Finite Group Databasebeta

← #32996649

Subgroups — #32996649

Order 18688.

486 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 87 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#22610008C211yesmaximal, characteristic, complemented
2#22608265C211yesmaximal, characteristic, complemented
3#22607931C212yesmaximal, complemented
4#22607928C212yesmaximal
5C73×#2392C211yesmaximal, characteristic
6#17988644—21no—
7#17988450—21no—
8#17988354—21no—
9#17988354C2212yes—
10#17988313—21no—
11#17988311C2212yes—
12C2×C4×D4×C73—21no—
13C2×C4×D4×C73C411yescharacteristic, complemented
14C2×#117×C73C2211yescharacteristic
15#398×C73C411yescharacteristic, complemented
16#397×C73C2211yescharacteristic
17#396×C73—21no—
18#394×C73—21no—
19#394×C73C411yescharacteristic, complemented
20#392×C73C2211yescharacteristic
21C4×C4⋊C4×C73C411yescharacteristic, complemented
22C4×C22⋊C4×C73—21no—
23C2×C22⋊Dic73—22no—
24C4⋊#10596831—41no—
25C22⋊#10596831—22no—
26#14556502—41no—
27#14556498—41no—
28#14556498—41no—
29C22×D4×C73—21no—
30C22×D4×C73C2×C411yescharacteristic
31C23×C4×C73—21no—
32C23×C4×C73D411yescharacteristic
33C4×D4×C73—41no—
34C4×D4×C73—41no—
35C4×D4×C73—44no—
36#117×C73—41no—
37#117×C73—41no—
38C2×C4⋊C4×C73—21no—
39C2×C4⋊C4×C73—21no—
40C2×C4⋊C4×C73C2×C411yescharacteristic
41C2×C4⋊C4×C73C2×C411yescharacteristic
42C2×C4⋊C4×C73D411yescharacteristic
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×C73D411yescharacteristic
49C2×C22⋊C4×C73C2311yescharacteristic
50C2×C42×C73—21no—
51C2×C42×C73C2×C411yescharacteristic
52C2×C42×C73D411yescharacteristic
53#95×C73—21no—
54#95×C73—21no—
55#95×C73C2×C411yescharacteristic
56#95×C73C2×C411yescharacteristic
57C22×Dic73—42no—
58C2×#10596831—42no—
59C22⋊Dic73—42no—
60C22⋊Dic73—42no—
61C24×C73—21no—
62C24×C73C2×D411yescharacteristic
63C2×D4×C73—41no—
64C2×D4×C73—42no—
65C2×D4×C73—42no—
66C2×D4×C73—42no—
67C2×D4×C73—42no—
68C22×C4×C73—21no—
69C22×C4×C73—22no—
70C22×C4×C73—21no—
71C22×C4×C73—21no—
72C22×C4×C73—21no—
73C22×C4×C73—21no—
74C22×C4×C73—41no—
75C22×C4×C73—42no—
76C22×C4×C73—42no—
77C22×C4×C73—41no—
78C22×C4×C73—41no—
79C22×C4×C73C22⋊C411yescharacteristic
80C22×C4×C73C22⋊C411yescharacteristic
81C22×C4×C73C22⋊C411yescharacteristic
82C22×C4×C73C22⋊C411yescharacteristic
83C22×C4×C73C22×C411yescharacteristic
84C22×C4×C73C2×D411yescharacteristic
85C4⋊C4×C73—41no—
86C4⋊C4×C73—41no—
87C4⋊C4×C73—41no—
88C4⋊C4×C73—42no—
89C4⋊C4×C73—42no—
90C22⋊C4×C73—41no—
91C22⋊C4×C73—41no—
92C22⋊C4×C73—42no—
93C22⋊C4×C73—42no—
94C22⋊C4×C73—42no—
95C22⋊C4×C73—42no—
96C22⋊C4×C73—42no—
97C42×C73—21no—
98C42×C73—21no—
99C42×C73—21no—
100C42×C73—21no—
101C42×C73—42no—
102C23×C73—21no—
103C23×C73—21no—
104C23×C73—22no—
105C23×C73—41no—
106C23×C73—41no—
107C23×C73—41no—
108C23×C73—41no—
109C23×C73—41no—
110C23×C73—41no—
111C23×C73—41no—
112C23×C73—41no—
113C23×C73C2×C22⋊C411yescharacteristic
114C23×C73AGL(1,ℤ/8)12yes—
115D4×C73—42no—
116D4×C73—48no—
117D4×C73—42no—
118C2×C4×C73—21no—
119C2×C4×C73—21no—
120C2×C4×C73—22no—
121C2×C4×C73—21no—
122C2×C4×C73—21no—
123C2×C4×C73—21no—
124C2×C4×C73—21no—
125C2×C4×C73—21no—
126C2×C4×C73—21no—
127C2×C4×C73—21no—
128C2×C4×C73—42no—
129C2×C4×C73—42no—
130C2×C4×C73—41no—
131C2×C4×C73—41no—
132C2×C4×C73—41no—
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—82no—
143C2×C4×C73—82no—
144C2×C4×C73#9911yescharacteristic
145C2×C4×C73#9911yescharacteristic
146C2×Dic73—82no—
147C2×Dic73—82no—
148C2×Dic73—82no—
149#10596831—82no—
150C22×C73—21no—
151C22×C73—21no—
152C22×C73—22no—
153C22×C73—42no—
154C22×C73—42no—
155C22×C73—42no—
156C22×C73—41no—
157C22×C73—41no—
158C22×C73—42no—
159C22×C73—81no—
160C22×C73—81no—
161C22×C73—81no—
162C22×C73—81no—
163C22×C73—81no—
164C22×C73C2×#9911yescharacteristic
165C22×C73#41811yescharacteristic
166C22×C73#42111yescharacteristic
167C4×C73—21no—
168C4×C73—21no—
169C4×C73—42no—
170C4×C73—41no—
171C4×C73—41no—
172C4×C73—41no—
173C4×C73—41no—
174C4×C73—41no—
175C4×C73—81no—
176C4×C73—81no—
177Dic73—162no—
178#11798—731nomaximal, Sylow
179C2×C73—21no—
180C2×C73—21no—
181C2×C73—42no—
182C2×C73—82no—
183C2×C73#151011yescharacteristic
184C2×C73#212011yescharacteristic
185C2×C73#212611yescharacteristic
186#2392D7311yescharacteristic
187#1820—731no—
188#1464—731no—
189#1344—732no—
190#1343—732no—
191C73#1179811yescharacteristic, complemented, Sylow
192C2×C4×D4—21no—
193C2×C4×D4Dic7311yescharacteristic, complemented
194C2×#117C2×D7311yescharacteristic
195#432—1461no—
196#398Dic7311yescharacteristic, complemented
197#397C2×D7311yescharacteristic
198#396—21no—
199#394—21no—
200#394Dic7311yescharacteristic, complemented
201#392C2×D7311yescharacteristic
202#381—1461no—
203C4×C4⋊C4Dic7311yescharacteristic, complemented
204C4×C22⋊C4—21no—
205#342—732no—
206#342—1461no—
207#324—1461no—
208#323—732no—
209C22×D4—21no—
210C22×D4C2×Dic7311yescharacteristic
211C23×C4—21no—
212C23×C4C22⋊D7311yescharacteristic
213C4×D4—41no—
214C4×D4—41no—
215C4×D4—44no—
216#117—41no—
217#117—41no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4C2×Dic7311yescharacteristic
221C2×C4⋊C4C2×Dic7311yescharacteristic
222C2×C4⋊C4C22⋊D7311yescharacteristic
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—1462no—
229C2×C22⋊C4C22⋊D7311yescharacteristic
230C2×C22⋊C4C22×D7311yescharacteristic
231C2×C42—21no—
232C2×C42C2×Dic7311yescharacteristic
233C2×C42C22⋊D7311yescharacteristic
234C4⋊C8—2921no—
235C22⋊C8—1462no—
236#97—2921no—
237#95—21no—
238#95—21no—
239#95—2921no—
240#95—2921no—
241#95C2×Dic7311yescharacteristic
242#95C2×Dic7311yescharacteristic
243C24—21no—
244C24C2×C22⋊D7311yescharacteristic
245C2×D4—41no—
246C2×D4—42no—
247C2×D4—42no—
248C2×D4—42no—
249C2×D4—42no—
250C22×C4—21no—
251C22×C4—22no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—41no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—2922no—
262C22×C4C22⋊Dic7311yescharacteristic
263C22×C4C22⋊Dic7311yescharacteristic
264C22×C4C22⋊Dic7311yescharacteristic
265C22×C4C22⋊Dic7311yescharacteristic
266C22×C4C22×Dic7311yescharacteristic
267C22×C4C2×C22⋊D7311yescharacteristic
268C2×C8—2922no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C22⋊C4—41no—
275C22⋊C4—41no—
276C22⋊C4—42no—
277C22⋊C4—42no—
278C22⋊C4—42no—
279C22⋊C4—42no—
280C22⋊C4—42no—
281C22⋊C4—2922no—
282C22⋊C4—2922no—
283C42—21no—
284C42—21no—
285C42—21no—
286C42—21no—
287C42—42no—
288C23—21no—
289C23—21no—
290C23—22no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23C2×C22⋊Dic7311yescharacteristic
300C23#1455661712yes—
301D4—42no—
302D4—48no—
303D4—42no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—22no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4—5842no—
331C2×C4—5842no—
332C2×C4—5842no—
333C2×C4#1455650611yescharacteristic
334C2×C4#1455650611yescharacteristic
335C8—5842no—
336C22—21no—
337C22—21no—
338C22—22no—
339C22—42no—
340C22—42no—
341C22—42no—
342C22—41no—
343C22—41no—
344C22—42no—
345C22—81no—
346C22—81no—
347C22—81no—
348C22—81no—
349C22—81no—
350C22C2×#1455650611yescharacteristic
351C22#1798859211yescharacteristic
352C22#1798860411yescharacteristic
353C4—21no—
354C4—21no—
355C4—42no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—41no—
361C4—81no—
362C4—81no—
363C4—11682no—
364C2—21no—
365C2—21no—
366C2—42no—
367C2—82no—
368C2#2260846011yescharacteristic
369C2#2261157211yescharacteristic
370C2#2261159611yescharacteristic
subgroupG/H
centerC22C2×#14556506
commutatorC22×C4×C73C22×C4
FrattiniC2×C22⋊C4C22×D73
FittingC73×#2392C2
socleC22×C73C2×#99
prime radical#392×C73C22
cosocleC2×C22⋊C4×C73C23
hypercenter#2392D73
hypocenterC73#11798
LCSG ▷ C22×C4×C73 ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2392

Complements

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

NH ≅ G/Ntype
1. #22610008C2semidirect
2. #22608265C2semidirect
3. #22607931C2semidirect
13. C2×C4×D4×C73C4semidirect
15. #398×C73C4semidirect
19. #394×C73C4semidirect
21. C4×C4⋊C4×C73C4semidirect
191. C73#11798semidirect
193. C2×C4×D4Dic73semidirect
196. #398Dic73semidirect
200. #394Dic73semidirect
203. C4×C4⋊C4Dic73semidirect

Maximal subgroups

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

subgroupindex
#226100082
#226082652
#226079312
#226079282
C73×#23922
#1179873