C₂³

yes

characteristic

#107

—

C₄⋊C₈

D₄

yes

—

#103

—

#102

—

#102

—

C₂²⋊C₈

D₄

yes

—

C₄×C₈

D₄

yes

—

C₄×C₈

C₂³

yes

characteristic

C₄×C₈

C₂³

yes

—

C₄×C₈

C₂³

yes

—

C₄×C₈

C₂³

yes

characteristic

#41

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

—

C₂²×C₄

C₂×D₄

yes

characteristic

SD₁₆

—

D₈

—

D₈

—

C₂×C₈

—

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×D₄

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×D₄

yes

—

C₄²

C₂⁴

yes

characteristic

C₂³

—

C₂³

#127

yes

characteristic

Q₈

—

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#127

yes

characteristic

C₂×C₄

#127

yes

—

C₂×C₄

C₂²×D₄

yes

characteristic

C₂×C₄

C₂²×D₄

yes

—

C₈

—

C₈

—

C₈

—

C₈

#135

yes

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

C₂×#127

yes

characteristic

C₄

—

C₄

—

100

C₄

—

101

C₄

C₂×#135

yes

—

102

C₄

#576

yes

—

103

C₂

—

104

C₂

—

105

C₂

#3147

yes

characteristic

106

C₂

#3155

yes

—

	subgroup	G/H
center, socle	C₂²	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴
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

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ G

LECS

G ▷ C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
1. #3207	C₂	semidirect
2. #3169	C₂	semidirect
3. #1717	C₂	semidirect
4. #1712	C₂	semidirect
5. #1710	C₂	semidirect
6. #1706	C₂	semidirect
7. #1450	C₂	semidirect

H ≅ G/N

type

1. #3207

C₂

semidirect

2. #3169

C₂

semidirect

3. #1717

C₂

semidirect

4. #1712

C₂

semidirect

5. #1710

C₂

semidirect

6. #1706

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
#3207	2
#3169	2
#1717	2
#1712	2
#1710	2
#1706	2
#1450	2

subgroup

index

Subgroups — #19667

Conjugacy classes of subgroups

Complements

Maximal subgroups