Finite Group Databasebeta

← #31885467

Subgroups — #31885467

Order 18176.

568 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22590823C211yesmaximal, characteristic
2#22588408C211yesmaximal, characteristic
3#22586164C212yesmaximal, complemented
4#22586162C212yesmaximal, complemented
5C2×#336×C71C211yesmaximal, characteristic
6C2×#14556087C411yescharacteristic
7C2×#14556087C411yescharacteristic
8C2×C22⋊(C71⋊C8)—21no—
9#17985897C411yescharacteristic
10#17985897C411yescharacteristic
11#17985897C2211yescharacteristic
12#17985897C2211yescharacteristic
13#17985872—21no—
14#17985872—21no—
15C4×Dic71⋊C4—21no—
16C4×Dic71⋊C4—21no—
17#17985754—21no—
18#17985754C2212yes—
19C23×C8×C71—21no—
20C22×C42×C71C2211yescharacteristic
21#336×C71—21no—
22#336×C71C2212yes—
23C22×C71⋊C8—22no—
24#14556087—22no—
25#14556087—42no—
26#14556087C814yescomplemented
27C2×C4⋊Dic71C2×C411yescharacteristic
28C2×C4⋊Dic71C2×C411yescharacteristic
29C2×C4⋊Dic71C2×C411yescharacteristic
30C2×C4⋊Dic71C2×C411yescharacteristic
31C2×Dic71⋊C4—21no—
32C2×Dic71⋊C4—21no—
33C2×Dic71⋊C4—21no—
34C2×Dic71⋊C4—21no—
35C2×C22⋊Dic71—21no—
36C2×C22⋊Dic71—21no—
37C2×C4×Dic71—21no—
38C2×C4×Dic71—21no—
39C22⋊(C71⋊C8)—42no—
40C71⋊#95—21no—
41C71⋊#95—21no—
42C71⋊#95—21no—
43C71⋊#95—21no—
44C23×C4×C71C2×C411yescharacteristic
45C23×C4×C71D411yescharacteristic
46C23×C4×C71D411yescharacteristic
47C22×C8×C71—24no—
48C22×C8×C71—22no—
49C22×C8×C71—42no—
50C22×C8×C71—42no—
51C2×C42×C71—42no—
52C2×C42×C71C2×C411yescharacteristic
53C2×C42×C71D411yescharacteristic
54C2×C42×C71D411yescharacteristic
55C2×C42×C71C2311yescharacteristic
56C22×Dic71—21no—
57C22×Dic71—21no—
58C22×Dic71—21no—
59C22×Dic71—21no—
60C2×C71⋊C8—42no—
61C2×C71⋊C8—42no—
62C2×C71⋊C8—42no—
63C4⋊Dic71—21no—
64C4⋊Dic71—21no—
65C4⋊Dic71—41no—
66C4⋊Dic71—41no—
67C4⋊Dic71C2×C812yescomplemented
68C4⋊Dic71C2×C812yescomplemented
69Dic71⋊C4—42no—
70Dic71⋊C4—42no—
71Dic71⋊C4—42no—
72Dic71⋊C4—41no—
73Dic71⋊C4—41no—
74C22⋊Dic71—24no—
75C22⋊Dic71—42no—
76C4×Dic71—24no—
77C4×Dic71—42no—
78C24×C71C22⋊C411yescharacteristic
79C22×C4×C71—22no—
80C22×C4×C71—22no—
81C22×C4×C71—22no—
82C22×C4×C71—22no—
83C22×C4×C71—21no—
84C22×C4×C71—21no—
85C22×C4×C71—22no—
86C22×C4×C71—22no—
87C22×C4×C71—41no—
88C22×C4×C71C22⋊C411yescharacteristic
89C22×C4×C71C22⋊C411yescharacteristic
90C22×C4×C71C22⋊C411yescharacteristic
91C22×C4×C71C2×C812yes—
92C22×C4×C71M1612yes—
93C22×C4×C71C22×C411yescharacteristic
94C22×C4×C71C2×D411yescharacteristic
95C22×C4×C71C2×D411yescharacteristic
96C2×C8×C71—24no—
97C2×C8×C71—48no—
98C2×C8×C71—42no—
99C2×C8×C71—42no—
100C42×C71—41no—
101C42×C71—41no—
102C42×C71—41no—
103C42×C71—41no—
104C23×C71—21no—
105C23×C71—21no—
106C23×C71—21no—
107C23×C71—21no—
108C23×C71C22⋊C812yes—
109C23×C71#9912yes—
110C23×C71#10112yes—
111C23×C71C2×C22⋊C411yescharacteristic
112C2×C4×C71—24no—
113C2×C4×C71—24no—
114C2×C4×C71—24no—
115C2×C4×C71—24no—
116C2×C4×C71—21no—
117C2×C4×C71—21no—
118C2×C4×C71—21no—
119C2×C4×C71—21no—
120C2×C4×C71—21no—
121C2×C4×C71—21no—
122C2×C4×C71—21no—
123C2×C4×C71—21no—
124C2×C4×C71—21no—
125C2×C4×C71—21no—
126C2×C4×C71—21no—
127C2×C4×C71—21no—
128C2×C4×C71—21no—
129C2×C4×C71—44no—
130C2×C4×C71C22⋊C812yes—
131C2×C4×C71#10412yes—
132C2×C4×C71C22×C811yescharacteristic
133C2×C4×C71C2×M1611yescharacteristic
134C2×C4×C71#13512yes—
135C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
136C2×C4×C71#13711yescharacteristic
137C2×Dic71—22no—
138C2×Dic71—22no—
139C2×Dic71—41no—
140C2×Dic71—41no—
141C2×Dic71—41no—
142C2×Dic71—41no—
143C2×Dic71—41no—
144C2×Dic71—41no—
145C2×Dic71—41no—
146C2×Dic71—41no—
147C2×Dic71—41no—
148C2×Dic71—41no—
149C8×C71—44no—
150C71⋊C8—82no—
151C22×C71—22no—
152C22×C71—22no—
153C22×C71—22no—
154C22×C71—22no—
155C22×C71—22no—
156C22×C71—22no—
157C22×C71#32414yes—
158C22×C71C2×C22⋊C811yescharacteristic
159C22×C71C2×#9911yescharacteristic
160C22×C71C2×#10111yescharacteristic
161C22×C71#41611yescharacteristic
162C22×C71#41711yescharacteristic
163C22×C71C2×#10411yescharacteristic
164C22×C71#42111yescharacteristic
165C4×C71—22no—
166C4×C71—21no—
167C4×C71—21no—
168C4×C71—21no—
169C4×C71—22no—
170C4×C71—21no—
171C4×C71—22no—
172C4×C71—41no—
173Dic71—42no—
174Dic71—42no—
175Dic71—81no—
176Dic71—81no—
177#7579—711nomaximal, Sylow
178C2×C71—24no—
179C2×C71C2×#32411yescharacteristic
180C2×C71#147611yescharacteristic
181C2×C71#147711yescharacteristic
182C2×C71#150411yescharacteristic
183C2×C71#151611yescharacteristic
184C2×C71#153111yescharacteristic
185C2×C71#155211yescharacteristic
186#2296—711no—
187#1817—711no—
188C2×#336D7111yescharacteristic
189#1271—712no—
190#1271—712no—
191C71#757911yescharacteristic, complemented, Sylow
192C23×C8—21no—
193C2×#117—711no—
194C2×#117—711no—
195C22×C42C2×D7111yescharacteristic
196C2×C22⋊C8—1421no—
197#389—711no—
198#389—711no—
199#389—711no—
200#389—711no—
201#385—1421no—
202#385—1421no—
203C4×C4⋊C4—1421no—
204C4×C4⋊C4—1421no—
205#336—21no—
206#336—712no—
207#336—1421no—
208#336C2×D7112yes—
209C23×C4C4×D7111yescharacteristic
210C23×C4D28411yescharacteristic
211C23×C4C22⋊D7111yescharacteristic
212C22×C8—24no—
213C22×C8—22no—
214C22×C8—42no—
215C22×C8—42no—
216C22×C8—1422no—
217#117—2842no—
218#117—714no—
219#117—1422no—
220C2×C4⋊C4—711no—
221C2×C4⋊C4—711no—
222C2×C4⋊C4—711no—
223C2×C4⋊C4—711no—
224C2×C4⋊C4—1421no—
225C2×C4⋊C4—1421no—
226C2×C4⋊C4—1421no—
227C2×C4⋊C4—1421no—
228C2×C22⋊C4—1421no—
229C2×C22⋊C4—1421no—
230C2×C42—42no—
231C2×C42—1421no—
232C2×C42—1421no—
233C2×C42C4×D7111yescharacteristic
234C2×C42D28411yescharacteristic
235C2×C42C22⋊D7111yescharacteristic
236C2×C42C22×D7111yescharacteristic
237C22⋊C8—2842no—
238#95—1421no—
239#95—1421no—
240#95—1421no—
241#95—1421no—
242C24(C2×D71)⋊C411yescharacteristic
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—41no—
252C22×C4—1421no—
253C22×C4—1421no—
254C22×C4—1421no—
255C22×C4—1421no—
256C22×C4(C2×D71)⋊C411yescharacteristic
257C22×C4(C2×D71)⋊C411yescharacteristic
258C22×C4(C2×D71)⋊C411yescharacteristic
259C22×C4C8×D7112yes—
260C22×C4#1176584812yes—
261C22×C4C2×C4×D7111yescharacteristic
262C22×C4C2×C22⋊D7111yescharacteristic
263C22×C4C2×D28411yescharacteristic
264C2×C8—24no—
265C2×C8—48no—
266C2×C8—42no—
267C2×C8—42no—
268C2×C8—2842no—
269C2×C8—2842no—
270C2×C8—2842no—
271C4⋊C4—2841no—
272C4⋊C4—2841no—
273C4⋊C4—2842no—
274C4⋊C4—2842no—
275C4⋊C4—2842no—
276C4⋊C4—2841no—
277C4⋊C4—2841no—
278C4⋊C4—712no—
279C4⋊C4—712no—
280C4⋊C4—1421no—
281C4⋊C4—1421no—
282C22⋊C4—2842no—
283C22⋊C4—1424no—
284C42—41no—
285C42—41no—
286C42—41no—
287C42—41no—
288C42—2842no—
289C42—1424no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23(C2×D71)⋊C812yes—
295C23#1455604212yes—
296C23#1455604612yes—
297C23C2×(C2×D71)⋊C411yescharacteristic
298C2×C4—24no—
299C2×C4—24no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—24no—
308C2×C4—24no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—44no—
316C2×C4—2841no—
317C2×C4—2841no—
318C2×C4—2841no—
319C2×C4—2841no—
320C2×C4—2841no—
321C2×C4—2841no—
322C2×C4—2841no—
323C2×C4—2841no—
324C2×C4—2841no—
325C2×C4—2841no—
326C2×C4—1422no—
327C2×C4—1422no—
328C2×C4(C2×D71)⋊C812yes—
329C2×C4#1455605612yes—
330C2×C4C2×C8×D7111yescharacteristic
331C2×C4C2×#1176584811yescharacteristic
332C2×C4#1455615012yes—
333C2×C4#1455615411yescharacteristic
334C2×C4#1455615911yescharacteristic
335C8—44no—
336C8—5682no—
337C22—22no—
338C22—22no—
339C22—22no—
340C22—22no—
341C22—22no—
342C22—22no—
343C22#1798572414yes—
344C22C2×(C2×D71)⋊C811yescharacteristic
345C22C2×#1455604211yescharacteristic
346C22C2×#1455604611yescharacteristic
347C22#1798599611yescharacteristic
348C22#1798599811yescharacteristic
349C22C2×#1455605611yescharacteristic
350C22#1798601511yescharacteristic
351C4—22no—
352C4—22no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—22no—
357C4—21no—
358C4—41no—
359C4—2842no—
360C4—2842no—
361C4—5681no—
362C4—5681no—
363C2—24no—
364C2C2×#1798572411yescharacteristic
365C2#2258672311yescharacteristic
366C2#2258672511yescharacteristic
367C2#2258684211yescharacteristic
368C2#2258689711yescharacteristic
369C2#2258695411yescharacteristic
370C2#2258703811yescharacteristic
subgroupG/H
centerC23C2×(C2×D71)⋊C4
commutatorC2×C4×C71C22×C8
FrattiniC2×C42C22×D71
FittingC2×#336×C71C2
socleC23×C71C2×C22⋊C4
prime radicalC24×C71C22⋊C4
cosocleC2×C42×C71C23
hypercenterC2×#336D71
hypocenterC71#7579
LCSG ▷ C2×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#336

Complements

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

NH ≅ G/Ntype
3. #22586164C2semidirect
4. #22586162C2semidirect
26. #14556087C8semidirect
67. C4⋊Dic71C2×C8semidirect
68. C4⋊Dic71C2×C8semidirect
191. C71#7579semidirect

Maximal subgroups

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

subgroupindex
#225908232
#225884082
#225861642
#225861622
C2×#336×C712
#757971