Finite Group Databasebeta

← #30751392

Subgroups — #30751392

Order 17152.

632 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16863449C211yesmaximal, characteristic
2C2×#16863133C212yesmaximal
3#22558436C211yesmaximal, characteristic
4#22558087C212yesmaximal
5C67×#2855C211yesmaximal, characteristic
6#16863449—24no—
7C2×C4⋊(C67⋊C8)—21no—
8C2×C22⋊(C67⋊C8)C2212yes—
9C2×C4×C67⋊C8—21no—
10C2×C67⋊#95—21no—
11C2×C67⋊#95—21no—
12C2×C67⋊#95C2212yes—
13#16863133—24no—
14C2×C22⋊Q8×C67C2211yescharacteristic
15C22×C42×C67C2211yescharacteristic
16#391×C67C411yescharacteristic, complemented
17#391×C67C411yescharacteristic, complemented
18#389×C67C411yescharacteristic, complemented
19#389×C67C411yescharacteristic, complemented
20#387×C67—21no—
21#387×C67—21no—
22#385×C67—21no—
23#381×C67—21no—
24#380×C67C2211yescharacteristic
25C23×Dic67—22no—
26C22×C67⋊C8—22no—
27C4⋊(C67⋊C8)—24no—
28C22⋊(C67⋊C8)—22no—
29C22⋊(C67⋊C8)D414yes—
30C4×C67⋊C8—24no—
31C67⋊#95—24no—
32C67⋊#95—22no—
33C67⋊#95—22no—
34C67⋊#95—42no—
35C22×Q8×C67C2×C411yescharacteristic
36C23×C4×C67D411yescharacteristic
37C23×C4×C67D411yescharacteristic
38C23×C4×C67C2311yescharacteristic
39C22⋊Q8×C67—22no—
40C22⋊Q8×C67—22no—
41C2×C4⋊C4×C67—21no—
42C2×C4⋊C4×C67—21no—
43C2×C4⋊C4×C67—21no—
44C2×C4⋊C4×C67C2×C411yescharacteristic
45C2×C4⋊C4×C67C2×C411yescharacteristic
46C2×C4⋊C4×C67C2×C411yescharacteristic
47C2×C22⋊C4×C67—21no—
48C2×C22⋊C4×C67—21no—
49C2×C42×C67—22no—
50C2×C42×C67—22no—
51C2×C42×C67C2×C411yescharacteristic
52C2×C42×C67C2×C411yescharacteristic
53C2×C42×C67D411yescharacteristic
54C2×C42×C67D411yescharacteristic
55#95×C67—21no—
56#95×C67—21no—
57#95×C67—21no—
58#95×C67—21no—
59#95×C67—21no—
60#95×C67—21no—
61#95×C67—21no—
62C22×Dic67—24no—
63C22×Dic67—42no—
64C22×Dic67—42no—
65C22×Dic67—42no—
66C22×Dic67—42no—
67C22×Dic67—42no—
68C22×Dic67—42no—
69C2×C67⋊C8—24no—
70C2×C67⋊C8—42no—
71C2×C67⋊C8—42no—
72C24×C67C2×D411yescharacteristic
73C2×Q8×C67—22no—
74C2×Q8×C67—41no—
75C2×Q8×C67—41no—
76C22×C4×C67—22no—
77C22×C4×C67—22no—
78C22×C4×C67—22no—
79C22×C4×C67—22no—
80C22×C4×C67—21no—
81C22×C4×C67—21no—
82C22×C4×C67—21no—
83C22×C4×C67—21no—
84C22×C4×C67—41no—
85C22×C4×C67—41no—
86C22×C4×C67—41no—
87C22×C4×C67C22⋊C411yescharacteristic
88C22×C4×C67C22⋊C411yescharacteristic
89C22×C4×C67C22⋊C411yescharacteristic
90C22×C4×C67C22⋊C411yescharacteristic
91C22×C4×C67SD1612yes—
92C22×C4×C67Q1612yes—
93C22×C4×C67C22×C411yescharacteristic
94C22×C4×C67C2×D412yes—
95C22×C4×C67C2×D411yescharacteristic
96C22×C4×C67#4112yes—
97C4⋊C4×C67—22no—
98C4⋊C4×C67—41no—
99C4⋊C4×C67—41no—
100C4⋊C4×C67—42no—
101C4⋊C4×C67—42no—
102C4⋊C4×C67—41no—
103C4⋊C4×C67—41no—
104C22⋊C4×C67—42no—
105C22⋊C4×C67—42no—
106C42×C67—22no—
107C42×C67—22no—
108C42×C67—21no—
109C42×C67—21no—
110C42×C67—21no—
111C42×C67—21no—
112C23×C67—21no—
113C23×C67—21no—
114C23×C67—21no—
115C23×C67—21no—
116C23×C67C2×C22⋊C411yescharacteristic
117C23×C67#12112yes—
118C23×C67C2×SD1611yescharacteristic
119C23×C67C2×Q1611yescharacteristic
120C23×C67#13512yes—
121Q8×C67—82no—
122C2×C4×C67—21no—
123C2×C4×C67—21no—
124C2×C4×C67—21no—
125C2×C4×C67—21no—
126C2×C4×C67—21no—
127C2×C4×C67—21no—
128C2×C4×C67—21no—
129C2×C4×C67—21no—
130C2×C4×C67—21no—
131C2×C4×C67—21no—
132C2×C4×C67—24no—
133C2×C4×C67—24no—
134C2×C4×C67—21no—
135C2×C4×C67—21no—
136C2×C4×C67—21no—
137C2×C4×C67—42no—
138C2×C4×C67—42no—
139C2×C4×C67—44no—
140C2×C4×C67—41no—
141C2×C4×C67—41no—
142C2×C4×C67—41no—
143C2×C4×C67—41no—
144C2×C4×C67—41no—
145C2×C4×C67—41no—
146C2×C4×C67—41no—
147C2×C4×C67—41no—
148C2×C4×C67—41no—
149C2×C4×C67—41no—
150C2×C4×C67—41no—
151C2×C4×C67—41no—
152C2×C4×C67#10312yes—
153C2×C4×C67#10312yes—
154C2×C4×C67#10412yes—
155C2×C4×C67C4×D412yes—
156C2×C4×C67#12011yescharacteristic
157C2×C4×C67#12311yescharacteristic
158C2×Dic67—44no—
159C2×Dic67—44no—
160C2×Dic67—44no—
161C2×Dic67—82no—
162C2×Dic67—82no—
163C2×Dic67—82no—
164C2×Dic67—82no—
165C67⋊C8—44no—
166C22×C67—22no—
167C22×C67—22no—
168C22×C67—22no—
169C22×C67—21no—
170C22×C67—21no—
171C22×C67—21no—
172C22×C67—21no—
173C22×C67—22no—
174C22×C67#38611yescharacteristic
175C22×C67#41011yescharacteristic
176C22×C67#41011yescharacteristic
177C22×C67C2×#10311yescharacteristic
178C22×C67#41611yescharacteristic
179C22×C67C2×#10411yescharacteristic
180C22×C67#42111yescharacteristic
181C22×C67#46512yes—
182C22×C67#46712yes—
183C4×C67—22no—
184C4×C67—22no—
185C4×C67—22no—
186C4×C67—21no—
187C4×C67—22no—
188C4×C67—21no—
189C4×C67—41no—
190C4×C67—81no—
191C4×C67—81no—
192C4×C67—81no—
193C4×C67—81no—
194Dic67—84no—
195#9112—671nomaximal, Sylow
196C2×C67—22no—
197C2×C67—22no—
198C2×C67#150011yescharacteristic
199C2×C67#150311yescharacteristic
200C2×C67#151811yescharacteristic
201C2×C67#152111yescharacteristic
202C2×C67#187111yescharacteristic
203C2×C67#189211yescharacteristic
204C2×C67#189911yescharacteristic
205C2×#431—671no—
206#2855D6711yescharacteristic
207C2×#328—672no—
208#1434—671no—
209#1294—672no—
210C67#911211yescharacteristic, complemented, Sylow
211C2×C22⋊Q8C2×D6711yescharacteristic
212C22×C42C2×D6711yescharacteristic
213#431—1344no—
214C2×C4⋊C8—1341no—
215C2×C22⋊C8—672no—
216C2×C4×C8—1341no—
217#391Dic6711yescharacteristic, complemented
218#391Dic6711yescharacteristic, complemented
219#389Dic6711yescharacteristic, complemented
220#389Dic6711yescharacteristic, complemented
221#387—21no—
222#387—21no—
223#385—21no—
224#381—21no—
225#380C2×D6711yescharacteristic
226C2×#95—672no—
227C2×#95—1341no—
228C2×#95—1341no—
229#328—1344no—
230C22×Q8C2×Dic6711yescharacteristic
231C23×C4—1342no—
232C23×C4C22⋊D6711yescharacteristic
233C23×C4C22⋊D6711yescharacteristic
234C23×C4C22×D6711yescharacteristic
235C22×C8—1342no—
236C22⋊Q8—22no—
237C22⋊Q8—22no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4C2×Dic6711yescharacteristic
242C2×C4⋊C4C2×Dic6711yescharacteristic
243C2×C4⋊C4C2×Dic6711yescharacteristic
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42C2×Dic6711yescharacteristic
249C2×C42C2×Dic6711yescharacteristic
250C2×C42C22⋊D6711yescharacteristic
251C2×C42C22⋊D6711yescharacteristic
252C4⋊C8—1344no—
253C22⋊C8—674no—
254C22⋊C8—1342no—
255C4×C8—1344no—
256#95—21no—
257#95—21no—
258#95—21no—
259#95—21no—
260#95—21no—
261#95—21no—
262#95—21no—
263#95—2682no—
264#95—1344no—
265#95—1342no—
266#95—1342no—
267C24C2×C22⋊D6711yescharacteristic
268C2×Q8—22no—
269C2×Q8—41no—
270C2×Q8—41no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—2682no—
283C22×C4—2682no—
284C22×C4—2682no—
285C22×C4—2682no—
286C22×C4—2682no—
287C22×C4—2682no—
288C22×C4—1344no—
289C22×C4C22⋊Dic6711yescharacteristic
290C22×C4C22⋊Dic6711yescharacteristic
291C22×C4C22⋊Dic6711yescharacteristic
292C22×C4C22⋊Dic6711yescharacteristic
293C22×C4#1176161512yes—
294C22×C4#1176161812yes—
295C22×C4C22×Dic6711yescharacteristic
296C22×C4D4×D6712yes—
297C22×C4C2×C22⋊D6711yescharacteristic
298C22×C4#1176162712yes—
299C2×C8—2682no—
300C2×C8—2682no—
301C2×C8—1344no—
302C4⋊C4—22no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C22⋊C4—42no—
310C22⋊C4—42no—
311C42—22no—
312C42—22no—
313C42—21no—
314C42—21no—
315C42—21no—
316C42—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23C2×C22⋊Dic6711yescharacteristic
322C23#1452401512yes—
323C23C2×#1176161511yescharacteristic
324C23C2×#1176161811yescharacteristic
325C23#1452406312yes—
326Q8—82no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—24no—
335C2×C4—24no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—44no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—2684no—
358C2×C4—2684no—
359C2×C4—2684no—
360C2×C4—5362no—
361C2×C4—5362no—
362C2×C4—5362no—
363C2×C4—5362no—
364C2×C4#1452396712yes—
365C2×C4#1452396712yes—
366C2×C4#1452397012yes—
367C2×C4D4×Dic6712yes—
368C2×C4#1452401111yescharacteristic
369C2×C4#1452402411yescharacteristic
370C8—2684no—
371C22—22no—
372C22—21no—
373C22—21no—
374C22—22no—
375C22—22no—
376C22—22no—
377C22—21no—
378C22—21no—
379C22#1686327611yescharacteristic
380C22#1686337011yescharacteristic
381C22#1686337011yescharacteristic
382C22C2×#1452396711yescharacteristic
383C22#1686339211yescharacteristic
384C22C2×#1452397011yescharacteristic
385C22#1686341311yescharacteristic
386C22#1686362712yes—
387C22#1686363512yes—
388C4—22no—
389C4—22no—
390C4—22no—
391C4—21no—
392C4—22no—
393C4—21no—
394C4—41no—
395C4—81no—
396C4—81no—
397C4—81no—
398C4—81no—
399C4—5364no—
400C2—22no—
401C2—22no—
402C2#2255869511yescharacteristic
403C2#2255870811yescharacteristic
404C2#2255877311yescharacteristic
405C2#2255878611yescharacteristic
406C2#2256051211yescharacteristic
407C2#2256063311yescharacteristic
408C2#2256066811yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic67
commutatorC22×C4×C67C22×C4
FrattiniC23×C4C22×D67
FittingC67×#2855C2
socleC23×C67C2×C22⋊C4
prime radicalC24×C67C2×D4
cosocleC23×C4×C67C23
hypercenter#2855D67
hypocenterC67#9112
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2855

Complements

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

NH ≅ G/Ntype
16. #391×C67C4semidirect
17. #391×C67C4semidirect
18. #389×C67C4semidirect
19. #389×C67C4semidirect
210. C67#9112semidirect
217. #391Dic67semidirect
218. #391Dic67semidirect
219. #389Dic67semidirect
220. #389Dic67semidirect

Maximal subgroups

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

subgroupindex
C2×#168634492
C2×#168631332
#225584362
#225580872
C67×#28552
#911267