Finite Group Databasebeta

← #31889959

Subgroups — #31889959

Order 18176.

754 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 754 conjugacy classes of subgroups — too many to draw. Only its 131 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22588423C211yesmaximal, characteristic, complemented
2#22588417C211yesmaximal, characteristic, complemented
3#22586873C212yesmaximal, complemented
4#22586869C212yesmaximal
5C2×#542×C71C211yesmaximal, characteristic
6#17985963C2212yes—
7#17985862C2212yes—
8#17985768—21no—
9#17985768—21no—
10#17985766—21no—
11#17985766—21no—
12#17985736—24no—
13#17985733—24no—
14#542×C71—24no—
15#542×C71—24no—
16C2×C4⋊Q8×C71C411yescharacteristic, complemented
17C2×#126×C71—21no—
18C2×#124×C71—21no—
19C2×#124×C71C411yescharacteristic, complemented
20C2×#123×C71—21no—
21C2×C22⋊Q8×C71—21no—
22C2×C22⋊Q8×C71C2211yescharacteristic
23C2×#121×C71C2211yescharacteristic
24C2×C4×Q8×C71C411yescharacteristic, complemented
25C2×C4×D4×C71C411yescharacteristic, complemented
26C2×#117×C71C2211yescharacteristic
27C2×#11765849—22no—
28C2×C22⋊Dic71—22no—
29C22⋊(C71⋊C8)—22no—
30C22⋊(C71⋊C8)—22no—
31C71⋊#95—22no—
32C71⋊#95—22no—
33C22×Q8×C71—21no—
34C22×Q8×C71C2×C411yescharacteristic
35C22×D4×C71—21no—
36C22×D4×C71C2×C411yescharacteristic
37C23×C4×C71—21no—
38C23×C4×C71C2311yescharacteristic
39C4⋊Q8×C71—22no—
40C4⋊Q8×C71—22no—
41#126×C71—28no—
42#124×C71—28no—
43#124×C71—22no—
44#124×C71—22no—
45#123×C71—28no—
46C22⋊Q8×C71—28no—
47C22⋊Q8×C71—22no—
48C22⋊Q8×C71D414yes—
49#121×C71—22no—
50#121×C71D414yes—
51C4×Q8×C71—22no—
52C4×Q8×C71—22no—
53C4×D4×C71—22no—
54C4×D4×C71—22no—
55#117×C71—22no—
56#117×C71—22no—
57C2×C4⋊C4×C71—21no—
58C2×C4⋊C4×C71—21no—
59C2×C4⋊C4×C71—21no—
60C2×C4⋊C4×C71C2×C411yescharacteristic
61C2×C4⋊C4×C71C2×C411yescharacteristic
62C2×C4⋊C4×C71D411yescharacteristic
63C2×C4⋊C4×C71D411yescharacteristic
64C2×C22⋊C4×C71—21no—
65C2×C22⋊C4×C71—21no—
66C2×C22⋊C4×C71—21no—
67C2×C22⋊C4×C71—21no—
68C2×C22⋊C4×C71—21no—
69C2×C22⋊C4×C71D411yescharacteristic
70C2×C22⋊C4×C71D411yescharacteristic
71C2×C42×C71—21no—
72C2×C42×C71C2×C411yescharacteristic
73C2×C42×C71C2×C411yescharacteristic
74C22×Dic71—42no—
75#11765849—44no—
76C2×C71⋊C8—42no—
77C22⋊Dic71—44no—
78C24×C71—21no—
79C24×C71C2×D411yescharacteristic
80C2×Q8×C71—24no—
81C2×Q8×C71—21no—
82C2×Q8×C71—42no—
83C2×Q8×C71—82no—
84C2×Q8×C71C22⋊C412yes—
85C2×D4×C71—24no—
86C2×D4×C71—21no—
87C2×D4×C71—42no—
88C2×D4×C71—82no—
89C2×D4×C71C22⋊C412yes—
90C22×C4×C71—21no—
91C22×C4×C71—21no—
92C22×C4×C71—21no—
93C22×C4×C71—24no—
94C22×C4×C71—42no—
95C22×C4×C71—82no—
96C22×C4×C71C22⋊C411yescharacteristic
97C22×C4×C71C22⋊C411yescharacteristic
98C22×C4×C71C22⋊C411yescharacteristic
99C22×C4×C71C22⋊C411yescharacteristic
100C22×C4×C71C22×C411yescharacteristic
101C22×C4×C71C2×D411yescharacteristic
102C22×C4×C71C2×D412yes—
103C22×C4×C71C2×D412yes—
104C4⋊C4×C71—24no—
105C4⋊C4×C71—24no—
106C4⋊C4×C71—21no—
107C4⋊C4×C71—21no—
108C4⋊C4×C71—24no—
109C4⋊C4×C71—22no—
110C4⋊C4×C71—22no—
111C4⋊C4×C71C22⋊C412yes—
112C4⋊C4×C71C22⋊C412yes—
113C22⋊C4×C71—24no—
114C22⋊C4×C71—24no—
115C22⋊C4×C71—24no—
116C22⋊C4×C71—24no—
117C22⋊C4×C71—22no—
118C22⋊C4×C71—22no—
119C22⋊C4×C71—24no—
120C42×C71—24no—
121C42×C71—21no—
122C42×C71—21no—
123C42×C71—21no—
124C42×C71—21no—
125C23×C71—22no—
126C23×C71—22no—
127C23×C71—41no—
128C23×C71—41no—
129C23×C71—42no—
130C23×C71—42no—
131C23×C71—42no—
132C23×C71C2×C22⋊C411yescharacteristic
133C23×C71#12012yes—
134Q8×C71—42no—
135Q8×C71—42no—
136Q8×C71—84no—
137D4×C71—42no—
138D4×C71—42no—
139D4×C71—84no—
140C2×C4×C71—22no—
141C2×C4×C71—22no—
142C2×C4×C71—21no—
143C2×C4×C71—21no—
144C2×C4×C71—22no—
145C2×C4×C71—21no—
146C2×C4×C71—21no—
147C2×C4×C71—22no—
148C2×C4×C71—22no—
149C2×C4×C71—42no—
150C2×C4×C71—42no—
151C2×C4×C71—42no—
152C2×C4×C71—42no—
153C2×C4×C71—42no—
154C2×C4×C71—41no—
155C2×C4×C71—41no—
156C2×C4×C71—41no—
157C2×C4×C71—41no—
158C2×C4×C71—84no—
159C2×C4×C71C2×C22⋊C411yescharacteristic
160C2×C4×C71C2×C22⋊C411yescharacteristic
161C2×C4×C71#12012yes—
162C2×Dic71—82no—
163C2×Dic71—82no—
164C2×Dic71—82no—
165C71⋊C8—82no—
166C22×C71—21no—
167C22×C71—21no—
168C22×C71—22no—
169C22×C71—22no—
170C22×C71—22no—
171C22×C71—42no—
172C22×C71—42no—
173C22×C71—42no—
174C22×C71—82no—
175C22×C71—84no—
176C22×C71#37911yescharacteristic
177C22×C71#41012yes—
178C4×C71—22no—
179C4×C71—21no—
180C4×C71—21no—
181C4×C71—42no—
182C4×C71—42no—
183C4×C71—41no—
184C4×C71—41no—
185C4×C71—41no—
186C4×C71—41no—
187C4×C71—42no—
188Dic71—162no—
189#8519—711nomaximal, Sylow
190C2×C71—22no—
191C2×C71—42no—
192C2×C71—82no—
193C2×C71#178211yescharacteristic
194C2×C71#179511yescharacteristic
195C2×C71#179611yescharacteristic
196C2×#542D7111yescharacteristic
197#1822—711no—
198#1820—711no—
199#1511—712no—
200#1510—712no—
201C71#851911yescharacteristic, complemented, Sylow
202#542—24no—
203#542—24no—
204C2×C4⋊Q8Dic7111yescharacteristic, complemented
205C2×#126—21no—
206C2×#124—21no—
207C2×#124Dic7111yescharacteristic, complemented
208C2×#123—21no—
209C2×C22⋊Q8—21no—
210C2×C22⋊Q8C2×D7111yescharacteristic
211C2×#121C2×D7111yescharacteristic
212C2×C4×Q8Dic7111yescharacteristic, complemented
213C2×C4×D4Dic7111yescharacteristic, complemented
214C2×#117C2×D7111yescharacteristic
215#407—712no—
216#381—712no—
217#343—1421no—
218#343—1421no—
219#342—1421no—
220#342—1421no—
221#328—1424no—
222#327—1424no—
223C22×Q8—21no—
224C22×Q8C2×Dic7111yescharacteristic
225C22×D4—21no—
226C22×D4C2×Dic7111yescharacteristic
227C23×C4—21no—
228C23×C4C22×D7111yescharacteristic
229C2×M16—1422no—
230C4⋊Q8—22no—
231C4⋊Q8—22no—
232#126—28no—
233#124—28no—
234#124—22no—
235#124—22no—
236#123—28no—
237C22⋊Q8—28no—
238C22⋊Q8—22no—
239C22⋊Q8C22⋊D7114yes—
240#121—22no—
241#121C22⋊D7114yes—
242C4×Q8—22no—
243C4×Q8—22no—
244C4×D4—22no—
245C4×D4—22no—
246#117—22no—
247#117—22no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4C2×Dic7111yescharacteristic
252C2×C4⋊C4C2×Dic7111yescharacteristic
253C2×C4⋊C4C22⋊D7111yescharacteristic
254C2×C4⋊C4C22⋊D7111yescharacteristic
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—1422no—
261C2×C22⋊C4C22⋊D7111yescharacteristic
262C2×C22⋊C4C22⋊D7111yescharacteristic
263C2×C42—21no—
264C2×C42C2×Dic7111yescharacteristic
265C2×C42C2×Dic7111yescharacteristic
266C22⋊C8—1422no—
267C22⋊C8—1422no—
268#95—1422no—
269#95—1422no—
270C24—21no—
271C24C2×C22⋊D7111yescharacteristic
272C2×Q8—24no—
273C2×Q8—21no—
274C2×Q8—42no—
275C2×Q8—82no—
276C2×Q8C22⋊Dic7112yes—
277C2×D4—24no—
278C2×D4—21no—
279C2×D4—42no—
280C2×D4—82no—
281C2×D4C22⋊Dic7112yes—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—24no—
286C22×C4—42no—
287C22×C4—82no—
288C22×C4—2842no—
289C22×C4C22⋊Dic7111yescharacteristic
290C22×C4C22⋊Dic7111yescharacteristic
291C22×C4C22⋊Dic7111yescharacteristic
292C22×C4C22⋊Dic7111yescharacteristic
293C22×C4C22×Dic7111yescharacteristic
294C22×C4C2×C22⋊D7111yescharacteristic
295C22×C4C2×C22⋊D7112yes—
296C22×C4C2×C22⋊D7112yes—
297M16—2844no—
298C2×C8—2842no—
299C4⋊C4—24no—
300C4⋊C4—24no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—24no—
304C4⋊C4—22no—
305C4⋊C4—22no—
306C4⋊C4C22⋊Dic7112yes—
307C4⋊C4C22⋊Dic7112yes—
308C22⋊C4—24no—
309C22⋊C4—24no—
310C22⋊C4—24no—
311C22⋊C4—24no—
312C22⋊C4—22no—
313C22⋊C4—22no—
314C22⋊C4—24no—
315C22⋊C4—2844no—
316C42—24no—
317C42—21no—
318C42—21no—
319C42—21no—
320C42—21no—
321C23—22no—
322C23—22no—
323C23—41no—
324C23—41no—
325C23—42no—
326C23—42no—
327C23—42no—
328C23C2×C22⋊Dic7111yescharacteristic
329C23#1455609712yes—
330Q8—42no—
331Q8—42no—
332Q8—84no—
333D4—42no—
334D4—42no—
335D4—84no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—84no—
355C2×C4—5682no—
356C2×C4—5682no—
357C2×C4—5682no—
358C2×C4C2×C22⋊Dic7111yescharacteristic
359C2×C4C2×C22⋊Dic7111yescharacteristic
360C2×C4#1455609712yes—
361C8—5682no—
362C22—21no—
363C22—21no—
364C22—22no—
365C22—22no—
366C22—22no—
367C22—42no—
368C22—42no—
369C22—42no—
370C22—82no—
371C22—84no—
372C22#1798585511yescharacteristic
373C22#1798597312yes—
374C4—22no—
375C4—21no—
376C4—21no—
377C4—42no—
378C4—42no—
379C4—42no—
380C4—41no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—11362no—
385C2—22no—
386C2—42no—
387C2—82no—
388C2#2258825811yescharacteristic
389C2#2258831011yescharacteristic
390C2#2258831511yescharacteristic
subgroupG/H
centerC22#17985855
commutatorC22×C4×C71C22×C4
FrattiniC23×C4C22×D71
FittingC2×#542×C71C2
socleC22×C71#379
prime radicalC2×#121×C71C22
cosocleC23×C4×C71C23
hypercenterC2×#542D71
hypocenterC71#8519
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#542

Complements

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

NH ≅ G/Ntype
1. #22588423C2semidirect
2. #22588417C2semidirect
3. #22586873C2semidirect
16. C2×C4⋊Q8×C71C4semidirect
19. C2×#124×C71C4semidirect
24. C2×C4×Q8×C71C4semidirect
25. C2×C4×D4×C71C4semidirect
201. C71#8519semidirect
204. C2×C4⋊Q8Dic71semidirect
207. C2×#124Dic71semidirect
212. C2×C4×Q8Dic71semidirect
213. C2×C4×D4Dic71semidirect

Maximal subgroups

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

subgroupindex
#225884232
#225884172
#225868732
#225868692
C2×#542×C712
#851971