Finite Group Databasebeta

← #30748818

Subgroups — #30748818

Order 17152.

456 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22560247C211yesmaximal, characteristic, complemented
2#22560234C211yesmaximal, characteristic, complemented
3#22558752C212yesmaximal
4#22558114C212yesmaximal, complemented
5C67×#2929C211yesmaximal, characteristic
6C2×#14523979—21no—
7C2×#14523975—21no—
8#16863360C2212yes—
9#16863256C2212yes—
10#16863161—21no—
11#16863159—21no—
12#16863133—22no—
13#16863130—22no—
14C2×#131×C67C411yescharacteristic, complemented
15C23×C8×C67C2211yescharacteristic
16#525×C67C2211yescharacteristic
17#435×C67—21no—
18#435×C67—21no—
19C8×D4×C67—21no—
20C8×D4×C67—21no—
21#424×C67C411yescharacteristic, complemented
22#408×C67C411yescharacteristic, complemented
23#407×C67C2211yescharacteristic
24#405×C67C411yescharacteristic, complemented
25C2×#11761612—22no—
26C2×C4⋊Dic67—22no—
27C2×C22⋊Dic67—22no—
28#14523979—42no—
29#14523975—42no—
30C22⋊(C67⋊C8)—22no—
31C22⋊(C67⋊C8)—22no—
32C67⋊#95—22no—
33C2×#41×C67C2×C411yescharacteristic
34C23×C4×C67C2311yescharacteristic
35#131×C67—42no—
36C2×M16×C67—21no—
37C2×M16×C67C2×C411yescharacteristic
38C2×M16×C67C2×C411yescharacteristic
39C22×C8×C67—21no—
40C22×C8×C67—42no—
41C22×C8×C67C2×C411yescharacteristic
42C22×C8×C67C2×C411yescharacteristic
43C22×C8×C67D411yescharacteristic
44C22×C8×C67D411yescharacteristic
45C22×C8×C67D411yescharacteristic
46C22×C8×C67D411yescharacteristic
47#123×C67—21no—
48C22⋊Q8×C67D411yescharacteristic
49C22⋊Q8×C67Q811yescharacteristic
50#121×C67D411yescharacteristic
51#121×C67Q811yescharacteristic
52#120×C67—21no—
53C4×D4×C67—21no—
54C4×D4×C67—21no—
55#117×C67C2×C411yescharacteristic
56C4⋊C8×C67—21no—
57C4⋊C8×C67—21no—
58C22⋊C8×C67—21no—
59C22⋊C8×C67—21no—
60C22⋊C8×C67—21no—
61C22⋊C8×C67—21no—
62#97×C67—21no—
63C4×C8×C67—21no—
64C22×Dic67—42no—
65#11761612—42no—
66#11761612—42no—
67C2×C67⋊C8—42no—
68C4⋊Dic67—42no—
69C4⋊Dic67—42no—
70C22⋊Dic67—82no—
71C24×C67#4111yescharacteristic
72#41×C67—41no—
73#41×C67—41no—
74C2×Q8×C67C4⋊C411yescharacteristic
75C2×D4×C67—21no—
76C2×D4×C67—21no—
77C2×D4×C67—21no—
78C2×D4×C67C4⋊C411yescharacteristic
79C22×C4×C67—21no—
80C22×C4×C67—41no—
81C22×C4×C67—41no—
82C22×C4×C67C22×C411yescharacteristic
83C22×C4×C67C2×D411yescharacteristic
84C22×C4×C67C2×D411yescharacteristic
85C22×C4×C67C2×D411yescharacteristic
86C22×C4×C67C2×Q811yescharacteristic
87C22×C4×C67#4111yescharacteristic
88M16×C67—42no—
89M16×C67—41no—
90M16×C67—41no—
91C2×C8×C67—22no—
92C2×C8×C67—22no—
93C2×C8×C67—22no—
94C2×C8×C67—21no—
95C2×C8×C67—21no—
96C2×C8×C67—41no—
97C2×C8×C67—41no—
98C2×C8×C67—41no—
99C2×C8×C67—42no—
100C2×C8×C67C22⋊C411yescharacteristic
101C2×C8×C67C22⋊C411yescharacteristic
102C2×C8×C67C22⋊C411yescharacteristic
103C2×C8×C67C22⋊C411yescharacteristic
104C4⋊C4×C67—21no—
105C4⋊C4×C67—21no—
106C4⋊C4×C67C4⋊C411yescharacteristic
107C4⋊C4×C67C4⋊C411yescharacteristic
108C22⋊C4×C67—21no—
109C22⋊C4×C67—21no—
110C22⋊C4×C67—21no—
111C22⋊C4×C67—21no—
112C22⋊C4×C67—21no—
113C42×C67—21no—
114C23×C67—22no—
115C23×C67—21no—
116C23×C67—41no—
117C23×C67—41no—
118C23×C67#11711yescharacteristic
119C23×C67#12111yescharacteristic
120C23×C67C22⋊Q811yescharacteristic
121Q8×C67—41no—
122D4×C67—41no—
123D4×C67—42no—
124D4×C67—42no—
125D4×C67—42no—
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—21no—
133C2×C4×C67—21no—
134C2×C4×C67—21no—
135C2×C4×C67—41no—
136C2×C4×C67—41no—
137C2×C4×C67—41no—
138C2×C4×C67—41no—
139C2×C4×C67—41no—
140C2×C4×C67C2×C22⋊C411yescharacteristic
141C2×C4×C67C2×C4⋊C411yescharacteristic
142C2×C4×C67#12111yescharacteristic
143C2×C4×C67C22⋊Q811yescharacteristic
144C2×Dic67—82no—
145C2×Dic67—82no—
146C2×Dic67—82no—
147C8×C67—21no—
148C8×C67—21no—
149C8×C67—21no—
150C8×C67—41no—
151C8×C67—41no—
152C8×C67#9912yes—
153C67⋊C8—82no—
154C22×C67—21no—
155C22×C67—21no—
156C22×C67—21no—
157C22×C67—21no—
158C22×C67—21no—
159C22×C67—21no—
160C22×C67—21no—
161C22×C67—21no—
162C22×C67—44no—
163C22×C67—41no—
164C22×C67—41no—
165C22×C67—41no—
166C22×C67#38011yescharacteristic
167C22×C67#47112yes—
168C4×C67—21no—
169C4×C67—21no—
170C4×C67—21no—
171C4×C67—41no—
172C4×C67—41no—
173C4×C67—41no—
174C4×C67C2×#9911yescharacteristic
175C4×C67#41011yescharacteristic
176Dic67—162no—
177#8629—671nomaximal, Sylow
178C2×C67—22no—
179C2×C67—41no—
180C2×C67—41no—
181C2×C67—81no—
182C2×C67#179911yescharacteristic
183C2×C67#181411yescharacteristic
184C2×C67#181511yescharacteristic
185#2929D6711yescharacteristic
186#1810—671no—
187#1807—671no—
188#1513—672no—
189#1305—672no—
190C67#862911yescharacteristic, complemented, Sylow
191C2×#131Dic6711yescharacteristic, complemented
192C23×C8C2×D6711yescharacteristic
193#525C2×D6711yescharacteristic
194#435—21no—
195#435—21no—
196C8×D4—21no—
197C8×D4—21no—
198C2×#108—1341no—
199C2×#106—1341no—
200#424Dic6711yescharacteristic, complemented
201#408Dic6711yescharacteristic, complemented
202#407—672no—
203#407C2×D6711yescharacteristic
204#405Dic6711yescharacteristic, complemented
205#380—672no—
206#341—1341no—
207#340—1341no—
208#328—1342no—
209#327—1342no—
210C2×#41C2×Dic6711yescharacteristic
211C23×C4C22×D6711yescharacteristic
212#131—42no—
213C2×M16—21no—
214C2×M16—1342no—
215C2×M16C2×Dic6711yescharacteristic
216C2×M16C2×Dic6711yescharacteristic
217C22×C8—21no—
218C22×C8—42no—
219C22×C8C2×Dic6711yescharacteristic
220C22×C8C2×Dic6711yescharacteristic
221C22×C8C22⋊D6711yescharacteristic
222C22×C8C22⋊D6711yescharacteristic
223C22×C8C22⋊D6711yescharacteristic
224C22×C8C22⋊D6711yescharacteristic
225#123—21no—
226C22⋊Q8Dic13411yescharacteristic
227C22⋊Q8D26811yescharacteristic
228#121Dic13411yescharacteristic
229#121D26811yescharacteristic
230#120—21no—
231C4×D4—21no—
232C4×D4—21no—
233#117C2×Dic6711yescharacteristic
234C2×C4⋊C4—1342no—
235C2×C22⋊C4—1342no—
236#108—2682no—
237#106—2682no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C22⋊C8—21no—
241C22⋊C8—21no—
242C22⋊C8—21no—
243C22⋊C8—21no—
244C22⋊C8—1342no—
245C22⋊C8—1342no—
246#97—21no—
247C4×C8—21no—
248#95—1342no—
249C24#1176162811yescharacteristic
250#41—41no—
251#41—41no—
252C2×Q8C4⋊Dic6711yescharacteristic
253C2×D4—21no—
254C2×D4—21no—
255C2×D4—21no—
256C2×D4C4⋊Dic6711yescharacteristic
257C22×C4—21no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—2682no—
261C22×C4C22×Dic6711yescharacteristic
262C22×C4C2×C22⋊D6711yescharacteristic
263C22×C4C2×C22⋊D6711yescharacteristic
264C22×C4C2×D26811yescharacteristic
265C22×C4C2×Dic13411yescharacteristic
266C22×C4#1176162811yescharacteristic
267M16—42no—
268M16—41no—
269M16—41no—
270M16—2682no—
271M16—2682no—
272C2×C8—22no—
273C2×C8—22no—
274C2×C8—22no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—41no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—42no—
281C2×C8—2682no—
282C2×C8C22⋊Dic6711yescharacteristic
283C2×C8C22⋊Dic6711yescharacteristic
284C2×C8C22⋊Dic6711yescharacteristic
285C2×C8C22⋊Dic6711yescharacteristic
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—2682no—
289C4⋊C4—2682no—
290C4⋊C4C4⋊Dic6711yescharacteristic
291C4⋊C4C4⋊Dic6711yescharacteristic
292C22⋊C4—21no—
293C22⋊C4—21no—
294C22⋊C4—21no—
295C22⋊C4—21no—
296C22⋊C4—21no—
297C22⋊C4—5362no—
298C42—21no—
299C23—22no—
300C23—21no—
301C23—41no—
302C23—41no—
303C23#1452400211yescharacteristic
304C23#1452401711yescharacteristic
305C23#1452402211yescharacteristic
306Q8—41no—
307D4—41no—
308D4—42no—
309D4—42no—
310D4—42no—
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—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—5362no—
326C2×C4—5362no—
327C2×C4—5362no—
328C2×C4C2×C22⋊Dic6711yescharacteristic
329C2×C4C2×C4⋊Dic6711yescharacteristic
330C2×C4#1452401711yescharacteristic
331C2×C4#1452402211yescharacteristic
332C8—21no—
333C8—21no—
334C8—21no—
335C8—41no—
336C8—41no—
337C8—5362no—
338C8#1452395812yes—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—44no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22#1686325611yescharacteristic
352C22#1686365912yes—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—10722no—
360C4C2×#1452395811yescharacteristic
361C4#1686337011yescharacteristic
362C2—22no—
363C2—41no—
364C2—41no—
365C2—81no—
366C2#2256019711yescharacteristic
367C2#2256026311yescharacteristic
368C2#2256026811yescharacteristic
subgroupG/H
centerC22#16863256
commutatorC22×C4×C67C22×C4
FrattiniC23×C4C22×D67
FittingC67×#2929C2
socleC22×C67#380
prime radical#525×C67C22
cosocleC23×C4×C67C23
hypercenter#2929D67
hypocenterC67#8629
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2929

Complements

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

NH ≅ G/Ntype
1. #22560247C2semidirect
2. #22560234C2semidirect
4. #22558114C2semidirect
14. C2×#131×C67C4semidirect
21. #424×C67C4semidirect
22. #408×C67C4semidirect
24. #405×C67C4semidirect
190. C67#8629semidirect
191. C2×#131Dic67semidirect
200. #424Dic67semidirect
201. #408Dic67semidirect
204. #405Dic67semidirect

Maximal subgroups

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

subgroupindex
#225602472
#225602342
#225587522
#225581142
C67×#29292
#862967