Group #47983323 — subgroups

Subgroups — #47983323

Order 35456.

280 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 149 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×#22579811	C₂	1	4	yes	maximal
2	C₂×C₄×#14555660	C₂	1	1	yes	maximal, characteristic
3	#31799872	C₂	1	1	yes	maximal, characteristic
4	#384×C₂₇₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#14555660	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₂₇₇	—	2	2	no	—
7	C₂×C₄⋊Dic₂₇₇	C₂²	1	2	yes	—
8	#22579811	—	2	8	no	—
9	C₄×#14555660	C₄	1	4	yes	complemented
10	C₂×C₄⋊C₄×C₂₇₇	—	2	2	no	—
11	C₂×C₄⋊C₄×C₂₇₇	C₂²	1	2	yes	—
12	C₂×C₄²×C₂₇₇	C₂²	1	1	yes	characteristic
13	C₂²×Dic₂₇₇	—	2	2	no	—
14	C₂×#14555660	C₂×C₄	1	4	yes	—
15	C₂×#14555660	D₄	1	4	yes	—
16	C₂×#14555660	Q₈	1	4	yes	—
17	C₄⋊Dic₂₇₇	—	2	4	no	—
18	C₄⋊Dic₂₇₇	—	4	4	no	—
19	C₂²×C₄×C₂₇₇	—	2	2	no	—
20	C₂²×C₄×C₂₇₇	D₄	1	2	yes	—
21	C₂²×C₄×C₂₇₇	C₂³	1	1	yes	characteristic
22	C₄⋊C₄×C₂₇₇	—	2	4	no	—
23	C₄⋊C₄×C₂₇₇	—	4	4	no	—
24	C₄²×C₂₇₇	C₂×C₄	1	2	yes	—
25	C₄²×C₂₇₇	Q₈	1	2	yes	—
26	C₂×Dic₂₇₇	—	4	2	no	—
27	C₂×Dic₂₇₇	—	4	4	no	—
28	#14555660	C₄⋊C₄	1	8	yes	complemented
29	C₂³×C₂₇₇	C₂×D₄	1	1	yes	characteristic
30	C₂×C₄×C₂₇₇	—	4	2	no	—
31	C₂×C₄×C₂₇₇	—	4	4	no	—
32	C₂×C₄×C₂₇₇	C₄⋊C₄	1	4	yes	—
33	C₂×C₄×C₂₇₇	SD₁₆	1	4	yes	—
34	C₂×C₄×C₂₇₇	SD₁₆	1	4	yes	—
35	C₂×C₄×C₂₇₇	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₂₇₇	C₂×D₄	1	2	yes	—
37	C₂×C₄×C₂₇₇	C₂×Q₈	1	2	yes	—
38	C₂×C₄×C₂₇₇	C₂×Q₈	1	1	yes	characteristic
39	C₂²×C₂₇₇	C₂×C₄⋊C₄	1	1	yes	characteristic
40	C₂²×C₂₇₇	#127	1	1	yes	characteristic
41	C₂²×C₂₇₇	C₄⋊Q₈	1	1	yes	characteristic
42	C₂²×C₂₇₇	C₂×SD₁₆	1	2	yes	—
43	C₂²×C₂₇₇	C₂×SD₁₆	1	2	yes	—
44	C₄×C₂₇₇	—	8	2	no	—
45	C₄×C₂₇₇	#106	1	8	yes	complemented
46	C₄×C₂₇₇	C₂×C₄⋊C₄	1	2	yes	—
47	C₄×C₂₇₇	C₄⋊Q₈	1	2	yes	—
48	Dic₂₇₇	—	8	2	no	—
49	C₂×C₂₇₇	#384	1	1	yes	characteristic
50	C₂×C₂₇₇	C₂×#106	1	2	yes	—
51	C₂×C₂₇₇	#492	1	1	yes	characteristic
52	C₂×C₂₇₇	#492	1	1	yes	characteristic
53	C₂×C₂₇₇	#498	1	2	yes	—
54	C₂₇₇	#1844	1	1	yes	characteristic, complemented, Sylow
55	#1844	—	277	1	no	maximal, Sylow
56	C₂×#106	—	277	4	no	—
57	C₂×C₄×C₈	—	277	1	no	—
58	#384	—	277	1	no	—
59	#384	D₂₇₇	1	1	yes	characteristic
60	C₂²×C₈	—	277	2	no	—
61	C₂×C₄⋊C₄	—	2	2	no	—
62	C₂×C₄⋊C₄	—	277	2	no	—
63	C₂×C₄⋊C₄	—	554	2	no	—
64	C₂×C₄⋊C₄	C₂×D₂₇₇	1	2	yes	—
65	C₂×C₄²	C₂×D₂₇₇	1	1	yes	characteristic
66	#106	—	554	8	no	—
67	C₄×C₈	—	277	4	no	—
68	C₂²×C₄	—	2	2	no	—
69	C₂²×C₄	—	554	2	no	—
70	C₂²×C₄	C₂²⋊D₂₇₇	1	2	yes	—
71	C₂²×C₄	C₂²×D₂₇₇	1	1	yes	characteristic
72	C₂×C₈	—	277	4	no	—
73	C₂×C₈	—	277	4	no	—
74	C₂×C₈	—	277	4	no	—
75	C₄⋊C₄	—	2	4	no	—
76	C₄⋊C₄	—	4	4	no	—
77	C₄⋊C₄	—	554	4	no	—
78	C₄⋊C₄	—	1108	4	no	—
79	C₄²	C₄×D₂₇₇	1	2	yes	—
80	C₄²	Dic₅₅₄	1	2	yes	—
81	C₂³	C₂×C₂²⋊D₂₇₇	1	1	yes	characteristic
82	C₂×C₄	—	4	2	no	—
83	C₂×C₄	—	4	4	no	—
84	C₂×C₄	—	1108	2	no	—
85	C₂×C₄	—	1108	4	no	—
86	C₂×C₄	#17984542	1	4	yes	—
87	C₂×C₄	#17984550	1	4	yes	—
88	C₂×C₄	#17984551	1	4	yes	—
89	C₂×C₄	C₂×C₄×D₂₇₇	1	1	yes	characteristic
90	C₂×C₄	D₄×D₂₇₇	1	2	yes	—
91	C₂×C₄	Q₈×D₂₇₇	1	2	yes	—
92	C₂×C₄	C₂×Dic₅₅₄	1	1	yes	characteristic
93	C₈	—	277	8	no	—
94	C₂²	C₂×#17984542	1	1	yes	characteristic
95	C₂²	#22579876	1	1	yes	characteristic
96	C₂²	#22579878	1	1	yes	characteristic
97	C₂²	C₂×#17984550	1	2	yes	—
98	C₂²	C₂×#17984551	1	2	yes	—
99	C₄	—	8	2	no	—
100	C₄	—	2216	2	no	—
101	C₄	#22579811	1	8	yes	complemented
102	C₄	C₄⋊C₄×D₂₇₇	1	2	yes	—
103	C₄	#22579879	1	2	yes	—
104	C₂	#31799871	1	1	yes	characteristic
105	C₂	C₂×#22579811	1	2	yes	—
106	C₂	#31800374	1	1	yes	characteristic
107	C₂	#31800376	1	1	yes	characteristic
108	C₂	#31800398	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22579811

C₂

yes

maximal

C₂×C₄×#14555660

C₂

yes

maximal, characteristic

#31799872

C₂

yes

maximal, characteristic

#384×C₂₇₇

C₂

yes

maximal, characteristic

C₂²×#14555660

C₂²

yes

—

C₂×C₄⋊Dic₂₇₇

—

C₂×C₄⋊Dic₂₇₇

C₂²

yes

—

#22579811

—

C₄×#14555660

C₄

yes

complemented

C₂×C₄⋊C₄×C₂₇₇

—

C₂×C₄⋊C₄×C₂₇₇

C₂²

yes

—

C₂×C₄²×C₂₇₇

C₂²

yes

characteristic

C₂²×Dic₂₇₇

—

C₂×#14555660

C₂×C₄

yes

—

C₂×#14555660

D₄

yes

—

C₂×#14555660

Q₈

yes

—

C₄⋊Dic₂₇₇

—

C₄⋊Dic₂₇₇

—

C₂²×C₄×C₂₇₇

—

C₂²×C₄×C₂₇₇

D₄

yes

—

C₂²×C₄×C₂₇₇

C₂³

yes

characteristic

C₄⋊C₄×C₂₇₇

—

C₄⋊C₄×C₂₇₇

—

C₄²×C₂₇₇

C₂×C₄

yes

—

C₄²×C₂₇₇

Q₈

yes

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

—

#14555660

C₄⋊C₄

yes

complemented

C₂³×C₂₇₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

C₄⋊C₄

yes

—

C₂×C₄×C₂₇₇

SD₁₆

yes

—

C₂×C₄×C₂₇₇

SD₁₆

yes

—

C₂×C₄×C₂₇₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₇₇

C₂×D₄

yes

—

C₂×C₄×C₂₇₇

C₂×Q₈

yes

—

C₂×C₄×C₂₇₇

C₂×Q₈

yes

characteristic

C₂²×C₂₇₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₇₇

#127

yes

characteristic

C₂²×C₂₇₇

C₄⋊Q₈

yes

characteristic

C₂²×C₂₇₇

C₂×SD₁₆

yes

—

C₂²×C₂₇₇

C₂×SD₁₆

yes

—

C₄×C₂₇₇

—

C₄×C₂₇₇

#106

yes

complemented

C₄×C₂₇₇

C₂×C₄⋊C₄

yes

—

C₄×C₂₇₇

C₄⋊Q₈

yes

—

Dic₂₇₇

—

C₂×C₂₇₇

#384

yes

characteristic

C₂×C₂₇₇

C₂×#106

yes

—

C₂×C₂₇₇

yes

C₂×C₂₇₇

yes

C₂×C₂₇₇

yes

—

C₂₇₇

#1844

yes

characteristic, complemented, Sylow

#1844

—

277

maximal, Sylow

C₂×#106

—

277

—

C₂×C₄×C₈

—

277

—

#384

—

277

—

#384

D₂₇₇

yes

characteristic

C₂²×C₈

—

277

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

277

—

C₂×C₄⋊C₄

—

554

—

C₂×C₄⋊C₄

C₂×D₂₇₇

yes

—

C₂×C₄²

C₂×D₂₇₇

yes

characteristic

#106

—

554

—

C₄×C₈

—

277

—

C₂²×C₄

—

C₂²×C₄

—

554

—

C₂²×C₄

C₂²⋊D₂₇₇

yes

—

C₂²×C₄

C₂²×D₂₇₇

yes

characteristic

C₂×C₈

—

277

—

C₂×C₈

—

277

—

C₂×C₈

—

277

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

554

—

C₄⋊C₄

—

1108

—

C₄²

C₄×D₂₇₇

yes

—

C₄²

Dic₅₅₄

yes

—

C₂³

C₂×C₂²⋊D₂₇₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

1108

—

C₂×C₄

—

1108

—

C₂×C₄

#17984542

yes

—

C₂×C₄

#17984550

yes

—

C₂×C₄

#17984551

yes

—

C₂×C₄

C₂×C₄×D₂₇₇

yes

characteristic

C₂×C₄

D₄×D₂₇₇

yes

—

C₂×C₄

Q₈×D₂₇₇

yes

—

C₂×C₄

C₂×Dic₅₅₄

yes

characteristic

C₈

—

277

—

C₂²

yes

C₂²

yes

C₂²

yes

C₂²

yes

—

C₂²

C₂×#17984551

yes

—

C₄

—

100

C₄

—

2216

—

101

C₄

#22579811

yes

complemented

102

C₄

C₄⋊C₄×D₂₇₇

yes

—

103

C₄

#22579879

yes

—

104

C₂

#31799871

yes

characteristic

105

C₂

C₂×#22579811

yes

—

106

C₂

#31800374

yes

characteristic

107

C₂

#31800376

yes

characteristic

108

C₂

#31800398

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₂₇₇
commutator	C₂×C₄×C₂₇₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₇₇
Fitting	#384×C₂₇₇	C₂
socle, prime radical	C₂³×C₂₇₇	C₂×D₄
cosocle	C₂²×C₄×C₂₇₇	C₂³
hypercenter	#384	D₂₇₇
hypocenter	C₂₇₇	#1844
LCS	G ▷ C₂×C₄×C₂₇₇ ▷ C₂×C₂₇₇ ▷ C₂₇₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₂₇₇

commutator

C₂×C₄×C₂₇₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₇₇

Fitting

#384×C₂₇₇

C₂

socle, prime radical

C₂³×C₂₇₇

C₂×D₄

cosocle

C₂²×C₄×C₂₇₇

C₂³

D₂₇₇

C₂₇₇

G ▷ C₂×C₄×C₂₇₇ ▷ C₂×C₂₇₇ ▷ C₂₇₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
9. C₄×#14555660	C₄	semidirect
28. #14555660	C₄⋊C₄	semidirect
45. C₄×C₂₇₇	#106	semidirect
54. C₂₇₇	#1844	semidirect
101. C₄	#22579811	semidirect

H ≅ G/N

type

9. C₄×#14555660

C₄

semidirect

28. #14555660

C₄⋊C₄

semidirect

45. C₄×C₂₇₇

#106

semidirect

54. C₂₇₇

#1844

semidirect

101. C₄

#22579811

semidirect

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

subgroup	index
C₂×#22579811	2
C₂×C₄×#14555660	2
#31799872	2
#384×C₂₇₇	2
#1844	277

subgroup

index

#384×C₂₇₇

277

Subgroups — #47983323

Conjugacy classes of subgroups

Complements

Maximal subgroups