Finite Group Databasebeta

← #31897498

Subgroups — #31897498

Order 18176.

572 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17985883C211yesmaximal, characteristic
2C4×C22⋊(C71⋊C8)C211yesmaximal, characteristic
3#22586182C212yesmaximal
4#22586182C212yesmaximal
5C71×#1797C211yesmaximal, characteristic
6C22×Dic71⋊C4C411yescharacteristic
7C22×Dic71⋊C4C411yescharacteristic
8C2×C22⋊(C71⋊C8)C2212yes—
9C2×C4×C71⋊C8—21no—
10#17985883—21no—
11#17985883—21no—
12#17985883—21no—
13#17985883—21no—
14C2×C71⋊#95C411yescharacteristic
15C2×C71⋊#95C411yescharacteristic
16C2×C71⋊#95C2211yescharacteristic
17C2×C71⋊#95C2211yescharacteristic
18#17985754—21no—
19C22×C42×C71C2211yescharacteristic
20C2×C4⋊C8×C71—21no—
21C2×C22⋊C8×C71C2212yes—
22#336×C71—21no—
23C23×Dic71C2×C411yescharacteristic
24C23×Dic71C2×C411yescharacteristic
25C23×Dic71C2×C411yescharacteristic
26C23×Dic71C2×C411yescharacteristic
27C22×C71⋊C8—22no—
28C2×Dic71⋊C4—21no—
29C2×Dic71⋊C4—21no—
30C2×Dic71⋊C4—21no—
31C2×Dic71⋊C4—21no—
32C2×Dic71⋊C4—22no—
33C2×Dic71⋊C4—42no—
34C2×Dic71⋊C4C814yescomplemented
35C22⋊(C71⋊C8)—22no—
36C22⋊(C71⋊C8)Q814yes—
37C4×C71⋊C8—42no—
38C71⋊#95—21no—
39C71⋊#95—21no—
40C71⋊#95—21no—
41C71⋊#95—21no—
42C71⋊#95—21no—
43C71⋊#95—21no—
44C71⋊#95—21no—
45C71⋊#95—21no—
46C23×C4×C71C2×C411yescharacteristic
47C23×C4×C71C2×C411yescharacteristic
48C23×C4×C71C2311yescharacteristic
49C22×C8×C71—22no—
50C2×C42×C71—42no—
51C2×C42×C71D411yescharacteristic
52C2×C42×C71D411yescharacteristic
53C2×C42×C71D411yescharacteristic
54C2×C42×C71D411yescharacteristic
55C4⋊C8×C71—42no—
56C22⋊C8×C71—24no—
57C22×Dic71—21no—
58C22×Dic71—21no—
59C22×Dic71—21no—
60C22×Dic71—21no—
61C22×Dic71—21no—
62C22×Dic71—21no—
63C22×Dic71—22no—
64C22×Dic71—22no—
65C22×Dic71—41no—
66C22×Dic71—41no—
67C22×Dic71—41no—
68C22×Dic71—41no—
69C22×Dic71—41no—
70C22×Dic71—41no—
71C22×Dic71—41no—
72C22×Dic71—41no—
73C22×Dic71C2×C812yes—
74C22×Dic71C2×C812yes—
75C2×C71⋊C8—24no—
76C2×C71⋊C8—42no—
77C2×C71⋊C8—42no—
78Dic71⋊C4—24no—
79Dic71⋊C4—24no—
80Dic71⋊C4—42no—
81Dic71⋊C4—42no—
82C24×C71C22×C411yescharacteristic
83C22×C4×C71—22no—
84C22×C4×C71—22no—
85C22×C4×C71—41no—
86C22×C4×C71—41no—
87C22×C4×C71—41no—
88C22×C4×C71—41no—
89C22×C4×C71—41no—
90C22×C4×C71C22⋊C411yescharacteristic
91C22×C4×C71C22⋊C411yescharacteristic
92C22×C4×C71C22⋊C411yescharacteristic
93C22×C4×C71C22⋊C411yescharacteristic
94C22×C4×C71C2×C812yes—
95C22×C4×C71M1612yes—
96C22×C4×C71C2×D411yescharacteristic
97C22×C4×C71C2×D411yescharacteristic
98C22×C4×C71C2×Q812yes—
99C22×C4×C71#4112yes—
100C2×C8×C71—42no—
101C2×C8×C71—42no—
102C2×C8×C71—42no—
103C42×C71—22no—
104C42×C71—22no—
105C42×C71—41no—
106C42×C71—41no—
107C23×C71—21no—
108C23×C71—21no—
109C23×C71—21no—
110C23×C71—21no—
111C23×C71C2×C22⋊C411yescharacteristic
112C23×C71C4×Q812yes—
113C23×C71C22×C811yescharacteristic
114C23×C71C2×M1611yescharacteristic
115C23×C71#13112yes—
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—42no—
130C2×C4×C71—42no—
131C2×C4×C71—42no—
132C2×C4×C71—42no—
133C2×C4×C71—44no—
134C2×C4×C71C22⋊C812yes—
135C2×C4×C71C22⋊C812yes—
136C2×C4×C71#9912yes—
137C2×C4×C71C22⋊Q812yes—
138C2×C4×C71#12411yescharacteristic
139C2×C4×C71C4⋊Q811yescharacteristic
140C2×Dic71—22no—
141C2×Dic71—22no—
142C2×Dic71—42no—
143C2×Dic71—42no—
144C2×Dic71—41no—
145C2×Dic71—41no—
146C2×Dic71—41no—
147C2×Dic71—41no—
148C2×Dic71—42no—
149C2×Dic71—42no—
150C2×Dic71—41no—
151C2×Dic71—41no—
152C2×Dic71—41no—
153C2×Dic71—41no—
154C2×Dic71—41no—
155C2×Dic71—41no—
156C2×Dic71—82no—
157C2×Dic71—82no—
158C8×C71—82no—
159C71⋊C8—44no—
160C22×C71—21no—
161C22×C71—21no—
162C22×C71—21no—
163C22×C71—21no—
164C22×C71—22no—
165C22×C71—22no—
166C22×C71—22no—
167C22×C71—22no—
168C22×C71#39111yescharacteristic
169C22×C71C2×C22⋊C811yescharacteristic
170C22×C71#40811yescharacteristic
171C22×C71C2×#9911yescharacteristic
172C22×C71#41011yescharacteristic
173C22×C71#41311yescharacteristic
174C22×C71#41311yescharacteristic
175C22×C71C8×Q812yes—
176C22×C71#44612yes—
177C4×C71—22no—
178C4×C71—22no—
179C4×C71—22no—
180C4×C71—22no—
181C4×C71—21no—
182C4×C71—21no—
183C4×C71—41no—
184Dic71—42no—
185Dic71—42no—
186Dic71—81no—
187Dic71—81no—
188#9880—711nomaximal, Sylow
189C2×C71—22no—
190C2×C71—22no—
191C2×C71#145811yescharacteristic
192C2×C71#145911yescharacteristic
193C2×C71#146811yescharacteristic
194C2×C71#147311yescharacteristic
195C2×C71#198411yescharacteristic
196C2×C71#199611yescharacteristic
197C2×C71#199811yescharacteristic
198C2×#387—711no—
199#1797D7111yescharacteristic
200C4×C22⋊C8—711no—
201#1280—712no—
202#1280—712no—
203C71#988011yescharacteristic, complemented, Sylow
204C22×C4⋊C4—711no—
205C22×C4⋊C4—711no—
206C22×C42C2×D7111yescharacteristic
207C2×C4⋊C8—21no—
208C2×C22⋊C8—712no—
209C2×C22⋊C8C2×D7112yes—
210C2×C4×C8—1421no—
211#387—1421no—
212#387—1421no—
213#387—1421no—
214#387—1421no—
215C2×#95—711no—
216C2×#95—711no—
217C2×#95—711no—
218C2×#95—711no—
219#336—21no—
220#336—1421no—
221C23×C4—711no—
222C23×C4—711no—
223C23×C4—711no—
224C23×C4—711no—
225C23×C4C4×D7111yescharacteristic
226C23×C4C4×D7111yescharacteristic
227C23×C4C22×D7111yescharacteristic
228C22×C8—22no—
229C22×C8—1422no—
230C2×C4⋊C4—2842no—
231C2×C4⋊C4—714no—
232C2×C4⋊C4—1421no—
233C2×C4⋊C4—1421no—
234C2×C4⋊C4—1421no—
235C2×C4⋊C4—1421no—
236C2×C4⋊C4—1422no—
237C2×C42—42no—
238C2×C42D28411yescharacteristic
239C2×C42D28411yescharacteristic
240C2×C42C22⋊D7111yescharacteristic
241C2×C42C22⋊D7111yescharacteristic
242C4⋊C8—42no—
243C22⋊C8—24no—
244C22⋊C8—714no—
245C22⋊C8—1422no—
246C4×C8—2842no—
247#95—1421no—
248#95—1421no—
249#95—1421no—
250#95—1421no—
251#95—1421no—
252#95—1421no—
253#95—1421no—
254#95—1421no—
255C24C2×C4×D7111yescharacteristic
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—2841no—
264C22×C4—2841no—
265C22×C4—2841no—
266C22×C4—2841no—
267C22×C4—2841no—
268C22×C4—2841no—
269C22×C4—2841no—
270C22×C4—2841no—
271C22×C4—712no—
272C22×C4—712no—
273C22×C4—1421no—
274C22×C4—1421no—
275C22×C4—1421no—
276C22×C4—1421no—
277C22×C4—1421no—
278C22×C4—1421no—
279C22×C4—1422no—
280C22×C4—1422no—
281C22×C4(C2×D71)⋊C411yescharacteristic
282C22×C4(C2×D71)⋊C411yescharacteristic
283C22×C4(C2×D71)⋊C411yescharacteristic
284C22×C4(C2×D71)⋊C411yescharacteristic
285C22×C4C8×D7112yes—
286C22×C4#1176584812yes—
287C22×C4C2×C22⋊D7111yescharacteristic
288C22×C4C2×D28411yescharacteristic
289C22×C4Q8×D7112yes—
290C22×C4#1176586412yes—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—2842no—
295C2×C8—2842no—
296C2×C8—1424no—
297C4⋊C4—2842no—
298C4⋊C4—2842no—
299C4⋊C4—1424no—
300C4⋊C4—1424no—
301C42—22no—
302C42—22no—
303C42—41no—
304C42—41no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23C2×(C2×D71)⋊C411yescharacteristic
310C23#1455609312yes—
311C23C2×C8×D7111yescharacteristic
312C23C2×#1176584811yescharacteristic
313C23#1455613612yes—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—44no—
332C2×C4—2842no—
333C2×C4—2842no—
334C2×C4—2841no—
335C2×C4—2841no—
336C2×C4—2841no—
337C2×C4—2841no—
338C2×C4—2842no—
339C2×C4—2842no—
340C2×C4—2841no—
341C2×C4—2841no—
342C2×C4—2841no—
343C2×C4—2841no—
344C2×C4—2841no—
345C2×C4—2841no—
346C2×C4—5682no—
347C2×C4—5682no—
348C2×C4—1422no—
349C2×C4—1422no—
350C2×C4(C2×D71)⋊C812yes—
351C2×C4(C2×D71)⋊C812yes—
352C2×C4#1455604212yes—
353C2×C4#1455610812yes—
354C2×C4#1455611511yescharacteristic
355C2×C4#1455612611yescharacteristic
356C8—82no—
357C8—2844no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—22no—
363C22—22no—
364C22—22no—
365C22—22no—
366C22#1798590411yescharacteristic
367C22C2×(C2×D71)⋊C811yescharacteristic
368C22#1798596711yescharacteristic
369C22C2×#1455604211yescharacteristic
370C22#1798597211yescharacteristic
371C22#1798598111yescharacteristic
372C22#1798598111yescharacteristic
373C22#1798612312yes—
374C22#1798612612yes—
375C4—22no—
376C4—22no—
377C4—22no—
378C4—21no—
379C4—22no—
380C4—21no—
381C4—41no—
382C4—2842no—
383C4—2842no—
384C4—5681no—
385C4—5681no—
386C2—22no—
387C2—22no—
388C2#2258665211yescharacteristic
389C2#2258665611yescharacteristic
390C2#2258668911yescharacteristic
391C2#2258671011yescharacteristic
392C2#2258927511yescharacteristic
393C2#2258932411yescharacteristic
394C2#2258933211yescharacteristic
subgroupG/H
centerC23C2×(C2×D71)⋊C4
commutatorC23×C71C22×C8
FrattiniC23×C4C22×D71
FittingC71×#1797C2
socleC23×C71C2×C22⋊C4
prime radicalC24×C71C22×C4
cosocleC23×C4×C71C23
hypercenter#1797D71
hypocenterC71#9880
LCSG ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1797

Complements

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

NH ≅ G/Ntype
34. C2×Dic71⋊C4C8semidirect
203. C71#9880semidirect

Maximal subgroups

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

subgroupindex
C2×#179858832
C4×C22⋊(C71⋊C8)2
#225861822
C71×#17972
#988071