Group #28362889 — subgroups

Subgroups — #28362889

Order 13952.

260 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 141 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⋊#11724784	C₂	1	2	yes	maximal
2	C₂×#16812610	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×#11724784	C₂	1	1	yes	maximal, characteristic
4	#20294781	C₂	1	2	yes	maximal
5	#376×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11724784	C₂²	1	2	yes	—
7	C₂²×#11724784	C₂²	1	2	yes	—
8	C₄⋊#11724784	C₄	1	8	yes	complemented
9	#16812610	—	2	2	no	—
10	C₄×#11724784	—	2	2	no	—
11	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
12	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
13	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
14	#95×C₁₀₉	C₄	1	4	yes	—
15	C₂×#11724784	—	2	2	no	—
16	C₂×#11724784	—	2	2	no	—
17	C₂×#11724784	—	2	2	no	—
18	C₂×#11724784	—	2	2	no	—
19	C₂×#11724784	C₂×C₄	1	4	yes	—
20	C₂×#11724784	C₂×C₄	1	4	yes	—
21	C₂²×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
25	C₄²×C₁₀₉	—	2	2	no	—
26	C₄²×C₁₀₉	C₂×C₄	1	4	yes	—
27	C₄²×C₁₀₉	D₄	1	2	yes	—
28	C₄²×C₁₀₉	Q₈	1	2	yes	—
29	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
30	C₂×C₄×C₁₀₉	—	2	4	no	—
31	C₂×C₄×C₁₀₉	—	2	4	no	—
32	C₂×C₄×C₁₀₉	—	2	2	no	—
33	C₂×C₄×C₁₀₉	—	2	4	no	—
34	C₂×C₄×C₁₀₉	C₄²	1	4	yes	—
35	C₂×C₄×C₁₀₉	C₄⋊C₄	1	4	yes	—
36	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
37	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₄×C₁₀₉	#41	1	2	yes	—
40	#11724784	—	2	4	no	—
41	#11724784	—	2	4	no	—
42	C₂²×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
43	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₁₀₉	#117	1	1	yes	characteristic
45	C₂²×C₁₀₉	#131	1	4	yes	—
46	C₄×C₁₀₉	—	2	4	no	—
47	C₄×C₁₀₉	—	2	4	no	—
48	C₄×C₁₀₉	—	4	2	no	—
49	C₄×C₁₀₉	C₄×D₄	1	2	yes	—
50	C₄×C₁₀₉	C₄×Q₈	1	2	yes	—
51	C₂×C₁₀₉	C₄×C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₁₀₉	#405	1	2	yes	—
53	C₂×C₁₀₉	#424	1	2	yes	—
54	C₂×C₁₀₉	#433	1	2	yes	—
55	#1768	—	109	1	no	maximal, Sylow
56	C₁₀₉	#1768	1	1	yes	characteristic, complemented, Sylow
57	C₂×C₄⋊C₈	—	109	2	no	—
58	C₂×#97	—	109	1	no	—
59	C₂×C₄×C₈	—	109	1	no	—
60	#376	D₁₀₉	1	1	yes	characteristic
61	#336	—	109	2	no	—
62	C₂²×C₈	—	109	2	no	—
63	C₂²×C₈	—	109	2	no	—
64	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
65	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
66	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
67	C₄⋊C₈	—	109	8	no	—
68	#97	—	218	2	no	—
69	C₄×C₈	—	218	2	no	—
70	#95	Dic₁₀₉	1	4	yes	—
71	C₂²×C₄	C₂×Dic₁₀₉	1	2	yes	—
72	C₂²×C₄	C₂×Dic₁₀₉	1	2	yes	—
73	C₂²×C₄	C₂×Dic₁₀₉	1	2	yes	—
74	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
75	C₂×C₈	—	109	4	no	—
76	C₂×C₈	—	109	4	no	—
77	C₂×C₈	—	218	2	no	—
78	C₂×C₈	—	218	2	no	—
79	C₂×C₈	—	218	2	no	—
80	C₂×C₈	—	218	2	no	—
81	C₄²	—	2	2	no	—
82	C₄²	Dic₂₁₈	1	2	yes	—
83	C₄²	C₄×D₁₀₉	1	4	yes	—
84	C₄²	D₄₃₆	1	2	yes	—
85	C₂³	C₂²×Dic₁₀₉	1	1	yes	characteristic
86	C₂×C₄	—	2	4	no	—
87	C₂×C₄	—	2	4	no	—
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	2	4	no	—
90	C₂×C₄	C₄×Dic₁₀₉	1	4	yes	—
91	C₂×C₄	C₄⋊Dic₁₀₉	1	4	yes	—
92	C₂×C₄	C₂×C₄×D₁₀₉	1	2	yes	—
93	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
94	C₂×C₄	C₂×Dic₂₁₈	1	1	yes	characteristic
95	C₂×C₄	#13182029	1	2	yes	—
96	C₈	—	218	4	no	—
97	C₈	—	218	4	no	—
98	C₂²	C₂×C₄×Dic₁₀₉	1	1	yes	characteristic
99	C₂²	C₂×C₄⋊Dic₁₀₉	1	1	yes	characteristic
100	C₂²	#16812658	1	1	yes	characteristic
101	C₂²	#16812706	1	4	yes	—
102	C₄	—	2	4	no	—
103	C₄	—	2	4	no	—
104	C₄	—	4	2	no	—
105	C₄	C₄×D₄₃₆	1	2	yes	—
106	C₄	C₄×Dic₂₁₈	1	2	yes	—
107	C₂	C₄×C₄⋊Dic₁₀₉	1	1	yes	characteristic
108	C₂	#20294982	1	2	yes	—
109	C₂	#20295053	1	2	yes	—
110	C₂	#20295087	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#11724784

C₂

yes

maximal

C₂×#16812610

C₂

yes

maximal, characteristic

C₂×C₄×#11724784

C₂

yes

maximal, characteristic

#20294781

C₂

yes

maximal

#376×C₁₀₉

C₂

yes

maximal, characteristic

C₂²×#11724784

C₂²

yes

—

C₂²×#11724784

C₂²

yes

—

C₄⋊#11724784

C₄

yes

complemented

#16812610

—

C₄×#11724784

—

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

#95×C₁₀₉

C₄

yes

—

—

—

—

—

C₂×C₄

yes

—

C₂×#11724784

C₂×C₄

yes

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₄²×C₁₀₉

—

C₄²×C₁₀₉

C₂×C₄

yes

—

C₄²×C₁₀₉

D₄

yes

—

C₄²×C₁₀₉

Q₈

yes

—

C₂³×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₄²

yes

—

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₉

#41

yes

—

#11724784

—

#11724784

—

C₂²×C₁₀₉

C₂×C₄²

yes

characteristic

C₂²×C₁₀₉

C₂×C₄⋊C₄

yes

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₄×D₄

yes

—

C₄×C₁₀₉

C₄×Q₈

yes

—

C₂×C₁₀₉

C₄×C₄⋊C₄

yes

characteristic

C₂×C₁₀₉

#405

yes

—

C₂×C₁₀₉

#424

yes

—

C₂×C₁₀₉

#433

yes

—

#1768

—

109

maximal, Sylow

C₁₀₉

#1768

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

109

—

C₂×#97

—

109

—

C₂×C₄×C₈

—

109

—

#376

D₁₀₉

yes

characteristic

#336

—

109

—

C₂²×C₈

—

109

—

C₂²×C₈

—

109

—

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

C₄⋊C₈

—

109

—

#97

—

218

—

C₄×C₈

—

218

—

#95

Dic₁₀₉

yes

—

C₂²×C₄

C₂×Dic₁₀₉

yes

—

C₂²×C₄

C₂×Dic₁₀₉

yes

—

C₂²×C₄

C₂×Dic₁₀₉

yes

—

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₄²

—

C₄²

Dic₂₁₈

yes

—

C₄²

C₄×D₁₀₉

yes

—

C₄²

D₄₃₆

yes

—

C₂³

C₂²×Dic₁₀₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×Dic₁₀₉

yes

—

C₂×C₄

C₄⋊Dic₁₀₉

yes

—

C₂×C₄

C₂×C₄×D₁₀₉

yes

—

C₂×C₄

C₂×D₄₃₆

yes

characteristic

C₂×C₄

C₂×Dic₂₁₈

yes

characteristic

C₂×C₄

#13182029

yes

—

C₈

—

218

—

C₈

—

218

—

C₂²

C₂×C₄×Dic₁₀₉

yes

characteristic

C₂²

C₂×C₄⋊Dic₁₀₉

yes

characteristic

100

C₂²

#16812658

yes

characteristic

101

C₂²

#16812706

yes

—

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

C₄×D₄₃₆

yes

—

106

C₄

C₄×Dic₂₁₈

yes

—

107

C₂

C₄×C₄⋊Dic₁₀₉

yes

characteristic

108

C₂

#20294982

yes

—

109

C₂

#20295053

yes

—

110

C₂

#20295087

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₀₉
commutator	C₂²×C₁₀₉	C₂×C₄²
Fitting	#376×C₁₀₉	C₂
socle, prime radical	C₂³×C₁₀₉	C₂²×C₄
cosocle	C₂²×C₄×C₁₀₉	C₂³
hypercenter	#376	D₁₀₉
hypocenter	C₁₀₉	#1768
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂²×C₄ ◁ #376

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₀₉

commutator

C₂²×C₁₀₉

C₂×C₄²

Fitting

#376×C₁₀₉

C₂

socle, prime radical

C₂³×C₁₀₉

C₂²×C₄

cosocle

C₂²×C₄×C₁₀₉

C₂³

D₁₀₉

C₁₀₉

G ▷ C₂²×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂²×C₄ ◁ #376

N	H ≅ G/N	type
8. C₄⋊#11724784	C₄	semidirect
56. C₁₀₉	#1768	semidirect

H ≅ G/N

type

8. C₄⋊#11724784

C₄

semidirect

56. C₁₀₉

#1768

semidirect

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

subgroup	index
C₂×C₄⋊#11724784	2
C₂×#16812610	2
C₂×C₄×#11724784	2
#20294781	2
#376×C₁₀₉	2
#1768	109

subgroup

index

#376×C₁₀₉

109

Subgroups — #28362889

Conjugacy classes of subgroups

Complements

Maximal subgroups