Finite Group Databasebeta

← #32001736

Subgroups — #32001736

Order 18176.

766 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#11765855C211yesmaximal, characteristic
2#22590992C211yesmaximal, characteristic
3#22589210C212yesmaximal
4#22589075C212yesmaximal
5#22589019C212yesmaximal
6#22589015C212yesmaximal
7C4×#14556059C211yesmaximal, characteristic
8#22588173C211yesmaximal, characteristic
9#22588170C211yesmaximal, characteristic
10#22588027C211yesmaximal, characteristic
11C71×#2320C211yesmaximal, characteristic
12C22×#11765855C2211yescharacteristic
13C2×C4×Dic142C2211yescharacteristic
14C4×#11765855C418yescomplemented
15C2×#14556061C2211yescharacteristic
16C2×#14556059C2212yes—
17C2×#14556052C2212yes—
18C2×#14556052C2211yescharacteristic
19C2×#14556051C2212yes—
20C2×#14556051C2211yescharacteristic
21C2×#14556039C2211yescharacteristic
22C2×#14556039C2212yes—
23C2×C4×C71⋊C8C2212yes—
24C2×C4×C71⋊C8C2211yescharacteristic
25#17985905C2212yes—
26#17985897C2212yes—
27#17985881—22no—
28#17985866—21no—
29#17985863—21no—
30C4×C4⋊Dic71C2211yescharacteristic
31#17985761C2212yes—
32#17985761C2212yes—
33#17985761C2212yes—
34C2×C4×Q8×C71C2211yescharacteristic
35#391×C71C2212yes—
36#389×C71C2212yes—
37#387×C71—22no—
38#383×C71—21no—
39#382×C71—21no—
40C4×C4⋊C4×C71C2211yescharacteristic
41#376×C71C2211yescharacteristic
42C22×Dic142C2311yescharacteristic
43C2×#11765855—24no—
44C2×#11765855C2×C414yes—
45C22×C71⋊C8C2311yescharacteristic
46C22×C71⋊C8C2312yes—
47C22×C71⋊C8C2311yescharacteristic
48C4×Dic142—22no—
49C4×Dic142C2×C414yes—
50C2×C4⋊Dic71C2312yes—
51C2×C4⋊Dic71C2311yescharacteristic
52C2×Dic71⋊C4—21no—
53C2×Dic71⋊C4—22no—
54C2×C4×Dic71—21no—
55C2×C4×Dic71—21no—
56#14556061C2×C414yes—
57#14556059—24no—
58#14556052—24no—
59#14556052C2×C414yes—
60#14556051—24no—
61#14556051C2×C414yes—
62#14556039—24no—
63#14556039—22no—
64C4×C71⋊C8—24no—
65C4×C71⋊C8C2×C414yes—
66C71⋊#95—21no—
67C71⋊#95—22no—
68C71⋊#95—22no—
69C71⋊#95—22no—
70C22×Q8×C71C2311yescharacteristic
71C4×Q8×C71—22no—
72C4×Q8×C71C2×C414yes—
73C2×C4⋊C4×C71—21no—
74C2×C4⋊C4×C71—22no—
75C2×C4⋊C4×C71C2312yes—
76C2×C4⋊C4×C71C2311yescharacteristic
77C2×C42×C71—21no—
78C2×C42×C71—21no—
79C2×C42×C71C2312yes—
80C2×C42×C71C2311yescharacteristic
81#95×C71—21no—
82#95×C71—22no—
83#95×C71—22no—
84#95×C71—22no—
85#95×C71D411yescharacteristic
86#95×C71D411yescharacteristic
87#95×C71D412yes—
88C2×Dic142—22no—
89C2×Dic142—22no—
90C2×Dic142—21no—
91C2×Dic142C22×C412yes—
92C22×Dic71—21no—
93C22×Dic71—21no—
94C22×Dic71—22no—
95#11765855—28no—
96C2×C71⋊C8—22no—
97C2×C71⋊C8—22no—
98C2×C71⋊C8—22no—
99C2×C71⋊C8C22×C412yes—
100C2×C71⋊C8C22×C412yes—
101C2×C71⋊C8#4114yes—
102C2×C71⋊C8#4114yes—
103C4⋊Dic71—21no—
104C4⋊Dic71—22no—
105C4⋊Dic71—22no—
106C4⋊Dic71C22×C412yes—
107Dic71⋊C4—24no—
108Dic71⋊C4—82no—
109C4×Dic71—24no—
110C4×Dic71—81no—
111C2×Q8×C71—21no—
112C2×Q8×C71—22no—
113C2×Q8×C71—22no—
114C2×Q8×C71C22×C412yes—
115C22×C4×C71—21no—
116C22×C4×C71—21no—
117C22×C4×C71—22no—
118C22×C4×C71C2×D412yes—
119C22×C4×C71C2×D412yes—
120C22×C4×C71C2×D411yescharacteristic
121C22×C4×C71C2×D411yescharacteristic
122C22×C4×C71C2411yescharacteristic
123C4⋊C4×C71—24no—
124C4⋊C4×C71—21no—
125C4⋊C4×C71—22no—
126C4⋊C4×C71—22no—
127C4⋊C4×C71—82no—
128C4⋊C4×C71C22×C412yes—
129C42×C71—24no—
130C42×C71—22no—
131C42×C71—22no—
132C42×C71—81no—
133C42×C71C22×C412yes—
134C42×C71#4112yes—
135C23×C71C22×D411yescharacteristic
136Q8×C71—22no—
137Q8×C71—22no—
138Q8×C71—42no—
139C2×C4×C71—21no—
140C2×C4×C71—21no—
141C2×C4×C71—22no—
142C2×C4×C71—22no—
143C2×C4×C71—22no—
144C2×C4×C71—22no—
145C2×C4×C71—22no—
146C2×C4×C71—22no—
147C2×C4×C71—22no—
148C2×C4×C71—22no—
149C2×C4×C71—22no—
150C2×C4×C71—42no—
151C2×C4×C71—41no—
152C2×C4×C71—41no—
153C2×C4×C71—42no—
154C2×C4×C71—42no—
155C2×C4×C71—42no—
156C2×C4×C71C4×D412yes—
157C2×C4×C71C4×D412yes—
158C2×C4×C71C23×C411yescharacteristic
159C2×C4×C71C2×#4112yes—
160C2×C4×C71C2×#4112yes—
161C2×C4×C71C2×#4111yescharacteristic
162C2×Dic71—22no—
163C2×Dic71—22no—
164C2×Dic71—42no—
165C2×Dic71—41no—
166C2×Dic71—41no—
167C2×Dic71—42no—
168C2×Dic71—42no—
169C2×Dic71—42no—
170Dic142—22no—
171Dic142—22no—
172Dic142—42no—
173C71⋊C8—22no—
174C71⋊C8—24no—
175C71⋊C8#11714yes—
176C22×C71C2×C4×D411yescharacteristic
177C22×C71#53212yes—
178C22×C71#57711yescharacteristic
179C22×C71#57711yescharacteristic
180C22×C71#57812yes—
181C4×C71—22no—
182C4×C71—22no—
183C4×C71—42no—
184C4×C71—42no—
185C4×C71—42no—
186C4×C71—42no—
187C4×C71—82no—
188C4×C71C2×#11711yescharacteristic
189C4×C71C4×#4111yescharacteristic
190C4×C71#52912yes—
191Dic71—42no—
192Dic71—42no—
193Dic71—82no—
194#21283—711nomaximal, Sylow
195C2×C71#229311yescharacteristic
196C2×C71#295011yescharacteristic
197C2×C71#295211yescharacteristic
198C2×C71#317812yes—
199C2×C71#317812yes—
200C2×C4×Q16—711no—
201#2320—711no—
202#2320D7111yescharacteristic
203#1971—712no—
204#1943—712no—
205#1933—712no—
206#1933—712no—
207C4×#106—711no—
208#1763—711no—
209#1763—711no—
210#1726—711no—
211C71#2128311yescharacteristic, complemented, Sylow
212C22×Q16—711no—
213C2×C4×Q8—711no—
214C2×C4×Q8C2×D7111yescharacteristic
215C4×Q16—718no—
216C2×#107—711no—
217C2×#106—712no—
218C2×#103—711no—
219C2×#103—711no—
220C2×#103—712no—
221C2×#103—712no—
222C2×#97—712no—
223C2×#97—711no—
224C2×C4×C8—711no—
225C2×C4×C8—712no—
226#391—712no—
227#391C2×D7112yes—
228#389—712no—
229#389C2×D7112yes—
230#387—22no—
231#387—1422no—
232#383—21no—
233#383—1421no—
234#382—21no—
235#382—1421no—
236C4×C4⋊C4—711no—
237C4×C4⋊C4C2×D7111yescharacteristic
238#376C2×D7111yescharacteristic
239#340—712no—
240#340—712no—
241#340—712no—
242C22×Q8—711no—
243C22×Q8C22×D7111yescharacteristic
244C2×Q16—714no—
245C2×Q16—1424no—
246C22×C8—711no—
247C22×C8—711no—
248C22×C8—712no—
249C4×Q8—22no—
250C4×Q8—714no—
251C4×Q8—1422no—
252C4×Q8C4×D7114yes—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—711no—
256C2×C4⋊C4—712no—
257C2×C4⋊C4—1422no—
258C2×C4⋊C4—1421no—
259C2×C4⋊C4C22×D7112yes—
260C2×C4⋊C4C22×D7111yescharacteristic
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—1421no—
264C2×C42—1421no—
265C2×C42C22×D7112yes—
266C2×C42C22×D7111yescharacteristic
267#107—714no—
268#106—1424no—
269#103—714no—
270#103—714no—
271#103—1424no—
272#103—1424no—
273#97—1424no—
274#97—1422no—
275C4×C8—714no—
276C4×C8—1424no—
277#95—21no—
278#95—22no—
279#95—22no—
280#95—22no—
281#95—1421no—
282#95—1422no—
283#95—1422no—
284#95—1422no—
285#95C22⋊D7111yescharacteristic
286#95C22⋊D7111yescharacteristic
287#95C22⋊D7112yes—
288C2×Q8—21no—
289C2×Q8—22no—
290C2×Q8—22no—
291C2×Q8—712no—
292C2×Q8—1422no—
293C2×Q8—1422no—
294C2×Q8—1421no—
295C2×Q8C2×C4×D7112yes—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—1422no—
300C22×C4—1421no—
301C22×C4—1421no—
302C22×C4C2×C22⋊D7112yes—
303C22×C4C2×C22⋊D7112yes—
304C22×C4C2×C22⋊D7111yescharacteristic
305C22×C4C2×C22⋊D7111yescharacteristic
306C22×C4C23×D7111yescharacteristic
307Q16—1428no—
308C2×C8—712no—
309C2×C8—714no—
310C2×C8—712no—
311C2×C8—714no—
312C2×C8—1422no—
313C2×C8—1422no—
314C2×C8—1422no—
315C4⋊C4—21no—
316C4⋊C4—24no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—82no—
320C4⋊C4—5682no—
321C4⋊C4—712no—
322C4⋊C4—1422no—
323C4⋊C4—1422no—
324C4⋊C4—1421no—
325C4⋊C4—1424no—
326C4⋊C4C2×C4×D7112yes—
327C42—22no—
328C42—22no—
329C42—24no—
330C42—81no—
331C42—5681no—
332C42—1424no—
333C42C2×C4×D7112yes—
334C42#1176586512yes—
335C23C22×C22⋊D7111yescharacteristic
336Q8—22no—
337Q8—22no—
338Q8—42no—
339Q8—2842no—
340Q8—1422no—
341Q8—1422no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—42no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—2842no—
360C2×C4—2841no—
361C2×C4—2841no—
362C2×C4—2842no—
363C2×C4—2842no—
364C2×C4—2842no—
365C2×C4—1422no—
366C2×C4—1422no—
367C2×C4C4×C22⋊D7112yes—
368C2×C4C4×C22⋊D7112yes—
369C2×C4C22×C4×D7111yescharacteristic
370C2×C4#41×D7112yes—
371C2×C4#41×D7112yes—
372C2×C4C2×#1176586511yescharacteristic
373C8—714no—
374C8—1422no—
375C8—1424no—
376C22C2×C4×C22⋊D7111yescharacteristic
377C22#1798654512yes—
378C22#1798680711yescharacteristic
379C22#1798681011yescharacteristic
380C22#1798681412yes—
381C4—22no—
382C4—22no—
383C4—42no—
384C4—42no—
385C4—42no—
386C4—42no—
387C4—82no—
388C4—2842no—
389C4—2842no—
390C4—5682no—
391C4#117×D7111yescharacteristic
392C4#1798646711yescharacteristic
393C4#1798652712yes—
394C2#2259079911yescharacteristic
395C2#2259720611yescharacteristic
396C2#2259722111yescharacteristic
397C2#2259900212yes—
398C2#2259901012yes—
subgroupG/H
centerC23C22×C22⋊D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC71×#2320C2
socle, prime radicalC23×C71C22×D4
cosocleC22×C4×C71C24
hypercenter#2320D71
hypocenterC71#21283
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ #376 ◁ #2320

Complements

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

NH ≅ G/Ntype
14. C4×#11765855C4semidirect
211. C71#21283semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#117658552
#225909922
#225892102
#225890752
#225890192
#225890152
C4×#145560592
#225881732
#225881702
#225880272
C71×#23202
#2128371