Group #39908768 — subgroups

Subgroups — #389×C₄₀₉ #39908768

Order 26176.

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

normal subgroups only

This group has 170 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	C₂×C₄⋊C₄×C₄₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊C₄×C₄₀₉	C₂	1	2	yes	maximal
3	C₂×C₄⋊C₄×C₄₀₉	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄²×C₄₀₉	C₂	1	1	yes	maximal, characteristic
5	#95×C₄₀₉	C₂	1	2	yes	maximal
6	C₂²×C₄×C₄₀₉	C₂²	1	2	yes	—
7	C₂²×C₄×C₄₀₉	C₂²	1	2	yes	—
8	C₂²×C₄×C₄₀₉	C₂²	1	1	yes	characteristic
9	C₂²×C₄×C₄₀₉	C₂²	1	1	yes	characteristic
10	C₂²×C₄×C₄₀₉	C₂²	1	1	yes	characteristic
11	C₄⋊C₄×C₄₀₉	—	2	4	no	—
12	C₄⋊C₄×C₄₀₉	—	2	2	no	—
13	C₄⋊C₄×C₄₀₉	C₄	1	4	yes	complemented
14	C₄²×C₄₀₉	—	2	2	no	—
15	C₂³×C₄₀₉	C₂³	1	1	yes	characteristic
16	C₂×C₄×C₄₀₉	—	2	1	no	—
17	C₂×C₄×C₄₀₉	—	2	1	no	—
18	C₂×C₄×C₄₀₉	—	2	2	no	—
19	C₂×C₄×C₄₀₉	—	2	2	no	—
20	C₂×C₄×C₄₀₉	—	2	1	no	—
21	C₂×C₄×C₄₀₉	—	2	1	no	—
22	C₂×C₄×C₄₀₉	—	2	2	no	—
23	C₂×C₄×C₄₀₉	—	2	2	no	—
24	C₂×C₄×C₄₀₉	—	2	2	no	—
25	C₂×C₄×C₄₀₉	C₂×C₄	1	4	yes	—
26	C₂×C₄×C₄₀₉	C₂×C₄	1	2	yes	—
27	C₂×C₄×C₄₀₉	D₄	1	2	yes	—
28	C₂×C₄×C₄₀₉	D₄	1	2	yes	—
29	C₂×C₄×C₄₀₉	Q₈	1	2	yes	—
30	C₂×C₄×C₄₀₉	Q₈	1	2	yes	—
31	C₂²×C₄₀₉	C₂²×C₄	1	1	yes	characteristic
32	C₂²×C₄₀₉	C₂×D₄	1	1	yes	characteristic
33	C₂²×C₄₀₉	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₄₀₉	C₂×Q₈	1	1	yes	characteristic
35	C₂²×C₄₀₉	C₂×Q₈	1	1	yes	characteristic
36	C₂²×C₄₀₉	#41	1	1	yes	characteristic
37	C₂²×C₄₀₉	#41	1	1	yes	characteristic
38	C₄×C₄₀₉	—	2	4	no	—
39	C₄×C₄₀₉	—	2	2	no	—
40	C₄×C₄₀₉	—	2	2	no	—
41	C₄×C₄₀₉	—	4	2	no	—
42	C₄×C₄₀₉	C₄⋊C₄	1	4	yes	complemented
43	C₂×C₄₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₄₀₉	C₄×D₄	1	1	yes	characteristic
45	C₂×C₄₀₉	C₄×Q₈	1	1	yes	characteristic
46	C₂×C₄₀₉	#121	1	1	yes	characteristic
47	C₂×C₄₀₉	C₂²⋊Q₈	1	1	yes	characteristic
48	C₂×C₄₀₉	#125	1	1	yes	characteristic
49	C₂×C₄₀₉	C₄⋊Q₈	1	1	yes	characteristic
50	C₄₀₉	#389	1	1	yes	characteristic, direct product factor, Sylow
51	#389	C₄₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	C₂×C₄⋊C₄	C₂×C₄₀₉	1	1	yes	characteristic
53	C₂×C₄⋊C₄	C₂×C₄₀₉	1	2	yes	—
54	C₂×C₄⋊C₄	C₂×C₄₀₉	1	1	yes	characteristic
55	C₂×C₄²	C₂×C₄₀₉	1	1	yes	characteristic
56	#95	C₂×C₄₀₉	1	2	yes	—
57	C₂²×C₄	C₂²×C₄₀₉	1	2	yes	—
58	C₂²×C₄	C₂²×C₄₀₉	1	2	yes	—
59	C₂²×C₄	C₂²×C₄₀₉	1	1	yes	characteristic
60	C₂²×C₄	C₂²×C₄₀₉	1	1	yes	characteristic
61	C₂²×C₄	C₂²×C₄₀₉	1	1	yes	characteristic
62	C₄⋊C₄	—	2	4	no	—
63	C₄⋊C₄	—	2	2	no	—
64	C₄⋊C₄	C₄×C₄₀₉	1	4	yes	complemented
65	C₄²	—	2	2	no	—
66	C₂³	C₂³×C₄₀₉	1	1	yes	characteristic
67	C₂×C₄	—	2	1	no	—
68	C₂×C₄	—	2	1	no	—
69	C₂×C₄	—	2	2	no	—
70	C₂×C₄	—	2	2	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	2	1	no	—
73	C₂×C₄	—	2	2	no	—
74	C₂×C₄	—	2	2	no	—
75	C₂×C₄	—	2	2	no	—
76	C₂×C₄	C₂×C₄×C₄₀₉	1	4	yes	—
77	C₂×C₄	C₂×C₄×C₄₀₉	1	2	yes	—
78	C₂×C₄	D₄×C₄₀₉	1	2	yes	—
79	C₂×C₄	D₄×C₄₀₉	1	2	yes	—
80	C₂×C₄	Q₈×C₄₀₉	1	2	yes	—
81	C₂×C₄	Q₈×C₄₀₉	1	2	yes	—
82	C₂²	C₂²×C₄×C₄₀₉	1	1	yes	characteristic
83	C₂²	C₂×D₄×C₄₀₉	1	1	yes	characteristic
84	C₂²	C₂×D₄×C₄₀₉	1	1	yes	characteristic
85	C₂²	C₂×Q₈×C₄₀₉	1	1	yes	characteristic
86	C₂²	C₂×Q₈×C₄₀₉	1	1	yes	characteristic
87	C₂²	#41×C₄₀₉	1	1	yes	characteristic
88	C₂²	#41×C₄₀₉	1	1	yes	characteristic
89	C₄	—	2	4	no	—
90	C₄	—	2	2	no	—
91	C₄	—	2	2	no	—
92	C₄	—	4	2	no	—
93	C₄	C₄⋊C₄×C₄₀₉	1	4	yes	complemented
94	C₂	C₂×C₄⋊C₄×C₄₀₉	1	1	yes	characteristic
95	C₂	C₄×D₄×C₄₀₉	1	1	yes	characteristic
96	C₂	C₄×Q₈×C₄₀₉	1	1	yes	characteristic
97	C₂	#121×C₄₀₉	1	1	yes	characteristic
98	C₂	C₂²⋊Q₈×C₄₀₉	1	1	yes	characteristic
99	C₂	#125×C₄₀₉	1	1	yes	characteristic
100	C₂	C₄⋊Q₈×C₄₀₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊C₄×C₄₀₉

C₂

yes

maximal, characteristic

C₂×C₄⋊C₄×C₄₀₉

C₂

yes

maximal

C₂×C₄⋊C₄×C₄₀₉

C₂

yes

maximal, characteristic

C₂×C₄²×C₄₀₉

C₂

yes

maximal, characteristic

#95×C₄₀₉

C₂

yes

maximal

C₂²×C₄×C₄₀₉

C₂²

yes

—

C₂²×C₄×C₄₀₉

C₂²

yes

—

C₂²×C₄×C₄₀₉

C₂²

yes

characteristic

C₂²×C₄×C₄₀₉

C₂²

yes

characteristic

C₂²×C₄×C₄₀₉

C₂²

yes

characteristic

C₄⋊C₄×C₄₀₉

—

C₄⋊C₄×C₄₀₉

—

C₄⋊C₄×C₄₀₉

C₄

yes

complemented

C₄²×C₄₀₉

—

C₂³×C₄₀₉

C₂³

yes

characteristic

C₂×C₄×C₄₀₉

—

C₂×C₄×C₄₀₉

—

C₂×C₄×C₄₀₉

—

C₂×C₄×C₄₀₉

—

C₂×C₄×C₄₀₉

—

C₂×C₄×C₄₀₉

—

C₂×C₄×C₄₀₉

—

C₂×C₄×C₄₀₉

—

C₂×C₄×C₄₀₉

—

C₂×C₄×C₄₀₉

C₂×C₄

yes

—

C₂×C₄×C₄₀₉

C₂×C₄

yes

—

C₂×C₄×C₄₀₉

D₄

yes

—

C₂×C₄×C₄₀₉

D₄

yes

—

C₂×C₄×C₄₀₉

Q₈

yes

—

C₂×C₄×C₄₀₉

Q₈

yes

—

C₂²×C₄₀₉

C₂²×C₄

yes

characteristic

C₂²×C₄₀₉

C₂×D₄

yes

characteristic

C₂²×C₄₀₉

C₂×D₄

yes

characteristic

C₂²×C₄₀₉

C₂×Q₈

yes

characteristic

C₂²×C₄₀₉

C₂×Q₈

yes

C₂²×C₄₀₉

yes

C₂²×C₄₀₉

yes

C₄×C₄₀₉

—

C₄×C₄₀₉

—

C₄×C₄₀₉

—

C₄×C₄₀₉

—

C₄×C₄₀₉

C₄⋊C₄

yes

complemented

C₂×C₄₀₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄₀₉

C₄×D₄

yes

characteristic

C₂×C₄₀₉

C₄×Q₈

yes

characteristic

C₂×C₄₀₉

#121

yes

characteristic

C₂×C₄₀₉

C₂²⋊Q₈

yes

characteristic

C₂×C₄₀₉

#125

yes

characteristic

C₂×C₄₀₉

C₄⋊Q₈

yes

characteristic

C₄₀₉

#389

yes

characteristic, direct product factor, Sylow

#389

C₄₀₉

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₄⋊C₄

C₂×C₄₀₉

yes

characteristic

C₂×C₄⋊C₄

C₂×C₄₀₉

yes

—

C₂×C₄⋊C₄

C₂×C₄₀₉

yes

characteristic

C₂×C₄²

C₂×C₄₀₉

yes

characteristic

#95

C₂×C₄₀₉

yes

—

C₂²×C₄

C₂²×C₄₀₉

yes

—

C₂²×C₄

C₂²×C₄₀₉

yes

—

C₂²×C₄

C₂²×C₄₀₉

yes

characteristic

C₂²×C₄

C₂²×C₄₀₉

yes

characteristic

C₂²×C₄

C₂²×C₄₀₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₄×C₄₀₉

yes

complemented

C₄²

—

C₂³

C₂³×C₄₀₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₄×C₄₀₉

yes

—

C₂×C₄

C₂×C₄×C₄₀₉

yes

—

C₂×C₄

D₄×C₄₀₉

yes

—

C₂×C₄

D₄×C₄₀₉

yes

—

C₂×C₄

Q₈×C₄₀₉

yes

—

C₂×C₄

Q₈×C₄₀₉

yes

—

C₂²

C₂²×C₄×C₄₀₉

yes

characteristic

C₂²

C₂×D₄×C₄₀₉

yes

characteristic

C₂²

C₂×D₄×C₄₀₉

yes

characteristic

C₂²

C₂×Q₈×C₄₀₉

yes

characteristic

C₂²

C₂×Q₈×C₄₀₉

yes

characteristic

C₂²

#41×C₄₀₉

yes

characteristic

C₂²

#41×C₄₀₉

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

C₄

C₄⋊C₄×C₄₀₉

yes

complemented

C₂

C₂×C₄⋊C₄×C₄₀₉

yes

characteristic

C₂

C₄×D₄×C₄₀₉

yes

characteristic

C₂

C₄×Q₈×C₄₀₉

yes

characteristic

C₂

#121×C₄₀₉

yes

characteristic

C₂

C₂²⋊Q₈×C₄₀₉

yes

characteristic

C₂

#125×C₄₀₉

yes

characteristic

100

C₂

C₄⋊Q₈×C₄₀₉

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₄₀₉	C₂³
commutator	C₂²	C₂²×C₄×C₄₀₉
Frattini, cosocle	C₂³	C₂³×C₄₀₉
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄₀₉ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₄₀₉

C₂³

commutator

C₂²

C₂²×C₄×C₄₀₉

Frattini, cosocle

C₂³

C₂³×C₄₀₉

LCS

G ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³×C₄₀₉ ◁ G

N	H ≅ G/N	type
13. C₄⋊C₄×C₄₀₉	C₄	semidirect
42. C₄×C₄₀₉	C₄⋊C₄	semidirect
50. C₄₀₉	#389	direct (N × H)
51. #389	C₄₀₉	direct (N × H)
64. C₄⋊C₄	C₄×C₄₀₉	semidirect
93. C₄	C₄⋊C₄×C₄₀₉	semidirect

H ≅ G/N

type

13. C₄⋊C₄×C₄₀₉

C₄

semidirect

42. C₄×C₄₀₉

C₄⋊C₄

semidirect

50. C₄₀₉

#389

direct (N × H)

51. #389

C₄₀₉

direct (N × H)

64. C₄⋊C₄

C₄×C₄₀₉

semidirect

93. C₄

C₄⋊C₄×C₄₀₉

semidirect

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

subgroup	index
C₂×C₄⋊C₄×C₄₀₉	2
C₂×C₄²×C₄₀₉	2
#95×C₄₀₉	2
#389	409

subgroup

index

C₂×C₄⋊C₄×C₄₀₉

C₂×C₄²×C₄₀₉

#95×C₄₀₉

#389

409

Subgroups — #389×C₄₀₉ #39908768

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #389×C409 #39908768

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #389×C₄₀₉ #39908768