Group #28378849 — subgroups

Subgroups — C₄×#16812769 #28378849

Order 13952.

228 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 98 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×#16812769	C₂	1	2	yes	maximal
2	C₄×#13182039	C₂	1	2	yes	maximal
3	#20295902	C₂	1	2	yes	maximal
4	C₄×C₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#13182039	C₂²	1	4	yes	—
6	#16812769	C₄	1	8	yes	direct product factor
7	#16812754	—	2	2	no	—
8	C₄×#11724786	—	2	2	no	—
9	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
10	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
11	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
12	C₄×#11724784	C₄	1	1	yes	characteristic
13	C₈×Dic₁₀₉	C₄	1	1	yes	characteristic
14	C₈×Dic₁₀₉	C₄	1	1	yes	characteristic
15	C₄×C₈×C₁₀₉	C₄	1	1	yes	characteristic
16	#13182039	—	2	4	no	—
17	#13182039	C₂×C₄	1	8	yes	—
18	C₂×#11724786	—	2	4	no	—
19	C₂×C₄×D₁₀₉	Q₈	1	1	yes	characteristic
20	C₂×C₄×D₁₀₉	Q₈	1	1	yes	characteristic
21	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
22	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
23	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
24	C₈×D₁₀₉	—	2	2	no	—
25	C₈×D₁₀₉	C₂×C₄	1	4	yes	—
26	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
27	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
28	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
29	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
31	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
32	C₂²×D₁₀₉	C₂×Q₈	1	1	yes	characteristic
33	C₂×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
36	C₂×Dic₁₀₉	C₄⋊C₄	1	1	yes	characteristic
37	C₂×Dic₁₀₉	C₄⋊C₄	1	1	yes	characteristic
38	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
39	C₄×D₁₀₉	—	2	2	no	—
40	C₄×D₁₀₉	—	2	2	no	—
41	C₄×D₁₀₉	C₂²×C₄	1	2	yes	—
42	C₄×D₁₀₉	#41	1	2	yes	—
43	#11724786	—	2	8	no	—
44	C₈×C₁₀₉	—	2	1	no	—
45	C₈×C₁₀₉	C₄²	1	2	yes	—
46	#11724784	—	2	1	no	—
47	#11724784	C₄²	1	2	yes	—
48	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂×D₁₀₉	—	2	1	no	—
50	C₂×D₁₀₉	—	2	1	no	—
51	C₂×D₁₀₉	C₄×Q₈	1	2	yes	—
52	C₄×C₁₀₉	—	2	1	no	—
53	C₄×C₁₀₉	#108	1	2	yes	complemented
54	C₄×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
55	C₄×C₁₀₉	#117	1	1	yes	characteristic
56	Dic₁₀₉	—	2	1	no	—
57	Dic₁₀₉	#108	1	2	yes	complemented
58	Dic₁₀₉	C₄×D₄	1	2	yes	—
59	C₂×C₁₀₉	C₄×C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₁₀₉	C₂×#108	1	1	yes	characteristic
61	C₂×C₁₀₉	C₂×#108	1	1	yes	characteristic
62	D₁₀₉	—	2	2	no	—
63	C₄×#108	—	109	1	no	maximal, Sylow
64	C₁₀₉	C₄×#108	1	1	yes	characteristic, complemented, Sylow
65	C₂×#108	—	109	2	no	—
66	C₄×M₁₆	—	109	2	no	—
67	C₂×C₄×C₈	—	109	1	no	—
68	#341	—	109	2	no	—
69	C₂×M₁₆	—	109	4	no	—
70	C₂²×C₈	—	109	1	no	—
71	C₂²×C₈	—	109	1	no	—
72	C₂×C₄²	—	109	1	no	—
73	#108	—	109	8	no	—
74	#97	—	218	2	no	—
75	C₄×C₈	—	109	1	no	—
76	C₄×C₈	—	109	1	no	—
77	C₄×C₈	—	109	1	no	—
78	C₄×C₈	—	218	2	no	—
79	C₄×C₈	#89137	1	1	yes	characteristic
80	C₂²×C₄	—	109	1	no	—
81	C₂²×C₄	—	109	1	no	—
82	C₂²×C₄	—	109	1	no	—
83	M₁₆	—	109	8	no	—
84	M₁₆	—	218	4	no	—
85	C₂×C₈	—	109	1	no	—
86	C₂×C₈	—	109	4	no	—
87	C₂×C₈	—	109	1	no	—
88	C₂×C₈	—	218	2	no	—
89	C₂×C₈	—	218	4	no	—
90	C₂×C₈	C₂×#89137	1	1	yes	characteristic
91	C₂×C₈	C₂×#89137	1	1	yes	characteristic
92	C₄²	—	109	1	no	—
93	C₄²	—	109	1	no	—
94	C₄²	—	109	1	no	—
95	C₄²	C₂×#89137	1	1	yes	characteristic
96	C₂³	—	109	1	no	—
97	C₂×C₄	—	109	1	no	—
98	C₂×C₄	—	109	1	no	—
99	C₂×C₄	—	109	2	no	—
100	C₂×C₄	—	109	2	no	—
101	C₂×C₄	—	109	1	no	—
102	C₂×C₄	—	218	2	no	—
103	C₂×C₄	—	218	2	no	—
104	C₂×C₄	C₄⋊#89137	1	1	yes	characteristic
105	C₂×C₄	C₄⋊#89137	1	1	yes	characteristic
106	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
107	C₈	—	2	1	no	—
108	C₈	—	109	2	no	—
109	C₈	—	218	8	no	—
110	C₈	—	218	1	no	—
111	C₈	C₄×#89137	1	2	yes	—
112	C₂²	—	109	2	no	—
113	C₂²	—	218	1	no	—
114	C₂²	—	218	1	no	—
115	C₂²	C₂×C₄⋊#89137	1	1	yes	characteristic
116	C₄	—	2	1	no	—
117	C₄	—	109	2	no	—
118	C₄	—	109	2	no	—
119	C₄	—	218	1	no	—
120	C₄	#16812769	1	2	yes	direct product factor
121	C₄	C₂×C₄×#89137	1	1	yes	characteristic
122	C₄	#16812778	1	1	yes	characteristic
123	C₂	—	218	2	no	—
124	C₂	C₄×C₄⋊#89137	1	1	yes	characteristic
125	C₂	C₂×#16812769	1	1	yes	characteristic
126	C₂	D₁₀₉⋊#108	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812769

C₂

yes

maximal

C₄×#13182039

C₂

yes

maximal

#20295902

C₂

yes

maximal

C₄×C₈×D₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182039

C₂²

yes

—

#16812769

C₄

yes

direct product factor

#16812754

—

C₄×#11724786

—

C₂×C₈×D₁₀₉

C₂²

yes

characteristic

C₂×C₈×D₁₀₉

C₂²

yes

characteristic

C₄²×D₁₀₉

C₂²

yes

characteristic

C₄×#11724784

C₄

yes

characteristic

C₈×Dic₁₀₉

C₄

yes

characteristic

C₈×Dic₁₀₉

C₄

yes

characteristic

C₄×C₈×C₁₀₉

C₄

yes

characteristic

#13182039

—

#13182039

C₂×C₄

yes

—

C₂×#11724786

—

C₂×C₄×D₁₀₉

Q₈

yes

characteristic

C₂×C₄×D₁₀₉

Q₈

yes

characteristic

C₂×C₄×D₁₀₉

C₂³

yes

characteristic

C₂×#11724784

C₂×C₄

yes

characteristic

C₂×#11724784

C₂×C₄

yes

characteristic

C₈×D₁₀₉

—

C₈×D₁₀₉

C₂×C₄

yes

—

C₄×Dic₁₀₉

C₂×C₄

yes

characteristic

C₄×Dic₁₀₉

D₄

yes

characteristic

C₄×Dic₁₀₉

D₄

yes

characteristic

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×D₁₀₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₉

C₄⋊C₄

yes

characteristic

C₂×Dic₁₀₉

C₄⋊C₄

yes

characteristic

C₂×Dic₁₀₉

C₂×D₄

yes

characteristic

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

C₂²×C₄

yes

—

C₄×D₁₀₉

#41

yes

—

#11724786

—

C₈×C₁₀₉

—

C₈×C₁₀₉

C₄²

yes

—

#11724784

—

#11724784

C₄²

yes

—

C₂²×C₁₀₉

C₂×C₄⋊C₄

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

C₄×Q₈

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#108

yes

complemented

C₄×C₁₀₉

C₂×C₄²

yes

characteristic

C₄×C₁₀₉

#117

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

#108

yes

complemented

Dic₁₀₉

C₄×D₄

yes

—

C₂×C₁₀₉

C₄×C₄⋊C₄

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

D₁₀₉

—

C₄×#108

—

109

maximal, Sylow

C₁₀₉

C₄×#108

yes

characteristic, complemented, Sylow

C₂×#108

—

109

—

C₄×M₁₆

—

109

—

C₂×C₄×C₈

—

109

—

#341

—

109

—

C₂×M₁₆

—

109

—

C₂²×C₈

—

109

—

C₂²×C₈

—

109

—

C₂×C₄²

—

109

—

#108

—

109

—

#97

—

218

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₄×C₈

—

218

—

C₄×C₈

#89137

yes

characteristic

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

M₁₆

—

109

—

M₁₆

—

218

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₂×C₈

yes

C₂×C₈

yes

C₄²

—

109

—

C₄²

—

109

—

C₄²

—

109

—

C₄²

C₂×#89137

yes

characteristic

C₂³

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

100

C₂×C₄

—

109

—

101

C₂×C₄

—

109

—

102

C₂×C₄

—

218

—

103

C₂×C₄

—

218

—

104

C₂×C₄

C₄⋊#89137

yes

characteristic

105

C₂×C₄

C₄⋊#89137

yes

characteristic

106

C₂×C₄

C₂²×#89137

yes

characteristic

107

C₈

—

108

C₈

—

109

—

109

C₈

—

218

—

110

C₈

—

218

—

111

C₈

C₄×#89137

yes

—

112

C₂²

—

109

—

113

C₂²

—

218

—

114

C₂²

—

218

—

115

C₂²

C₂×C₄⋊#89137

yes

characteristic

116

C₄

—

117

C₄

—

109

—

118

C₄

—

109

—

119

C₄

—

218

—

120

C₄

#16812769

yes

direct product factor

121

C₄

C₂×C₄×#89137

yes

characteristic

122

C₄

#16812778

yes

characteristic

123

C₂

—

218

—

124

C₂

C₄×C₄⋊#89137

yes

characteristic

125

C₂

C₂×#16812769

yes

characteristic

126

C₂

D₁₀₉⋊#108

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₄⋊#89137
commutator	C₄×C₁₀₉	C₂×C₄²
Frattini	C₂×C₄	C₂²×#89137
Fitting	C₄×C₈×C₁₀₉	C₄
socle	C₂²×C₁₀₉	C₂×C₄⋊C₄
prime radical	C₂²×D₁₀₉	C₂×Q₈
cosocle	C₂×C₄×D₁₀₉	C₂³
hypercenter	C₄×C₈	#89137
hypocenter	C₁₀₉	C₄×#108
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈

subgroup

G/H

center

C₂×C₄

C₄⋊#89137

commutator

C₄×C₁₀₉

C₂×C₄²

Frattini

C₂×C₄

C₂²×#89137

Fitting

C₄×C₈×C₁₀₉

C₄

socle

C₂²×C₁₀₉

C₂×C₄⋊C₄

prime radical

C₂²×D₁₀₉

C₂×Q₈

cosocle

C₂×C₄×D₁₀₉

C₂³

C₄×C₈

C₁₀₉

G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈

N	H ≅ G/N	type
6. #16812769	C₄	direct (N × H)
53. C₄×C₁₀₉	#108	semidirect
57. Dic₁₀₉	#108	semidirect
64. C₁₀₉	C₄×#108	semidirect
120. C₄	#16812769	direct (N × H)

H ≅ G/N

type

6. #16812769

C₄

direct (N × H)

53. C₄×C₁₀₉

#108

semidirect

57. Dic₁₀₉

#108

semidirect

64. C₁₀₉

C₄×#108

semidirect

120. C₄

#16812769

direct (N × H)

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

subgroup	index
C₂×#16812769	2
C₄×#13182039	2
#20295902	2
C₄×C₈×D₁₀₉	2
C₄×#108	109

subgroup

index

C₄×C₈×D₁₀₉

109

Subgroups — C₄×#16812769 #28378849

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C4×#16812769 #28378849

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16812769 #28378849