Finite Group Databasebeta

← #31968123

Subgroups — #31968123

Order 18176.

600 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#22598899C211yesmaximal, characteristic
2#22597434C211yesmaximal, characteristic, complemented
3#22597415C211yesmaximal, characteristic
4#22587244C212yesmaximal, complemented
5#22587232C212yesmaximal, complemented
6#22587154C212yesmaximal, complemented
7C8×#11765854C212yesmaximal, complemented
8#22587103C211yesmaximal, characteristic, complemented
9#22586814C211yesmaximal, characteristic, complemented
10#22586810C211yesmaximal, characteristic
11C71×#1450C211yesmaximal, characteristic, complemented
12#17986556C2211yescharacteristic
13#17986549C2211yescharacteristic
14#17986397C411yescharacteristic
15#17986372C411yescharacteristic
16#17986350C411yescharacteristic
17#17986346C411yescharacteristic
18#17986228C412yes—
19#17986228C412yes—
20#17986129C2212yes—
21#17986128—22no—
22C8×Dic142C2212yes—
23C4×#11765854C2212yes—
24C4×#11765854C2212yes—
25#17986078C2212yes—
26#17986069—22no—
27C8×D284C2212yes—
28#17986057—22no—
29#17986056C2211yescharacteristic
30#17986056C2211yescharacteristic
31#17985964—22no—
32#17985752C2212yes—
33#17985750C2212yes—
34#17985731C2212yes—
35#17985731C2212yes—
36#17985728C2212yes—
37#17985728C2212yes—
38#431×C71C2211yescharacteristic
39#431×C71C2211yescharacteristic
40C2×C4×C8×C71C2211yescharacteristic
41#322×C71C2211yescharacteristic
42#322×C71C2211yescharacteristic
43C82×C71C2211yescharacteristic
44C82×C71C2211yescharacteristic
45C2×#11765865—21no—
46C2×#11765854C2×C412yes—
47C2×#11765854C2×C412yes—
48C2×#11765848—22no—
49C2×C8×D71—22no—
50#14556128C2×C411yescharacteristic
51#14556128C2×C411yescharacteristic
52#14556128C2×C411yescharacteristic
53#14556125C2×C411yescharacteristic
54#14556120—21no—
55#14556116—21no—
56#14556107—21no—
57#14556107C2×C412yes—
58#14556102—21no—
59#14556102C2×C412yes—
60C4×Dic142—21no—
61C4×Dic142C2312yes—
62C4×D284—21no—
63C4×D284C2312yes—
64#14556087—21no—
65#14556060C2×C412yes—
66#14556060C2×C412yes—
67C4⋊(C71⋊C8)C2312yes—
68C4⋊(C71⋊C8)C2312yes—
69Dic71⋊C8—22no—
70Dic71⋊C8—22no—
71#14556053C2×C412yes—
72#14556053C2×C412yes—
73#14556050C2×C412yes—
74#14556050C2×C412yes—
75C22⋊(C71⋊C8)—21no—
76C22⋊(C71⋊C8)—21no—
77(C2×D71)⋊C8—22no—
78(C2×D71)⋊C8—22no—
79#14556039—21no—
80#14556039—21no—
81#14556038—22no—
82C8×Dic71—22no—
83C22×C8×C71C2×C411yescharacteristic
84C22×C8×C71C2×C411yescharacteristic
85C2×C42×C71C2311yescharacteristic
86C4⋊C8×C71—21no—
87C4⋊C8×C71D412yes—
88C22⋊C8×C71—21no—
89C22⋊C8×C71D412yes—
90C4×C8×C71C2×C411yescharacteristic
91C4×C8×C71C2×C411yescharacteristic
92C4×C8×C71C2311yescharacteristic
93C4×C8×C71C2311yescharacteristic
94C4×C8×C71C2311yescharacteristic
95C4×C8×C71C2311yescharacteristic
96C4×C8×C71C2311yescharacteristic
97C4×C8×C71C2311yescharacteristic
98#11765865—82no—
99C2×Dic142—21no—
100C2×Dic142—21no—
101C2×Dic142C22×C412yes—
102C2×D284—21no—
103C2×D284—21no—
104C2×D284C22×C412yes—
105C2×C22⋊D71—22no—
106C2×C4×D71—22no—
107#11765854—24no—
108#11765854—24no—
109#11765848—82no—
110C2×C71⋊C8—22no—
111C2×C71⋊C8—22no—
112C8×D71—82no—
113C4⋊Dic71—21no—
114C4⋊Dic71—21no—
115C4⋊Dic71—21no—
116C4⋊Dic71—21no—
117C4⋊Dic71C22×C412yes—
118C4⋊Dic71C22×C412yes—
119Dic71⋊C4—22no—
120Dic71⋊C4—22no—
121C22⋊Dic71—22no—
122(C2×D71)⋊C4—22no—
123(C2×D71)⋊C4—22no—
124C4×Dic71—22no—
125C22×C4×C71C22×C411yescharacteristic
126C22×C4×C71C2×D411yescharacteristic
127C22×C4×C71#4111yescharacteristic
128C2×C8×C71—22no—
129C2×C8×C71—22no—
130C2×C8×C71—21no—
131C2×C8×C71—21no—
132C2×C8×C71C22×C411yescharacteristic
133C2×C8×C71C22×C411yescharacteristic
134C2×C8×C71C22×C411yescharacteristic
135C2×C8×C71C22×C411yescharacteristic
136C2×C8×C71C2×D412yes—
137C2×C8×C71C2×D412yes—
138C42×C71C22×C411yescharacteristic
139C42×C71C2×D411yescharacteristic
140C42×C71#4111yescharacteristic
141C42×C71C2411yescharacteristic
142C23×C71C4×D411yescharacteristic
143C22×D71—22no—
144C2×C4×C71—21no—
145C2×C4×C71—21no—
146C2×C4×C71—22no—
147C2×C4×C71—41no—
148C2×C4×C71C4×D411yescharacteristic
149C2×C4×C71C4×D411yescharacteristic
150C2×C4×C71C4×D411yescharacteristic
151C2×C4×C71C23×C411yescharacteristic
152C2×C4×C71C22×D411yescharacteristic
153C2×C4×C71C2×#4111yescharacteristic
154C22⋊D71—82no—
155C2×Dic71—22no—
156C2×Dic71—22no—
157C2×Dic71—22no—
158D284—22no—
159D284—22no—
160D284—81no—
161D284—81no—
162C4×D71—42no—
163C4×D71—42no—
164Dic142—22no—
165Dic142—22no—
166Dic142—81no—
167Dic142—81no—
168C8×C71—21no—
169C8×C71—21no—
170C8×C71—22no—
171C8×C71—22no—
172C8×C71—41no—
173C8×C71—41no—
174C8×C71#13112yes—
175C8×C71#13112yes—
176C71⋊C8—42no—
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×C71C2×#13111yescharacteristic
190C4×C71C2×#13111yescharacteristic
191C4×C71#57712yes—
192Dic71—42no—
193Dic71—42no—
194Dic71—42no—
195#17971—711nomaximal, Sylow
196C2×C71—41no—
197C2×C71#293111yescharacteristic
198C2×C71#294911yescharacteristic
199C2×C71#295711yescharacteristic
200D71—82no—
201#3167—711no—
202#2985—711no—
203#2981—711no—
204#1592—712no—
205#1590—712no—
206SD16⋊C8—712no—
207C8×SD16—712no—
208#1568—711no—
209#1497—711no—
210#1496—711no—
211#1450D7111yescharacteristic, complemented
212C71#1797111yescharacteristic, complemented, Sylow
213#533—711no—
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#432—711no—
232#431C2×D7111yescharacteristic
233#431C2×D7111yescharacteristic
234#408—1422no—
235C2×C4×C8C2×D7111yescharacteristic
236#335—712no—
237#334—712no—
238Q8⋊C8—712no—
239Q8⋊C8—712no—
240D4⋊C8—712no—
241D4⋊C8—712no—
242#322C2×D7111yescharacteristic
243#322C2×D7111yescharacteristic
244C82C2×D7111yescharacteristic
245C82C2×D7111yescharacteristic
246C2×#41—1421no—
247C2×SD16—712no—
248C2×SD16—712no—
249C2×M16—1422no—
250C22×C8—1422no—
251C22×C8C4×D7111yescharacteristic
252C22×C8C4×D7111yescharacteristic
253C4⋊Q8—711no—
254C4⋊Q8—711no—
255C4⋊Q8—711no—
256#127—711no—
257#125—1421no—
258#124—1421no—
259C22⋊Q8—712no—
260C22⋊Q8—1421no—
261#121—712no—
262#121—1421no—
263C4×Q8—712no—
264C4×Q8—1421no—
265C4×D4—712no—
266C4×D4—1421no—
267#117—1421no—
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⋊C8D28412yes—
277#103—712no—
278#103—712no—
279#102—712no—
280#102—712no—
281C22⋊C8—21no—
282C22⋊C8—1421no—
283C22⋊C8—1421no—
284C22⋊C8—1422no—
285C22⋊C8—1422no—
286C22⋊C8D28412yes—
287#97—1422no—
288#97—1421no—
289#97—1421no—
290C4×C8—1422no—
291C4×C8C4×D7111yescharacteristic
292C4×C8C4×D7111yescharacteristic
293C4×C8C22×D7111yescharacteristic
294C4×C8C22×D7111yescharacteristic
295C4×C8C22×D7111yescharacteristic
296C4×C8C22×D7111yescharacteristic
297C4×C8C22×D7111yescharacteristic
298C4×C8C22×D7111yescharacteristic
299#41—5682no—
300C2×Q8—712no—
301C2×Q8—1421no—
302C2×Q8—1421no—
303C2×D4—712no—
304C2×D4—1422no—
305C2×D4—1421no—
306C2×D4—1421no—
307C22×C4—1422no—
308C22×C4C2×C4×D7111yescharacteristic
309C22×C4C2×D28411yescharacteristic
310C22×C4#1176586511yescharacteristic
311SD16—1424no—
312SD16—1424no—
313M16—5682no—
314C2×C8—22no—
315C2×C8—22no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—5682no—
319C2×C8—1422no—
320C2×C8—1422no—
321C2×C8C2×C4×D7111yescharacteristic
322C2×C8C2×C4×D7111yescharacteristic
323C2×C8C2×C4×D7111yescharacteristic
324C2×C8C2×C4×D7111yescharacteristic
325C2×C8C2×D28412yes—
326C2×C8C2×D28412yes—
327C4⋊C4—712no—
328C4⋊C4—712no—
329C4⋊C4—1422no—
330C4⋊C4—1422no—
331C4⋊C4—1421no—
332C4⋊C4—1421no—
333C4⋊C4—1421no—
334C4⋊C4—1421no—
335C22⋊C4—1422no—
336C22⋊C4—1422no—
337C22⋊C4—1422no—
338C42—1422no—
339C42C2×C4×D7111yescharacteristic
340C42C2×D28411yescharacteristic
341C42#1176586511yescharacteristic
342C42C23×D7111yescharacteristic
343C23—1422no—
344C23C4×D28411yescharacteristic
345Q8—5681no—
346Q8—5681no—
347Q8—1422no—
348Q8—1422no—
349D4—5681no—
350D4—5681no—
351D4—5682no—
352D4—1422no—
353D4—1422no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—22no—
357C2×C4—41no—
358C2×C4—2842no—
359C2×C4—2842no—
360C2×C4—1422no—
361C2×C4—1422no—
362C2×C4—1422no—
363C2×C4C4×D28411yescharacteristic
364C2×C4C4×D28411yescharacteristic
365C2×C4C4×D28411yescharacteristic
366C2×C4C22×C4×D7111yescharacteristic
367C2×C4C22×D28411yescharacteristic
368C2×C4C2×#1176586511yescharacteristic
369C8—21no—
370C8—21no—
371C8—22no—
372C8—22no—
373C8—41no—
374C8—41no—
375C8—2842no—
376C8—2842no—
377C8#1455613412yes—
378C8#1455613412yes—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—2842no—
383C22—2842no—
384C22—2842no—
385C22C2×C4×D28411yescharacteristic
386C4—21no—
387C4—21no—
388C4—21no—
389C4—21no—
390C4—2842no—
391C4—2842no—
392C4—2842no—
393C4C2×#1455613411yescharacteristic
394C4C2×#1455613411yescharacteristic
395C4#1798681112yes—
396C2—41no—
397C2—5682no—
398C2#2259705311yescharacteristic
399C2#2259720111yescharacteristic
400C2#2259725811yescharacteristic
subgroupG/H
centerC2×C4C22×D284
commutatorC2×C4×C71C23×C4
FrattiniC42C23×D71
FittingC71×#1450C2
socleC22×C71C2×C4×D4
prime radical#17986549C22
cosocleC42×C71C24
hypercenter#1450D71
hypocenterC71#17971
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ #431 ◁ #1450

Complements

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

NH ≅ G/Ntype
2. #22597434C2semidirect
4. #22587244C2semidirect
5. #22587232C2semidirect
6. #22587154C2semidirect
7. C8×#11765854C2semidirect
8. #22587103C2semidirect
9. #22586814C2semidirect
11. C71×#1450C2semidirect
211. #1450D71semidirect
212. C71#17971semidirect

Maximal subgroups

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

subgroupindex
#225988992
#225974342
#225974152
#225872442
#225872322
#225871542
C8×#117658542
#225871032
#225868142
#225868102
C71×#14502
#1797171