Finite Group Databasebeta

← #30765208

Subgroups — #30765208

Order 17152.

814 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 814 conjugacy classes of subgroups — too many to draw. Only its 107 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#22558539C212yesmaximal, complemented
2#22558141C214yesmaximal, complemented
3C2×#390×C67C211yesmaximal, characteristic
4#16863360—22no—
5#16863122—22no—
6#16863122C2214yescomplemented
7C23×D4×C67C411yescharacteristic
8C22×C4⋊C4×C67C2211yescharacteristic
9C22×C22⋊C4×C67—22no—
10C22×C42×C67C411yescharacteristic
11#390×C67—22no—
12#390×C67C412yes—
13#390×C67C2212yes—
14C2×#11761612—42no—
15C22⋊(C67⋊C8)—24no—
16C22⋊(C67⋊C8)—42no—
17C25×C67—21no—
18C22×D4×C67—24no—
19C22×D4×C67—21no—
20C22×D4×C67—41no—
21C22×D4×C67—41no—
22C22×D4×C67—42no—
23C22×D4×C67C2×C412yes—
24C23×C4×C67—21no—
25C23×C4×C67C2×C411yescharacteristic
26C23×C4×C67D412yes—
27C2×C4⋊C4×C67—41no—
28C2×C4⋊C4×C67—41no—
29C2×C4⋊C4×C67C2×C411yescharacteristic
30C2×C4⋊C4×C67D412yes—
31C2×C4⋊C4×C67C2311yescharacteristic
32C2×C22⋊C4×C67—24no—
33C2×C22⋊C4×C67—24no—
34C2×C22⋊C4×C67—48no—
35C2×C42×C67—21no—
36C2×C42×C67—42no—
37C2×C42×C67C2×C412yes—
38#11761612—84no—
39C2×C67⋊C8—44no—
40C24×C67—24no—
41C24×C67—22no—
42C24×C67—44no—
43C24×C67—42no—
44C24×C67—44no—
45C24×C67—41no—
46C24×C67—41no—
47C24×C67C22⋊C411yescharacteristic
48C2×D4×C67—24no—
49C2×D4×C67—22no—
50C2×D4×C67—42no—
51C2×D4×C67—42no—
52C2×D4×C67—48no—
53C2×D4×C67—48no—
54C2×D4×C67—42no—
55C2×D4×C67—42no—
56C2×D4×C67M1614yescomplemented
57C22×C4×C67—21no—
58C22×C4×C67—21no—
59C22×C4×C67—22no—
60C22×C4×C67—22no—
61C22×C4×C67—24no—
62C22×C4×C67—22no—
63C22×C4×C67—22no—
64C22×C4×C67—22no—
65C22×C4×C67—42no—
66C22×C4×C67—42no—
67C22×C4×C67—44no—
68C22×C4×C67C22⋊C411yescharacteristic
69C22×C4×C67C22⋊C412yes—
70C22×C4×C67C22×C411yescharacteristic
71C22×C4×C67C2×D412yes—
72C4⋊C4×C67—41no—
73C4⋊C4×C67—42no—
74C4⋊C4×C67—41no—
75C22⋊C4×C67—48no—
76C22⋊C4×C67—48no—
77C42×C67—42no—
78C42×C67—42no—
79C23×C67—22no—
80C23×C67—22no—
81C23×C67—21no—
82C23×C67—21no—
83C23×C67—24no—
84C23×C67—44no—
85C23×C67—44no—
86C23×C67—44no—
87C23×C67—416no—
88C23×C67—44no—
89C23×C67—42no—
90C23×C67—42no—
91C23×C67—88no—
92C23×C67—82no—
93C23×C67—82no—
94C23×C67—82no—
95C23×C67—82no—
96C23×C67#9911yescharacteristic
97C23×C67#9911yescharacteristic
98C23×C67C2×C22⋊C411yescharacteristic
99D4×C67—44no—
100D4×C67—48no—
101D4×C67—44no—
102C2×C4×C67—24no—
103C2×C4×C67—24no—
104C2×C4×C67—21no—
105C2×C4×C67—21no—
106C2×C4×C67—21no—
107C2×C4×C67—21no—
108C2×C4×C67—24no—
109C2×C4×C67—22no—
110C2×C4×C67—22no—
111C2×C4×C67—48no—
112C2×C4×C67—42no—
113C2×C4×C67—44no—
114C2×C4×C67—42no—
115C2×C4×C67—42no—
116C2×C4×C67—42no—
117C2×C4×C67—82no—
118C2×C4×C67#10012yes—
119C2×C4×C67C2×M1612yes—
120C67⋊C8—84no—
121C22×C67—24no—
122C22×C67—24no—
123C22×C67—21no—
124C22×C67—21no—
125C22×C67—21no—
126C22×C67—21no—
127C22×C67—21no—
128C22×C67—21no—
129C22×C67—48no—
130C22×C67—44no—
131C22×C67—84no—
132C22×C67—88no—
133C22×C67—88no—
134C22×C67—82no—
135C22×C67—82no—
136C22×C67#35312yes—
137C22×C67#35512yes—
138C22×C67#40711yescharacteristic
139C22×C67C2×#9911yescharacteristic
140C22×C67C2×#10011yescharacteristic
141C4×C67—22no—
142C4×C67—22no—
143C4×C67—44no—
144C4×C67—42no—
145C4×C67—42no—
146#11505—671nomaximal, Sylow
147C2×C67—21no—
148C2×C67—21no—
149C2×C67—22no—
150C2×C67—22no—
151C2×C67—84no—
152C2×C67#146211yescharacteristic
153C2×C67C2×#35311yescharacteristic
154C2×C67C2×#35511yescharacteristic
155C2×#390D6711yescharacteristic
156#1463—672no—
157#1318—674no—
158C67#1150511yescharacteristic, complemented, Sylow
159C23×D4Dic6711yescharacteristic
160C22×C4⋊C4C2×D6711yescharacteristic
161C22×C22⋊C4—22no—
162C22×C42Dic6711yescharacteristic
163#407—1342no—
164#390—22no—
165#390Dic6712yes—
166#390C2×D6712yes—
167#324—674no—
168#324—1342no—
169C25—21no—
170C22×D4—24no—
171C22×D4—21no—
172C22×D4—41no—
173C22×D4—41no—
174C22×D4—42no—
175C22×D4C2×Dic6712yes—
176C23×C4—21no—
177C23×C4C2×Dic6711yescharacteristic
178C23×C4C22⋊D6712yes—
179C2×M16—2682no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4C2×Dic6711yescharacteristic
183C2×C4⋊C4C22⋊D6712yes—
184C2×C4⋊C4C22×D6711yescharacteristic
185C2×C22⋊C4—24no—
186C2×C22⋊C4—24no—
187C2×C22⋊C4—48no—
188C2×C42—21no—
189C2×C42—42no—
190C2×C42C2×Dic6712yes—
191C22⋊C8—2682no—
192C22⋊C8—1344no—
193C24—24no—
194C24—22no—
195C24—44no—
196C24—42no—
197C24—44no—
198C24—41no—
199C24—41no—
200C24C22⋊Dic6711yescharacteristic
201C2×D4—24no—
202C2×D4—22no—
203C2×D4—42no—
204C2×D4—42no—
205C2×D4—48no—
206C2×D4—48no—
207C2×D4—42no—
208C2×D4—42no—
209C2×D4#1176161214yescomplemented
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—24no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—44no—
221C22×C4C22⋊Dic6711yescharacteristic
222C22×C4C22⋊Dic6712yes—
223C22×C4C22×Dic6711yescharacteristic
224C22×C4C2×C22⋊D6712yes—
225M16—5364no—
226C2×C8—2684no—
227C4⋊C4—41no—
228C4⋊C4—42no—
229C4⋊C4—41no—
230C22⋊C4—48no—
231C22⋊C4—48no—
232C42—42no—
233C42—42no—
234C23—21no—
235C23—21no—
236C23—22no—
237C23—22no—
238C23—24no—
239C23—44no—
240C23—44no—
241C23—44no—
242C23—416no—
243C23—44no—
244C23—42no—
245C23—42no—
246C23—88no—
247C23—82no—
248C23—82no—
249C23—82no—
250C23—82no—
251C23#1452395811yescharacteristic
252C23#1452395811yescharacteristic
253C23C2×C22⋊Dic6711yescharacteristic
254D4—44no—
255D4—48no—
256D4—44no—
257C2×C4—24no—
258C2×C4—24no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—24no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—48no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—44no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—82no—
273C2×C4#1452396012yes—
274C2×C4C2×#1176161212yes—
275C8—5364no—
276C22—24no—
277C22—24no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—21no—
284C22—48no—
285C22—44no—
286C22—84no—
287C22—88no—
288C22—88no—
289C22—82no—
290C22—82no—
291C22#1686318912yes—
292C22#1686319312yes—
293C22#1686336011yescharacteristic
294C22C2×#1452395811yescharacteristic
295C22C2×#1452396011yescharacteristic
296C4—22no—
297C4—22no—
298C4—44no—
299C4—42no—
300C4—42no—
301C2—21no—
302C2—21no—
303C2—22no—
304C2—22no—
305C2—84no—
306C2#2255853611yescharacteristic
307C2C2×#1686318911yescharacteristic
308C2C2×#1686319311yescharacteristic
subgroupG/H
centerC22C2×#14523958
commutatorC22×C4×C67C22×C4
FrattiniC2×C4⋊C4C22×D67
FittingC2×#390×C67C2
socleC22×C67C2×#99
prime radicalC23×D4×C67C4
cosocleC2×C4⋊C4×C67C23
hypercenterC2×#390D67
hypocenterC67#11505
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
1. #22558539C2semidirect
2. #22558141C2semidirect
6. #16863122C22semidirect
56. C2×D4×C67M16semidirect
158. C67#11505semidirect
209. C2×D4#11761612semidirect

Maximal subgroups

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

subgroupindex
#225585392
#225581412
C2×#390×C672
#1150567