Finite Group Databasebeta

← #31907628

Subgroups — #31907628

Order 18176.

552 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 115 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#22586654C211yesmaximal, characteristic, complemented
3#22586340C212yesmaximal, complemented
4#22586337C212yesmaximal
5C71×#2392C211yesmaximal, characteristic
6C2×#14556087—21no—
7#17986052—21no—
8#17985766—21no—
9#17985766C2212yes—
10#17985725—21no—
11#17985723C2212yes—
12C2×C4×D4×C71—21no—
13C2×C4×D4×C71C411yescharacteristic, complemented
14C2×#117×C71C2211yescharacteristic
15#398×C71C411yescharacteristic, complemented
16#397×C71C2211yescharacteristic
17#396×C71—21no—
18#394×C71—21no—
19#394×C71C411yescharacteristic, complemented
20#392×C71C2211yescharacteristic
21C4×C4⋊C4×C71C411yescharacteristic, complemented
22C4×C22⋊C4×C71—21no—
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—
30C4×C71⋊C8—41no—
31C22×D4×C71—21no—
32C22×D4×C71C2×C411yescharacteristic
33C23×C4×C71—21no—
34C23×C4×C71D411yescharacteristic
35C4×D4×C71—41no—
36C4×D4×C71—41no—
37C4×D4×C71—42no—
38C4×D4×C71—42no—
39#117×C71—41no—
40#117×C71—41no—
41C2×C4⋊C4×C71—21no—
42C2×C4⋊C4×C71—21no—
43C2×C4⋊C4×C71C2×C411yescharacteristic
44C2×C4⋊C4×C71C2×C411yescharacteristic
45C2×C4⋊C4×C71D411yescharacteristic
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×C71C2×C411yescharacteristic
55C2×C42×C71D411yescharacteristic
56#95×C71—21no—
57#95×C71—21no—
58#95×C71C2×C411yescharacteristic
59#95×C71C2×C411yescharacteristic
60C22×Dic71—42no—
61C2×C71⋊C8—42no—
62C4⋊Dic71—44no—
63C22⋊Dic71—24no—
64C22⋊Dic71—42no—
65C4×Dic71—44no—
66C24×C71—21no—
67C24×C71C2×D411yescharacteristic
68C2×D4×C71—41no—
69C2×D4×C71—42no—
70C2×D4×C71—42no—
71C2×D4×C71—41no—
72C2×D4×C71—41no—
73C2×D4×C71—42no—
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—41no—
81C22×C4×C71—41no—
82C22×C4×C71—42no—
83C22×C4×C71—42no—
84C22×C4×C71C22⋊C411yescharacteristic
85C22×C4×C71C22⋊C411yescharacteristic
86C22×C4×C71C22⋊C411yescharacteristic
87C22×C4×C71C22⋊C411yescharacteristic
88C22×C4×C71D812yes—
89C22×C4×C71SD1612yes—
90C22×C4×C71C22×C411yescharacteristic
91C22×C4×C71C2×D411yescharacteristic
92C4⋊C4×C71—41no—
93C4⋊C4×C71—41no—
94C4⋊C4×C71—41no—
95C4⋊C4×C71—42no—
96C4⋊C4×C71—42no—
97C22⋊C4×C71—41no—
98C22⋊C4×C71—41no—
99C22⋊C4×C71—42no—
100C22⋊C4×C71—42no—
101C22⋊C4×C71—42no—
102C22⋊C4×C71—41no—
103C22⋊C4×C71—41no—
104C22⋊C4×C71—42no—
105C42×C71—21no—
106C42×C71—21no—
107C42×C71—21no—
108C42×C71—21no—
109C42×C71—42no—
110C23×C71—21no—
111C23×C71—21no—
112C23×C71—22no—
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×C71C2×C22⋊C411yescharacteristic
122C23×C71C2×D811yescharacteristic
123C23×C71C2×SD1611yescharacteristic
124D4×C71—42no—
125D4×C71—44no—
126D4×C71—44no—
127D4×C71—42no—
128C2×C4×C71—21no—
129C2×C4×C71—21no—
130C2×C4×C71—22no—
131C2×C4×C71—22no—
132C2×C4×C71—22no—
133C2×C4×C71—22no—
134C2×C4×C71—22no—
135C2×C4×C71—21no—
136C2×C4×C71—21no—
137C2×C4×C71—21no—
138C2×C4×C71—21no—
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—82no—
152C2×C4×C71—82no—
153C2×C4×C71#9911yescharacteristic
154C2×C4×C71#9911yescharacteristic
155C2×C4×C71#10212yes—
156C2×C4×C71#10212yes—
157C2×C4×C71#10412yes—
158C2×Dic71—44no—
159C2×Dic71—82no—
160C2×Dic71—82no—
161C71⋊C8—82no—
162C22×C71—21no—
163C22×C71—21no—
164C22×C71—21no—
165C22×C71—21no—
166C22×C71—41no—
167C22×C71—41no—
168C22×C71—42no—
169C22×C71—42no—
170C22×C71—41no—
171C22×C71—41no—
172C22×C71—42no—
173C22×C71—81no—
174C22×C71—81no—
175C22×C71—81no—
176C22×C71—81no—
177C22×C71—81no—
178C22×C71C2×#9911yescharacteristic
179C22×C71C2×#10211yescharacteristic
180C22×C71C2×#10411yescharacteristic
181C4×C71—22no—
182C4×C71—42no—
183C4×C71—41no—
184C4×C71—41no—
185C4×C71—41no—
186C4×C71—41no—
187C4×C71—81no—
188C4×C71—81no—
189C4×C71#32712yes—
190C4×C71#32712yes—
191Dic71—84no—
192#11691—711nomaximal, Sylow
193C2×C71—21no—
194C2×C71—21no—
195C2×C71—41no—
196C2×C71—41no—
197C2×C71—82no—
198C2×C71C2×#32711yescharacteristic
199C2×C71#212011yescharacteristic
200C2×C71#212611yescharacteristic
201#2392D7111yescharacteristic
202#1740—711no—
203#1458—711no—
204#1344—712no—
205#1343—712no—
206C71#1169111yescharacteristic, complemented, Sylow
207C2×C4×D4—21no—
208C2×C4×D4Dic7111yescharacteristic, complemented
209C2×#117—1421no—
210C2×#117C2×D7111yescharacteristic
211#431—1421no—
212#398Dic7111yescharacteristic, complemented
213#397C2×D7111yescharacteristic
214#396—21no—
215#394—21no—
216#394Dic7111yescharacteristic, complemented
217#392C2×D7111yescharacteristic
218C4×C4⋊C4Dic7111yescharacteristic, complemented
219C4×C22⋊C4—21no—
220#342—712no—
221#342—1421no—
222#324—1421no—
223#323—712no—
224C22×D4—21no—
225C22×D4C2×Dic7111yescharacteristic
226C23×C4—21no—
227C23×C4C22⋊D7111yescharacteristic
228C4×D4—41no—
229C4×D4—41no—
230C4×D4—42no—
231C4×D4—42no—
232#117—41no—
233#117—41no—
234#117—2842no—
235#117—1424no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—2841no—
239C2×C4⋊C4C2×Dic7111yescharacteristic
240C2×C4⋊C4C2×Dic7111yescharacteristic
241C2×C4⋊C4C22⋊D7111yescharacteristic
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×C42—21no—
251C2×C42—2841no—
252C2×C42C2×Dic7111yescharacteristic
253C2×C42C22⋊D7111yescharacteristic
254C4⋊C8—2841no—
255C22⋊C8—1422no—
256C4×C8—2841no—
257#95—21no—
258#95—21no—
259#95C2×Dic7111yescharacteristic
260#95C2×Dic7111yescharacteristic
261C24—21no—
262C24C2×C22⋊D7111yescharacteristic
263C2×D4—41no—
264C2×D4—42no—
265C2×D4—42no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—42no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—2842no—
280C22×C4C22⋊Dic7111yescharacteristic
281C22×C4C22⋊Dic7111yescharacteristic
282C22×C4C22⋊Dic7111yescharacteristic
283C22×C4C22⋊Dic7111yescharacteristic
284C22×C4#1176585012yes—
285C22×C4#1176585312yes—
286C22×C4C22×Dic7111yescharacteristic
287C22×C4C2×C22⋊D7111yescharacteristic
288C2×C8—2842no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—2844no—
295C22⋊C4—41no—
296C22⋊C4—41no—
297C22⋊C4—42no—
298C22⋊C4—42no—
299C22⋊C4—42no—
300C22⋊C4—41no—
301C22⋊C4—41no—
302C22⋊C4—42no—
303C22⋊C4—2842no—
304C22⋊C4—1424no—
305C42—21no—
306C42—21no—
307C42—21no—
308C42—21no—
309C42—42no—
310C42—2844no—
311C23—21no—
312C23—21no—
313C23—22no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23C2×C22⋊Dic7111yescharacteristic
323C23C2×#1176585011yescharacteristic
324C23C2×#1176585311yescharacteristic
325D4—42no—
326D4—44no—
327D4—44no—
328D4—42no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
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—82no—
353C2×C4—82no—
354C2×C4—2844no—
355C2×C4—5682no—
356C2×C4—5682no—
357C2×C4#1455604311yescharacteristic
358C2×C4#1455604311yescharacteristic
359C2×C4#1455604912yes—
360C2×C4#1455604912yes—
361C2×C4#1455605512yes—
362C8—5682no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—41no—
368C22—41no—
369C22—42no—
370C22—42no—
371C22—41no—
372C22—41no—
373C22—42no—
374C22—81no—
375C22—81no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22C2×#1455604311yescharacteristic
380C22C2×#1455604911yescharacteristic
381C22C2×#1455605511yescharacteristic
382C4—22no—
383C4—42no—
384C4—41no—
385C4—41no—
386C4—41no—
387C4—41no—
388C4—81no—
389C4—81no—
390C4—5684no—
391C4#1798573312yes—
392C4#1798573312yes—
393C2—21no—
394C2—21no—
395C2—41no—
396C2—41no—
397C2—82no—
398C2C2×#1798573311yescharacteristic
399C2#2258998111yescharacteristic
400C2#2259000511yescharacteristic
subgroupG/H
centerC2×C4#14556043
commutatorC22×C4×C71C22×C4
FrattiniC2×C22⋊C4C22×D71
FittingC71×#2392C2
socleC22×C71C2×#99
prime radical#392×C71C22
cosocleC2×C22⋊C4×C71C23
hypercenter#2392D71
hypocenterC71#11691
LCSG ▷ C22×C4×C71 ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#117 ◁ #2392

Complements

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

NH ≅ G/Ntype
1. #22588073C2semidirect
2. #22586654C2semidirect
3. #22586340C2semidirect
13. C2×C4×D4×C71C4semidirect
15. #398×C71C4semidirect
19. #394×C71C4semidirect
21. C4×C4⋊C4×C71C4semidirect
206. C71#11691semidirect
208. C2×C4×D4Dic71semidirect
212. #398Dic71semidirect
216. #394Dic71semidirect
218. C4×C4⋊C4Dic71semidirect

Maximal subgroups

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

subgroupindex
#225880732
#225866542
#225863402
#225863372
C71×#23922
#1169171