Finite Group Databasebeta

← #16835478

Subgroups — #16835478

Order 3712.

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

normal subgroups only

This group has 360 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.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11745345C211yesmaximal, characteristic
2C2×#11745225C212yesmaximal, complemented
3D29⋊(C22⋊C8)C211yesmaximal, characteristic
4#14272036C212yesmaximal, complemented
5C22⋊Q8×D29C211yesmaximal, characteristic
6C2×Q8×D29C2211yescharacteristic
7C22×C4×D29C2211yescharacteristic
8C2×C22⋊#1181—21no—
9#11745345—24no—
10C2×C4⋊#1181C2212yes—
11C2×C4×#1181—21no—
12C2×D29⋊C8C2212yes—
13C29×C22⋊Q8C411yescharacteristic, complemented
14#11745308C411yescharacteristic, complemented
15#11745286C411yescharacteristic, complemented
16#11745259C411yescharacteristic, complemented
17#11745258—21no—
18C4⋊C4×D29—21no—
19C4⋊C4×D29C2211yescharacteristic
20C22⋊C4×D29—21no—
21#11745240—21no—
22C22⋊#6599—21no—
23#11745225—24no—
24#11745219—21no—
25C23×D29D411yescharacteristic
26C22×#1181—22no—
27C2×Q8×C29C2×C411yescharacteristic
28C22×C4×C29C2×C411yescharacteristic
29C22×Dic29D411yescharacteristic
30Q8×D29—22no—
31Q8×D29—22no—
32C2×C4×D29—22no—
33C2×C4×D29—22no—
34C2×C4×D29—21no—
35C2×C4×D29—21no—
36C2×C4×D29C2×C411yescharacteristic
37C2×C4×D29D411yescharacteristic
38C2×C4×D29D411yescharacteristic
39C2×C4×D29C2311yescharacteristic
40C2×Dic58—21no—
41C2×Dic58C2×C411yescharacteristic
42C22⋊#1181—24no—
43C2×#6599—22no—
44C4⋊#1181—22no—
45C4⋊#1181D414yes—
46C4×#1181—24no—
47D29⋊C8—22no—
48D29⋊C8—22no—
49C4⋊C4×C29—21no—
50C4⋊C4×C29C2×C411yescharacteristic
51C22⋊C4×C29—21no—
52C22⋊Dic29—21no—
53#90781—21no—
54#90781—21no—
55C4⋊Dic29—21no—
56C4⋊Dic29C2×C411yescharacteristic
57#90779—21no—
58#90779—21no—
59#90779—21no—
60C23×C29C22⋊C411yescharacteristic
61C22×D29—21no—
62C22×D29—21no—
63C22×D29—22no—
64C22×D29—22no—
65C22×D29C22⋊C411yescharacteristic
66C22×D29C2×D411yescharacteristic
67C2×#1181—24no—
68C2×#1181—42no—
69C2×#1181—42no—
70Q8×C29—22no—
71C2×C4×C29—22no—
72C2×C4×C29—21no—
73C2×C4×C29—21no—
74C2×C4×C29C22⋊C411yescharacteristic
75C2×C4×C29C22×C411yescharacteristic
76C2×Dic29—22no—
77C2×Dic29—21no—
78C2×Dic29—21no—
79C2×Dic29C22⋊C411yescharacteristic
80C2×Dic29C2×D411yescharacteristic
81C4×D29—21no—
82C4×D29—21no—
83C4×D29—22no—
84C4×D29—22no—
85C4×D29—41no—
86C4×D29—41no—
87C4×D29—42no—
88C4×D29C2×D412yes—
89C4×D29#4112yes—
90Dic58—22no—
91Dic58—42no—
92#6599—42no—
93#1873—291nomaximal, Sylow
94C22×C29—21no—
95C22×C29—21no—
96C22×C29—21no—
97C22×C29C2×C22⋊C411yescharacteristic
98C2×D29—21no—
99C2×D29—21no—
100C2×D29—21no—
101C2×D29—42no—
102C2×D29—42no—
103C2×D29#12011yescharacteristic
104C2×D29#12311yescharacteristic
105C2×D29#13512yes—
106C2×D29#13712yes—
107#1181—44no—
108C4×C29—21no—
109C4×C29—41no—
110C4×C29—41no—
111C4×C29C4×D412yes—
112Dic29—21no—
113Dic29—41no—
114Dic29—41no—
115Dic29#12112yes—
116C2×C22⋊Q8—291no—
117C2×#117—291no—
118C2×#103—292no—
119C2×C22⋊C8—291no—
120#340—292no—
121C2×C29—41no—
122C2×C29#38611yescharacteristic
123C2×C29#41611yescharacteristic
124C2×C29#41911yescharacteristic
125D29—41no—
126D29#45212yes—
127D29#48512yes—
128C22×Q8—291no—
129C23×C4—291no—
130C22×C8—292no—
131C22⋊Q8—291no—
132C22⋊Q8—291no—
133C22⋊Q8—291no—
134C22⋊Q8—581no—
135C22⋊Q8—581no—
136C22⋊Q8#118111yescharacteristic, complemented
137#117—584no—
138C2×C4⋊C4—292no—
139C2×C4⋊C4—291no—
140C2×C4⋊C4—581no—
141C2×C22⋊C4—581no—
142C2×C22⋊C4—581no—
143C2×C42—581no—
144#103—584no—
145C22⋊C8—581no—
146C22⋊C8—581no—
147C29#187311yescharacteristic, complemented, Sylow
148C24—291no—
149C2×Q8—291no—
150C2×Q8—581no—
151C2×Q8—582no—
152C2×Q8—582no—
153C2×Q8C2×#118111yescharacteristic
154C22×C4—291no—
155C22×C4—291no—
156C22×C4—291no—
157C22×C4—291no—
158C22×C4—291no—
159C22×C4—582no—
160C22×C4—582no—
161C22×C4—581no—
162C22×C4—581no—
163C22×C4—582no—
164C22×C4C2×#118111yescharacteristic
165C2×C8—582no—
166C2×C8—582no—
167C2×C8—582no—
168C4⋊C4—21no—
169C4⋊C4—294no—
170C4⋊C4—291no—
171C4⋊C4—581no—
172C4⋊C4—581no—
173C4⋊C4—581no—
174C4⋊C4—582no—
175C4⋊C4—581no—
176C4⋊C4C2×#118111yescharacteristic
177C22⋊C4—21no—
178C22⋊C4—581no—
179C22⋊C4—581no—
180C22⋊C4—584no—
181C22⋊C4—581no—
182C42—584no—
183C23—291no—
184C23—291no—
185C23—581no—
186C23—581no—
187C23—582no—
188C23—582no—
189C23C22⋊#118111yescharacteristic
190Q8—22no—
191Q8—582no—
192Q8—1162no—
193C2×C4—22no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—291no—
197C2×C4—292no—
198C2×C4—292no—
199C2×C4—291no—
200C2×C4—582no—
201C2×C4—581no—
202C2×C4—581no—
203C2×C4—582no—
204C2×C4—582no—
205C2×C4—584no—
206C2×C4—581no—
207C2×C4—581no—
208C2×C4—1161no—
209C2×C4—1161no—
210C2×C4—1162no—
211C2×C4—1162no—
212C2×C4—1162no—
213C2×C4C22⋊#118111yescharacteristic
214C2×C4C22×#118111yescharacteristic
215C8—1162no—
216C22—21no—
217C22—21no—
218C22—21no—
219C22—291no—
220C22—291no—
221C22—292no—
222C22—292no—
223C22—581no—
224C22—581no—
225C22—581no—
226C22—1162no—
227C22—1162no—
228C22C2×C22⋊#118111yescharacteristic
229C4—21no—
230C4—41no—
231C4—41no—
232C4—292no—
233C4—581no—
234C4—1164no—
235C4—1161no—
236C4—1161no—
237C4D4×#118112yes—
238C2—41no—
239C2—292no—
240C2—292no—
241C2—1161no—
242C2#1427211111yescharacteristic
243C2#1427215311yescharacteristic
244C2#1427215611yescharacteristic
subgroupG/H
centerC22C2×C22⋊#1181
commutatorC2×C4×C29C22×C4
FrattiniC2×C4C22×#1181
FittingC29×C22⋊Q8C4
socleC22×C29C2×C22⋊C4
prime radicalC23×D29D4
cosocleC2×C4×D29C23
hypercenterC22⋊Q8#1181
hypocenterC29#1873
LCSG ▷ C2×C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#11745225C2semidirect
4. #14272036C2semidirect
13. C29×C22⋊Q8C4semidirect
14. #11745308C4semidirect
15. #11745286C4semidirect
16. #11745259C4semidirect
136. C22⋊Q8#1181semidirect
147. C29#1873semidirect

Maximal subgroups

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

subgroupindex
C2×#117453452
C2×#117452252
D29⋊(C22⋊C8)2
#142720362
C22⋊Q8×D292
#187329