Group #44483688 — subgroups

Subgroups — #44483688

Order 30592.

206 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 65 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	#29505151	C₂	1	1	yes	maximal, characteristic
2	#29505067	C₂	1	2	yes	maximal
3	#29504521	C₂	1	1	yes	maximal, characteristic
4	Q₈⋊(C₂₃₉⋊C₈)	C₂	1	2	yes	maximal
5	Q₈²×C₂₃₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#16838538	—	2	2	no	—
7	#21439960	C₂²	1	1	yes	characteristic
8	#21439893	—	2	2	no	—
9	C₄⋊(C₂₃₉⋊C₈)	C₂²	1	2	yes	—
10	#21439884	—	2	2	no	—
11	C₄×C₂₃₉⋊C₈	C₂²	1	1	yes	characteristic
12	C₄⋊Q₈×C₂₃₉	—	2	2	no	—
13	C₄⋊Q₈×C₂₃₉	—	2	2	no	—
14	C₄⋊Q₈×C₂₃₉	C₂²	1	1	yes	characteristic
15	C₄×Q₈×C₂₃₉	—	2	2	no	—
16	C₄×Q₈×C₂₃₉	C₂²	1	2	yes	—
17	C₂×Dic₄₇₈	—	2	1	no	—
18	#16838538	—	4	4	no	—
19	C₂×C₂₃₉⋊C₈	—	2	1	no	—
20	C₂×C₂₃₉⋊C₈	—	2	2	no	—
21	C₄⋊Dic₂₃₉	—	2	2	no	—
22	C₂×Q₈×C₂₃₉	—	2	2	no	—
23	C₂×Q₈×C₂₃₉	D₄	1	2	yes	—
24	C₂×Q₈×C₂₃₉	D₄	1	2	yes	—
25	C₄⋊C₄×C₂₃₉	—	2	4	no	—
26	C₄⋊C₄×C₂₃₉	—	2	2	no	—
27	C₄⋊C₄×C₂₃₉	—	2	2	no	—
28	C₄⋊C₄×C₂₃₉	D₄	1	2	yes	—
29	C₄²×C₂₃₉	—	2	2	no	—
30	C₄²×C₂₃₉	—	2	2	no	—
31	C₄²×C₂₃₉	C₂³	1	1	yes	characteristic
32	Dic₄₇₈	—	4	2	no	—
33	C₂×Dic₂₃₉	—	4	1	no	—
34	C₂₃₉⋊C₈	—	2	2	no	—
35	C₂₃₉⋊C₈	—	4	2	no	—
36	Q₈×C₂₃₉	—	2	2	no	—
37	Q₈×C₂₃₉	—	4	2	no	—
38	Q₈×C₂₃₉	—	8	2	no	—
39	Q₈×C₂₃₉	Q₁₆	1	4	yes	complemented
40	C₂×C₄×C₂₃₉	—	2	2	no	—
41	C₂×C₄×C₂₃₉	—	2	2	no	—
42	C₂×C₄×C₂₃₉	—	2	2	no	—
43	C₂×C₄×C₂₃₉	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₂₃₉	C₂×D₄	1	2	yes	—
45	C₂²×C₂₃₉	#120	1	1	yes	characteristic
46	C₄×C₂₃₉	—	2	4	no	—
47	C₄×C₂₃₉	—	2	1	no	—
48	C₄×C₂₃₉	—	4	2	no	—
49	C₄×C₂₃₉	—	4	2	no	—
50	C₄×C₂₃₉	C₂×Q₁₆	1	2	yes	—
51	C₄×C₂₃₉	#137	1	2	yes	—
52	Dic₂₃₉	—	8	1	no	—
53	C₂×C₂₃₉	#451	1	2	yes	—
54	C₂×C₂₃₉	#453	1	1	yes	characteristic
55	C₂₃₉	#1633	1	1	yes	characteristic, complemented, Sylow
56	#1633	—	239	1	no	maximal, Sylow
57	Q₈²	D₂₃₉	1	1	yes	characteristic
58	#494	—	239	1	no	—
59	#477	—	239	2	no	—
60	#333	—	239	1	no	—
61	Q₈⋊C₈	—	239	2	no	—
62	C₂×Q₁₆	—	478	2	no	—
63	C₄⋊Q₈	—	2	2	no	—
64	C₄⋊Q₈	—	2	2	no	—
65	C₄⋊Q₈	—	239	1	no	—
66	C₄⋊Q₈	C₂×D₂₃₉	1	1	yes	characteristic
67	C₄×Q₈	—	2	2	no	—
68	C₄×Q₈	C₂×D₂₃₉	1	2	yes	—
69	#107	—	478	2	no	—
70	C₄⋊C₈	—	239	2	no	—
71	#103	—	478	2	no	—
72	C₄×C₈	—	239	1	no	—
73	C₂×Q₈	—	2	2	no	—
74	C₂×Q₈	—	478	1	no	—
75	C₂×Q₈	C₂²⋊D₂₃₉	1	2	yes	—
76	C₂×Q₈	C₂²⋊D₂₃₉	1	2	yes	—
77	Q₁₆	—	956	4	no	—
78	C₂×C₈	—	478	2	no	—
79	C₂×C₈	—	478	1	no	—
80	C₄⋊C₄	—	2	4	no	—
81	C₄⋊C₄	—	2	2	no	—
82	C₄⋊C₄	—	2	2	no	—
83	C₄⋊C₄	—	478	2	no	—
84	C₄⋊C₄	C₂²⋊D₂₃₉	1	2	yes	—
85	C₄²	—	2	2	no	—
86	C₄²	—	2	2	no	—
87	C₄²	C₂²×D₂₃₉	1	1	yes	characteristic
88	Q₈	—	2	2	no	—
89	Q₈	—	4	2	no	—
90	Q₈	—	8	2	no	—
91	Q₈	—	956	2	no	—
92	Q₈	#16838538	1	4	yes	complemented
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	956	1	no	—
97	C₂×C₄	C₂×C₂²⋊D₂₃₉	1	1	yes	characteristic
98	C₂×C₄	C₂×C₂²⋊D₂₃₉	1	2	yes	—
99	C₈	—	956	2	no	—
100	C₈	—	478	2	no	—
101	C₂²	#21439929	1	1	yes	characteristic
102	C₄	—	2	4	no	—
103	C₄	—	2	1	no	—
104	C₄	—	4	2	no	—
105	C₄	—	4	2	no	—
106	C₄	—	1912	1	no	—
107	C₄	C₂×#16838538	1	2	yes	—
108	C₄	#21439991	1	2	yes	—
109	C₂	#29504928	1	2	yes	—
110	C₂	#29504940	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#29505151

C₂

yes

maximal, characteristic

#29505067

C₂

yes

maximal

#29504521

C₂

yes

maximal, characteristic

Q₈⋊(C₂₃₉⋊C₈)

C₂

yes

maximal

Q₈²×C₂₃₉

C₂

yes

maximal, characteristic

C₂×#16838538

—

#21439960

C₂²

yes

characteristic

#21439893

—

C₄⋊(C₂₃₉⋊C₈)

C₂²

yes

—

#21439884

—

C₄×C₂₃₉⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₂₃₉

—

C₄⋊Q₈×C₂₃₉

—

C₄⋊Q₈×C₂₃₉

C₂²

yes

characteristic

C₄×Q₈×C₂₃₉

—

C₄×Q₈×C₂₃₉

C₂²

yes

—

C₂×Dic₄₇₈

—

#16838538

—

C₂×C₂₃₉⋊C₈

—

C₂×C₂₃₉⋊C₈

—

C₄⋊Dic₂₃₉

—

C₂×Q₈×C₂₃₉

—

C₂×Q₈×C₂₃₉

D₄

yes

—

C₂×Q₈×C₂₃₉

D₄

yes

—

C₄⋊C₄×C₂₃₉

—

C₄⋊C₄×C₂₃₉

—

C₄⋊C₄×C₂₃₉

—

C₄⋊C₄×C₂₃₉

D₄

yes

—

C₄²×C₂₃₉

—

C₄²×C₂₃₉

—

C₄²×C₂₃₉

C₂³

yes

characteristic

Dic₄₇₈

—

C₂×Dic₂₃₉

—

C₂₃₉⋊C₈

—

C₂₃₉⋊C₈

—

Q₈×C₂₃₉

—

Q₈×C₂₃₉

—

Q₈×C₂₃₉

—

Q₈×C₂₃₉

Q₁₆

yes

complemented

C₂×C₄×C₂₃₉

—

C₂×C₄×C₂₃₉

—

C₂×C₄×C₂₃₉

—

C₂×C₄×C₂₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃₉

C₂×D₄

yes

—

C₂²×C₂₃₉

#120

yes

characteristic

C₄×C₂₃₉

—

C₄×C₂₃₉

—

C₄×C₂₃₉

—

C₄×C₂₃₉

—

C₄×C₂₃₉

C₂×Q₁₆

yes

—

C₄×C₂₃₉

#137

yes

—

Dic₂₃₉

—

C₂×C₂₃₉

#451

yes

—

C₂×C₂₃₉

#453

yes

characteristic

C₂₃₉

#1633

yes

characteristic, complemented, Sylow

#1633

—

239

maximal, Sylow

Q₈²

D₂₃₉

yes

characteristic

#494

—

239

—

#477

—

239

—

#333

—

239

—

Q₈⋊C₈

—

239

—

C₂×Q₁₆

—

478

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

239

—

C₄⋊Q₈

C₂×D₂₃₉

yes

characteristic

C₄×Q₈

—

C₄×Q₈

C₂×D₂₃₉

yes

—

#107

—

478

—

C₄⋊C₈

—

239

—

#103

—

478

—

C₄×C₈

—

239

—

C₂×Q₈

—

C₂×Q₈

—

478

—

C₂×Q₈

C₂²⋊D₂₃₉

yes

—

C₂×Q₈

C₂²⋊D₂₃₉

yes

—

Q₁₆

—

956

—

C₂×C₈

—

478

—

C₂×C₈

—

478

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

478

—

C₄⋊C₄

C₂²⋊D₂₃₉

yes

—

C₄²

—

C₄²

—

C₄²

C₂²×D₂₃₉

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

Q₈

—

956

—

Q₈

#16838538

yes

complemented

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

956

—

C₂×C₄

C₂×C₂²⋊D₂₃₉

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₂₃₉

yes

—

C₈

—

956

—

100

C₈

—

478

—

101

C₂²

#21439929

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

—

106

C₄

—

1912

—

107

C₄

C₂×#16838538

yes

—

108

C₄

#21439991

yes

—

109

C₂

#29504928

yes

—

110

C₂

#29504940

yes

characteristic

	subgroup	G/H
center	C₂²	#21439929
commutator, cosocle	C₄²×C₂₃₉	C₂³
Frattini	C₄²	C₂²×D₂₃₉
Fitting	Q₈²×C₂₃₉	C₂
socle, prime radical	C₂²×C₂₃₉	#120
hypercenter	Q₈²	D₂₃₉
hypocenter	C₂₃₉	#1633
LCS	G ▷ C₄²×C₂₃₉ ▷ C₂²×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ Q₈²

subgroup

G/H

center

C₂²

#21439929

commutator, cosocle

C₄²×C₂₃₉

C₂³

Frattini

C₄²

C₂²×D₂₃₉

Fitting

Q₈²×C₂₃₉

C₂

socle, prime radical

C₂²×C₂₃₉

#120

hypercenter

Q₈²

D₂₃₉

hypocenter

C₂₃₉

#1633

LCS

G ▷ C₄²×C₂₃₉ ▷ C₂²×C₂₃₉ ▷ C₂₃₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ Q₈²

N	H ≅ G/N	type
39. Q₈×C₂₃₉	Q₁₆	semidirect
55. C₂₃₉	#1633	semidirect
92. Q₈	#16838538	semidirect

H ≅ G/N

type

39. Q₈×C₂₃₉

Q₁₆

semidirect

55. C₂₃₉

#1633

semidirect

92. Q₈

#16838538

semidirect

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

subgroup	index
#29505151	2
#29505067	2
#29504521	2
Q₈⋊(C₂₃₉⋊C₈)	2
Q₈²×C₂₃₉	2
#1633	239

subgroup

index

#29505151

#29505067

#29504521

Q₈⋊(C₂₃₉⋊C₈)

Q₈²×C₂₃₉

#1633

239

Subgroups — #44483688

Conjugacy classes of subgroups

Complements

Maximal subgroups