Finite Group Databasebeta

← #31907781

Subgroups — #31907781

Order 18176.

606 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 606 conjugacy classes of subgroups — too many to draw. Only its 95 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
1C4×#14556043C211yesmaximal, characteristic, complemented
2#22586658C211yesmaximal, characteristic, complemented
3#22586364C212yesmaximal, complemented
4#22586337C212yesmaximal, complemented
5C71×#2397C211yesmaximal, characteristic
6C2×#14556043—21no—
7C2×C22⋊(C71⋊C8)—21no—
8C4×C22⋊Dic71—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
23C22×C71⋊C8—41no—
24C2×C22⋊Dic71—22no—
25C2×C4×Dic71—41no—
26#14556043—24no—
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—
60C2×C71⋊C8—42no—
61C2×C71⋊C8—82no—
62C22⋊Dic71—24no—
63C22⋊Dic71—42no—
64C4×Dic71—82no—
65C24×C71—21no—
66C24×C71C22⋊C411yescharacteristic
67C24×C71C2×D411yescharacteristic
68C2×D4×C71—22no—
69C2×D4×C71—22no—
70C2×D4×C71—41no—
71C2×D4×C71—41no—
72C2×D4×C71—42no—
73C2×D4×C71—42no—
74C2×D4×C71—42no—
75C2×D4×C71—42no—
76C2×D4×C71—42no—
77C2×D4×C71—42no—
78C2×D4×C71—42no—
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—21no—
86C22×C4×C71—42no—
87C22×C4×C71—42no—
88C22×C4×C71—42no—
89C22×C4×C71—42no—
90C22×C4×C71—41no—
91C22×C4×C71—41no—
92C22×C4×C71C22⋊C411yescharacteristic
93C22×C4×C71C22⋊C411yescharacteristic
94C22×C4×C71C22⋊C411yescharacteristic
95C22×C4×C71C22×C411yescharacteristic
96C22×C4×C71C2×D411yescharacteristic
97C4⋊C4×C71—41no—
98C4⋊C4×C71—41no—
99C4⋊C4×C71—42no—
100C4⋊C4×C71—42no—
101C22⋊C4×C71—41no—
102C22⋊C4×C71—41no—
103C22⋊C4×C71—42no—
104C22⋊C4×C71—42no—
105C22⋊C4×C71—42no—
106C22⋊C4×C71—42no—
107C22⋊C4×C71—42no—
108C42×C71—24no—
109C42×C71—21no—
110C42×C71—21no—
111C23×C71—21no—
112C23×C71—21no—
113C23×C71—21no—
114C23×C71—21no—
115C23×C71—22no—
116C23×C71—41no—
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#10412yes—
127C23×C71C2×C22⋊C411yescharacteristic
128C23×C71#13512yes—
129D4×C71—44no—
130D4×C71—42no—
131D4×C71—48no—
132D4×C71—48no—
133D4×C71—42no—
134C2×C4×C71—22no—
135C2×C4×C71—22no—
136C2×C4×C71—22no—
137C2×C4×C71—21no—
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—41no—
145C2×C4×C71—41no—
146C2×C4×C71—41no—
147C2×C4×C71—42no—
148C2×C4×C71—42no—
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—81no—
156C2×C4×C71—81no—
157C2×C4×C71—82no—
158C2×C4×C71—82no—
159C2×C4×C71#9911yescharacteristic
160C2×C4×C71#9911yescharacteristic
161C2×Dic71—44no—
162C2×Dic71—82no—
163C2×Dic71—82no—
164C71⋊C8—82no—
165C22×C71—21no—
166C22×C71—21no—
167C22×C71—22no—
168C22×C71—22no—
169C22×C71—42no—
170C22×C71—42no—
171C22×C71—42no—
172C22×C71—42no—
173C22×C71—41no—
174C22×C71—41no—
175C22×C71—41no—
176C22×C71—41no—
177C22×C71—42no—
178C22×C71—81no—
179C22×C71—81no—
180C22×C71—81no—
181C22×C71—81no—
182C22×C71—81no—
183C22×C71—81no—
184C22×C71C2×#9911yescharacteristic
185C22×C71#41611yescharacteristic
186C22×C71C2×#10411yescharacteristic
187C4×C71—21no—
188C4×C71—42no—
189C4×C71—42no—
190C4×C71—41no—
191C4×C71—41no—
192C4×C71—41no—
193C4×C71—41no—
194C4×C71—81no—
195C4×C71#35312yes—
196Dic71—84no—
197#11726—711nomaximal, Sylow
198C2×C71—21no—
199C2×C71—21no—
200C2×C71—42no—
201C2×C71—42no—
202C2×C71—82no—
203C2×C71#150111yescharacteristic
204C2×C71#212011yescharacteristic
205C2×C71C2×#35311yescharacteristic
206#2397D7111yescharacteristic
207C4×#99—711no—
208#1459—711no—
209#1352—712no—
210#1343—712no—
211C71#1172611yescharacteristic, complemented, Sylow
212C2×C4×D4—21no—
213C2×C4×D4—21no—
214C2×C4×D4Dic7111yescharacteristic, complemented
215C2×C4×D4C2×D7111yescharacteristic
216C2×#99—1421no—
217C2×C22⋊C8—1421no—
218#399—21no—
219#399Dic7111yescharacteristic, complemented
220#399C2×D7111yescharacteristic
221#392—21no—
222#392Dic7111yescharacteristic, complemented
223#392C2×D7111yescharacteristic
224C4×C22⋊C4—1421no—
225#376Dic7111yescharacteristic, complemented
226#342—712no—
227#324—1421no—
228#323—712no—
229C22×D4—21no—
230C22×D4—21no—
231C22×D4C2×Dic7111yescharacteristic
232C22×D4C22⋊D7111yescharacteristic
233C23×C4—21no—
234C23×C4C22⋊D7111yescharacteristic
235C23×C4C22⋊D7111yescharacteristic
236C22×C8—2841no—
237C4×D4—41no—
238C4×D4—41no—
239C4×D4—44no—
240C4×D4—44no—
241C4×D4—41no—
242C4×D4—41no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4C2×Dic7111yescharacteristic
246C2×C4⋊C4C22⋊D7111yescharacteristic
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—1422no—
253C2×C22⋊C4C2×Dic7111yescharacteristic
254C2×C22⋊C4C22×D7111yescharacteristic
255C2×C42—21no—
256C2×C42—2841no—
257C2×C42C2×Dic7111yescharacteristic
258#99—1424no—
259C22⋊C8—2841no—
260C22⋊C8—1422no—
261#95—21no—
262#95—2841no—
263#95C2×Dic7111yescharacteristic
264#95C2×Dic7111yescharacteristic
265C24—21no—
266C24C22⋊Dic7111yescharacteristic
267C24C2×C22⋊D7111yescharacteristic
268C2×D4—22no—
269C2×D4—22no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—42no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—42no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—2842no—
293C22×C4C22⋊Dic7111yescharacteristic
294C22×C4C22⋊Dic7111yescharacteristic
295C22×C4C22⋊Dic7111yescharacteristic
296C22×C4C22×Dic7111yescharacteristic
297C22×C4C2×C22⋊D7111yescharacteristic
298C2×C8—2842no—
299C2×C8—5682no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C22⋊C4—41no—
305C22⋊C4—41no—
306C22⋊C4—42no—
307C22⋊C4—42no—
308C22⋊C4—42no—
309C22⋊C4—42no—
310C22⋊C4—42no—
311C22⋊C4—2842no—
312C22⋊C4—1424no—
313C42—24no—
314C42—21no—
315C42—21no—
316C42—5682no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—22no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23#1455605512yes—
333C23C2×C22⋊Dic7111yescharacteristic
334C23#1455614812yes—
335D4—44no—
336D4—42no—
337D4—48no—
338D4—48no—
339D4—42no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—2844no—
366C2×C4—5682no—
367C2×C4—5682no—
368C2×C4#1455604311yescharacteristic
369C2×C4#1455604311yescharacteristic
370C8—5682no—
371C22—21no—
372C22—21no—
373C22—22no—
374C22—22no—
375C22—42no—
376C22—42no—
377C22—42no—
378C22—42no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—42no—
384C22—81no—
385C22—81no—
386C22—81no—
387C22—81no—
388C22—81no—
389C22—81no—
390C22C2×#1455604311yescharacteristic
391C22#1798599511yescharacteristic
392C22C2×#1455605511yescharacteristic
393C4—21no—
394C4—42no—
395C4—42no—
396C4—41no—
397C4—41no—
398C4—41no—
399C4—41no—
400C4—81no—
401C4—5684no—
402C4#1798579212yes—
403C2—21no—
404C2—21no—
405C2—42no—
406C2—42no—
407C2—82no—
408C2#2258683111yescharacteristic
409C2#2258998111yescharacteristic
410C2C2×#1798579211yescharacteristic
subgroupG/H
centerC2×C4#14556043
commutatorC22×C4×C71C22×C4
FrattiniC2×C22⋊C4C22×D71
Fitting, prime radicalC71×#2397C2
socleC22×C71C2×#99
cosocleC2×C22⋊C4×C71C23
hypercenter#2397D71
hypocenterC71#11726
LCSG ▷ C22×C4×C71 ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2397

Complements

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

NH ≅ G/Ntype
1. C4×#14556043C2semidirect
2. #22586658C2semidirect
3. #22586364C2semidirect
4. #22586337C2semidirect
14. C2×C4×D4×C71C4semidirect
17. #399×C71C4semidirect
20. #392×C71C4semidirect
22. #376×C71C4semidirect
211. C71#11726semidirect
214. C2×C4×D4Dic71semidirect
219. #399Dic71semidirect
222. #392Dic71semidirect
225. #376Dic71semidirect

Maximal subgroups

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

subgroupindex
C4×#145560432
#225866582
#225863642
#225863372
C71×#23972
#1172671