Group #28378578 — subgroups

Subgroups — C₂×#20295890 #28378578

Order 13952.

200 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 106 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×C₄⋊#11724786	C₂	1	2	yes	maximal
2	#20295890	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₈×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11724786	—	2	2	no	—
5	C₄⋊#11724786	—	2	2	no	—
6	C₄⋊#11724786	C₂²	1	4	yes	—
7	C₂²×#11724784	C₄	1	1	yes	characteristic
8	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
9	C₈×Dic₁₀₉	C₄	1	2	yes	—
10	C₈×Dic₁₀₉	C₂²	1	2	yes	—
11	C₂²×C₈×C₁₀₉	C₄	1	1	yes	characteristic
12	C₂×#11724786	—	2	4	no	—
13	C₂×#11724786	—	4	4	no	—
14	C₂²×Dic₁₀₉	D₄	1	1	yes	characteristic
15	C₂²×Dic₁₀₉	Q₈	1	1	yes	characteristic
16	C₂×#11724784	C₈	1	4	yes	complemented
17	C₂×#11724784	C₂×C₄	1	2	yes	—
18	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
19	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
20	C₄×Dic₁₀₉	Q₈	1	1	yes	characteristic
21	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
22	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₈×C₁₀₉	—	2	2	no	—
24	C₂×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
25	C₂³×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
26	C₂×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
27	C₂×C₄×C₁₀₉	C₂×C₈	1	2	yes	—
28	C₂×C₄×C₁₀₉	M₁₆	1	2	yes	—
29	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
30	C₂×Dic₁₀₉	—	2	2	no	—
31	C₂×Dic₁₀₉	C₄⋊C₄	1	1	yes	characteristic
32	C₂×Dic₁₀₉	C₄⋊C₄	1	1	yes	characteristic
33	C₂×Dic₁₀₉	D₈	1	2	yes	—
34	C₂×Dic₁₀₉	Q₁₆	1	2	yes	—
35	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
36	C₂×Dic₁₀₉	C₂×Q₈	1	1	yes	characteristic
37	#11724786	—	4	4	no	—
38	C₈×C₁₀₉	—	2	2	no	—
39	#11724784	C₂×C₈	1	4	yes	complemented
40	C₂²×C₁₀₉	C₄⋊C₈	1	2	yes	—
41	C₂²×C₁₀₉	#107	1	2	yes	—
42	C₂²×C₁₀₉	#108	1	2	yes	—
43	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₄×C₁₀₉	C₄⋊C₈	1	2	yes	—
45	C₄×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
46	C₄×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
47	Dic₁₀₉	—	2	1	no	—
48	Dic₁₀₉	—	2	1	no	—
49	Dic₁₀₉	#107	1	2	yes	—
50	Dic₁₀₉	C₂×D₈	1	1	yes	characteristic
51	Dic₁₀₉	C₂×Q₁₆	1	1	yes	characteristic
52	C₂×C₁₀₉	#335	1	4	yes	complemented
53	C₂×C₁₀₉	C₂×C₄⋊C₈	1	1	yes	characteristic
54	C₂×C₁₀₉	C₂×#107	1	1	yes	characteristic
55	C₂×C₁₀₉	C₂×#108	1	1	yes	characteristic
56	C₂×#335	—	109	1	no	maximal, Sylow
57	C₁₀₉	C₂×#335	1	1	yes	characteristic, complemented, Sylow
58	C₂×C₄⋊C₈	—	109	2	no	—
59	C₂×C₄×C₈	—	109	1	no	—
60	#335	—	109	4	no	—
61	C₂²×C₈	—	109	1	no	—
62	C₂²×C₈	—	218	2	no	—
63	C₂²×C₈	#89137	1	1	yes	characteristic
64	C₂×C₄²	—	109	1	no	—
65	C₄⋊C₈	—	109	4	no	—
66	C₄⋊C₈	—	218	2	no	—
67	C₄×C₈	—	109	2	no	—
68	C₄×C₈	—	109	2	no	—
69	C₂²×C₄	—	109	1	no	—
70	C₂²×C₄	—	109	1	no	—
71	C₂²×C₄	C₂×#89137	1	1	yes	characteristic
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	—	109	4	no	—
74	C₂×C₈	—	109	2	no	—
75	C₂×C₈	—	436	4	no	—
76	C₂×C₈	—	218	4	no	—
77	C₂×C₈	C₂×#89137	1	2	yes	—
78	C₄²	—	109	1	no	—
79	C₄²	—	109	1	no	—
80	C₄²	—	109	1	no	—
81	C₄²	—	109	1	no	—
82	C₂³	C₄⋊#89137	1	1	yes	characteristic
83	C₂×C₄	—	109	2	no	—
84	C₂×C₄	—	109	2	no	—
85	C₂×C₄	—	109	1	no	—
86	C₂×C₄	—	109	1	no	—
87	C₂×C₄	—	109	1	no	—
88	C₂×C₄	—	109	1	no	—
89	C₂×C₄	—	218	2	no	—
90	C₂×C₄	C₄⋊#89137	1	1	yes	characteristic
91	C₂×C₄	D₁₀₉⋊C₈	1	2	yes	—
92	C₂×C₄	#13182038	1	2	yes	—
93	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
94	C₈	—	2	2	no	—
95	C₈	—	109	4	no	—
96	C₈	—	436	4	no	—
97	C₂²	#16812766	1	2	yes	—
98	C₂²	#16812768	1	2	yes	—
99	C₂²	#16812769	1	2	yes	—
100	C₂²	C₂×C₄⋊#89137	1	1	yes	characteristic
101	C₄	—	109	1	no	—
102	C₄	—	109	2	no	—
103	C₄	—	109	1	no	—
104	C₄	—	218	1	no	—
105	C₄	—	218	1	no	—
106	C₄	#16812766	1	2	yes	—
107	C₄	C₂×D₁₀₉⋊C₈	1	1	yes	characteristic
108	C₄	C₂×#13182038	1	1	yes	characteristic
109	C₂	#20295890	1	4	yes	direct product factor
110	C₂	C₂×#16812766	1	1	yes	characteristic
111	C₂	C₂×#16812768	1	1	yes	characteristic
112	C₂	C₂×#16812769	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#11724786

C₂

yes

maximal

#20295890

C₂

yes

maximal, direct product factor

C₂×C₈×Dic₁₀₉

C₂

yes

maximal, characteristic

C₂²×#11724786

—

C₄⋊#11724786

—

C₄⋊#11724786

C₂²

yes

—

C₂²×#11724784

C₄

yes

characteristic

C₂×C₄×Dic₁₀₉

C₂²

yes

characteristic

C₈×Dic₁₀₉

C₄

yes

—

C₈×Dic₁₀₉

C₂²

yes

—

C₂²×C₈×C₁₀₉

C₄

yes

characteristic

C₂×#11724786

—

C₂×#11724786

—

C₂²×Dic₁₀₉

D₄

yes

characteristic

C₂²×Dic₁₀₉

Q₈

yes

characteristic

C₂×#11724784

C₈

yes

complemented

C₂×#11724784

C₂×C₄

yes

—

C₄×Dic₁₀₉

C₂×C₄

yes

characteristic

C₄×Dic₁₀₉

D₄

yes

characteristic

C₄×Dic₁₀₉

Q₈

yes

characteristic

C₄×Dic₁₀₉

C₂³

yes

characteristic

C₂²×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

—

C₂³×C₁₀₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×C₈

yes

—

C₂×C₄×C₁₀₉

M₁₆

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₄⋊C₄

yes

characteristic

C₂×Dic₁₀₉

C₄⋊C₄

yes

characteristic

C₂×Dic₁₀₉

D₈

yes

—

C₂×Dic₁₀₉

Q₁₆

yes

—

C₂×Dic₁₀₉

C₂×D₄

yes

characteristic

C₂×Dic₁₀₉

C₂×Q₈

yes

characteristic

#11724786

—

C₈×C₁₀₉

—

#11724784

C₂×C₈

yes

complemented

C₂²×C₁₀₉

C₄⋊C₈

yes

—

C₂²×C₁₀₉

#107

yes

—

C₂²×C₁₀₉

#108

yes

—

C₂²×C₁₀₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₀₉

C₄⋊C₈

yes

—

C₄×C₁₀₉

C₂²×C₈

yes

characteristic

C₄×C₁₀₉

C₂×M₁₆

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

#107

yes

—

Dic₁₀₉

C₂×D₈

yes

characteristic

Dic₁₀₉

C₂×Q₁₆

yes

characteristic

C₂×C₁₀₉

#335

yes

complemented

C₂×C₁₀₉

C₂×C₄⋊C₈

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

109

maximal, Sylow

C₁₀₉

C₂×#335

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

109

—

C₂×C₄×C₈

—

109

—

#335

—

109

—

C₂²×C₈

—

109

—

C₂²×C₈

—

218

—

C₂²×C₈

#89137

yes

characteristic

C₂×C₄²

—

109

—

C₄⋊C₈

—

109

—

C₄⋊C₈

—

218

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

C₂×#89137

yes

characteristic

C₂×C₈

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

436

—

C₂×C₈

—

218

—

C₂×C₈

C₂×#89137

yes

—

C₄²

—

109

—

C₄²

—

109

—

C₄²

—

109

—

C₄²

—

109

—

C₂³

C₄⋊#89137

yes

characteristic

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

218

—

C₂×C₄

C₄⋊#89137

yes

characteristic

C₂×C₄

D₁₀₉⋊C₈

yes

—

C₂×C₄

#13182038

yes

—

C₂×C₄

C₂²×#89137

yes

characteristic

C₈

—

C₈

—

109

—

C₈

—

436

—

C₂²

#16812766

yes

—

C₂²

#16812768

yes

—

C₂²

#16812769

yes

—

100

C₂²

C₂×C₄⋊#89137

yes

characteristic

101

C₄

—

109

—

102

C₄

—

109

—

103

C₄

—

109

—

104

C₄

—

218

—

105

C₄

—

218

—

106

C₄

#16812766

yes

—

107

C₄

C₂×D₁₀₉⋊C₈

yes

characteristic

108

C₄

C₂×#13182038

yes

characteristic

109

C₂

#20295890

yes

direct product factor

110

C₂

C₂×#16812766

yes

characteristic

111

C₂

C₂×#16812768

yes

characteristic

112

C₂

C₂×#16812769

yes

characteristic

	subgroup	G/H
center	C₂³	C₄⋊#89137
commutator	C₄×C₁₀₉	C₂²×C₈
Frattini	C₂×C₄	C₂²×#89137
Fitting	C₂²×C₈×C₁₀₉	C₄
socle, prime radical	C₂³×C₁₀₉	C₄⋊C₄
cosocle	C₄×Dic₁₀₉	C₂³
hypercenter	C₂²×C₈	#89137
hypocenter	C₁₀₉	C₂×#335
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈

subgroup

G/H

center

C₂³

C₄⋊#89137

commutator

C₄×C₁₀₉

C₂²×C₈

Frattini

C₂×C₄

C₂²×#89137

Fitting

C₂²×C₈×C₁₀₉

C₄

socle, prime radical

C₂³×C₁₀₉

C₄⋊C₄

cosocle

C₄×Dic₁₀₉

C₂³

C₂²×C₈

C₁₀₉

G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈

N	H ≅ G/N	type
2. #20295890	C₂	direct (N × H)
16. C₂×#11724784	C₈	semidirect
39. #11724784	C₂×C₈	semidirect
52. C₂×C₁₀₉	#335	semidirect
57. C₁₀₉	C₂×#335	semidirect
109. C₂	#20295890	direct (N × H)

H ≅ G/N

type

2. #20295890

C₂

direct (N × H)

16. C₂×#11724784

C₈

semidirect

39. #11724784

C₂×C₈

semidirect

52. C₂×C₁₀₉

#335

semidirect

57. C₁₀₉

C₂×#335

semidirect

109. C₂

#20295890

direct (N × H)

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

subgroup	index
C₂×C₄⋊#11724786	2
#20295890	2
C₂×C₈×Dic₁₀₉	2
C₂×#335	109

subgroup

index

C₂×C₄⋊#11724786

#20295890

C₂×C₈×Dic₁₀₉

C₂×#335

109

Subgroups — C₂×#20295890 #28378578

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#20295890 #28378578

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20295890 #28378578