Group #28378070 — subgroups

Subgroups — #28378070

Order 13952.

160 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 90 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	C₂²×#13182032	C₂	1	2	yes	maximal
2	C₂×C₄×#11724784	C₂	1	1	yes	maximal, characteristic
3	C₂×#13182032	—	2	2	no	—
4	C₂×#13182032	—	2	2	no	—
5	C₂×#13182032	C₄	1	4	yes	complemented
6	C₂²×#11724784	C₄	1	1	yes	characteristic
7	C₂²×#11724784	C₂²	1	1	yes	characteristic
8	C₄×#11724784	—	2	2	no	—
9	C₂×C₄²×C₁₀₉	C₄	1	1	yes	characteristic
10	#13182032	—	2	4	no	—
11	C₂×#11724784	—	2	1	no	—
12	C₂×#11724784	—	2	1	no	—
13	C₂×#11724784	C₈	1	2	yes	—
14	C₂×#11724784	C₂×C₄	1	2	yes	—
15	C₂×#11724784	D₄	1	1	yes	characteristic
16	C₂×#11724784	D₄	1	2	yes	—
17	C₂×#11724784	Q₈	1	1	yes	characteristic
18	C₂²×C₄×C₁₀₉	C₈	1	2	yes	—
19	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
20	C₄²×C₁₀₉	—	2	1	no	—
21	C₄²×C₁₀₉	—	2	1	no	—
22	C₂³×C₁₀₉	C₂×C₈	1	1	yes	characteristic
23	C₂×C₄×C₁₀₉	—	2	2	no	—
24	C₂×C₄×C₁₀₉	—	2	2	no	—
25	C₂×C₄×C₁₀₉	C₁₆	1	4	yes	complemented
26	C₂×C₄×C₁₀₉	C₄²	1	1	yes	characteristic
27	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
28	C₂×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
29	C₂×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
30	C₂×C₄×C₁₀₉	M₁₆	1	1	yes	characteristic
31	C₂×C₄×C₁₀₉	M₁₆	1	1	yes	characteristic
32	#11724784	—	2	2	no	—
33	#11724784	C₂²⋊C₄	1	2	yes	—
34	#11724784	C₄⋊C₄	1	2	yes	—
35	C₂²×C₁₀₉	C₄×C₈	1	1	yes	characteristic
36	C₂²×C₁₀₉	C₂²⋊C₈	1	1	yes	characteristic
37	C₂²×C₁₀₉	C₄⋊C₈	1	1	yes	characteristic
38	C₂²×C₁₀₉	C₂×C₁₆	1	2	yes	—
39	C₂²×C₁₀₉	M₃₂	1	2	yes	—
40	C₄×C₁₀₉	—	2	4	no	—
41	C₄×C₁₀₉	#95	1	1	yes	characteristic
42	C₄×C₁₀₉	#97	1	1	yes	characteristic
43	C₄×C₁₀₉	C₂²⋊C₈	1	1	yes	characteristic
44	C₄×C₁₀₉	C₄⋊C₈	1	1	yes	characteristic
45	C₂×C₁₀₉	#336	1	1	yes	characteristic
46	C₂×C₁₀₉	C₄×C₁₆	1	1	yes	characteristic
47	C₂×C₁₀₉	#346	1	1	yes	characteristic
48	C₂×C₁₀₉	C₂²⋊C₁₆	1	2	yes	—
49	C₂×C₁₀₉	C₄⋊C₁₆	1	2	yes	—
50	#1374	—	109	1	no	maximal, Sylow
51	C₁₀₉	#1374	1	1	yes	characteristic, complemented, Sylow
52	C₂²×C₁₆	—	109	2	no	—
53	C₂×C₄×C₈	—	109	1	no	—
54	C₂²×C₈	—	109	1	no	—
55	C₂²×C₈	—	109	1	no	—
56	C₂×C₄²	#89137	1	1	yes	characteristic
57	C₂×C₁₆	—	109	4	no	—
58	C₂×C₁₆	—	218	2	no	—
59	C₂×C₁₆	—	218	2	no	—
60	C₄×C₈	—	218	2	no	—
61	C₂²×C₄	#11724786	1	2	yes	—
62	C₂²×C₄	C₂×#89137	1	1	yes	characteristic
63	C₂×C₈	—	109	2	no	—
64	C₂×C₈	—	109	2	no	—
65	C₂×C₈	—	109	1	no	—
66	C₂×C₈	—	109	1	no	—
67	C₂×C₈	—	109	2	no	—
68	C₂×C₈	—	218	1	no	—
69	C₂×C₈	—	218	1	no	—
70	C₄²	—	2	1	no	—
71	C₄²	—	2	1	no	—
72	C₁₆	—	218	4	no	—
73	C₂³	C₂×#11724786	1	1	yes	characteristic
74	C₂×C₄	—	2	2	no	—
75	C₂×C₄	—	2	2	no	—
76	C₂×C₄	#13182032	1	4	yes	complemented
77	C₂×C₄	C₄×#89137	1	1	yes	characteristic
78	C₂×C₄	C₂²⋊#89137	1	1	yes	characteristic
79	C₂×C₄	C₄⋊#89137	1	1	yes	characteristic
80	C₂×C₄	D₁₀₉⋊C₈	1	1	yes	characteristic
81	C₂×C₄	#13182038	1	1	yes	characteristic
82	C₂×C₄	#13182039	1	1	yes	characteristic
83	C₈	—	109	2	no	—
84	C₈	—	109	2	no	—
85	C₈	—	218	2	no	—
86	C₂²	C₄×#11724786	1	1	yes	characteristic
87	C₂²	C₂²⋊#11724786	1	1	yes	characteristic
88	C₂²	C₄⋊#11724786	1	1	yes	characteristic
89	C₂²	C₂×#13182032	1	2	yes	—
90	C₂²	#16812773	1	2	yes	—
91	C₄	—	2	4	no	—
92	C₄	(C₂×#89137)⋊C₄	1	1	yes	characteristic
93	C₄	#16812754	1	1	yes	characteristic
94	C₄	#16812756	1	1	yes	characteristic
95	C₄	#16812766	1	1	yes	characteristic
96	C₂	#20295892	1	1	yes	characteristic
97	C₂	C₄×#13182032	1	1	yes	characteristic
98	C₂	#20295914	1	1	yes	characteristic
99	C₂	C₂²⋊#13182032	1	2	yes	—
100	C₂	C₄⋊#13182032	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#13182032

C₂

yes

maximal

C₂×C₄×#11724784

C₂

yes

maximal, characteristic

C₂×#13182032

—

C₂×#13182032

—

C₂×#13182032

C₄

yes

complemented

C₂²×#11724784

C₄

yes

characteristic

C₂²×#11724784

C₂²

yes

characteristic

C₄×#11724784

—

C₂×C₄²×C₁₀₉

C₄

yes

—

—

—

C₈

yes

—

C₂×#11724784

C₂×C₄

yes

—

C₂×#11724784

D₄

yes

characteristic

C₂×#11724784

D₄

yes

—

C₂×#11724784

Q₈

yes

characteristic

C₂²×C₄×C₁₀₉

C₈

yes

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₄²×C₁₀₉

—

C₄²×C₁₀₉

—

C₂³×C₁₀₉

C₂×C₈

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₁₆

yes

complemented

C₂×C₄×C₁₀₉

C₄²

yes

characteristic

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×C₈

yes

characteristic

C₂×C₄×C₁₀₉

M₁₆

yes

characteristic

C₂×C₄×C₁₀₉

M₁₆

yes

characteristic

#11724784

—

#11724784

C₂²⋊C₄

yes

—

#11724784

C₄⋊C₄

yes

—

C₂²×C₁₀₉

C₄×C₈

yes

characteristic

C₂²×C₁₀₉

C₂²⋊C₈

yes

characteristic

C₂²×C₁₀₉

C₄⋊C₈

yes

characteristic

C₂²×C₁₀₉

C₂×C₁₆

yes

—

C₂²×C₁₀₉

M₃₂

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

C₄×C₁₀₉

C₂²⋊C₈

yes

characteristic

C₄×C₁₀₉

C₄⋊C₈

yes

characteristic

C₂×C₁₀₉

#336

yes

characteristic

C₂×C₁₀₉

C₄×C₁₆

yes

characteristic

C₂×C₁₀₉

#346

yes

characteristic

C₂×C₁₀₉

C₂²⋊C₁₆

yes

—

C₂×C₁₀₉

C₄⋊C₁₆

yes

—

#1374

—

109

maximal, Sylow

C₁₀₉

#1374

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

109

—

C₂×C₄×C₈

—

109

—

C₂²×C₈

—

109

—

C₂²×C₈

—

109

—

C₂×C₄²

#89137

yes

characteristic

C₂×C₁₆

—

109

—

C₂×C₁₆

—

218

—

C₂×C₁₆

—

218

—

C₄×C₈

—

218

—

C₂²×C₄

#11724786

yes

—

C₂²×C₄

C₂×#89137

yes

characteristic

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₄²

—

C₄²

—

C₁₆

—

218

—

C₂³

C₂×#11724786

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

D₁₀₉⋊C₈

yes

C₂×C₄

yes

C₂×C₄

yes

C₈

—

109

—

C₈

—

109

—

C₈

—

218

—

C₂²

yes

C₂²

yes

C₂²

yes

C₂²

yes

—

C₂²

#16812773

yes

—

C₄

—

C₄

yes

C₄

yes

C₄

yes

C₄

yes

C₂

yes

C₂

yes

C₂

yes

C₂

yes

—

100

C₂

C₄⋊#13182032

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂×#89137
commutator	C₂×C₁₀₉	C₄×C₁₆
Fitting	C₂×C₄²×C₁₀₉	C₄
socle, prime radical	C₂³×C₁₀₉	C₂×C₈
cosocle	C₂²×#11724784	C₂²
hypercenter	C₂×C₄²	#89137
hypocenter	C₁₀₉	#1374
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄²

subgroup

G/H

center, Frattini

C₂²×C₄

C₂×#89137

commutator

C₂×C₁₀₉

C₄×C₁₆

Fitting

C₂×C₄²×C₁₀₉

C₄

socle, prime radical

C₂³×C₁₀₉

C₂×C₈

C₂²

C₂×C₄²

C₁₀₉

G ▷ C₂×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂²×C₄ ◁ C₂×C₄²

N	H ≅ G/N	type
5. C₂×#13182032	C₄	semidirect
25. C₂×C₄×C₁₀₉	C₁₆	semidirect
51. C₁₀₉	#1374	semidirect
76. C₂×C₄	#13182032	semidirect

H ≅ G/N

type

5. C₂×#13182032

C₄

semidirect

25. C₂×C₄×C₁₀₉

C₁₆

semidirect

51. C₁₀₉

#1374

semidirect

76. C₂×C₄

#13182032

semidirect

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

subgroup	index
C₂²×#13182032	2
C₂×C₄×#11724784	2
#1374	109

subgroup

index

C₂²×#13182032

C₂×C₄×#11724784

#1374

109

Subgroups — #28378070

Conjugacy classes of subgroups

Complements

Maximal subgroups