Finite Group Databasebeta

← #31908476

Subgroups — #31908476

Order 18176.

536 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 103 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#22586658C211yesmaximal, characteristic, complemented
3#22586340C212yesmaximal, complemented
4#22586337C212yesmaximal, complemented
5C71×#2535C211yesmaximal, characteristic
6C2×#14556043—21no—
7C2×C22⋊(C71⋊C8)—21no—
8#17985859—21no—
9#17985766C2212yes—
10#17985725—21no—
11#17985723C2212yes—
12C2×#121×C71—21no—
13C2×C4×D4×C71C411yescharacteristic, complemented
14C2×C4×D4×C71C2211yescharacteristic
15#397×C71C411yescharacteristic, complemented
16#397×C71C2211yescharacteristic
17#392×C71C411yescharacteristic, complemented
18#392×C71C2211yescharacteristic
19#386×C71—21no—
20#386×C71—21no—
21#382×C71C411yescharacteristic, complemented
22#380×C71—21no—
23C22×C71⋊C8—41no—
24C2×C4⋊Dic71—41no—
25C2×C22⋊Dic71—22no—
26#14556043—42no—
27C22⋊(C71⋊C8)—22no—
28C22⋊(C71⋊C8)—41no—
29C71⋊#95—41no—
30C22×D4×C71—21no—
31C22×D4×C71—21no—
32C22×D4×C71C2×C411yescharacteristic
33C22×D4×C71D411yescharacteristic
34C23×C4×C71—21no—
35C23×C4×C71D411yescharacteristic
36C23×C4×C71D411yescharacteristic
37#121×C71—42no—
38#121×C71—42no—
39C4×D4×C71—41no—
40C4×D4×C71—41no—
41C4×D4×C71—41no—
42C4×D4×C71—41no—
43C2×C4⋊C4×C71—21no—
44C2×C4⋊C4×C71C2×C411yescharacteristic
45C2×C4⋊C4×C71D411yescharacteristic
46C2×C22⋊C4×C71—21no—
47C2×C22⋊C4×C71—21no—
48C2×C22⋊C4×C71—21no—
49C2×C22⋊C4×C71—21no—
50C2×C22⋊C4×C71—21no—
51C2×C22⋊C4×C71C2×C411yescharacteristic
52C2×C22⋊C4×C71C2311yescharacteristic
53C2×C42×C71C2×C411yescharacteristic
54#95×C71—21no—
55#95×C71—21no—
56#95×C71—21no—
57#95×C71C2×C411yescharacteristic
58#95×C71C2×C411yescharacteristic
59C22×Dic71—42no—
60C2×C71⋊C8—42no—
61C2×C71⋊C8—82no—
62C4⋊Dic71—82no—
63C22⋊Dic71—42no—
64C22⋊Dic71—42no—
65C24×C71—21no—
66C24×C71C22⋊C411yescharacteristic
67C24×C71C2×D411yescharacteristic
68C2×D4×C71—41no—
69C2×D4×C71—41no—
70C2×D4×C71—41no—
71C2×D4×C71—42no—
72C2×D4×C71—42no—
73C2×D4×C71—42no—
74C2×D4×C71—42no—
75C2×D4×C71—41no—
76C2×D4×C71—41no—
77C2×D4×C71—81no—
78C2×D4×C71—81no—
79C2×D4×C71D812yes—
80C2×D4×C71SD1612yes—
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—42no—
89C22×C4×C71—42no—
90C22×C4×C71—42no—
91C22×C4×C71—81no—
92C22×C4×C71—81no—
93C22×C4×C71C22⋊C411yescharacteristic
94C22×C4×C71C22⋊C411yescharacteristic
95C22×C4×C71C22⋊C411yescharacteristic
96C22×C4×C71C22×C411yescharacteristic
97C22×C4×C71C2×D411yescharacteristic
98C4⋊C4×C71—41no—
99C4⋊C4×C71—41no—
100C4⋊C4×C71—42no—
101C22⋊C4×C71—41no—
102C22⋊C4×C71—41no—
103C22⋊C4×C71—42no—
104C22⋊C4×C71—41no—
105C22⋊C4×C71—41no—
106C22⋊C4×C71—41no—
107C22⋊C4×C71—41no—
108C22⋊C4×C71—81no—
109C22⋊C4×C71—81no—
110C42×C71—21no—
111C42×C71—21no—
112C23×C71—21no—
113C23×C71—21no—
114C23×C71—21no—
115C23×C71—21no—
116C23×C71—22no—
117C23×C71—41no—
118C23×C71—41no—
119C23×C71—41no—
120C23×C71—41no—
121C23×C71—41no—
122C23×C71—41no—
123C23×C71—41no—
124C23×C71—41no—
125C23×C71—41no—
126C23×C71—41no—
127C23×C71#10212yes—
128C23×C71C2×C22⋊C411yescharacteristic
129C23×C71C2×D811yescharacteristic
130C23×C71C2×SD1611yescharacteristic
131D4×C71—42no—
132D4×C71—42no—
133D4×C71—42no—
134D4×C71—42no—
135D4×C71—84no—
136D4×C71—82no—
137D4×C71—82no—
138C2×C4×C71—21no—
139C2×C4×C71—21no—
140C2×C4×C71—21no—
141C2×C4×C71—21no—
142C2×C4×C71—21no—
143C2×C4×C71—41no—
144C2×C4×C71—42no—
145C2×C4×C71—42no—
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—41no—
155C2×C4×C71—41no—
156C2×C4×C71—41no—
157C2×C4×C71—41no—
158C2×C4×C71—81no—
159C2×C4×C71—81no—
160C2×C4×C71—84no—
161C2×C4×C71#9911yescharacteristic
162C2×C4×C71#9911yescharacteristic
163C2×C4×C71#10212yes—
164C2×Dic71—82no—
165C2×Dic71—82no—
166C2×Dic71—82no—
167C71⋊C8—82no—
168C22×C71—21no—
169C22×C71—21no—
170C22×C71—21no—
171C22×C71—22no—
172C22×C71—21no—
173C22×C71—42no—
174C22×C71—41no—
175C22×C71—41no—
176C22×C71—42no—
177C22×C71—42no—
178C22×C71—41no—
179C22×C71—41no—
180C22×C71—41no—
181C22×C71—41no—
182C22×C71—42no—
183C22×C71—81no—
184C22×C71—81no—
185C22×C71—81no—
186C22×C71—81no—
187C22×C71—81no—
188C22×C71—81no—
189C22×C71C2×#9911yescharacteristic
190C22×C71C2×#10211yescharacteristic
191C22×C71#42111yescharacteristic
192C4×C71—21no—
193C4×C71—21no—
194C4×C71—22no—
195C4×C71—41no—
196C4×C71—41no—
197C4×C71—41no—
198C4×C71—81no—
199C4×C71—81no—
200C4×C71—81no—
201Dic71—162no—
202#11875—711nomaximal, Sylow
203C2×C71—21no—
204C2×C71—21no—
205C2×C71—41no—
206C2×C71—42no—
207C2×C71—41no—
208C2×C71—82no—
209C2×C71#151911yescharacteristic
210C2×C71#212011yescharacteristic
211C2×C71#212711yescharacteristic
212#2535D7111yescharacteristic
213#1799—711no—
214#1459—711no—
215#1344—712no—
216#1343—712no—
217C71#1187511yescharacteristic, complemented, Sylow
218C2×#121—21no—
219C2×C4×D4Dic7111yescharacteristic, complemented
220C2×C4×D4C2×D7111yescharacteristic
221C2×#99—1421no—
222C2×C22⋊C8—1421no—
223#397Dic7111yescharacteristic, complemented
224#397C2×D7111yescharacteristic
225#392Dic7111yescharacteristic, complemented
226#392C2×D7111yescharacteristic
227#386—21no—
228#386—21no—
229#382Dic7111yescharacteristic, complemented
230#380—21no—
231#380—1421no—
232#342—712no—
233#324—1421no—
234#323—712no—
235C22×D4—21no—
236C22×D4—21no—
237C22×D4C2×Dic7111yescharacteristic
238C22×D4C22⋊D7111yescharacteristic
239C23×C4—21no—
240C23×C4C22⋊D7111yescharacteristic
241C23×C4C22⋊D7111yescharacteristic
242C22×C8—2841no—
243#121—42no—
244#121—42no—
245C4×D4—41no—
246C4×D4—41no—
247C4×D4—41no—
248C4×D4—41no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—2841no—
251C2×C4⋊C4C2×Dic7111yescharacteristic
252C2×C4⋊C4C22⋊D7111yescharacteristic
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—1422no—
259C2×C22⋊C4C2×Dic7111yescharacteristic
260C2×C22⋊C4C22×D7111yescharacteristic
261C2×C42C2×Dic7111yescharacteristic
262#99—2842no—
263C22⋊C8—2841no—
264C22⋊C8—1422no—
265#95—21no—
266#95—21no—
267#95—21no—
268#95—2841no—
269#95C2×Dic7111yescharacteristic
270#95C2×Dic7111yescharacteristic
271C24—21no—
272C24C22⋊Dic7111yescharacteristic
273C24C2×C22⋊D7111yescharacteristic
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—42no—
278C2×D4—42no—
279C2×D4—42no—
280C2×D4—42no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4—81no—
284C2×D4—81no—
285C2×D4#1176585012yes—
286C2×D4#1176585312yes—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—81no—
298C22×C4—81no—
299C22×C4—2842no—
300C22×C4C22⋊Dic7111yescharacteristic
301C22×C4C22⋊Dic7111yescharacteristic
302C22×C4C22⋊Dic7111yescharacteristic
303C22×C4C22×Dic7111yescharacteristic
304C22×C4C2×C22⋊D7111yescharacteristic
305C2×C8—2842no—
306C2×C8—5682no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—42no—
310C4⋊C4—5682no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—42no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C22⋊C4—81no—
319C22⋊C4—81no—
320C22⋊C4—2842no—
321C22⋊C4—2842no—
322C42—21no—
323C42—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—22no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23#1455604912yes—
340C23C2×C22⋊Dic7111yescharacteristic
341C23C2×#1176585011yescharacteristic
342C23C2×#1176585311yescharacteristic
343D4—42no—
344D4—42no—
345D4—42no—
346D4—42no—
347D4—84no—
348D4—82no—
349D4—82no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—84no—
373C2×C4—5682no—
374C2×C4—5682no—
375C2×C4—5682no—
376C2×C4#1455604311yescharacteristic
377C2×C4#1455604311yescharacteristic
378C2×C4#1455604912yes—
379C8—5682no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—22no—
384C22—21no—
385C22—42no—
386C22—41no—
387C22—41no—
388C22—42no—
389C22—42no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—42no—
395C22—81no—
396C22—81no—
397C22—81no—
398C22—81no—
399C22—81no—
400C22—81no—
401C22C2×#1455604311yescharacteristic
402C22C2×#1455604911yescharacteristic
403C22#1798601611yescharacteristic
404C4—22no—
405C4—21no—
406C4—21no—
407C4—41no—
408C4—41no—
409C4—41no—
410C4—81no—
411C4—81no—
412C4—81no—
413C4—11362no—
414C2—21no—
415C2—21no—
416C2—41no—
417C2—42no—
418C2—41no—
419C2—82no—
420C2#2258690811yescharacteristic
421C2#2258998111yescharacteristic
422C2#2259000911yescharacteristic
subgroupG/H
centerC22C2×#14556043
commutatorC22×C4×C71C22×C4
FrattiniC2×C22⋊C4C22×D71
Fitting, prime radicalC71×#2535C2
socleC22×C71C2×#99
cosocleC2×C22⋊C4×C71C23
hypercenter#2535D71
hypocenterC71#11875
LCSG ▷ C22×C4×C71 ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2535

Complements

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

NH ≅ G/Ntype
1. #22588328C2semidirect
2. #22586658C2semidirect
3. #22586340C2semidirect
4. #22586337C2semidirect
13. C2×C4×D4×C71C4semidirect
15. #397×C71C4semidirect
17. #392×C71C4semidirect
21. #382×C71C4semidirect
217. C71#11875semidirect
219. C2×C4×D4Dic71semidirect
223. #397Dic71semidirect
225. #392Dic71semidirect
229. #382Dic71semidirect

Maximal subgroups

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

subgroupindex
#225883282
#225866582
#225863402
#225863372
C71×#25352
#1187571