Finite Group Databasebeta

← #31908218

Subgroups — #31908218

Order 18176.

582 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 91 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#22586364C212yesmaximal, complemented
4#22586337C212yesmaximal, complemented
5C71×#2397C211yesmaximal, characteristic
6C2×#14556043—21no—
7#17985963—21no—
8#17985859—21no—
9#17985766C2212yes—
10#17985725—21no—
11#17985723C2212yes—
12C2×C4×D4×C71—21no—
13C2×C4×D4×C71—21no—
14C2×C4×D4×C71C411yescharacteristic, complemented
15C2×C4×D4×C71C2211yescharacteristic
16#399×C71—21no—
17#399×C71C411yescharacteristic, complemented
18#399×C71C2211yescharacteristic
19#392×C71—21no—
20#392×C71C411yescharacteristic, complemented
21#392×C71C2211yescharacteristic
22#376×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×D4×C71—21no—
31C22×D4×C71—21no—
32C22×D4×C71C2×C411yescharacteristic
33C22×D4×C71D411yescharacteristic
34C23×C4×C71—21no—
35C23×C4×C71D411yescharacteristic
36C23×C4×C71D411yescharacteristic
37C4×D4×C71—41no—
38C4×D4×C71—41no—
39C4×D4×C71—44no—
40C4×D4×C71—44no—
41C4×D4×C71—41no—
42C4×D4×C71—41no—
43C2×C4⋊C4×C71—21no—
44C2×C4⋊C4×C71—21no—
45C2×C4⋊C4×C71C2×C411yescharacteristic
46C2×C4⋊C4×C71D411yescharacteristic
47C2×C22⋊C4×C71—21no—
48C2×C22⋊C4×C71—21no—
49C2×C22⋊C4×C71—21no—
50C2×C22⋊C4×C71—21no—
51C2×C22⋊C4×C71—21no—
52C2×C22⋊C4×C71C2×C411yescharacteristic
53C2×C22⋊C4×C71C2311yescharacteristic
54C2×C42×C71—21no—
55C2×C42×C71C2×C411yescharacteristic
56#95×C71—21no—
57#95×C71C2×C411yescharacteristic
58#95×C71C2×C411yescharacteristic
59C22×Dic71—42no—
60#11765849—82no—
61C2×C71⋊C8—42no—
62C4⋊Dic71—82no—
63C22⋊Dic71—42no—
64C22⋊Dic71—42no—
65C24×C71—21no—
66C24×C71C22⋊C411yescharacteristic
67C24×C71C2×D411yescharacteristic
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—42no—
75C2×D4×C71—42no—
76C2×D4×C71—42no—
77C2×D4×C71—42no—
78C2×D4×C71—41no—
79C2×D4×C71—41no—
80C2×D4×C71—42no—
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—42no—
92C22×C4×C71—41no—
93C22×C4×C71—41no—
94C22×C4×C71C22⋊C411yescharacteristic
95C22×C4×C71C22⋊C411yescharacteristic
96C22×C4×C71C22⋊C411yescharacteristic
97C22×C4×C71C22×C411yescharacteristic
98C22×C4×C71C2×D411yescharacteristic
99C4⋊C4×C71—41no—
100C4⋊C4×C71—41no—
101C4⋊C4×C71—42no—
102C4⋊C4×C71—42no—
103C22⋊C4×C71—41no—
104C22⋊C4×C71—41no—
105C22⋊C4×C71—42no—
106C22⋊C4×C71—42no—
107C22⋊C4×C71—42no—
108C22⋊C4×C71—42no—
109C22⋊C4×C71—42no—
110C42×C71—24no—
111C42×C71—21no—
112C42×C71—21no—
113C23×C71—21no—
114C23×C71—21no—
115C23×C71—22no—
116C23×C71—21no—
117C23×C71—21no—
118C23×C71—21no—
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—41no—
128C23×C71—41no—
129C23×C71C2×C22⋊C411yescharacteristic
130C23×C71AGL(1,ℤ/8)11yescharacteristic
131C23×C71#13711yescharacteristic
132D4×C71—42no—
133D4×C71—42no—
134D4×C71—42no—
135D4×C71—44no—
136D4×C71—44no—
137D4×C71—48no—
138D4×C71—42no—
139C2×C4×C71—22no—
140C2×C4×C71—22no—
141C2×C4×C71—22no—
142C2×C4×C71—21no—
143C2×C4×C71—21no—
144C2×C4×C71—21no—
145C2×C4×C71—21no—
146C2×C4×C71—21no—
147C2×C4×C71—21no—
148C2×C4×C71—41no—
149C2×C4×C71—41no—
150C2×C4×C71—41no—
151C2×C4×C71—41no—
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—41no—
159C2×C4×C71—41no—
160C2×C4×C71—81no—
161C2×C4×C71—81no—
162C2×C4×C71—82no—
163C2×C4×C71—82no—
164C2×C4×C71#9911yescharacteristic
165C2×C4×C71#9911yescharacteristic
166C2×Dic71—82no—
167C2×Dic71—82no—
168C2×Dic71—82no—
169C71⋊C8—82no—
170C22×C71—21no—
171C22×C71—21no—
172C22×C71—21no—
173C22×C71—21no—
174C22×C71—41no—
175C22×C71—41no—
176C22×C71—42no—
177C22×C71—42no—
178C22×C71—42no—
179C22×C71—41no—
180C22×C71—41no—
181C22×C71—42no—
182C22×C71—41no—
183C22×C71—41no—
184C22×C71—41no—
185C22×C71—81no—
186C22×C71—81no—
187C22×C71—81no—
188C22×C71—81no—
189C22×C71—81no—
190C22×C71—81no—
191C22×C71C2×#9911yescharacteristic
192C22×C71#41711yescharacteristic
193C22×C71#42111yescharacteristic
194C4×C71—21no—
195C4×C71—42no—
196C4×C71—42no—
197C4×C71—41no—
198C4×C71—41no—
199C4×C71—41no—
200C4×C71—41no—
201C4×C71—81no—
202C4×C71#35312yes—
203Dic71—162no—
204#11821—711nomaximal, Sylow
205C2×C71—21no—
206C2×C71—21no—
207C2×C71—41no—
208C2×C71—41no—
209C2×C71—81no—
210C2×C71—82no—
211C2×C71#151311yescharacteristic
212C2×C71#212011yescharacteristic
213C2×C71C2×#35311yescharacteristic
214#2397D7111yescharacteristic
215#1799—711no—
216#1465—711no—
217#1352—712no—
218#1343—712no—
219C71#1182111yescharacteristic, complemented, Sylow
220C2×C4×D4—21no—
221C2×C4×D4—21no—
222C2×C4×D4Dic7111yescharacteristic, complemented
223C2×C4×D4C2×D7111yescharacteristic
224C2×#99—1421no—
225#407—1421no—
226#399—21no—
227#399Dic7111yescharacteristic, complemented
228#399C2×D7111yescharacteristic
229#392—21no—
230#392Dic7111yescharacteristic, complemented
231#392C2×D7111yescharacteristic
232#380—1421no—
233#376Dic7111yescharacteristic, complemented
234#342—712no—
235#324—1421no—
236#323—712no—
237C22×D4—21no—
238C22×D4—21no—
239C22×D4C2×Dic7111yescharacteristic
240C22×D4C22⋊D7111yescharacteristic
241C23×C4—21no—
242C23×C4C22⋊D7111yescharacteristic
243C23×C4C22⋊D7111yescharacteristic
244C2×M16—2841no—
245C4×D4—41no—
246C4×D4—41no—
247C4×D4—44no—
248C4×D4—44no—
249C4×D4—41no—
250C4×D4—41no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—2841no—
254C2×C4⋊C4C2×Dic7111yescharacteristic
255C2×C4⋊C4C22⋊D7111yescharacteristic
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—1422no—
262C2×C22⋊C4C2×Dic7111yescharacteristic
263C2×C22⋊C4C22×D7111yescharacteristic
264C2×C42—21no—
265C2×C42C2×Dic7111yescharacteristic
266#99—2842no—
267C22⋊C8—2841no—
268C22⋊C8—1422no—
269#95—21no—
270#95—2841no—
271#95C2×Dic7111yescharacteristic
272#95C2×Dic7111yescharacteristic
273C24—21no—
274C24C22⋊Dic7111yescharacteristic
275C24C2×C22⋊D7111yescharacteristic
276C2×D4—21no—
277C2×D4—21no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—42no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—42no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—2842no—
303C22×C4C22⋊Dic7111yescharacteristic
304C22×C4C22⋊Dic7111yescharacteristic
305C22×C4C22⋊Dic7111yescharacteristic
306C22×C4C22×Dic7111yescharacteristic
307C22×C4C2×C22⋊D7111yescharacteristic
308M16—5682no—
309C2×C8—2842no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—5682no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C22⋊C4—42no—
318C22⋊C4—42no—
319C22⋊C4—42no—
320C22⋊C4—42no—
321C22⋊C4—42no—
322C22⋊C4—2842no—
323C22⋊C4—2842no—
324C42—24no—
325C42—21no—
326C42—21no—
327C23—21no—
328C23—21no—
329C23—22no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23C2×C22⋊Dic7111yescharacteristic
344C23#1455615311yescharacteristic
345C23#1455615811yescharacteristic
346D4—42no—
347D4—42no—
348D4—42no—
349D4—44no—
350D4—44no—
351D4—48no—
352D4—42no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—5682no—
379C2×C4—5682no—
380C2×C4—5682no—
381C2×C4#1455604311yescharacteristic
382C2×C4#1455604311yescharacteristic
383C8—5682no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—41no—
389C22—41no—
390C22—42no—
391C22—42no—
392C22—42no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—42no—
397C22—41no—
398C22—41no—
399C22—81no—
400C22—81no—
401C22—81no—
402C22—81no—
403C22—81no—
404C22—81no—
405C22C2×#1455604311yescharacteristic
406C22#1798600011yescharacteristic
407C22#1798601611yescharacteristic
408C4—21no—
409C4—42no—
410C4—42no—
411C4—41no—
412C4—41no—
413C4—41no—
414C4—41no—
415C4—81no—
416C4—11362no—
417C4#1798579212yes—
418C2—21no—
419C2—21no—
420C2—41no—
421C2—41no—
422C2—81no—
423C2—82no—
424C2#2258688311yescharacteristic
425C2#2258998111yescharacteristic
426C2C2×#1798579211yescharacteristic
subgroupG/H
centerC22C2×#14556043
commutatorC22×C4×C71C22×C4
FrattiniC2×C22⋊C4C22×D71
Fitting, prime radicalC71×#2397C2
socleC22×C71C2×#99
cosocleC2×C22⋊C4×C71C23
hypercenter#2397D71
hypocenterC71#11821
LCSG ▷ C22×C4×C71 ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2397

Complements

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

NH ≅ G/Ntype
1. #22588328C2semidirect
2. #22586678C2semidirect
3. #22586364C2semidirect
4. #22586337C2semidirect
14. C2×C4×D4×C71C4semidirect
17. #399×C71C4semidirect
20. #392×C71C4semidirect
22. #376×C71C4semidirect
219. C71#11821semidirect
222. C2×C4×D4Dic71semidirect
227. #399Dic71semidirect
230. #392Dic71semidirect
233. #376Dic71semidirect

Maximal subgroups

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

subgroupindex
#225883282
#225866782
#225863642
#225863372
C71×#23972
#1182171