Group #28378228 — subgroups

Subgroups — #28378228

Order 13952.

248 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 126 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	C₄×D₁₀₉⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₄×D₁₀₉⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#11724786⋊C₈	C₂	1	4	yes	maximal, complemented
4	C₄×C₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₂×D₁₀₉⋊C₈	C₄	1	2	yes	—
6	C₂×D₁₀₉⋊C₈	C₄	1	2	yes	—
7	C₄×#11724786	C₄	1	2	yes	—
8	C₄×#11724786	C₄	1	2	yes	—
9	C₄×#11724786	C₂²	1	2	yes	—
10	C₄×#11724786	C₂²	1	2	yes	—
11	C₂×C₈×D₁₀₉	C₄	1	2	yes	—
12	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
13	C₄×#11724784	C₄	1	1	yes	characteristic
14	C₈×Dic₁₀₉	C₂²	1	2	yes	—
15	C₄×C₈×C₁₀₉	C₄	1	1	yes	characteristic
16	D₁₀₉⋊C₈	—	2	4	no	—
17	D₁₀₉⋊C₈	C₈	1	8	yes	complemented
18	C₂×#11724786	C₂×C₄	1	4	yes	—
19	C₂×#11724786	C₂×C₄	1	4	yes	—
20	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
23	C₂×#11724784	C₂×C₄	1	2	yes	—
24	C₈×D₁₀₉	—	2	4	no	—
25	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
26	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
27	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
28	C₂×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
29	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
30	C₂²×D₁₀₉	C₄²	1	1	yes	characteristic
31	C₂×C₄×C₁₀₉	C₄²	1	1	yes	characteristic
32	C₂×C₄×C₁₀₉	C₄²	1	1	yes	characteristic
33	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
34	C₂×Dic₁₀₉	C₄²	1	1	yes	characteristic
35	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
36	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
37	C₄×D₁₀₉	C₂×C₈	1	4	yes	—
38	C₄×D₁₀₉	M₁₆	1	4	yes	—
39	C₄×D₁₀₉	M₁₆	1	4	yes	—
40	#11724786	—	2	4	no	—
41	#11724786	C₂×C₈	1	8	yes	complemented
42	C₈×C₁₀₉	—	2	2	no	—
43	#11724784	—	2	2	no	—
44	C₂²×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
45	C₂×D₁₀₉	C₄×C₈	1	2	yes	—
46	C₂×D₁₀₉	#97	1	2	yes	—
47	C₂×D₁₀₉	#97	1	2	yes	—
48	C₄×C₁₀₉	C₄×C₈	1	2	yes	—
49	C₄×C₁₀₉	#97	1	2	yes	—
50	C₄×C₁₀₉	C₂×M₁₆	1	2	yes	—
51	Dic₁₀₉	#97	1	2	yes	—
52	Dic₁₀₉	C₂²×C₈	1	2	yes	—
53	Dic₁₀₉	C₂×M₁₆	1	2	yes	—
54	C₂×C₁₀₉	C₂×C₄×C₈	1	1	yes	characteristic
55	C₂×C₁₀₉	C₂×#97	1	1	yes	characteristic
56	C₂×C₁₀₉	C₂×#97	1	1	yes	characteristic
57	D₁₀₉	#322	1	4	yes	complemented
58	C₂×#322	—	109	1	no	maximal, Sylow
59	C₁₀₉	C₂×#322	1	1	yes	characteristic, complemented, Sylow
60	C₂×C₄×C₈	—	109	1	no	—
61	C₂×C₄×C₈	—	109	1	no	—
62	C₂×C₄×C₈	—	109	1	no	—
63	#322	—	109	4	no	—
64	C₂²×C₈	—	109	2	no	—
65	C₂²×C₈	—	109	2	no	—
66	C₂²×C₈	—	109	2	no	—
67	C₂×C₄²	—	109	1	no	—
68	C₄×C₈	—	109	2	no	—
69	C₄×C₈	—	109	2	no	—
70	C₄×C₈	—	109	2	no	—
71	C₄×C₈	—	109	2	no	—
72	C₄×C₈	—	109	2	no	—
73	C₄×C₈	—	109	1	no	—
74	C₄×C₈	#89137	1	1	yes	characteristic
75	C₂²×C₄	—	109	1	no	—
76	C₂²×C₄	—	109	1	no	—
77	C₂²×C₄	—	109	1	no	—
78	C₂×C₈	—	109	4	no	—
79	C₂×C₈	—	109	8	no	—
80	C₂×C₈	—	109	4	no	—
81	C₂×C₈	—	109	2	no	—
82	C₂×C₈	—	218	4	no	—
83	C₂×C₈	—	218	4	no	—
84	C₂×C₈	C₂×#89137	1	2	yes	—
85	C₄²	—	109	1	no	—
86	C₄²	—	109	1	no	—
87	C₄²	—	109	1	no	—
88	C₄²	C₂×#89137	1	1	yes	characteristic
89	C₂³	—	109	1	no	—
90	C₂×C₄	—	109	1	no	—
91	C₂×C₄	—	109	1	no	—
92	C₂×C₄	—	109	1	no	—
93	C₂×C₄	—	109	4	no	—
94	C₂×C₄	—	109	4	no	—
95	C₂×C₄	—	109	4	no	—
96	C₂×C₄	C₄×#89137	1	1	yes	characteristic
97	C₂×C₄	C₄×#89137	1	1	yes	characteristic
98	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
99	C₈	—	2	2	no	—
100	C₈	—	109	8	no	—
101	C₈	—	218	4	no	—
102	C₈	—	218	2	no	—
103	C₂²	—	109	2	no	—
104	C₂²	—	109	2	no	—
105	C₂²	—	109	2	no	—
106	C₂²	C₂×C₄×#89137	1	1	yes	characteristic
107	C₄	—	109	2	no	—
108	C₄	—	109	2	no	—
109	C₄	—	109	2	no	—
110	C₄	C₈×#89137	1	2	yes	—
111	C₄	#16812753	1	2	yes	—
112	C₄	D₁₀₉⋊M₁₆	1	2	yes	—
113	C₂	—	109	4	no	—
114	C₂	C₂×C₈×#89137	1	1	yes	characteristic
115	C₂	C₂×#16812753	1	1	yes	characteristic
116	C₂	#20295985	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×D₁₀₉⋊C₈

C₂

yes

maximal, characteristic

C₄×D₁₀₉⋊C₈

C₂

yes

maximal, characteristic

#11724786⋊C₈

C₂

yes

maximal, complemented

C₄×C₈×D₁₀₉

C₂

yes

maximal, characteristic

C₂×D₁₀₉⋊C₈

C₄

yes

—

C₂×D₁₀₉⋊C₈

C₄

yes

—

C₄×#11724786

C₄

yes

—

C₄×#11724786

C₄

yes

—

C₄×#11724786

C₂²

yes

—

C₄×#11724786

C₂²

yes

—

C₂×C₈×D₁₀₉

C₄

yes

—

C₄²×D₁₀₉

C₂²

yes

characteristic

C₄×#11724784

C₄

yes

characteristic

C₈×Dic₁₀₉

C₂²

yes

—

C₄×C₈×C₁₀₉

C₄

yes

characteristic

D₁₀₉⋊C₈

—

D₁₀₉⋊C₈

C₈

yes

complemented

C₂×#11724786

C₂×C₄

yes

—

C₂×#11724786

C₂×C₄

yes

—

C₂×C₄×D₁₀₉

C₂×C₄

yes

characteristic

C₂×C₄×D₁₀₉

C₂×C₄

yes

characteristic

C₂×C₄×D₁₀₉

C₂×C₄

yes

characteristic

C₂×#11724784

C₂×C₄

yes

—

C₈×D₁₀₉

—

C₄×Dic₁₀₉

C₂×C₄

yes

characteristic

C₄×Dic₁₀₉

C₂×C₄

yes

characteristic

C₄×Dic₁₀₉

C₂³

yes

characteristic

C₂×C₈×C₁₀₉

C₂×C₄

yes

—

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×D₁₀₉

C₄²

yes

characteristic

C₂×C₄×C₁₀₉

C₄²

yes

characteristic

C₂×C₄×C₁₀₉

C₄²

yes

characteristic

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₉

C₄²

yes

characteristic

C₂×Dic₁₀₉

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₉

C₂²×C₄

yes

characteristic

C₄×D₁₀₉

C₂×C₈

yes

—

C₄×D₁₀₉

M₁₆

yes

—

C₄×D₁₀₉

M₁₆

yes

—

#11724786

—

#11724786

C₂×C₈

yes

complemented

C₈×C₁₀₉

—

#11724784

—

C₂²×C₁₀₉

C₂×C₄²

yes

characteristic

C₂×D₁₀₉

C₄×C₈

yes

—

C₂×D₁₀₉

#97

yes

—

C₂×D₁₀₉

#97

yes

—

C₄×C₁₀₉

C₄×C₈

yes

—

C₄×C₁₀₉

#97

yes

—

C₄×C₁₀₉

C₂×M₁₆

yes

—

Dic₁₀₉

#97

yes

—

Dic₁₀₉

C₂²×C₈

yes

—

Dic₁₀₉

C₂×M₁₆

yes

—

C₂×C₁₀₉

C₂×C₄×C₈

yes

characteristic

C₂×C₁₀₉

C₂×#97

yes

characteristic

C₂×C₁₀₉

C₂×#97

yes

D₁₀₉

yes

—

109

maximal, Sylow

C₁₀₉

C₂×#322

yes

characteristic, complemented, Sylow

C₂×C₄×C₈

—

109

—

C₂×C₄×C₈

—

109

—

C₂×C₄×C₈

—

109

—

#322

—

109

—

C₂²×C₈

—

109

—

C₂²×C₈

—

109

—

C₂²×C₈

—

109

—

C₂×C₄²

—

109

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₄×C₈

#89137

yes

characteristic

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₂×C₈

C₂×#89137

yes

—

C₄²

—

109

—

C₄²

—

109

—

C₄²

—

109

—

C₄²

C₂×#89137

yes

characteristic

C₂³

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂²×#89137

yes

characteristic

C₈

—

100

C₈

—

109

—

101

C₈

—

218

—

102

C₈

—

218

—

103

C₂²

—

109

—

104

C₂²

—

109

—

105

C₂²

—

109

—

106

C₂²

C₂×C₄×#89137

yes

characteristic

107

C₄

—

109

—

108

C₄

—

109

—

109

C₄

—

109

—

110

C₄

C₈×#89137

yes

—

111

C₄

#16812753

yes

—

112

C₄

D₁₀₉⋊M₁₆

yes

—

113

C₂

—

109

—

114

C₂

C₂×C₈×#89137

yes

characteristic

115

C₂

C₂×#16812753

yes

characteristic

116

C₂

#20295985

yes

characteristic

	subgroup	G/H
center	C₄²	C₂×#89137
commutator	C₂×C₁₀₉	C₂×C₄×C₈
Frattini	C₂×C₄	C₂²×#89137
Fitting	C₄×C₈×C₁₀₉	C₄
socle	C₂²×C₁₀₉	C₂×C₄²
prime radical	C₂²×D₁₀₉	C₄²
cosocle	C₄×Dic₁₀₉	C₂³
hypercenter	C₄×C₈	#89137
hypocenter	C₁₀₉	C₂×#322
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₄² ◁ C₄×C₈

subgroup

G/H

center

C₄²

C₂×#89137

commutator

C₂×C₁₀₉

C₂×C₄×C₈

Frattini

C₂×C₄

C₂²×#89137

Fitting

C₄×C₈×C₁₀₉

C₄

socle

C₂²×C₁₀₉

C₂×C₄²

prime radical

C₂²×D₁₀₉

C₄²

cosocle

C₄×Dic₁₀₉

C₂³

C₄×C₈

C₁₀₉

G ▷ C₂×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₄² ◁ C₄×C₈

N	H ≅ G/N	type
3. #11724786⋊C₈	C₂	semidirect
17. D₁₀₉⋊C₈	C₈	semidirect
41. #11724786	C₂×C₈	semidirect
57. D₁₀₉	#322	semidirect
59. C₁₀₉	C₂×#322	semidirect

H ≅ G/N

type

3. #11724786⋊C₈

C₂

semidirect

17. D₁₀₉⋊C₈

C₈

semidirect

41. #11724786

C₂×C₈

semidirect

57. D₁₀₉

#322

semidirect

59. C₁₀₉

C₂×#322

semidirect

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

subgroup	index
C₄×D₁₀₉⋊C₈	2
#11724786⋊C₈	2
C₄×C₈×D₁₀₉	2
C₂×#322	109

subgroup

index

C₄×D₁₀₉⋊C₈

#11724786⋊C₈

C₄×C₈×D₁₀₉

C₂×#322

109

Subgroups — #28378228

Conjugacy classes of subgroups

Complements

Maximal subgroups