Finite Group Databasebeta

← #31908320

Subgroups — #31908320

Order 18176.

506 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 506 conjugacy classes of subgroups — too many to draw. Only its 91 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22588414C211yesmaximal, characteristic
2C2×#17985723C211yesmaximal, characteristic
3#22586364C212yesmaximal, complemented
4#22586340C212yesmaximal, complemented
5C71×#2482C211yesmaximal, characteristic
6C2×C22⋊(C71⋊C8)—21no—
7#17985855—21no—
8#17985766—21no—
9#17985766C2212yes—
10#17985723—21no—
11#17985723C2212yes—
12C2×#117×C71C411yescharacteristic, complemented
13C22×C22⋊C4×C71C2211yescharacteristic
14#399×C71C411yescharacteristic, complemented
15#399×C71C2211yescharacteristic
16#397×C71C411yescharacteristic, complemented
17#397×C71C2211yescharacteristic
18#388×C71—21no—
19#388×C71—21no—
20#385×C71—21no—
21#381×C71C411yescharacteristic, complemented
22#379×C71—21no—
23C22×C71⋊C8—41no—
24C2×C22⋊Dic71—22no—
25C2×C22⋊Dic71—41no—
26C2×C22⋊Dic71—41no—
27C22⋊(C71⋊C8)—22no—
28C22⋊(C71⋊C8)—41no—
29C25×C71D411yescharacteristic
30C23×C4×C71C2×C411yescharacteristic
31C23×C4×C71D411yescharacteristic
32#117×C71—41no—
33#117×C71—41no—
34C2×C4⋊C4×C71—21no—
35C2×C4⋊C4×C71—21no—
36C2×C4⋊C4×C71C2×C411yescharacteristic
37C2×C4⋊C4×C71C2×C411yescharacteristic
38C2×C22⋊C4×C71—21no—
39C2×C22⋊C4×C71—21no—
40C2×C22⋊C4×C71—21no—
41C2×C22⋊C4×C71—21no—
42C2×C22⋊C4×C71—21no—
43C2×C22⋊C4×C71—21no—
44C2×C22⋊C4×C71—41no—
45C2×C22⋊C4×C71—41no—
46C2×C22⋊C4×C71C2×C411yescharacteristic
47C2×C22⋊C4×C71D411yescharacteristic
48C2×C22⋊C4×C71D411yescharacteristic
49C2×C22⋊C4×C71C2311yescharacteristic
50C2×C42×C71—21no—
51#95×C71—21no—
52#95×C71—21no—
53#95×C71—21no—
54#95×C71C2×C411yescharacteristic
55#95×C71C2×C411yescharacteristic
56C22×Dic71—42no—
57C2×C71⋊C8—42no—
58C2×C71⋊C8—82no—
59C22⋊Dic71—42no—
60C22⋊Dic71—42no—
61C22⋊Dic71—82no—
62C22⋊Dic71—82no—
63C24×C71—21no—
64C24×C71—21no—
65C24×C71—22no—
66C24×C71—41no—
67C24×C71—42no—
68C24×C71—81no—
69C24×C71C22⋊C411yescharacteristic
70C24×C71C22⋊C411yescharacteristic
71C24×C71C2×D411yescharacteristic
72C22×C4×C71—21no—
73C22×C4×C71—21no—
74C22×C4×C71—21no—
75C22×C4×C71—21no—
76C22×C4×C71—21no—
77C22×C4×C71—21no—
78C22×C4×C71—21no—
79C22×C4×C71—21no—
80C22×C4×C71—42no—
81C22×C4×C71—81no—
82C22×C4×C71C22⋊C411yescharacteristic
83C22×C4×C71C22⋊C411yescharacteristic
84C22×C4×C71C22×C411yescharacteristic
85C22×C4×C71C2×D411yescharacteristic
86C4⋊C4×C71—41no—
87C4⋊C4×C71—41no—
88C4⋊C4×C71—81no—
89C4⋊C4×C71—81no—
90C22⋊C4×C71—41no—
91C22⋊C4×C71—41no—
92C22⋊C4×C71—41no—
93C22⋊C4×C71—41no—
94C22⋊C4×C71—42no—
95C22⋊C4×C71—42no—
96C22⋊C4×C71—41no—
97C22⋊C4×C71—41no—
98C22⋊C4×C71—42no—
99C22⋊C4×C71—81no—
100C22⋊C4×C71—81no—
101C42×C71—42no—
102C23×C71—22no—
103C23×C71—21no—
104C23×C71—21no—
105C23×C71—21no—
106C23×C71—21no—
107C23×C71—21no—
108C23×C71—21no—
109C23×C71—21no—
110C23×C71—41no—
111C23×C71—42no—
112C23×C71—42no—
113C23×C71—41no—
114C23×C71—41no—
115C23×C71—41no—
116C23×C71—41no—
117C23×C71—42no—
118C23×C71—42no—
119C23×C71—81no—
120C23×C71—81no—
121C23×C71—84no—
122C23×C71—84no—
123C23×C71#9911yescharacteristic
124C23×C71#9911yescharacteristic
125C23×C71C2×C22⋊C411yescharacteristic
126C23×C71#13512yes—
127C2×C4×C71—21no—
128C2×C4×C71—21no—
129C2×C4×C71—21no—
130C2×C4×C71—21no—
131C2×C4×C71—42no—
132C2×C4×C71—42no—
133C2×C4×C71—42no—
134C2×C4×C71—41no—
135C2×C4×C71—41no—
136C2×C4×C71—41no—
137C2×C4×C71—41no—
138C2×C4×C71—41no—
139C2×C4×C71—41no—
140C2×C4×C71—41no—
141C2×C4×C71—41no—
142C2×C4×C71—41no—
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—81no—
151C2×Dic71—82no—
152C2×Dic71—82no—
153C2×Dic71—82no—
154C71⋊C8—82no—
155C22×C71—21no—
156C22×C71—21no—
157C22×C71—21no—
158C22×C71—21no—
159C22×C71—21no—
160C22×C71—21no—
161C22×C71—22no—
162C22×C71—21no—
163C22×C71—42no—
164C22×C71—42no—
165C22×C71—42no—
166C22×C71—42no—
167C22×C71—41no—
168C22×C71—41no—
169C22×C71—41no—
170C22×C71—41no—
171C22×C71—41no—
172C22×C71—81no—
173C22×C71—84no—
174C22×C71—81no—
175C22×C71—84no—
176C22×C71#35212yes—
177C22×C71C2×#9911yescharacteristic
178C22×C71#41611yescharacteristic
179C22×C71#42111yescharacteristic
180C4×C71—41no—
181C4×C71—41no—
182C4×C71—41no—
183C4×C71—41no—
184C4×C71—81no—
185C4×C71—81no—
186C4×C71—81no—
187C4×C71—81no—
188Dic71—162no—
189#11841—711nomaximal, Sylow
190C2×C71—21no—
191C2×C71—21no—
192C2×C71—22no—
193C2×C71—42no—
194C2×C71—41no—
195C2×C71—81no—
196C2×C71#151611yescharacteristic
197C2×C71C2×#35211yescharacteristic
198C2×C71#212711yescharacteristic
199#2482D7111yescharacteristic
200#1819—711no—
201C2×#323—711no—
202#1352—712no—
203#1344—712no—
204C71#1184111yescharacteristic, complemented, Sylow
205C2×#117Dic7111yescharacteristic, complemented
206C22×C22⋊C4C2×D7111yescharacteristic
207C2×C22⋊C8—1421no—
208#399Dic7111yescharacteristic, complemented
209#399C2×D7111yescharacteristic
210#397Dic7111yescharacteristic, complemented
211#397C2×D7111yescharacteristic
212#388—21no—
213#388—21no—
214#385—21no—
215#381Dic7111yescharacteristic, complemented
216#379—21no—
217#379—1421no—
218#342—712no—
219#342—1421no—
220#323—712no—
221#323—1421no—
222C25C22⋊D7111yescharacteristic
223C23×C4C2×Dic7111yescharacteristic
224C23×C4C22⋊D7111yescharacteristic
225C22×C8—2841no—
226#117—41no—
227#117—41no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4C2×Dic7111yescharacteristic
231C2×C4⋊C4C2×Dic7111yescharacteristic
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—41no—
239C2×C22⋊C4—41no—
240C2×C22⋊C4—2841no—
241C2×C22⋊C4—2841no—
242C2×C22⋊C4—1422no—
243C2×C22⋊C4C2×Dic7111yescharacteristic
244C2×C22⋊C4C22⋊D7111yescharacteristic
245C2×C22⋊C4C22⋊D7111yescharacteristic
246C2×C22⋊C4C22×D7111yescharacteristic
247C2×C42—21no—
248C22⋊C8—2841no—
249C22⋊C8—1422no—
250#95—21no—
251#95—21no—
252#95—21no—
253#95C2×Dic7111yescharacteristic
254#95C2×Dic7111yescharacteristic
255C24—21no—
256C24—21no—
257C24—22no—
258C24—41no—
259C24—42no—
260C24—81no—
261C24C22⋊Dic7111yescharacteristic
262C24C22⋊Dic7111yescharacteristic
263C24C2×C22⋊D7111yescharacteristic
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—42no—
273C22×C4—81no—
274C22×C4—2842no—
275C22×C4C22⋊Dic7111yescharacteristic
276C22×C4C22⋊Dic7111yescharacteristic
277C22×C4C22×Dic7111yescharacteristic
278C22×C4C2×C22⋊D7111yescharacteristic
279C2×C8—2842no—
280C2×C8—5682no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—81no—
284C4⋊C4—81no—
285C22⋊C4—41no—
286C22⋊C4—41no—
287C22⋊C4—41no—
288C22⋊C4—41no—
289C22⋊C4—42no—
290C22⋊C4—42no—
291C22⋊C4—41no—
292C22⋊C4—41no—
293C22⋊C4—42no—
294C22⋊C4—81no—
295C22⋊C4—81no—
296C22⋊C4—2842no—
297C22⋊C4—2842no—
298C22⋊C4—5682no—
299C22⋊C4—5682no—
300C42—42no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—22no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—41no—
310C23—42no—
311C23—42no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—42no—
317C23—42no—
318C23—81no—
319C23—81no—
320C23—84no—
321C23—84no—
322C23#1455604311yescharacteristic
323C23#1455604311yescharacteristic
324C23C2×C22⋊Dic7111yescharacteristic
325C23#1455614812yes—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—81no—
350C2×C4—5682no—
351C2×C4—5682no—
352C2×C4—5682no—
353C8—5682no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—22no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—42no—
363C22—42no—
364C22—42no—
365C22—42no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—81no—
372C22—84no—
373C22—81no—
374C22—84no—
375C22#1798578912yes—
376C22C2×#1455604311yescharacteristic
377C22#1798599511yescharacteristic
378C22#1798601611yescharacteristic
379C4—41no—
380C4—41no—
381C4—41no—
382C4—41no—
383C4—81no—
384C4—81no—
385C4—81no—
386C4—81no—
387C4—11362no—
388C2—21no—
389C2—21no—
390C2—22no—
391C2—42no—
392C2—41no—
393C2—81no—
394C2#2258689611yescharacteristic
395C2C2×#1798578911yescharacteristic
396C2#2259000911yescharacteristic
subgroupG/H
centerC22C2×#14556043
commutatorC22×C4×C71C22×C4
FrattiniC2×C22⋊C4C22×D71
FittingC71×#2482C2
socleC22×C71C2×#99
prime radicalC25×C71D4
cosocleC2×C22⋊C4×C71C23
hypercenter#2482D71
hypocenterC71#11841
LCSG ▷ C22×C4×C71 ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ #2482

Complements

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

NH ≅ G/Ntype
3. #22586364C2semidirect
4. #22586340C2semidirect
12. C2×#117×C71C4semidirect
14. #399×C71C4semidirect
16. #397×C71C4semidirect
21. #381×C71C4semidirect
204. C71#11841semidirect
205. C2×#117Dic71semidirect
208. #399Dic71semidirect
210. #397Dic71semidirect
215. #381Dic71semidirect

Maximal subgroups

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

subgroupindex
#225884142
C2×#179857232
#225863642
#225863402
C71×#24822
#1184171