Group #28379393 — subgroups

Subgroups — #28379393

Order 13952.

152 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 78 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⋊#13182032	C₂	1	2	yes	maximal, complemented
2	C₂²⋊#13182032	C₂	1	2	yes	maximal
3	#20295914	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×#11724784	C₂	1	1	yes	maximal, characteristic
5	C₂×#13182032	C₂²	1	4	yes	—
6	C₂²×#11724784	C₄	1	1	yes	characteristic
7	C₂²×#11724784	C₂²	1	1	yes	characteristic
8	C₄×#11724784	C₄	1	1	yes	characteristic
9	C₄×#11724784	C₄	1	1	yes	characteristic
10	C₄×#11724784	C₂²	1	1	yes	characteristic
11	C₄×#11724784	C₂²	1	1	yes	characteristic
12	C₂×C₄²×C₁₀₉	C₄	1	1	yes	characteristic
13	#13182032	—	2	4	no	—
14	C₂×#11724784	—	2	2	no	—
15	C₂×#11724784	—	2	2	no	—
16	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
17	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
18	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
19	C₂×#11724784	C₂³	1	1	yes	characteristic
20	C₂²×C₄×C₁₀₉	C₈	1	1	yes	characteristic
21	C₂²×C₄×C₁₀₉	C₈	1	1	yes	characteristic
22	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
23	C₄²×C₁₀₉	C₈	1	1	yes	characteristic
24	C₄²×C₁₀₉	C₈	1	1	yes	characteristic
25	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
26	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₁₀₉	C₂×C₈	1	1	yes	characteristic
28	C₂×C₄×C₁₀₉	—	2	2	no	—
29	C₂×C₄×C₁₀₉	—	2	2	no	—
30	C₂×C₄×C₁₀₉	—	2	1	no	—
31	C₂×C₄×C₁₀₉	—	2	1	no	—
32	C₂×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
33	C₂×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
34	C₂×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
35	C₂×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
36	C₂×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
37	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
38	#11724784	—	2	1	no	—
39	#11724784	—	2	1	no	—
40	#11724784	M₁₆	1	2	yes	—
41	#11724784	#41	1	2	yes	—
42	C₂²×C₁₀₉	—	2	1	no	—
43	C₂²×C₁₀₉	—	2	1	no	—
44	C₂²×C₁₀₉	M₃₂	1	2	yes	—
45	C₂²×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
46	C₄×C₁₀₉	—	2	1	no	—
47	C₄×C₁₀₉	—	2	1	no	—
48	C₄×C₁₀₉	—	2	1	no	—
49	C₄×C₁₀₉	—	2	1	no	—
50	C₄×C₁₀₉	M₃₂	1	2	yes	—
51	C₄×C₁₀₉	#117	1	1	yes	characteristic
52	C₄×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
53	C₂×C₁₀₉	—	2	2	no	—
54	C₂×C₁₀₉	#431	1	1	yes	characteristic
55	C₂×C₁₀₉	C₂×M₃₂	1	1	yes	characteristic
56	C₂×C₁₀₉	C₂×M₃₂	1	1	yes	characteristic
57	#2163	—	109	1	no	maximal, Sylow
58	C₁₀₉	#2163	1	1	yes	characteristic, complemented, Sylow
59	C₂×C₄×C₈	—	109	1	no	—
60	C₄⋊C₁₆	—	109	2	no	—
61	C₂²⋊C₁₆	—	109	2	no	—
62	#346	—	109	2	no	—
63	C₂²×C₈	—	109	1	no	—
64	C₂²×C₈	—	109	1	no	—
65	C₂×C₄²	#89137	1	1	yes	characteristic
66	C₂×C₁₆	—	109	4	no	—
67	C₄×C₈	—	109	1	no	—
68	C₄×C₈	—	109	1	no	—
69	C₄×C₈	—	109	1	no	—
70	C₄×C₈	—	109	1	no	—
71	C₂²×C₄	#11724786	1	1	yes	characteristic
72	C₂²×C₄	#11724786	1	1	yes	characteristic
73	C₂²×C₄	C₂×#89137	1	1	yes	characteristic
74	C₂×C₈	—	109	1	no	—
75	C₂×C₈	—	109	1	no	—
76	C₂×C₈	—	109	1	no	—
77	C₂×C₈	—	109	1	no	—
78	C₂×C₈	—	218	2	no	—
79	C₂×C₈	—	218	2	no	—
80	C₄²	#11724786	1	1	yes	characteristic
81	C₄²	#11724786	1	1	yes	characteristic
82	C₄²	C₂×#89137	1	1	yes	characteristic
83	C₄²	C₂×#89137	1	1	yes	characteristic
84	C₁₆	—	218	4	no	—
85	C₂³	C₂×#11724786	1	1	yes	characteristic
86	C₂×C₄	—	2	2	no	—
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	2	1	no	—
90	C₂×C₄	C₂×#11724786	1	1	yes	characteristic
91	C₂×C₄	C₂×#11724786	1	1	yes	characteristic
92	C₂×C₄	C₂×#11724786	1	1	yes	characteristic
93	C₂×C₄	C₂×#11724786	1	1	yes	characteristic
94	C₂×C₄	C₂×#11724786	1	1	yes	characteristic
95	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
96	C₈	—	109	2	no	—
97	C₈	—	109	2	no	—
98	C₈	—	218	1	no	—
99	C₈	—	218	1	no	—
100	C₂²	—	2	1	no	—
101	C₂²	—	2	1	no	—
102	C₂²	#16812773	1	2	yes	—
103	C₂²	C₂²×#11724786	1	1	yes	characteristic
104	C₄	—	2	1	no	—
105	C₄	—	2	1	no	—
106	C₄	—	2	1	no	—
107	C₄	—	2	1	no	—
108	C₄	#16812773	1	2	yes	—
109	C₄	#16812778	1	1	yes	characteristic
110	C₄	D₁₀₉⋊M₁₆	1	1	yes	characteristic
111	C₂	—	2	2	no	—
112	C₂	#20296042	1	1	yes	characteristic
113	C₂	C₂×#16812773	1	1	yes	characteristic
114	C₂	C₂×#16812773	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄⋊#13182032

C₂

yes

maximal, complemented

C₂²⋊#13182032

C₂

yes

maximal

#20295914

C₂

yes

maximal, complemented

C₂×C₄×#11724784

C₂

yes

maximal, characteristic

C₂×#13182032

C₂²

yes

—

C₂²×#11724784

C₄

yes

characteristic

C₂²×#11724784

C₂²

yes

characteristic

C₄×#11724784

C₄

yes

characteristic

C₄×#11724784

C₄

yes

characteristic

C₄×#11724784

C₂²

yes

characteristic

C₄×#11724784

C₂²

yes

characteristic

C₂×C₄²×C₁₀₉

C₄

yes

—

—

—

C₂×C₄

yes

characteristic

C₂×#11724784

C₂×C₄

yes

characteristic

C₂×#11724784

C₂×C₄

yes

characteristic

C₂×#11724784

C₂³

yes

characteristic

C₂²×C₄×C₁₀₉

C₈

yes

characteristic

C₂²×C₄×C₁₀₉

C₈

yes

characteristic

C₂²×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₄²×C₁₀₉

C₈

yes

characteristic

C₄²×C₁₀₉

C₈

yes

characteristic

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₂³×C₁₀₉

C₂×C₈

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×C₈

yes

characteristic

C₂×C₄×C₁₀₉

C₂×C₈

yes

characteristic

C₂×C₄×C₁₀₉

C₂×C₈

yes

characteristic

C₂×C₄×C₁₀₉

C₂×C₈

yes

characteristic

C₂×C₄×C₁₀₉

C₂×C₈

yes

characteristic

C₂×C₄×C₁₀₉

C₂²×C₄

yes

—

—

M₁₆

yes

—

#11724784

#41

yes

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

M₃₂

yes

—

C₂²×C₁₀₉

C₂²×C₈

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

M₃₂

yes

—

C₄×C₁₀₉

#117

yes

characteristic

C₄×C₁₀₉

C₂×M₁₆

yes

characteristic

C₂×C₁₀₉

—

C₂×C₁₀₉

#431

yes

characteristic

C₂×C₁₀₉

C₂×M₃₂

yes

characteristic

C₂×C₁₀₉

C₂×M₃₂

yes

characteristic

#2163

—

109

maximal, Sylow

C₁₀₉

#2163

yes

characteristic, complemented, Sylow

C₂×C₄×C₈

—

109

—

C₄⋊C₁₆

—

109

—

C₂²⋊C₁₆

—

109

—

#346

—

109

—

C₂²×C₈

—

109

—

C₂²×C₈

—

109

—

C₂×C₄²

#89137

yes

characteristic

C₂×C₁₆

—

109

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

yes

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₄²

yes

C₄²

yes

C₄²

yes

C₄²

yes

C₁₆

—

218

—

C₂³

C₂×#11724786

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂²×#89137

yes

characteristic

C₈

—

109

—

C₈

—

109

—

C₈

—

218

—

C₈

—

218

—

100

C₂²

—

101

C₂²

—

102

C₂²

#16812773

yes

—

103

C₂²

C₂²×#11724786

yes

characteristic

104

C₄

—

105

C₄

—

106

C₄

—

107

C₄

—

108

C₄

#16812773

yes

—

109

C₄

#16812778

yes

characteristic

110

C₄

D₁₀₉⋊M₁₆

yes

characteristic

111

C₂

—

112

C₂

#20296042

yes

characteristic

113

C₂

C₂×#16812773

yes

characteristic

114

C₂

C₂×#16812773

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#89137
commutator, socle	C₂²×C₁₀₉	C₂²×C₈
Fitting	C₂×C₄²×C₁₀₉	C₄
prime radical	C₂³×C₁₀₉	C₂×C₈
cosocle	C₂×#11724784	C₂³
hypercenter	C₂×C₄²	#89137
hypocenter	C₁₀₉	#2163
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄²

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×#89137

commutator, socle

C₂²×C₁₀₉

C₂²×C₈

Fitting

C₂×C₄²×C₁₀₉

C₄

prime radical

C₂³×C₁₀₉

C₂×C₈

C₂³

C₂×C₄²

C₁₀₉

G ▷ C₂²×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂×C₄ ◁ C₂×C₄²

N	H ≅ G/N	type
1. C₄⋊#13182032	C₂	semidirect
3. #20295914	C₂	semidirect
58. C₁₀₉	#2163	semidirect

H ≅ G/N

type

1. C₄⋊#13182032

C₂

semidirect

3. #20295914

C₂

semidirect

58. C₁₀₉

#2163

semidirect

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

subgroup	index
C₄⋊#13182032	2
C₂²⋊#13182032	2
#20295914	2
C₂×C₄×#11724784	2
#2163	109

subgroup

index

109

Subgroups — #28379393

Conjugacy classes of subgroups

Complements

Maximal subgroups