Group #28379241 — subgroups

Subgroups — #28379241

Order 13952.

184 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 114 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×#16812773	C₂	1	1	yes	maximal, characteristic
2	C₂²×#13182032	C₂	1	1	yes	maximal, characteristic
3	#20295914	C₂	1	2	yes	maximal, complemented
4	C₄×#13182032	C₂	1	2	yes	maximal, complemented
5	#20294982	C₂	1	1	yes	maximal, characteristic
6	#16812773	C₄	1	4	yes	complemented
7	C₂×#13182032	C₄	1	4	yes	complemented
8	C₂×#13182032	C₂²	1	1	yes	characteristic
9	C₂×#13182032	C₂²	1	1	yes	characteristic
10	C₂×#13182032	C₂²	1	2	yes	—
11	C₂×#13182013	C₄	1	1	yes	characteristic
12	C₂²×#11724784	C₂²	1	1	yes	characteristic
13	#16812610	C₄	1	2	yes	—
14	C₄×#11724784	C₂²	1	2	yes	—
15	#117×C₁₀₉	C₄	1	1	yes	characteristic
16	#13182032	C₂×C₄	1	2	yes	—
17	#13182032	C₂×C₄	1	2	yes	—
18	#13182032	C₂×C₄	1	4	yes	—
19	#13182013	C₈	1	4	yes	—
20	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
21	C₂×#11724784	C₂×C₄	1	2	yes	—
22	C₂×#11724784	C₂×C₄	1	2	yes	—
23	C₂×#11724784	C₂×C₄	1	2	yes	—
24	C₂×#11724784	C₂³	1	1	yes	characteristic
25	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
26	C₄⋊C₄×C₁₀₉	C₈	1	2	yes	—
27	C₂²⋊C₄×C₁₀₉	C₈	1	2	yes	—
28	C₄²×C₁₀₉	C₂×C₄	1	2	yes	—
29	C₂³×C₁₀₉	C₂×C₈	1	1	yes	characteristic
30	C₂×C₄×C₁₀₉	C₄²	1	2	yes	—
31	C₂×C₄×C₁₀₉	C₂×C₈	1	4	yes	—
32	C₂×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
33	C₂×C₄×C₁₀₉	C₂×C₈	1	2	yes	—
34	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
35	#11724784	C₄²	1	2	yes	—
36	#11724784	C₂×C₈	1	4	yes	—
37	#11724784	C₂²×C₄	1	1	yes	characteristic
38	#11724784	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₁₀₉	—	2	2	no	—
40	C₂²×C₁₀₉	C₄×C₈	1	2	yes	—
41	C₂²×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
42	C₄×C₁₀₉	—	2	4	no	—
43	C₄×C₁₀₉	C₄×C₈	1	2	yes	—
44	C₄×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
45	C₄×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
46	C₂×C₁₀₉	—	2	2	no	—
47	C₂×C₁₀₉	C₂×C₄×C₈	1	1	yes	characteristic
48	C₂×C₁₀₉	#504	1	2	yes	—
49	#2108	—	109	1	no	maximal, Sylow
50	C₁₀₉	#2108	1	1	yes	characteristic, complemented, Sylow
51	C₂×M₃₂	—	109	1	no	—
52	C₂²×C₁₆	—	109	1	no	—
53	#405	—	109	1	no	—
54	#346	—	109	2	no	—
55	C₄×C₁₆	—	109	2	no	—
56	C₂×M₁₆	—	109	1	no	—
57	C₂²×C₈	—	109	1	no	—
58	#117	#89137	1	1	yes	characteristic
59	M₃₂	—	109	4	no	—
60	C₂×C₁₆	—	109	1	no	—
61	C₂×C₁₆	—	109	2	no	—
62	C₂×C₁₆	—	109	1	no	—
63	C₂×C₁₆	—	109	4	no	—
64	#97	—	109	2	no	—
65	C₄×C₈	—	109	2	no	—
66	C₂²×C₄	C₂×#89137	1	1	yes	characteristic
67	M₁₆	—	109	4	no	—
68	C₂×C₈	—	109	2	no	—
69	C₂×C₈	—	109	2	no	—
70	C₂×C₈	—	109	2	no	—
71	C₂×C₈	—	109	1	no	—
72	C₂×C₈	—	109	1	no	—
73	C₄⋊C₄	#11724786	1	2	yes	—
74	C₂²⋊C₄	#11724786	1	2	yes	—
75	C₄²	C₂×#89137	1	2	yes	—
76	C₁₆	—	109	2	no	—
77	C₁₆	—	109	2	no	—
78	C₁₆	—	109	4	no	—
79	C₂³	C₂×#11724786	1	1	yes	characteristic
80	C₂×C₄	C₄×#89137	1	2	yes	—
81	C₂×C₄	C₂×#11724786	1	4	yes	—
82	C₂×C₄	C₂×#11724786	1	1	yes	characteristic
83	C₂×C₄	D₁₀₉⋊C₈	1	2	yes	—
84	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
85	C₈	—	109	1	no	—
86	C₈	—	109	1	no	—
87	C₈	—	109	2	no	—
88	C₈	—	109	4	no	—
89	C₂²	—	2	2	no	—
90	C₂²	C₄×#11724786	1	2	yes	—
91	C₂²	C₂²×#11724786	1	1	yes	characteristic
92	C₄	—	2	4	no	—
93	C₄	C₄×#11724786	1	2	yes	—
94	C₄	C₂×C₄×#89137	1	1	yes	characteristic
95	C₄	C₂×D₁₀₉⋊C₈	1	1	yes	characteristic
96	C₂	—	2	2	no	—
97	C₂	C₂×C₄×#11724786	1	1	yes	characteristic
98	C₂	#20296084	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812773

C₂

yes

maximal, characteristic

C₂²×#13182032

C₂

yes

maximal, characteristic

#20295914

C₂

yes

maximal, complemented

C₄×#13182032

C₂

yes

maximal, complemented

#20294982

C₂

yes

maximal, characteristic

#16812773

C₄

yes

complemented

C₂×#13182032

C₄

yes

complemented

C₂×#13182032

C₂²

yes

characteristic

C₂×#13182032

C₂²

yes

characteristic

C₂×#13182032

C₂²

yes

—

C₂×#13182013

C₄

yes

characteristic

C₂²×#11724784

C₂²

yes

characteristic

#16812610

C₄

yes

—

C₄×#11724784

C₂²

yes

—

#117×C₁₀₉

C₄

yes

characteristic

#13182032

C₂×C₄

yes

—

#13182032

C₂×C₄

yes

—

#13182032

C₂×C₄

yes

—

#13182013

C₈

yes

—

C₂×#11724784

C₂×C₄

yes

characteristic

C₂×#11724784

C₂×C₄

yes

—

C₂×#11724784

C₂×C₄

yes

—

C₂×#11724784

C₂×C₄

yes

—

C₂×#11724784

C₂³

yes

characteristic

C₂²×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₄⋊C₄×C₁₀₉

C₈

yes

—

C₂²⋊C₄×C₁₀₉

C₈

yes

—

C₄²×C₁₀₉

C₂×C₄

yes

—

C₂³×C₁₀₉

C₂×C₈

yes

characteristic

C₂×C₄×C₁₀₉

C₄²

yes

—

C₂×C₄×C₁₀₉

C₂×C₈

yes

—

C₂×C₄×C₁₀₉

C₂×C₈

yes

characteristic

C₂×C₄×C₁₀₉

C₂×C₈

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

#11724784

C₄²

yes

—

#11724784

C₂×C₈

yes

—

#11724784

C₂²×C₄

yes

characteristic

#11724784

C₂²×C₄

yes

characteristic

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₄×C₈

yes

—

C₂²×C₁₀₉

C₂²×C₈

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

C₄×C₈

yes

—

C₄×C₁₀₉

C₂×C₄²

yes

characteristic

C₄×C₁₀₉

C₂²×C₈

yes

characteristic

C₂×C₁₀₉

—

C₂×C₁₀₉

C₂×C₄×C₈

yes

characteristic

C₂×C₁₀₉

#504

yes

—

#2108

—

109

maximal, Sylow

C₁₀₉

#2108

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

109

—

C₂²×C₁₆

—

109

—

#405

—

109

—

#346

—

109

—

C₄×C₁₆

—

109

—

C₂×M₁₆

—

109

—

C₂²×C₈

—

109

—

#117

#89137

yes

characteristic

M₃₂

—

109

—

C₂×C₁₆

—

109

—

C₂×C₁₆

—

109

—

C₂×C₁₆

—

109

—

C₂×C₁₆

—

109

—

#97

—

109

—

C₄×C₈

—

109

—

C₂²×C₄

C₂×#89137

yes

characteristic

M₁₆

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₄⋊C₄

#11724786

yes

—

C₂²⋊C₄

#11724786

yes

—

C₄²

C₂×#89137

yes

—

C₁₆

—

109

—

C₁₆

—

109

—

C₁₆

—

109

—

C₂³

C₂×#11724786

yes

characteristic

C₂×C₄

C₄×#89137

yes

—

C₂×C₄

C₂×#11724786

yes

—

C₂×C₄

C₂×#11724786

yes

characteristic

C₂×C₄

D₁₀₉⋊C₈

yes

—

C₂×C₄

C₂²×#89137

yes

characteristic

C₈

—

109

—

C₈

—

109

—

C₈

—

109

—

C₈

—

109

—

C₂²

—

C₂²

C₄×#11724786

yes

—

C₂²

C₂²×#11724786

yes

characteristic

C₄

—

C₄

C₄×#11724786

yes

—

C₄

C₂×C₄×#89137

yes

characteristic

C₄

C₂×D₁₀₉⋊C₈

yes

characteristic

C₂

—

C₂

C₂×C₄×#11724786

yes

characteristic

C₂

#20296084

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#89137
commutator	C₂×C₁₀₉	C₂×C₄×C₈
Fitting	#117×C₁₀₉	C₄
socle	C₂²×C₁₀₉	C₂²×C₈
prime radical	C₂³×C₁₀₉	C₂×C₈
cosocle	C₂×#11724784	C₂³
hypercenter	#117	#89137
hypocenter	C₁₀₉	#2108
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ #117

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×#89137

commutator

C₂×C₁₀₉

C₂×C₄×C₈

Fitting

#117×C₁₀₉

C₄

socle

C₂²×C₁₀₉

C₂²×C₈

prime radical

C₂³×C₁₀₉

C₂×C₈

C₂³

C₁₀₉

G ▷ C₂×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂×C₄ ◁ #117

N	H ≅ G/N	type
3. #20295914	C₂	semidirect
4. C₄×#13182032	C₂	semidirect
6. #16812773	C₄	semidirect
7. C₂×#13182032	C₄	semidirect
50. C₁₀₉	#2108	semidirect

H ≅ G/N

type

3. #20295914

C₂

semidirect

4. C₄×#13182032

C₂

semidirect

6. #16812773

C₄

semidirect

7. C₂×#13182032

C₄

semidirect

50. C₁₀₉

#2108

semidirect

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

subgroup	index
C₂×#16812773	2
C₂²×#13182032	2
#20295914	2
C₄×#13182032	2
#20294982	2
#2108	109

subgroup

index

109

Subgroups — #28379241

Conjugacy classes of subgroups

Complements

Maximal subgroups