Finite Group Databasebeta

← #30765492

Subgroups — #30765492

Order 17152.

416 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 91 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#22558547C211yesmaximal, characteristic
2#22558527C211yesmaximal, characteristic
3#22558155C212yesmaximal, complemented
4#22558153C212yesmaximal, complemented
5C67×#2566C211yesmaximal, characteristic
6#16863360—21no—
7C2×C22⋊(C67⋊C8)—21no—
8#16863122C2212yes—
9#16863122C2212yes—
10#16863120—21no—
11#16863120—21no—
12C2×#125×C67C411yescharacteristic
13C2×C4×D4×C67C2211yescharacteristic
14#396×C67—21no—
15#396×C67—21no—
16#390×C67C411yescharacteristic
17#390×C67C411yescharacteristic
18#385×C67—21no—
19#385×C67—21no—
20#384×C67C2211yescharacteristic
21#382×C67C2211yescharacteristic
22C4×C4⋊C4×C67C411yescharacteristic
23C2×#11761612—41no—
24C22×C67⋊C8—41no—
25C22⋊(C67⋊C8)—22no—
26C22⋊(C67⋊C8)—22no—
27C22⋊(C67⋊C8)—41no—
28C22⋊(C67⋊C8)—41no—
29C22×D4×C67C2×C411yescharacteristic
30C23×C4×C67D411yescharacteristic
31C23×C4×C67D411yescharacteristic
32#125×C67—41no—
33#125×C67—41no—
34C4×D4×C67—41no—
35C4×D4×C67—41no—
36C2×C4⋊C4×C67—21no—
37C2×C4⋊C4×C67—21no—
38C2×C4⋊C4×C67—21no—
39C2×C4⋊C4×C67—21no—
40C2×C4⋊C4×C67—21no—
41C2×C4⋊C4×C67C2×C411yescharacteristic
42C2×C4⋊C4×C67C2×C411yescharacteristic
43C2×C4⋊C4×C67C2311yescharacteristic
44C2×C22⋊C4×C67—21no—
45C2×C22⋊C4×C67—21no—
46C2×C22⋊C4×C67—21no—
47C2×C22⋊C4×C67—21no—
48C2×C22⋊C4×C67D411yescharacteristic
49C2×C22⋊C4×C67D411yescharacteristic
50C2×C42×C67C2×C411yescharacteristic
51C2×C42×C67C2×C411yescharacteristic
52C2×C42×C67C2×C411yescharacteristic
53#95×C67—21no—
54#95×C67—21no—
55#95×C67—21no—
56#11761612—82no—
57C2×C67⋊C8—42no—
58C2×C67⋊C8—42no—
59C2×C67⋊C8—82no—
60C24×C67C22⋊C411yescharacteristic
61C24×C67C22⋊C411yescharacteristic
62C2×D4×C67—22no—
63C2×D4×C67—22no—
64C2×D4×C67—41no—
65C22×C4×C67—21no—
66C22×C4×C67—21no—
67C22×C4×C67—21no—
68C22×C4×C67—21no—
69C22×C4×C67—21no—
70C22×C4×C67—21no—
71C22×C4×C67—21no—
72C22×C4×C67—21no—
73C22×C4×C67—81no—
74C22×C4×C67—81no—
75C22×C4×C67C22⋊C411yescharacteristic
76C22×C4×C67C22⋊C411yescharacteristic
77C22×C4×C67C22×C411yescharacteristic
78C22×C4×C67C2×D411yescharacteristic
79C22×C4×C67C2×D411yescharacteristic
80C4⋊C4×C67—41no—
81C4⋊C4×C67—41no—
82C4⋊C4×C67—41no—
83C4⋊C4×C67—42no—
84C4⋊C4×C67—42no—
85C4⋊C4×C67—41no—
86C4⋊C4×C67—41no—
87C4⋊C4×C67—81no—
88C4⋊C4×C67—81no—
89C22⋊C4×C67—41no—
90C22⋊C4×C67—41no—
91C22⋊C4×C67—42no—
92C22⋊C4×C67—42no—
93C22⋊C4×C67—81no—
94C22⋊C4×C67—81no—
95C42×C67—21no—
96C42×C67—21no—
97C42×C67—22no—
98C42×C67—22no—
99C23×C67—21no—
100C23×C67—21no—
101C23×C67—21no—
102C23×C67—21no—
103C23×C67—21no—
104C23×C67—41no—
105C23×C67—41no—
106C23×C67—41no—
107C23×C67—41no—
108C23×C67#10112yes—
109C23×C67C2×C22⋊C411yescharacteristic
110D4×C67—44no—
111C2×C4×C67—21no—
112C2×C4×C67—21no—
113C2×C4×C67—22no—
114C2×C4×C67—21no—
115C2×C4×C67—21no—
116C2×C4×C67—21no—
117C2×C4×C67—41no—
118C2×C4×C67—42no—
119C2×C4×C67—42no—
120C2×C4×C67—41no—
121C2×C4×C67—41no—
122C2×C4×C67—41no—
123C2×C4×C67—41no—
124C2×C4×C67—41no—
125C2×C4×C67—41no—
126C2×C4×C67—41no—
127C2×C4×C67—41no—
128C2×C4×C67—41no—
129C2×C4×C67—41no—
130C2×C4×C67—41no—
131C2×C4×C67—41no—
132C2×C4×C67—41no—
133C2×C4×C67—41no—
134C2×C4×C67—81no—
135C2×C4×C67—81no—
136C2×C4×C67#9911yescharacteristic
137C2×C4×C67#9911yescharacteristic
138C2×C4×C67#13112yes—
139C67⋊C8—82no—
140C67⋊C8—82no—
141C22×C67—21no—
142C22×C67—21no—
143C22×C67—22no—
144C22×C67—42no—
145C22×C67—42no—
146C22×C67—41no—
147C22×C67—41no—
148C22×C67—41no—
149C22×C67—41no—
150C22×C67—41no—
151C22×C67—81no—
152C22×C67—81no—
153C22×C67#40811yescharacteristic
154C22×C67C2×#9911yescharacteristic
155C22×C67C2×#10111yescharacteristic
156C4×C67—21no—
157C4×C67—21no—
158C4×C67—22no—
159C4×C67—42no—
160C4×C67—41no—
161C4×C67—41no—
162C4×C67—41no—
163C4×C67—81no—
164C4×C67—81no—
165C4×C67—81no—
166#11572—671nomaximal, Sylow
167C2×C67—21no—
168C2×C67—21no—
169C2×C67—42no—
170C2×C67—81no—
171C2×C67#147211yescharacteristic
172C2×C67#212711yescharacteristic
173C2×C67#213211yescharacteristic
174#2566D6711yescharacteristic
175#1465—671no—
176#1459—671no—
177#1325—672no—
178#1324—672no—
179C67#1157211yescharacteristic, complemented, Sylow
180C2×#125Dic6711yescharacteristic
181C2×C4×D4C2×D6711yescharacteristic
182#407—1341no—
183C2×C22⋊C8—1341no—
184#396—21no—
185#396—21no—
186#390Dic6711yescharacteristic
187#390Dic6711yescharacteristic
188#385—21no—
189#385—21no—
190#384C2×D6711yescharacteristic
191#382C2×D6711yescharacteristic
192C4×C4⋊C4Dic6711yescharacteristic
193#324—672no—
194#324—672no—
195#323—1341no—
196#323—1341no—
197C22×D4C2×Dic6711yescharacteristic
198C23×C4C22⋊D6711yescharacteristic
199C23×C4C22⋊D6711yescharacteristic
200C2×M16—2681no—
201C22×C8—2681no—
202#125—41no—
203#125—41no—
204C4×D4—41no—
205C4×D4—41no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4C2×Dic6711yescharacteristic
212C2×C4⋊C4C2×Dic6711yescharacteristic
213C2×C4⋊C4C22×D6711yescharacteristic
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4C22⋊D6711yescharacteristic
219C2×C22⋊C4C22⋊D6711yescharacteristic
220C2×C42C2×Dic6711yescharacteristic
221C2×C42C2×Dic6711yescharacteristic
222C2×C42C2×Dic6711yescharacteristic
223C22⋊C8—2681no—
224C22⋊C8—2681no—
225C22⋊C8—1342no—
226C22⋊C8—1342no—
227#95—21no—
228#95—21no—
229#95—21no—
230C24C22⋊Dic6711yescharacteristic
231C24C22⋊Dic6711yescharacteristic
232C2×D4—22no—
233C2×D4—22no—
234C2×D4—41no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—81no—
244C22×C4—81no—
245C22×C4C22⋊Dic6711yescharacteristic
246C22×C4C22⋊Dic6711yescharacteristic
247C22×C4C22×Dic6711yescharacteristic
248C22×C4C2×C22⋊D6711yescharacteristic
249C22×C4C2×C22⋊D6711yescharacteristic
250M16—5362no—
251C2×C8—2682no—
252C2×C8—2682no—
253C2×C8—5362no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—81no—
262C4⋊C4—81no—
263C22⋊C4—41no—
264C22⋊C4—41no—
265C22⋊C4—42no—
266C22⋊C4—42no—
267C22⋊C4—81no—
268C22⋊C4—81no—
269C42—21no—
270C42—21no—
271C42—22no—
272C42—22no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23#1452396212yes—
283C23C2×C22⋊Dic6711yescharacteristic
284D4—44no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—22no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4#1452395811yescharacteristic
311C2×C4#1452395811yescharacteristic
312C2×C4#1452405012yes—
313C8—5362no—
314C8—5362no—
315C22—21no—
316C22—21no—
317C22—22no—
318C22—42no—
319C22—42no—
320C22—41no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22—81no—
326C22—81no—
327C22#1686336311yescharacteristic
328C22C2×#1452395811yescharacteristic
329C22C2×#1452396211yescharacteristic
330C4—21no—
331C4—21no—
332C4—22no—
333C4—42no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—81no—
338C4—81no—
339C4—81no—
340C2—21no—
341C2—21no—
342C2—42no—
343C2—81no—
344C2#2255857311yescharacteristic
345C2#2256187811yescharacteristic
346C2#2256189411yescharacteristic
subgroupG/H
centerC22C2×#14523958
commutatorC22×C4×C67C22×C4
FrattiniC2×C4⋊C4C22×D67
FittingC67×#2566C2
socleC22×C67C2×#99
prime radicalC22×D4×C67C2×C4
cosocleC2×C4⋊C4×C67C23
hypercenter#2566D67
hypocenterC67#11572
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2566

Complements

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

NH ≅ G/Ntype
3. #22558155C2semidirect
4. #22558153C2semidirect
179. C67#11572semidirect

Maximal subgroups

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

subgroupindex
#225585472
#225585272
#225581552
#225581532
C67×#25662
#1157267