Finite Group Databasebeta

← #32996473

Subgroups — #32996473

Order 18688.

436 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 83 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#22615629C211yesmaximal, characteristic
2#22608265C211yesmaximal, characteristic
3#22607954C212yesmaximal
4#22607930C212yesmaximal
5C73×#1740C211yesmaximal, characteristic
6C2×#14556583C411yescharacteristic, complemented
7#17988644—21no—
8#17988526C411yescharacteristic, complemented
9#17988523C2211yescharacteristic
10#17988517C411yescharacteristic, complemented
11#17988512C2211yescharacteristic
12#17988482—21no—
13#17988481—21no—
14#17988471—21no—
15#17988460—21no—
16#17988433C411yescharacteristic, complemented
17#17988313—21no—
18#17988311C2212yes—
19C2×#117×C73—21no—
20C2×#117×C73C2211yescharacteristic
21#342×C73—21no—
22#342×C73C2212yes—
23#14556583—41no—
24#14556583—41no—
25C2×C4⋊Dic73C2×C411yescharacteristic
26C2×C4⋊Dic73C2×C411yescharacteristic
27C2×#11924007—21no—
28C2×#11924007—21no—
29C2×#11924007—21no—
30C2×#11924007—21no—
31C2×C22⋊Dic73—21no—
32C2×C22⋊Dic73—21no—
33C2×C4×Dic73—21no—
34C4⋊#10596831—41no—
35C22⋊#10596831—22no—
36#14556502—41no—
37#14556498—21no—
38#14556498—21no—
39#14556498—21no—
40#14556498—21no—
41#14556498—21no—
42#14556498C2×C411yescharacteristic
43#14556498C2×C411yescharacteristic
44C23×C4×C73D411yescharacteristic
45#117×C73—24no—
46#117×C73—41no—
47#117×C73—41no—
48#117×C73—42no—
49C2×C4⋊C4×C73—41no—
50C2×C4⋊C4×C73C2×C411yescharacteristic
51C2×C4⋊C4×C73D411yescharacteristic
52C2×C22⋊C4×C73—22no—
53C2×C22⋊C4×C73D411yescharacteristic
54C2×C22⋊C4×C73C2311yescharacteristic
55C2×C42×C73—41no—
56C2×C42×C73C2×C411yescharacteristic
57C2×C42×C73D411yescharacteristic
58C22×Dic73—21no—
59C22×Dic73—21no—
60C22×Dic73—21no—
61C22×Dic73—21no—
62C2×#10596831—42no—
63C4⋊Dic73—41no—
64C4⋊Dic73—41no—
65#11924007—42no—
66#11924007—42no—
67#11924007—41no—
68#11924007—41no—
69#11924007—42no—
70C22⋊Dic73—42no—
71C22⋊Dic73—41no—
72C22⋊Dic73—41no—
73C4×Dic73—42no—
74C24×C73C2×D411yescharacteristic
75C22×C4×C73—21no—
76C22×C4×C73—21no—
77C22×C4×C73—21no—
78C22×C4×C73—21no—
79C22×C4×C73—22no—
80C22×C4×C73—42no—
81C22×C4×C73C22⋊C411yescharacteristic
82C22×C4×C73C22⋊C411yescharacteristic
83C22×C4×C73C22⋊C411yescharacteristic
84C22×C4×C73C22⋊C411yescharacteristic
85C22×C4×C73C22×C411yescharacteristic
86C22×C4×C73C2×D411yescharacteristic
87C4⋊C4×C73—41no—
88C4⋊C4×C73—41no—
89C4⋊C4×C73—44no—
90C22⋊C4×C73—24no—
91C22⋊C4×C73—41no—
92C22⋊C4×C73—41no—
93C22⋊C4×C73—42no—
94C42×C73—21no—
95C42×C73—21no—
96C42×C73—21no—
97C42×C73—21no—
98C42×C73—44no—
99C23×C73—21no—
100C23×C73—21no—
101C23×C73—41no—
102C23×C73—41no—
103C23×C73C2×C22⋊C411yescharacteristic
104C23×C73AGL(1,ℤ/8)11yescharacteristic
105C23×C73#13711yescharacteristic
106C2×C4×C73—22no—
107C2×C4×C73—22no—
108C2×C4×C73—22no—
109C2×C4×C73—22no—
110C2×C4×C73—21no—
111C2×C4×C73—21no—
112C2×C4×C73—21no—
113C2×C4×C73—21no—
114C2×C4×C73—21no—
115C2×C4×C73—21no—
116C2×C4×C73—21no—
117C2×C4×C73—21no—
118C2×C4×C73—41no—
119C2×C4×C73—41no—
120C2×C4×C73—41no—
121C2×C4×C73—41no—
122C2×C4×C73—44no—
123C2×C4×C73—82no—
124C2×C4×C73—82no—
125C2×C4×C73#9911yescharacteristic
126C2×C4×C73#9911yescharacteristic
127C2×C4×C73#10412yes—
128C2×Dic73—22no—
129C2×Dic73—41no—
130C2×Dic73—41no—
131C2×Dic73—41no—
132C2×Dic73—41no—
133C2×Dic73—41no—
134C2×Dic73—41no—
135C2×Dic73—41no—
136C2×Dic73—41no—
137C2×Dic73—41no—
138C2×Dic73—41no—
139C2×Dic73—41no—
140#10596831—82no—
141C22×C73—21no—
142C22×C73—21no—
143C22×C73—21no—
144C22×C73—21no—
145C22×C73—41no—
146C22×C73—41no—
147C22×C73—41no—
148C22×C73—41no—
149C22×C73—81no—
150C22×C73C2×#9911yescharacteristic
151C22×C73#41711yescharacteristic
152C22×C73C2×#10411yescharacteristic
153C4×C73—21no—
154C4×C73—21no—
155C4×C73—22no—
156C4×C73—41no—
157C4×C73—41no—
158C4×C73—41no—
159C4×C73—41no—
160C4×C73—84no—
161Dic73—42no—
162Dic73—81no—
163Dic73—81no—
164Dic73—81no—
165#11760—731nomaximal, Sylow
166C2×C73—21no—
167C2×C73—21no—
168C2×C73—41no—
169C2×C73—41no—
170C2×C73#150411yescharacteristic
171C2×C73#212311yescharacteristic
172C2×C73#212611yescharacteristic
173#2611—731no—
174#1740D7311yescharacteristic
175#1464—731no—
176#1352—732no—
177#1344—732no—
178C73#1176011yescharacteristic, complemented, Sylow
179C2×#125—731no—
180C2×#117—21no—
181C2×#117C2×D7311yescharacteristic
182#432—1461no—
183#400—731no—
184#399—731no—
185#398—731no—
186#397—731no—
187#389—1461no—
188#388—1461no—
189#387—1461no—
190#385—1461no—
191#376—731no—
192#342—21no—
193#342C2×D7312yes—
194#324—1461no—
195#323—732no—
196C23×C4C22⋊D7311yescharacteristic
197#125—2921no—
198#125—2921no—
199#117—24no—
200#117—41no—
201#117—41no—
202#117—42no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—731no—
205C2×C4⋊C4—731no—
206C2×C4⋊C4—1461no—
207C2×C4⋊C4—1461no—
208C2×C4⋊C4—1461no—
209C2×C4⋊C4—1461no—
210C2×C4⋊C4C4×D7311yescharacteristic
211C2×C4⋊C4D29211yescharacteristic
212C2×C22⋊C4—22no—
213C2×C22⋊C4—1461no—
214C2×C22⋊C4—1461no—
215C2×C22⋊C4C22⋊D7311yescharacteristic
216C2×C22⋊C4C22×D7311yescharacteristic
217C2×C42—41no—
218C2×C42—1461no—
219C2×C42C4×D7311yescharacteristic
220C2×C42D29211yescharacteristic
221C4⋊C8—2921no—
222C22⋊C8—1462no—
223#97—2921no—
224#95—731no—
225#95—731no—
226#95—1461no—
227#95—1461no—
228#95—1461no—
229#95—1461no—
230#95—1461no—
231C24C2×C22⋊D7311yescharacteristic
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—42no—
238C22×C4—1461no—
239C22×C4—1461no—
240C22×C4—1461no—
241C22×C4—1461no—
242C22×C4#1192400511yescharacteristic
243C22×C4#1192400511yescharacteristic
244C22×C4#1192400511yescharacteristic
245C22×C4#1192400511yescharacteristic
246C22×C4C2×C4×D7311yescharacteristic
247C22×C4C2×D29211yescharacteristic
248C2×C8—2922no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—44no—
252C4⋊C4—2921no—
253C4⋊C4—2921no—
254C4⋊C4—2922no—
255C4⋊C4—2922no—
256C4⋊C4—2921no—
257C4⋊C4—2921no—
258C4⋊C4—2922no—
259C22⋊C4—24no—
260C22⋊C4—41no—
261C22⋊C4—41no—
262C22⋊C4—42no—
263C22⋊C4—2921no—
264C22⋊C4—2921no—
265C22⋊C4—2922no—
266C42—21no—
267C42—21no—
268C42—21no—
269C42—21no—
270C42—44no—
271C42—2922no—
272C23—21no—
273C23—21no—
274C23—41no—
275C23—41no—
276C23C2×#1192400511yescharacteristic
277C23#1455661711yescharacteristic
278C23#1455662211yescharacteristic
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—44no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—2921no—
299C2×C4—2921no—
300C2×C4—2921no—
301C2×C4—2921no—
302C2×C4—2921no—
303C2×C4—2921no—
304C2×C4—2921no—
305C2×C4—2921no—
306C2×C4—2921no—
307C2×C4—2921no—
308C2×C4—2921no—
309C2×C4—1462no—
310C2×C4#1455650511yescharacteristic
311C2×C4#1455650511yescharacteristic
312C2×C4#1455651912yes—
313C8—5842no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22—41no—
322C22—81no—
323C22C2×#1455650511yescharacteristic
324C22#1798858611yescharacteristic
325C22C2×#1455651911yescharacteristic
326C4—21no—
327C4—21no—
328C4—22no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—84no—
334C4—2922no—
335C4—5841no—
336C4—5841no—
337C4—5841no—
338C2—21no—
339C2—21no—
340C2—41no—
341C2—41no—
342C2#2260843311yescharacteristic
343C2#2261158411yescharacteristic
344C2#2261159411yescharacteristic
subgroupG/H
centerC22C2×#14556505
commutatorC22×C4×C73C22×C4
FrattiniC2×C22⋊C4C22×D73
FittingC73×#1740C2
socleC22×C73C2×#99
prime radicalC24×C73C2×D4
cosocleC2×C22⋊C4×C73C23
hypercenter#1740D73
hypocenterC73#11760
LCSG ▷ C22×C4×C73 ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1740

Complements

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

NH ≅ G/Ntype
6. C2×#14556583C4semidirect
8. #17988526C4semidirect
10. #17988517C4semidirect
16. #17988433C4semidirect
178. C73#11760semidirect

Maximal subgroups

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

subgroupindex
#226156292
#226082652
#226079542
#226079302
C73×#17402
#1176073