Finite Group Databasebeta

← #30748813

Subgroups — #30748813

Order 17152.

456 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22560247C211yesmaximal, characteristic, complemented
2#22560238C211yesmaximal, characteristic, complemented
3#22558752C212yesmaximal
4#22558114C212yesmaximal, complemented
5C67×#2929C211yesmaximal, characteristic
6C2×#14523979—21no—
7C2×#14523977—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#14523977—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—21no—
115C23×C67—21no—
116C23×C67—21no—
117C23×C67—41no—
118C23×C67—41no—
119C23×C67#11711yescharacteristic
120C23×C67#12111yescharacteristic
121C23×C67C22⋊Q811yescharacteristic
122Q8×C67—41no—
123D4×C67—41no—
124D4×C67—42no—
125D4×C67—42no—
126D4×C67—41no—
127D4×C67—41no—
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—21no—
136C2×C4×C67—21no—
137C2×C4×C67—41no—
138C2×C4×C67—41no—
139C2×C4×C67—41no—
140C2×C4×C67—41no—
141C2×C4×C67—41no—
142C2×C4×C67C2×C22⋊C411yescharacteristic
143C2×C4×C67C2×C4⋊C411yescharacteristic
144C2×C4×C67#12111yescharacteristic
145C2×C4×C67C22⋊Q811yescharacteristic
146C2×Dic67—82no—
147C2×Dic67—82no—
148C2×Dic67—82no—
149C8×C67—21no—
150C8×C67—21no—
151C8×C67—21no—
152C8×C67—41no—
153C8×C67—41no—
154C8×C67#9912yes—
155C67⋊C8—82no—
156C22×C67—21no—
157C22×C67—21no—
158C22×C67—21no—
159C22×C67—21no—
160C22×C67—21no—
161C22×C67—21no—
162C22×C67—21no—
163C22×C67—21no—
164C22×C67—42no—
165C22×C67—42no—
166C22×C67—41no—
167C22×C67—41no—
168C22×C67—41no—
169C22×C67#38011yescharacteristic
170C22×C67#47211yescharacteristic
171C22×C67#47311yescharacteristic
172C4×C67—21no—
173C4×C67—21no—
174C4×C67—21no—
175C4×C67—41no—
176C4×C67—41no—
177C4×C67—41no—
178C4×C67C2×#9911yescharacteristic
179C4×C67#41011yescharacteristic
180Dic67—162no—
181#8628—671nomaximal, Sylow
182C2×C67—21no—
183C2×C67—21no—
184C2×C67—41no—
185C2×C67—41no—
186C2×C67—81no—
187C2×C67#179911yescharacteristic
188C2×C67#181411yescharacteristic
189C2×C67#181611yescharacteristic
190#2929D6711yescharacteristic
191#1810—671no—
192#1808—671no—
193#1513—672no—
194#1305—672no—
195C67#862811yescharacteristic, complemented, Sylow
196C2×#131Dic6711yescharacteristic, complemented
197C23×C8C2×D6711yescharacteristic
198#525C2×D6711yescharacteristic
199#435—21no—
200#435—21no—
201C8×D4—21no—
202C8×D4—21no—
203C2×#108—1341no—
204C2×#107—1341no—
205#424Dic6711yescharacteristic, complemented
206#408Dic6711yescharacteristic, complemented
207#407—672no—
208#407C2×D6711yescharacteristic
209#405Dic6711yescharacteristic, complemented
210#380—672no—
211#341—1341no—
212#340—1341no—
213#328—1342no—
214#327—1342no—
215C2×#41C2×Dic6711yescharacteristic
216C23×C4C22×D6711yescharacteristic
217#131—42no—
218C2×M16—21no—
219C2×M16—1342no—
220C2×M16C2×Dic6711yescharacteristic
221C2×M16C2×Dic6711yescharacteristic
222C22×C8—21no—
223C22×C8—42no—
224C22×C8C2×Dic6711yescharacteristic
225C22×C8C2×Dic6711yescharacteristic
226C22×C8C22⋊D6711yescharacteristic
227C22×C8C22⋊D6711yescharacteristic
228C22×C8C22⋊D6711yescharacteristic
229C22×C8C22⋊D6711yescharacteristic
230#123—21no—
231C22⋊Q8Dic13411yescharacteristic
232C22⋊Q8D26811yescharacteristic
233#121Dic13411yescharacteristic
234#121D26811yescharacteristic
235#120—21no—
236C4×D4—21no—
237C4×D4—21no—
238#117C2×Dic6711yescharacteristic
239C2×C4⋊C4—1342no—
240C2×C22⋊C4—1342no—
241#108—2682no—
242#107—2682no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C22⋊C8—21no—
246C22⋊C8—21no—
247C22⋊C8—21no—
248C22⋊C8—21no—
249C22⋊C8—1342no—
250C22⋊C8—1342no—
251#97—21no—
252C4×C8—21no—
253#95—1342no—
254C24#1176162811yescharacteristic
255#41—41no—
256#41—41no—
257C2×Q8C4⋊Dic6711yescharacteristic
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—21no—
261C2×D4C4⋊Dic6711yescharacteristic
262C22×C4—21no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—2682no—
266C22×C4C22×Dic6711yescharacteristic
267C22×C4C2×C22⋊D6711yescharacteristic
268C22×C4C2×C22⋊D6711yescharacteristic
269C22×C4C2×D26811yescharacteristic
270C22×C4C2×Dic13411yescharacteristic
271C22×C4#1176162811yescharacteristic
272M16—42no—
273M16—41no—
274M16—41no—
275M16—2682no—
276M16—2682no—
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—22no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—42no—
286C2×C8—2682no—
287C2×C8C22⋊Dic6711yescharacteristic
288C2×C8C22⋊Dic6711yescharacteristic
289C2×C8C22⋊Dic6711yescharacteristic
290C2×C8C22⋊Dic6711yescharacteristic
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4—2682no—
294C4⋊C4—2682no—
295C4⋊C4C4⋊Dic6711yescharacteristic
296C4⋊C4C4⋊Dic6711yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—21no—
301C22⋊C4—21no—
302C22⋊C4—5362no—
303C42—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—41no—
308C23—41no—
309C23#1452400211yescharacteristic
310C23#1452401711yescharacteristic
311C23#1452402211yescharacteristic
312Q8—41no—
313D4—41no—
314D4—42no—
315D4—42no—
316D4—41no—
317D4—41no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—5362no—
333C2×C4—5362no—
334C2×C4—5362no—
335C2×C4C2×C22⋊Dic6711yescharacteristic
336C2×C4C2×C4⋊Dic6711yescharacteristic
337C2×C4#1452401711yescharacteristic
338C2×C4#1452402211yescharacteristic
339C8—21no—
340C8—21no—
341C8—21no—
342C8—41no—
343C8—41no—
344C8—5362no—
345C8#1452395812yes—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—42no—
355C22—42no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22#1686325611yescharacteristic
360C22#1686366611yescharacteristic
361C22#1686367111yescharacteristic
362C4—21no—
363C4—21no—
364C4—21no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—10722no—
369C4C2×#1452395811yescharacteristic
370C4#1686337011yescharacteristic
371C2—21no—
372C2—21no—
373C2—41no—
374C2—41no—
375C2—81no—
376C2#2256019711yescharacteristic
377C2#2256026311yescharacteristic
378C2#2256027311yescharacteristic
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#8628
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. #22560238C2semidirect
4. #22558114C2semidirect
14. C2×#131×C67C4semidirect
21. #424×C67C4semidirect
22. #408×C67C4semidirect
24. #405×C67C4semidirect
195. C67#8628semidirect
196. C2×#131Dic67semidirect
205. #424Dic67semidirect
206. #408Dic67semidirect
209. #405Dic67semidirect

Maximal subgroups

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

subgroupindex
#225602472
#225602382
#225587522
#225581142
C67×#29292
#862867