Finite Group Databasebeta

← #31908238

Subgroups — #31908238

Order 18176.

476 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 87 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#22588328C211yesmaximal, characteristic, complemented
2#22586678C211yesmaximal, characteristic, complemented
3#22586361C212yesmaximal, complemented
4#22586337C212yesmaximal, complemented
5C71×#2631C211yesmaximal, characteristic
6C2×#14556043—21no—
7#17985963—21no—
8#17985859—21no—
9#17985766C2212yes—
10#17985725—21no—
11#17985723C2212yes—
12C2×#124×C71C411yescharacteristic, complemented
13C2×C4×D4×C71C2211yescharacteristic
14#397×C71—21no—
15#393×C71C411yescharacteristic, complemented
16#393×C71C2211yescharacteristic
17#392×C71—21no—
18#392×C71C411yescharacteristic, complemented
19#392×C71C2211yescharacteristic
20#386×C71—21no—
21#386×C71—21no—
22#383×C71C411yescharacteristic, complemented
23C2×#11765849—41no—
24C2×C4⋊Dic71—41no—
25C2×C22⋊Dic71—22no—
26#14556043—42no—
27C22⋊(C71⋊C8)—22no—
28C22⋊(C71⋊C8)—41no—
29C71⋊#95—41no—
30C22×Q8×C71C2×C411yescharacteristic
31C22×D4×C71—21no—
32C22×D4×C71—21no—
33C22×D4×C71C2×C411yescharacteristic
34C22×D4×C71D411yescharacteristic
35C23×C4×C71D411yescharacteristic
36C23×C4×C71D411yescharacteristic
37#124×C71—41no—
38#124×C71—41no—
39C4×D4×C71—41no—
40C4×D4×C71—41no—
41C2×C4⋊C4×C71D411yescharacteristic
42C2×C22⋊C4×C71—21no—
43C2×C22⋊C4×C71—21no—
44C2×C22⋊C4×C71—21no—
45C2×C22⋊C4×C71—21no—
46C2×C22⋊C4×C71—21no—
47C2×C22⋊C4×C71—21no—
48C2×C22⋊C4×C71C2×C411yescharacteristic
49C2×C22⋊C4×C71C2311yescharacteristic
50C2×C42×C71C2×C411yescharacteristic
51#95×C71—21no—
52#95×C71—21no—
53#95×C71—21no—
54#95×C71—21no—
55#95×C71C2×C411yescharacteristic
56#95×C71C2×C411yescharacteristic
57C22×Dic71—42no—
58#11765849—82no—
59C2×C71⋊C8—42no—
60C4⋊Dic71—82no—
61C22⋊Dic71—42no—
62C22⋊Dic71—42no—
63C24×C71—21no—
64C24×C71C22⋊C411yescharacteristic
65C24×C71C2×D411yescharacteristic
66C2×Q8×C71—41no—
67C2×Q8×C71—81no—
68C2×D4×C71—21no—
69C2×D4×C71—21no—
70C2×D4×C71—41no—
71C2×D4×C71—41no—
72C2×D4×C71—41no—
73C2×D4×C71—42no—
74C2×D4×C71—41no—
75C2×D4×C71—41no—
76C2×D4×C71—81no—
77C2×D4×C71—81no—
78C2×D4×C71—81no—
79C2×D4×C71—81no—
80C2×D4×C71—81no—
81C22×C4×C71—21no—
82C22×C4×C71—21no—
83C22×C4×C71—21no—
84C22×C4×C71—21no—
85C22×C4×C71—21no—
86C22×C4×C71—21no—
87C22×C4×C71—21no—
88C22×C4×C71—81no—
89C22×C4×C71—81no—
90C22×C4×C71C22⋊C411yescharacteristic
91C22×C4×C71C22⋊C411yescharacteristic
92C22×C4×C71C22⋊C411yescharacteristic
93C22×C4×C71C22×C411yescharacteristic
94C22×C4×C71C2×D411yescharacteristic
95C4⋊C4×C71—41no—
96C22⋊C4×C71—41no—
97C22⋊C4×C71—41no—
98C22⋊C4×C71—42no—
99C22⋊C4×C71—41no—
100C22⋊C4×C71—41no—
101C22⋊C4×C71—81no—
102C22⋊C4×C71—81no—
103C22⋊C4×C71—81no—
104C22⋊C4×C71—81no—
105C42×C71—21no—
106C42×C71—21no—
107C23×C71—21no—
108C23×C71—21no—
109C23×C71—21no—
110C23×C71—21no—
111C23×C71—21no—
112C23×C71—41no—
113C23×C71—41no—
114C23×C71—41no—
115C23×C71—41no—
116C23×C71—41no—
117C23×C71—41no—
118C23×C71—41no—
119C23×C71—41no—
120C23×C71—41no—
121C23×C71—41no—
122C23×C71—41no—
123C23×C71C2×C22⋊C411yescharacteristic
124C23×C71AGL(1,ℤ/8)11yescharacteristic
125C23×C71#13711yescharacteristic
126Q8×C71—82no—
127D4×C71—42no—
128D4×C71—42no—
129D4×C71—82no—
130D4×C71—82no—
131D4×C71—82no—
132D4×C71—84no—
133C2×C4×C71—21no—
134C2×C4×C71—21no—
135C2×C4×C71—21no—
136C2×C4×C71—21no—
137C2×C4×C71—21no—
138C2×C4×C71—21no—
139C2×C4×C71—41no—
140C2×C4×C71—42no—
141C2×C4×C71—42no—
142C2×C4×C71—41no—
143C2×C4×C71—41no—
144C2×C4×C71—41no—
145C2×C4×C71—41no—
146C2×C4×C71—41no—
147C2×C4×C71—41no—
148C2×C4×C71—41no—
149C2×C4×C71—41no—
150C2×C4×C71—41no—
151C2×C4×C71—41no—
152C2×C4×C71—41no—
153C2×C4×C71—41no—
154C2×C4×C71—81no—
155C2×C4×C71—81no—
156C2×C4×C71#9911yescharacteristic
157C2×C4×C71#9911yescharacteristic
158C2×Dic71—82no—
159C2×Dic71—82no—
160C2×Dic71—82no—
161C71⋊C8—82no—
162C22×C71—21no—
163C22×C71—21no—
164C22×C71—21no—
165C22×C71—21no—
166C22×C71—41no—
167C22×C71—41no—
168C22×C71—42no—
169C22×C71—41no—
170C22×C71—41no—
171C22×C71—41no—
172C22×C71—41no—
173C22×C71—41no—
174C22×C71—81no—
175C22×C71—81no—
176C22×C71—81no—
177C22×C71—81no—
178C22×C71—81no—
179C22×C71—81no—
180C22×C71—81no—
181C22×C71—81no—
182C22×C71—81no—
183C22×C71C2×#9911yescharacteristic
184C22×C71#41711yescharacteristic
185C22×C71#42111yescharacteristic
186C4×C71—21no—
187C4×C71—21no—
188C4×C71—41no—
189C4×C71—41no—
190C4×C71—41no—
191C4×C71—41no—
192C4×C71—81no—
193C4×C71—81no—
194C4×C71—81no—
195Dic71—162no—
196#11825—711nomaximal, Sylow
197C2×C71—21no—
198C2×C71—21no—
199C2×C71—41no—
200C2×C71—41no—
201C2×C71—81no—
202C2×C71—161no—
203C2×C71#151311yescharacteristic
204C2×C71#212211yescharacteristic
205C2×C71#212711yescharacteristic
206#2631D7111yescharacteristic
207#1799—711no—
208#1465—711no—
209#1351—712no—
210#1343—712no—
211C71#1182511yescharacteristic, complemented, Sylow
212C2×#124Dic7111yescharacteristic, complemented
213C2×C4×D4C2×D7111yescharacteristic
214C2×#99—1421no—
215#407—1421no—
216#397—21no—
217#393Dic7111yescharacteristic, complemented
218#393C2×D7111yescharacteristic
219#392—21no—
220#392Dic7111yescharacteristic, complemented
221#392C2×D7111yescharacteristic
222#386—21no—
223#386—21no—
224#383Dic7111yescharacteristic, complemented
225#380—1421no—
226#342—712no—
227#324—1421no—
228#323—712no—
229C22×Q8C2×Dic7111yescharacteristic
230C22×D4—21no—
231C22×D4—21no—
232C22×D4C2×Dic7111yescharacteristic
233C22×D4C22⋊D7111yescharacteristic
234C23×C4C22⋊D7111yescharacteristic
235C23×C4C22⋊D7111yescharacteristic
236C2×M16—2841no—
237#124—41no—
238#124—41no—
239C4×D4—41no—
240C4×D4—41no—
241C2×C4⋊C4—2841no—
242C2×C4⋊C4C22⋊D7111yescharacteristic
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—1422no—
250C2×C22⋊C4C2×Dic7111yescharacteristic
251C2×C22⋊C4C22×D7111yescharacteristic
252C2×C42C2×Dic7111yescharacteristic
253#99—2842no—
254C22⋊C8—2841no—
255C22⋊C8—1422no—
256#95—21no—
257#95—21no—
258#95—21no—
259#95—21no—
260#95—2841no—
261#95C2×Dic7111yescharacteristic
262#95C2×Dic7111yescharacteristic
263C24—21no—
264C24C22⋊Dic7111yescharacteristic
265C24C2×C22⋊D7111yescharacteristic
266C2×Q8—41no—
267C2×Q8—81no—
268C2×D4—21no—
269C2×D4—21no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—42no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—81no—
277C2×D4—81no—
278C2×D4—81no—
279C2×D4—81no—
280C2×D4—81no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—81no—
289C22×C4—81no—
290C22×C4—2842no—
291C22×C4C22⋊Dic7111yescharacteristic
292C22×C4C22⋊Dic7111yescharacteristic
293C22×C4C22⋊Dic7111yescharacteristic
294C22×C4C22×Dic7111yescharacteristic
295C22×C4C2×C22⋊D7111yescharacteristic
296M16—5682no—
297C2×C8—2842no—
298C4⋊C4—41no—
299C4⋊C4—5682no—
300C22⋊C4—41no—
301C22⋊C4—41no—
302C22⋊C4—42no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C22⋊C4—81no—
306C22⋊C4—81no—
307C22⋊C4—81no—
308C22⋊C4—81no—
309C22⋊C4—2842no—
310C22⋊C4—2842no—
311C42—21no—
312C42—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23C2×C22⋊Dic7111yescharacteristic
330C23#1455615311yescharacteristic
331C23#1455615811yescharacteristic
332Q8—82no—
333D4—42no—
334D4—42no—
335D4—82no—
336D4—82no—
337D4—82no—
338D4—84no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—5682no—
363C2×C4—5682no—
364C2×C4—5682no—
365C2×C4#1455604311yescharacteristic
366C2×C4#1455604311yescharacteristic
367C8—5682no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—41no—
373C22—41no—
374C22—42no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22—81no—
386C22—81no—
387C22—81no—
388C22—81no—
389C22C2×#1455604311yescharacteristic
390C22#1798600011yescharacteristic
391C22#1798601611yescharacteristic
392C4—21no—
393C4—21no—
394C4—41no—
395C4—41no—
396C4—41no—
397C4—41no—
398C4—81no—
399C4—81no—
400C4—81no—
401C4—11362no—
402C2—21no—
403C2—21no—
404C2—41no—
405C2—41no—
406C2—81no—
407C2—161no—
408C2#2258688311yescharacteristic
409C2#2258999011yescharacteristic
410C2#2259000911yescharacteristic
subgroupG/H
centerC22C2×#14556043
commutatorC22×C4×C71C22×C4
FrattiniC2×C22⋊C4C22×D71
Fitting, prime radicalC71×#2631C2
socleC22×C71C2×#99
cosocleC2×C22⋊C4×C71C23
hypercenter#2631D71
hypocenterC71#11825
LCSG ▷ C22×C4×C71 ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2631

Complements

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

NH ≅ G/Ntype
1. #22588328C2semidirect
2. #22586678C2semidirect
3. #22586361C2semidirect
4. #22586337C2semidirect
12. C2×#124×C71C4semidirect
15. #393×C71C4semidirect
18. #392×C71C4semidirect
22. #383×C71C4semidirect
211. C71#11825semidirect
212. C2×#124Dic71semidirect
217. #393Dic71semidirect
220. #392Dic71semidirect
224. #383Dic71semidirect

Maximal subgroups

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

subgroupindex
#225883282
#225866782
#225863612
#225863372
C71×#26312
#1182571