Group #36371013 — subgroups

Subgroups — #390×C₃₄₇ #36371013

Order 22208.

258 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 106 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×D₄×C₃₄₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊C₄×C₃₄₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂²⋊C₄×C₃₄₇	C₂	1	4	yes	maximal, complemented
4	C₂×C₄²×C₃₄₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂⁴×C₃₄₇	C₂²	1	2	yes	—
6	C₂×D₄×C₃₄₇	—	2	2	no	—
7	C₂×D₄×C₃₄₇	—	2	2	no	—
8	C₂×D₄×C₃₄₇	C₄	1	4	yes	complemented
9	C₂²×C₄×C₃₄₇	C₂²	1	2	yes	—
10	C₂²×C₄×C₃₄₇	C₂²	1	2	yes	—
11	C₂²×C₄×C₃₄₇	C₂²	1	1	yes	characteristic
12	C₄⋊C₄×C₃₄₇	—	2	2	no	—
13	C₂²⋊C₄×C₃₄₇	—	2	8	no	—
14	C₄²×C₃₄₇	—	2	2	no	—
15	C₂³×C₃₄₇	—	2	4	no	—
16	C₂³×C₃₄₇	—	2	2	no	—
17	C₂³×C₃₄₇	—	2	2	no	—
18	C₂³×C₃₄₇	—	2	4	no	—
19	C₂³×C₃₄₇	C₂×C₄	1	4	yes	—
20	C₂³×C₃₄₇	C₂³	1	1	yes	characteristic
21	D₄×C₃₄₇	—	2	8	no	—
22	C₂×C₄×C₃₄₇	—	2	2	no	—
23	C₂×C₄×C₃₄₇	—	2	2	no	—
24	C₂×C₄×C₃₄₇	—	2	4	no	—
25	C₂×C₄×C₃₄₇	—	2	2	no	—
26	C₂×C₄×C₃₄₇	C₂×C₄	1	2	yes	—
27	C₂×C₄×C₃₄₇	D₄	1	2	yes	—
28	C₂×C₄×C₃₄₇	D₄	1	2	yes	—
29	C₂×C₄×C₃₄₇	D₄	1	4	yes	complemented
30	C₂²×C₃₄₇	—	2	4	no	—
31	C₂²×C₃₄₇	—	2	8	no	—
32	C₂²×C₃₄₇	—	4	4	no	—
33	C₂²×C₃₄₇	—	4	2	no	—
34	C₂²×C₃₄₇	—	4	2	no	—
35	C₂²×C₃₄₇	C₂²×C₄	1	1	yes	characteristic
36	C₂²×C₃₄₇	C₂×D₄	1	1	yes	characteristic
37	C₂²×C₃₄₇	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₃₄₇	C₂×D₄	1	2	yes	—
39	C₂²×C₃₄₇	#41	1	2	yes	—
40	C₄×C₃₄₇	—	2	4	no	—
41	C₄×C₃₄₇	—	4	2	no	—
42	C₄×C₃₄₇	C₂²⋊C₄	1	4	yes	complemented
43	C₂×C₃₄₇	—	4	4	no	—
44	C₂×C₃₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₃₄₇	C₄×D₄	1	2	yes	—
46	C₂×C₃₄₇	#121	1	2	yes	—
47	C₂×C₃₄₇	#124	1	1	yes	characteristic
48	C₂×C₃₄₇	#127	1	1	yes	characteristic
49	C₃₄₇	#390	1	1	yes	characteristic, direct product factor, Sylow
50	#390	C₃₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
51	C₂²×D₄	C₂×C₃₄₇	1	1	yes	characteristic
52	C₂×C₄⋊C₄	C₂×C₃₄₇	1	1	yes	characteristic, complemented
53	C₂×C₂²⋊C₄	C₂×C₃₄₇	1	4	yes	complemented
54	C₂×C₄²	C₂×C₃₄₇	1	1	yes	characteristic, complemented
55	C₂⁴	C₂²×C₃₄₇	1	2	yes	—
56	C₂×D₄	—	2	2	no	—
57	C₂×D₄	—	2	2	no	—
58	C₂×D₄	C₄×C₃₄₇	1	4	yes	complemented
59	C₂²×C₄	C₂²×C₃₄₇	1	2	yes	—
60	C₂²×C₄	C₂²×C₃₄₇	1	2	yes	—
61	C₂²×C₄	C₂²×C₃₄₇	1	1	yes	characteristic
62	C₄⋊C₄	—	2	2	no	—
63	C₂²⋊C₄	—	2	8	no	—
64	C₄²	—	2	2	no	—
65	C₂³	—	2	4	no	—
66	C₂³	—	2	2	no	—
67	C₂³	—	2	2	no	—
68	C₂³	—	2	4	no	—
69	C₂³	C₂×C₄×C₃₄₇	1	4	yes	—
70	C₂³	C₂³×C₃₄₇	1	1	yes	characteristic
71	D₄	—	2	8	no	—
72	C₂×C₄	—	2	2	no	—
73	C₂×C₄	—	2	2	no	—
74	C₂×C₄	—	2	4	no	—
75	C₂×C₄	—	2	2	no	—
76	C₂×C₄	C₂×C₄×C₃₄₇	1	2	yes	—
77	C₂×C₄	D₄×C₃₄₇	1	2	yes	—
78	C₂×C₄	D₄×C₃₄₇	1	2	yes	—
79	C₂×C₄	D₄×C₃₄₇	1	4	yes	complemented
80	C₂²	—	2	4	no	—
81	C₂²	—	2	8	no	—
82	C₂²	—	4	4	no	—
83	C₂²	—	4	2	no	—
84	C₂²	—	4	2	no	—
85	C₂²	C₂²×C₄×C₃₄₇	1	1	yes	characteristic
86	C₂²	C₂×D₄×C₃₄₇	1	1	yes	characteristic
87	C₂²	C₂×D₄×C₃₄₇	1	1	yes	characteristic
88	C₂²	C₂×D₄×C₃₄₇	1	2	yes	—
89	C₂²	#41×C₃₄₇	1	2	yes	—
90	C₄	—	2	4	no	—
91	C₄	—	4	2	no	—
92	C₄	C₂²⋊C₄×C₃₄₇	1	4	yes	complemented
93	C₂	—	4	4	no	—
94	C₂	C₂×C₂²⋊C₄×C₃₄₇	1	1	yes	characteristic
95	C₂	C₄×D₄×C₃₄₇	1	2	yes	—
96	C₂	#121×C₃₄₇	1	2	yes	—
97	C₂	#124×C₃₄₇	1	1	yes	characteristic
98	C₂	#127×C₃₄₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×D₄×C₃₄₇

C₂

yes

maximal, characteristic

C₂×C₄⋊C₄×C₃₄₇

C₂

yes

maximal, characteristic, complemented

C₂×C₂²⋊C₄×C₃₄₇

C₂

yes

maximal, complemented

C₂×C₄²×C₃₄₇

C₂

yes

maximal, characteristic, complemented

C₂⁴×C₃₄₇

C₂²

yes

—

C₂×D₄×C₃₄₇

—

C₂×D₄×C₃₄₇

—

C₂×D₄×C₃₄₇

C₄

yes

complemented

C₂²×C₄×C₃₄₇

C₂²

yes

—

C₂²×C₄×C₃₄₇

C₂²

yes

—

C₂²×C₄×C₃₄₇

C₂²

yes

characteristic

C₄⋊C₄×C₃₄₇

—

C₂²⋊C₄×C₃₄₇

—

C₄²×C₃₄₇

—

C₂³×C₃₄₇

—

C₂³×C₃₄₇

—

C₂³×C₃₄₇

—

C₂³×C₃₄₇

—

C₂³×C₃₄₇

C₂×C₄

yes

—

C₂³×C₃₄₇

C₂³

yes

characteristic

D₄×C₃₄₇

—

C₂×C₄×C₃₄₇

—

C₂×C₄×C₃₄₇

—

C₂×C₄×C₃₄₇

—

C₂×C₄×C₃₄₇

—

C₂×C₄×C₃₄₇

C₂×C₄

yes

—

C₂×C₄×C₃₄₇

D₄

yes

—

C₂×C₄×C₃₄₇

D₄

yes

—

C₂×C₄×C₃₄₇

D₄

yes

complemented

C₂²×C₃₄₇

—

C₂²×C₃₄₇

—

C₂²×C₃₄₇

—

C₂²×C₃₄₇

—

C₂²×C₃₄₇

—

C₂²×C₃₄₇

C₂²×C₄

yes

characteristic

C₂²×C₃₄₇

C₂×D₄

yes

characteristic

C₂²×C₃₄₇

C₂×D₄

yes

characteristic

C₂²×C₃₄₇

C₂×D₄

yes

—

C₂²×C₃₄₇

#41

yes

—

C₄×C₃₄₇

—

C₄×C₃₄₇

—

C₄×C₃₄₇

C₂²⋊C₄

yes

complemented

C₂×C₃₄₇

—

C₂×C₃₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₃₄₇

C₄×D₄

yes

—

C₂×C₃₄₇

#121

yes

—

C₂×C₃₄₇

yes

C₂×C₃₄₇

yes

C₃₄₇

yes

characteristic, direct product factor, Sylow

#390

C₃₄₇

yes

maximal, characteristic, direct product factor, Sylow

C₂²×D₄

C₂×C₃₄₇

yes

characteristic

C₂×C₄⋊C₄

C₂×C₃₄₇

yes

characteristic, complemented

C₂×C₂²⋊C₄

C₂×C₃₄₇

yes

complemented

C₂×C₄²

C₂×C₃₄₇

yes

characteristic, complemented

C₂⁴

C₂²×C₃₄₇

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₄×C₃₄₇

yes

complemented

C₂²×C₄

C₂²×C₃₄₇

yes

—

C₂²×C₄

C₂²×C₃₄₇

yes

—

C₂²×C₄

C₂²×C₃₄₇

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂×C₄×C₃₄₇

yes

—

C₂³

C₂³×C₃₄₇

yes

characteristic

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₄×C₃₄₇

yes

—

C₂×C₄

D₄×C₃₄₇

yes

—

C₂×C₄

D₄×C₃₄₇

yes

—

C₂×C₄

D₄×C₃₄₇

yes

complemented

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

C₂²×C₄×C₃₄₇

yes

characteristic

C₂²

C₂×D₄×C₃₄₇

yes

characteristic

C₂²

C₂×D₄×C₃₄₇

yes

characteristic

C₂²

C₂×D₄×C₃₄₇

yes

—

C₂²

#41×C₃₄₇

yes

—

C₄

—

C₄

—

C₄

C₂²⋊C₄×C₃₄₇

yes

complemented

C₂

—

C₂

C₂×C₂²⋊C₄×C₃₄₇

yes

characteristic

C₂

C₄×D₄×C₃₄₇

yes

—

C₂

#121×C₃₄₇

yes

—

C₂

#124×C₃₄₇

yes

characteristic

C₂

#127×C₃₄₇

yes

characteristic

	subgroup	G/H
center, socle	C₂³×C₃₄₇	C₂³
commutator	C₂²	C₂²×C₄×C₃₄₇
Frattini, cosocle	C₂³	C₂³×C₃₄₇
prime radical	C₂²×D₄×C₃₄₇	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃₄₇ ◁ G

subgroup

G/H

center, socle

C₂³×C₃₄₇

C₂³

commutator

C₂²

C₂²×C₄×C₃₄₇

Frattini, cosocle

C₂³

C₂³×C₃₄₇

prime radical

C₂²×D₄×C₃₄₇

C₂

LCS

G ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³×C₃₄₇ ◁ G

N	H ≅ G/N	type
2. C₂×C₄⋊C₄×C₃₄₇	C₂	semidirect
3. C₂×C₂²⋊C₄×C₃₄₇	C₂	semidirect
4. C₂×C₄²×C₃₄₇	C₂	semidirect
8. C₂×D₄×C₃₄₇	C₄	semidirect
29. C₂×C₄×C₃₄₇	D₄	semidirect
42. C₄×C₃₄₇	C₂²⋊C₄	semidirect
49. C₃₄₇	#390	direct (N × H)
50. #390	C₃₄₇	direct (N × H)
52. C₂×C₄⋊C₄	C₂×C₃₄₇	semidirect
53. C₂×C₂²⋊C₄	C₂×C₃₄₇	semidirect
54. C₂×C₄²	C₂×C₃₄₇	semidirect
58. C₂×D₄	C₄×C₃₄₇	semidirect
79. C₂×C₄	D₄×C₃₄₇	semidirect
92. C₄	C₂²⋊C₄×C₃₄₇	semidirect

H ≅ G/N

type

2. C₂×C₄⋊C₄×C₃₄₇

C₂

semidirect

3. C₂×C₂²⋊C₄×C₃₄₇

C₂

semidirect

4. C₂×C₄²×C₃₄₇

C₂

semidirect

8. C₂×D₄×C₃₄₇

C₄

semidirect

29. C₂×C₄×C₃₄₇

D₄

semidirect

42. C₄×C₃₄₇

C₂²⋊C₄

semidirect

49. C₃₄₇

#390

direct (N × H)

50. #390

C₃₄₇

direct (N × H)

52. C₂×C₄⋊C₄

C₂×C₃₄₇

semidirect

53. C₂×C₂²⋊C₄

C₂×C₃₄₇

semidirect

54. C₂×C₄²

C₂×C₃₄₇

semidirect

58. C₂×D₄

C₄×C₃₄₇

semidirect

79. C₂×C₄

D₄×C₃₄₇

semidirect

92. C₄

C₂²⋊C₄×C₃₄₇

semidirect

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

subgroup	index
C₂²×D₄×C₃₄₇	2
C₂×C₄⋊C₄×C₃₄₇	2
C₂×C₂²⋊C₄×C₃₄₇	2
C₂×C₄²×C₃₄₇	2
#390	347

subgroup

index

C₂²×D₄×C₃₄₇

C₂×C₄⋊C₄×C₃₄₇

C₂×C₂²⋊C₄×C₃₄₇

C₂×C₄²×C₃₄₇

#390

347

Subgroups — #390×C₃₄₇ #36371013

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #390×C347 #36371013

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #390×C₃₄₇ #36371013