Finite Group Databasebeta

← #31997939

Subgroups — #31997939

Order 18176.

1110 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986055C212yesmaximal
2C43×D71C211yesmaximal, characteristic
3#22588492C211yesmaximal, characteristic, complemented
4#22588456C212yesmaximal, complemented
5#22588317C214yesmaximal
6#22588195C214yesmaximal, complemented
7C71×#1842C211yesmaximal, characteristic, complemented
8C2×C42×D71C411yescharacteristic
9C2×C42×D71C412yes—
10C2×C42×D71C411yescharacteristic
11C2×C42×D71C2212yes—
12C2×C42×D71C2211yescharacteristic
13#17986055—28no—
14C2×C4⋊(C71⋊C8)C2214yes—
15C2×C4⋊(C71⋊C8)C2212yes—
16C2×(C2×D71)⋊C8C2214yes—
17C2×#14556038C2214yes—
18C42×Dic71C412yes—
19C42×Dic71C411yescharacteristic
20C42×Dic71C411yescharacteristic
21C42×Dic71C2212yes—
22C42×Dic71C2211yescharacteristic
23#17985753C2218yes—
24C2×C4⋊C8×C71C2214yes—
25C2×C4⋊C8×C71C2212yes—
26C43×C71C2211yescharacteristic
27C22×C4×D71C2×C412yes—
28C22×C4×D71C2×C411yescharacteristic
29C22×C4×D71C2×C411yescharacteristic
30C22×C4×D71C2×C412yes—
31C22×C4×D71C2311yescharacteristic
32C22×C71⋊C8C2314yes—
33C2×C4×Dic71C2×C412yes—
34C2×C4×Dic71C2×C412yes—
35C2×C4×Dic71C2×C412yes—
36C2×C4×Dic71C2×C412yes—
37C2×C4×Dic71C2×C412yes—
38C2×C4×Dic71C2×C411yescharacteristic
39C2×C4×Dic71C2×C412yes—
40C2×C4×Dic71C2×C412yes—
41C2×C4×Dic71C2×C411yescharacteristic
42C2×C4×Dic71C2×C411yescharacteristic
43C2×C4×Dic71C2×C411yescharacteristic
44C2×C4×Dic71C2312yes—
45C2×C4×Dic71C2311yescharacteristic
46C42×D71—28no—
47C42×D71—24no—
48C42×D71—24no—
49C42×D71—24no—
50C42×D71—24no—
51C42×D71D414yes—
52C42×D71Q814yes—
53C4⋊(C71⋊C8)—24no—
54C4⋊(C71⋊C8)—28no—
55(C2×D71)⋊C8—28no—
56#14556038—28no—
57C22×C8×C71C2314yes—
58C2×C42×C71C2×C412yes—
59C2×C42×C71C2×C411yescharacteristic
60C2×C42×C71C2×C411yescharacteristic
61C2×C42×C71C2311yescharacteristic
62C2×C42×C71C2312yes—
63C4⋊C8×C71—24no—
64C4⋊C8×C71—28no—
65C23×D71C22×C411yescharacteristic
66C22×Dic71C22×C411yescharacteristic
67C22×Dic71C22×C411yescharacteristic
68C22×Dic71C22×C411yescharacteristic
69C22×Dic71C22×C412yes—
70C22×Dic71C22×C412yes—
71C2×C4×D71—24no—
72C2×C4×D71—24no—
73C2×C4×D71—24no—
74C2×C4×D71—24no—
75C2×C4×D71—22no—
76C2×C4×D71—22no—
77C2×C4×D71—24no—
78C2×C4×D71—22no—
79C2×C4×D71—22no—
80C2×C4×D71—24no—
81C2×C4×D71C4⋊C414yes—
82C2×C4×D71C4⋊C414yes—
83C2×C4×D71M1618yes—
84C2×C4×D71C2×D412yes—
85C2×C4×D71C2×Q812yes—
86C2×C71⋊C8—24no—
87C2×C71⋊C8—28no—
88C4×Dic71—24no—
89C4×Dic71—24no—
90C4×Dic71—24no—
91C4×Dic71—22no—
92C4×Dic71—24no—
93C4×Dic71—22no—
94C4×Dic71—22no—
95C4×Dic71—24no—
96C4×Dic71—24no—
97C4×Dic71—24no—
98C4×Dic71C4⋊C414yes—
99C4×Dic71C4⋊C414yes—
100C4×Dic71M1614yes—
101C4×Dic71C2×D412yes—
102C4×Dic71C2×Q812yes—
103C22×C4×C71C22×C411yescharacteristic
104C22×C4×C71C22×C412yes—
105C22×C4×C71C22×C411yescharacteristic
106C22×C4×C71C22×C412yes—
107C22×C4×C71C2411yescharacteristic
108C2×C8×C71—24no—
109C2×C8×C71—28no—
110C42×C71—22no—
111C42×C71—24no—
112C42×C71M1614yes—
113C42×C71C2×D412yes—
114C42×C71C2×Q812yes—
115C42×C71#4114yes—
116C42×C71#4114yes—
117C23×C71C23×C411yescharacteristic
118C22×D71—22no—
119C22×D71—21no—
120C22×D71—21no—
121C22×D71C2×C4⋊C412yes—
122C22×D71C2×M1614yes—
123C2×C4×C71—22no—
124C2×C4×C71—22no—
125C2×C4×C71—22no—
126C2×C4×C71C2×C4⋊C412yes—
127C2×C4×C71C2×C4⋊C412yes—
128C2×C4×C71#11714yes—
129C2×C4×C71#11714yes—
130C2×C4×C71C2×M1612yes—
131C2×C4×C71C2×M1614yes—
132C2×C4×C71C2×M1612yes—
133C2×C4×C71C2×M1614yes—
134C2×C4×C71C22×D411yescharacteristic
135C2×C4×C71C22×Q811yescharacteristic
136C2×C4×C71C2×#4112yes—
137C2×C4×C71C2×#4112yes—
138C2×Dic71—22no—
139C2×Dic71—22no—
140C2×Dic71—22no—
141C2×Dic71—22no—
142C2×Dic71—21no—
143C2×Dic71—22no—
144C2×Dic71—21no—
145C2×Dic71—22no—
146C2×Dic71C2×C4⋊C412yes—
147C2×Dic71C2×C4⋊C412yes—
148C2×Dic71C2×C4⋊C412yes—
149C2×Dic71C2×M1614yes—
150C2×Dic71C2×M1614yes—
151C4×D71—216no—
152C4×D71—28no—
153C4×D71—22no—
154C4×D71—22no—
155C4×D71—24no—
156C4×D71—28no—
157C4×D71—216no—
158C8×C71—44no—
159C71⋊C8—44no—
160C22×C71C22×C4⋊C411yescharacteristic
161C22×C71C2×#11712yes—
162C22×C71C22×M1612yes—
163C22×C71C22×M1611yescharacteristic
164C22×C71C22×M1611yescharacteristic
165C2×D71—22no—
166C2×D71—24no—
167C2×D71—24no—
168C2×D71—22no—
169C2×D71#42314yes—
170C4×C71—22no—
171C4×C71—24no—
172C4×C71#42314yes—
173C4×C71#43218yescomplemented
174C4×C71#53211yescharacteristic
175C4×C71#53211yescharacteristic
176C4×C71#53312yes—
177Dic71—22no—
178Dic71—22no—
179Dic71—24no—
180Dic71—24no—
181Dic71—22no—
182#20810—711nomaximal, Sylow
183C2×C71#228511yescharacteristic
184C2×C71C2×#42311yescharacteristic
185C2×C71C2×#42311yescharacteristic
186C2×C71C2×#43212yes—
187C2×C71#292012yes—
188D71—24no—
189C2×#432—712no—
190C2×C43—711no—
191#1842—711no—
192#1842D7111yescharacteristic, complemented
193#1831—712no—
194#1797—714no—
195#1767—714no—
196C71#2081011yescharacteristic, complemented, Sylow
197C22×C42—711no—
198C22×C42—711no—
199C22×C42—712no—
200C22×C42—711no—
201C22×C42—712no—
202#432—1428no—
203C2×C4⋊C8—714no—
204C2×C4⋊C8—712no—
205C2×C4⋊C8C2×D7114yes—
206C2×C4⋊C8C2×D7112yes—
207C2×C22⋊C8—714no—
208C2×#97—714no—
209C43—711no—
210C43—712no—
211C43—711no—
212C43—711no—
213C43—712no—
214C43C2×D7111yescharacteristic
215#336—718no—
216C23×C4—711no—
217C23×C4—711no—
218C23×C4—712no—
219C23×C4—711no—
220C23×C4—712no—
221C22×C8—714no—
222C22×C8C22×D7114yes—
223C2×C42—711no—
224C2×C42—711no—
225C2×C42—711no—
226C2×C42—711no—
227C2×C42—714no—
228C2×C42—714no—
229C2×C42—712no—
230C2×C42—712no—
231C2×C42—712no—
232C2×C42—712no—
233C2×C42—712no—
234C2×C42—712no—
235C2×C42—711no—
236C2×C42—712no—
237C2×C42—712no—
238C2×C42—1424no—
239C2×C42—1428no—
240C2×C42—1424no—
241C2×C42—1424no—
242C2×C42—1424no—
243C2×C42C4×D7112yes—
244C2×C42C4×D7111yescharacteristic
245C2×C42C4×D7111yescharacteristic
246C2×C42C22×D7111yescharacteristic
247C2×C42C22×D7112yes—
248C4⋊C8—24no—
249C4⋊C8—28no—
250C4⋊C8—1424no—
251C4⋊C8—1428no—
252C22⋊C8—1428no—
253#97—1428no—
254C24—711no—
255C22×C4—711no—
256C22×C4—711no—
257C22×C4—711no—
258C22×C4—712no—
259C22×C4—712no—
260C22×C4—712no—
261C22×C4—712no—
262C22×C4—714no—
263C22×C4—714no—
264C22×C4—718no—
265C22×C4—1424no—
266C22×C4—1424no—
267C22×C4—1424no—
268C22×C4—1422no—
269C22×C4—1422no—
270C22×C4—1424no—
271C22×C4—1424no—
272C22×C4—1422no—
273C22×C4—1424no—
274C22×C4—1422no—
275C22×C4C2×C4×D7111yescharacteristic
276C22×C4C2×C4×D7112yes—
277C22×C4C2×C4×D7111yescharacteristic
278C22×C4C2×C4×D7112yes—
279C22×C4C23×D7111yescharacteristic
280C2×C8—24no—
281C2×C8—28no—
282C2×C8—1424no—
283C2×C8—1428no—
284C42—24no—
285C42—22no—
286C42—712no—
287C42—712no—
288C42—714no—
289C42—714no—
290C42—714no—
291C42—1424no—
292C42—1424no—
293C42—1424no—
294C42—1422no—
295C42—1422no—
296C42—1424no—
297C42—1424no—
298C42—1424no—
299C42—1424no—
300C42—1422no—
301C42#1176584814yes—
302C42D4×D7112yes—
303C42Q8×D7112yes—
304C42#1176586414yes—
305C42#1176586614yes—
306C23—712no—
307C23—714no—
308C23—1422no—
309C23—1421no—
310C23—1421no—
311C23C22×C4×D7111yescharacteristic
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—712no—
316C2×C4—712no—
317C2×C4—712no—
318C2×C4—714no—
319C2×C4—714no—
320C2×C4—1428no—
321C2×C4—14216no—
322C2×C4—1422no—
323C2×C4—1422no—
324C2×C4—1422no—
325C2×C4—1422no—
326C2×C4—1421no—
327C2×C4—1422no—
328C2×C4—1421no—
329C2×C4—1422no—
330C2×C4—14216no—
331C2×C4—1428no—
332C2×C4—1422no—
333C2×C4—1422no—
334C2×C4—1424no—
335C2×C4C4⋊C4×D7112yes—
336C2×C4C4⋊C4×D7112yes—
337C2×C4#1455608614yes—
338C2×C4#1455608614yes—
339C2×C4M16×D7114yes—
340C2×C4M16×D7112yes—
341C2×C4C2×#1176584814yes—
342C2×C4C2×#1176584812yes—
343C2×C4C2×D4×D7111yescharacteristic
344C2×C4C2×Q8×D7111yescharacteristic
345C2×C4C2×#1176586412yes—
346C2×C4C2×#1176586612yes—
347C8—44no—
348C8—2844no—
349C22—714no—
350C22—1422no—
351C22—1422no—
352C22—1424no—
353C22—1424no—
354C22C2×C4⋊C4×D7111yescharacteristic
355C22C2×#1455608612yes—
356C22C2×M16×D7112yes—
357C22C2×M16×D7111yescharacteristic
358C22C22×#1176584811yescharacteristic
359C4—22no—
360C4—24no—
361C4—1422no—
362C4—1422no—
363C4—1422no—
364C4—1424no—
365C4—1424no—
366C4#1798602114yes—
367C4#1798605518yescomplemented
368C4#1798654711yescharacteristic
369C4#1798655011yescharacteristic
370C4#1798655412yes—
371C2—1424no—
372C2#2259074511yescharacteristic
373C2#423×D7111yescharacteristic
374C2C2×#1798602111yescharacteristic
375C2C2×#1798605512yes—
376C2#2259697212yes—
subgroupG/H
center, FrattiniC22×C4C23×D71
commutator, socleC23×C71C23×C4
FittingC71×#1842C2
prime radicalC23×D71C22×C4
cosocleC22×C4×C71C24
hypercenter#1842D71
hypocenterC71#20810
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C22×C4 ◁ #1842

Complements

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

NH ≅ G/Ntype
3. #22588492C2semidirect
4. #22588456C2semidirect
6. #22588195C2semidirect
7. C71×#1842C2semidirect
173. C4×C71#432semidirect
192. #1842D71semidirect
196. C71#20810semidirect
367. C4#17986055semidirect

Maximal subgroups

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

subgroupindex
C2×#179860552
C43×D712
#225884922
#225884562
#225883172
#225881952
C71×#18422
#2081071