Finite Group Databasebeta

← #31908233

Subgroups — #31908233

Order 18176.

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

normal subgroups only

This group has 464 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#22586681C211yesmaximal, characteristic, complemented
3#22586364C212yesmaximal, complemented
4#22586337C212yesmaximal, complemented
5C71×#2635C211yesmaximal, characteristic
6#17986056—21no—
7C2×#14556043—21no—
8#17985859—21no—
9#17985766C2212yes—
10#17985723—21no—
11#17985723C2212yes—
12C2×#121×C71C411yescharacteristic, complemented
13C2×C4×D4×C71C2211yescharacteristic
14#400×C71C411yescharacteristic, complemented
15#399×C71C2211yescharacteristic
16#394×C71—21no—
17#394×C71C411yescharacteristic, complemented
18#392×C71—21no—
19#392×C71C2211yescharacteristic
20#388×C71—21no—
21#386×C71—21no—
22#381×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—
30C22×D4×C71—21no—
31C22×D4×C71—21no—
32C22×D4×C71C2×C411yescharacteristic
33C22×D4×C71D411yescharacteristic
34C23×C4×C71C2×C411yescharacteristic
35C23×C4×C71D411yescharacteristic
36#121×C71—41no—
37#121×C71—41no—
38C4×D4×C71—41no—
39C4×D4×C71—41no—
40C2×C4⋊C4×C71—21no—
41C2×C4⋊C4×C71C2×C411yescharacteristic
42C2×C4⋊C4×C71C2×C411yescharacteristic
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×C71—21no—
49C2×C22⋊C4×C71D411yescharacteristic
50C2×C22⋊C4×C71C2311yescharacteristic
51C2×C42×C71D411yescharacteristic
52#95×C71—21no—
53#95×C71—21no—
54#95×C71—21no—
55#95×C71C2×C411yescharacteristic
56#95×C71C2×C411yescharacteristic
57C22×Dic71—42no—
58C2×C71⋊C8—42no—
59C4⋊Dic71—82no—
60C22⋊Dic71—42no—
61C22⋊Dic71—42no—
62C24×C71—21no—
63C24×C71C22⋊C411yescharacteristic
64C24×C71C2×D411yescharacteristic
65C2×D4×C71—21no—
66C2×D4×C71—21no—
67C2×D4×C71—41no—
68C2×D4×C71—41no—
69C2×D4×C71—41no—
70C2×D4×C71—42no—
71C2×D4×C71—41no—
72C2×D4×C71—41no—
73C2×D4×C71—81no—
74C2×D4×C71—81no—
75C2×D4×C71—81no—
76C2×D4×C71—81no—
77C2×D4×C71—81no—
78C22×C4×C71—21no—
79C22×C4×C71—21no—
80C22×C4×C71—21no—
81C22×C4×C71—21no—
82C22×C4×C71—21no—
83C22×C4×C71—21no—
84C22×C4×C71—21no—
85C22×C4×C71—81no—
86C22×C4×C71—81no—
87C22×C4×C71C22⋊C411yescharacteristic
88C22×C4×C71C22⋊C411yescharacteristic
89C22×C4×C71C22⋊C411yescharacteristic
90C22×C4×C71C22×C411yescharacteristic
91C22×C4×C71C2×D411yescharacteristic
92C4⋊C4×C71—41no—
93C4⋊C4×C71—41no—
94C4⋊C4×C71—81no—
95C22⋊C4×C71—41no—
96C22⋊C4×C71—41no—
97C22⋊C4×C71—42no—
98C22⋊C4×C71—41no—
99C22⋊C4×C71—41no—
100C22⋊C4×C71—81no—
101C22⋊C4×C71—81no—
102C22⋊C4×C71—81no—
103C22⋊C4×C71—81no—
104C42×C71—21no—
105C42×C71—21no—
106C23×C71—21no—
107C23×C71—21no—
108C23×C71—21no—
109C23×C71—21no—
110C23×C71—21no—
111C23×C71—41no—
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×C71C2×C22⋊C411yescharacteristic
123C23×C71AGL(1,ℤ/8)11yescharacteristic
124C23×C71#13711yescharacteristic
125D4×C71—42no—
126D4×C71—42no—
127D4×C71—82no—
128D4×C71—82no—
129D4×C71—82no—
130D4×C71—84no—
131C2×C4×C71—21no—
132C2×C4×C71—21no—
133C2×C4×C71—21no—
134C2×C4×C71—21no—
135C2×C4×C71—21no—
136C2×C4×C71—21no—
137C2×C4×C71—41no—
138C2×C4×C71—42no—
139C2×C4×C71—41no—
140C2×C4×C71—41no—
141C2×C4×C71—41no—
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—81no—
152C2×C4×C71—81no—
153C2×C4×C71—81no—
154C2×C4×C71#9911yescharacteristic
155C2×C4×C71#9911yescharacteristic
156C2×Dic71—82no—
157C2×Dic71—82no—
158C2×Dic71—82no—
159C71⋊C8—82no—
160C22×C71—21no—
161C22×C71—21no—
162C22×C71—21no—
163C22×C71—21no—
164C22×C71—41no—
165C22×C71—41no—
166C22×C71—41no—
167C22×C71—41no—
168C22×C71—41no—
169C22×C71—41no—
170C22×C71—41no—
171C22×C71—81no—
172C22×C71—81no—
173C22×C71—81no—
174C22×C71—81no—
175C22×C71—81no—
176C22×C71—81no—
177C22×C71—81no—
178C22×C71—81no—
179C22×C71—81no—
180C22×C71—81no—
181C22×C71C2×#9911yescharacteristic
182C22×C71#41711yescharacteristic
183C22×C71#42111yescharacteristic
184C4×C71—21no—
185C4×C71—21no—
186C4×C71—41no—
187C4×C71—41no—
188C4×C71—41no—
189C4×C71—41no—
190C4×C71—81no—
191C4×C71—81no—
192C4×C71—81no—
193Dic71—162no—
194#11824—711nomaximal, Sylow
195C2×C71—21no—
196C2×C71—21no—
197C2×C71—41no—
198C2×C71—41no—
199C2×C71—81no—
200C2×C71—161no—
201C2×C71#151311yescharacteristic
202C2×C71#212111yescharacteristic
203C2×C71#212811yescharacteristic
204#2635D7111yescharacteristic
205#1799—711no—
206#1466—711no—
207#1352—712no—
208#1343—712no—
209C71#1182411yescharacteristic, complemented, Sylow
210C2×#121Dic7111yescharacteristic, complemented
211C2×C4×D4C2×D7111yescharacteristic
212#432—1421no—
213C2×#99—1421no—
214#400Dic7111yescharacteristic, complemented
215#399C2×D7111yescharacteristic
216#394—21no—
217#394Dic7111yescharacteristic, complemented
218#392—21no—
219#392C2×D7111yescharacteristic
220#388—21no—
221#386—21no—
222#381Dic7111yescharacteristic, complemented
223#380—1421no—
224#342—712no—
225#323—712no—
226#323—1421no—
227C22×D4—21no—
228C22×D4—21no—
229C22×D4C2×Dic7111yescharacteristic
230C22×D4C22⋊D7111yescharacteristic
231C23×C4C2×Dic7111yescharacteristic
232C23×C4C22⋊D7111yescharacteristic
233#121—41no—
234#121—41no—
235C4×D4—41no—
236C4×D4—41no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—2841no—
239C2×C4⋊C4C2×Dic7111yescharacteristic
240C2×C4⋊C4C2×Dic7111yescharacteristic
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—1422no—
248C2×C22⋊C4C22⋊D7111yescharacteristic
249C2×C22⋊C4C22×D7111yescharacteristic
250C2×C42C22⋊D7111yescharacteristic
251C4⋊C8—2841no—
252#99—2842no—
253C22⋊C8—1422no—
254#97—2841no—
255#95—21no—
256#95—21no—
257#95—21no—
258#95—2841no—
259#95C2×Dic7111yescharacteristic
260#95C2×Dic7111yescharacteristic
261C24—21no—
262C24C22⋊Dic7111yescharacteristic
263C24C2×C22⋊D7111yescharacteristic
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—42no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—81no—
273C2×D4—81no—
274C2×D4—81no—
275C2×D4—81no—
276C2×D4—81no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—81no—
285C22×C4—81no—
286C22×C4—2842no—
287C22×C4C22⋊Dic7111yescharacteristic
288C22×C4C22⋊Dic7111yescharacteristic
289C22×C4C22⋊Dic7111yescharacteristic
290C22×C4C22×Dic7111yescharacteristic
291C22×C4C2×C22⋊D7111yescharacteristic
292C2×C8—2842no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—81no—
296C4⋊C4—5682no—
297C22⋊C4—41no—
298C22⋊C4—41no—
299C22⋊C4—42no—
300C22⋊C4—41no—
301C22⋊C4—41no—
302C22⋊C4—81no—
303C22⋊C4—81no—
304C22⋊C4—81no—
305C22⋊C4—81no—
306C22⋊C4—2842no—
307C22⋊C4—2842no—
308C42—21no—
309C42—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23C2×C22⋊Dic7111yescharacteristic
327C23#1455615311yescharacteristic
328C23#1455615811yescharacteristic
329D4—42no—
330D4—42no—
331D4—82no—
332D4—82no—
333D4—82no—
334D4—84no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—5682no—
359C2×C4—5682no—
360C2×C4—5682no—
361C2×C4#1455604311yescharacteristic
362C2×C4#1455604311yescharacteristic
363C8—5682no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—81no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22—81no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22C2×#1455604311yescharacteristic
386C22#1798600011yescharacteristic
387C22#1798601611yescharacteristic
388C4—21no—
389C4—21no—
390C4—41no—
391C4—41no—
392C4—41no—
393C4—41no—
394C4—81no—
395C4—81no—
396C4—81no—
397C4—11362no—
398C2—21no—
399C2—21no—
400C2—41no—
401C2—41no—
402C2—81no—
403C2—161no—
404C2#2258688311yescharacteristic
405C2#2258998611yescharacteristic
406C2#2259001211yescharacteristic
subgroupG/H
centerC22C2×#14556043
commutatorC22×C4×C71C22×C4
FrattiniC2×C22⋊C4C22×D71
Fitting, prime radicalC71×#2635C2
socleC22×C71C2×#99
cosocleC2×C22⋊C4×C71C23
hypercenter#2635D71
hypocenterC71#11824
LCSG ▷ C22×C4×C71 ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2635

Complements

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

NH ≅ G/Ntype
1. #22588328C2semidirect
2. #22586681C2semidirect
3. #22586364C2semidirect
4. #22586337C2semidirect
12. C2×#121×C71C4semidirect
14. #400×C71C4semidirect
17. #394×C71C4semidirect
22. #381×C71C4semidirect
209. C71#11824semidirect
210. C2×#121Dic71semidirect
214. #400Dic71semidirect
217. #394Dic71semidirect
222. #381Dic71semidirect

Maximal subgroups

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

subgroupindex
#225883282
#225866812
#225863642
#225863372
C71×#26352
#1182471