Finite Group Databasebeta

← #32015154

Subgroups — #32015154

Order 18176.

810 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#107×D71C211yesmaximal, characteristic
2#22591213C212yesmaximal
3#22589078C211yesmaximal, characteristic, complemented
4#22589076C211yesmaximal, characteristic, complemented
5#22588911C212yesmaximal, complemented
6#22588909C212yesmaximal, complemented
7#22588429C212yesmaximal
8#22588362C211yesmaximal, characteristic
9#22588239C211yesmaximal, characteristic, complemented
10#22588117C211yesmaximal, characteristic
11C71×#1776C211yesmaximal, characteristic, complemented
12C22×C8×D71C2211yescharacteristic
13C2×C4⋊C4×D71C2212yes—
14C2×#14556062C2211yescharacteristic
15C2×#14556061C2212yes—
16#107×D71C418yescomplemented
17C2×#14556060C2211yescharacteristic
18C2×(C2×D71)⋊C8C2211yescharacteristic
19C2×#14556038C2211yescharacteristic
20#17985897C2212yes—
21#17985895C2212yes—
22#17985883—22no—
23#17985881—22no—
24#17985858—22no—
25#17985856C2212yes—
26C4⋊C4×Dic71C2212yes—
27C4×(C2×D71)⋊C4—22no—
28C4×(C2×D71)⋊C4C2211yescharacteristic
29#17985762C2212yes—
30#17985761C2212yes—
31#17985761C2212yes—
32#17985761C2212yes—
33#17985754C2211yescharacteristic
34#17985753C2211yescharacteristic
35C2×#107×C71C2211yescharacteristic
36C2×#106×C71C2211yescharacteristic
37C2×#97×C71C2211yescharacteristic
38C4×C4⋊C4×C71C2212yes—
39#340×C71C2212yes—
40C22×C4×D71—22no—
41C22×C4×D71C2311yescharacteristic
42C22×C71⋊C8C2311yescharacteristic
43C22×C71⋊C8C2311yescharacteristic
44C2×C8×D71—22no—
45C2×C8×D71—22no—
46C2×C8×D71C2×C414yes—
47C2×C4⋊Dic71C2312yes—
48C2×C4⋊Dic71C2312yes—
49C2×Dic71⋊C4—22no—
50C2×Dic71⋊C4—22no—
51C2×Dic71⋊C4—22no—
52C4⋊C4×D71—24no—
53C4⋊C4×D71C2×C418yes—
54C2×(C2×D71)⋊C4—22no—
55C2×(C2×D71)⋊C4D412yes—
56C2×C4×Dic71—22no—
57C2×C4×Dic71C2311yescharacteristic
58#14556062C2×C414yes—
59#14556061C2×C418yes—
60#14556060—22no—
61(C2×D71)⋊C8—22no—
62#14556038—22no—
63C71⋊#95—22no—
64C71⋊#95—22no—
65C71⋊#95—22no—
66C71⋊#95—22no—
67C71⋊#95D412yes—
68C22×C8×C71C2311yescharacteristic
69C22×C8×C71C2311yescharacteristic
70C2×C4⋊C4×C71C2312yes—
71C2×C4⋊C4×C71C2312yes—
72C2×C42×C71—22no—
73C2×C42×C71C2311yescharacteristic
74#107×C71C2×C414yes—
75#106×C71—22no—
76#97×C71—22no—
77#95×C71—22no—
78C23×D71C2×D411yescharacteristic
79C22×Dic71—22no—
80C22×Dic71—22no—
81C22×Dic71C2×D412yes—
82C22×Dic71C2×D411yescharacteristic
83C2×C4×D71—21no—
84C2×C4×D71—21no—
85C2×C4×D71—21no—
86C2×C4×D71—21no—
87C2×C4×D71—28no—
88C2×C4×D71—82no—
89C2×C4×D71—82no—
90C2×C4×D71C22×C412yes—
91C2×C4×D71#4112yes—
92C2×C71⋊C8—21no—
93C2×C71⋊C8—21no—
94C2×C71⋊C8—21no—
95C2×C71⋊C8—21no—
96C2×C71⋊C8—21no—
97C2×C71⋊C8C22×C412yes—
98C8×D71—28no—
99C4⋊Dic71—22no—
100C4⋊Dic71—22no—
101C4⋊Dic71—22no—
102C4⋊Dic71C22×C414yes—
103Dic71⋊C4—28no—
104Dic71⋊C4—82no—
105Dic71⋊C4—82no—
106(C2×D71)⋊C4—24no—
107(C2×D71)⋊C4—44no—
108C4×Dic71—21no—
109C4×Dic71—21no—
110C4×Dic71—82no—
111C22×C4×C71—22no—
112C22×C4×C71—22no—
113C22×C4×C71C2×D412yes—
114C22×C4×C71C2411yescharacteristic
115C2×C8×C71—21no—
116C2×C8×C71—21no—
117C2×C8×C71—21no—
118C2×C8×C71C22×C412yes—
119C2×C8×C71#4112yes—
120C2×C8×C71#4112yes—
121C4⋊C4×C71—22no—
122C4⋊C4×C71—22no—
123C4⋊C4×C71—22no—
124C4⋊C4×C71C22×C414yes—
125C42×C71—21no—
126C42×C71—21no—
127C42×C71—82no—
128C23×C71C22×D411yescharacteristic
129C22×D71—21no—
130C22×D71—21no—
131C22×D71—22no—
132C22×D71C4×D412yes—
133C22×D71#13714yes—
134C2×C4×C71—24no—
135C2×C4×C71—22no—
136C2×C4×C71—22no—
137C2×C4×C71—22no—
138C2×C4×C71—42no—
139C2×C4×C71—42no—
140C2×C4×C71—42no—
141C2×C4×C71—42no—
142C2×C4×C71—42no—
143C2×C4×C71C23×C411yescharacteristic
144C2×C4×C71C2×#4111yescharacteristic
145C2×C4×C71C2×#4111yescharacteristic
146C2×C4×C71C2×#4111yescharacteristic
147C2×C4×C712−1+411yescharacteristic
148C2×C4×C712−1+411yescharacteristic
149C2×Dic71—21no—
150C2×Dic71—21no—
151C2×Dic71—24no—
152C2×Dic71—22no—
153C2×Dic71—22no—
154C2×Dic71—22no—
155C2×Dic71—42no—
156C2×Dic71—42no—
157C2×Dic71—42no—
158C2×Dic71—42no—
159C2×Dic71—42no—
160C2×Dic71C4×D412yes—
161C4×D71—22no—
162C4×D71—22no—
163C4×D71—22no—
164C4×D71—22no—
165C4×D71—88no—
166C8×C71—41no—
167C8×C71#11714yes—
168C71⋊C8—22no—
169C71⋊C8—41no—
170C22×C71C2×C4×D411yescharacteristic
171C22×C71#54711yescharacteristic
172C22×C71#54711yescharacteristic
173C22×C71C2×#13712yes—
174C22×C71#57612yes—
175C2×D71—24no—
176C2×D71—24no—
177C2×D71—22no—
178C2×D71—22no—
179C2×D71#44114yes—
180C4×C71—42no—
181C4×C71—44no—
182C4×C71—82no—
183C4×C71C2×#11711yescharacteristic
184C4×C71#51911yescharacteristic
185C4×C71#55211yescharacteristic
186C4×C71#55211yescharacteristic
187Dic71—22no—
188Dic71—42no—
189Dic71—44no—
190Dic71—82no—
191#22618—711nomaximal, Sylow
192C2×C71#234411yescharacteristic
193C2×C71C2×#44111yescharacteristic
194C2×C71#294811yescharacteristic
195C2×C71#334612yes—
196C2×C71#334612yes—
197D71—24no—
198C22×#107—711no—
199#2345—712no—
200#1943—711no—
201#1943—711no—
202#1918—712no—
203#1918—712no—
204#1824—712no—
205#1807—711no—
206#1776—711no—
207#1776D7111yescharacteristic, complemented
208#1752—711no—
209C71#2261811yescharacteristic, complemented, Sylow
210C23×C8—711no—
211C22×C4⋊C4—712no—
212C2×#107—711no—
213C2×#107—712no—
214C2×#107—718no—
215C2×#107C2×D7111yescharacteristic
216C2×#106—711no—
217C2×#106C2×D7111yescharacteristic
218C2×C22⋊C8—711no—
219C2×#97—711no—
220C2×#97C2×D7111yescharacteristic
221#389—712no—
222#389—712no—
223#387—1422no—
224#387—1422no—
225#380—712no—
226#380—1422no—
227C4×C4⋊C4—712no—
228C4×C4⋊C4C2×D7112yes—
229C4×C22⋊C4—711no—
230C4×C22⋊C4—1422no—
231#340—712no—
232#340—712no—
233#340—712no—
234#340—712no—
235#340C2×D7112yes—
236#336—711no—
237#336—711no—
238C23×C4—711no—
239C23×C4—1422no—
240C22×C8—711no—
241C22×C8—714no—
242C22×C8—711no—
243C22×C8—1422no—
244C22×C8—1422no—
245C22×C8C22×D7111yescharacteristic
246C22×C8C22×D7111yescharacteristic
247C2×C4⋊C4—712no—
248C2×C4⋊C4—718no—
249C2×C4⋊C4—712no—
250C2×C4⋊C4—1424no—
251C2×C4⋊C4—1422no—
252C2×C4⋊C4—1422no—
253C2×C4⋊C4—1422no—
254C2×C4⋊C4C22×D7112yes—
255C2×C4⋊C4C22×D7112yes—
256C2×C22⋊C4—712no—
257C2×C22⋊C4—1422no—
258C2×C42—22no—
259C2×C42—711no—
260C2×C42—1422no—
261C2×C42C22×D7111yescharacteristic
262#107—714no—
263#107—718no—
264#107C4×D7114yes—
265#106—22no—
266#106—1422no—
267C22⋊C8—1422no—
268#97—22no—
269#97—1422no—
270#95—22no—
271#95—712no—
272#95—1422no—
273#95—1422no—
274#95—1422no—
275#95—1422no—
276C24—711no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—5682no—
280C22×C4—5682no—
281C22×C4—711no—
282C22×C4—712no—
283C22×C4—712no—
284C22×C4—712no—
285C22×C4—1428no—
286C22×C4—1422no—
287C22×C4—1422no—
288C22×C4—1421no—
289C22×C4—1421no—
290C22×C4—1421no—
291C22×C4—1421no—
292C22×C4D4×D7112yes—
293C22×C4C23×D7111yescharacteristic
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—712no—
298C2×C8—1421no—
299C2×C8—1421no—
300C2×C8—1421no—
301C2×C8—1421no—
302C2×C8—1421no—
303C2×C8—1428no—
304C2×C8C2×C4×D7112yes—
305C2×C8#1176586512yes—
306C2×C8#1176586512yes—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—5682no—
311C4⋊C4—5682no—
312C4⋊C4—714no—
313C4⋊C4—1428no—
314C4⋊C4—1422no—
315C4⋊C4—1422no—
316C4⋊C4—1422no—
317C4⋊C4C2×C4×D7114yes—
318C22⋊C4—2844no—
319C22⋊C4—1424no—
320C42—21no—
321C42—21no—
322C42—82no—
323C42—5682no—
324C42—1421no—
325C42—1421no—
326C23—712no—
327C23—714no—
328C23—1421no—
329C23—1421no—
330C23—1422no—
331C23C2×D4×D7111yescharacteristic
332C2×C4—22no—
333C2×C4—24no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—2842no—
342C2×C4—2842no—
343C2×C4—2842no—
344C2×C4—2842no—
345C2×C4—2842no—
346C2×C4—5688no—
347C2×C4—712no—
348C2×C4—1422no—
349C2×C4—1424no—
350C2×C4—1422no—
351C2×C4—1422no—
352C2×C4—1421no—
353C2×C4—1421no—
354C2×C4—1422no—
355C2×C4—1422no—
356C2×C4—1422no—
357C2×C4—1422no—
358C2×C4C22×C4×D7111yescharacteristic
359C2×C4#41×D7111yescharacteristic
360C2×C4C2×#1176586511yescharacteristic
361C2×C4C2×#1176586511yescharacteristic
362C2×C4#1455617511yescharacteristic
363C2×C4#1455617611yescharacteristic
364C8—41no—
365C8—2841no—
366C8—1422no—
367C8#1455608514yes—
368C22—714no—
369C22—1422no—
370C22—1424no—
371C22—1424no—
372C22—1422no—
373C22C4×D4×D7111yescharacteristic
374C22#1798665011yescharacteristic
375C22#1798665411yescharacteristic
376C22#137×D7112yes—
377C22#1798680312yes—
378C4—44no—
379C4—42no—
380C4—82no—
381C4—2844no—
382C4—2842no—
383C4—5682no—
384C4—1422no—
385C4C2×#1455608511yescharacteristic
386C4#1798647511yescharacteristic
387C4#1798668811yescharacteristic
388C4#1798669011yescharacteristic
389C2—1424no—
390C2#2259120511yescharacteristic
391C2#441×D7111yescharacteristic
392C2#2259718911yescharacteristic
393C2#2260082612yes—
394C2#2260083012yes—
subgroupG/H
centerC23C2×D4×D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC71×#1776C2
socleC23×C71C22×D4
prime radicalC23×D71C2×D4
cosocleC22×C4×C71C24
hypercenter#1776D71
hypocenterC71#22618
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1776

Complements

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

NH ≅ G/Ntype
3. #22589078C2semidirect
4. #22589076C2semidirect
5. #22588911C2semidirect
6. #22588909C2semidirect
9. #22588239C2semidirect
11. C71×#1776C2semidirect
16. #107×D71C4semidirect
207. #1776D71semidirect
209. C71#22618semidirect

Maximal subgroups

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

subgroupindex
C2×#107×D712
#225912132
#225890782
#225890762
#225889112
#225889092
#225884292
#225883622
#225882392
#225881172
C71×#17762
#2261871