Group #27171035 — subgroups

Subgroups — C₄×#15692190 #27171035

Order 12928.

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×#15692190	C₂	1	2	yes	maximal
2	C₄×#13108058	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×#11717579	C₂	1	1	yes	maximal, characteristic
4	#20264941	C₂	1	2	yes	maximal
5	C₄×M₁₆×C₁₀₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#13108058	C₂²	1	2	yes	—
7	C₂²×#11717579	C₂²	1	1	yes	characteristic
8	C₂²×#11717579	C₂²	1	1	yes	characteristic
9	#15692190	C₄	1	8	yes	direct product factor
10	#15692166	—	2	1	no	—
11	C₄×#11717579	—	2	1	no	—
12	C₄×#11717579	C₄	1	4	yes	—
13	C₂×M₁₆×C₁₀₁	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
15	#97×C₁₀₁	—	2	1	no	—
16	C₄×C₈×C₁₀₁	—	2	1	no	—
17	#13108058	—	2	2	no	—
18	#13108058	C₂×C₄	1	4	yes	—
19	C₂×#11717579	—	2	2	no	—
20	C₂×#11717579	—	2	2	no	—
21	C₂×#11717579	C₂×C₄	1	2	yes	—
22	C₂×#11717579	C₂×C₄	1	2	yes	—
23	C₂×#11717579	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₁₀₁	Q₈	1	2	yes	—
25	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
26	M₁₆×C₁₀₁	—	2	2	no	—
27	M₁₆×C₁₀₁	C₂×C₄	1	4	yes	—
28	C₂×C₈×C₁₀₁	—	2	2	no	—
29	C₄²×C₁₀₁	C₂×C₄	1	2	yes	—
30	C₄²×C₁₀₁	D₄	1	1	yes	characteristic
31	C₄²×C₁₀₁	D₄	1	1	yes	characteristic
32	C₂³×C₁₀₁	C₂×Q₈	1	1	yes	characteristic
33	C₂×C₄×C₁₀₁	—	2	4	no	—
34	C₂×C₄×C₁₀₁	C₄⋊C₄	1	2	yes	—
35	C₂×C₄×C₁₀₁	C₄⋊C₄	1	2	yes	—
36	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
38	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₁₀₁	#41	1	2	yes	—
40	C₈×C₁₀₁	—	2	4	no	—
41	#11717579	—	2	2	no	—
42	#11717579	—	2	4	no	—
43	#11717579	C₄²	1	4	yes	—
44	C₂²×C₁₀₁	—	2	2	no	—
45	C₂²×C₁₀₁	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₂²×C₁₀₁	C₄×Q₈	1	2	yes	—
47	C₄×C₁₀₁	—	2	2	no	—
48	C₄×C₁₀₁	#108	1	4	yes	complemented
49	C₄×C₁₀₁	C₂×C₄²	1	1	yes	characteristic
50	C₄×C₁₀₁	#117	1	1	yes	characteristic
51	C₄×C₁₀₁	C₄×D₄	1	2	yes	—
52	C₂×C₁₀₁	—	2	2	no	—
53	C₂×C₁₀₁	C₄×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₁₀₁	C₂×#108	1	2	yes	—
55	C₄×#108	—	101	1	no	maximal, Sylow
56	C₁₀₁	C₄×#108	1	1	yes	characteristic, complemented, Sylow
57	C₂×#108	—	101	2	no	—
58	C₄×M₁₆	—	101	1	no	—
59	C₄×M₁₆	D₁₀₁	1	1	yes	characteristic
60	C₂×C₄×C₈	—	101	1	no	—
61	#341	—	101	2	no	—
62	C₂×M₁₆	—	101	2	no	—
63	C₂×M₁₆	C₂×D₁₀₁	1	2	yes	—
64	C₂²×C₈	—	101	1	no	—
65	C₂²×C₈	—	101	1	no	—
66	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
67	#108	—	101	8	no	—
68	#97	—	2	1	no	—
69	#97	—	202	1	no	—
70	C₄×C₈	—	2	1	no	—
71	C₄×C₈	—	101	4	no	—
72	C₄×C₈	—	202	1	no	—
73	C₂²×C₄	Dic₂₀₂	1	2	yes	—
74	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
75	M₁₆	—	2	2	no	—
76	M₁₆	—	101	4	no	—
77	M₁₆	—	202	2	no	—
78	M₁₆	C₄×D₁₀₁	1	4	yes	—
79	C₂×C₈	—	2	2	no	—
80	C₂×C₈	—	101	2	no	—
81	C₂×C₈	—	101	4	no	—
82	C₂×C₈	—	101	2	no	—
83	C₂×C₈	—	202	2	no	—
84	C₂×C₈	—	202	2	no	—
85	C₄²	C₄×D₁₀₁	1	2	yes	—
86	C₄²	C₂²⋊D₁₀₁	1	1	yes	characteristic
87	C₄²	C₂²⋊D₁₀₁	1	1	yes	characteristic
88	C₂³	C₂×Dic₂₀₂	1	1	yes	characteristic
89	C₂×C₄	—	2	4	no	—
90	C₂×C₄	#13108053	1	2	yes	—
91	C₂×C₄	#13108053	1	2	yes	—
92	C₂×C₄	C₂×C₄×D₁₀₁	1	1	yes	characteristic
93	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
94	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
95	C₂×C₄	#13108074	1	2	yes	—
96	C₈	—	2	4	no	—
97	C₈	—	101	4	no	—
98	C₈	—	202	2	no	—
99	C₈	—	202	4	no	—
100	C₂²	—	2	2	no	—
101	C₂²	C₂×#13108053	1	1	yes	characteristic
102	C₂²	C₄×Dic₂₀₂	1	2	yes	—
103	C₄	—	2	2	no	—
104	C₄	#15692190	1	4	yes	direct product factor
105	C₄	C₄²×D₁₀₁	1	1	yes	characteristic
106	C₄	#15692212	1	1	yes	characteristic
107	C₄	C₄×C₂²⋊D₁₀₁	1	2	yes	—
108	C₂	—	2	2	no	—
109	C₂	C₄×#13108053	1	1	yes	characteristic
110	C₂	C₂×#15692190	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15692190

C₂

yes

maximal

C₄×#13108058

C₂

yes

maximal, characteristic

C₂×C₄×#11717579

C₂

yes

maximal, characteristic

#20264941

C₂

yes

maximal

C₄×M₁₆×C₁₀₁

C₂

yes

maximal, characteristic

C₂×#13108058

C₂²

yes

—

C₂²×#11717579

C₂²

yes

characteristic

C₂²×#11717579

C₂²

yes

characteristic

#15692190

C₄

yes

direct product factor

#15692166

—

C₄×#11717579

—

C₄×#11717579

C₄

yes

—

C₂×M₁₆×C₁₀₁

C₂²

yes

—

C₂×C₄²×C₁₀₁

C₂²

yes

characteristic

#97×C₁₀₁

—

C₄×C₈×C₁₀₁

—

#13108058

—

#13108058

C₂×C₄

yes

—

C₂×#11717579

—

C₂×#11717579

—

C₂×#11717579

C₂×C₄

yes

—

C₂×#11717579

C₂×C₄

yes

—

C₂×#11717579

C₂×C₄

yes

—

C₂²×C₄×C₁₀₁

Q₈

yes

—

C₂²×C₄×C₁₀₁

C₂³

yes

characteristic

M₁₆×C₁₀₁

—

M₁₆×C₁₀₁

C₂×C₄

yes

—

C₂×C₈×C₁₀₁

—

C₄²×C₁₀₁

C₂×C₄

yes

—

C₄²×C₁₀₁

D₄

yes

characteristic

C₄²×C₁₀₁

D₄

yes

characteristic

C₂³×C₁₀₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₄⋊C₄

yes

—

C₂×C₄×C₁₀₁

C₄⋊C₄

yes

—

C₂×C₄×C₁₀₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₁

C₂²×C₄

yes

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₁

#41

yes

—

C₈×C₁₀₁

—

#11717579

—

#11717579

—

#11717579

C₄²

yes

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₀₁

C₄×Q₈

yes

—

C₄×C₁₀₁

—

C₄×C₁₀₁

#108

yes

complemented

C₄×C₁₀₁

C₂×C₄²

yes

characteristic

C₄×C₁₀₁

#117

yes

characteristic

C₄×C₁₀₁

C₄×D₄

yes

—

C₂×C₁₀₁

—

C₂×C₁₀₁

C₄×C₄⋊C₄

yes

characteristic

C₂×C₁₀₁

C₂×#108

yes

—

C₄×#108

—

101

maximal, Sylow

C₁₀₁

C₄×#108

yes

characteristic, complemented, Sylow

C₂×#108

—

101

—

C₄×M₁₆

—

101

—

C₄×M₁₆

D₁₀₁

yes

characteristic

C₂×C₄×C₈

—

101

—

#341

—

101

—

C₂×M₁₆

—

101

—

C₂×M₁₆

C₂×D₁₀₁

yes

—

C₂²×C₈

—

101

—

C₂²×C₈

—

101

—

C₂×C₄²

C₂×D₁₀₁

yes

characteristic

#108

—

101

—

#97

—

#97

—

202

—

C₄×C₈

—

C₄×C₈

—

101

—

C₄×C₈

—

202

—

C₂²×C₄

Dic₂₀₂

yes

—

C₂²×C₄

C₂²×D₁₀₁

yes

characteristic

M₁₆

—

M₁₆

—

101

—

M₁₆

—

202

—

M₁₆

C₄×D₁₀₁

yes

—

C₂×C₈

—

C₂×C₈

—

101

—

C₂×C₈

—

101

—

C₂×C₈

—

101

—

C₂×C₈

—

202

—

C₂×C₈

—

202

—

C₄²

C₄×D₁₀₁

yes

—

C₄²

C₂²⋊D₁₀₁

yes

characteristic

C₄²

C₂²⋊D₁₀₁

yes

characteristic

C₂³

C₂×Dic₂₀₂

yes

characteristic

C₂×C₄

—

C₂×C₄

#13108053

yes

—

C₂×C₄

#13108053

yes

—

C₂×C₄

C₂×C₄×D₁₀₁

yes

characteristic

C₂×C₄

C₂×C₄×D₁₀₁

yes

—

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

characteristic

C₂×C₄

#13108074

yes

—

C₈

—

C₈

—

101

—

C₈

—

202

—

C₈

—

202

—

100

C₂²

—

101

C₂²

C₂×#13108053

yes

characteristic

102

C₂²

C₄×Dic₂₀₂

yes

—

103

C₄

—

104

C₄

#15692190

yes

direct product factor

105

C₄

C₄²×D₁₀₁

yes

characteristic

106

C₄

#15692212

yes

characteristic

107

C₄

C₄×C₂²⋊D₁₀₁

yes

—

108

C₂

—

109

C₂

C₄×#13108053

yes

characteristic

110

C₂

C₂×#15692190

yes

—

	subgroup	G/H
center	C₄²	C₂²⋊D₁₀₁
commutator	C₄×C₁₀₁	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₀₁
Fitting	C₄×M₁₆×C₁₀₁	C₂
socle	C₂²×C₁₀₁	C₂×C₄⋊C₄
prime radical	C₂³×C₁₀₁	C₂×Q₈
cosocle	C₂²×C₄×C₁₀₁	C₂³
hypercenter	C₄×M₁₆	D₁₀₁
hypocenter	C₁₀₁	C₄×#108
LCS	G ▷ C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₄²

C₂²⋊D₁₀₁

commutator

C₄×C₁₀₁

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₁₀₁

Fitting

C₄×M₁₆×C₁₀₁

C₂

socle

C₂²×C₁₀₁

C₂×C₄⋊C₄

prime radical

C₂³×C₁₀₁

C₂×Q₈

cosocle

C₂²×C₄×C₁₀₁

C₂³

hypercenter

C₄×M₁₆

D₁₀₁

hypocenter

C₁₀₁

C₄×#108

LCS

G ▷ C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
9. #15692190	C₄	direct (N × H)
48. C₄×C₁₀₁	#108	semidirect
56. C₁₀₁	C₄×#108	semidirect
104. C₄	#15692190	direct (N × H)

H ≅ G/N

type

9. #15692190

C₄

direct (N × H)

48. C₄×C₁₀₁

#108

semidirect

56. C₁₀₁

C₄×#108

semidirect

104. C₄

#15692190

direct (N × H)

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

subgroup	index
C₂×#15692190	2
C₄×#13108058	2
C₂×C₄×#11717579	2
#20264941	2
C₄×M₁₆×C₁₀₁	2
C₄×#108	101

subgroup

index

C₄×M₁₆×C₁₀₁

101

Subgroups — C₄×#15692190 #27171035

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C4×#15692190 #27171035

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#15692190 #27171035