Finite Group Databasebeta

← #31893033

Subgroups — #31893033

Order 18176.

612 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986056C211yesmaximal, characteristic
2C2×#17985736C212yesmaximal
3#22586567C211yesmaximal, characteristic
4#22586218C212yesmaximal
5C71×#2855C211yesmaximal, characteristic
6#17986056—24no—
7C2×C4⋊(C71⋊C8)—21no—
8C2×C22⋊(C71⋊C8)C2212yes—
9C2×#14556039—21no—
10C2×C71⋊#95—21no—
11C2×C71⋊#95—21no—
12C2×C71⋊#95C2212yes—
13#17985736—24no—
14C2×C22⋊Q8×C71C2211yescharacteristic
15C22×C42×C71C2211yescharacteristic
16#391×C71C411yescharacteristic, complemented
17#391×C71C411yescharacteristic, complemented
18#389×C71C411yescharacteristic, complemented
19#389×C71C411yescharacteristic, complemented
20#387×C71—21no—
21#387×C71—21no—
22#385×C71—21no—
23#381×C71—21no—
24#380×C71C2211yescharacteristic
25C23×Dic71—22no—
26C22×C71⋊C8—22no—
27C4⋊(C71⋊C8)—24no—
28C22⋊(C71⋊C8)—22no—
29C22⋊(C71⋊C8)D414yes—
30#14556039—24no—
31C71⋊#95—24no—
32C71⋊#95—22no—
33C71⋊#95—22no—
34C71⋊#95—42no—
35C22×Q8×C71C2×C411yescharacteristic
36C23×C4×C71D411yescharacteristic
37C23×C4×C71D411yescharacteristic
38C23×C4×C71C2311yescharacteristic
39C22⋊Q8×C71—22no—
40C22⋊Q8×C71—22no—
41C2×C4⋊C4×C71—21no—
42C2×C4⋊C4×C71—21no—
43C2×C4⋊C4×C71—21no—
44C2×C4⋊C4×C71C2×C411yescharacteristic
45C2×C4⋊C4×C71C2×C411yescharacteristic
46C2×C4⋊C4×C71C2×C411yescharacteristic
47C2×C22⋊C4×C71—21no—
48C2×C22⋊C4×C71—21no—
49C2×C42×C71—22no—
50C2×C42×C71—22no—
51C2×C42×C71C2×C411yescharacteristic
52C2×C42×C71C2×C411yescharacteristic
53C2×C42×C71D411yescharacteristic
54C2×C42×C71D411yescharacteristic
55#95×C71—21no—
56#95×C71—21no—
57#95×C71—21no—
58#95×C71—21no—
59#95×C71—21no—
60#95×C71—21no—
61#95×C71—21no—
62C22×Dic71—24no—
63C22×Dic71—42no—
64C22×Dic71—42no—
65C22×Dic71—42no—
66C22×Dic71—42no—
67C22×Dic71—42no—
68C22×Dic71—42no—
69C2×C71⋊C8—24no—
70C2×C71⋊C8—42no—
71C2×C71⋊C8—42no—
72C24×C71C2×D411yescharacteristic
73C2×Q8×C71—22no—
74C2×Q8×C71—41no—
75C2×Q8×C71—41no—
76C22×C4×C71—22no—
77C22×C4×C71—22no—
78C22×C4×C71—21no—
79C22×C4×C71—21no—
80C22×C4×C71—22no—
81C22×C4×C71—22no—
82C22×C4×C71—21no—
83C22×C4×C71—21no—
84C22×C4×C71—21no—
85C22×C4×C71—21no—
86C22×C4×C71—41no—
87C22×C4×C71—41no—
88C22×C4×C71—41no—
89C22×C4×C71C22⋊C411yescharacteristic
90C22×C4×C71C22⋊C411yescharacteristic
91C22×C4×C71C22⋊C411yescharacteristic
92C22×C4×C71C22⋊C411yescharacteristic
93C22×C4×C71C22×C411yescharacteristic
94C22×C4×C71C2×D412yes—
95C22×C4×C71C2×D411yescharacteristic
96C22×C4×C71#4112yes—
97C4⋊C4×C71—22no—
98C4⋊C4×C71—41no—
99C4⋊C4×C71—41no—
100C4⋊C4×C71—42no—
101C4⋊C4×C71—42no—
102C4⋊C4×C71—41no—
103C4⋊C4×C71—41no—
104C22⋊C4×C71—42no—
105C22⋊C4×C71—41no—
106C22⋊C4×C71—41no—
107C42×C71—22no—
108C42×C71—22no—
109C42×C71—21no—
110C42×C71—21no—
111C42×C71—21no—
112C42×C71—21no—
113C23×C71—21no—
114C23×C71—21no—
115C23×C71—21no—
116C23×C71—21no—
117C23×C71C2×C22⋊C411yescharacteristic
118C23×C71#12112yes—
119C23×C71AGL(1,ℤ/8)11yescharacteristic
120C23×C71#13711yescharacteristic
121C23×C71#13712yes—
122Q8×C71—82no—
123C2×C4×C71—21no—
124C2×C4×C71—21no—
125C2×C4×C71—21no—
126C2×C4×C71—21no—
127C2×C4×C71—21no—
128C2×C4×C71—21no—
129C2×C4×C71—21no—
130C2×C4×C71—21no—
131C2×C4×C71—21no—
132C2×C4×C71—21no—
133C2×C4×C71—21no—
134C2×C4×C71—21no—
135C2×C4×C71—21no—
136C2×C4×C71—21no—
137C2×C4×C71—21no—
138C2×C4×C71—42no—
139C2×C4×C71—42no—
140C2×C4×C71—44no—
141C2×C4×C71—42no—
142C2×C4×C71—42no—
143C2×C4×C71—41no—
144C2×C4×C71—41no—
145C2×C4×C71—41no—
146C2×C4×C71—41no—
147C2×C4×C71—41no—
148C2×C4×C71—41no—
149C2×C4×C71—41no—
150C2×C4×C71—41no—
151C2×C4×C71—41no—
152C2×C4×C71—41no—
153C2×C4×C71—41no—
154C2×C4×C71—41no—
155C2×C4×C71#10412yes—
156C2×C4×C71C4×D412yes—
157C2×C4×C71#12011yescharacteristic
158C2×C4×C71#12311yescharacteristic
159C2×Dic71—44no—
160C2×Dic71—44no—
161C2×Dic71—44no—
162C2×Dic71—82no—
163C2×Dic71—82no—
164C2×Dic71—82no—
165C2×Dic71—82no—
166C71⋊C8—44no—
167C22×C71—22no—
168C22×C71—22no—
169C22×C71—22no—
170C22×C71—21no—
171C22×C71—21no—
172C22×C71—21no—
173C22×C71—21no—
174C22×C71—22no—
175C22×C71#38611yescharacteristic
176C22×C71#41011yescharacteristic
177C22×C71#41011yescharacteristic
178C22×C71#41711yescharacteristic
179C22×C71#41911yescharacteristic
180C22×C71C2×#10411yescharacteristic
181C22×C71#42111yescharacteristic
182C22×C71#46812yes—
183C22×C71#47012yes—
184C4×C71—22no—
185C4×C71—21no—
186C4×C71—22no—
187C4×C71—21no—
188C4×C71—41no—
189C4×C71—41no—
190C4×C71—41no—
191C4×C71—81no—
192C4×C71—81no—
193C4×C71—81no—
194C4×C71—81no—
195Dic71—84no—
196#9116—711nomaximal, Sylow
197C2×C71—22no—
198C2×C71—22no—
199C2×C71#150611yescharacteristic
200C2×C71#150911yescharacteristic
201C2×C71#151211yescharacteristic
202C2×C71#151511yescharacteristic
203C2×C71#187111yescharacteristic
204C2×C71#189411yescharacteristic
205C2×C71#189911yescharacteristic
206C2×#432—711no—
207#2855D7111yescharacteristic
208C2×#328—712no—
209#1434—711no—
210#1294—712no—
211C71#911611yescharacteristic, complemented, Sylow
212C2×C22⋊Q8C2×D7111yescharacteristic
213C22×C42C2×D7111yescharacteristic
214#432—1424no—
215C2×C4⋊C8—1421no—
216C2×C22⋊C8—712no—
217C2×#97—1421no—
218#391Dic7111yescharacteristic, complemented
219#391Dic7111yescharacteristic, complemented
220#389Dic7111yescharacteristic, complemented
221#389Dic7111yescharacteristic, complemented
222#387—21no—
223#387—21no—
224#385—21no—
225#381—21no—
226#380C2×D7111yescharacteristic
227C2×#95—712no—
228C2×#95—1421no—
229C2×#95—1421no—
230#328—1424no—
231C22×Q8C2×Dic7111yescharacteristic
232C23×C4—1422no—
233C23×C4C22⋊D7111yescharacteristic
234C23×C4C22⋊D7111yescharacteristic
235C23×C4C22×D7111yescharacteristic
236C22×C8—1422no—
237C22⋊Q8—22no—
238C22⋊Q8—22no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4C2×Dic7111yescharacteristic
243C2×C4⋊C4C2×Dic7111yescharacteristic
244C2×C4⋊C4C2×Dic7111yescharacteristic
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42C2×Dic7111yescharacteristic
250C2×C42C2×Dic7111yescharacteristic
251C2×C42C22⋊D7111yescharacteristic
252C2×C42C22⋊D7111yescharacteristic
253C4⋊C8—1424no—
254C22⋊C8—714no—
255C22⋊C8—1422no—
256#97—1424no—
257#95—21no—
258#95—21no—
259#95—21no—
260#95—21no—
261#95—21no—
262#95—21no—
263#95—21no—
264#95—2842no—
265#95—1424no—
266#95—1422no—
267#95—1422no—
268C24C2×C22⋊D7111yescharacteristic
269C2×Q8—22no—
270C2×Q8—41no—
271C2×Q8—41no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—2842no—
286C22×C4—2842no—
287C22×C4—2842no—
288C22×C4—2842no—
289C22×C4—2842no—
290C22×C4—2842no—
291C22×C4—1424no—
292C22×C4C22⋊Dic7111yescharacteristic
293C22×C4C22⋊Dic7111yescharacteristic
294C22×C4C22⋊Dic7111yescharacteristic
295C22×C4C22⋊Dic7111yescharacteristic
296C22×C4C22×Dic7111yescharacteristic
297C22×C4D4×D7112yes—
298C22×C4C2×C22⋊D7111yescharacteristic
299C22×C4#1176586412yes—
300C2×C8—2842no—
301C2×C8—2842no—
302C2×C8—1424no—
303C4⋊C4—22no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C22⋊C4—42no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C42—22no—
314C42—22no—
315C42—21no—
316C42—21no—
317C42—21no—
318C42—21no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23—21no—
323C23C2×C22⋊Dic7111yescharacteristic
324C23#1455610012yes—
325C23#1455615311yescharacteristic
326C23#1455615811yescharacteristic
327C23#1455615912yes—
328Q8—82no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—44no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—2844no—
362C2×C4—2844no—
363C2×C4—2844no—
364C2×C4—5682no—
365C2×C4—5682no—
366C2×C4—5682no—
367C2×C4—5682no—
368C2×C4#1455605512yes—
369C2×C4D4×Dic7112yes—
370C2×C4#1455609611yescharacteristic
371C2×C4#1455610911yescharacteristic
372C8—2844no—
373C22—22no—
374C22—22no—
375C22—22no—
376C22—22no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22#1798587911yescharacteristic
382C22#1798597311yescharacteristic
383C22#1798597311yescharacteristic
384C22#1798600011yescharacteristic
385C22#1798600811yescharacteristic
386C22C2×#1455605511yescharacteristic
387C22#1798601611yescharacteristic
388C22#1798624712yes—
389C22#1798625512yes—
390C4—21no—
391C4—22no—
392C4—21no—
393C4—22no—
394C4—41no—
395C4—41no—
396C4—41no—
397C4—81no—
398C4—81no—
399C4—81no—
400C4—81no—
401C4—5684no—
402C2—22no—
403C2—22no—
404C2#2258685211yescharacteristic
405C2#2258686511yescharacteristic
406C2#2258687811yescharacteristic
407C2#2258689111yescharacteristic
408C2#2258864311yescharacteristic
409C2#2258877411yescharacteristic
410C2#2258879911yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic71
commutatorC22×C4×C71C22×C4
FrattiniC23×C4C22×D71
FittingC71×#2855C2
socleC23×C71C2×C22⋊C4
prime radicalC24×C71C2×D4
cosocleC23×C4×C71C23
hypercenter#2855D71
hypocenterC71#9116
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2855

Complements

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

NH ≅ G/Ntype
16. #391×C71C4semidirect
17. #391×C71C4semidirect
18. #389×C71C4semidirect
19. #389×C71C4semidirect
211. C71#9116semidirect
218. #391Dic71semidirect
219. #391Dic71semidirect
220. #389Dic71semidirect
221. #389Dic71semidirect

Maximal subgroups

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

subgroupindex
C2×#179860562
C2×#179857362
#225865672
#225862182
C71×#28552
#911671