Group #36374715 — subgroups

Subgroups — #36374715

Order 22336.

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×Dic₆₉₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#26013637	C₂	1	2	yes	maximal, complemented
3	#26013626	C₂	1	2	yes	maximal, complemented
4	#26013620	C₂	1	4	yes	maximal, complemented
5	#26013618	C₂	1	4	yes	maximal, complemented
6	#26013598	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#41×C₃₄₉	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₆₉₈	—	2	4	no	—
9	C₂×Dic₆₉₈	C₂²	1	2	yes	—
10	C₂×Dic₆₉₈	C₂²	1	2	yes	—
11	C₂²×Dic₃₄₉	C₂²	1	2	yes	—
12	C₄⋊Dic₃₄₉	C₂²	1	2	yes	—
13	#19150212	C₂²	1	8	yes	—
14	C₂²⋊Dic₃₄₉	C₂²	1	2	yes	—
15	C₂²⋊Dic₃₄₉	C₂²	1	8	yes	—
16	C₄×Dic₃₄₉	C₂²	1	2	yes	—
17	#41×C₃₄₉	—	2	4	no	—
18	C₂×Q₈×C₃₄₉	C₂²	1	1	yes	characteristic
19	C₂×D₄×C₃₄₉	C₂²	1	2	yes	—
20	C₂×D₄×C₃₄₉	C₂²	1	1	yes	characteristic
21	C₂²×C₄×C₃₄₉	C₂²	1	2	yes	—
22	C₂²×C₄×C₃₄₉	C₂²	1	1	yes	characteristic
23	Dic₆₉₈	—	2	4	no	—
24	Dic₆₉₈	—	2	4	no	—
25	C₂×Dic₃₄₉	—	2	4	no	—
26	C₂×Dic₃₄₉	C₂³	1	4	yes	—
27	C₂×Dic₃₄₉	C₂³	1	4	yes	—
28	C₂³×C₃₄₉	C₂³	1	2	yes	—
29	C₂³×C₃₄₉	C₂³	1	1	yes	characteristic
30	Q₈×C₃₄₉	—	2	2	no	—
31	D₄×C₃₄₉	—	2	4	no	—
32	D₄×C₃₄₉	—	2	2	no	—
33	C₂×C₄×C₃₄₉	—	2	4	no	—
34	C₂×C₄×C₃₄₉	D₄	1	4	yes	—
35	C₂×C₄×C₃₄₉	C₂³	1	2	yes	—
36	C₂×C₄×C₃₄₉	C₂³	1	1	yes	characteristic
37	C₂×C₄×C₃₄₉	C₂³	1	1	yes	characteristic
38	C₂²×C₃₄₉	—	2	2	no	—
39	C₂²×C₃₄₉	—	2	2	no	—
40	C₂²×C₃₄₉	—	2	4	no	—
41	C₂²×C₃₄₉	C₂×D₄	1	2	yes	—
42	C₂²×C₃₄₉	C₂⁴	1	1	yes	characteristic
43	C₄×C₃₄₉	—	2	2	no	—
44	C₄×C₃₄₉	C₂×D₄	1	2	yes	—
45	C₄×C₃₄₉	C₂×D₄	1	2	yes	—
46	Dic₃₄₉	—	2	4	no	—
47	Dic₃₄₉	—	2	4	no	—
48	C₂×C₃₄₉	—	2	2	no	—
49	C₂×C₃₄₉	—	4	2	no	—
50	C₂×C₃₄₉	C₂²×D₄	1	1	yes	characteristic
51	C₂×C₃₄₉	2₋¹⁺⁴	1	2	yes	—
52	C₃₄₉	#536	1	1	yes	characteristic, complemented, Sylow
53	#536	—	349	1	no	maximal, Sylow
54	C₂×#41	D₃₄₉	1	1	yes	characteristic
55	C₂²×Q₈	—	349	1	no	—
56	C₄⋊Q₈	—	349	2	no	—
57	#124	—	349	2	no	—
58	#123	—	349	4	no	—
59	C₂²⋊Q₈	—	349	4	no	—
60	#117	—	349	1	no	—
61	#41	—	2	4	no	—
62	C₂×Q₈	—	349	2	no	—
63	C₂×Q₈	—	349	2	no	—
64	C₂×Q₈	—	698	4	no	—
65	C₂×Q₈	C₂×D₃₄₉	1	1	yes	characteristic
66	C₂×D₄	C₂×D₃₄₉	1	2	yes	—
67	C₂×D₄	C₂×D₃₄₉	1	1	yes	characteristic
68	C₂²×C₄	—	349	2	no	—
69	C₂²×C₄	C₂×D₃₄₉	1	2	yes	—
70	C₂²×C₄	C₂×D₃₄₉	1	1	yes	characteristic
71	C₄⋊C₄	—	349	2	no	—
72	C₄⋊C₄	—	349	8	no	—
73	C₂²⋊C₄	—	349	2	no	—
74	C₂²⋊C₄	—	349	8	no	—
75	C₄²	—	349	2	no	—
76	C₂³	C₂²×D₃₄₉	1	2	yes	—
77	C₂³	C₂²×D₃₄₉	1	1	yes	characteristic
78	Q₈	—	2	2	no	—
79	Q₈	—	698	4	no	—
80	Q₈	—	698	4	no	—
81	D₄	—	2	4	no	—
82	D₄	—	2	2	no	—
83	C₂×C₄	—	2	4	no	—
84	C₂×C₄	—	349	4	no	—
85	C₂×C₄	—	349	4	no	—
86	C₂×C₄	—	698	4	no	—
87	C₂×C₄	C₂²⋊D₃₄₉	1	4	yes	—
88	C₂×C₄	C₂²×D₃₄₉	1	2	yes	—
89	C₂×C₄	C₂²×D₃₄₉	1	1	yes	characteristic
90	C₂×C₄	C₂²×D₃₄₉	1	1	yes	characteristic
91	C₂²	—	2	2	no	—
92	C₂²	—	2	2	no	—
93	C₂²	—	2	4	no	—
94	C₂²	C₂×C₂²⋊D₃₄₉	1	2	yes	—
95	C₂²	C₂³×D₃₄₉	1	1	yes	characteristic
96	C₄	—	2	2	no	—
97	C₄	—	698	4	no	—
98	C₄	—	698	4	no	—
99	C₄	C₂×C₂²⋊D₃₄₉	1	2	yes	—
100	C₄	C₂×C₂²⋊D₃₄₉	1	2	yes	—
101	C₂	—	2	2	no	—
102	C₂	—	4	2	no	—
103	C₂	C₂²×C₂²⋊D₃₄₉	1	1	yes	characteristic
104	C₂	#26013686	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₆₉₈

C₂

yes

maximal, characteristic, complemented

#26013637

C₂

yes

maximal, complemented

#26013626

C₂

yes

maximal, complemented

#26013620

C₂

yes

maximal, complemented

#26013618

C₂

yes

maximal, complemented

#26013598

C₂

yes

maximal, characteristic, complemented

C₂×#41×C₃₄₉

C₂

yes

maximal, characteristic

C₂×Dic₆₉₈

—

C₂×Dic₆₉₈

C₂²

yes

—

C₂×Dic₆₉₈

C₂²

yes

—

C₂²×Dic₃₄₉

C₂²

yes

—

C₄⋊Dic₃₄₉

C₂²

yes

—

#19150212

C₂²

yes

—

C₂²⋊Dic₃₄₉

C₂²

yes

—

C₂²⋊Dic₃₄₉

C₂²

yes

—

C₄×Dic₃₄₉

C₂²

yes

—

#41×C₃₄₉

—

C₂×Q₈×C₃₄₉

C₂²

yes

characteristic

C₂×D₄×C₃₄₉

C₂²

yes

—

C₂×D₄×C₃₄₉

C₂²

yes

characteristic

C₂²×C₄×C₃₄₉

C₂²

yes

—

C₂²×C₄×C₃₄₉

C₂²

yes

characteristic

Dic₆₉₈

—

Dic₆₉₈

—

C₂×Dic₃₄₉

—

C₂×Dic₃₄₉

C₂³

yes

—

C₂×Dic₃₄₉

C₂³

yes

—

C₂³×C₃₄₉

C₂³

yes

—

C₂³×C₃₄₉

C₂³

yes

characteristic

Q₈×C₃₄₉

—

D₄×C₃₄₉

—

D₄×C₃₄₉

—

C₂×C₄×C₃₄₉

—

C₂×C₄×C₃₄₉

D₄

yes

—

C₂×C₄×C₃₄₉

C₂³

yes

—

C₂×C₄×C₃₄₉

C₂³

yes

characteristic

C₂×C₄×C₃₄₉

C₂³

yes

characteristic

C₂²×C₃₄₉

—

C₂²×C₃₄₉

—

C₂²×C₃₄₉

—

C₂²×C₃₄₉

C₂×D₄

yes

—

C₂²×C₃₄₉

C₂⁴

yes

characteristic

C₄×C₃₄₉

—

C₄×C₃₄₉

C₂×D₄

yes

—

C₄×C₃₄₉

C₂×D₄

yes

—

Dic₃₄₉

—

Dic₃₄₉

—

C₂×C₃₄₉

—

C₂×C₃₄₉

—

C₂×C₃₄₉

C₂²×D₄

yes

characteristic

C₂×C₃₄₉

2₋¹⁺⁴

yes

—

C₃₄₉

#536

yes

characteristic, complemented, Sylow

#536

—

349

maximal, Sylow

C₂×#41

D₃₄₉

yes

characteristic

C₂²×Q₈

—

349

—

C₄⋊Q₈

—

349

—

#124

—

349

—

#123

—

349

—

C₂²⋊Q₈

—

349

—

#117

—

349

—

#41

—

C₂×Q₈

—

349

—

C₂×Q₈

—

349

—

C₂×Q₈

—

698

—

C₂×Q₈

C₂×D₃₄₉

yes

characteristic

C₂×D₄

C₂×D₃₄₉

yes

—

C₂×D₄

C₂×D₃₄₉

yes

characteristic

C₂²×C₄

—

349

—

C₂²×C₄

C₂×D₃₄₉

yes

—

C₂²×C₄

C₂×D₃₄₉

yes

characteristic

C₄⋊C₄

—

349

—

C₄⋊C₄

—

349

—

C₂²⋊C₄

—

349

—

C₂²⋊C₄

—

349

—

C₄²

—

349

—

C₂³

C₂²×D₃₄₉

yes

—

C₂³

C₂²×D₃₄₉

yes

characteristic

Q₈

—

Q₈

—

698

—

Q₈

—

698

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

349

—

C₂×C₄

—

349

—

C₂×C₄

—

698

—

C₂×C₄

C₂²⋊D₃₄₉

yes

—

C₂×C₄

C₂²×D₃₄₉

yes

—

C₂×C₄

C₂²×D₃₄₉

yes

characteristic

C₂×C₄

C₂²×D₃₄₉

yes

characteristic

C₂²

—

C₂²

—

C₂²

—

C₂²

C₂×C₂²⋊D₃₄₉

yes

—

C₂²

C₂³×D₃₄₉

yes

characteristic

C₄

—

C₄

—

698

—

C₄

—

698

—

C₄

C₂×C₂²⋊D₃₄₉

yes

—

100

C₄

C₂×C₂²⋊D₃₄₉

yes

—

101

C₂

—

102

C₂

—

103

C₂

C₂²×C₂²⋊D₃₄₉

yes

characteristic

104

C₂

#26013686

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₄₉
commutator, socle, cosocle	C₂²×C₃₄₉	C₂⁴
Fitting, prime radical	C₂×#41×C₃₄₉	C₂
hypercenter	C₂×#41	D₃₄₉
hypocenter	C₃₄₉	#536
LCS	G ▷ C₂²×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₂² ◁ C₂×#41

subgroup

G/H

center, Frattini

C₂²

C₂³×D₃₄₉

commutator, socle, cosocle

C₂²×C₃₄₉

C₂⁴

Fitting, prime radical

C₂×#41×C₃₄₉

C₂

hypercenter

C₂×#41

D₃₄₉

C₃₄₉

G ▷ C₂²×C₃₄₉ ▷ C₃₄₉

C₁ ◁ C₂² ◁ C₂×#41

N	H ≅ G/N	type
1. C₂²×Dic₆₉₈	C₂	semidirect
2. #26013637	C₂	semidirect
3. #26013626	C₂	semidirect
4. #26013620	C₂	semidirect
5. #26013618	C₂	semidirect
6. #26013598	C₂	semidirect
52. C₃₄₉	#536	semidirect

H ≅ G/N

type

1. C₂²×Dic₆₉₈

C₂

semidirect

2. #26013637

C₂

semidirect

3. #26013626

C₂

semidirect

4. #26013620

C₂

semidirect

5. #26013618

C₂

semidirect

6. #26013598

C₂

semidirect

52. C₃₄₉

#536

semidirect

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

subgroup	index
C₂²×Dic₆₉₈	2
#26013637	2
#26013626	2
#26013620	2
#26013618	2
#26013598	2
C₂×#41×C₃₄₉	2
#536	349

subgroup

index

C₂²×Dic₆₉₈

C₂×#41×C₃₄₉

349

Subgroups — #36374715

Conjugacy classes of subgroups

Complements

Maximal subgroups