Finite Group Databasebeta

← #31896299

Subgroups — #31896299

Order 18176.

640 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22588320C211yesmaximal, characteristic
2#22588102C211yesmaximal, characteristic
3C2×#17985725C214yesmaximal
4C2×#389×C71C211yesmaximal, characteristic
5C2×C4⋊(C71⋊C8)—21no—
6C2×C22⋊(C71⋊C8)C2212yes—
7C2×C22⋊(C71⋊C8)C2212yes—
8C2×#14556039—21no—
9#17985754—21no—
10#17985754—21no—
11#17985725—28no—
12C22×C4⋊C4×C71C411yescharacteristic
13C22×C4⋊C4×C71C411yescharacteristic
14C22×C4⋊C4×C71C2212yes—
15C22×C42×C71C2211yescharacteristic
16#389×C71—21no—
17#389×C71—21no—
18#389×C71—21no—
19#389×C71—21no—
20C2×#95×C71C412yes—
21C22×C71⋊C8—22no—
22C22×C71⋊C8—22no—
23C4⋊(C71⋊C8)—42no—
24C22⋊(C71⋊C8)—24no—
25C22⋊(C71⋊C8)—22no—
26C22⋊(C71⋊C8)D414yes—
27#14556039—42no—
28C23×C4×C71C2×C411yescharacteristic
29C23×C4×C71C2×C412yes—
30C23×C4×C71C2×C411yescharacteristic
31C23×C4×C71C2×C412yes—
32C23×C4×C71C2311yescharacteristic
33C2×C4⋊C4×C71—21no—
34C2×C4⋊C4×C71—21no—
35C2×C4⋊C4×C71—24no—
36C2×C4⋊C4×C71—21no—
37C2×C4⋊C4×C71—21no—
38C2×C4⋊C4×C71—24no—
39C2×C4⋊C4×C71—22no—
40C2×C4⋊C4×C71—22no—
41C2×C4⋊C4×C71—24no—
42C2×C4⋊C4×C71—42no—
43C2×C42×C71—42no—
44C2×C42×C71D411yescharacteristic
45C2×C42×C71D411yescharacteristic
46C2×C42×C71D411yescharacteristic
47C2×C42×C71D411yescharacteristic
48#95×C71—22no—
49#95×C71—22no—
50C2×C71⋊C8—24no—
51C2×C71⋊C8—42no—
52C2×C71⋊C8—42no—
53C2×C71⋊C8—42no—
54C2×C71⋊C8—42no—
55C2×C71⋊C8—42no—
56C24×C71C22×C411yescharacteristic
57C22×C4×C71—22no—
58C22×C4×C71—22no—
59C22×C4×C71—22no—
60C22×C4×C71—22no—
61C22×C4×C71—24no—
62C22×C4×C71—22no—
63C22×C4×C71—22no—
64C22×C4×C71—24no—
65C22×C4×C71—22no—
66C22×C4×C71—22no—
67C22×C4×C71—24no—
68C22×C4×C71—41no—
69C22×C4×C71—41no—
70C22×C4×C71—41no—
71C22×C4×C71—42no—
72C22×C4×C71—42no—
73C22×C4×C71C22⋊C411yescharacteristic
74C22×C4×C71C22⋊C411yescharacteristic
75C22×C4×C71C22⋊C411yescharacteristic
76C22×C4×C71C22⋊C411yescharacteristic
77C22×C4×C71M1614yes—
78C22×C4×C71C2×D412yes—
79C22×C4×C71C2×D411yescharacteristic
80C22×C4×C71C2×D411yescharacteristic
81C22×C4×C71#4112yes—
82C4⋊C4×C71—24no—
83C4⋊C4×C71—24no—
84C4⋊C4×C71—42no—
85C4⋊C4×C71—42no—
86C4⋊C4×C71—44no—
87C4⋊C4×C71—42no—
88C4⋊C4×C71—42no—
89C42×C71—22no—
90C42×C71—22no—
91C42×C71—41no—
92C42×C71—41no—
93C23×C71—21no—
94C23×C71—21no—
95C23×C71—21no—
96C23×C71—21no—
97C23×C71C2×C22⋊C411yescharacteristic
98C23×C71C4×D412yes—
99C23×C71C2×M1612yes—
100C23×C71#13112yes—
101C2×C4×C71—28no—
102C2×C4×C71—21no—
103C2×C4×C71—21no—
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—24no—
113C2×C4×C71—21no—
114C2×C4×C71—21no—
115C2×C4×C71—21no—
116C2×C4×C71—28no—
117C2×C4×C71—42no—
118C2×C4×C71—42no—
119C2×C4×C71—44no—
120C2×C4×C71—42no—
121C2×C4×C71—42no—
122C2×C4×C71—44no—
123C2×C4×C71—42no—
124C2×C4×C71—42no—
125C2×C4×C71—42no—
126C2×C4×C71—84no—
127C2×C4×C71#9912yes—
128C2×C4×C71#10112yes—
129C2×C4×C71#10112yes—
130C2×C4×C71#12112yes—
131C2×C4×C71#12411yescharacteristic
132C2×C4×C71#12711yescharacteristic
133C71⋊C8—44no—
134C71⋊C8—82no—
135C22×C71—22no—
136C22×C71—22no—
137C22×C71—22no—
138C22×C71—22no—
139C22×C71—22no—
140C22×C71—22no—
141C22×C71#39011yescharacteristic
142C22×C71#40711yescharacteristic
143C22×C71#40811yescharacteristic
144C22×C71C2×#9911yescharacteristic
145C22×C71#41011yescharacteristic
146C22×C71C2×#10111yescharacteristic
147C22×C71C2×#10111yescharacteristic
148C22×C71#43514yes—
149C4×C71—22no—
150C4×C71—21no—
151C4×C71—21no—
152C4×C71—22no—
153C4×C71—22no—
154C4×C71—22no—
155C4×C71—44no—
156C4×C71—41no—
157C4×C71—82no—
158#9654—711nomaximal, Sylow
159C2×C71—24no—
160C2×C71#146311yescharacteristic
161C2×C71#146711yescharacteristic
162C2×C71#147011yescharacteristic
163C2×C71#147211yescharacteristic
164C2×C71#195511yescharacteristic
165C2×C71#196411yescharacteristic
166C2×C71#196611yescharacteristic
167C2×#389D7111yescharacteristic
168#1797—711no—
169#1749—711no—
170C2×#324—714no—
171C71#965411yescharacteristic, complemented, Sylow
172C22×C4⋊C4Dic7111yescharacteristic
173C22×C4⋊C4Dic7111yescharacteristic
174C22×C4⋊C4C2×D7112yes—
175C22×C42C2×D7111yescharacteristic
176C2×C4⋊C8—1421no—
177C2×C22⋊C8—712no—
178C2×C22⋊C8—712no—
179C2×#97—1421no—
180#389—21no—
181#389—21no—
182#389—21no—
183#389—21no—
184C2×#95Dic7112yes—
185#336—1421no—
186#336—1421no—
187#324—1428no—
188C23×C4C2×Dic7111yescharacteristic
189C23×C4C2×Dic7112yes—
190C23×C4C2×Dic7111yescharacteristic
191C23×C4C2×Dic7112yes—
192C23×C4C22×D7111yescharacteristic
193C22×C8—1422no—
194C22×C8—1422no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—42no—
205C2×C42—42no—
206C2×C42C22⋊D7111yescharacteristic
207C2×C42C22⋊D7111yescharacteristic
208C2×C42C22⋊D7111yescharacteristic
209C2×C42C22⋊D7111yescharacteristic
210C4⋊C8—2842no—
211C22⋊C8—714no—
212C22⋊C8—1424no—
213C22⋊C8—1422no—
214#97—2842no—
215#95—22no—
216#95—22no—
217C24C22×Dic7111yescharacteristic
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—24no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—24no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—24no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4C22⋊Dic7111yescharacteristic
235C22×C4C22⋊Dic7111yescharacteristic
236C22×C4C22⋊Dic7111yescharacteristic
237C22×C4C22⋊Dic7111yescharacteristic
238C22×C4#1176584914yes—
239C22×C4D4×D7112yes—
240C22×C4C2×C22⋊D7111yescharacteristic
241C22×C4C2×C22⋊D7111yescharacteristic
242C22×C4#1176586412yes—
243C2×C8—2842no—
244C2×C8—2842no—
245C2×C8—2842no—
246C2×C8—2842no—
247C2×C8—2842no—
248C2×C8—1424no—
249C4⋊C4—24no—
250C4⋊C4—24no—
251C4⋊C4—42no—
252C4⋊C4—42no—
253C4⋊C4—44no—
254C4⋊C4—42no—
255C4⋊C4—42no—
256C42—22no—
257C42—22no—
258C42—41no—
259C42—41no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23C2×C22⋊Dic7111yescharacteristic
265C23D4×Dic7112yes—
266C23C2×#1176584912yes—
267C23#1455613512yes—
268C2×C4—28no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—28no—
280C2×C4—24no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—44no—
287C2×C4—44no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—84no—
294C2×C4#1455604312yes—
295C2×C4#1455604712yes—
296C2×C4#1455604712yes—
297C2×C4#1455609912yes—
298C2×C4#1455611511yescharacteristic
299C2×C4#1455612411yescharacteristic
300C8—2844no—
301C8—5682no—
302C22—22no—
303C22—22no—
304C22—22no—
305C22—22no—
306C22—22no—
307C22—22no—
308C22#1798590211yescharacteristic
309C22#1798596311yescharacteristic
310C22#1798596611yescharacteristic
311C22C2×#1455604311yescharacteristic
312C22#1798597311yescharacteristic
313C22C2×#1455604711yescharacteristic
314C22C2×#1455604711yescharacteristic
315C22#1798607114yes—
316C4—22no—
317C4—22no—
318C4—22no—
319C4—22no—
320C4—21no—
321C4—21no—
322C4—41no—
323C4—44no—
324C4—82no—
325C2—24no—
326C2#2258667011yescharacteristic
327C2#2258668411yescharacteristic
328C2#2258669611yescharacteristic
329C2#2258670411yescharacteristic
330C2#2258913911yescharacteristic
331C2#2258917711yescharacteristic
332C2#2258918511yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic71
commutator, prime radicalC24×C71C22×C4
FrattiniC23×C4C22×D71
FittingC2×#389×C71C2
socleC23×C71C2×C22⋊C4
cosocleC23×C4×C71C23
hypercenterC2×#389D71
hypocenterC71#9654
LCSG ▷ C24×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
171. C71#9654semidirect

Maximal subgroups

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

subgroupindex
#225883202
#225881022
C2×#179857252
C2×#389×C712
#965471