Group #26039010 — subgroups

Subgroups — #26039010

Order 11456.

270 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 111 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	C₂²×Dic₃₅₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#19153111	C₂	1	2	yes	maximal, complemented
3	#19153100	C₂	1	2	yes	maximal, complemented
4	#19153094	C₂	1	4	yes	maximal, complemented
5	#19153092	C₂	1	4	yes	maximal, complemented
6	#19153072	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#41×C₁₇₉	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₃₅₈	—	2	4	no	—
9	C₂×Dic₃₅₈	C₂²	1	2	yes	—
10	C₂×Dic₃₅₈	C₂²	1	2	yes	—
11	C₂²×Dic₁₇₉	C₂²	1	2	yes	—
12	C₄⋊Dic₁₇₉	C₂²	1	2	yes	—
13	Dic₁₇₉⋊C₄	C₂²	1	8	yes	—
14	C₂²⋊Dic₁₇₉	C₂²	1	2	yes	—
15	C₂²⋊Dic₁₇₉	C₂²	1	8	yes	—
16	C₄×Dic₁₇₉	C₂²	1	2	yes	—
17	#41×C₁₇₉	—	2	4	no	—
18	C₂×Q₈×C₁₇₉	C₂²	1	1	yes	characteristic
19	C₂×D₄×C₁₇₉	C₂²	1	2	yes	—
20	C₂×D₄×C₁₇₉	C₂²	1	1	yes	characteristic
21	C₂²×C₄×C₁₇₉	C₂²	1	2	yes	—
22	C₂²×C₄×C₁₇₉	C₂²	1	1	yes	characteristic
23	Dic₃₅₈	—	2	4	no	—
24	Dic₃₅₈	—	2	4	no	—
25	C₂×Dic₁₇₉	—	2	4	no	—
26	C₂×Dic₁₇₉	C₂³	1	4	yes	—
27	C₂×Dic₁₇₉	C₂³	1	4	yes	—
28	C₂³×C₁₇₉	C₂³	1	2	yes	—
29	C₂³×C₁₇₉	C₂³	1	1	yes	characteristic
30	Q₈×C₁₇₉	—	2	2	no	—
31	D₄×C₁₇₉	—	2	4	no	—
32	D₄×C₁₇₉	—	2	2	no	—
33	C₂×C₄×C₁₇₉	—	2	4	no	—
34	C₂×C₄×C₁₇₉	D₄	1	4	yes	—
35	C₂×C₄×C₁₇₉	C₂³	1	2	yes	—
36	C₂×C₄×C₁₇₉	C₂³	1	1	yes	characteristic
37	C₂×C₄×C₁₇₉	C₂³	1	1	yes	characteristic
38	C₂²×C₁₇₉	—	2	2	no	—
39	C₂²×C₁₇₉	—	2	2	no	—
40	C₂²×C₁₇₉	—	2	4	no	—
41	C₂²×C₁₇₉	C₂×D₄	1	2	yes	—
42	C₂²×C₁₇₉	C₂⁴	1	1	yes	characteristic
43	C₄×C₁₇₉	—	2	2	no	—
44	C₄×C₁₇₉	C₂×D₄	1	2	yes	—
45	C₄×C₁₇₉	C₂×D₄	1	2	yes	—
46	Dic₁₇₉	—	2	4	no	—
47	Dic₁₇₉	—	2	4	no	—
48	C₂×C₁₇₉	—	2	2	no	—
49	C₂×C₁₇₉	—	4	2	no	—
50	C₂×C₁₇₉	C₂²×D₄	1	1	yes	characteristic
51	C₂×C₁₇₉	2₋¹⁺⁴	1	2	yes	—
52	C₁₇₉	#536	1	1	yes	characteristic, complemented, Sylow
53	#536	—	179	1	no	maximal, Sylow
54	C₂×#41	D₁₇₉	1	1	yes	characteristic
55	C₂²×Q₈	—	179	1	no	—
56	C₄⋊Q₈	—	179	2	no	—
57	#124	—	179	2	no	—
58	#123	—	179	4	no	—
59	C₂²⋊Q₈	—	179	4	no	—
60	#117	—	179	1	no	—
61	#41	—	2	4	no	—
62	C₂×Q₈	—	358	4	no	—
63	C₂×Q₈	—	179	2	no	—
64	C₂×Q₈	—	179	2	no	—
65	C₂×Q₈	C₂×D₁₇₉	1	1	yes	characteristic
66	C₂×D₄	C₂×D₁₇₉	1	2	yes	—
67	C₂×D₄	C₂×D₁₇₉	1	1	yes	characteristic
68	C₂²×C₄	—	179	2	no	—
69	C₂²×C₄	C₂×D₁₇₉	1	2	yes	—
70	C₂²×C₄	C₂×D₁₇₉	1	1	yes	characteristic
71	C₄⋊C₄	—	179	8	no	—
72	C₄⋊C₄	—	179	2	no	—
73	C₂²⋊C₄	—	179	2	no	—
74	C₂²⋊C₄	—	179	8	no	—
75	C₄²	—	179	2	no	—
76	C₂³	C₂²×D₁₇₉	1	2	yes	—
77	C₂³	C₂²×D₁₇₉	1	1	yes	characteristic
78	Q₈	—	2	2	no	—
79	Q₈	—	358	4	no	—
80	Q₈	—	358	4	no	—
81	D₄	—	2	4	no	—
82	D₄	—	2	2	no	—
83	C₂×C₄	—	2	4	no	—
84	C₂×C₄	—	358	4	no	—
85	C₂×C₄	—	179	4	no	—
86	C₂×C₄	—	179	4	no	—
87	C₂×C₄	C₂²⋊D₁₇₉	1	4	yes	—
88	C₂×C₄	C₂²×D₁₇₉	1	2	yes	—
89	C₂×C₄	C₂²×D₁₇₉	1	1	yes	characteristic
90	C₂×C₄	C₂²×D₁₇₉	1	1	yes	characteristic
91	C₂²	—	2	2	no	—
92	C₂²	—	2	4	no	—
93	C₂²	—	2	2	no	—
94	C₂²	C₂×C₂²⋊D₁₇₉	1	2	yes	—
95	C₂²	C₂³×D₁₇₉	1	1	yes	characteristic
96	C₄	—	2	2	no	—
97	C₄	—	358	4	no	—
98	C₄	—	358	4	no	—
99	C₄	C₂×C₂²⋊D₁₇₉	1	2	yes	—
100	C₄	C₂×C₂²⋊D₁₇₉	1	2	yes	—
101	C₂	—	2	2	no	—
102	C₂	—	4	2	no	—
103	C₂	C₂²×C₂²⋊D₁₇₉	1	1	yes	characteristic
104	C₂	#19153160	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₃₅₈

C₂

yes

maximal, characteristic, complemented

#19153111

C₂

yes

maximal, complemented

#19153100

C₂

yes

maximal, complemented

#19153094

C₂

yes

maximal, complemented

#19153092

C₂

yes

maximal, complemented

#19153072

C₂

yes

maximal, characteristic, complemented

C₂×#41×C₁₇₉

C₂

yes

maximal, characteristic

C₂×Dic₃₅₈

—

C₂×Dic₃₅₈

C₂²

yes

—

C₂×Dic₃₅₈

C₂²

yes

—

C₂²×Dic₁₇₉

C₂²

yes

—

C₄⋊Dic₁₇₉

C₂²

yes

—

Dic₁₇₉⋊C₄

C₂²

yes

—

C₂²⋊Dic₁₇₉

C₂²

yes

—

C₂²⋊Dic₁₇₉

C₂²

yes

—

C₄×Dic₁₇₉

C₂²

yes

—

#41×C₁₇₉

—

C₂×Q₈×C₁₇₉

C₂²

yes

characteristic

C₂×D₄×C₁₇₉

C₂²

yes

—

C₂×D₄×C₁₇₉

C₂²

yes

characteristic

C₂²×C₄×C₁₇₉

C₂²

yes

—

C₂²×C₄×C₁₇₉

C₂²

yes

characteristic

Dic₃₅₈

—

Dic₃₅₈

—

C₂×Dic₁₇₉

—

C₂×Dic₁₇₉

C₂³

yes

—

C₂×Dic₁₇₉

C₂³

yes

—

C₂³×C₁₇₉

C₂³

yes

—

C₂³×C₁₇₉

C₂³

yes

characteristic

Q₈×C₁₇₉

—

D₄×C₁₇₉

—

D₄×C₁₇₉

—

C₂×C₄×C₁₇₉

—

C₂×C₄×C₁₇₉

D₄

yes

—

C₂×C₄×C₁₇₉

C₂³

yes

—

C₂×C₄×C₁₇₉

C₂³

yes

characteristic

C₂×C₄×C₁₇₉

C₂³

yes

characteristic

C₂²×C₁₇₉

—

C₂²×C₁₇₉

—

C₂²×C₁₇₉

—

C₂²×C₁₇₉

C₂×D₄

yes

—

C₂²×C₁₇₉

C₂⁴

yes

characteristic

C₄×C₁₇₉

—

C₄×C₁₇₉

C₂×D₄

yes

—

C₄×C₁₇₉

C₂×D₄

yes

—

Dic₁₇₉

—

Dic₁₇₉

—

C₂×C₁₇₉

—

C₂×C₁₇₉

—

C₂×C₁₇₉

C₂²×D₄

yes

characteristic

C₂×C₁₇₉

2₋¹⁺⁴

yes

—

C₁₇₉

#536

yes

characteristic, complemented, Sylow

#536

—

179

maximal, Sylow

C₂×#41

D₁₇₉

yes

characteristic

C₂²×Q₈

—

179

—

C₄⋊Q₈

—

179

—

#124

—

179

—

#123

—

179

—

C₂²⋊Q₈

—

179

—

#117

—

179

—

#41

—

C₂×Q₈

—

358

—

C₂×Q₈

—

179

—

C₂×Q₈

—

179

—

C₂×Q₈

C₂×D₁₇₉

yes

characteristic

C₂×D₄

C₂×D₁₇₉

yes

—

C₂×D₄

C₂×D₁₇₉

yes

characteristic

C₂²×C₄

—

179

—

C₂²×C₄

C₂×D₁₇₉

yes

—

C₂²×C₄

C₂×D₁₇₉

yes

characteristic

C₄⋊C₄

—

179

—

C₄⋊C₄

—

179

—

C₂²⋊C₄

—

179

—

C₂²⋊C₄

—

179

—

C₄²

—

179

—

C₂³

C₂²×D₁₇₉

yes

—

C₂³

C₂²×D₁₇₉

yes

characteristic

Q₈

—

Q₈

—

358

—

Q₈

—

358

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

358

—

C₂×C₄

—

179

—

C₂×C₄

—

179

—

C₂×C₄

C₂²⋊D₁₇₉

yes

—

C₂×C₄

C₂²×D₁₇₉

yes

—

C₂×C₄

C₂²×D₁₇₉

yes

characteristic

C₂×C₄

C₂²×D₁₇₉

yes

characteristic

C₂²

—

C₂²

—

C₂²

—

C₂²

C₂×C₂²⋊D₁₇₉

yes

—

C₂²

C₂³×D₁₇₉

yes

characteristic

C₄

—

C₄

—

358

—

C₄

—

358

—

C₄

C₂×C₂²⋊D₁₇₉

yes

—

100

C₄

C₂×C₂²⋊D₁₇₉

yes

—

101

C₂

—

102

C₂

—

103

C₂

C₂²×C₂²⋊D₁₇₉

yes

characteristic

104

C₂

#19153160

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₇₉
commutator, socle, cosocle	C₂²×C₁₇₉	C₂⁴
Fitting, prime radical	C₂×#41×C₁₇₉	C₂
hypercenter	C₂×#41	D₁₇₉
hypocenter	C₁₇₉	#536
LCS	G ▷ C₂²×C₁₇₉ ▷ C₁₇₉
UCS	C₁ ◁ C₂² ◁ C₂×#41

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₇₉

commutator, socle, cosocle

C₂²×C₁₇₉

C₂⁴

Fitting, prime radical

C₂×#41×C₁₇₉

C₂

hypercenter

C₂×#41

D₁₇₉

C₁₇₉

G ▷ C₂²×C₁₇₉ ▷ C₁₇₉

C₁ ◁ C₂² ◁ C₂×#41

N	H ≅ G/N	type
1. C₂²×Dic₃₅₈	C₂	semidirect
2. #19153111	C₂	semidirect
3. #19153100	C₂	semidirect
4. #19153094	C₂	semidirect
5. #19153092	C₂	semidirect
6. #19153072	C₂	semidirect
52. C₁₇₉	#536	semidirect

H ≅ G/N

type

1. C₂²×Dic₃₅₈

C₂

semidirect

2. #19153111

C₂

semidirect

3. #19153100

C₂

semidirect

4. #19153094

C₂

semidirect

5. #19153092

C₂

semidirect

6. #19153072

C₂

semidirect

52. C₁₇₉

#536

semidirect

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

subgroup	index
C₂²×Dic₃₅₈	2
#19153111	2
#19153100	2
#19153094	2
#19153092	2
#19153072	2
C₂×#41×C₁₇₉	2
#536	179

subgroup

index

C₂²×Dic₃₅₈

C₂×#41×C₁₇₉

179

Subgroups — #26039010

Conjugacy classes of subgroups

Complements

Maximal subgroups