Finite Group Databasebeta

← #31908213

Subgroups — #31908213

Order 18176.

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

normal subgroups only

This group has 626 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#22586681C211yesmaximal, characteristic, complemented
3#22586361C212yesmaximal, complemented
4#22586337C212yesmaximal, complemented
5C71×#2388C211yesmaximal, characteristic
6#17986056—21no—
7C2×#14556043—21no—
8#17985859—21no—
9#17985766C2212yes—
10#17985723—21no—
11#17985723C2212yes—
12C22×#41×C71C411yescharacteristic, complemented
13C2×C4×D4×C71—21no—
14C2×C4×D4×C71C2211yescharacteristic
15#395×C71C411yescharacteristic, complemented
16#394×C71—21no—
17#394×C71C411yescharacteristic, complemented
18#393×C71C2211yescharacteristic
19#392×C71—21no—
20#392×C71C2211yescharacteristic
21C4×C22⋊C4×C71—21no—
22C4×C22⋊C4×C71C411yescharacteristic, complemented
23C2×C4⋊Dic71—41no—
24C2×C22⋊Dic71—22no—
25C4⋊(C71⋊C8)—41no—
26#14556043—42no—
27C22⋊(C71⋊C8)—22no—
28#14556039—41no—
29C71⋊#95—41no—
30C2×#41×C71—44no—
31C2×#41×C71—41no—
32C2×#41×C71—41no—
33C22×Q8×C71C2×C411yescharacteristic
34C22×D4×C71—21no—
35C22×D4×C71—21no—
36C22×D4×C71C2×C411yescharacteristic
37C22×D4×C71D411yescharacteristic
38C23×C4×C71—21no—
39C23×C4×C71C2×C411yescharacteristic
40C23×C4×C71D411yescharacteristic
41C4×D4×C71—41no—
42C4×D4×C71—41no—
43C4×D4×C71—44no—
44C2×C4⋊C4×C71—21no—
45C2×C4⋊C4×C71C2×C411yescharacteristic
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×C71D411yescharacteristic
52C2×C22⋊C4×C71C2311yescharacteristic
53C2×C42×C71—21no—
54C2×C42×C71D411yescharacteristic
55#95×C71—21no—
56#95×C71—21no—
57#95×C71C2×C411yescharacteristic
58#95×C71C2×C411yescharacteristic
59C22×Dic71—42no—
60C2×C71⋊C8—42no—
61C4⋊Dic71—82no—
62C22⋊Dic71—42no—
63C22⋊Dic71—42no—
64C24×C71—21no—
65C24×C71C22⋊C411yescharacteristic
66C24×C71C2×D411yescharacteristic
67#41×C71—44no—
68#41×C71—44no—
69#41×C71—44no—
70#41×C71—44no—
71C2×Q8×C71—42no—
72C2×Q8×C71—41no—
73C2×D4×C71—21no—
74C2×D4×C71—21no—
75C2×D4×C71—41no—
76C2×D4×C71—41no—
77C2×D4×C71—41no—
78C2×D4×C71—42no—
79C2×D4×C71—42no—
80C2×D4×C71—42no—
81C2×D4×C71—42no—
82C2×D4×C71—42no—
83C2×D4×C71—41no—
84C2×D4×C71—41no—
85C2×D4×C71—42no—
86C22×C4×C71—24no—
87C22×C4×C71—21no—
88C22×C4×C71—21no—
89C22×C4×C71—21no—
90C22×C4×C71—21no—
91C22×C4×C71—21no—
92C22×C4×C71—21no—
93C22×C4×C71—21no—
94C22×C4×C71—42no—
95C22×C4×C71—42no—
96C22×C4×C71—42no—
97C22×C4×C71—41no—
98C22×C4×C71—41no—
99C22×C4×C71C22⋊C411yescharacteristic
100C22×C4×C71C22⋊C411yescharacteristic
101C22×C4×C71C22⋊C411yescharacteristic
102C22×C4×C71C22×C411yescharacteristic
103C22×C4×C71C2×D411yescharacteristic
104C4⋊C4×C71—41no—
105C4⋊C4×C71—42no—
106C22⋊C4×C71—41no—
107C22⋊C4×C71—41no—
108C22⋊C4×C71—42no—
109C22⋊C4×C71—42no—
110C22⋊C4×C71—42no—
111C22⋊C4×C71—42no—
112C22⋊C4×C71—42no—
113C42×C71—21no—
114C42×C71—21no—
115C42×C71—42no—
116C23×C71—21no—
117C23×C71—21no—
118C23×C71—22no—
119C23×C71—21no—
120C23×C71—21no—
121C23×C71—21no—
122C23×C71—41no—
123C23×C71—41no—
124C23×C71—41no—
125C23×C71—41no—
126C23×C71—41no—
127C23×C71—41no—
128C23×C71—41no—
129C23×C71—41no—
130C23×C71—41no—
131C23×C71—41no—
132C23×C71C2×C22⋊C411yescharacteristic
133C23×C71AGL(1,ℤ/8)11yescharacteristic
134C23×C71#13711yescharacteristic
135Q8×C71—42no—
136Q8×C71—42no—
137D4×C71—42no—
138D4×C71—42no—
139D4×C71—42no—
140D4×C71—44no—
141D4×C71—44no—
142D4×C71—48no—
143D4×C71—42no—
144C2×C4×C71—22no—
145C2×C4×C71—22no—
146C2×C4×C71—22no—
147C2×C4×C71—21no—
148C2×C4×C71—21no—
149C2×C4×C71—21no—
150C2×C4×C71—21no—
151C2×C4×C71—21no—
152C2×C4×C71—21no—
153C2×C4×C71—44no—
154C2×C4×C71—42no—
155C2×C4×C71—41no—
156C2×C4×C71—41no—
157C2×C4×C71—41no—
158C2×C4×C71—41no—
159C2×C4×C71—44no—
160C2×C4×C71—42no—
161C2×C4×C71—42no—
162C2×C4×C71—41no—
163C2×C4×C71—41no—
164C2×C4×C71—41no—
165C2×C4×C71—41no—
166C2×C4×C71—41no—
167C2×C4×C71—41no—
168C2×C4×C71—81no—
169C2×C4×C71#9911yescharacteristic
170C2×C4×C71#9911yescharacteristic
171C2×Dic71—82no—
172C2×Dic71—82no—
173C2×Dic71—82no—
174C71⋊C8—82no—
175C22×C71—21no—
176C22×C71—21no—
177C22×C71—21no—
178C22×C71—21no—
179C22×C71—41no—
180C22×C71—41no—
181C22×C71—42no—
182C22×C71—42no—
183C22×C71—41no—
184C22×C71—41no—
185C22×C71—41no—
186C22×C71—42no—
187C22×C71—41no—
188C22×C71—41no—
189C22×C71—81no—
190C22×C71—81no—
191C22×C71—81no—
192C22×C71—81no—
193C22×C71—81no—
194C22×C71—81no—
195C22×C71—81no—
196C22×C71C2×#9911yescharacteristic
197C22×C71#41711yescharacteristic
198C22×C71#42111yescharacteristic
199C4×C71—21no—
200C4×C71—42no—
201C4×C71—42no—
202C4×C71—41no—
203C4×C71—41no—
204C4×C71—41no—
205C4×C71—41no—
206C4×C71—81no—
207C4×C71AΣL(1,𝔽16)12yes—
208Dic71—162no—
209#11820—711nomaximal, Sylow
210C2×C71—21no—
211C2×C71—21no—
212C2×C71—41no—
213C2×C71—41no—
214C2×C71—81no—
215C2×C71—82no—
216C2×C71#151311yescharacteristic
217C2×C71C2×AΣL(1,𝔽16)11yescharacteristic
218C2×C71#212611yescharacteristic
219#2388D7111yescharacteristic
220#1799—711no—
221#1466—711no—
222#1351—712no—
223#1343—712no—
224C71#1182011yescharacteristic, complemented, Sylow
225C22×#41Dic7111yescharacteristic, complemented
226C2×C4×D4—21no—
227C2×C4×D4C2×D7111yescharacteristic
228#432—1421no—
229C2×#99—1421no—
230#395Dic7111yescharacteristic, complemented
231#394—21no—
232#394Dic7111yescharacteristic, complemented
233#393C2×D7111yescharacteristic
234#392—21no—
235#392C2×D7111yescharacteristic
236#380—1421no—
237C4×C22⋊C4—21no—
238C4×C22⋊C4Dic7111yescharacteristic, complemented
239#342—712no—
240#323—712no—
241#323—1421no—
242C2×#41—44no—
243C2×#41—41no—
244C2×#41—41no—
245C22×Q8C2×Dic7111yescharacteristic
246C22×D4—21no—
247C22×D4—21no—
248C22×D4C2×Dic7111yescharacteristic
249C22×D4C22⋊D7111yescharacteristic
250C23×C4—21no—
251C23×C4C2×Dic7111yescharacteristic
252C23×C4C22⋊D7111yescharacteristic
253C4×D4—41no—
254C4×D4—41no—
255C4×D4—44no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—2841no—
258C2×C4⋊C4C2×Dic7111yescharacteristic
259C2×C22⋊C4—21no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—1422no—
265C2×C22⋊C4C22⋊D7111yescharacteristic
266C2×C22⋊C4C22×D7111yescharacteristic
267C2×C42—21no—
268C2×C42C22⋊D7111yescharacteristic
269C4⋊C8—2841no—
270#99—2842no—
271C22⋊C8—1422no—
272#97—2841no—
273#95—21no—
274#95—21no—
275#95—2841no—
276#95C2×Dic7111yescharacteristic
277#95C2×Dic7111yescharacteristic
278C24—21no—
279C24C22⋊Dic7111yescharacteristic
280C24C2×C22⋊D7111yescharacteristic
281#41—44no—
282#41—44no—
283#41—44no—
284#41—44no—
285C2×Q8—42no—
286C2×Q8—41no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—42no—
300C22×C4—24no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—2842no—
314C22×C4C22⋊Dic7111yescharacteristic
315C22×C4C22⋊Dic7111yescharacteristic
316C22×C4C22⋊Dic7111yescharacteristic
317C22×C4C22×Dic7111yescharacteristic
318C22×C4C2×C22⋊D7111yescharacteristic
319C2×C8—2842no—
320C4⋊C4—41no—
321C4⋊C4—42no—
322C4⋊C4—5682no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C22⋊C4—42no—
326C22⋊C4—42no—
327C22⋊C4—42no—
328C22⋊C4—42no—
329C22⋊C4—42no—
330C22⋊C4—2842no—
331C22⋊C4—2842no—
332C42—21no—
333C42—21no—
334C42—42no—
335C23—21no—
336C23—21no—
337C23—22no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23—41no—
350C23—41no—
351C23C2×C22⋊Dic7111yescharacteristic
352C23#1455615311yescharacteristic
353C23#1455615811yescharacteristic
354Q8—42no—
355Q8—42no—
356D4—42no—
357D4—42no—
358D4—42no—
359D4—48no—
360D4—42no—
361D4—44no—
362D4—44no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—44no—
373C2×C4—42no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—44no—
380C2×C4—42no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—81no—
388C2×C4—5682no—
389C2×C4—5682no—
390C2×C4—5682no—
391C2×C4#1455604311yescharacteristic
392C2×C4#1455604311yescharacteristic
393C8—5682no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—41no—
399C22—41no—
400C22—42no—
401C22—42no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—42no—
407C22—41no—
408C22—81no—
409C22—81no—
410C22—81no—
411C22—81no—
412C22—81no—
413C22—81no—
414C22—81no—
415C22C2×#1455604311yescharacteristic
416C22#1798600011yescharacteristic
417C22#1798601611yescharacteristic
418C4—21no—
419C4—42no—
420C4—41no—
421C4—42no—
422C4—41no—
423C4—41no—
424C4—41no—
425C4—81no—
426C4—11362no—
427C4#1798578612yes—
428C2—21no—
429C2—21no—
430C2—41no—
431C2—41no—
432C2—82no—
433C2—81no—
434C2#2258688311yescharacteristic
435C2C2×#1798578611yescharacteristic
436C2#2259000511yescharacteristic
subgroupG/H
centerC22C2×#14556043
commutatorC22×C4×C71C22×C4
FrattiniC2×C22⋊C4C22×D71
Fitting, prime radicalC71×#2388C2
socleC22×C71C2×#99
cosocleC2×C22⋊C4×C71C23
hypercenter#2388D71
hypocenterC71#11820
LCSG ▷ C22×C4×C71 ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2388

Complements

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

NH ≅ G/Ntype
1. #22588328C2semidirect
2. #22586681C2semidirect
3. #22586361C2semidirect
4. #22586337C2semidirect
12. C22×#41×C71C4semidirect
15. #395×C71C4semidirect
17. #394×C71C4semidirect
22. C4×C22⋊C4×C71C4semidirect
224. C71#11820semidirect
225. C22×#41Dic71semidirect
230. #395Dic71semidirect
232. #394Dic71semidirect
238. C4×C22⋊C4Dic71semidirect

Maximal subgroups

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

subgroupindex
#225883282
#225866812
#225863612
#225863372
C71×#23882
#1182071