Finite Group Databasebeta

← #31907913

Subgroups — #31907913

Order 18176.

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

normal subgroups only

This group has 500 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#22588073C211yesmaximal, characteristic, complemented
2#22586674C211yesmaximal, characteristic, complemented
3#22586364C212yesmaximal, complemented
4#22586337C212yesmaximal
5C71×#2598C211yesmaximal, characteristic
6C2×#14556087—21no—
7#17986056—21no—
8#17985766—21no—
9#17985766C2212yes—
10#17985725—21no—
11#17985723C2212yes—
12C2×C4×D4×C71C411yescharacteristic, complemented
13C2×#117×C71C2211yescharacteristic
14#400×C71C411yescharacteristic, complemented
15#399×C71C2211yescharacteristic
16#394×C71C411yescharacteristic, complemented
17#392×C71C2211yescharacteristic
18#388×C71—21no—
19#386×C71—21no—
20#386×C71—21no—
21#385×C71—21no—
22#383×C71C411yescharacteristic, complemented
23#14556087—24no—
24#14556087—42no—
25C2×C4⋊Dic71—41no—
26C2×C22⋊Dic71—22no—
27C2×C4×Dic71—41no—
28C4⋊(C71⋊C8)—41no—
29C22⋊(C71⋊C8)—22no—
30#14556039—41no—
31C22×D4×C71—21no—
32C22×D4×C71C2×C411yescharacteristic
33C23×C4×C71—21no—
34C23×C4×C71D411yescharacteristic
35C4×D4×C71—41no—
36C4×D4×C71—41no—
37#117×C71—41no—
38#117×C71—41no—
39C2×C4⋊C4×C71—21no—
40C2×C4⋊C4×C71C2×C411yescharacteristic
41C2×C4⋊C4×C71C2×C411yescharacteristic
42C2×C4⋊C4×C71D411yescharacteristic
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×C71D411yescharacteristic
49C2×C22⋊C4×C71C2311yescharacteristic
50C2×C42×C71C2×C411yescharacteristic
51C2×C42×C71D411yescharacteristic
52#95×C71—21no—
53#95×C71—21no—
54#95×C71—21no—
55#95×C71—21no—
56#95×C71C2×C411yescharacteristic
57#95×C71C2×C411yescharacteristic
58C22×Dic71—42no—
59C2×C71⋊C8—42no—
60C4⋊Dic71—44no—
61C22⋊Dic71—24no—
62C22⋊Dic71—42no—
63C4×Dic71—44no—
64C24×C71—21no—
65C24×C71C2×D411yescharacteristic
66C2×D4×C71—41no—
67C2×D4×C71—42no—
68C2×D4×C71—42no—
69C2×D4×C71—41no—
70C2×D4×C71—41no—
71C2×D4×C71—81no—
72C22×C4×C71—21no—
73C22×C4×C71—21no—
74C22×C4×C71—21no—
75C22×C4×C71—21no—
76C22×C4×C71—22no—
77C22×C4×C71—21no—
78C22×C4×C71—21no—
79C22×C4×C71—21no—
80C22×C4×C71—42no—
81C22×C4×C71—81no—
82C22×C4×C71—81no—
83C22×C4×C71C22⋊C411yescharacteristic
84C22×C4×C71C22⋊C411yescharacteristic
85C22×C4×C71C22⋊C411yescharacteristic
86C22×C4×C71C22⋊C411yescharacteristic
87C22×C4×C71C22×C411yescharacteristic
88C22×C4×C71C2×D411yescharacteristic
89C4⋊C4×C71—41no—
90C4⋊C4×C71—41no—
91C4⋊C4×C71—41no—
92C4⋊C4×C71—81no—
93C22⋊C4×C71—41no—
94C22⋊C4×C71—41no—
95C22⋊C4×C71—42no—
96C22⋊C4×C71—41no—
97C22⋊C4×C71—41no—
98C22⋊C4×C71—42no—
99C22⋊C4×C71—81no—
100C22⋊C4×C71—81no—
101C42×C71—21no—
102C42×C71—21no—
103C42×C71—21no—
104C42×C71—21no—
105C23×C71—21no—
106C23×C71—21no—
107C23×C71—22no—
108C23×C71—41no—
109C23×C71—41no—
110C23×C71—41no—
111C23×C71—41no—
112C23×C71—41no—
113C23×C71—41no—
114C23×C71—41no—
115C23×C71—41no—
116C23×C71C2×C22⋊C411yescharacteristic
117C23×C71AGL(1,ℤ/8)11yescharacteristic
118C23×C71#13711yescharacteristic
119D4×C71—42no—
120D4×C71—42no—
121D4×C71—82no—
122D4×C71—82no—
123C2×C4×C71—22no—
124C2×C4×C71—22no—
125C2×C4×C71—22no—
126C2×C4×C71—22no—
127C2×C4×C71—21no—
128C2×C4×C71—21no—
129C2×C4×C71—21no—
130C2×C4×C71—21no—
131C2×C4×C71—21no—
132C2×C4×C71—21no—
133C2×C4×C71—21no—
134C2×C4×C71—21no—
135C2×C4×C71—41no—
136C2×C4×C71—41no—
137C2×C4×C71—41no—
138C2×C4×C71—41no—
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—84no—
149C2×C4×C71#9911yescharacteristic
150C2×C4×C71#9911yescharacteristic
151C2×C4×C71#10412yes—
152C2×Dic71—44no—
153C2×Dic71—82no—
154C2×Dic71—82no—
155C71⋊C8—82no—
156C22×C71—21no—
157C22×C71—21no—
158C22×C71—21no—
159C22×C71—21no—
160C22×C71—41no—
161C22×C71—41no—
162C22×C71—42no—
163C22×C71—42no—
164C22×C71—41no—
165C22×C71—41no—
166C22×C71—42no—
167C22×C71—81no—
168C22×C71—81no—
169C22×C71—81no—
170C22×C71—81no—
171C22×C71—81no—
172C22×C71C2×#9911yescharacteristic
173C22×C71#41711yescharacteristic
174C22×C71C2×#10411yescharacteristic
175C4×C71—21no—
176C4×C71—21no—
177C4×C71—22no—
178C4×C71—41no—
179C4×C71—41no—
180C4×C71—41no—
181C4×C71—41no—
182C4×C71—81no—
183C4×C71—81no—
184C4×C71—81no—
185Dic71—84no—
186#11754—711nomaximal, Sylow
187C2×C71—21no—
188C2×C71—21no—
189C2×C71—41no—
190C2×C71—41no—
191C2×C71—82no—
192C2×C71#150411yescharacteristic
193C2×C71#212011yescharacteristic
194C2×C71#212811yescharacteristic
195#2598D7111yescharacteristic
196#1740—711no—
197#1464—711no—
198#1352—712no—
199#1343—712no—
200C71#1175411yescharacteristic, complemented, Sylow
201C2×C4×D4Dic7111yescharacteristic, complemented
202C2×#117—1421no—
203C2×#117C2×D7111yescharacteristic
204#432—1421no—
205#400Dic7111yescharacteristic, complemented
206#399C2×D7111yescharacteristic
207#394Dic7111yescharacteristic, complemented
208#392C2×D7111yescharacteristic
209#388—21no—
210#386—21no—
211#386—21no—
212#385—21no—
213#383Dic7111yescharacteristic, complemented
214#342—712no—
215#342—1421no—
216#324—1421no—
217#323—712no—
218C22×D4—21no—
219C22×D4C2×Dic7111yescharacteristic
220C23×C4—21no—
221C23×C4C22⋊D7111yescharacteristic
222C4×D4—41no—
223C4×D4—41no—
224#117—41no—
225#117—41no—
226#117—2842no—
227#117—1424no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—2841no—
230C2×C4⋊C4C2×Dic7111yescharacteristic
231C2×C4⋊C4C2×Dic7111yescharacteristic
232C2×C4⋊C4C22⋊D7111yescharacteristic
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—1422no—
239C2×C22⋊C4C22⋊D7111yescharacteristic
240C2×C22⋊C4C22×D7111yescharacteristic
241C2×C42—2841no—
242C2×C42C2×Dic7111yescharacteristic
243C2×C42C22⋊D7111yescharacteristic
244C4⋊C8—2841no—
245C22⋊C8—1422no—
246#97—2841no—
247#95—21no—
248#95—21no—
249#95—21no—
250#95—21no—
251#95C2×Dic7111yescharacteristic
252#95C2×Dic7111yescharacteristic
253C24—21no—
254C24C2×C22⋊D7111yescharacteristic
255C2×D4—41no—
256C2×D4—42no—
257C2×D4—42no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—81no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—42no—
270C22×C4—81no—
271C22×C4—81no—
272C22×C4—2842no—
273C22×C4C22⋊Dic7111yescharacteristic
274C22×C4C22⋊Dic7111yescharacteristic
275C22×C4C22⋊Dic7111yescharacteristic
276C22×C4C22⋊Dic7111yescharacteristic
277C22×C4C22×Dic7111yescharacteristic
278C22×C4C2×C22⋊D7111yescharacteristic
279C2×C8—2842no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—81no—
284C4⋊C4—2844no—
285C22⋊C4—41no—
286C22⋊C4—41no—
287C22⋊C4—42no—
288C22⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—42no—
291C22⋊C4—81no—
292C22⋊C4—81no—
293C22⋊C4—2842no—
294C22⋊C4—1424no—
295C42—21no—
296C42—21no—
297C42—21no—
298C42—21no—
299C42—2844no—
300C23—21no—
301C23—21no—
302C23—22no—
303C23—41no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23—41no—
310C23—41no—
311C23C2×C22⋊Dic7111yescharacteristic
312C23#1455615311yescharacteristic
313C23#1455615811yescharacteristic
314D4—42no—
315D4—42no—
316D4—82no—
317D4—82no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—84no—
344C2×C4—2844no—
345C2×C4—5682no—
346C2×C4—5682no—
347C2×C4#1455604311yescharacteristic
348C2×C4#1455604311yescharacteristic
349C2×C4#1455605512yes—
350C8—5682no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—41no—
356C22—41no—
357C22—42no—
358C22—42no—
359C22—41no—
360C22—41no—
361C22—42no—
362C22—81no—
363C22—81no—
364C22—81no—
365C22—81no—
366C22—81no—
367C22C2×#1455604311yescharacteristic
368C22#1798600011yescharacteristic
369C22C2×#1455605511yescharacteristic
370C4—21no—
371C4—21no—
372C4—22no—
373C4—41no—
374C4—41no—
375C4—41no—
376C4—41no—
377C4—81no—
378C4—81no—
379C4—81no—
380C4—5684no—
381C2—21no—
382C2—21no—
383C2—41no—
384C2—41no—
385C2—82no—
386C2#2258684411yescharacteristic
387C2#2258998111yescharacteristic
388C2#2259001211yescharacteristic
subgroupG/H
centerC22C2×#14556043
commutatorC22×C4×C71C22×C4
FrattiniC2×C22⋊C4C22×D71
FittingC71×#2598C2
socleC22×C71C2×#99
prime radical#392×C71C22
cosocleC2×C22⋊C4×C71C23
hypercenter#2598D71
hypocenterC71#11754
LCSG ▷ C22×C4×C71 ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2598

Complements

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

NH ≅ G/Ntype
1. #22588073C2semidirect
2. #22586674C2semidirect
3. #22586364C2semidirect
12. C2×C4×D4×C71C4semidirect
14. #400×C71C4semidirect
16. #394×C71C4semidirect
22. #383×C71C4semidirect
200. C71#11754semidirect
201. C2×C4×D4Dic71semidirect
205. #400Dic71semidirect
207. #394Dic71semidirect
213. #383Dic71semidirect

Maximal subgroups

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

subgroupindex
#225880732
#225866742
#225863642
#225863372
C71×#25982
#1175471