Finite Group Databasebeta

← #31908340

Subgroups — #31908340

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#22586361C212yesmaximal, complemented
4#22586340C212yesmaximal, complemented
5C71×#2561C211yesmaximal, characteristic
6C2×C22⋊(C71⋊C8)—21no—
7#17985855—21no—
8#17985766—21no—
9#17985766C2212yes—
10#17985723—21no—
11#17985723C2212yes—
12C2×C22⋊Q8×C71C411yescharacteristic, complemented
13C22×C22⋊C4×C71C2211yescharacteristic
14#397×C71C411yescharacteristic, complemented
15#397×C71C2211yescharacteristic
16#393×C71C411yescharacteristic, complemented
17#393×C71C2211yescharacteristic
18#388×C71—21no—
19#388×C71—21no—
20#381×C71—21no—
21#379×C71—21no—
22C4×C22⋊C4×C71C411yescharacteristic, complemented
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
30C22×Q8×C71C2×C411yescharacteristic
31C23×C4×C71C2×C411yescharacteristic
32C23×C4×C71D411yescharacteristic
33C22⋊Q8×C71—41no—
34C22⋊Q8×C71—41no—
35C2×C4⋊C4×C71—21no—
36C2×C4⋊C4×C71C2×C411yescharacteristic
37C2×C22⋊C4×C71—21no—
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—41no—
44C2×C22⋊C4×C71—41no—
45C2×C22⋊C4×C71C2×C411yescharacteristic
46C2×C22⋊C4×C71D411yescharacteristic
47C2×C22⋊C4×C71D411yescharacteristic
48C2×C22⋊C4×C71C2311yescharacteristic
49C2×C42×C71—21no—
50#95×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
72C2×Q8×C71—41no—
73C2×Q8×C71—81no—
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—21no—
81C22×C4×C71—21no—
82C22×C4×C71—42no—
83C22×C4×C71—81no—
84C22×C4×C71C22⋊C411yescharacteristic
85C22×C4×C71C22⋊C411yescharacteristic
86C22×C4×C71C22×C411yescharacteristic
87C22×C4×C71C2×D411yescharacteristic
88C4⋊C4×C71—41no—
89C4⋊C4×C71—42no—
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—81no—
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—
127Q8×C71—82no—
128C2×C4×C71—21no—
129C2×C4×C71—21no—
130C2×C4×C71—21no—
131C2×C4×C71—21no—
132C2×C4×C71—42no—
133C2×C4×C71—41no—
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—81no—
150C2×C4×C71—81no—
151C2×C4×C71—81no—
152C2×Dic71—82no—
153C2×Dic71—82no—
154C2×Dic71—82no—
155C71⋊C8—82no—
156C22×C71—21no—
157C22×C71—21no—
158C22×C71—21no—
159C22×C71—21no—
160C22×C71—21no—
161C22×C71—21no—
162C22×C71—22no—
163C22×C71—21no—
164C22×C71—42no—
165C22×C71—42no—
166C22×C71—42no—
167C22×C71—42no—
168C22×C71—41no—
169C22×C71—41no—
170C22×C71—41no—
171C22×C71—41no—
172C22×C71—41no—
173C22×C71—81no—
174C22×C71—84no—
175C22×C71—81no—
176C22×C71—84no—
177C22×C71#35412yes—
178C22×C71C2×#9911yescharacteristic
179C22×C71#41611yescharacteristic
180C22×C71#42111yescharacteristic
181C4×C71—41no—
182C4×C71—41no—
183C4×C71—41no—
184C4×C71—41no—
185C4×C71—81no—
186C4×C71—81no—
187C4×C71—81no—
188C4×C71—81no—
189Dic71—162no—
190#11845—711nomaximal, Sylow
191C2×C71—21no—
192C2×C71—21no—
193C2×C71—22no—
194C2×C71—42no—
195C2×C71—41no—
196C2×C71—81no—
197C2×C71#151611yescharacteristic
198C2×C71#212111yescharacteristic
199C2×C71C2×#35411yescharacteristic
200#2561D7111yescharacteristic
201#1819—711no—
202C2×#323—711no—
203#1351—712no—
204#1344—712no—
205C71#1184511yescharacteristic, complemented, Sylow
206C2×C22⋊Q8Dic7111yescharacteristic, complemented
207C22×C22⋊C4C2×D7111yescharacteristic
208C2×C22⋊C8—1421no—
209#397Dic7111yescharacteristic, complemented
210#397C2×D7111yescharacteristic
211#393Dic7111yescharacteristic, complemented
212#393C2×D7111yescharacteristic
213#388—21no—
214#388—21no—
215#381—21no—
216#379—21no—
217#379—1421no—
218C4×C22⋊C4Dic7111yescharacteristic, complemented
219#342—712no—
220#342—1421no—
221#323—712no—
222#323—1421no—
223C25C22⋊D7111yescharacteristic
224C22×Q8C2×Dic7111yescharacteristic
225C23×C4C2×Dic7111yescharacteristic
226C23×C4C22⋊D7111yescharacteristic
227C22×C8—2841no—
228C22⋊Q8—41no—
229C22⋊Q8—41no—
230C2×C4⋊C4—21no—
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#95—21no—
254#95C2×Dic7111yescharacteristic
255#95C2×Dic7111yescharacteristic
256C24—21no—
257C24—21no—
258C24—22no—
259C24—41no—
260C24—42no—
261C24—81no—
262C24C22⋊Dic7111yescharacteristic
263C24C22⋊Dic7111yescharacteristic
264C24C2×C22⋊D7111yescharacteristic
265C2×Q8—41no—
266C2×Q8—81no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—42no—
276C22×C4—81no—
277C22×C4—2842no—
278C22×C4C22⋊Dic7111yescharacteristic
279C22×C4C22⋊Dic7111yescharacteristic
280C22×C4C22×Dic7111yescharacteristic
281C22×C4C2×C22⋊D7111yescharacteristic
282C2×C8—2842no—
283C2×C8—5682no—
284C4⋊C4—41no—
285C4⋊C4—42no—
286C22⋊C4—41no—
287C22⋊C4—41no—
288C22⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—42no—
291C22⋊C4—42no—
292C22⋊C4—41no—
293C22⋊C4—41no—
294C22⋊C4—42no—
295C22⋊C4—81no—
296C22⋊C4—81no—
297C22⋊C4—2842no—
298C22⋊C4—2842no—
299C22⋊C4—5682no—
300C22⋊C4—5682no—
301C42—81no—
302C23—22no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—41no—
311C23—42no—
312C23—42no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—42no—
318C23—42no—
319C23—81no—
320C23—81no—
321C23—84no—
322C23—84no—
323C23#1455604311yescharacteristic
324C23#1455604311yescharacteristic
325C23C2×C22⋊Dic7111yescharacteristic
326C23#1455614812yes—
327Q8—82no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
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—81no—
351C2×C4—81no—
352C2×C4—5682no—
353C2×C4—5682no—
354C2×C4—5682no—
355C8—5682no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—22no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—42no—
365C22—42no—
366C22—42no—
367C22—42no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—81no—
374C22—84no—
375C22—81no—
376C22—84no—
377C22#1798579412yes—
378C22C2×#1455604311yescharacteristic
379C22#1798599511yescharacteristic
380C22#1798601611yescharacteristic
381C4—41no—
382C4—41no—
383C4—41no—
384C4—41no—
385C4—81no—
386C4—81no—
387C4—81no—
388C4—81no—
389C4—11362no—
390C2—21no—
391C2—21no—
392C2—22no—
393C2—42no—
394C2—41no—
395C2—81no—
396C2#2258689611yescharacteristic
397C2#2258998611yescharacteristic
398C2C2×#1798579411yescharacteristic
subgroupG/H
centerC22C2×#14556043
commutatorC22×C4×C71C22×C4
FrattiniC2×C22⋊C4C22×D71
FittingC71×#2561C2
socleC22×C71C2×#99
prime radicalC25×C71D4
cosocleC2×C22⋊C4×C71C23
hypercenter#2561D71
hypocenterC71#11845
LCSG ▷ C22×C4×C71 ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ #2561

Complements

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

NH ≅ G/Ntype
3. #22586361C2semidirect
4. #22586340C2semidirect
12. C2×C22⋊Q8×C71C4semidirect
14. #397×C71C4semidirect
16. #393×C71C4semidirect
22. C4×C22⋊C4×C71C4semidirect
205. C71#11845semidirect
206. C2×C22⋊Q8Dic71semidirect
209. #397Dic71semidirect
211. #393Dic71semidirect
218. C4×C22⋊C4Dic71semidirect

Maximal subgroups

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

subgroupindex
#225884142
C2×#179857232
#225863612
#225863402
C71×#25612
#1184571