Finite Group Databasebeta

← #30874659

Subgroups — #30874659

Order 17152.

750 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
1C2×C8×Dic134C211yesmaximal, characteristic
2#22563161C212yesmaximal
3#22560975C212yesmaximal
4#22560974C212yesmaximal
5#22560964C212yesmaximal
6#22560370C211yesmaximal, characteristic
7#22560364C211yesmaximal, characteristic
8#106×Dic67C211yesmaximal, characteristic
9#106×Dic67C211yesmaximal, characteristic
10C4×#14523975C211yesmaximal, characteristic
11C67×#1846C211yesmaximal, characteristic
12C2×C4×Dic134C2211yescharacteristic
13C8×Dic134C418yescomplemented
14C2×#14523976C2212yes—
15C2×#14523975C2211yescharacteristic
16C2×#14523975C2211yescharacteristic
17C2×#14523974C2212yes—
18C2×C4⋊(C67⋊C8)C2211yescharacteristic
19C2×Dic67⋊C8C2211yescharacteristic
20C2×Dic67⋊C8C2211yescharacteristic
21C2×C8×Dic67C2211yescharacteristic
22C2×C8×Dic67C2211yescharacteristic
23#16863300—22no—
24#16863281—22no—
25#16863278—22no—
26#16863265C2212yes—
27#16863261C2212yes—
28C4⋊C4×Dic67C2212yes—
29C4⋊C4×Dic67C2212yes—
30C4×C4⋊Dic67C2212yes—
31#16863247—22no—
32#16863159C2212yes—
33#16863158C2212yes—
34#16863158C2212yes—
35C2×#107×C67C2212yes—
36C2×#106×C67C2211yescharacteristic
37C2×#106×C67C2211yescharacteristic
38C2×C4×C8×C67C2211yescharacteristic
39#382×C67C2212yes—
40C22×Dic134D411yescharacteristic
41C22×C67⋊C8C2311yescharacteristic
42C22×C67⋊C8C2311yescharacteristic
43C4×Dic134C2×C414yes—
44C4×Dic134Q814yes—
45C2×C4⋊Dic67D411yescharacteristic
46C2×C4⋊Dic67C2312yes—
47C2×C4⋊Dic67C2312yes—
48C2×Dic67⋊C4—22no—
49C2×Dic67⋊C4—22no—
50C2×Dic67⋊C4D411yescharacteristic
51C2×Dic67⋊C4D411yescharacteristic
52C2×C4×Dic67—22no—
53C2×C4×Dic67—22no—
54C2×C4×Dic67—22no—
55C2×C4×Dic67C2311yescharacteristic
56C2×C4×Dic67C2311yescharacteristic
57#14523976—42no—
58#14523975—22no—
59#14523975Q814yescomplemented
60#14523974—42no—
61C4⋊(C67⋊C8)C2×C414yes—
62Dic67⋊C8C2×C414yes—
63Dic67⋊C8C2×C414yes—
64C8×Dic67C2×C414yes—
65C8×Dic67C2×C414yes—
66C67⋊#95—22no—
67C67⋊#95—22no—
68C67⋊#95—22no—
69C67⋊#95—22no—
70C67⋊#95—22no—
71C22×C8×C67C2311yescharacteristic
72C22×C8×C67C2311yescharacteristic
73C2×C4⋊C4×C67C2312yes—
74C2×C4⋊C4×C67C2312yes—
75C2×C42×C67C2311yescharacteristic
76#107×C67—42no—
77#106×C67—22no—
78#106×C67—22no—
79C4×C8×C67C2×C414yes—
80#95×C67—22no—
81#95×C67—22no—
82C2×Dic134—24no—
83C2×Dic134C4⋊C414yes—
84C22×Dic67—22no—
85C22×Dic67—22no—
86C22×Dic67C2×D411yescharacteristic
87C22×Dic67C2×D411yescharacteristic
88C22×Dic67C2×D411yescharacteristic
89C22×Dic67C2×D411yescharacteristic
90C2×C67⋊C8—21no—
91C2×C67⋊C8—21no—
92C2×C67⋊C8—21no—
93C2×C67⋊C8—21no—
94C2×C67⋊C8C22×C412yes—
95C2×C67⋊C8C22×C412yes—
96C4⋊Dic67—22no—
97C4⋊Dic67—22no—
98C4⋊Dic67—22no—
99C4⋊Dic67C4⋊C414yes—
100C4⋊Dic67C2×Q814yes—
101Dic67⋊C4—44no—
102Dic67⋊C4—82no—
103Dic67⋊C4C4⋊C414yes—
104Dic67⋊C4C4⋊C414yes—
105C4×Dic67—44no—
106C4×Dic67—82no—
107C4×Dic67—82no—
108C4×Dic67C22×C412yes—
109C4×Dic67C22×C412yes—
110C4×Dic67C2×Q812yes—
111C4×Dic67C2×Q812yes—
112C22×C4×C67—22no—
113C22×C4×C67—22no—
114C22×C4×C67C2×D411yescharacteristic
115C22×C4×C67C2×D411yescharacteristic
116C22×C4×C67C2411yescharacteristic
117C2×C8×C67—21no—
118C2×C8×C67—21no—
119C2×C8×C67C22×C412yes—
120C2×C8×C67C22×C412yes—
121C2×C8×C67C2×Q812yes—
122C2×C8×C67#4112yes—
123C4⋊C4×C67—22no—
124C4⋊C4×C67—22no—
125C4⋊C4×C67—22no—
126C4⋊C4×C67—22no—
127C42×C67C22×C412yes—
128C42×C67C2×Q812yes—
129C23×C67C22×D411yescharacteristic
130C2×C4×C67—22no—
131C2×C4×C67—22no—
132C2×C4×C67—42no—
133C2×C4×C67—42no—
134C2×C4×C67—42no—
135C2×C4×C67—42no—
136C2×C4×C67—42no—
137C2×C4×C67—42no—
138C2×C4×C67C2×C4⋊C412yes—
139C2×C4×C67C2×C4⋊C412yes—
140C2×C4×C67C23×C411yescharacteristic
141C2×C4×C67C22×Q811yescharacteristic
142C2×C4×C67C22×Q811yescharacteristic
143C2×C4×C67C2×#4111yescharacteristic
144C2×C4×C67H(5,𝔽2)11yescharacteristic
145C2×C4×C672−1+411yescharacteristic
146C2×Dic67—22no—
147C2×Dic67—22no—
148C2×Dic67—22no—
149C2×Dic67—24no—
150C2×Dic67—42no—
151C2×Dic67—42no—
152C2×Dic67—42no—
153C2×Dic67—42no—
154C2×Dic67—42no—
155C2×Dic67C2×C4⋊C412yes—
156C2×Dic67C2×C4⋊C412yes—
157C2×Dic67C2×C4⋊C412yes—
158C2×Dic67C2×C4⋊C412yes—
159C2×Dic67#13514yes—
160Dic134—28no—
161C8×C67—22no—
162C8×C67C4×Q814yes—
163C67⋊C8—22no—
164C67⋊C8—22no—
165C22×C67C22×C4⋊C411yescharacteristic
166C22×C67D4×Q811yescharacteristic
167C22×C67#55011yescharacteristic
168C22×C67C2×#13512yes—
169C22×C67#57712yes—
170C4×C67—22no—
171C4×C67—22no—
172C4×C67—82no—
173C4×C67—82no—
174C4×C67C2×C4×Q811yescharacteristic
175C4×C67#52011yescharacteristic
176C4×C67#55711yescharacteristic
177C4×C67Q8211yescharacteristic
178Dic67—22no—
179Dic67—22no—
180Dic67—22no—
181Dic67—44no—
182Dic67—82no—
183Dic67#42714yes—
184#22725—671nomaximal, Sylow
185C2×C67Q8×C4⋊C411yescharacteristic
186C2×C67C2×#42711yescharacteristic
187C2×C67#291211yescharacteristic
188C2×C67#338112yes—
189C2×C67#339412yes—
190C2×C8×Q8—671no—
191#2355—672no—
192#1949—672no—
193#1949—672no—
194#1947—672no—
195#1846—671no—
196#1846D6711yescharacteristic
197#1844—671no—
198C4×#106—671no—
199C4×#106—671no—
200C4×#106—671no—
201C67#2272511yescharacteristic, complemented, Sylow
202C2×C4×Q8—671no—
203C8×Q8—678no—
204C2×#107—672no—
205C2×#107C2×D6712yes—
206C2×#106—672no—
207C2×#106—671no—
208C2×#106—671no—
209C2×#106C2×D6711yescharacteristic
210C2×#106C2×D6711yescharacteristic
211C2×C4⋊C8—671no—
212C2×C4⋊C8—671no—
213C2×C4⋊C8—671no—
214C2×C4×C8—671no—
215C2×C4×C8—671no—
216C2×C4×C8C2×D6711yescharacteristic
217#391—1342no—
218#387—1342no—
219#387—1342no—
220#384—672no—
221#382—672no—
222#382C2×D6712yes—
223C4×C4⋊C4—672no—
224C4×C4⋊C4—672no—
225C4×C4⋊C4—672no—
226C4×C4⋊C4—1342no—
227#340—672no—
228#340—672no—
229#340—672no—
230C22×Q8—671no—
231C22×C8—671no—
232C22×C8—671no—
233C22×C8C22×D6711yescharacteristic
234C22×C8C22×D6711yescharacteristic
235C4×Q8—674no—
236C4×Q8—674no—
237C2×C4⋊C4—671no—
238C2×C4⋊C4—671no—
239C2×C4⋊C4—671no—
240C2×C4⋊C4—672no—
241C2×C4⋊C4—672no—
242C2×C4⋊C4—1342no—
243C2×C4⋊C4—1342no—
244C2×C4⋊C4C22×D6712yes—
245C2×C4⋊C4C22×D6712yes—
246C2×C42—671no—
247C2×C42—671no—
248C2×C42—1342no—
249C2×C42—1342no—
250C2×C42—1342no—
251C2×C42C22×D6711yescharacteristic
252#107—42no—
253#107—2682no—
254#106—22no—
255#106—22no—
256#106—2682no—
257#106—674no—
258#106—1342no—
259C4⋊C8—674no—
260C4⋊C8—674no—
261C4⋊C8—674no—
262C4×C8—674no—
263C4×C8—674no—
264C4×C8C4×D6714yes—
265#95—22no—
266#95—22no—
267#95—1342no—
268#95—1342no—
269#95—1342no—
270#95—1342no—
271#95—1342no—
272C2×Q8—674no—
273C2×Q8—1344no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—671no—
277C22×C4—671no—
278C22×C4—671no—
279C22×C4—671no—
280C22×C4—1342no—
281C22×C4—1342no—
282C22×C4D4×D6711yescharacteristic
283C22×C4D4×D6711yescharacteristic
284C22×C4C23×D6711yescharacteristic
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—672no—
288C2×C8—672no—
289C2×C8—1341no—
290C2×C8—1341no—
291C2×C8—1341no—
292C2×C8—1341no—
293C2×C8C2×C4×D6712yes—
294C2×C8C2×C4×D6712yes—
295C2×C8Q8×D6712yes—
296C2×C8#1176162712yes—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—2684no—
302C4⋊C4—5362no—
303C4⋊C4—674no—
304C4⋊C4—674no—
305C4⋊C4—674no—
306C4⋊C4—674no—
307C4⋊C4—1342no—
308C4⋊C4—1342no—
309C4⋊C4—1342no—
310C42—2684no—
311C42—5362no—
312C42—5362no—
313C42—672no—
314C42—672no—
315C42—672no—
316C42—672no—
317C42C2×C4×D6712yes—
318C42Q8×D6712yes—
319C23C2×D4×D6711yescharacteristic
320Q8—1348no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—2682no—
330C2×C4—2682no—
331C2×C4—2682no—
332C2×C4—2682no—
333C2×C4—2682no—
334C2×C4—672no—
335C2×C4—674no—
336C2×C4—672no—
337C2×C4—672no—
338C2×C4—672no—
339C2×C4—1344no—
340C2×C4—1342no—
341C2×C4—1342no—
342C2×C4—1342no—
343C2×C4C4⋊C4×D6712yes—
344C2×C4C4⋊C4×D6712yes—
345C2×C4C22×C4×D6711yescharacteristic
346C2×C4C2×Q8×D6711yescharacteristic
347C2×C4C2×Q8×D6711yescharacteristic
348C2×C4C2×#1176162711yescharacteristic
349C2×C4#1452408811yescharacteristic
350C2×C4#1452409111yescharacteristic
351C8—22no—
352C8—1342no—
353C8—1342no—
354C8#1452400814yes—
355C22C2×C4⋊C4×D6711yescharacteristic
356C22#1686406111yescharacteristic
357C22#1686406611yescharacteristic
358C22#135×D6712yes—
359C22#1686420312yes—
360C4—22no—
361C4—22no—
362C4—82no—
363C4—82no—
364C4—2684no—
365C4—5362no—
366C4—674no—
367C4—1342no—
368C4—1342no—
369C4—1342no—
370C4C2×#1452400811yescharacteristic
371C4#1686387711yescharacteristic
372C4#1686412111yescharacteristic
373C4#1686412611yescharacteristic
374C2#2256312111yescharacteristic
375C2#427×D6711yescharacteristic
376C2#2256880111yescharacteristic
377C2#2257308112yes—
378C2#2257320212yes—
subgroupG/H
centerC23C2×D4×D67
commutatorC2×C4×C67C23×C4
FrattiniC22×C4C23×D67
FittingC67×#1846C2
socle, prime radicalC23×C67C22×D4
cosocleC22×C4×C67C24
hypercenter#1846D67
hypocenterC67#22725
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1846

Complements

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

NH ≅ G/Ntype
13. C8×Dic134C4semidirect
59. #14523975Q8semidirect
201. C67#22725semidirect

Maximal subgroups

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

subgroupindex
C2×C8×Dic1342
#225631612
#225609752
#225609742
#225609642
#225603702
#225603642
#106×Dic672
C4×#145239752
C67×#18462
#2272567