Group #44484176 — subgroups

Subgroups — #44484176

Order 30592.

220 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 69 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	#29505170	C₂	1	1	yes	maximal, characteristic, complemented
2	#29505149	C₂	1	1	yes	maximal, characteristic
3	#29505145	C₂	1	2	yes	maximal, complemented
4	#29505119	C₂	1	2	yes	maximal, complemented
5	#322×C₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16838537	—	2	4	no	—
7	C₂×D₁₉₁₂	—	2	2	no	—
8	#21439960	C₂²	1	2	yes	—
9	#21439957	C₂²	1	1	yes	characteristic
10	#21439957	C₂²	1	1	yes	characteristic
11	#21439892	—	2	2	no	—
12	#21439882	—	2	4	no	—
13	C₄×C₈×C₂₃₉	C₂²	1	2	yes	—
14	C₄×C₈×C₂₃₉	C₂²	1	1	yes	characteristic
15	C₂×Dic₄₇₈	—	2	2	no	—
16	C₂×D₉₅₆	—	2	2	no	—
17	C₂×D₉₅₆	—	2	1	no	—
18	C₂×D₉₅₆	—	2	1	no	—
19	C₂×D₉₅₆	—	2	2	no	—
20	#16838537	—	4	8	no	—
21	D₁₉₁₂	—	4	4	no	—
22	C₄⋊Dic₂₃₉	—	2	2	no	—
23	C₄⋊Dic₂₃₉	—	2	2	no	—
24	C₂×C₈×C₂₃₉	D₄	1	4	yes	—
25	C₂×C₈×C₂₃₉	D₄	1	2	yes	—
26	C₄²×C₂₃₉	C₂³	1	1	yes	characteristic
27	C₂²×D₂₃₉	—	4	1	no	—
28	C₂²×D₂₃₉	—	4	1	no	—
29	Dic₄₇₈	—	4	4	no	—
30	D₉₅₆	—	4	2	no	—
31	D₉₅₆	—	4	2	no	—
32	D₉₅₆	—	4	4	no	—
33	D₉₅₆	—	4	4	no	—
34	C₂×Dic₂₃₉	—	4	2	no	—
35	C₂×C₄×C₂₃₉	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₂₃₉	C₂×D₄	1	2	yes	—
37	C₈×C₂₃₉	—	2	4	no	—
38	C₈×C₂₃₉	SD₁₆	1	4	yes	complemented
39	C₂²×C₂₃₉	#127	1	1	yes	characteristic
40	C₂×D₂₃₉	—	8	1	no	—
41	C₂×D₂₃₉	—	8	1	no	—
42	C₂×D₂₃₉	—	8	2	no	—
43	C₂×D₂₃₉	—	8	2	no	—
44	C₄×C₂₃₉	C₂×SD₁₆	1	2	yes	—
45	C₄×C₂₃₉	AGL(1,ℤ/8)	1	4	yes	—
46	Dic₂₃₉	—	8	2	no	—
47	C₂×C₂₃₉	#492	1	1	yes	characteristic
48	C₂×C₂₃₉	#496	1	2	yes	—
49	D₂₃₉	—	16	1	no	—
50	D₂₃₉	—	16	1	no	—
51	C₂₃₉	#1715	1	1	yes	characteristic, complemented, Sylow
52	#1715	—	239	1	no	maximal, Sylow
53	#498	—	239	1	no	—
54	#493	—	239	1	no	—
55	#492	—	239	2	no	—
56	#486	—	239	2	no	—
57	#322	D₂₃₉	1	1	yes	characteristic, complemented
58	C₂×SD₁₆	—	478	4	no	—
59	C₂×D₈	—	478	2	no	—
60	C₄⋊Q₈	—	239	2	no	—
61	#127	—	239	1	no	—
62	#127	—	239	1	no	—
63	#106	—	478	2	no	—
64	#102	—	478	4	no	—
65	C₄×C₈	C₂×D₂₃₉	1	2	yes	—
66	C₄×C₈	C₂×D₂₃₉	1	1	yes	characteristic
67	C₂×Q₈	—	478	2	no	—
68	C₂×D₄	—	478	2	no	—
69	C₂×D₄	—	478	1	no	—
70	C₂×D₄	—	478	1	no	—
71	C₂×D₄	—	478	2	no	—
72	SD₁₆	—	956	8	no	—
73	D₈	—	956	4	no	—
74	C₂×C₈	D₉₅₆	1	4	yes	—
75	C₂×C₈	D₉₅₆	1	2	yes	—
76	C₄⋊C₄	—	478	2	no	—
77	C₄⋊C₄	—	478	2	no	—
78	C₄²	C₂²×D₂₃₉	1	1	yes	characteristic
79	C₂³	—	956	1	no	—
80	C₂³	—	956	1	no	—
81	Q₈	—	956	4	no	—
82	D₄	—	956	2	no	—
83	D₄	—	956	2	no	—
84	D₄	—	956	4	no	—
85	D₄	—	956	4	no	—
86	C₂×C₄	—	956	2	no	—
87	C₂×C₄	C₂×D₉₅₆	1	1	yes	characteristic
88	C₂×C₄	C₂×D₉₅₆	1	2	yes	—
89	C₈	—	2	4	no	—
90	C₈	#16838537	1	4	yes	complemented
91	C₂²	—	1912	1	no	—
92	C₂²	—	1912	1	no	—
93	C₂²	—	1912	2	no	—
94	C₂²	—	1912	2	no	—
95	C₂²	#21439957	1	1	yes	characteristic
96	C₄	—	1912	2	no	—
97	C₄	C₂×#16838537	1	2	yes	—
98	C₄	#21439988	1	4	yes	—
99	C₂	—	3824	1	no	—
100	C₂	—	3824	1	no	—
101	C₂	#29505145	1	1	yes	characteristic
102	C₂	#29505161	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#29505170

C₂

yes

maximal, characteristic, complemented

#29505149

C₂

yes

maximal, characteristic

#29505145

C₂

yes

maximal, complemented

#29505119

C₂

yes

maximal, complemented

#322×C₂₃₉

C₂

yes

maximal, characteristic, complemented

C₂×#16838537

—

C₂×D₁₉₁₂

—

#21439960

C₂²

yes

—

#21439957

C₂²

yes

characteristic

#21439957

C₂²

yes

characteristic

#21439892

—

#21439882

—

C₄×C₈×C₂₃₉

C₂²

yes

—

C₄×C₈×C₂₃₉

C₂²

yes

characteristic

C₂×Dic₄₇₈

—

C₂×D₉₅₆

—

C₂×D₉₅₆

—

C₂×D₉₅₆

—

C₂×D₉₅₆

—

#16838537

—

D₁₉₁₂

—

C₄⋊Dic₂₃₉

—

C₄⋊Dic₂₃₉

—

C₂×C₈×C₂₃₉

D₄

yes

—

C₂×C₈×C₂₃₉

D₄

yes

—

C₄²×C₂₃₉

C₂³

yes

characteristic

C₂²×D₂₃₉

—

C₂²×D₂₃₉

—

Dic₄₇₈

—

D₉₅₆

—

D₉₅₆

—

D₉₅₆

—

D₉₅₆

—

C₂×Dic₂₃₉

—

C₂×C₄×C₂₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃₉

C₂×D₄

yes

—

C₈×C₂₃₉

—

C₈×C₂₃₉

SD₁₆

yes

complemented

C₂²×C₂₃₉

#127

yes

characteristic

C₂×D₂₃₉

—

C₂×D₂₃₉

—

C₂×D₂₃₉

—

C₂×D₂₃₉

—

C₄×C₂₃₉

C₂×SD₁₆

yes

—

C₄×C₂₃₉

AGL(1,ℤ/8)

yes

—

Dic₂₃₉

—

C₂×C₂₃₉

#492

yes

characteristic

C₂×C₂₃₉

#496

yes

—

D₂₃₉

—

D₂₃₉

—

C₂₃₉

#1715

yes

characteristic, complemented, Sylow

#1715

—

239

maximal, Sylow

#498

—

239

—

#493

—

239

—

#492

—

239

—

#486

—

239

—

#322

D₂₃₉

yes

characteristic, complemented

C₂×SD₁₆

—

478

—

C₂×D₈

—

478

—

C₄⋊Q₈

—

239

—

#127

—

239

—

#127

—

239

—

#106

—

478

—

#102

—

478

—

C₄×C₈

C₂×D₂₃₉

yes

—

C₄×C₈

C₂×D₂₃₉

yes

characteristic

C₂×Q₈

—

478

—

C₂×D₄

—

478

—

C₂×D₄

—

478

—

C₂×D₄

—

478

—

C₂×D₄

—

478

—

SD₁₆

—

956

—

D₈

—

956

—

C₂×C₈

D₉₅₆

yes

—

C₂×C₈

D₉₅₆

yes

—

C₄⋊C₄

—

478

—

C₄⋊C₄

—

478

—

C₄²

C₂²×D₂₃₉

yes

characteristic

C₂³

—

956

—

C₂³

—

956

—

Q₈

—

956

—

D₄

—

956

—

D₄

—

956

—

D₄

—

956

—

D₄

—

956

—

C₂×C₄

—

956

—

C₂×C₄

C₂×D₉₅₆

yes

characteristic

C₂×C₄

C₂×D₉₅₆

yes

—

C₈

—

C₈

#16838537

yes

complemented

C₂²

—

1912

—

C₂²

—

1912

—

C₂²

—

1912

—

C₂²

—

1912

—

C₂²

#21439957

yes

characteristic

C₄

—

1912

—

C₄

C₂×#16838537

yes

—

C₄

#21439988

yes

—

C₂

—

3824

—

100

C₂

—

3824

—

101

C₂

#29505145

yes

characteristic

102

C₂

#29505161

yes

—

	subgroup	G/H
center	C₂²	#21439957
commutator, cosocle	C₄²×C₂₃₉	C₂³
Frattini	C₄²	C₂²×D₂₃₉
Fitting	#322×C₂₃₉	C₂
socle	C₂²×C₂₃₉	#127
prime radical	#29505149	C₂
hypercenter	#322	D₂₃₉
hypocenter	C₂₃₉	#1715
LCS	G ▷ C₄²×C₂₃₉ ▷ C₂²×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322

subgroup

G/H

center

C₂²

#21439957

commutator, cosocle

C₄²×C₂₃₉

C₂³

Frattini

C₄²

C₂²×D₂₃₉

Fitting

#322×C₂₃₉

C₂

C₂²×C₂₃₉

C₂

D₂₃₉

C₂₃₉

G ▷ C₄²×C₂₃₉ ▷ C₂²×C₂₃₉ ▷ C₂₃₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #322

N	H ≅ G/N	type
1. #29505170	C₂	semidirect
3. #29505145	C₂	semidirect
4. #29505119	C₂	semidirect
5. #322×C₂₃₉	C₂	semidirect
38. C₈×C₂₃₉	SD₁₆	semidirect
51. C₂₃₉	#1715	semidirect
57. #322	D₂₃₉	semidirect
90. C₈	#16838537	semidirect

H ≅ G/N

type

1. #29505170

C₂

semidirect

3. #29505145

C₂

semidirect

4. #29505119

C₂

semidirect

5. #322×C₂₃₉

C₂

semidirect

38. C₈×C₂₃₉

SD₁₆

semidirect

51. C₂₃₉

#1715

semidirect

57. #322

D₂₃₉

semidirect

90. C₈

#16838537

semidirect

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

subgroup	index
#29505170	2
#29505149	2
#29505145	2
#29505119	2
#322×C₂₃₉	2
#1715	239

subgroup

index

#322×C₂₃₉

239

Subgroups — #44484176

Conjugacy classes of subgroups

Complements

Maximal subgroups