Finite Group Databasebeta

← #32006228

Subgroups — #32006228

Order 18176.

750 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986403C211yesmaximal, characteristic
2#22591084C212yesmaximal
3#22589311C212yesmaximal
4#22589077C212yesmaximal
5#22589074C211yesmaximal, characteristic
6#22589071C211yesmaximal, characteristic
7#22588503C211yesmaximal, characteristic
8#22588238C212yesmaximal
9C4×#14556062C211yesmaximal, characteristic
10#22588026C211yesmaximal, characteristic
11C4×#107×C71C211yesmaximal, characteristic
12C2×#14556127C2212yes—
13#17986403C418yescomplemented
14C2×#14556062C2211yescharacteristic
15C2×#14556062C2211yescharacteristic
16C2×#14556061C2212yes—
17C2×#14556059C2212yes—
18C2×#14556039C2211yescharacteristic
19C2×#14556038C2211yescharacteristic
20C2×#14556038C2211yescharacteristic
21C2×C8×Dic71C2211yescharacteristic
22C2×C8×Dic71C2211yescharacteristic
23#17985905—22no—
24#17985903—22no—
25#17985898C2212yes—
26#17985895C2212yes—
27#17985886—22no—
28#17985883—22no—
29#17985864C2212yes—
30C4×C4⋊Dic71C2212yes—
31#17985845C2211yescharacteristic
32#17985762C2212yes—
33#17985761C2212yes—
34#17985761C2212yes—
35C2×#107×C71C2211yescharacteristic
36C2×#107×C71C2211yescharacteristic
37C2×C4×C8×C71C2211yescharacteristic
38C4×C4⋊C4×C71C2212yes—
39#340×C71C2212yes—
40C22×Dic142—22no—
41C22×C71⋊C8C2311yescharacteristic
42C22×C71⋊C8C2311yescharacteristic
43#14556127—24no—
44#14556127C2×C418yes—
45C2×C4⋊Dic71—22no—
46C2×C4⋊Dic71C2312yes—
47C2×C4⋊Dic71C2312yes—
48C2×Dic71⋊C4—22no—
49C2×Dic71⋊C4—22no—
50C2×C4×Dic71—22no—
51C2×C4×Dic71C2311yescharacteristic
52C2×C4×Dic71C2311yescharacteristic
53#14556062—22no—
54#14556062C2×C414yes—
55#14556061C2×C418yes—
56#14556059—24no—
57#14556039Q814yescomplemented
58#14556038—22no—
59#14556038—22no—
60C8×Dic71—22no—
61C8×Dic71C2×C414yes—
62C71⋊#95—22no—
63C71⋊#95—22no—
64C71⋊#95—22no—
65C71⋊#95—22no—
66C71⋊#95—22no—
67C71⋊#95D411yescharacteristic
68C71⋊#95D411yescharacteristic
69C71⋊#95D411yescharacteristic
70C71⋊#95D411yescharacteristic
71C22×C8×C71C2311yescharacteristic
72C22×C8×C71C2311yescharacteristic
73C2×C4⋊C4×C71C2312yes—
74C2×C4⋊C4×C71C2312yes—
75C2×C42×C71—22no—
76C2×C42×C71C2311yescharacteristic
77#107×C71—22no—
78#107×C71C2×C414yes—
79C4×C8×C71Q814yescomplemented
80#95×C71—22no—
81C2×Dic142—28no—
82C2×Dic142—82no—
83C2×Dic142—82no—
84C22×Dic71—22no—
85C22×Dic71—22no—
86C22×Dic71C2×D411yescharacteristic
87C22×Dic71C2×D411yescharacteristic
88C22×Dic71C2×D411yescharacteristic
89C22×Dic71C2×D411yescharacteristic
90C2×C71⋊C8—21no—
91C2×C71⋊C8—21no—
92C2×C71⋊C8C22×C412yes—
93C2×C71⋊C8C2×Q812yes—
94C2×C71⋊C8C2×Q812yes—
95C2×C71⋊C8#4112yes—
96C4⋊Dic71—22no—
97C4⋊Dic71—22no—
98C4⋊Dic71—22no—
99C4⋊Dic71—82no—
100C4⋊Dic71C22×C414yes—
101Dic71⋊C4—28no—
102Dic71⋊C4—82no—
103C4×Dic71—21no—
104C4×Dic71—21no—
105C4×Dic71—82no—
106C4×Dic71C22×C412yes—
107C4×Dic71#4112yes—
108C22×C4×C71—22no—
109C22×C4×C71—22no—
110C22×C4×C71C2×D411yescharacteristic
111C22×C4×C71C2×D411yescharacteristic
112C22×C4×C71C2411yescharacteristic
113C2×C8×C71—21no—
114C2×C8×C71—21no—
115C2×C8×C71C22×C412yes—
116C2×C8×C71C2×Q812yes—
117C2×C8×C71C2×Q812yes—
118C2×C8×C71#4112yes—
119C4⋊C4×C71—22no—
120C4⋊C4×C71—22no—
121C4⋊C4×C71—22no—
122C4⋊C4×C71C22×C414yes—
123C42×C71—82no—
124C42×C71C2×Q812yes—
125C42×C71C2×Q812yes—
126C23×C71C22×D411yescharacteristic
127C2×C4×C71—21no—
128C2×C4×C71—21no—
129C2×C4×C71—21no—
130C2×C4×C71—21no—
131C2×C4×C71—24no—
132C2×C4×C71—42no—
133C2×C4×C71—42no—
134C2×C4×C71—42no—
135C2×C4×C71—42no—
136C2×C4×C71—42no—
137C2×C4×C71C4⋊Q812yes—
138C2×C4×C71C4⋊Q812yes—
139C2×C4×C71C23×C411yescharacteristic
140C2×C4×C71C22×Q811yescharacteristic
141C2×C4×C71C22×Q811yescharacteristic
142C2×C4×C71C2×#4111yescharacteristic
143C2×C4×C71C2×#4111yescharacteristic
144C2×C4×C71C2×#4111yescharacteristic
145C2×Dic71—21no—
146C2×Dic71—21no—
147C2×Dic71—21no—
148C2×Dic71—21no—
149C2×Dic71—24no—
150C2×Dic71—21no—
151C2×Dic71—21no—
152C2×Dic71—21no—
153C2×Dic71—21no—
154C2×Dic71—21no—
155C2×Dic71—21no—
156C2×Dic71—42no—
157C2×Dic71—42no—
158C2×Dic71—42no—
159C2×Dic71—42no—
160C2×Dic71—42no—
161C2×Dic71C4×D412yes—
162C2×Dic71C4×D412yes—
163Dic142—88no—
164C8×C71—22no—
165C8×C71C4×Q814yes—
166C71⋊C8—22no—
167C71⋊C8C4×Q814yes—
168C22×C71C2×C4×D411yescharacteristic
169C22×C71C2×C4⋊Q811yescharacteristic
170C22×C71#53211yescharacteristic
171C22×C71#57611yescharacteristic
172C22×C71#57611yescharacteristic
173C22×C71#57811yescharacteristic
174C22×C71#57811yescharacteristic
175C4×C71—22no—
176C4×C71—22no—
177C4×C71—44no—
178C4×C71—82no—
179C4×C71C2×C4×Q811yescharacteristic
180C4×C71C2×C4×Q811yescharacteristic
181C4×C71#53311yescharacteristic
182C4×C71#53311yescharacteristic
183Dic71—22no—
184Dic71—22no—
185Dic71—41no—
186Dic71—41no—
187Dic71—44no—
188Dic71—82no—
189#21725—711nomaximal, Sylow
190C2×C71C4×C4⋊Q811yescharacteristic
191C2×C71#294811yescharacteristic
192C2×C71#295011yescharacteristic
193C2×C71#316411yescharacteristic
194C2×C71#316411yescharacteristic
195C2×C71#317511yescharacteristic
196C2×C71#317511yescharacteristic
197C2×#500—711no—
198#2331—712no—
199#1993—712no—
200#1943—712no—
201#1942—711no—
202#1942—711no—
203#1846—711no—
204#1776—712no—
205C4×#107—711no—
206C4×#107D7111yescharacteristic
207#1726—711no—
208C71#2172511yescharacteristic, complemented, Sylow
209C2×C4⋊Q8—712no—
210#500—718no—
211C2×#107—711no—
212C2×#107—712no—
213C2×#107—711no—
214C2×#107C2×D7111yescharacteristic
215C2×#107C2×D7111yescharacteristic
216C2×#106—712no—
217C2×#97—711no—
218C2×#97—711no—
219C2×#97—711no—
220C2×C4×C8—711no—
221C2×C4×C8—711no—
222C2×C4×C8C2×D7111yescharacteristic
223#391—1422no—
224#391—1422no—
225#389—712no—
226#389—712no—
227#387—1422no—
228#387—1422no—
229#382—712no—
230C4×C4⋊C4—712no—
231C4×C4⋊C4C2×D7112yes—
232#376—711no—
233#340—712no—
234#340—712no—
235#340—712no—
236#340C2×D7112yes—
237C22×Q8—1422no—
238C22×C8—711no—
239C22×C8—711no—
240C22×C8C22×D7111yescharacteristic
241C22×C8C22×D7111yescharacteristic
242C4⋊Q8—718no—
243C4⋊Q8—1424no—
244C2×C4⋊C4—712no—
245C2×C4⋊C4—712no—
246C2×C4⋊C4—1422no—
247C2×C4⋊C4—1422no—
248C2×C4⋊C4—1422no—
249C2×C4⋊C4C22×D7112yes—
250C2×C4⋊C4C22×D7112yes—
251C2×C42—22no—
252C2×C42—711no—
253C2×C42—711no—
254C2×C42—1422no—
255C2×C42C22×D7111yescharacteristic
256#107—22no—
257#107—714no—
258#107—718no—
259#107—1422no—
260#107C4×D7114yes—
261#106—1424no—
262#97—714no—
263#97—1422no—
264#97—1422no—
265C4×C8—714no—
266C4×C8—1422no—
267C4×C8Dic14214yescomplemented
268#95—22no—
269#95—711no—
270#95—711no—
271#95—711no—
272#95—711no—
273#95—1422no—
274#95—1422no—
275#95—1422no—
276#95—1422no—
277#95—1422no—
278C2×Q8—5682no—
279C2×Q8—5682no—
280C2×Q8—1428no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—711no—
284C22×C4—711no—
285C22×C4—711no—
286C22×C4—711no—
287C22×C4—1422no—
288C22×C4—1422no—
289C22×C4D4×D7111yescharacteristic
290C22×C4D4×D7111yescharacteristic
291C22×C4C23×D7111yescharacteristic
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—712no—
295C2×C8—712no—
296C2×C8—712no—
297C2×C8—712no—
298C2×C8—1421no—
299C2×C8—1421no—
300C2×C8C2×C4×D7112yes—
301C2×C8C2×Dic14212yes—
302C2×C8C2×Dic14212yes—
303C2×C8#1176586512yes—
304C4⋊C4—22no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—5682no—
308C4⋊C4—5682no—
309C4⋊C4—714no—
310C4⋊C4—1428no—
311C4⋊C4—1422no—
312C4⋊C4—1422no—
313C4⋊C4—1422no—
314C4⋊C4C2×C4×D7114yes—
315C42—82no—
316C42—5682no—
317C42—712no—
318C42—712no—
319C42—1421no—
320C42—1421no—
321C42Q8×D7112yes—
322C42C2×Dic14212yes—
323C23C2×D4×D7111yescharacteristic
324Q8—5688no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—24no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—2842no—
336C2×C4—2842no—
337C2×C4—2842no—
338C2×C4—2842no—
339C2×C4—2842no—
340C2×C4—712no—
341C2×C4—712no—
342C2×C4—1424no—
343C2×C4—1421no—
344C2×C4—1421no—
345C2×C4—1421no—
346C2×C4—1421no—
347C2×C4—1421no—
348C2×C4—1421no—
349C2×C4—1421no—
350C2×C4—1421no—
351C2×C4—1421no—
352C2×C4—1421no—
353C2×C4#1455612712yes—
354C2×C4#1455612712yes—
355C2×C4C22×C4×D7111yescharacteristic
356C2×C4C2×Q8×D7111yescharacteristic
357C2×C4C22×Dic14211yescharacteristic
358C2×C4#41×D7111yescharacteristic
359C2×C4#41×D7111yescharacteristic
360C2×C4C2×#1176586511yescharacteristic
361C8—22no—
362C8—714no—
363C8—1422no—
364C8C4×Dic14214yes—
365C22C4×D4×D7111yescharacteristic
366C22C2×#1455612711yescharacteristic
367C22#1798654611yescharacteristic
368C22#1798680011yescharacteristic
369C22#1798680111yescharacteristic
370C22#1798681211yescharacteristic
371C22#1798681311yescharacteristic
372C4—22no—
373C4—22no—
374C4—44no—
375C4—82no—
376C4—2841no—
377C4—2844no—
378C4—2841no—
379C4—5682no—
380C4—1422no—
381C4—1422no—
382C4C4×Q8×D7111yescharacteristic
383C4C2×C4×Dic14211yescharacteristic
384C4#1798655211yescharacteristic
385C4#1798655311yescharacteristic
386C2C4×#1455612711yescharacteristic
387C2#2259718711yescharacteristic
388C2#2259720311yescharacteristic
389C2#2259887611yescharacteristic
390C2#2259888011yescharacteristic
391C2#2259897511yescharacteristic
392C2#2259898111yescharacteristic
subgroupG/H
centerC23C2×D4×D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC4×#107×C71C2
socle, prime radicalC23×C71C22×D4
cosocleC22×C4×C71C24
hypercenterC4×#107D71
hypocenterC71#21725
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#107

Complements

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

NH ≅ G/Ntype
13. #17986403C4semidirect
57. #14556039Q8semidirect
79. C4×C8×C71Q8semidirect
208. C71#21725semidirect
267. C4×C8Dic142semidirect

Maximal subgroups

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

subgroupindex
C2×#179864032
#225910842
#225893112
#225890772
#225890742
#225890712
#225885032
#225882382
C4×#145560622
#225880262
C4×#107×C712
#2172571