Finite Group Databasebeta

← #31908325

Subgroups — #31908325

Order 18176.

390 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 87 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#22588417C211yesmaximal, characteristic
2#22586654C211yesmaximal, characteristic
3#22586361C212yesmaximal
4#22586340C212yesmaximal
5C71×#2599C211yesmaximal, characteristic
6#17986052—21no—
7#17985862—21no—
8#17985766—21no—
9#17985766C2212yes—
10#17985725—21no—
11#17985723C2212yes—
12C2×C4×Q8×C71C411yescharacteristic, complemented
13C2×#117×C71C2211yescharacteristic
14#398×C71C411yescharacteristic, complemented
15#397×C71C2211yescharacteristic
16#395×C71C411yescharacteristic, complemented
17#393×C71C2211yescharacteristic
18#388×C71—21no—
19#387×C71—21no—
20#387×C71—21no—
21#385×C71—21no—
22#383×C71C411yescharacteristic, complemented
23C2×C22⋊Dic71—22no—
24C4⋊(C71⋊C8)—41no—
25C22⋊(C71⋊C8)—22no—
26C4×C71⋊C8—41no—
27C71⋊#95—41no—
28C71⋊#95—41no—
29C22×Q8×C71C2×C411yescharacteristic
30C23×C4×C71D411yescharacteristic
31C4×Q8×C71—41no—
32C4×Q8×C71—41no—
33#117×C71—41no—
34#117×C71—41no—
35C2×C4⋊C4×C71—21no—
36C2×C4⋊C4×C71—21no—
37C2×C4⋊C4×C71—21no—
38C2×C4⋊C4×C71C2×C411yescharacteristic
39C2×C4⋊C4×C71C2×C411yescharacteristic
40C2×C4⋊C4×C71D411yescharacteristic
41C2×C22⋊C4×C71—21no—
42C2×C22⋊C4×C71—21no—
43C2×C22⋊C4×C71D411yescharacteristic
44C2×C22⋊C4×C71C2311yescharacteristic
45C2×C42×C71—21no—
46C2×C42×C71C2×C411yescharacteristic
47C2×C42×C71D411yescharacteristic
48#95×C71—21no—
49#95×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—
58C22⋊Dic71—42no—
59C22⋊Dic71—42no—
60C24×C71C2×D411yescharacteristic
61C2×Q8×C71—42no—
62C2×Q8×C71—41no—
63C22×C4×C71—21no—
64C22×C4×C71—22no—
65C22×C4×C71—21no—
66C22×C4×C71—21no—
67C22×C4×C71—21no—
68C22×C4×C71—21no—
69C22×C4×C71—21no—
70C22×C4×C71—41no—
71C22×C4×C71C22⋊C411yescharacteristic
72C22×C4×C71C22⋊C411yescharacteristic
73C22×C4×C71C22⋊C411yescharacteristic
74C22×C4×C71C22⋊C411yescharacteristic
75C22×C4×C71C22×C411yescharacteristic
76C22×C4×C71C2×D411yescharacteristic
77C4⋊C4×C71—41no—
78C4⋊C4×C71—41no—
79C4⋊C4×C71—41no—
80C4⋊C4×C71—42no—
81C4⋊C4×C71—42no—
82C4⋊C4×C71—41no—
83C4⋊C4×C71—41no—
84C22⋊C4×C71—41no—
85C22⋊C4×C71—41no—
86C22⋊C4×C71—42no—
87C22⋊C4×C71—42no—
88C42×C71—21no—
89C42×C71—21no—
90C42×C71—21no—
91C42×C71—21no—
92C42×C71—42no—
93C23×C71—21no—
94C23×C71—21no—
95C23×C71—41no—
96C23×C71—41no—
97C23×C71C2×C22⋊C411yescharacteristic
98C23×C71#13512yes—
99Q8×C71—42no—
100Q8×C71—42no—
101C2×C4×C71—21no—
102C2×C4×C71—21no—
103C2×C4×C71—22no—
104C2×C4×C71—21no—
105C2×C4×C71—21no—
106C2×C4×C71—21no—
107C2×C4×C71—21no—
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—41no—
117C2×C4×C71—41no—
118C2×C4×C71—41no—
119C2×C4×C71—41no—
120C2×C4×C71—41no—
121C2×C4×C71—41no—
122C2×C4×C71—41no—
123C2×C4×C71—41no—
124C2×C4×C71—41no—
125C2×C4×C71—41no—
126C2×C4×C71—41no—
127C2×C4×C71—41no—
128C2×C4×C71#9911yescharacteristic
129C2×C4×C71#9911yescharacteristic
130C2×Dic71—82no—
131C2×Dic71—82no—
132C2×Dic71—82no—
133C71⋊C8—82no—
134C22×C71—21no—
135C22×C71—21no—
136C22×C71—22no—
137C22×C71—42no—
138C22×C71—41no—
139C22×C71—41no—
140C22×C71—81no—
141C22×C71C2×#9911yescharacteristic
142C22×C71#41611yescharacteristic
143C22×C71#42111yescharacteristic
144C4×C71—21no—
145C4×C71—21no—
146C4×C71—41no—
147C4×C71—42no—
148C4×C71—41no—
149C4×C71—41no—
150C4×C71—41no—
151C4×C71—41no—
152C4×C71—81no—
153C4×C71—81no—
154C4×C71—81no—
155Dic71—162no—
156#11842—711nomaximal, Sylow
157C2×C71—21no—
158C2×C71—21no—
159C2×C71—42no—
160C2×C71#151611yescharacteristic
161C2×C71#212011yescharacteristic
162C2×C71#212911yescharacteristic
163#2599D7111yescharacteristic
164#1820—711no—
165#1458—711no—
166#1351—712no—
167#1344—712no—
168C71#1184211yescharacteristic, complemented, Sylow
169C2×C4×Q8Dic7111yescharacteristic, complemented
170C2×#117C2×D7111yescharacteristic
171#431—1421no—
172#398Dic7111yescharacteristic, complemented
173#397C2×D7111yescharacteristic
174#395Dic7111yescharacteristic, complemented
175#393C2×D7111yescharacteristic
176#388—21no—
177#387—21no—
178#387—21no—
179#385—21no—
180#383Dic7111yescharacteristic, complemented
181#381—1421no—
182#342—712no—
183#342—1421no—
184#324—1421no—
185#323—712no—
186C22×Q8C2×Dic7111yescharacteristic
187C23×C4C22⋊D7111yescharacteristic
188C4×Q8—41no—
189C4×Q8—41no—
190#117—41no—
191#117—41no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4C2×Dic7111yescharacteristic
196C2×C4⋊C4C2×Dic7111yescharacteristic
197C2×C4⋊C4C22⋊D7111yescharacteristic
198C2×C22⋊C4—21no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—1422no—
201C2×C22⋊C4C22⋊D7111yescharacteristic
202C2×C22⋊C4C22×D7111yescharacteristic
203C2×C42—21no—
204C2×C42C2×Dic7111yescharacteristic
205C2×C42C22⋊D7111yescharacteristic
206C4⋊C8—2841no—
207C22⋊C8—1422no—
208C4×C8—2841no—
209#95—21no—
210#95—21no—
211#95—21no—
212#95—21no—
213#95—21no—
214#95—21no—
215#95—2841no—
216#95—2841no—
217#95C2×Dic7111yescharacteristic
218#95C2×Dic7111yescharacteristic
219C24C2×C22⋊D7111yescharacteristic
220C2×Q8—42no—
221C2×Q8—41no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—41no—
230C22×C4—2842no—
231C22×C4C22⋊Dic7111yescharacteristic
232C22×C4C22⋊Dic7111yescharacteristic
233C22×C4C22⋊Dic7111yescharacteristic
234C22×C4C22⋊Dic7111yescharacteristic
235C22×C4C22×Dic7111yescharacteristic
236C22×C4C2×C22⋊D7111yescharacteristic
237C2×C8—2842no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C22⋊C4—41no—
246C22⋊C4—41no—
247C22⋊C4—42no—
248C22⋊C4—42no—
249C22⋊C4—2842no—
250C22⋊C4—2842no—
251C42—21no—
252C42—21no—
253C42—21no—
254C42—21no—
255C42—42no—
256C23—21no—
257C23—21no—
258C23—41no—
259C23—41no—
260C23C2×C22⋊Dic7111yescharacteristic
261C23#1455614812yes—
262Q8—42no—
263Q8—42no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—5682no—
292C2×C4—5682no—
293C2×C4—5682no—
294C2×C4#1455604311yescharacteristic
295C2×C4#1455604311yescharacteristic
296C8—5682no—
297C22—21no—
298C22—21no—
299C22—22no—
300C22—42no—
301C22—41no—
302C22—41no—
303C22—81no—
304C22C2×#1455604311yescharacteristic
305C22#1798599511yescharacteristic
306C22#1798601611yescharacteristic
307C4—21no—
308C4—21no—
309C4—41no—
310C4—42no—
311C4—41no—
312C4—41no—
313C4—41no—
314C4—41no—
315C4—81no—
316C4—81no—
317C4—81no—
318C4—11362no—
319C2—21no—
320C2—21no—
321C2—42no—
322C2#2258689611yescharacteristic
323C2#2258998111yescharacteristic
324C2#2259001511yescharacteristic
subgroupG/H
centerC22C2×#14556043
commutatorC22×C4×C71C22×C4
FrattiniC2×C22⋊C4C22×D71
FittingC71×#2599C2
socleC22×C71C2×#99
prime radicalC24×C71C2×D4
cosocleC2×C22⋊C4×C71C23
hypercenter#2599D71
hypocenterC71#11842
LCSG ▷ C22×C4×C71 ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2599

Complements

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

NH ≅ G/Ntype
12. C2×C4×Q8×C71C4semidirect
14. #398×C71C4semidirect
16. #395×C71C4semidirect
22. #383×C71C4semidirect
168. C71#11842semidirect
169. C2×C4×Q8Dic71semidirect
172. #398Dic71semidirect
174. #395Dic71semidirect
180. #383Dic71semidirect

Maximal subgroups

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

subgroupindex
#225884172
#225866542
#225863612
#225863402
C71×#25992
#1184271