Finite Group Databasebeta

← #31890424

Subgroups — #31890424

Order 18176.

456 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22588378C211yesmaximal, characteristic, complemented
2#22588365C211yesmaximal, characteristic, complemented
3#22586883C212yesmaximal
4#22586245C212yesmaximal, complemented
5C71×#2929C211yesmaximal, characteristic
6C2×#14556064—21no—
7C2×#14556060—21no—
8#17985963C2212yes—
9#17985859C2212yes—
10#17985764—21no—
11#17985762—21no—
12#17985736—22no—
13#17985733—22no—
14C2×#131×C71C411yescharacteristic, complemented
15C23×C8×C71C2211yescharacteristic
16#525×C71C2211yescharacteristic
17#435×C71—21no—
18#435×C71—21no—
19C8×D4×C71—21no—
20C8×D4×C71—21no—
21#424×C71C411yescharacteristic, complemented
22#408×C71C411yescharacteristic, complemented
23#407×C71C2211yescharacteristic
24#405×C71C411yescharacteristic, complemented
25C2×#11765849—22no—
26C2×C4⋊Dic71—22no—
27C2×C22⋊Dic71—22no—
28#14556064—42no—
29#14556060—42no—
30C22⋊(C71⋊C8)—22no—
31C22⋊(C71⋊C8)—22no—
32C71⋊#95—22no—
33C2×#41×C71C2×C411yescharacteristic
34C23×C4×C71C2311yescharacteristic
35#131×C71—42no—
36C2×M16×C71—21no—
37C2×M16×C71C2×C411yescharacteristic
38C2×M16×C71C2×C411yescharacteristic
39C22×C8×C71—21no—
40C22×C8×C71—42no—
41C22×C8×C71C2×C411yescharacteristic
42C22×C8×C71C2×C411yescharacteristic
43C22×C8×C71D411yescharacteristic
44C22×C8×C71D411yescharacteristic
45C22×C8×C71D411yescharacteristic
46C22×C8×C71D411yescharacteristic
47#123×C71—21no—
48C22⋊Q8×C71D411yescharacteristic
49C22⋊Q8×C71Q811yescharacteristic
50#121×C71D411yescharacteristic
51#121×C71Q811yescharacteristic
52#120×C71—21no—
53C4×D4×C71—21no—
54C4×D4×C71—21no—
55#117×C71C2×C411yescharacteristic
56C4⋊C8×C71—21no—
57C4⋊C8×C71—21no—
58C22⋊C8×C71—21no—
59C22⋊C8×C71—21no—
60C22⋊C8×C71—21no—
61C22⋊C8×C71—21no—
62#97×C71—21no—
63C4×C8×C71—21no—
64C22×Dic71—42no—
65#11765849—42no—
66#11765849—42no—
67C2×C71⋊C8—42no—
68C4⋊Dic71—42no—
69C4⋊Dic71—42no—
70C22⋊Dic71—82no—
71C24×C71#4111yescharacteristic
72#41×C71—41no—
73#41×C71—41no—
74C2×Q8×C71C4⋊C411yescharacteristic
75C2×D4×C71—21no—
76C2×D4×C71—21no—
77C2×D4×C71—21no—
78C2×D4×C71C4⋊C411yescharacteristic
79C22×C4×C71—21no—
80C22×C4×C71—41no—
81C22×C4×C71—41no—
82C22×C4×C71C22×C411yescharacteristic
83C22×C4×C71C2×D411yescharacteristic
84C22×C4×C71C2×D411yescharacteristic
85C22×C4×C71C2×D411yescharacteristic
86C22×C4×C71C2×Q811yescharacteristic
87C22×C4×C71#4111yescharacteristic
88M16×C71—42no—
89M16×C71—41no—
90M16×C71—41no—
91C2×C8×C71—22no—
92C2×C8×C71—22no—
93C2×C8×C71—22no—
94C2×C8×C71—21no—
95C2×C8×C71—21no—
96C2×C8×C71—41no—
97C2×C8×C71—41no—
98C2×C8×C71—41no—
99C2×C8×C71—42no—
100C2×C8×C71C22⋊C411yescharacteristic
101C2×C8×C71C22⋊C411yescharacteristic
102C2×C8×C71C22⋊C411yescharacteristic
103C2×C8×C71C22⋊C411yescharacteristic
104C4⋊C4×C71—21no—
105C4⋊C4×C71—21no—
106C4⋊C4×C71C4⋊C411yescharacteristic
107C4⋊C4×C71C4⋊C411yescharacteristic
108C22⋊C4×C71—21no—
109C22⋊C4×C71—21no—
110C22⋊C4×C71—21no—
111C22⋊C4×C71—21no—
112C22⋊C4×C71—21no—
113C42×C71—21no—
114C23×C71—22no—
115C23×C71—21no—
116C23×C71—41no—
117C23×C71—41no—
118C23×C71#11711yescharacteristic
119C23×C71#12111yescharacteristic
120C23×C71C22⋊Q811yescharacteristic
121Q8×C71—41no—
122D4×C71—41no—
123D4×C71—42no—
124D4×C71—42no—
125D4×C71—42no—
126C2×C4×C71—21no—
127C2×C4×C71—21no—
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—21no—
135C2×C4×C71—41no—
136C2×C4×C71—41no—
137C2×C4×C71—41no—
138C2×C4×C71—41no—
139C2×C4×C71—41no—
140C2×C4×C71C2×C22⋊C411yescharacteristic
141C2×C4×C71C2×C4⋊C411yescharacteristic
142C2×C4×C71#12111yescharacteristic
143C2×C4×C71C22⋊Q811yescharacteristic
144C2×Dic71—82no—
145C2×Dic71—82no—
146C2×Dic71—82no—
147C8×C71—21no—
148C8×C71—21no—
149C8×C71—21no—
150C8×C71—41no—
151C8×C71—41no—
152C8×C71#9912yes—
153C71⋊C8—82no—
154C22×C71—21no—
155C22×C71—21no—
156C22×C71—21no—
157C22×C71—21no—
158C22×C71—21no—
159C22×C71—21no—
160C22×C71—21no—
161C22×C71—21no—
162C22×C71—44no—
163C22×C71—41no—
164C22×C71—41no—
165C22×C71—41no—
166C22×C71#38011yescharacteristic
167C22×C71#47112yes—
168C4×C71—21no—
169C4×C71—21no—
170C4×C71—21no—
171C4×C71—41no—
172C4×C71—41no—
173C4×C71—41no—
174C4×C71C2×#9911yescharacteristic
175C4×C71#41011yescharacteristic
176Dic71—162no—
177#8626—711nomaximal, Sylow
178C2×C71—22no—
179C2×C71—41no—
180C2×C71—41no—
181C2×C71—81no—
182C2×C71#179911yescharacteristic
183C2×C71#181411yescharacteristic
184C2×C71#181511yescharacteristic
185#2929D7111yescharacteristic
186#1810—711no—
187#1807—711no—
188#1513—712no—
189#1305—712no—
190C71#862611yescharacteristic, complemented, Sylow
191C2×#131Dic7111yescharacteristic, complemented
192C23×C8C2×D7111yescharacteristic
193#525C2×D7111yescharacteristic
194#435—21no—
195#435—21no—
196C8×D4—21no—
197C8×D4—21no—
198C2×#108—1421no—
199C2×#106—1421no—
200#424Dic7111yescharacteristic, complemented
201#408Dic7111yescharacteristic, complemented
202#407—712no—
203#407C2×D7111yescharacteristic
204#405Dic7111yescharacteristic, complemented
205#380—712no—
206#341—1421no—
207#340—1421no—
208#328—1422no—
209#327—1422no—
210C2×#41C2×Dic7111yescharacteristic
211C23×C4C22×D7111yescharacteristic
212#131—42no—
213C2×M16—21no—
214C2×M16—1422no—
215C2×M16C2×Dic7111yescharacteristic
216C2×M16C2×Dic7111yescharacteristic
217C22×C8—21no—
218C22×C8—42no—
219C22×C8C2×Dic7111yescharacteristic
220C22×C8C2×Dic7111yescharacteristic
221C22×C8C22⋊D7111yescharacteristic
222C22×C8C22⋊D7111yescharacteristic
223C22×C8C22⋊D7111yescharacteristic
224C22×C8C22⋊D7111yescharacteristic
225#123—21no—
226C22⋊Q8Dic14211yescharacteristic
227C22⋊Q8D28411yescharacteristic
228#121Dic14211yescharacteristic
229#121D28411yescharacteristic
230#120—21no—
231C4×D4—21no—
232C4×D4—21no—
233#117C2×Dic7111yescharacteristic
234C2×C4⋊C4—1422no—
235C2×C22⋊C4—1422no—
236#108—2842no—
237#106—2842no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C22⋊C8—21no—
241C22⋊C8—21no—
242C22⋊C8—21no—
243C22⋊C8—21no—
244C22⋊C8—1422no—
245C22⋊C8—1422no—
246#97—21no—
247C4×C8—21no—
248#95—1422no—
249C24#1176586511yescharacteristic
250#41—41no—
251#41—41no—
252C2×Q8C4⋊Dic7111yescharacteristic
253C2×D4—21no—
254C2×D4—21no—
255C2×D4—21no—
256C2×D4C4⋊Dic7111yescharacteristic
257C22×C4—21no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—2842no—
261C22×C4C22×Dic7111yescharacteristic
262C22×C4C2×C22⋊D7111yescharacteristic
263C22×C4C2×C22⋊D7111yescharacteristic
264C22×C4C2×D28411yescharacteristic
265C22×C4C2×Dic14211yescharacteristic
266C22×C4#1176586511yescharacteristic
267M16—42no—
268M16—41no—
269M16—41no—
270M16—2842no—
271M16—2842no—
272C2×C8—22no—
273C2×C8—22no—
274C2×C8—22no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—41no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—42no—
281C2×C8—2842no—
282C2×C8C22⋊Dic7111yescharacteristic
283C2×C8C22⋊Dic7111yescharacteristic
284C2×C8C22⋊Dic7111yescharacteristic
285C2×C8C22⋊Dic7111yescharacteristic
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—2842no—
289C4⋊C4—2842no—
290C4⋊C4C4⋊Dic7111yescharacteristic
291C4⋊C4C4⋊Dic7111yescharacteristic
292C22⋊C4—21no—
293C22⋊C4—21no—
294C22⋊C4—21no—
295C22⋊C4—21no—
296C22⋊C4—21no—
297C22⋊C4—5682no—
298C42—21no—
299C23—22no—
300C23—21no—
301C23—41no—
302C23—41no—
303C23#1455608711yescharacteristic
304C23#1455610211yescharacteristic
305C23#1455610711yescharacteristic
306Q8—41no—
307D4—41no—
308D4—42no—
309D4—42no—
310D4—42no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—5682no—
326C2×C4—5682no—
327C2×C4—5682no—
328C2×C4C2×C22⋊Dic7111yescharacteristic
329C2×C4C2×C4⋊Dic7111yescharacteristic
330C2×C4#1455610211yescharacteristic
331C2×C4#1455610711yescharacteristic
332C8—21no—
333C8—21no—
334C8—21no—
335C8—41no—
336C8—41no—
337C8—5682no—
338C8#1455604312yes—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—44no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22#1798585911yescharacteristic
352C22#1798626212yes—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—11362no—
360C4C2×#1455604311yescharacteristic
361C4#1798597311yescharacteristic
362C2—22no—
363C2—41no—
364C2—41no—
365C2—81no—
366C2#2258832811yescharacteristic
367C2#2258839411yescharacteristic
368C2#2258839911yescharacteristic
subgroupG/H
centerC22#17985859
commutatorC22×C4×C71C22×C4
FrattiniC23×C4C22×D71
FittingC71×#2929C2
socleC22×C71#380
prime radical#525×C71C22
cosocleC23×C4×C71C23
hypercenter#2929D71
hypocenterC71#8626
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2929

Complements

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

NH ≅ G/Ntype
1. #22588378C2semidirect
2. #22588365C2semidirect
4. #22586245C2semidirect
14. C2×#131×C71C4semidirect
21. #424×C71C4semidirect
22. #408×C71C4semidirect
24. #405×C71C4semidirect
190. C71#8626semidirect
191. C2×#131Dic71semidirect
200. #424Dic71semidirect
201. #408Dic71semidirect
204. #405Dic71semidirect

Maximal subgroups

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

subgroupindex
#225883782
#225883652
#225868832
#225862452
C71×#29292
#862671