Finite Group Databasebeta

← #31885881

Subgroups — #31885881

Order 18176.

488 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22588102C211yesmaximal, characteristic
2C4×C22⋊(C71⋊C8)C211yesmaximal, characteristic
3#22586166C212yesmaximal, complemented
4#22586163C212yesmaximal, complemented
5C71×#2855C211yesmaximal, characteristic
6C2×C22⋊(C71⋊C8)—21no—
7C2×C22⋊(C71⋊C8)—21no—
8C2×#14556039—21no—
9C2×C4×C71⋊C8—21no—
10#17985754C2212yes—
11#17985754C2212yes—
12C2×C22⋊Q8×C71C411yescharacteristic
13C22×C42×C71C2211yescharacteristic
14#391×C71C411yescharacteristic
15#391×C71C2211yescharacteristic
16#389×C71C411yescharacteristic
17#389×C71C2211yescharacteristic
18#387×C71—21no—
19#387×C71—21no—
20#385×C71—21no—
21#381×C71—21no—
22#380×C71C411yescharacteristic
23C22×C71⋊C8—22no—
24C22×C71⋊C8—22no—
25C22⋊(C71⋊C8)—24no—
26C22⋊(C71⋊C8)—24no—
27#14556039—42no—
28C4×C71⋊C8—42no—
29C22×Q8×C71C2×C411yescharacteristic
30C23×C4×C71C2×C411yescharacteristic
31C23×C4×C71D411yescharacteristic
32C23×C4×C71D411yescharacteristic
33C22⋊Q8×C71—42no—
34C2×C4⋊C4×C71—21no—
35C2×C4⋊C4×C71—21no—
36C2×C4⋊C4×C71—21no—
37C2×C4⋊C4×C71C2×C411yescharacteristic
38C2×C4⋊C4×C71C2×C411yescharacteristic
39C2×C4⋊C4×C71C2×C411yescharacteristic
40C2×C22⋊C4×C71—21no—
41C2×C22⋊C4×C71—21no—
42C2×C42×C71—42no—
43C2×C42×C71C2×C411yescharacteristic
44C2×C42×C71D411yescharacteristic
45C2×C42×C71D411yescharacteristic
46C2×C42×C71C2311yescharacteristic
47#95×C71—21no—
48#95×C71—21no—
49#95×C71—21no—
50#95×C71—21no—
51#95×C71—21no—
52#95×C71—21no—
53#95×C71—21no—
54C2×C71⋊C8—24no—
55C2×C71⋊C8—24no—
56C2×C71⋊C8—42no—
57C2×C71⋊C8—42no—
58C2×C71⋊C8—42no—
59C2×C71⋊C8—42no—
60C24×C71C22⋊C411yescharacteristic
61C2×Q8×C71—22no—
62C2×Q8×C71—41no—
63C2×Q8×C71—41no—
64C22×C4×C71—22no—
65C22×C4×C71—22no—
66C22×C4×C71—22no—
67C22×C4×C71—22no—
68C22×C4×C71—22no—
69C22×C4×C71—22no—
70C22×C4×C71—21no—
71C22×C4×C71—21no—
72C22×C4×C71—21no—
73C22×C4×C71—21no—
74C22×C4×C71—41no—
75C22×C4×C71—41no—
76C22×C4×C71—41no—
77C22×C4×C71C22⋊C411yescharacteristic
78C22×C4×C71C22⋊C411yescharacteristic
79C22×C4×C71C22⋊C411yescharacteristic
80C22×C4×C71C22×C411yescharacteristic
81C22×C4×C71C2×D411yescharacteristic
82C22×C4×C71C2×D411yescharacteristic
83C4⋊C4×C71—22no—
84C4⋊C4×C71—41no—
85C4⋊C4×C71—41no—
86C4⋊C4×C71—42no—
87C4⋊C4×C71—42no—
88C4⋊C4×C71—42no—
89C22⋊C4×C71—42no—
90C22⋊C4×C71—42no—
91C42×C71—22no—
92C42×C71—22no—
93C42×C71—41no—
94C42×C71—41no—
95C23×C71—21no—
96C23×C71—21no—
97C23×C71—21no—
98C23×C71—21no—
99C23×C71—21no—
100C23×C71—21no—
101C23×C71#10412yes—
102C23×C71C2×C22⋊C411yescharacteristic
103Q8×C71—82no—
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—21no—
112C2×C4×C71—21no—
113C2×C4×C71—21no—
114C2×C4×C71—21no—
115C2×C4×C71—21no—
116C2×C4×C71—21no—
117C2×C4×C71—44no—
118C2×C4×C71—44no—
119C2×C4×C71—44no—
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—41no—
129C2×C4×C71—41no—
130C2×C4×C71—41no—
131C2×C4×C71—41no—
132C2×C4×C71#9912yes—
133C2×C4×C71#10412yes—
134C2×C4×C71#13112yes—
135C2×C4×C71#13512yes—
136C2×C4×C71#13712yes—
137C71⋊C8—44no—
138C71⋊C8—44no—
139C22×C71—22no—
140C22×C71—22no—
141C22×C71—22no—
142C22×C71—41no—
143C22×C71—41no—
144C22×C71—41no—
145C22×C71—41no—
146C22×C71#40811yescharacteristic
147C22×C71C2×#9911yescharacteristic
148C22×C71#41311yescharacteristic
149C22×C71#41611yescharacteristic
150C22×C71#41911yescharacteristic
151C22×C71C2×#10411yescharacteristic
152C22×C71C2×#10411yescharacteristic
153C4×C71—22no—
154C4×C71—22no—
155C4×C71—22no—
156C4×C71—22no—
157C4×C71—22no—
158C4×C71—41no—
159C4×C71—81no—
160C4×C71—81no—
161C4×C71—81no—
162C4×C71—81no—
163#7675—711nomaximal, Sylow
164C2×C71—42no—
165C2×C71#146811yescharacteristic
166C2×C71#148511yescharacteristic
167C2×C71#149411yescharacteristic
168C2×C71#150111yescharacteristic
169C2×C71#150811yescharacteristic
170C2×C71#152411yescharacteristic
171C2×C71#155411yescharacteristic
172#2855D7111yescharacteristic
173#1749—711no—
174C4×C22⋊C8—711no—
175#1272—712no—
176#1271—712no—
177C71#767511yescharacteristic, complemented, Sylow
178C2×C22⋊Q8Dic7111yescharacteristic
179C22×C42C2×D7111yescharacteristic
180C2×C22⋊C8—1421no—
181C2×C22⋊C8—1421no—
182C2×#97—1421no—
183C2×C4×C8—1421no—
184#391Dic7111yescharacteristic
185#391C2×D7111yescharacteristic
186#389Dic7111yescharacteristic
187#389C2×D7111yescharacteristic
188#387—21no—
189#387—21no—
190#385—21no—
191#381—21no—
192#380Dic7111yescharacteristic
193#336—712no—
194#336—712no—
195C22×Q8C2×Dic7111yescharacteristic
196C23×C4C2×Dic7111yescharacteristic
197C23×C4C22⋊D7111yescharacteristic
198C23×C4C22⋊D7111yescharacteristic
199C22×C8—1422no—
200C22×C8—1422no—
201C22⋊Q8—42no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4C2×Dic7111yescharacteristic
206C2×C4⋊C4C2×Dic7111yescharacteristic
207C2×C4⋊C4C2×Dic7111yescharacteristic
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C42—42no—
211C2×C42C2×Dic7111yescharacteristic
212C2×C42C22⋊D7111yescharacteristic
213C2×C42C22⋊D7111yescharacteristic
214C2×C42C22×D7111yescharacteristic
215C22⋊C8—1424no—
216C22⋊C8—1424no—
217#97—2842no—
218C4×C8—2842no—
219#95—21no—
220#95—21no—
221#95—21no—
222#95—21no—
223#95—21no—
224#95—21no—
225#95—21no—
226C24C22⋊Dic7111yescharacteristic
227C2×Q8—22no—
228C2×Q8—41no—
229C2×Q8—41no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4C22⋊Dic7111yescharacteristic
244C22×C4C22⋊Dic7111yescharacteristic
245C22×C4C22⋊Dic7111yescharacteristic
246C22×C4C22×Dic7111yescharacteristic
247C22×C4C2×C22⋊D7111yescharacteristic
248C22×C4C2×C22⋊D7111yescharacteristic
249C2×C8—2842no—
250C2×C8—2842no—
251C2×C8—2842no—
252C2×C8—2842no—
253C2×C8—1424no—
254C2×C8—1424no—
255C4⋊C4—22no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C22⋊C4—42no—
262C22⋊C4—42no—
263C42—22no—
264C42—22no—
265C42—41no—
266C42—41no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23#1455605512yes—
274C23C2×C22⋊Dic7111yescharacteristic
275Q8—82no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—44no—
290C2×C4—44no—
291C2×C4—44no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4#1455604312yes—
305C2×C4#1455605512yes—
306C2×C4#1455613512yes—
307C2×C4#1455614812yes—
308C2×C4#1455615912yes—
309C8—2844no—
310C8—2844no—
311C22—22no—
312C22—22no—
313C22—22no—
314C22—41no—
315C22—41no—
316C22—41no—
317C22—41no—
318C22#1798596611yescharacteristic
319C22C2×#1455604311yescharacteristic
320C22#1798598311yescharacteristic
321C22#1798599511yescharacteristic
322C22#1798600811yescharacteristic
323C22C2×#1455605511yescharacteristic
324C22C2×#1455605511yescharacteristic
325C4—22no—
326C4—22no—
327C4—22no—
328C4—22no—
329C4—22no—
330C4—41no—
331C4—81no—
332C4—81no—
333C4—81no—
334C4—81no—
335C2—42no—
336C2#2258668811yescharacteristic
337C2#2258676111yescharacteristic
338C2#2258680011yescharacteristic
339C2#2258683111yescharacteristic
340C2#2258686011yescharacteristic
341C2#2258692811yescharacteristic
342C2#2258705011yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic71
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#2855C2
socleC23×C71C2×C22⋊C4
prime radicalC24×C71C22⋊C4
cosocleC2×C42×C71C23
hypercenter#2855D71
hypocenterC71#7675
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2855

Complements

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

NH ≅ G/Ntype
3. #22586166C2semidirect
4. #22586163C2semidirect
177. C71#7675semidirect

Maximal subgroups

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

subgroupindex
#225881022
C4×C22⋊(C71⋊C8)2
#225861662
#225861632
C71×#28552
#767571