Finite Group Databasebeta

← #31907801

Subgroups — #31907801

Order 18176.

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

normal subgroups only

This group has 500 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
1C4×#14556043C211yesmaximal, characteristic, complemented
2#22586658C211yesmaximal, characteristic, complemented
3#22586361C212yesmaximal, complemented
4#22586337C212yesmaximal, complemented
5C71×#2631C211yesmaximal, characteristic
6C2×#14556043—21no—
7C2×C22⋊(C71⋊C8)—21no—
8C4×C22⋊Dic71—21no—
9#17985766C2212yes—
10#17985725—21no—
11#17985723C2212yes—
12C2×#124×C71C411yescharacteristic, complemented
13C2×C4×D4×C71C2211yescharacteristic
14#397×C71—21no—
15#393×C71C411yescharacteristic, complemented
16#393×C71C2211yescharacteristic
17#392×C71—21no—
18#392×C71C411yescharacteristic, complemented
19#392×C71C2211yescharacteristic
20#386×C71—21no—
21#386×C71—21no—
22#383×C71C411yescharacteristic, complemented
23C22×C71⋊C8—41no—
24C2×C22⋊Dic71—22no—
25C2×C4×Dic71—41no—
26#14556043—24no—
27C22⋊(C71⋊C8)—22no—
28C22⋊(C71⋊C8)—41no—
29C71⋊#95—41no—
30C22×Q8×C71C2×C411yescharacteristic
31C22×D4×C71—21no—
32C22×D4×C71—21no—
33C22×D4×C71C2×C411yescharacteristic
34C22×D4×C71D411yescharacteristic
35C23×C4×C71D411yescharacteristic
36C23×C4×C71D411yescharacteristic
37#124×C71—41no—
38#124×C71—41no—
39C4×D4×C71—41no—
40C4×D4×C71—41no—
41C2×C4⋊C4×C71D411yescharacteristic
42C2×C22⋊C4×C71—21no—
43C2×C22⋊C4×C71—21no—
44C2×C22⋊C4×C71—21no—
45C2×C22⋊C4×C71—21no—
46C2×C22⋊C4×C71—21no—
47C2×C22⋊C4×C71—21no—
48C2×C22⋊C4×C71C2×C411yescharacteristic
49C2×C22⋊C4×C71C2311yescharacteristic
50C2×C42×C71C2×C411yescharacteristic
51#95×C71—21no—
52#95×C71—21no—
53#95×C71—21no—
54#95×C71—21no—
55#95×C71C2×C411yescharacteristic
56#95×C71C2×C411yescharacteristic
57C22×Dic71—42no—
58C2×C71⋊C8—42no—
59C2×C71⋊C8—82no—
60C22⋊Dic71—24no—
61C22⋊Dic71—42no—
62C4×Dic71—82no—
63C24×C71—21no—
64C24×C71C22⋊C411yescharacteristic
65C24×C71C2×D411yescharacteristic
66C2×Q8×C71—41no—
67C2×Q8×C71—81no—
68C2×D4×C71—22no—
69C2×D4×C71—22no—
70C2×D4×C71—41no—
71C2×D4×C71—41no—
72C2×D4×C71—42no—
73C2×D4×C71—42no—
74C2×D4×C71—81no—
75C2×D4×C71—81no—
76C2×D4×C71—81no—
77C2×D4×C71—81no—
78C2×D4×C71—81no—
79C22×C4×C71—21no—
80C22×C4×C71—21no—
81C22×C4×C71—21no—
82C22×C4×C71—21no—
83C22×C4×C71—21no—
84C22×C4×C71—21no—
85C22×C4×C71—21no—
86C22×C4×C71—81no—
87C22×C4×C71—81no—
88C22×C4×C71C22⋊C411yescharacteristic
89C22×C4×C71C22⋊C411yescharacteristic
90C22×C4×C71C22⋊C411yescharacteristic
91C22×C4×C71C22×C411yescharacteristic
92C22×C4×C71C2×D411yescharacteristic
93C4⋊C4×C71—41no—
94C22⋊C4×C71—41no—
95C22⋊C4×C71—41no—
96C22⋊C4×C71—42no—
97C22⋊C4×C71—41no—
98C22⋊C4×C71—41no—
99C22⋊C4×C71—81no—
100C22⋊C4×C71—81no—
101C22⋊C4×C71—81no—
102C22⋊C4×C71—81no—
103C42×C71—21no—
104C42×C71—21no—
105C23×C71—21no—
106C23×C71—21no—
107C23×C71—21no—
108C23×C71—21no—
109C23×C71—41no—
110C23×C71—41no—
111C23×C71—41no—
112C23×C71—41no—
113C23×C71—41no—
114C23×C71—41no—
115C23×C71—41no—
116C23×C71—41no—
117C23×C71—41no—
118C23×C71—41no—
119C23×C71—41no—
120C23×C71#10412yes—
121C23×C71C2×C22⋊C411yescharacteristic
122C23×C71#13512yes—
123Q8×C71—82no—
124D4×C71—44no—
125D4×C71—82no—
126D4×C71—84no—
127D4×C71—84no—
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—41no—
135C2×C4×C71—42no—
136C2×C4×C71—42no—
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#9911yescharacteristic
152C2×C4×C71#9911yescharacteristic
153C2×Dic71—44no—
154C2×Dic71—82no—
155C2×Dic71—82no—
156C71⋊C8—82no—
157C22×C71—21no—
158C22×C71—21no—
159C22×C71—22no—
160C22×C71—22no—
161C22×C71—42no—
162C22×C71—42no—
163C22×C71—41no—
164C22×C71—41no—
165C22×C71—41no—
166C22×C71—41no—
167C22×C71—81no—
168C22×C71—81no—
169C22×C71—81no—
170C22×C71—81no—
171C22×C71—81no—
172C22×C71—81no—
173C22×C71—81no—
174C22×C71—81no—
175C22×C71—81no—
176C22×C71C2×#9911yescharacteristic
177C22×C71#41611yescharacteristic
178C22×C71C2×#10411yescharacteristic
179C4×C71—21no—
180C4×C71—21no—
181C4×C71—41no—
182C4×C71—41no—
183C4×C71—41no—
184C4×C71—41no—
185C4×C71—81no—
186C4×C71—81no—
187C4×C71—81no—
188Dic71—84no—
189#11730—711nomaximal, Sylow
190C2×C71—21no—
191C2×C71—21no—
192C2×C71—42no—
193C2×C71—42no—
194C2×C71—161no—
195C2×C71#150111yescharacteristic
196C2×C71#212211yescharacteristic
197C2×C71#212711yescharacteristic
198#2631D7111yescharacteristic
199C4×#99—711no—
200#1459—711no—
201#1351—712no—
202#1343—712no—
203C71#1173011yescharacteristic, complemented, Sylow
204C2×#124Dic7111yescharacteristic, complemented
205C2×C4×D4C2×D7111yescharacteristic
206C2×#99—1421no—
207C2×C22⋊C8—1421no—
208#397—21no—
209#393Dic7111yescharacteristic, complemented
210#393C2×D7111yescharacteristic
211#392—21no—
212#392Dic7111yescharacteristic, complemented
213#392C2×D7111yescharacteristic
214#386—21no—
215#386—21no—
216#383Dic7111yescharacteristic, complemented
217C4×C22⋊C4—1421no—
218#342—712no—
219#324—1421no—
220#323—712no—
221C22×Q8C2×Dic7111yescharacteristic
222C22×D4—21no—
223C22×D4—21no—
224C22×D4C2×Dic7111yescharacteristic
225C22×D4C22⋊D7111yescharacteristic
226C23×C4C22⋊D7111yescharacteristic
227C23×C4C22⋊D7111yescharacteristic
228C22×C8—2841no—
229#124—41no—
230#124—41no—
231C4×D4—41no—
232C4×D4—41no—
233C2×C4⋊C4C22⋊D7111yescharacteristic
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—1422no—
241C2×C22⋊C4C2×Dic7111yescharacteristic
242C2×C22⋊C4C22×D7111yescharacteristic
243C2×C42—2841no—
244C2×C42C2×Dic7111yescharacteristic
245#99—1424no—
246C22⋊C8—2841no—
247C22⋊C8—1422no—
248#95—21no—
249#95—21no—
250#95—21no—
251#95—21no—
252#95—2841no—
253#95C2×Dic7111yescharacteristic
254#95C2×Dic7111yescharacteristic
255C24—21no—
256C24C22⋊Dic7111yescharacteristic
257C24C2×C22⋊D7111yescharacteristic
258C2×Q8—41no—
259C2×Q8—81no—
260C2×D4—22no—
261C2×D4—22no—
262C2×D4—41no—
263C2×D4—41no—
264C2×D4—42no—
265C2×D4—42no—
266C2×D4—81no—
267C2×D4—81no—
268C2×D4—81no—
269C2×D4—81no—
270C2×D4—81no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—81no—
279C22×C4—81no—
280C22×C4—2842no—
281C22×C4C22⋊Dic7111yescharacteristic
282C22×C4C22⋊Dic7111yescharacteristic
283C22×C4C22⋊Dic7111yescharacteristic
284C22×C4C22×Dic7111yescharacteristic
285C22×C4C2×C22⋊D7111yescharacteristic
286C2×C8—2842no—
287C2×C8—5682no—
288C4⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—41no—
291C22⋊C4—42no—
292C22⋊C4—41no—
293C22⋊C4—41no—
294C22⋊C4—81no—
295C22⋊C4—81no—
296C22⋊C4—81no—
297C22⋊C4—81no—
298C22⋊C4—2842no—
299C22⋊C4—1424no—
300C42—21no—
301C42—21no—
302C42—5682no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—41no—
308C23—41no—
309C23—41no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23#1455605512yes—
319C23C2×C22⋊Dic7111yescharacteristic
320C23#1455614812yes—
321Q8—82no—
322D4—44no—
323D4—82no—
324D4—84no—
325D4—84no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—42no—
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—81no—
348C2×C4—81no—
349C2×C4—2844no—
350C2×C4—5682no—
351C2×C4—5682no—
352C2×C4#1455604311yescharacteristic
353C2×C4#1455604311yescharacteristic
354C8—5682no—
355C22—21no—
356C22—21no—
357C22—22no—
358C22—22no—
359C22—42no—
360C22—42no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22—81no—
369C22—81no—
370C22—81no—
371C22—81no—
372C22—81no—
373C22—81no—
374C22C2×#1455604311yescharacteristic
375C22#1798599511yescharacteristic
376C22C2×#1455605511yescharacteristic
377C4—21no—
378C4—21no—
379C4—41no—
380C4—41no—
381C4—41no—
382C4—41no—
383C4—81no—
384C4—81no—
385C4—81no—
386C4—5684no—
387C2—21no—
388C2—21no—
389C2—42no—
390C2—42no—
391C2—161no—
392C2#2258683111yescharacteristic
393C2#2258999011yescharacteristic
394C2#2259000911yescharacteristic
subgroupG/H
centerC22C2×#14556043
commutatorC22×C4×C71C22×C4
FrattiniC2×C22⋊C4C22×D71
Fitting, prime radicalC71×#2631C2
socleC22×C71C2×#99
cosocleC2×C22⋊C4×C71C23
hypercenter#2631D71
hypocenterC71#11730
LCSG ▷ C22×C4×C71 ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2631

Complements

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

NH ≅ G/Ntype
1. C4×#14556043C2semidirect
2. #22586658C2semidirect
3. #22586361C2semidirect
4. #22586337C2semidirect
12. C2×#124×C71C4semidirect
15. #393×C71C4semidirect
18. #392×C71C4semidirect
22. #383×C71C4semidirect
203. C71#11730semidirect
204. C2×#124Dic71semidirect
209. #393Dic71semidirect
212. #392Dic71semidirect
216. #383Dic71semidirect

Maximal subgroups

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

subgroupindex
C4×#145560432
#225866582
#225863612
#225863372
C71×#26312
#1173071