Finite Group Databasebeta

← #31968298

Subgroups — #31968298

Order 18176.

584 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22598900C211yesmaximal, characteristic
2#22597435C211yesmaximal, characteristic, complemented
3#22597418C211yesmaximal, characteristic
4#22587215C212yesmaximal, complemented
5#22587200C212yesmaximal, complemented
6#22587180C212yesmaximal, complemented
7#22587179C212yesmaximal, complemented
8#22587101C211yesmaximal, characteristic, complemented
9#22586810C211yesmaximal, characteristic
10#22586633C211yesmaximal, characteristic, complemented
11C71×#1497C211yesmaximal, characteristic, complemented
12#17986549C2211yescharacteristic
13#17986395C411yescharacteristic
14#17986373C411yescharacteristic
15#17986349C411yescharacteristic
16#17986347C411yescharacteristic
17#17986230C412yes—
18#17986230C412yes—
19#17986130—22no—
20#17986130C2212yes—
21Q8×C71⋊C8C2212yes—
22C4×#11765852C2212yes—
23C4×#11765852C2212yes—
24#17986076C2212yes—
25#17986072—22no—
26#17986064C2212yes—
27#17986060—22no—
28#17986056C2211yescharacteristic
29#17986052C2211yescharacteristic
30#17985964—22no—
31C2×C4×C71⋊C8C2211yescharacteristic
32#17985751C2212yes—
33#17985749C2212yes—
34#17985728C2212yes—
35#17985728C2212yes—
36#17985721C2211yescharacteristic
37#17985721C2211yescharacteristic
38#17985720C2211yescharacteristic
39#17985720C2211yescharacteristic
40#533×C71C2211yescharacteristic
41#432×C71C2211yescharacteristic
42#431×C71C2211yescharacteristic
43Q8⋊C8×C71C2212yes—
44Q8⋊C8×C71C2212yes—
45C2×#11765865—21no—
46C2×#11765852C2×C412yes—
47C2×#11765852C2×C412yes—
48C2×#11765848—22no—
49C22×C71⋊C8C2×C411yescharacteristic
50C22×C71⋊C8C2×C411yescharacteristic
51C2×C8×D71—22no—
52#14556128C2×C411yescharacteristic
53#14556125C2×C411yescharacteristic
54#14556116—21no—
55#14556102—21no—
56#14556102C2×C412yes—
57C4×D284—21no—
58C4×D284C2312yes—
59#14556059C2×C412yes—
60#14556059C2×C412yes—
61C4⋊(C71⋊C8)—22no—
62C4⋊(C71⋊C8)—22no—
63C4⋊(C71⋊C8)D412yes—
64C4⋊(C71⋊C8)C2312yes—
65#14556052C2×C412yes—
66#14556052C2×C412yes—
67#14556048C2×C412yes—
68#14556048C2×C412yes—
69C22⋊(C71⋊C8)—21no—
70C22⋊(C71⋊C8)D412yes—
71(C2×D71)⋊C8—22no—
72(C2×D71)⋊C8—22no—
73#14556039—22no—
74#14556039—21no—
75C4×C71⋊C8—22no—
76C4×C71⋊C8C2×C411yescharacteristic
77C4×C71⋊C8C2×C411yescharacteristic
78C4×C71⋊C8C2311yescharacteristic
79C4×C71⋊C8C2311yescharacteristic
80C4×C71⋊C8C2311yescharacteristic
81C4×C71⋊C8C2311yescharacteristic
82C4⋊Q8×C71C2×C411yescharacteristic
83C4⋊Q8×C71C2×C411yescharacteristic
84#125×C71—21no—
85C22⋊Q8×C71—21no—
86C22⋊Q8×C71C2×C412yes—
87C4×Q8×C71—21no—
88C4×Q8×C71C2312yes—
89#117×C71—21no—
90C2×C42×C71C2311yescharacteristic
91C4⋊C8×C71—21no—
92C4⋊C8×C71C2312yes—
93C22⋊C8×C71—21no—
94C22⋊C8×C71—21no—
95#97×C71—21no—
96C4×C8×C71C2311yescharacteristic
97C4×C8×C71C2311yescharacteristic
98#11765865—82no—
99C2×Dic142—21no—
100C2×D284—21no—
101C2×D284—21no—
102C2×D284C22×C412yes—
103C2×C22⋊D71—22no—
104C2×C4×D71—22no—
105#11765852—42no—
106#11765852—42no—
107#11765848—82no—
108C2×C71⋊C8—22no—
109C2×C71⋊C8—41no—
110C2×C71⋊C8—41no—
111C2×C71⋊C8C22×C411yescharacteristic
112C2×C71⋊C8C22×C411yescharacteristic
113C2×C71⋊C8C22×C411yescharacteristic
114C2×C71⋊C8C22×C411yescharacteristic
115C2×C71⋊C8C2×D412yes—
116C2×C71⋊C8C2×D412yes—
117C8×D71—82no—
118C4⋊Dic71—21no—
119C4⋊Dic71C22×C412yes—
120(C2×D71)⋊C4—22no—
121(C2×D71)⋊C4—22no—
122C2×Q8×C71—21no—
123C2×Q8×C71C22×C412yes—
124C22×C4×C71C22×C411yescharacteristic
125C22×C4×C71C2×D411yescharacteristic
126C22×C4×C71#4111yescharacteristic
127C2×C8×C71—22no—
128C2×C8×C71—21no—
129C2×C8×C71—21no—
130C4⋊C4×C71—21no—
131C4⋊C4×C71—21no—
132C4⋊C4×C71—21no—
133C4⋊C4×C71—22no—
134C4⋊C4×C71—22no—
135C4⋊C4×C71C22×C412yes—
136C22⋊C4×C71—22no—
137C42×C71—22no—
138C42×C71C22×C411yescharacteristic
139C42×C71C2×D411yescharacteristic
140C42×C71#4111yescharacteristic
141C42×C71C2411yescharacteristic
142C23×C71C4×D411yescharacteristic
143C22×D71—22no—
144Q8×C71—22no—
145Q8×C71—22no—
146Q8×C71—81no—
147C2×C4×C71—21no—
148C2×C4×C71—21no—
149C2×C4×C71—22no—
150C2×C4×C71—22no—
151C2×C4×C71—22no—
152C2×C4×C71—41no—
153C2×C4×C71C4×D411yescharacteristic
154C2×C4×C71C4×D411yescharacteristic
155C2×C4×C71C4×D411yescharacteristic
156C2×C4×C71C23×C411yescharacteristic
157C2×C4×C71C22×D411yescharacteristic
158C2×C4×C71C2×#4111yescharacteristic
159C22⋊D71—82no—
160C2×Dic71—22no—
161D284—22no—
162D284—22no—
163D284—81no—
164D284—81no—
165C4×D71—42no—
166C4×D71—42no—
167Dic142—81no—
168C8×C71—41no—
169C8×C71—41no—
170C8×C71—42no—
171C71⋊C8—21no—
172C71⋊C8—21no—
173C71⋊C8—21no—
174C71⋊C8—21no—
175C71⋊C8—22no—
176C71⋊C8—22no—
177C71⋊C8—42no—
178C22×C71—21no—
179C22×C71—21no—
180C22×C71—21no—
181C22×C71C2×C4×D411yescharacteristic
182C2×D71—42no—
183C2×D71—42no—
184C2×D71—42no—
185C4×C71—21no—
186C4×C71—21no—
187C4×C71—21no—
188C4×C71—21no—
189C4×C71—42no—
190C4×C71—42no—
191C4×C71#56811yescharacteristic
192C4×C71#56811yescharacteristic
193C4×C71#57712yes—
194Dic71—42no—
195#17990—711nomaximal, Sylow
196C2×C71—41no—
197C2×C71#293211yescharacteristic
198C2×C71#294911yescharacteristic
199C2×C71#295711yescharacteristic
200D71—82no—
201#3167—711no—
202#2985—711no—
203#2981—711no—
204#1588—712no—
205#1585—712no—
206#1583—712no—
207#1582—712no—
208#1568—711no—
209#1497D7111yescharacteristic, complemented
210#1496—711no—
211#1452—711no—
212C71#1799011yescharacteristic, complemented, Sylow
213#533C2×D7111yescharacteristic
214#532—711no—
215#498—711no—
216#492—711no—
217#487—711no—
218#486—711no—
219#465—712no—
220#465—712no—
221#446—712no—
222#446—1422no—
223C8×Q8—712no—
224C4×SD16—712no—
225C4×SD16—712no—
226#436—712no—
227#435—1422no—
228C8×D4—712no—
229#433—1422no—
230#432—711no—
231#432C2×D7111yescharacteristic
232#431—711no—
233#431C2×D7111yescharacteristic
234#408—1422no—
235C2×C4×C8—711no—
236#335—712no—
237#334—712no—
238Q8⋊C8C2×D7112yes—
239Q8⋊C8C2×D7112yes—
240D4⋊C8—712no—
241D4⋊C8—712no—
242#322—711no—
243#322—711no—
244#322—711no—
245#322—711no—
246C2×#41—1421no—
247C2×SD16—712no—
248C2×SD16—712no—
249C2×M16—1422no—
250C22×C8—711no—
251C22×C8—711no—
252C22×C8—1422no—
253C4⋊Q8—711no—
254C4⋊Q8C4×D7111yescharacteristic
255C4⋊Q8C4×D7111yescharacteristic
256#127—711no—
257#125—21no—
258#124—1421no—
259C22⋊Q8—21no—
260C22⋊Q8C4×D7112yes—
261#121—712no—
262#121—1421no—
263C4×Q8—21no—
264C4×Q8C22×D7112yes—
265C4×D4—712no—
266C4×D4—1421no—
267#117—21no—
268C2×C42C22×D7111yescharacteristic
269#106—712no—
270#106—712no—
271C4⋊C8—21no—
272C4⋊C8—712no—
273C4⋊C8—712no—
274C4⋊C8—1422no—
275C4⋊C8—1422no—
276C4⋊C8C22×D7112yes—
277#103—712no—
278#103—712no—
279#102—712no—
280#102—712no—
281C22⋊C8—21no—
282C22⋊C8—21no—
283C22⋊C8—712no—
284C22⋊C8—1422no—
285C22⋊C8—1422no—
286C22⋊C8—1421no—
287#97—21no—
288#97—1422no—
289#97—1421no—
290C4×C8—711no—
291C4×C8—711no—
292C4×C8—711no—
293C4×C8—711no—
294C4×C8—711no—
295C4×C8—711no—
296C4×C8—1422no—
297C4×C8C22×D7111yescharacteristic
298C4×C8C22×D7111yescharacteristic
299#41—5682no—
300C2×Q8—21no—
301C2×Q8—1421no—
302C2×Q8C2×C4×D7112yes—
303C2×D4—712no—
304C2×D4—1422no—
305C2×D4—1421no—
306C2×D4—1421no—
307C22×C4—1422no—
308C22×C4C2×C4×D7111yescharacteristic
309C22×C4D4×D7111yescharacteristic
310C22×C4#1176586411yescharacteristic
311SD16—2842no—
312SD16—2842no—
313M16—5682no—
314C2×C8—22no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—2841no—
318C2×C8—2841no—
319C2×C8—5682no—
320C2×C8—711no—
321C2×C8—711no—
322C2×C8—711no—
323C2×C8—711no—
324C2×C8—712no—
325C2×C8—712no—
326C2×C8—1422no—
327C4⋊C4—22no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—22no—
332C4⋊C4—712no—
333C4⋊C4—1421no—
334C4⋊C4C2×C4×D7112yes—
335C22⋊C4—22no—
336C22⋊C4—1422no—
337C22⋊C4—1422no—
338C42—22no—
339C42C2×C4×D7111yescharacteristic
340C42D4×D7111yescharacteristic
341C42#1176586411yescharacteristic
342C42C23×D7111yescharacteristic
343C23—1422no—
344C23(C22⋊D71)⋊C411yescharacteristic
345Q8—22no—
346Q8—22no—
347Q8—81no—
348Q8—5681no—
349D4—5681no—
350D4—5681no—
351D4—5682no—
352D4—1422no—
353D4—1422no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—41no—
360C2×C4—2842no—
361C2×C4—2842no—
362C2×C4—1422no—
363C2×C4(C22⋊D71)⋊C411yescharacteristic
364C2×C4(C22⋊D71)⋊C411yescharacteristic
365C2×C4(C22⋊D71)⋊C411yescharacteristic
366C2×C4C22×C4×D7111yescharacteristic
367C2×C4C2×D4×D7111yescharacteristic
368C2×C4C2×#1176586411yescharacteristic
369C8—41no—
370C8—41no—
371C8—42no—
372C8—2842no—
373C8—1421no—
374C8—1421no—
375C8—1421no—
376C8—1421no—
377C8—1422no—
378C8—1422no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—2842no—
383C22—2842no—
384C22—2842no—
385C22C2×(C22⋊D71)⋊C411yescharacteristic
386C4—21no—
387C4—21no—
388C4—21no—
389C4—21no—
390C4—42no—
391C4—42no—
392C4—2842no—
393C4#1798676411yescharacteristic
394C4#1798676411yescharacteristic
395C4#1798680812yes—
396C2—41no—
397C2—5682no—
398C2#2259705511yescharacteristic
399C2#2259719511yescharacteristic
400C2#2259725311yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D71
commutatorC2×C4×C71C23×C4
FrattiniC42C23×D71
FittingC71×#1497C2
socleC22×C71C2×C4×D4
prime radical#17986549C22
cosocleC42×C71C24
hypercenter#1497D71
hypocenterC71#17990
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1497

Complements

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

NH ≅ G/Ntype
2. #22597435C2semidirect
4. #22587215C2semidirect
5. #22587200C2semidirect
6. #22587180C2semidirect
7. #22587179C2semidirect
8. #22587101C2semidirect
10. #22586633C2semidirect
11. C71×#1497C2semidirect
209. #1497D71semidirect
212. C71#17990semidirect

Maximal subgroups

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

subgroupindex
#225989002
#225974352
#225974182
#225872152
#225872002
#225871802
#225871792
#225871012
#225868102
#225866332
C71×#14972
#1799071