Finite Group Databasebeta

← #30765307

Subgroups — #30765307

Order 17152.

460 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 99 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
1#22558547C212yesmaximal
2#22558153C212yesmaximal, complemented
3#22558143C212yesmaximal, complemented
4C4×#124×C67C211yesmaximal, characteristic
5#16863360—22no—
6#16863122C2214yes—
7#16863120—22no—
8C2×#124×C67C411yescharacteristic
9C2×C4×Q8×C67C411yescharacteristic
10C2×C4×D4×C67C2211yescharacteristic
11#391×C67C2211yescharacteristic
12#390×C67C411yescharacteristic
13#388×C67—22no—
14#382×C67C2211yescharacteristic
15C4×C22⋊C4×C67—22no—
16C43×C67C411yescharacteristic
17C2×#11761612—42no—
18C22⋊(C67⋊C8)—24no—
19C22⋊(C67⋊C8)—42no—
20C22×Q8×C67C2×C411yescharacteristic
21C22×D4×C67C2×C411yescharacteristic
22C23×C4×C67D412yes—
23#124×C67—22no—
24#124×C67—22no—
25C4×Q8×C67—41no—
26C4×Q8×C67—41no—
27C4×D4×C67—41no—
28C4×D4×C67—41no—
29C2×C4⋊C4×C67—21no—
30C2×C4⋊C4×C67C2×C411yescharacteristic
31C2×C4⋊C4×C67C2311yescharacteristic
32C2×C22⋊C4×C67—22no—
33C2×C22⋊C4×C67—22no—
34C2×C22⋊C4×C67D412yes—
35C2×C42×C67—21no—
36C2×C42×C67—21no—
37C2×C42×C67—21no—
38C2×C42×C67C2×C411yescharacteristic
39C2×C42×C67C2×C411yescharacteristic
40C2×C42×C67C2×C411yescharacteristic
41#95×C67—22no—
42#95×C67—22no—
43#11761612—84no—
44C2×C67⋊C8—44no—
45C24×C67C22⋊C412yes—
46C2×Q8×C67—21no—
47C2×Q8×C67—41no—
48C2×Q8×C67—41no—
49C2×Q8×C67M1612yescomplemented
50C2×D4×C67—21no—
51C2×D4×C67—21no—
52C2×D4×C67—41no—
53C2×D4×C67—41no—
54C22×C4×C67—21no—
55C22×C4×C67—21no—
56C22×C4×C67—21no—
57C22×C4×C67—22no—
58C22×C4×C67—22no—
59C22×C4×C67—21no—
60C22×C4×C67—44no—
61C22×C4×C67C22⋊C411yescharacteristic
62C22×C4×C67C22⋊C411yescharacteristic
63C22×C4×C67C22×C411yescharacteristic
64C22×C4×C67C2×D412yes—
65C4⋊C4×C67—41no—
66C4⋊C4×C67—41no—
67C4⋊C4×C67—42no—
68C22⋊C4×C67—28no—
69C22⋊C4×C67—42no—
70C22⋊C4×C67—44no—
71C42×C67—21no—
72C42×C67—21no—
73C42×C67—21no—
74C42×C67—41no—
75C42×C67—42no—
76C42×C67—42no—
77C42×C67—41no—
78C42×C67—41no—
79C42×C67M1612yescomplemented
80C23×C67—22no—
81C23×C67—22no—
82C23×C67—22no—
83C23×C67—42no—
84C23×C67—42no—
85C23×C67C2×C22⋊C411yescharacteristic
86Q8×C67—42no—
87Q8×C67—42no—
88D4×C67—42no—
89D4×C67—42no—
90C2×C4×C67—21no—
91C2×C4×C67—21no—
92C2×C4×C67—22no—
93C2×C4×C67—22no—
94C2×C4×C67—22no—
95C2×C4×C67—22no—
96C2×C4×C67—21no—
97C2×C4×C67—22no—
98C2×C4×C67—41no—
99C2×C4×C67—41no—
100C2×C4×C67—41no—
101C2×C4×C67—41no—
102C2×C4×C67—41no—
103C2×C4×C67—41no—
104C2×C4×C67—44no—
105C2×C4×C67—41no—
106C2×C4×C67—42no—
107C2×C4×C67—42no—
108C2×C4×C67—41no—
109C2×C4×C67—82no—
110C2×C4×C67#9911yescharacteristic
111C2×C4×C67#9911yescharacteristic
112C2×C4×C67C2×M1611yescharacteristic
113C2×C4×C67C2×M1611yescharacteristic
114C67⋊C8—84no—
115C22×C67—21no—
116C22×C67—21no—
117C22×C67—44no—
118C22×C67—42no—
119C22×C67—42no—
120C22×C67—42no—
121C22×C67—82no—
122C22×C67#40711yescharacteristic
123C22×C67C2×#9911yescharacteristic
124C22×C67#41311yescharacteristic
125C4×C67—21no—
126C4×C67—21no—
127C4×C67—21no—
128C4×C67—42no—
129C4×C67—42no—
130C4×C67—42no—
131C4×C67—41no—
132C4×C67—42no—
133C4×C67—81no—
134C4×C67#35412yes—
135#11531—671nomaximal, Sylow
136C2×C67—21no—
137C2×C67—21no—
138C2×C67—82no—
139C2×C67#146511yescharacteristic
140C2×C67C2×#35411yescharacteristic
141C2×C67#213211yescharacteristic
142C4×#124D6711yescharacteristic
143#1465—672no—
144#1324—672no—
145#1319—672no—
146C67#1153111yescharacteristic, complemented, Sylow
147C2×#124Dic6711yescharacteristic
148C2×C4×Q8Dic6711yescharacteristic
149C2×C4×D4C2×D6711yescharacteristic
150#407—1342no—
151#391C2×D6711yescharacteristic
152#390Dic6711yescharacteristic
153#388—22no—
154#382C2×D6711yescharacteristic
155C4×C22⋊C4—22no—
156C43Dic6711yescharacteristic
157#324—674no—
158#323—1342no—
159C22×Q8C2×Dic6711yescharacteristic
160C22×D4C2×Dic6711yescharacteristic
161C23×C4C22⋊D6712yes—
162C2×M16—2682no—
163#124—22no—
164#124—22no—
165C4×Q8—41no—
166C4×Q8—41no—
167C4×D4—41no—
168C4×D4—41no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4C2×Dic6711yescharacteristic
171C2×C4⋊C4C22×D6711yescharacteristic
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4C22⋊D6712yes—
175C2×C42—21no—
176C2×C42—21no—
177C2×C42—21no—
178C2×C42C2×Dic6711yescharacteristic
179C2×C42C2×Dic6711yescharacteristic
180C2×C42C2×Dic6711yescharacteristic
181C22⋊C8—2682no—
182C22⋊C8—1344no—
183#95—22no—
184#95—22no—
185C24C22⋊Dic6712yes—
186C2×Q8—21no—
187C2×Q8—41no—
188C2×Q8—41no—
189C2×Q8#1176161212yescomplemented
190C2×D4—21no—
191C2×D4—21no—
192C2×D4—41no—
193C2×D4—41no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—21no—
200C22×C4—44no—
201C22×C4C22⋊Dic6711yescharacteristic
202C22×C4C22⋊Dic6711yescharacteristic
203C22×C4C22×Dic6711yescharacteristic
204C22×C4C2×C22⋊D6712yes—
205M16—5364no—
206C2×C8—2684no—
207C4⋊C4—41no—
208C4⋊C4—41no—
209C4⋊C4—42no—
210C22⋊C4—28no—
211C22⋊C4—42no—
212C22⋊C4—44no—
213C42—21no—
214C42—21no—
215C42—21no—
216C42—41no—
217C42—42no—
218C42—42no—
219C42—41no—
220C42—41no—
221C42#1176161212yescomplemented
222C23—22no—
223C23—22no—
224C23—22no—
225C23—42no—
226C23—42no—
227C23C2×C22⋊Dic6711yescharacteristic
228Q8—42no—
229Q8—42no—
230D4—42no—
231D4—42no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—44no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—41no—
251C2×C4—82no—
252C2×C4#1452395811yescharacteristic
253C2×C4#1452395811yescharacteristic
254C2×C4C2×#1176161211yescharacteristic
255C2×C4C2×#1176161211yescharacteristic
256C8—5364no—
257C22—21no—
258C22—21no—
259C22—44no—
260C22—42no—
261C22—42no—
262C22—42no—
263C22—82no—
264C22#1686336011yescharacteristic
265C22C2×#1452395811yescharacteristic
266C22#1686338011yescharacteristic
267C4—21no—
268C4—21no—
269C4—21no—
270C4—42no—
271C4—42no—
272C4—42no—
273C4—42no—
274C4—41no—
275C4—81no—
276C4#1686319112yes—
277C2—21no—
278C2—21no—
279C2—82no—
280C2#2255854711yescharacteristic
281C2C2×#1686319111yescharacteristic
282C2#2256189411yescharacteristic
subgroupG/H
centerC22C2×#14523958
commutatorC22×C4×C67C22×C4
FrattiniC2×C4⋊C4C22×D67
FittingC4×#124×C67C2
socleC22×C67C2×#99
prime radicalC22×D4×C67C2×C4
cosocleC2×C4⋊C4×C67C23
hypercenterC4×#124D67
hypocenterC67#11531
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#124

Complements

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

NH ≅ G/Ntype
2. #22558153C2semidirect
3. #22558143C2semidirect
49. C2×Q8×C67M16semidirect
79. C42×C67M16semidirect
146. C67#11531semidirect
189. C2×Q8#11761612semidirect
221. C42#11761612semidirect

Maximal subgroups

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

subgroupindex
#225585472
#225581532
#225581432
C4×#124×C672
#1153167