Group #16845614 — subgroups

Subgroups — C₄×#11758410 #16845614

Order 3968.

228 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 121 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×#11758410	C₂	1	2	yes	maximal
2	C₄×#92474	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₃₁⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#14520813	C₂	1	2	yes	maximal
5	C₄×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
6	C₂×M₁₆×C₃₁	C₂²	1	2	yes	—
7	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
8	C₂×#92474	C₂²	1	2	yes	—
9	C₂²×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
10	C₂²×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
11	C₃₁×#97	—	2	1	no	—
12	C₄×C₈×C₃₁	—	2	1	no	—
13	#11758410	C₄	1	8	yes	direct product factor
14	#11758391	—	2	1	no	—
15	C₄×C₃₁⋊C₈	—	2	1	no	—
16	C₄×C₃₁⋊C₈	C₄	1	4	yes	—
17	C₂²×C₄×C₃₁	Q₈	1	2	yes	—
18	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
19	M₁₆×C₃₁	—	2	2	no	—
20	M₁₆×C₃₁	C₂×C₄	1	4	yes	—
21	C₂×C₈×C₃₁	—	2	2	no	—
22	C₄²×C₃₁	C₂×C₄	1	2	yes	—
23	C₄²×C₃₁	D₄	1	1	yes	characteristic
24	C₄²×C₃₁	D₄	1	1	yes	characteristic
25	#92474	—	2	2	no	—
26	#92474	C₂×C₄	1	4	yes	—
27	C₂×C₃₁⋊C₈	—	2	2	no	—
28	C₂×C₃₁⋊C₈	—	2	2	no	—
29	C₂×C₃₁⋊C₈	C₂×C₄	1	2	yes	—
30	C₂×C₃₁⋊C₈	C₂×C₄	1	2	yes	—
31	C₂×C₃₁⋊C₈	C₂×C₄	1	4	yes	—
32	C₂³×C₃₁	C₂×Q₈	1	1	yes	characteristic
33	C₂×C₄×C₃₁	—	2	4	no	—
34	C₂×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
35	C₂×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
36	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
38	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₃₁	#41	1	2	yes	—
40	C₈×C₃₁	—	2	4	no	—
41	C₃₁⋊C₈	—	2	2	no	—
42	C₃₁⋊C₈	—	2	4	no	—
43	C₃₁⋊C₈	C₄²	1	4	yes	—
44	C₄×#108	—	31	1	no	maximal, Sylow
45	C₂²×C₃₁	—	2	2	no	—
46	C₂²×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₃₁	C₄×Q₈	1	2	yes	—
48	C₄×C₃₁	—	2	2	no	—
49	C₄×C₃₁	#108	1	4	yes	complemented
50	C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
51	C₄×C₃₁	#117	1	1	yes	characteristic
52	C₄×C₃₁	C₄×D₄	1	2	yes	—
53	C₂×#108	—	31	2	no	—
54	C₄×M₁₆	—	31	1	no	—
55	C₄×M₁₆	D₃₁	1	1	yes	characteristic
56	C₂×C₄×C₈	—	31	1	no	—
57	#341	—	31	2	no	—
58	C₂×C₃₁	—	2	2	no	—
59	C₂×C₃₁	C₄×C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₃₁	C₂×#108	1	2	yes	—
61	C₂×M₁₆	—	31	2	no	—
62	C₂×M₁₆	C₂×D₃₁	1	2	yes	—
63	C₂²×C₈	—	31	1	no	—
64	C₂²×C₈	—	31	1	no	—
65	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
66	#108	—	31	8	no	—
67	#97	—	2	1	no	—
68	#97	—	62	1	no	—
69	C₄×C₈	—	2	1	no	—
70	C₄×C₈	—	31	4	no	—
71	C₄×C₈	—	62	1	no	—
72	C₃₁	C₄×#108	1	1	yes	characteristic, complemented, Sylow
73	C₂²×C₄	Dic₆₂	1	2	yes	—
74	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
75	M₁₆	—	2	2	no	—
76	M₁₆	—	31	4	no	—
77	M₁₆	—	62	2	no	—
78	M₁₆	C₄×D₃₁	1	4	yes	—
79	C₂×C₈	—	2	2	no	—
80	C₂×C₈	—	31	2	no	—
81	C₂×C₈	—	31	4	no	—
82	C₂×C₈	—	31	2	no	—
83	C₂×C₈	—	62	2	no	—
84	C₂×C₈	—	62	2	no	—
85	C₄²	C₄×D₃₁	1	2	yes	—
86	C₄²	C₂²⋊D₃₁	1	1	yes	characteristic
87	C₄²	C₂²⋊D₃₁	1	1	yes	characteristic
88	C₂³	C₂×Dic₆₂	1	1	yes	characteristic
89	C₂×C₄	—	2	4	no	—
90	C₂×C₄	Dic₃₁⋊C₄	1	2	yes	—
91	C₂×C₄	Dic₃₁⋊C₄	1	2	yes	—
92	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
93	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
94	C₂×C₄	#92495	1	2	yes	—
95	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
96	C₈	—	2	4	no	—
97	C₈	—	31	4	no	—
98	C₈	—	62	2	no	—
99	C₈	—	62	4	no	—
100	C₂²	—	2	2	no	—
101	C₂²	C₄×Dic₆₂	1	2	yes	—
102	C₂²	C₂×Dic₃₁⋊C₄	1	1	yes	characteristic
103	C₄	—	2	2	no	—
104	C₄	#11758410	1	4	yes	direct product factor
105	C₄	C₄²×D₃₁	1	1	yes	characteristic
106	C₄	#11758449	1	1	yes	characteristic
107	C₄	C₄×C₂²⋊D₃₁	1	2	yes	—
108	C₂	—	2	2	no	—
109	C₂	C₄×Dic₃₁⋊C₄	1	1	yes	characteristic
110	C₂	C₂×#11758410	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11758410

C₂

yes

maximal

C₄×#92474

C₂

yes

maximal, characteristic

C₂×C₄×C₃₁⋊C₈

C₂

yes

maximal, characteristic

#14520813

C₂

yes

maximal

C₄×M₁₆×C₃₁

C₂

yes

maximal, characteristic

C₂×M₁₆×C₃₁

C₂²

yes

—

C₂×C₄²×C₃₁

C₂²

yes

characteristic

C₂×#92474

C₂²

yes

—

C₂²×C₃₁⋊C₈

C₂²

yes

characteristic

C₂²×C₃₁⋊C₈

C₂²

yes

characteristic

C₃₁×#97

—

C₄×C₈×C₃₁

—

#11758410

C₄

yes

direct product factor

#11758391

—

C₄×C₃₁⋊C₈

—

C₄×C₃₁⋊C₈

C₄

yes

—

C₂²×C₄×C₃₁

Q₈

yes

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

M₁₆×C₃₁

—

M₁₆×C₃₁

C₂×C₄

yes

—

C₂×C₈×C₃₁

—

C₄²×C₃₁

C₂×C₄

yes

—

C₄²×C₃₁

D₄

yes

characteristic

C₄²×C₃₁

D₄

yes

characteristic

#92474

—

#92474

C₂×C₄

yes

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

C₂×C₄

yes

—

C₂×C₃₁⋊C₈

C₂×C₄

yes

—

C₂×C₃₁⋊C₈

C₂×C₄

yes

—

C₂³×C₃₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₄⋊C₄

yes

—

C₂×C₄×C₃₁

C₄⋊C₄

yes

—

C₂×C₄×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₁

C₂²×C₄

yes

—

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

#41

yes

—

C₈×C₃₁

—

C₃₁⋊C₈

—

C₃₁⋊C₈

—

C₃₁⋊C₈

C₄²

yes

—

C₄×#108

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₃₁

C₄×Q₈

yes

—

C₄×C₃₁

—

C₄×C₃₁

#108

yes

complemented

C₄×C₃₁

C₂×C₄²

yes

characteristic

C₄×C₃₁

#117

yes

characteristic

C₄×C₃₁

C₄×D₄

yes

—

C₂×#108

—

C₄×M₁₆

—

C₄×M₁₆

D₃₁

yes

characteristic

C₂×C₄×C₈

—

#341

—

C₂×C₃₁

—

C₂×C₃₁

C₄×C₄⋊C₄

yes

characteristic

C₂×C₃₁

C₂×#108

yes

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₃₁

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

C₂×D₃₁

yes

—

—

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₃₁

C₄×#108

yes

characteristic, complemented, Sylow

C₂²×C₄

Dic₆₂

yes

—

C₂²×C₄

C₂²×D₃₁

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

C₄×D₃₁

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄²

C₄×D₃₁

yes

—

C₄²

C₂²⋊D₃₁

yes

characteristic

C₄²

C₂²⋊D₃₁

yes

characteristic

C₂³

C₂×Dic₆₂

yes

characteristic

C₂×C₄

—

C₂×C₄

Dic₃₁⋊C₄

yes

—

C₂×C₄

Dic₃₁⋊C₄

yes

—

C₂×C₄

C₂×C₄×D₃₁

yes

characteristic

C₂×C₄

C₂×C₄×D₃₁

yes

—

C₂×C₄

#92495

yes

—

C₂×C₄

C₂×C₂²⋊D₃₁

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

—

100

C₂²

—

101

C₂²

C₄×Dic₆₂

yes

—

102

C₂²

C₂×Dic₃₁⋊C₄

yes

characteristic

103

C₄

—

104

C₄

#11758410

yes

direct product factor

105

C₄

C₄²×D₃₁

yes

characteristic

106

C₄

#11758449

yes

characteristic

107

C₄

C₄×C₂²⋊D₃₁

yes

—

108

C₂

—

109

C₂

C₄×Dic₃₁⋊C₄

yes

characteristic

110

C₂

C₂×#11758410

yes

—

	subgroup	G/H
center	C₄²	C₂²⋊D₃₁
commutator	C₄×C₃₁	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₃₁
Fitting	C₄×M₁₆×C₃₁	C₂
socle	C₂²×C₃₁	C₂×C₄⋊C₄
prime radical	C₂³×C₃₁	C₂×Q₈
cosocle	C₂²×C₄×C₃₁	C₂³
hypercenter	C₄×M₁₆	D₃₁
hypocenter	C₃₁	C₄×#108
LCS	G ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₄²

C₂²⋊D₃₁

commutator

C₄×C₃₁

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₃₁

Fitting

C₄×M₁₆×C₃₁

C₂

socle

C₂²×C₃₁

C₂×C₄⋊C₄

prime radical

C₂³×C₃₁

C₂×Q₈

cosocle

C₂²×C₄×C₃₁

C₂³

hypercenter

C₄×M₁₆

D₃₁

hypocenter

C₃₁

C₄×#108

LCS

G ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
13. #11758410	C₄	direct (N × H)
49. C₄×C₃₁	#108	semidirect
72. C₃₁	C₄×#108	semidirect
104. C₄	#11758410	direct (N × H)

H ≅ G/N

type

13. #11758410

C₄

direct (N × H)

49. C₄×C₃₁

#108

semidirect

72. C₃₁

C₄×#108

semidirect

104. C₄

#11758410

direct (N × H)

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

subgroup	index
C₂×#11758410	2
C₄×#92474	2
C₂×C₄×C₃₁⋊C₈	2
#14520813	2
C₄×M₁₆×C₃₁	2
C₄×#108	31

subgroup

index

C₂×C₄×C₃₁⋊C₈

C₄×M₁₆×C₃₁

Subgroups — C₄×#11758410 #16845614

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C4×#11758410 #16845614

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#11758410 #16845614