Finite Group Databasebeta

← #31908488

Subgroups — #31908488

Order 18176.

472 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 99 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#22593332C211yesmaximal, characteristic
2#22586658C211yesmaximal, characteristic
3#22586339C214yesmaximal, complemented
4C71×#1799C211yesmaximal, characteristic
5C2×#14556128C411yescharacteristic, complemented
6C2×#14556115—21no—
7C2×C22⋊(C71⋊C8)—21no—
8#17985924C412yescomplemented
9#17985924C2212yes—
10#17985904—21no—
11#17985859—21no—
12#17985859—21no—
13C4×C4⋊Dic71C411yescharacteristic, complemented
14#17985725—21no—
15#17985723C2212yes—
16C2×C4×D4×C71C2211yescharacteristic
17C2×#99×C71—21no—
18#380×C71—21no—
19#342×C71C2212yes—
20C22×Dic142—21no—
21C22×C71⋊C8—41no—
22#14556128—41no—
23#14556128—41no—
24#14556115—42no—
25#14556115—42no—
26C2×C4⋊Dic71—21no—
27C2×C4⋊Dic71—21no—
28C2×C4⋊Dic71C2×C412yes—
29C2×C22⋊Dic71—22no—
30C2×C22⋊Dic71—22no—
31C2×C4×Dic71—21no—
32C22⋊(C71⋊C8)—22no—
33C22⋊(C71⋊C8)—41no—
34C71⋊#95—22no—
35C71⋊#95—22no—
36C71⋊#95C2×C412yes—
37C22×D4×C71D411yescharacteristic
38C23×C4×C71D411yescharacteristic
39C23×C4×C71D411yescharacteristic
40C4×D4×C71—41no—
41C4×D4×C71—41no—
42C2×C4⋊C4×C71—41no—
43C2×C4⋊C4×C71D411yescharacteristic
44C2×C22⋊C4×C71—22no—
45C2×C22⋊C4×C71C2×C411yescharacteristic
46C2×C22⋊C4×C71C2311yescharacteristic
47C2×C42×C71C2×C411yescharacteristic
48#99×C71—42no—
49#95×C71—41no—
50C2×Dic142—42no—
51C2×Dic142—42no—
52C2×Dic142—42no—
53C22×Dic71—22no—
54C22×Dic71—22no—
55C2×C71⋊C8—42no—
56C2×C71⋊C8—82no—
57C4⋊Dic71—42no—
58C4⋊Dic71—41no—
59C4⋊Dic71—41no—
60C4⋊Dic71—42no—
61C22⋊Dic71—44no—
62C22⋊Dic71—42no—
63C22⋊Dic71—42no—
64C4×Dic71—42no—
65C24×C71C22⋊C411yescharacteristic
66C24×C71C2×D411yescharacteristic
67C2×D4×C71—41no—
68C2×D4×C71—41no—
69C2×D4×C71D812yes—
70C2×D4×C71SD1612yes—
71C22×C4×C71—21no—
72C22×C4×C71—21no—
73C22×C4×C71—21no—
74C22×C4×C71—21no—
75C22×C4×C71—42no—
76C22×C4×C71—42no—
77C22×C4×C71—42no—
78C22×C4×C71C22⋊C411yescharacteristic
79C22×C4×C71C22⋊C411yescharacteristic
80C22×C4×C71C22⋊C411yescharacteristic
81C22×C4×C71C22×C411yescharacteristic
82C22×C4×C71C2×D411yescharacteristic
83C4⋊C4×C71—41no—
84C4⋊C4×C71—82no—
85C22⋊C4×C71—41no—
86C22⋊C4×C71—41no—
87C22⋊C4×C71—42no—
88C22⋊C4×C71—42no—
89C42×C71—21no—
90C42×C71—21no—
91C23×C71—21no—
92C23×C71—21no—
93C23×C71—21no—
94C23×C71—21no—
95C23×C71—41no—
96C23×C71—41no—
97C23×C71—41no—
98C23×C71—41no—
99C23×C71#10212yes—
100C23×C71C2×C22⋊C411yescharacteristic
101C23×C71C2×D811yescharacteristic
102C23×C71C2×SD1611yescharacteristic
103D4×C71—42no—
104D4×C71—42no—
105C2×C4×C71—21no—
106C2×C4×C71—21no—
107C2×C4×C71—22no—
108C2×C4×C71—21no—
109C2×C4×C71—21no—
110C2×C4×C71—21no—
111C2×C4×C71—42no—
112C2×C4×C71—42no—
113C2×C4×C71—41no—
114C2×C4×C71—41no—
115C2×C4×C71—41no—
116C2×C4×C71—81no—
117C2×C4×C71—81no—
118C2×C4×C71—82no—
119C2×C4×C71—82no—
120C2×C4×C71—82no—
121C2×C4×C71#9911yescharacteristic
122C2×C4×C71#9911yescharacteristic
123C2×C4×C71#10212yes—
124C2×Dic71—42no—
125C2×Dic71—42no—
126C2×Dic71—42no—
127C2×Dic71—42no—
128C2×Dic71—42no—
129C2×Dic71—42no—
130Dic142—84no—
131C71⋊C8—82no—
132C22×C71—21no—
133C22×C71—21no—
134C22×C71—21no—
135C22×C71—22no—
136C22×C71—21no—
137C22×C71—42no—
138C22×C71—41no—
139C22×C71—41no—
140C22×C71—41no—
141C22×C71—41no—
142C22×C71—41no—
143C22×C71—41no—
144C22×C71—81no—
145C22×C71—81no—
146C22×C71C2×#9911yescharacteristic
147C22×C71C2×#10211yescharacteristic
148C22×C71#42111yescharacteristic
149C4×C71—21no—
150C4×C71—22no—
151C4×C71—41no—
152C4×C71—41no—
153C4×C71—41no—
154C4×C71—162no—
155C4×C71#35412yes—
156Dic71—82no—
157Dic71—82no—
158#11878—711nomaximal, Sylow
159C2×C71—21no—
160C2×C71—21no—
161C2×C71—41no—
162C2×C71—42no—
163C2×C71—41no—
164C2×C71#151911yescharacteristic
165C2×C71#212311yescharacteristic
166C2×C71C2×#35411yescharacteristic
167#2547—711no—
168#1799D7111yescharacteristic
169#1459—711no—
170#1344—714no—
171C71#1187811yescharacteristic, complemented, Sylow
172C2×C4⋊Q8—711no—
173C2×#124—1421no—
174C2×C4×D4C2×D7111yescharacteristic
175C2×#99—21no—
176C2×C22⋊C8—1421no—
177#397—712no—
178#397—712no—
179#391—1421no—
180#380—21no—
181#380—1421no—
182#380—1421no—
183C4×C4⋊C4—711no—
184#342C2×D7112yes—
185#324—1421no—
186#323—712no—
187C22×Q8—1421no—
188C22×D4C22⋊D7111yescharacteristic
189C23×C4D28411yescharacteristic
190C23×C4C22⋊D7111yescharacteristic
191C22×C8—2841no—
192C4⋊Q8—2841no—
193C4⋊Q8—2841no—
194#124—2842no—
195#124—2842no—
196C4×D4—41no—
197C4×D4—41no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—712no—
200C2×C4⋊C4—1421no—
201C2×C4⋊C4—1421no—
202C2×C4⋊C4D28411yescharacteristic
203C2×C22⋊C4—22no—
204C2×C22⋊C4—1422no—
205C2×C22⋊C4—1422no—
206C2×C22⋊C4C4×D7111yescharacteristic
207C2×C22⋊C4C22×D7111yescharacteristic
208C2×C42—1421no—
209C2×C42C4×D7111yescharacteristic
210#99—42no—
211C22⋊C8—2841no—
212C22⋊C8—1422no—
213#95—41no—
214#95—712no—
215#95—1422no—
216#95—1422no—
217C24(C2×D71)⋊C411yescharacteristic
218C24C2×C22⋊D7111yescharacteristic
219C2×Q8—2842no—
220C2×Q8—2842no—
221C2×Q8—2842no—
222C2×D4—41no—
223C2×D4—41no—
224C2×D4#1176585012yes—
225C2×D4#1176585212yes—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—1422no—
234C22×C4—1422no—
235C22×C4(C2×D71)⋊C411yescharacteristic
236C22×C4(C2×D71)⋊C411yescharacteristic
237C22×C4(C2×D71)⋊C411yescharacteristic
238C22×C4C2×C4×D7111yescharacteristic
239C22×C4C2×D28411yescharacteristic
240C2×C8—2842no—
241C2×C8—5682no—
242C4⋊C4—41no—
243C4⋊C4—82no—
244C4⋊C4—2842no—
245C4⋊C4—2842no—
246C4⋊C4—2841no—
247C4⋊C4—2841no—
248C22⋊C4—41no—
249C22⋊C4—41no—
250C22⋊C4—42no—
251C22⋊C4—42no—
252C22⋊C4—2844no—
253C22⋊C4—2842no—
254C22⋊C4—2842no—
255C42—21no—
256C42—21no—
257C42—2842no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—41no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23#1455604812yes—
267C23C2×(C2×D71)⋊C411yescharacteristic
268C23C2×#1176585011yescharacteristic
269C23C2×#1176585211yescharacteristic
270Q8—5684no—
271D4—42no—
272D4—42no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4—82no—
287C2×C4—82no—
288C2×C4—82no—
289C2×C4—2842no—
290C2×C4—2842no—
291C2×C4—2842no—
292C2×C4—2842no—
293C2×C4—2842no—
294C2×C4—2842no—
295C2×C4#1455604211yescharacteristic
296C2×C4#1455604211yescharacteristic
297C2×C4#1455604812yes—
298C8—5682no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—22no—
303C22—21no—
304C22—42no—
305C22—41no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22—41no—
310C22—41no—
311C22—81no—
312C22—81no—
313C22C2×#1455604211yescharacteristic
314C22C2×#1455604811yescharacteristic
315C22#1798601711yescharacteristic
316C4—22no—
317C4—21no—
318C4—41no—
319C4—41no—
320C4—41no—
321C4—162no—
322C4—5682no—
323C4—5682no—
324C4#1798579312yes—
325C2—21no—
326C2—21no—
327C2—41no—
328C2—42no—
329C2—41no—
330C2#2258690711yescharacteristic
331C2#2258999311yescharacteristic
332C2C2×#1798579311yescharacteristic
subgroupG/H
centerC22C2×#14556042
commutatorC22×C4×C71C22×C4
FrattiniC2×C22⋊C4C22×D71
FittingC71×#1799C2
socleC22×C71C2×#99
prime radicalC22×D4×C71D4
cosocleC2×C22⋊C4×C71C23
hypercenter#1799D71
hypocenterC71#11878
LCSG ▷ C22×C4×C71 ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1799

Complements

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

NH ≅ G/Ntype
3. #22586339C2semidirect
5. C2×#14556128C4semidirect
8. #17985924C4semidirect
13. C4×C4⋊Dic71C4semidirect
171. C71#11878semidirect

Maximal subgroups

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

subgroupindex
#225933322
#225866582
#225863392
C71×#17992
#1187871