Finite Group Databasebeta

← #30756023

Subgroups — #30756023

Order 17152.

548 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22560189C211yesmaximal, characteristic
2#22558436C211yesmaximal, characteristic
3#22558087C212yesmaximal
4#22558087C212yesmaximal
5C67×#2851C211yesmaximal, characteristic
6C2×C4⋊(C67⋊C8)—21no—
7C2×C22⋊(C67⋊C8)C2212yes—
8C2×C67⋊#95—21no—
9C2×C67⋊#95—21no—
10C2×C67⋊#95C2212yes—
11#16863151—21no—
12C2×#125×C67C411yescharacteristic, complemented
13C22×C42×C67C2211yescharacteristic
14#389×C67C411yescharacteristic, complemented
15#389×C67C411yescharacteristic, complemented
16#387×C67—21no—
17#387×C67—21no—
18#385×C67—21no—
19#385×C67—21no—
20#382×C67C411yescharacteristic, complemented
21#380×C67C2211yescharacteristic
22#380×C67C2211yescharacteristic
23C23×Dic67—22no—
24C22×C67⋊C8—22no—
25C4⋊(C67⋊C8)—42no—
26C22⋊(C67⋊C8)—24no—
27C67⋊#95—22no—
28C67⋊#95—42no—
29C67⋊#95—42no—
30C67⋊#95Q814yes—
31C23×C4×C67D411yescharacteristic
32C23×C4×C67D411yescharacteristic
33C23×C4×C67C2311yescharacteristic
34#125×C67—42no—
35C2×C4⋊C4×C67—21no—
36C2×C4⋊C4×C67—21no—
37C2×C4⋊C4×C67—21no—
38C2×C4⋊C4×C67—21no—
39C2×C4⋊C4×C67C2×C411yescharacteristic
40C2×C4⋊C4×C67C2×C411yescharacteristic
41C2×C4⋊C4×C67C2×C411yescharacteristic
42C2×C4⋊C4×C67C2×C411yescharacteristic
43C2×C22⋊C4×C67—21no—
44C2×C22⋊C4×C67—21no—
45C2×C42×C67—42no—
46C2×C42×C67C2×C411yescharacteristic
47C2×C42×C67C2×C411yescharacteristic
48C2×C42×C67D411yescharacteristic
49C2×C42×C67D411yescharacteristic
50#95×C67—21no—
51#95×C67—21no—
52#95×C67—21no—
53#95×C67—21no—
54#95×C67—21no—
55#95×C67—21no—
56C22×Dic67—24no—
57C22×Dic67—42no—
58C22×Dic67—42no—
59C22×Dic67—42no—
60C22×Dic67—42no—
61C22×Dic67—42no—
62C22×Dic67—42no—
63C2×C67⋊C8—42no—
64C2×C67⋊C8—42no—
65C2×C67⋊C8—42no—
66C24×C67C2×D411yescharacteristic
67C22×C4×C67—22no—
68C22×C4×C67—21no—
69C22×C4×C67—21no—
70C22×C4×C67—22no—
71C22×C4×C67—21no—
72C22×C4×C67—21no—
73C22×C4×C67—21no—
74C22×C4×C67—21no—
75C22×C4×C67—41no—
76C22×C4×C67—41no—
77C22×C4×C67—41no—
78C22×C4×C67—41no—
79C22×C4×C67—41no—
80C22×C4×C67C22⋊C411yescharacteristic
81C22×C4×C67C22⋊C411yescharacteristic
82C22×C4×C67C22⋊C411yescharacteristic
83C22×C4×C67C22⋊C411yescharacteristic
84C22×C4×C67C22×C411yescharacteristic
85C22×C4×C67C2×D411yescharacteristic
86C22×C4×C67C2×Q812yes—
87C22×C4×C67#4112yes—
88C4⋊C4×C67—41no—
89C4⋊C4×C67—41no—
90C4⋊C4×C67—41no—
91C4⋊C4×C67—41no—
92C4⋊C4×C67—42no—
93C4⋊C4×C67—42no—
94C4⋊C4×C67—42no—
95C4⋊C4×C67—42no—
96C22⋊C4×C67—42no—
97C22⋊C4×C67—41no—
98C22⋊C4×C67—41no—
99C42×C67—22no—
100C42×C67—22no—
101C42×C67—41no—
102C42×C67—41no—
103C23×C67—21no—
104C23×C67—21no—
105C23×C67—21no—
106C23×C67—21no—
107C23×C67C2×C22⋊C411yescharacteristic
108C23×C67C22⋊Q812yes—
109C23×C67#13512yes—
110C23×C67AGL(1,ℤ/8)11yescharacteristic
111C23×C67#13711yescharacteristic
112C2×C4×C67—21no—
113C2×C4×C67—21no—
114C2×C4×C67—21no—
115C2×C4×C67—21no—
116C2×C4×C67—21no—
117C2×C4×C67—21no—
118C2×C4×C67—21no—
119C2×C4×C67—21no—
120C2×C4×C67—21no—
121C2×C4×C67—21no—
122C2×C4×C67—21no—
123C2×C4×C67—21no—
124C2×C4×C67—21no—
125C2×C4×C67—21no—
126C2×C4×C67—21no—
127C2×C4×C67—42no—
128C2×C4×C67—42no—
129C2×C4×C67—44no—
130C2×C4×C67—42no—
131C2×C4×C67—42no—
132C2×C4×C67—41no—
133C2×C4×C67—41no—
134C2×C4×C67—41no—
135C2×C4×C67—41no—
136C2×C4×C67—41no—
137C2×C4×C67—41no—
138C2×C4×C67—41no—
139C2×C4×C67—41no—
140C2×C4×C67—41no—
141C2×C4×C67—41no—
142C2×C4×C67—41no—
143C2×C4×C67—41no—
144C2×C4×C67#9912yes—
145C2×C4×C67C4×Q812yes—
146C2×C4×C67#12411yescharacteristic
147C2×C4×C67C4⋊Q811yescharacteristic
148C2×Dic67—44no—
149C2×Dic67—44no—
150C2×Dic67—44no—
151C2×Dic67—82no—
152C2×Dic67—82no—
153C2×Dic67—82no—
154C2×Dic67—82no—
155C67⋊C8—82no—
156C22×C67—22no—
157C22×C67—22no—
158C22×C67—21no—
159C22×C67—21no—
160C22×C67—21no—
161C22×C67—21no—
162C22×C67—22no—
163C22×C67—22no—
164C22×C67#39111yescharacteristic
165C22×C67C2×#9911yescharacteristic
166C22×C67#41011yescharacteristic
167C22×C67#41611yescharacteristic
168C22×C67#41711yescharacteristic
169C22×C67#42111yescharacteristic
170C22×C67#42111yescharacteristic
171C22×C67#47812yes—
172C22×C67#47912yes—
173C4×C67—21no—
174C4×C67—22no—
175C4×C67—21no—
176C4×C67—22no—
177C4×C67—41no—
178C4×C67—41no—
179C4×C67—41no—
180C4×C67—81no—
181C4×C67—81no—
182C4×C67—81no—
183C4×C67—81no—
184Dic67—84no—
185#9913—671nomaximal, Sylow
186C2×C67—22no—
187C2×C67—22no—
188C2×C67#151211yescharacteristic
189C2×C67#151311yescharacteristic
190C2×C67#151611yescharacteristic
191C2×C67#152111yescharacteristic
192C2×C67#198911yescharacteristic
193C2×C67#199511yescharacteristic
194C2×C67#199611yescharacteristic
195#2851D6711yescharacteristic
196#1797—671no—
197#1434—671no—
198#1294—672no—
199#1294—672no—
200C67#991311yescharacteristic, complemented, Sylow
201C2×#125Dic6711yescharacteristic, complemented
202C22×C42C2×D6711yescharacteristic
203C2×C4⋊C8—1341no—
204C2×C22⋊C8—672no—
205#389Dic6711yescharacteristic, complemented
206#389Dic6711yescharacteristic, complemented
207#387—21no—
208#387—21no—
209#385—21no—
210#385—21no—
211#382Dic6711yescharacteristic, complemented
212#380C2×D6711yescharacteristic
213#380C2×D6711yescharacteristic
214C2×#95—672no—
215C2×#95—1341no—
216C2×#95—1341no—
217#336—1341no—
218C23×C4—1342no—
219C23×C4C22⋊D6711yescharacteristic
220C23×C4C22⋊D6711yescharacteristic
221C23×C4C22×D6711yescharacteristic
222C22×C8—1342no—
223#125—42no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4C2×Dic6711yescharacteristic
229C2×C4⋊C4C2×Dic6711yescharacteristic
230C2×C4⋊C4C2×Dic6711yescharacteristic
231C2×C4⋊C4C2×Dic6711yescharacteristic
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C42—42no—
235C2×C42C2×Dic6711yescharacteristic
236C2×C42C2×Dic6711yescharacteristic
237C2×C42C22⋊D6711yescharacteristic
238C2×C42C22⋊D6711yescharacteristic
239C4⋊C8—2682no—
240C22⋊C8—1344no—
241#95—21no—
242#95—21no—
243#95—21no—
244#95—21no—
245#95—21no—
246#95—21no—
247#95—2682no—
248#95—2682no—
249#95—674no—
250#95—1342no—
251C24C2×C22⋊D6711yescharacteristic
252C22×C4—22no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—22no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—2682no—
266C22×C4—2682no—
267C22×C4—2682no—
268C22×C4—2682no—
269C22×C4—2682no—
270C22×C4—2682no—
271C22×C4—1344no—
272C22×C4C22⋊Dic6711yescharacteristic
273C22×C4C22⋊Dic6711yescharacteristic
274C22×C4C22⋊Dic6711yescharacteristic
275C22×C4C22⋊Dic6711yescharacteristic
276C22×C4C22×Dic6711yescharacteristic
277C22×C4C2×C22⋊D6711yescharacteristic
278C22×C4Q8×D6712yes—
279C22×C4#1176162912yes—
280C2×C8—2682no—
281C2×C8—2682no—
282C2×C8—2682no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C22⋊C4—42no—
292C22⋊C4—41no—
293C22⋊C4—41no—
294C42—22no—
295C42—22no—
296C42—41no—
297C42—41no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23C2×C22⋊Dic6711yescharacteristic
303C23#1452401912yes—
304C23#1452406312yes—
305C23#1452406811yescharacteristic
306C23#1452407311yescharacteristic
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—44no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—2684no—
340C2×C4—2684no—
341C2×C4—2684no—
342C2×C4—5362no—
343C2×C4—5362no—
344C2×C4—5362no—
345C2×C4—5362no—
346C2×C4#1452395812yes—
347C2×C4Q8×Dic6712yes—
348C2×C4#1452403211yescharacteristic
349C2×C4#1452404111yescharacteristic
350C8—5362no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—22no—
356C22—22no—
357C22—22no—
358C22—22no—
359C22#1686330311yescharacteristic
360C22C2×#1452395811yescharacteristic
361C22#1686337011yescharacteristic
362C22#1686339211yescharacteristic
363C22#1686339711yescharacteristic
364C22#1686341311yescharacteristic
365C22#1686341311yescharacteristic
366C22#1686369312yes—
367C22#1686370012yes—
368C4—22no—
369C4—21no—
370C4—22no—
371C4—21no—
372C4—41no—
373C4—41no—
374C4—41no—
375C4—81no—
376C4—81no—
377C4—81no—
378C4—81no—
379C4—5364no—
380C2—22no—
381C2—22no—
382C2#2255874711yescharacteristic
383C2#2255875211yescharacteristic
384C2#2255876511yescharacteristic
385C2#2255878611yescharacteristic
386C2#2256116711yescharacteristic
387C2#2256119111yescharacteristic
388C2#2256119511yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic67
commutatorC22×C4×C67C22×C4
FrattiniC23×C4C22×D67
FittingC67×#2851C2
socleC23×C67C2×C22⋊C4
prime radicalC24×C67C2×D4
cosocleC23×C4×C67C23
hypercenter#2851D67
hypocenterC67#9913
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2851

Complements

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

NH ≅ G/Ntype
12. C2×#125×C67C4semidirect
14. #389×C67C4semidirect
15. #389×C67C4semidirect
20. #382×C67C4semidirect
200. C67#9913semidirect
201. C2×#125Dic67semidirect
205. #389Dic67semidirect
206. #389Dic67semidirect
211. #382Dic67semidirect

Maximal subgroups

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

subgroupindex
#225601892
#225584362
#225580872
C67×#28512
#991367