Finite Group Databasebeta

← #16819430

Subgroups — #16819430

Order 3712.

612 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 117 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#90805C211yesmaximal, characteristic
2C2×#11745158C212yesmaximal, complemented
3C2×#11745157C212yesmaximal, complemented
4C2×C22⋊#6597C211yesmaximal, characteristic, complemented
5C29×#380C211yesmaximal, characteristic, complemented
6C23×C4×C29C2211yescharacteristic
7C22×C22⋊D29—21no—
8C2×#90805—22no—
9C2×#90805—24no—
10C2×#90805—24no—
11C2×#90805C414yescomplemented
12C22×D116C2211yescharacteristic
13C22×C4×D29—21no—
14C22×Dic58C2211yescharacteristic
15C2×C4⋊C4×C29C2212yes—
16C2×C22⋊C4×C29—21no—
17C22×#6597C2212yes—
18C29×#95—21no—
19C22⋊#6597—22no—
20#11745158—24no—
21#11745157—24no—
22C24×C29D411yescharacteristic
23C23×D29—21no—
24C22×C4×C29—22no—
25C22×C4×C29—22no—
26C22×C4×C29—22no—
27C22×C4×C29C2×C412yes—
28C22×C4×C29D412yes—
29C22×C4×C29D411yescharacteristic
30C22×C4×C29C2311yescharacteristic
31C2×C22⋊D29—28no—
32C2×C22⋊D29—24no—
33C22×Dic29—21no—
34#90805—216no—
35#90805—28no—
36#90805—28no—
37C2×D116—21no—
38C2×D116—22no—
39C2×D116C2×C412yes—
40C2×D116D414yes—
41C2×C4×D29—28no—
42C2×C4×D29—24no—
43C2×Dic58—21no—
44C2×Dic58—22no—
45C2×Dic58C2×C412yes—
46C2×Dic58D414yes—
47C4⋊C4×C29—22no—
48C4⋊C4×C29—22no—
49C22⋊C4×C29—42no—
50C2×#6597—22no—
51C2×#6597—22no—
52C2×#6597—22no—
53C23×C29—22no—
54C23×C29—22no—
55C23×C29—21no—
56C23×C29—21no—
57C23×C29C22⋊C412yes—
58C23×C29C2×D411yescharacteristic
59C22×D29—22no—
60C22×D29—24no—
61C22×D29—42no—
62C22×D29—41no—
63C22×D29—41no—
64C2×C4×C29—22no—
65C2×C4×C29—22no—
66C2×C4×C29—22no—
67C2×C4×C29—24no—
68C2×C4×C29—42no—
69C2×C4×C29—42no—
70C2×C4×C29—42no—
71C2×C4×C29C22⋊C412yes—
72C2×C4×C29C22×C411yescharacteristic
73C2×C4×C29C2×D411yescharacteristic
74C2×C4×C29C2×D412yes—
75C2×C4×C29C2×D412yes—
76C22⋊D29—216no—
77C2×Dic29—22no—
78C2×Dic29—24no—
79D116—22no—
80D116—24no—
81D116C22⋊C414yescomplemented
82C4×D29—216no—
83Dic58—22no—
84Dic58—24no—
85Dic58C22⋊C414yescomplemented
86#6597—42no—
87#1788—291nomaximal, Sylow
88C22×C29—22no—
89C22×C29—22no—
90C22×C29—22no—
91C22×C29—42no—
92C22×C29—42no—
93C22×C29C2×C22⋊C411yescharacteristic
94C22×C29#12012yes—
95C22×C29#13512yes—
96C22×C29AGL(1,ℤ/8)11yescharacteristic
97C22×C29#13711yescharacteristic
98C2×D29—24no—
99C2×D29—42no—
100C2×D29—42no—
101C2×D29—44no—
102C2×D29—44no—
103C4×C29—22no—
104C4×C29—82no—
105C4×C29C2×C22⋊C411yescharacteristic
106C4×C29C2×C22⋊C411yescharacteristic
107C4×C29#12012yes—
108Dic29—24no—
109C22×#41—291no—
110C2×#103—292no—
111C2×#102—292no—
112C2×C22⋊C8—291no—
113#380D2911yescharacteristic, complemented
114C2×C29—42no—
115C2×C29#37911yescharacteristic
116C2×C29#41611yescharacteristic
117C2×C29#41711yescharacteristic
118C2×C29#44912yes—
119C2×C29#45212yes—
120D29—44no—
121C2×#41—294no—
122C2×#41—584no—
123C2×#41—584no—
124C2×#41—582no—
125C22×Q8—291no—
126C22×D4—291no—
127C22×D4—581no—
128C23×C4—581no—
129C23×C4C2×D2911yescharacteristic
130C22×C8—292no—
131C2×C4⋊C4C2×D2912yes—
132C2×C22⋊C4—21no—
133#103—584no—
134#102—584no—
135C22⋊C8—582no—
136#95—21no—
137C29#178811yescharacteristic, complemented, Sylow
138C24—581no—
139C24C22⋊D2911yescharacteristic
140#41—588no—
141#41—588no—
142#41—5816no—
143C2×Q8—292no—
144C2×Q8—294no—
145C2×Q8—581no—
146C2×Q8—582no—
147C2×D4—292no—
148C2×D4—294no—
149C2×D4—588no—
150C2×D4—584no—
151C2×D4—581no—
152C2×D4—582no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—588no—
157C22×C4—581no—
158C22×C4—584no—
159C22×C4C4×D2912yes—
160C22×C4D11612yes—
161C22×C4C22⋊D2911yescharacteristic
162C22×C4C22×D2911yescharacteristic
163C2×C8—582no—
164C2×C8—582no—
165C2×C8—582no—
166C4⋊C4—22no—
167C4⋊C4—22no—
168C22⋊C4—42no—
169C23—21no—
170C23—22no—
171C23—21no—
172C23—22no—
173C23—582no—
174C23—584no—
175C23—1161no—
176C23—1161no—
177C23—1162no—
178C23#9078112yes—
179C23C2×C22⋊D2911yescharacteristic
180Q8—294no—
181Q8—582no—
182Q8—584no—
183D4—294no—
184D4—5816no—
185D4—582no—
186D4—584no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—24no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—5816no—
195C2×C4—582no—
196C2×C4—584no—
197C2×C4#9078112yes—
198C2×C4C2×C4×D2911yescharacteristic
199C2×C4C2×D11611yescharacteristic
200C2×C4D4×D2912yes—
201C2×C4D4×D2912yes—
202C8—1162no—
203C22—22no—
204C22—22no—
205C22—22no—
206C22—42no—
207C22—42no—
208C22—584no—
209C22—1164no—
210C22—1164no—
211C22—1162no—
212C22—1162no—
213C22C2×#9078111yescharacteristic
214C22#1174529411yescharacteristic
215C22#1174529912yes—
216C22#1174530411yescharacteristic
217C22#1174531212yes—
218C4—22no—
219C4—82no—
220C4—584no—
221C4C22⋊C4×D2911yescharacteristic
222C4C22⋊C4×D2911yescharacteristic
223C4#1174524412yes—
224C2—42no—
225C2—1164no—
226C2#1427101811yescharacteristic
227C2#1427115711yescharacteristic
228C2#1427116211yescharacteristic
229C2#1427130612yes—
230C2#1427132312yes—
subgroupG/H
centerC23C2×C22⋊D29
commutatorC2×C4×C29C22×C4
FrattiniC22×C4C22×D29
FittingC29×#380C2
socleC23×C29C2×D4
prime radicalC22×#90805C2
cosocleC22×C4×C29C23
hypercenter#380D29
hypocenterC29#1788
LCSG ▷ C2×C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C23 ◁ C23×C4 ◁ #380

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#11745158C2semidirect
3. C2×#11745157C2semidirect
4. C2×C22⋊#6597C2semidirect
5. C29×#380C2semidirect
11. C2×#90805C4semidirect
81. D116C22⋊C4semidirect
85. Dic58C22⋊C4semidirect
113. #380D29semidirect
137. C29#1788semidirect

Maximal subgroups

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

subgroupindex
C22×#908052
C2×#117451582
C2×#117451572
C2×C22⋊#65972
C29×#3802
#178829