Group #16777424 — subgroups

Subgroups — #16777424

Order 3328.

148 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 148 conjugacy classes of subgroups — too many to draw. Only its 50 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	#13131210	C₂	1	2	yes	maximal
2	#13116594	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₁₃×C₁₆	C₄	1	1	yes	characteristic
4	#11718762	C₂²	1	1	yes	characteristic
5	C₂×#88052	C₄	1	1	yes	characteristic
6	#11718284	—	2	2	no	—
7	#11718262	C₄	1	4	yes	—
8	#11718261	—	2	2	no	—
9	#11718093	—	2	2	no	—
10	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
11	#88180	Q₈	1	1	yes	characteristic
12	C₂×#6099	D₄	1	1	yes	characteristic
13	#88106	—	2	4	no	—
14	C₂×C₁₃×C₁₆	—	2	1	no	—
15	C₂×C₁₃×C₁₆	—	4	1	no	—
16	#88055	C₂×C₄	1	2	yes	—
17	C₈×Dic₁₃	D₄	1	2	yes	—
18	#88052	—	4	1	no	—
19	C₂×#6090	—	2	1	no	—
20	#7240	—	13	1	no	maximal, Sylow
21	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
22	C₂×C₈×C₁₃	—	2	1	no	—
23	C₂×C₈×C₁₃	—	2	1	no	—
24	C₂×C₈×C₁₃	C₄²	1	1	yes	characteristic
25	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
26	C₂²⋊Dic₁₃	—	2	1	no	—
27	C₄⋊Dic₁₃	—	2	1	no	—
28	C₄×Dic₁₃	C₄⋊C₄	1	2	yes	—
29	#6099	—	4	1	no	—
30	C₂×#1055	C₂²⋊C₄	1	2	yes	—
31	#6092	—	4	4	no	—
32	C₁₃×C₁₆	—	4	1	no	—
33	#6090	—	4	1	no	—
34	#2166	—	13	2	no	—
35	#2151	—	13	1	no	—
36	C₂³×C₁₃	#108	1	1	yes	characteristic
37	C₂×C₄×C₁₃	—	2	1	no	—
38	C₂×C₄×C₁₃	—	2	1	no	—
39	C₂×C₄×C₁₃	#95	1	1	yes	characteristic
40	C₂×C₄×C₁₃	#108	1	1	yes	characteristic
41	C₂×Dic₁₃	—	2	2	no	—
42	C₈×C₁₃	—	2	1	no	—
43	C₈×C₁₃	#100	1	1	yes	characteristic
44	C₈×C₁₃	#101	1	1	yes	characteristic
45	#1055	—	2	2	no	—
46	C₂×M₃₂	—	13	1	no	—
47	C₂²×C₁₆	#261	1	1	yes	characteristic
48	#405	—	13	1	no	—
49	#364	—	26	2	no	—
50	C₄⋊C₁₆	—	26	2	no	—
51	#349	—	26	2	no	—
52	#347	—	13	4	no	—
53	C₂²×C₁₃	—	2	1	no	—
54	C₂²×C₁₃	—	4	1	no	—
55	C₂²×C₁₃	#341	1	1	yes	characteristic
56	C₄×C₁₃	—	2	1	no	—
57	C₄×C₁₃	H(3,ℤ/4)	1	1	yes	characteristic
58	C₄×C₁₃	#343	1	1	yes	characteristic
59	Dic₁₃	—	4	2	no	—
60	C₂×M₁₆	—	13	1	no	—
61	C₂²×C₈	C₂×#261	1	1	yes	characteristic
62	#117	—	13	1	no	—
63	M₃₂	—	26	4	no	—
64	M₃₂	—	52	1	no	—
65	C₂×C₁₆	—	2	1	no	—
66	C₂×C₁₆	—	4	1	no	—
67	C₂×C₁₆	—	26	1	no	—
68	#97	—	13	2	no	—
69	C₄×C₈	—	13	2	no	—
70	C₂×C₁₃	—	2	1	no	—
71	C₂×C₁₃	—	4	1	no	—
72	C₂×C₁₃	#1290	1	1	yes	characteristic
73	C₂²×C₄	C₄⋊#261	1	1	yes	characteristic
74	M₁₆	—	52	1	no	—
75	C₂×C₈	—	2	1	no	—
76	C₂×C₈	—	2	1	no	—
77	C₂×C₈	—	13	2	no	—
78	C₂×C₈	C₄×#261	1	1	yes	characteristic
79	C₂×C₈	C₂²⋊#261	1	1	yes	characteristic
80	C₄⋊C₄	—	26	1	no	—
81	C₂²⋊C₄	—	26	1	no	—
82	C₄²	—	13	2	no	—
83	C₁₆	—	4	1	no	—
84	C₁₆	—	52	4	no	—
85	C₁₆	—	52	1	no	—
86	C₁₃	#7240	1	1	yes	characteristic, complemented, Sylow
87	C₂³	#88103	1	1	yes	characteristic
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	2	1	no	—
90	C₂×C₄	—	26	2	no	—
91	C₂×C₄	#88104	1	1	yes	characteristic
92	C₂×C₄	(C₂×#261)⋊C₄	1	1	yes	characteristic
93	C₈	—	2	1	no	—
94	C₈	—	26	2	no	—
95	C₈	#88113	1	1	yes	characteristic
96	C₈	#88121	1	1	yes	characteristic
97	C₂²	—	2	1	no	—
98	C₂²	—	4	1	no	—
99	C₂²	#11718268	1	1	yes	characteristic
100	C₄	—	2	1	no	—
101	C₄	—	52	2	no	—
102	C₄	#11718267	1	1	yes	characteristic
103	C₄	#11718291	1	1	yes	characteristic
104	C₂	—	2	1	no	—
105	C₂	—	4	1	no	—
106	C₂	#13129701	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13131210

C₂

yes

maximal

#13116594

C₂

yes

maximal, characteristic

C₂²×C₁₃×C₁₆

C₄

yes

characteristic

#11718762

C₂²

yes

characteristic

C₂×#88052

C₄

yes

characteristic

#11718284

—

#11718262

C₄

yes

—

#11718261

—

#11718093

—

C₂²×C₈×C₁₃

C₂×C₄

yes

characteristic

#88180

Q₈

yes

characteristic

C₂×#6099

D₄

yes

characteristic

#88106

—

C₂×C₁₃×C₁₆

—

C₂×C₁₃×C₁₆

—

#88055

C₂×C₄

yes

—

C₈×Dic₁₃

D₄

yes

—

—

—

—

C₂²×C₄×C₁₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₄²

yes

characteristic

C₂×C₈×C₁₃

C₂²⋊C₄

yes

characteristic

C₂²⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄×Dic₁₃

C₄⋊C₄

yes

—

#6099

—

C₂×#1055

C₂²⋊C₄

yes

—

#6092

—

C₁₃×C₁₆

—

#6090

—

#2166

—

#2151

—

C₂³×C₁₃

#108

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

C₂×Dic₁₃

—

C₈×C₁₃

—

C₈×C₁₃

yes

C₈×C₁₃

yes

—

C₂×M₃₂

—

C₂²×C₁₆

yes

—

—

C₄⋊C₁₆

—

#349

—

#347

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

#341

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

yes

C₄×C₁₃

yes

Dic₁₃

—

C₂×M₁₆

—

C₂²×C₈

C₂×#261

yes

characteristic

#117

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#97

—

C₄×C₈

—

C₂×C₁₃

—

C₂×C₁₃

—

C₂×C₁₃

yes

C₂²×C₄

yes

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×#261

yes

characteristic

C₂×C₈

C₂²⋊#261

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₁₆

—

C₁₆

—

C₁₆

—

C₁₃

#7240

yes

characteristic, complemented, Sylow

C₂³

#88103

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#88104

yes

characteristic

C₂×C₄

(C₂×#261)⋊C₄

yes

characteristic

C₈

—

C₈

—

C₈

yes

C₈

yes

C₂²

—

C₂²

—

C₂²

#11718268

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

#11718267

yes

characteristic

103

C₄

#11718291

yes

characteristic

104

C₂

—

105

C₂

—

106

C₂

#13129701

yes

characteristic

	subgroup	G/H
center	C₄	#11718267
commutator	C₂×C₈×C₁₃	C₄²
Frattini	C₂²×C₈	C₂×#261
Fitting	C₂²×C₁₃×C₁₆	C₄
socle	C₂×C₁₃	#1290
prime radical	C₂³×C₁₃	#108
cosocle	#11718762	C₂²
hypercenter	C₂²×C₁₆	#261
hypocenter	C₁₃	#7240
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₂²×C₁₆

subgroup

G/H

center

C₄

#11718267

commutator

C₂×C₈×C₁₃

C₄²

Frattini

C₂²×C₈

C₂×#261

Fitting

C₂²×C₁₃×C₁₆

C₄

C₂×C₁₃

C₂³×C₁₃

C₂²

C₂²×C₁₆

C₁₃

G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₂²×C₁₆

N	H ≅ G/N	type
86. C₁₃	#7240	semidirect

H ≅ G/N

type

86. C₁₃

#7240

semidirect

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

subgroup	index
#13131210	2
#13116594	2
#7240	13

subgroup

index

#13131210

#13116594

#7240

Subgroups — #16777424

Conjugacy classes of subgroups

Complements

Maximal subgroups