Group #38733621 — subgroups

Subgroups — #390×C₃₈₉ #38733621

Order 24896.

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×C₃₈₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊C₄×C₃₈₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂²⋊C₄×C₃₈₉	C₂	1	4	yes	maximal, complemented
4	C₂×C₄²×C₃₈₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂⁴×C₃₈₉	C₂²	1	2	yes	—
6	C₂×D₄×C₃₈₉	—	2	2	no	—
7	C₂×D₄×C₃₈₉	—	2	2	no	—
8	C₂×D₄×C₃₈₉	C₄	1	4	yes	complemented
9	C₂²×C₄×C₃₈₉	C₂²	1	2	yes	—
10	C₂²×C₄×C₃₈₉	C₂²	1	2	yes	—
11	C₂²×C₄×C₃₈₉	C₂²	1	1	yes	characteristic
12	C₄⋊C₄×C₃₈₉	—	2	2	no	—
13	C₂²⋊C₄×C₃₈₉	—	2	8	no	—
14	C₄²×C₃₈₉	—	2	2	no	—
15	C₂³×C₃₈₉	—	2	4	no	—
16	C₂³×C₃₈₉	—	2	2	no	—
17	C₂³×C₃₈₉	—	2	2	no	—
18	C₂³×C₃₈₉	—	2	4	no	—
19	C₂³×C₃₈₉	C₂×C₄	1	4	yes	—
20	C₂³×C₃₈₉	C₂³	1	1	yes	characteristic
21	D₄×C₃₈₉	—	2	8	no	—
22	C₂×C₄×C₃₈₉	—	2	2	no	—
23	C₂×C₄×C₃₈₉	—	2	2	no	—
24	C₂×C₄×C₃₈₉	—	2	4	no	—
25	C₂×C₄×C₃₈₉	—	2	2	no	—
26	C₂×C₄×C₃₈₉	C₂×C₄	1	2	yes	—
27	C₂×C₄×C₃₈₉	D₄	1	2	yes	—
28	C₂×C₄×C₃₈₉	D₄	1	2	yes	—
29	C₂×C₄×C₃₈₉	D₄	1	4	yes	complemented
30	C₂²×C₃₈₉	—	2	4	no	—
31	C₂²×C₃₈₉	—	2	8	no	—
32	C₂²×C₃₈₉	—	4	4	no	—
33	C₂²×C₃₈₉	—	4	2	no	—
34	C₂²×C₃₈₉	—	4	2	no	—
35	C₂²×C₃₈₉	C₂²×C₄	1	1	yes	characteristic
36	C₂²×C₃₈₉	C₂×D₄	1	1	yes	characteristic
37	C₂²×C₃₈₉	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₃₈₉	C₂×D₄	1	2	yes	—
39	C₂²×C₃₈₉	#41	1	2	yes	—
40	C₄×C₃₈₉	—	2	4	no	—
41	C₄×C₃₈₉	—	4	2	no	—
42	C₄×C₃₈₉	C₂²⋊C₄	1	4	yes	complemented
43	C₂×C₃₈₉	—	4	4	no	—
44	C₂×C₃₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₃₈₉	C₄×D₄	1	2	yes	—
46	C₂×C₃₈₉	#121	1	2	yes	—
47	C₂×C₃₈₉	#124	1	1	yes	characteristic
48	C₂×C₃₈₉	#127	1	1	yes	characteristic
49	C₃₈₉	#390	1	1	yes	characteristic, direct product factor, Sylow
50	#390	C₃₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
51	C₂²×D₄	C₂×C₃₈₉	1	1	yes	characteristic
52	C₂×C₄⋊C₄	C₂×C₃₈₉	1	1	yes	characteristic, complemented
53	C₂×C₂²⋊C₄	C₂×C₃₈₉	1	4	yes	complemented
54	C₂×C₄²	C₂×C₃₈₉	1	1	yes	characteristic, complemented
55	C₂⁴	C₂²×C₃₈₉	1	2	yes	—
56	C₂×D₄	—	2	2	no	—
57	C₂×D₄	—	2	2	no	—
58	C₂×D₄	C₄×C₃₈₉	1	4	yes	complemented
59	C₂²×C₄	C₂²×C₃₈₉	1	2	yes	—
60	C₂²×C₄	C₂²×C₃₈₉	1	2	yes	—
61	C₂²×C₄	C₂²×C₃₈₉	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×C₃₈₉	1	4	yes	—
70	C₂³	C₂³×C₃₈₉	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×C₃₈₉	1	2	yes	—
77	C₂×C₄	D₄×C₃₈₉	1	2	yes	—
78	C₂×C₄	D₄×C₃₈₉	1	2	yes	—
79	C₂×C₄	D₄×C₃₈₉	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×C₃₈₉	1	1	yes	characteristic
86	C₂²	C₂×D₄×C₃₈₉	1	1	yes	characteristic
87	C₂²	C₂×D₄×C₃₈₉	1	1	yes	characteristic
88	C₂²	C₂×D₄×C₃₈₉	1	2	yes	—
89	C₂²	#41×C₃₈₉	1	2	yes	—
90	C₄	—	2	4	no	—
91	C₄	—	4	2	no	—
92	C₄	C₂²⋊C₄×C₃₈₉	1	4	yes	complemented
93	C₂	—	4	4	no	—
94	C₂	C₂×C₂²⋊C₄×C₃₈₉	1	1	yes	characteristic
95	C₂	C₄×D₄×C₃₈₉	1	2	yes	—
96	C₂	#121×C₃₈₉	1	2	yes	—
97	C₂	#124×C₃₈₉	1	1	yes	characteristic
98	C₂	#127×C₃₈₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×D₄×C₃₈₉

C₂

yes

maximal, characteristic

C₂×C₄⋊C₄×C₃₈₉

C₂

yes

maximal, characteristic, complemented

C₂×C₂²⋊C₄×C₃₈₉

C₂

yes

maximal, complemented

C₂×C₄²×C₃₈₉

C₂

yes

maximal, characteristic, complemented

C₂⁴×C₃₈₉

C₂²

yes

—

C₂×D₄×C₃₈₉

—

C₂×D₄×C₃₈₉

—

C₂×D₄×C₃₈₉

C₄

yes

complemented

C₂²×C₄×C₃₈₉

C₂²

yes

—

C₂²×C₄×C₃₈₉

C₂²

yes

—

C₂²×C₄×C₃₈₉

C₂²

yes

characteristic

C₄⋊C₄×C₃₈₉

—

C₂²⋊C₄×C₃₈₉

—

C₄²×C₃₈₉

—

C₂³×C₃₈₉

—

C₂³×C₃₈₉

—

C₂³×C₃₈₉

—

C₂³×C₃₈₉

—

C₂³×C₃₈₉

C₂×C₄

yes

—

C₂³×C₃₈₉

C₂³

yes

characteristic

D₄×C₃₈₉

—

C₂×C₄×C₃₈₉

—

C₂×C₄×C₃₈₉

—

C₂×C₄×C₃₈₉

—

C₂×C₄×C₃₈₉

—

C₂×C₄×C₃₈₉

C₂×C₄

yes

—

C₂×C₄×C₃₈₉

D₄

yes

—

C₂×C₄×C₃₈₉

D₄

yes

—

C₂×C₄×C₃₈₉

D₄

yes

complemented

C₂²×C₃₈₉

—

C₂²×C₃₈₉

—

C₂²×C₃₈₉

—

C₂²×C₃₈₉

—

C₂²×C₃₈₉

—

C₂²×C₃₈₉

C₂²×C₄

yes

characteristic

C₂²×C₃₈₉

C₂×D₄

yes

characteristic

C₂²×C₃₈₉

C₂×D₄

yes

characteristic

C₂²×C₃₈₉

C₂×D₄

yes

—

C₂²×C₃₈₉

#41

yes

—

C₄×C₃₈₉

—

C₄×C₃₈₉

—

C₄×C₃₈₉

C₂²⋊C₄

yes

complemented

C₂×C₃₈₉

—

C₂×C₃₈₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₃₈₉

C₄×D₄

yes

—

C₂×C₃₈₉

#121

yes

—

C₂×C₃₈₉

yes

C₂×C₃₈₉

yes

C₃₈₉

yes

characteristic, direct product factor, Sylow

#390

C₃₈₉

yes

maximal, characteristic, direct product factor, Sylow

C₂²×D₄

C₂×C₃₈₉

yes

characteristic

C₂×C₄⋊C₄

C₂×C₃₈₉

yes

characteristic, complemented

C₂×C₂²⋊C₄

C₂×C₃₈₉

yes

complemented

C₂×C₄²

C₂×C₃₈₉

yes

characteristic, complemented

C₂⁴

C₂²×C₃₈₉

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₄×C₃₈₉

yes

complemented

C₂²×C₄

C₂²×C₃₈₉

yes

—

C₂²×C₄

C₂²×C₃₈₉

yes

—

C₂²×C₄

C₂²×C₃₈₉

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂×C₄×C₃₈₉

yes

—

C₂³

C₂³×C₃₈₉

yes

characteristic

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₄×C₃₈₉

yes

—

C₂×C₄

D₄×C₃₈₉

yes

—

C₂×C₄

D₄×C₃₈₉

yes

—

C₂×C₄

D₄×C₃₈₉

yes

complemented

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

C₂²×C₄×C₃₈₉

yes

characteristic

C₂²

C₂×D₄×C₃₈₉

yes

characteristic

C₂²

C₂×D₄×C₃₈₉

yes

characteristic

C₂²

C₂×D₄×C₃₈₉

yes

—

C₂²

#41×C₃₈₉

yes

—

C₄

—

C₄

—

C₄

C₂²⋊C₄×C₃₈₉

yes

complemented

C₂

—

C₂

C₂×C₂²⋊C₄×C₃₈₉

yes

characteristic

C₂

C₄×D₄×C₃₈₉

yes

—

C₂

#121×C₃₈₉

yes

—

C₂

#124×C₃₈₉

yes

characteristic

C₂

#127×C₃₈₉

yes

characteristic

	subgroup	G/H
center, socle	C₂³×C₃₈₉	C₂³
commutator	C₂²	C₂²×C₄×C₃₈₉
Frattini, cosocle	C₂³	C₂³×C₃₈₉
prime radical	C₂²×D₄×C₃₈₉	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃₈₉ ◁ G

subgroup

G/H

center, socle

C₂³×C₃₈₉

C₂³

commutator

C₂²

C₂²×C₄×C₃₈₉

Frattini, cosocle

C₂³

C₂³×C₃₈₉

prime radical

C₂²×D₄×C₃₈₉

C₂

LCS

G ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³×C₃₈₉ ◁ G

N	H ≅ G/N	type
2. C₂×C₄⋊C₄×C₃₈₉	C₂	semidirect
3. C₂×C₂²⋊C₄×C₃₈₉	C₂	semidirect
4. C₂×C₄²×C₃₈₉	C₂	semidirect
8. C₂×D₄×C₃₈₉	C₄	semidirect
29. C₂×C₄×C₃₈₉	D₄	semidirect
42. C₄×C₃₈₉	C₂²⋊C₄	semidirect
49. C₃₈₉	#390	direct (N × H)
50. #390	C₃₈₉	direct (N × H)
52. C₂×C₄⋊C₄	C₂×C₃₈₉	semidirect
53. C₂×C₂²⋊C₄	C₂×C₃₈₉	semidirect
54. C₂×C₄²	C₂×C₃₈₉	semidirect
58. C₂×D₄	C₄×C₃₈₉	semidirect
79. C₂×C₄	D₄×C₃₈₉	semidirect
92. C₄	C₂²⋊C₄×C₃₈₉	semidirect

H ≅ G/N

type

2. C₂×C₄⋊C₄×C₃₈₉

C₂

semidirect

3. C₂×C₂²⋊C₄×C₃₈₉

C₂

semidirect

4. C₂×C₄²×C₃₈₉

C₂

semidirect

8. C₂×D₄×C₃₈₉

C₄

semidirect

29. C₂×C₄×C₃₈₉

D₄

semidirect

42. C₄×C₃₈₉

C₂²⋊C₄

semidirect

49. C₃₈₉

#390

direct (N × H)

50. #390

C₃₈₉

direct (N × H)

52. C₂×C₄⋊C₄

C₂×C₃₈₉

semidirect

53. C₂×C₂²⋊C₄

C₂×C₃₈₉

semidirect

54. C₂×C₄²

C₂×C₃₈₉

semidirect

58. C₂×D₄

C₄×C₃₈₉

semidirect

79. C₂×C₄

D₄×C₃₈₉

semidirect

92. C₄

C₂²⋊C₄×C₃₈₉

semidirect

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

subgroup	index
C₂²×D₄×C₃₈₉	2
C₂×C₄⋊C₄×C₃₈₉	2
C₂×C₂²⋊C₄×C₃₈₉	2
C₂×C₄²×C₃₈₉	2
#390	389

subgroup

index

C₂²×D₄×C₃₈₉

C₂×C₄⋊C₄×C₃₈₉

C₂×C₂²⋊C₄×C₃₈₉

C₂×C₄²×C₃₈₉

#390

389

Subgroups — #390×C₃₈₉ #38733621

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #390×C389 #38733621

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #390×C₃₈₉ #38733621