Finite Group Databasebeta

← #32996328

Subgroups — #32996328

Order 18688.

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

normal subgroups only

This group has 486 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
1C4×#14556506C211yesmaximal, characteristic, complemented
2#22608252C211yesmaximal, characteristic, complemented
3#22607955C212yesmaximal, complemented
4#22607928C212yesmaximal, complemented
5C73×#2635C211yesmaximal, characteristic
6#17988640—21no—
7C2×#14556506—21no—
8C4×C22⋊Dic73—21no—
9#17988354C2212yes—
10#17988311—21no—
11#17988311C2212yes—
12C2×#121×C73C411yescharacteristic, complemented
13C2×C4×D4×C73C2211yescharacteristic
14#400×C73C411yescharacteristic, complemented
15#399×C73C2211yescharacteristic
16#394×C73—21no—
17#394×C73C411yescharacteristic, complemented
18#392×C73—21no—
19#392×C73C2211yescharacteristic
20#388×C73—21no—
21#386×C73—21no—
22#381×C73C411yescharacteristic, complemented
23C2×C22⋊Dic73—22no—
24C2×C4×Dic73—41no—
25C4⋊#10596831—41no—
26#14556506—24no—
27C22⋊#10596831—22no—
28C4×#10596831—41no—
29#14556498—41no—
30C22×D4×C73—21no—
31C22×D4×C73—21no—
32C22×D4×C73C2×C411yescharacteristic
33C22×D4×C73D411yescharacteristic
34C23×C4×C73C2×C411yescharacteristic
35C23×C4×C73D411yescharacteristic
36#121×C73—41no—
37#121×C73—41no—
38C4×D4×C73—41no—
39C4×D4×C73—41no—
40C2×C4⋊C4×C73—21no—
41C2×C4⋊C4×C73C2×C411yescharacteristic
42C2×C4⋊C4×C73C2×C411yescharacteristic
43C2×C22⋊C4×C73—21no—
44C2×C22⋊C4×C73—21no—
45C2×C22⋊C4×C73—21no—
46C2×C22⋊C4×C73—21no—
47C2×C22⋊C4×C73—21no—
48C2×C22⋊C4×C73—21no—
49C2×C22⋊C4×C73D411yescharacteristic
50C2×C22⋊C4×C73C2311yescharacteristic
51C2×C42×C73D411yescharacteristic
52#95×C73—21no—
53#95×C73—21no—
54#95×C73—21no—
55#95×C73C2×C411yescharacteristic
56#95×C73C2×C411yescharacteristic
57C22×Dic73—42no—
58C2×#10596831—42no—
59C22⋊Dic73—24no—
60C22⋊Dic73—42no—
61C4×Dic73—82no—
62C24×C73—21no—
63C24×C73C22⋊C411yescharacteristic
64C24×C73C2×D411yescharacteristic
65C2×D4×C73—22no—
66C2×D4×C73—22no—
67C2×D4×C73—41no—
68C2×D4×C73—41no—
69C2×D4×C73—42no—
70C2×D4×C73—42no—
71C2×D4×C73—81no—
72C2×D4×C73—81no—
73C2×D4×C73—81no—
74C2×D4×C73—81no—
75C2×D4×C73—81no—
76C22×C4×C73—21no—
77C22×C4×C73—21no—
78C22×C4×C73—21no—
79C22×C4×C73—21no—
80C22×C4×C73—21no—
81C22×C4×C73—21no—
82C22×C4×C73—21no—
83C22×C4×C73—81no—
84C22×C4×C73—81no—
85C22×C4×C73C22⋊C411yescharacteristic
86C22×C4×C73C22⋊C411yescharacteristic
87C22×C4×C73C22⋊C411yescharacteristic
88C22×C4×C73C22×C411yescharacteristic
89C22×C4×C73C2×D411yescharacteristic
90C4⋊C4×C73—41no—
91C4⋊C4×C73—41no—
92C4⋊C4×C73—81no—
93C22⋊C4×C73—41no—
94C22⋊C4×C73—41no—
95C22⋊C4×C73—41no—
96C22⋊C4×C73—41no—
97C22⋊C4×C73—42no—
98C22⋊C4×C73—81no—
99C22⋊C4×C73—81no—
100C22⋊C4×C73—81no—
101C22⋊C4×C73—81no—
102C42×C73—21no—
103C42×C73—21no—
104C23×C73—21no—
105C23×C73—21no—
106C23×C73—21no—
107C23×C73—21no—
108C23×C73—21no—
109C23×C73—41no—
110C23×C73—41no—
111C23×C73—41no—
112C23×C73—41no—
113C23×C73—41no—
114C23×C73—41no—
115C23×C73—41no—
116C23×C73—41no—
117C23×C73—41no—
118C23×C73—41no—
119C23×C73—41no—
120C23×C73C2×C22⋊C411yescharacteristic
121C23×C73#13512yes—
122D4×C73—44no—
123D4×C73—82no—
124D4×C73—84no—
125D4×C73—84no—
126C2×C4×C73—21no—
127C2×C4×C73—21no—
128C2×C4×C73—21no—
129C2×C4×C73—21no—
130C2×C4×C73—21no—
131C2×C4×C73—41no—
132C2×C4×C73—42no—
133C2×C4×C73—41no—
134C2×C4×C73—41no—
135C2×C4×C73—41no—
136C2×C4×C73—41no—
137C2×C4×C73—41no—
138C2×C4×C73—41no—
139C2×C4×C73—41no—
140C2×C4×C73—41no—
141C2×C4×C73—41no—
142C2×C4×C73—41no—
143C2×C4×C73—41no—
144C2×C4×C73—41no—
145C2×C4×C73—81no—
146C2×C4×C73—81no—
147C2×C4×C73—81no—
148C2×C4×C73#9911yescharacteristic
149C2×C4×C73#9911yescharacteristic
150C2×C4×C73#10412yes—
151C2×Dic73—44no—
152C2×Dic73—82no—
153C2×Dic73—82no—
154#10596831—82no—
155C22×C73—21no—
156C22×C73—21no—
157C22×C73—22no—
158C22×C73—42no—
159C22×C73—41no—
160C22×C73—41no—
161C22×C73—41no—
162C22×C73—41no—
163C22×C73—41no—
164C22×C73—81no—
165C22×C73—81no—
166C22×C73—81no—
167C22×C73—81no—
168C22×C73—81no—
169C22×C73—81no—
170C22×C73—81no—
171C22×C73—81no—
172C22×C73—81no—
173C22×C73—81no—
174C22×C73C2×#9911yescharacteristic
175C22×C73#41611yescharacteristic
176C22×C73C2×#10411yescharacteristic
177C4×C73—21no—
178C4×C73—21no—
179C4×C73—22no—
180C4×C73—41no—
181C4×C73—41no—
182C4×C73—41no—
183C4×C73—81no—
184C4×C73—81no—
185C4×C73—81no—
186Dic73—84no—
187#11729—731nomaximal, Sylow
188C2×C73—21no—
189C2×C73—21no—
190C2×C73—42no—
191C2×C73—81no—
192C2×C73—161no—
193C2×C73#150111yescharacteristic
194C2×C73#212111yescharacteristic
195C2×C73#212811yescharacteristic
196#2635D7311yescharacteristic
197C4×#99—731no—
198#1460—731no—
199#1352—732no—
200#1343—732no—
201C73#1172911yescharacteristic, complemented, Sylow
202C2×#121Dic7311yescharacteristic, complemented
203C2×C4×D4C2×D7311yescharacteristic
204#431—1461no—
205C2×#99—1461no—
206#400Dic7311yescharacteristic, complemented
207#399C2×D7311yescharacteristic
208#394—21no—
209#394Dic7311yescharacteristic, complemented
210#392—21no—
211#392C2×D7311yescharacteristic
212#388—21no—
213#386—21no—
214#381Dic7311yescharacteristic, complemented
215C4×C22⋊C4—1461no—
216#342—732no—
217#323—732no—
218#323—1461no—
219C22×D4—21no—
220C22×D4—21no—
221C22×D4C2×Dic7311yescharacteristic
222C22×D4C22⋊D7311yescharacteristic
223C23×C4C2×Dic7311yescharacteristic
224C23×C4C22⋊D7311yescharacteristic
225#121—41no—
226#121—41no—
227C4×D4—41no—
228C4×D4—41no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4C2×Dic7311yescharacteristic
231C2×C4⋊C4C2×Dic7311yescharacteristic
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—1462no—
239C2×C22⋊C4C22⋊D7311yescharacteristic
240C2×C22⋊C4C22×D7311yescharacteristic
241C2×C42—2921no—
242C2×C42C22⋊D7311yescharacteristic
243C4⋊C8—2921no—
244#99—1464no—
245C22⋊C8—1462no—
246C4×C8—2921no—
247#95—21no—
248#95—21no—
249#95—21no—
250#95—2921no—
251#95C2×Dic7311yescharacteristic
252#95C2×Dic7311yescharacteristic
253C24—21no—
254C24C22⋊Dic7311yescharacteristic
255C24C2×C22⋊D7311yescharacteristic
256C2×D4—22no—
257C2×D4—22no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—42no—
261C2×D4—42no—
262C2×D4—81no—
263C2×D4—81no—
264C2×D4—81no—
265C2×D4—81no—
266C2×D4—81no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—81no—
275C22×C4—81no—
276C22×C4—2922no—
277C22×C4C22⋊Dic7311yescharacteristic
278C22×C4C22⋊Dic7311yescharacteristic
279C22×C4C22⋊Dic7311yescharacteristic
280C22×C4C22×Dic7311yescharacteristic
281C22×C4C2×C22⋊D7311yescharacteristic
282C2×C8—2922no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—81no—
286C22⋊C4—41no—
287C22⋊C4—41no—
288C22⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—42no—
291C22⋊C4—81no—
292C22⋊C4—81no—
293C22⋊C4—81no—
294C22⋊C4—81no—
295C22⋊C4—2922no—
296C22⋊C4—1464no—
297C42—21no—
298C42—21no—
299C42—5842no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23—41no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23C2×C22⋊Dic7311yescharacteristic
317C23#1455661112yes—
318D4—44no—
319D4—82no—
320D4—84no—
321D4—84no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—41no—
328C2×C4—42no—
329C2×C4—41no—
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—81no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—2924no—
345C2×C4—5842no—
346C2×C4—5842no—
347C2×C4#1455650611yescharacteristic
348C2×C4#1455650611yescharacteristic
349C2×C4#1455651812yes—
350C8—5842no—
351C22—21no—
352C22—21no—
353C22—22no—
354C22—42no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—81no—
361C22—81no—
362C22—81no—
363C22—81no—
364C22—81no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22—81no—
369C22—81no—
370C22C2×#1455650611yescharacteristic
371C22#1798858311yescharacteristic
372C22C2×#1455651811yescharacteristic
373C4—22no—
374C4—21no—
375C4—21no—
376C4—41no—
377C4—41no—
378C4—41no—
379C4—81no—
380C4—81no—
381C4—81no—
382C4—5844no—
383C2—21no—
384C2—21no—
385C2—42no—
386C2—81no—
387C2—161no—
388C2#2260842211yescharacteristic
389C2#2261157711yescharacteristic
390C2#2261160311yescharacteristic
subgroupG/H
centerC22C2×#14556506
commutatorC22×C4×C73C22×C4
FrattiniC2×C22⋊C4C22×D73
Fitting, prime radicalC73×#2635C2
socleC22×C73C2×#99
cosocleC2×C22⋊C4×C73C23
hypercenter#2635D73
hypocenterC73#11729
LCSG ▷ C22×C4×C73 ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2635

Complements

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

NH ≅ G/Ntype
1. C4×#14556506C2semidirect
2. #22608252C2semidirect
3. #22607955C2semidirect
4. #22607928C2semidirect
12. C2×#121×C73C4semidirect
14. #400×C73C4semidirect
17. #394×C73C4semidirect
22. #381×C73C4semidirect
201. C73#11729semidirect
202. C2×#121Dic73semidirect
206. #400Dic73semidirect
209. #394Dic73semidirect
214. #381Dic73semidirect

Maximal subgroups

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

subgroupindex
C4×#145565062
#226082522
#226079552
#226079282
C73×#26352
#1172973