Group #36374711 — subgroups

Subgroups — #36374711

Order 22336.

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

normal subgroups only

This group has 334 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	C₂×#19150233	C₂	1	1	yes	maximal, characteristic, complemented
2	#26013635	C₂	1	2	yes	maximal, complemented
3	#26013626	C₂	1	2	yes	maximal, complemented
4	#26013610	C₂	1	4	yes	maximal, complemented
5	#26013608	C₂	1	4	yes	maximal, complemented
6	#26013598	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₄×C₃₄₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#19150233	—	2	4	no	—
9	C₂×Dic₆₉₈	C₂²	1	1	yes	characteristic, complemented
10	C₂×D₁₃₉₆	C₂²	1	1	yes	characteristic, complemented
11	C₂×C₂²⋊D₃₄₉	C₂²	1	2	yes	—
12	C₂×C₂²⋊D₃₄₉	C₂²	1	8	yes	complemented
13	C₂×C₄×D₃₄₉	C₂²	1	2	yes	complemented
14	C₄⋊Dic₃₄₉	C₂²	1	2	yes	complemented
15	C₂²⋊Dic₃₄₉	C₂²	1	2	yes	—
16	C₂²⋊Dic₃₄₉	C₂²	1	8	yes	complemented
17	C₄×Dic₃₄₉	C₂²	1	2	yes	complemented
18	C₂⁴×C₃₄₉	C₂²	1	2	yes	—
19	C₂×D₄×C₃₄₉	—	2	4	no	—
20	C₂×D₄×C₃₄₉	C₂²	1	2	yes	—
21	C₂×D₄×C₃₄₉	C₂²	1	2	yes	—
22	C₂²×C₄×C₃₄₉	C₂²	1	1	yes	characteristic
23	C₂²×D₃₄₉	C₂³	1	2	yes	—
24	Dic₆₉₈	—	2	2	no	—
25	D₁₃₉₆	—	2	2	no	—
26	C₂²⋊D₃₄₉	—	2	4	no	—
27	C₂²⋊D₃₄₉	—	4	8	no	—
28	C₂×Dic₃₄₉	C₂³	1	4	yes	—
29	C₂×Dic₃₄₉	C₂³	1	2	yes	—
30	C₄×D₃₄₉	—	2	4	no	—
31	C₂³×C₃₄₉	—	2	4	no	—
32	C₂³×C₃₄₉	—	4	4	no	—
33	C₂³×C₃₄₉	C₂³	1	4	yes	—
34	C₂³×C₃₄₉	C₂³	1	1	yes	characteristic
35	D₄×C₃₄₉	—	2	4	no	—
36	D₄×C₃₄₉	—	2	4	no	—
37	C₂×C₄×C₃₄₉	D₄	1	4	yes	complemented
38	C₂×C₄×C₃₄₉	C₂³	1	1	yes	characteristic
39	C₂×C₄×C₃₄₉	C₂³	1	1	yes	characteristic
40	C₂²×C₃₄₉	—	2	2	no	—
41	C₂²×C₃₄₉	—	2	4	no	—
42	C₂²×C₃₄₉	—	2	8	no	—
43	C₂²×C₃₄₉	—	4	8	no	—
44	C₂²×C₃₄₉	C₂×D₄	1	2	yes	—
45	C₂²×C₃₄₉	C₂⁴	1	1	yes	characteristic
46	C₂×D₃₄₉	—	2	2	no	—
47	C₂×D₃₄₉	—	2	4	no	—
48	C₄×C₃₄₉	C₂×D₄	1	2	yes	—
49	C₄×C₃₄₉	C₂×D₄	1	2	yes	—
50	Dic₃₄₉	—	2	2	no	—
51	Dic₃₄₉	—	2	4	no	—
52	C₂×C₃₄₉	—	2	2	no	—
53	C₂×C₃₄₉	—	4	4	no	—
54	C₂×C₃₄₉	C₂²×D₄	1	1	yes	characteristic
55	C₂×C₃₄₉	H(5,𝔽₂)	1	2	yes	—
56	D₃₄₉	—	4	2	no	—
57	C₃₄₉	#535	1	1	yes	characteristic, complemented, Sylow
58	#535	—	349	1	no	maximal, Sylow
59	C₂×#41	—	349	1	no	—
60	C₂²×D₄	D₃₄₉	1	1	yes	characteristic, complemented
61	#127	—	349	2	no	—
62	#124	—	349	2	no	—
63	#121	—	349	4	no	—
64	#120	—	349	4	no	—
65	#117	—	349	1	no	—
66	C₂⁴	C₂×D₃₄₉	1	2	yes	—
67	#41	—	698	4	no	—
68	C₂×Q₈	—	349	1	no	—
69	C₂×D₄	—	2	4	no	—
70	C₂×D₄	—	349	8	no	—
71	C₂×D₄	—	349	2	no	—
72	C₂×D₄	—	349	1	no	—
73	C₂×D₄	C₂×D₃₄₉	1	2	yes	—
74	C₂×D₄	C₂×D₃₄₉	1	2	yes	—
75	C₂²×C₄	—	349	2	no	—
76	C₂²×C₄	C₂×D₃₄₉	1	1	yes	characteristic
77	C₄⋊C₄	—	349	2	no	—
78	C₂²⋊C₄	—	349	2	no	—
79	C₂²⋊C₄	—	349	8	no	—
80	C₄²	—	349	2	no	—
81	C₂³	—	2	4	no	—
82	C₂³	—	4	4	no	—
83	C₂³	—	349	2	no	—
84	C₂³	C₂²×D₃₄₉	1	4	yes	—
85	C₂³	C₂²×D₃₄₉	1	1	yes	characteristic
86	Q₈	—	698	2	no	—
87	D₄	—	2	4	no	—
88	D₄	—	2	4	no	—
89	D₄	—	1396	8	no	—
90	D₄	—	698	4	no	—
91	D₄	—	698	2	no	—
92	C₂×C₄	—	349	4	no	—
93	C₂×C₄	—	349	2	no	—
94	C₂×C₄	—	698	4	no	—
95	C₂×C₄	C₂²⋊D₃₄₉	1	4	yes	complemented
96	C₂×C₄	C₂²×D₃₄₉	1	1	yes	characteristic
97	C₂×C₄	C₂²×D₃₄₉	1	1	yes	characteristic
98	C₂²	—	2	2	no	—
99	C₂²	—	2	4	no	—
100	C₂²	—	2	8	no	—
101	C₂²	—	4	8	no	—
102	C₂²	—	698	2	no	—
103	C₂²	—	698	4	no	—
104	C₂²	C₂×C₂²⋊D₃₄₉	1	2	yes	—
105	C₂²	C₂³×D₃₄₉	1	1	yes	characteristic
106	C₄	—	698	2	no	—
107	C₄	—	698	4	no	—
108	C₄	C₂×C₂²⋊D₃₄₉	1	2	yes	—
109	C₄	C₂×C₂²⋊D₃₄₉	1	2	yes	—
110	C₂	—	2	2	no	—
111	C₂	—	4	4	no	—
112	C₂	—	1396	2	no	—
113	C₂	C₂²×C₂²⋊D₃₄₉	1	1	yes	characteristic
114	C₂	#26013684	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19150233

C₂

yes

maximal, characteristic, complemented

#26013635

C₂

yes

maximal, complemented

#26013626

C₂

yes

maximal, complemented

#26013610

C₂

yes

maximal, complemented

#26013608

C₂

yes

maximal, complemented

#26013598

C₂

yes

maximal, characteristic, complemented

C₂²×D₄×C₃₄₉

C₂

yes

maximal, characteristic, complemented

#19150233

—

C₂×Dic₆₉₈

C₂²

yes

characteristic, complemented

C₂×D₁₃₉₆

C₂²

yes

characteristic, complemented

C₂×C₂²⋊D₃₄₉

C₂²

yes

—

C₂×C₂²⋊D₃₄₉

C₂²

yes

complemented

C₂×C₄×D₃₄₉

C₂²

yes

complemented

C₄⋊Dic₃₄₉

C₂²

yes

complemented

C₂²⋊Dic₃₄₉

C₂²

yes

—

C₂²⋊Dic₃₄₉

C₂²

yes

complemented

C₄×Dic₃₄₉

C₂²

yes

complemented

C₂⁴×C₃₄₉

C₂²

yes

—

C₂×D₄×C₃₄₉

—

C₂×D₄×C₃₄₉

C₂²

yes

—

C₂×D₄×C₃₄₉

C₂²

yes

—

C₂²×C₄×C₃₄₉

C₂²

yes

characteristic

C₂²×D₃₄₉

C₂³

yes

—

Dic₆₉₈

—

D₁₃₉₆

—

C₂²⋊D₃₄₉

—

C₂²⋊D₃₄₉

—

C₂×Dic₃₄₉

C₂³

yes

—

C₂×Dic₃₄₉

C₂³

yes

—

C₄×D₃₄₉

—

C₂³×C₃₄₉

—

C₂³×C₃₄₉

—

C₂³×C₃₄₉

C₂³

yes

—

C₂³×C₃₄₉

C₂³

yes

characteristic

D₄×C₃₄₉

—

D₄×C₃₄₉

—

C₂×C₄×C₃₄₉

D₄

yes

complemented

C₂×C₄×C₃₄₉

C₂³

yes

characteristic

C₂×C₄×C₃₄₉

C₂³

yes

characteristic

C₂²×C₃₄₉

—

C₂²×C₃₄₉

—

C₂²×C₃₄₉

—

C₂²×C₃₄₉

—

C₂²×C₃₄₉

C₂×D₄

yes

—

C₂²×C₃₄₉

C₂⁴

yes

characteristic

C₂×D₃₄₉

—

C₂×D₃₄₉

—

C₄×C₃₄₉

C₂×D₄

yes

—

C₄×C₃₄₉

C₂×D₄

yes

—

Dic₃₄₉

—

Dic₃₄₉

—

C₂×C₃₄₉

—

C₂×C₃₄₉

—

C₂×C₃₄₉

C₂²×D₄

yes

characteristic

C₂×C₃₄₉

H(5,𝔽₂)

yes

—

D₃₄₉

—

C₃₄₉

#535

yes

characteristic, complemented, Sylow

#535

—

349

maximal, Sylow

C₂×#41

—

349

—

C₂²×D₄

D₃₄₉

yes

characteristic, complemented

#127

—

349

—

#124

—

349

—

#121

—

349

—

#120

—

349

—

#117

—

349

—

C₂⁴

C₂×D₃₄₉

yes

—

#41

—

698

—

C₂×Q₈

—

349

—

C₂×D₄

—

C₂×D₄

—

349

—

C₂×D₄

—

349

—

C₂×D₄

—

349

—

C₂×D₄

C₂×D₃₄₉

yes

—

C₂×D₄

C₂×D₃₄₉

yes

—

C₂²×C₄

—

349

—

C₂²×C₄

C₂×D₃₄₉

yes

characteristic

C₄⋊C₄

—

349

—

C₂²⋊C₄

—

349

—

C₂²⋊C₄

—

349

—

C₄²

—

349

—

C₂³

—

C₂³

—

C₂³

—

349

—

C₂³

C₂²×D₃₄₉

yes

—

C₂³

C₂²×D₃₄₉

yes

characteristic

Q₈

—

698

—

D₄

—

D₄

—

D₄

—

1396

—

D₄

—

698

—

D₄

—

698

—

C₂×C₄

—

349

—

C₂×C₄

—

349

—

C₂×C₄

—

698

—

C₂×C₄

C₂²⋊D₃₄₉

yes

complemented

C₂×C₄

C₂²×D₃₄₉

yes

characteristic

C₂×C₄

C₂²×D₃₄₉

yes

characteristic

C₂²

—

C₂²

—

100

C₂²

—

101

C₂²

—

102

C₂²

—

698

—

103

C₂²

—

698

—

104

C₂²

C₂×C₂²⋊D₃₄₉

yes

—

105

C₂²

C₂³×D₃₄₉

yes

characteristic

106

C₄

—

698

—

107

C₄

—

698

—

108

C₄

C₂×C₂²⋊D₃₄₉

yes

—

109

C₄

C₂×C₂²⋊D₃₄₉

yes

—

110

C₂

—

111

C₂

—

112

C₂

—

1396

—

113

C₂

C₂²×C₂²⋊D₃₄₉

yes

characteristic

114

C₂

#26013684

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₄₉
commutator, socle, cosocle	C₂²×C₃₄₉	C₂⁴
Fitting	C₂²×D₄×C₃₄₉	C₂
hypercenter	C₂²×D₄	D₃₄₉
hypocenter	C₃₄₉	#535
LCS	G ▷ C₂²×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₂² ◁ C₂²×D₄

subgroup

G/H

center, Frattini

C₂²

C₂³×D₃₄₉

commutator, socle, cosocle

C₂²×C₃₄₉

C₂⁴

Fitting

C₂²×D₄×C₃₄₉

C₂

hypercenter

C₂²×D₄

D₃₄₉

C₃₄₉

G ▷ C₂²×C₃₄₉ ▷ C₃₄₉

C₁ ◁ C₂² ◁ C₂²×D₄

N	H ≅ G/N	type
1. C₂×#19150233	C₂	semidirect
2. #26013635	C₂	semidirect
3. #26013626	C₂	semidirect
4. #26013610	C₂	semidirect
5. #26013608	C₂	semidirect
6. #26013598	C₂	semidirect
7. C₂²×D₄×C₃₄₉	C₂	semidirect
9. C₂×Dic₆₉₈	C₂²	semidirect
10. C₂×D₁₃₉₆	C₂²	semidirect
12. C₂×C₂²⋊D₃₄₉	C₂²	semidirect
13. C₂×C₄×D₃₄₉	C₂²	semidirect
14. C₄⋊Dic₃₄₉	C₂²	semidirect
16. C₂²⋊Dic₃₄₉	C₂²	semidirect
17. C₄×Dic₃₄₉	C₂²	semidirect
37. C₂×C₄×C₃₄₉	D₄	semidirect
57. C₃₄₉	#535	semidirect
60. C₂²×D₄	D₃₄₉	semidirect
95. C₂×C₄	C₂²⋊D₃₄₉	semidirect

H ≅ G/N

type

1. C₂×#19150233

C₂

semidirect

2. #26013635

C₂

semidirect

3. #26013626

C₂

semidirect

4. #26013610

C₂

semidirect

5. #26013608

C₂

semidirect

6. #26013598

C₂

semidirect

7. C₂²×D₄×C₃₄₉

C₂

semidirect

9. C₂×Dic₆₉₈

C₂²

semidirect

10. C₂×D₁₃₉₆

C₂²

semidirect

12. C₂×C₂²⋊D₃₄₉

C₂²

semidirect

13. C₂×C₄×D₃₄₉

C₂²

semidirect

14. C₄⋊Dic₃₄₉

C₂²

semidirect

16. C₂²⋊Dic₃₄₉

C₂²

semidirect

17. C₄×Dic₃₄₉

C₂²

semidirect

37. C₂×C₄×C₃₄₉

D₄

semidirect

57. C₃₄₉

#535

semidirect

60. C₂²×D₄

D₃₄₉

semidirect

95. C₂×C₄

C₂²⋊D₃₄₉

semidirect

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

subgroup	index
C₂×#19150233	2
#26013635	2
#26013626	2
#26013610	2
#26013608	2
#26013598	2
C₂²×D₄×C₃₄₉	2
#535	349

subgroup

index

C₂²×D₄×C₃₄₉

349

Subgroups — #36374711

Conjugacy classes of subgroups

Complements

Maximal subgroups