Finite Group Databasebeta

← #31894888

Subgroups — #31894888

Order 18176.

726 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 726 conjugacy classes of subgroups — too many to draw. Only its 115 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#22590693C211yesmaximal, characteristic
2#22586860C212yesmaximal, complemented
3#22586667C211yesmaximal, characteristic
4#22586226C212yesmaximal, complemented
5C71×#2561C211yesmaximal, characteristic
6C22×C22⋊Dic71—21no—
7#17985963C2212yes—
8C22⋊C4×Dic71—24no—
9C4×C22⋊Dic71—21no—
10C4×C22⋊Dic71C2212yes—
11#17985736—24no—
12#17985723—22no—
13C2×C22⋊Q8×C71C2211yescharacteristic
14C22×C22⋊C4×C71C2211yescharacteristic
15#397×C71C412yescomplemented
16#393×C71C412yescomplemented
17#388×C71—22no—
18#381×C71—21no—
19#379×C71—21no—
20C4×C22⋊C4×C71C2211yescharacteristic
21C23×Dic71—22no—
22C2×#11765849—22no—
23C2×C22⋊Dic71—22no—
24C2×C22⋊Dic71—28no—
25C2×C22⋊Dic71—22no—
26C2×C4×Dic71—22no—
27C2×C4×Dic71—22no—
28C22⋊(C71⋊C8)—22no—
29C22⋊(C71⋊C8)—22no—
30C71⋊#95—22no—
31C71⋊#95D414yes—
32C25×C71D411yescharacteristic
33C22×Q8×C71C2×C411yescharacteristic, complemented
34C23×C4×C71D411yescharacteristic
35C23×C4×C71C2311yescharacteristic
36C22⋊Q8×C71—22no—
37C22⋊Q8×C71—22no—
38C2×C4⋊C4×C71—21no—
39C2×C4⋊C4×C71C2×C411yescharacteristic, complemented
40C2×C22⋊C4×C71—22no—
41C2×C22⋊C4×C71—22no—
42C2×C22⋊C4×C71—22no—
43C2×C22⋊C4×C71—21no—
44C2×C22⋊C4×C71—21no—
45C2×C22⋊C4×C71—22no—
46C2×C22⋊C4×C71C2×C412yes—
47C2×C22⋊C4×C71D412yes—
48C2×C42×C71—21no—
49#95×C71—22no—
50#95×C71—22no—
51#95×C71C2×C412yescomplemented
52C22×Dic71—24no—
53C22×Dic71—28no—
54C22×Dic71—82no—
55C22×Dic71—82no—
56#11765849—82no—
57C2×C71⋊C8—42no—
58C22⋊Dic71—28no—
59C22⋊Dic71—28no—
60C4×Dic71—82no—
61C4×Dic71—82no—
62C24×C71—21no—
63C24×C71—24no—
64C24×C71—21no—
65C24×C71—42no—
66C24×C71—81no—
67C24×C71C22⋊C412yes—
68C24×C71C2×D411yescharacteristic
69C2×Q8×C71—22no—
70C2×Q8×C71—81no—
71C22×C4×C71—21no—
72C22×C4×C71—21no—
73C22×C4×C71—22no—
74C22×C4×C71—21no—
75C22×C4×C71—21no—
76C22×C4×C71—42no—
77C22×C4×C71—81no—
78C22×C4×C71C22⋊C412yes—
79C22×C4×C71C22×C411yescharacteristic
80C22×C4×C71C2×D412yes—
81C22×C4×C71C2×D411yescharacteristic
82C22×C4×C71#4112yes—
83C4⋊C4×C71—22no—
84C4⋊C4×C71—42no—
85C22⋊C4×C71—24no—
86C22⋊C4×C71—24no—
87C22⋊C4×C71—42no—
88C22⋊C4×C71—44no—
89C22⋊C4×C71—41no—
90C22⋊C4×C71—41no—
91C22⋊C4×C71—82no—
92C42×C71—81no—
93C23×C71—22no—
94C23×C71—22no—
95C23×C71—24no—
96C23×C71—22no—
97C23×C71—22no—
98C23×C71—22no—
99C23×C71—22no—
100C23×C71—41no—
101C23×C71—42no—
102C23×C71—42no—
103C23×C71—42no—
104C23×C71—42no—
105C23×C71—81no—
106C23×C71—81no—
107C23×C71—88no—
108C23×C71#10414yes—
109C23×C71C2×C22⋊C411yescharacteristic
110C23×C71#12011yescharacteristic
111C23×C71#12311yescharacteristic
112Q8×C71—82no—
113C2×C4×C71—22no—
114C2×C4×C71—41no—
115C2×C4×C71—41no—
116C2×C4×C71—42no—
117C2×C4×C71—42no—
118C2×C4×C71—42no—
119C2×C4×C71—42no—
120C2×C4×C71—42no—
121C2×C4×C71—42no—
122C2×C4×C71—42no—
123C2×C4×C71—41no—
124C2×C4×C71—41no—
125C2×C4×C71—42no—
126C2×C4×C71—42no—
127C2×C4×C71—81no—
128C2×C4×C71C4×D412yes—
129C2×C4×C71#12112yes—
130C2×Dic71—28no—
131C2×Dic71—44no—
132C2×Dic71—44no—
133C2×Dic71—88no—
134C71⋊C8—82no—
135C22×C71—21no—
136C22×C71—21no—
137C22×C71—21no—
138C22×C71—21no—
139C22×C71—21no—
140C22×C71—24no—
141C22×C71—21no—
142C22×C71—21no—
143C22×C71—44no—
144C22×C71—44no—
145C22×C71—44no—
146C22×C71—44no—
147C22×C71—42no—
148C22×C71—42no—
149C22×C71—82no—
150C22×C71—84no—
151C22×C71#38611yescharacteristic
152C22×C71C2×#10412yes—
153C22×C71#45512yes—
154C4×C71—22no—
155C4×C71—22no—
156C4×C71—42no—
157C4×C71—82no—
158C4×C71—81no—
159C4×C71—81no—
160Dic71—48no—
161#9407—711nomaximal, Sylow
162C2×C71—22no—
163C2×C71—22no—
164C2×C71—41no—
165C2×C71—44no—
166C2×C71#188511yescharacteristic
167C2×C71#189611yescharacteristic
168C2×C71#191311yescharacteristic
169#2561D7111yescharacteristic
170#2273—711no—
171#1508—712no—
172#1462—711no—
173#1297—712no—
174C71#940711yescharacteristic, complemented, Sylow
175C2×C22⋊Q8C2×D7111yescharacteristic
176C22×C22⋊C4—1421no—
177C22×C22⋊C4C2×D7111yescharacteristic
178#407—712no—
179#397Dic7112yescomplemented
180#393Dic7112yescomplemented
181#388—22no—
182#381—21no—
183#379—21no—
184C4×C22⋊C4—712no—
185C4×C22⋊C4—1424no—
186C4×C22⋊C4—1421no—
187C4×C22⋊C4C2×D7111yescharacteristic
188#328—1424no—
189#323—1422no—
190C25C22⋊D7111yescharacteristic
191C22×Q8C2×Dic7111yescharacteristic, complemented
192C23×C4—1422no—
193C23×C4C22⋊D7111yescharacteristic
194C23×C4C22×D7111yescharacteristic
195C2×M16—1422no—
196C22⋊Q8—22no—
197C22⋊Q8—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4C2×Dic7111yescharacteristic, complemented
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—1422no—
207C2×C22⋊C4—1428no—
208C2×C22⋊C4—1422no—
209C2×C22⋊C4C2×Dic7112yes—
210C2×C22⋊C4C22⋊D7112yes—
211C2×C42—21no—
212C2×C42—1422no—
213C2×C42—1422no—
214C22⋊C8—1422no—
215C22⋊C8—1422no—
216#95—22no—
217#95—22no—
218#95—714no—
219#95—1422no—
220#95C2×Dic7112yescomplemented
221C24—21no—
222C24—24no—
223C24—21no—
224C24—42no—
225C24—81no—
226C24C22⋊Dic7112yes—
227C24C2×C22⋊D7111yescharacteristic
228C2×Q8—22no—
229C2×Q8—81no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—22no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—42no—
236C22×C4—81no—
237C22×C4—5682no—
238C22×C4—5682no—
239C22×C4—1424no—
240C22×C4—1428no—
241C22×C4C22⋊Dic7112yes—
242C22×C4C22×Dic7111yescharacteristic
243C22×C4D4×D7112yes—
244C22×C4C2×C22⋊D7111yescharacteristic
245C22×C4#1176586412yes—
246M16—5682no—
247C2×C8—2842no—
248C4⋊C4—22no—
249C4⋊C4—42no—
250C22⋊C4—24no—
251C22⋊C4—24no—
252C22⋊C4—42no—
253C22⋊C4—44no—
254C22⋊C4—41no—
255C22⋊C4—41no—
256C22⋊C4—82no—
257C22⋊C4—1428no—
258C22⋊C4—1428no—
259C42—81no—
260C42—5682no—
261C42—5682no—
262C23—22no—
263C23—22no—
264C23—24no—
265C23—22no—
266C23—22no—
267C23—22no—
268C23—22no—
269C23—41no—
270C23—42no—
271C23—42no—
272C23—42no—
273C23—42no—
274C23—81no—
275C23—81no—
276C23—88no—
277C23#1455605514yes—
278C23C2×C22⋊Dic7111yescharacteristic
279C23#1455609611yescharacteristic
280C23#1455610911yescharacteristic
281Q8—82no—
282C2×C4—22no—
283C2×C4—41no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—81no—
297C2×C4—2844no—
298C2×C4—2844no—
299C2×C4—5688no—
300C2×C4—1428no—
301C2×C4D4×Dic7112yes—
302C2×C4#1455610012yes—
303C8—5682no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—24no—
310C22—21no—
311C22—21no—
312C22—44no—
313C22—44no—
314C22—44no—
315C22—44no—
316C22—42no—
317C22—42no—
318C22—82no—
319C22—84no—
320C22#1798587911yescharacteristic
321C22C2×#1455605512yes—
322C22#1798617612yes—
323C4—22no—
324C4—22no—
325C4—42no—
326C4—82no—
327C4—81no—
328C4—81no—
329C4—2848no—
330C2—22no—
331C2—22no—
332C2—41no—
333C2—44no—
334C2#2258872911yescharacteristic
335C2#2258878411yescharacteristic
336C2#2258888311yescharacteristic
subgroupG/H
centerC22#17985879
commutatorC22×C4×C71C22×C4
FrattiniC23×C4C22×D71
FittingC71×#2561C2
socleC22×C71#386
prime radicalC25×C71D4
cosocleC23×C4×C71C23
hypercenter#2561D71
hypocenterC71#9407
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2561

Complements

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

NH ≅ G/Ntype
2. #22586860C2semidirect
4. #22586226C2semidirect
15. #397×C71C4semidirect
16. #393×C71C4semidirect
33. C22×Q8×C71C2×C4semidirect
39. C2×C4⋊C4×C71C2×C4semidirect
51. #95×C71C2×C4semidirect
174. C71#9407semidirect
179. #397Dic71semidirect
180. #393Dic71semidirect
191. C22×Q8C2×Dic71semidirect
199. C2×C4⋊C4C2×Dic71semidirect
220. #95C2×Dic71semidirect

Maximal subgroups

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

subgroupindex
#225906932
#225868602
#225866672
#225862262
C71×#25612
#940771