Finite Group Databasebeta

← #32977682

Subgroups — #32977682

Order 18688.

780 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22610017C212yesmaximal, complemented
2#22609919C211yesmaximal, characteristic
3#22608503C212yesmaximal, complemented
4#22608249C211yesmaximal, characteristic
5C2×#552×C73C211yesmaximal, characteristic
6C2×#14556506—21no—
7C2×C22⋊#10596831C2212yes—
8#17988447C2212yes—
9#17988354—21no—
10#17988349—24no—
11#17988324—24no—
12#17988313—21no—
13#17988311—21no—
14#552×C73—24no—
15#552×C73—24no—
16C2×#126×C73—21no—
17C2×#125×C73—21no—
18C2×#125×C73C411yescharacteristic, complemented
19C2×#123×C73—21no—
20C2×C22⋊Q8×C73C411yescharacteristic, complemented
21C2×C22⋊Q8×C73C2211yescharacteristic
22C2×C4×Q8×C73C411yescharacteristic, complemented
23C2×C4×D4×C73C2211yescharacteristic
24C2×#117×C73—21no—
25C2×#117×C73C411yescharacteristic, complemented
26C22×C4⋊C4×C73C2211yescharacteristic
27C22×#10596831—22no—
28C2×C4⋊Dic73—22no—
29C2×C22⋊Dic73—22no—
30#14556506—42no—
31C22⋊#10596831—22no—
32C22⋊#10596831—22no—
33#14556498—22no—
34C22×Q8×C73C2×C411yescharacteristic
35C22×D4×C73D411yescharacteristic
36C23×C4×C73—21no—
37C23×C4×C73C2×C411yescharacteristic
38C23×C4×C73C2311yescharacteristic
39#126×C73—28no—
40#125×C73—28no—
41#125×C73—22no—
42#125×C73—22no—
43#123×C73—28no—
44C22⋊Q8×C73—22no—
45C22⋊Q8×C73—22no—
46C22⋊Q8×C73—22no—
47C22⋊Q8×C73D414yes—
48C4×Q8×C73—22no—
49C4×Q8×C73—22no—
50C4×D4×C73—22no—
51C4×D4×C73—22no—
52#117×C73—28no—
53#117×C73—22no—
54#117×C73—22no—
55C2×C4⋊C4×C73—21no—
56C2×C4⋊C4×C73—21no—
57C2×C4⋊C4×C73—21no—
58C2×C4⋊C4×C73—21no—
59C2×C4⋊C4×C73—21no—
60C2×C4⋊C4×C73—22no—
61C2×C4⋊C4×C73—21no—
62C2×C4⋊C4×C73C2×C411yescharacteristic
63C2×C4⋊C4×C73C2×C411yescharacteristic
64C2×C4⋊C4×C73C2×C411yescharacteristic
65C2×C4⋊C4×C73D411yescharacteristic
66C2×C4⋊C4×C73D414yes—
67C2×C22⋊C4×C73—21no—
68C2×C22⋊C4×C73—21no—
69C2×C22⋊C4×C73—21no—
70C2×C22⋊C4×C73D411yescharacteristic
71C2×C22⋊C4×C73D411yescharacteristic
72C2×C42×C73—21no—
73C2×C42×C73—21no—
74C2×C42×C73C2×C411yescharacteristic
75C22×Dic73—42no—
76C2×#10596831—44no—
77C2×#10596831—42no—
78C4⋊Dic73—44no—
79C22⋊Dic73—42no—
80C22⋊Dic73—42no—
81C24×C73C22⋊C411yescharacteristic
82C24×C73C2×D411yescharacteristic
83C2×Q8×C73—21no—
84C2×Q8×C73—42no—
85C2×Q8×C73C22⋊C412yes—
86C2×D4×C73—22no—
87C2×D4×C73—22no—
88C2×D4×C73—21no—
89C2×D4×C73—21no—
90C22×C4×C73—21no—
91C22×C4×C73—21no—
92C22×C4×C73—21no—
93C22×C4×C73—21no—
94C22×C4×C73—24no—
95C22×C4×C73—21no—
96C22×C4×C73—21no—
97C22×C4×C73—42no—
98C22×C4×C73—42no—
99C22×C4×C73—42no—
100C22×C4×C73—41no—
101C22×C4×C73—41no—
102C22×C4×C73C22⋊C411yescharacteristic
103C22×C4×C73C22⋊C411yescharacteristic
104C22×C4×C73C22⋊C411yescharacteristic
105C22×C4×C73C22×C411yescharacteristic
106C22×C4×C73C2×D412yes—
107C22×C4×C73C2×D412yes—
108C22×C4×C73C2×D411yescharacteristic
109C4⋊C4×C73—24no—
110C4⋊C4×C73—24no—
111C4⋊C4×C73—24no—
112C4⋊C4×C73—24no—
113C4⋊C4×C73—21no—
114C4⋊C4×C73—21no—
115C4⋊C4×C73—21no—
116C4⋊C4×C73—24no—
117C4⋊C4×C73—24no—
118C4⋊C4×C73—22no—
119C4⋊C4×C73C22⋊C412yes—
120C4⋊C4×C73C22⋊C412yes—
121C4⋊C4×C73C22⋊C412yes—
122C22⋊C4×C73—24no—
123C22⋊C4×C73—24no—
124C22⋊C4×C73—24no—
125C22⋊C4×C73—22no—
126C22⋊C4×C73—22no—
127C42×C73—24no—
128C42×C73—24no—
129C42×C73—21no—
130C42×C73—21no—
131C23×C73—21no—
132C23×C73—21no—
133C23×C73—21no—
134C23×C73—41no—
135C23×C73—41no—
136C23×C73—41no—
137C23×C73—41no—
138C23×C73C2×C22⋊C411yescharacteristic
139C23×C73#12012yes—
140C23×C73#13512yes—
141C23×C73AGL(1,ℤ/8)11yescharacteristic
142C23×C73#13711yescharacteristic
143Q8×C73—42no—
144Q8×C73—42no—
145D4×C73—42no—
146D4×C73—42no—
147C2×C4×C73—22no—
148C2×C4×C73—24no—
149C2×C4×C73—21no—
150C2×C4×C73—21no—
151C2×C4×C73—22no—
152C2×C4×C73—22no—
153C2×C4×C73—22no—
154C2×C4×C73—21no—
155C2×C4×C73—22no—
156C2×C4×C73—21no—
157C2×C4×C73—21no—
158C2×C4×C73—41no—
159C2×C4×C73—42no—
160C2×C4×C73—42no—
161C2×C4×C73—42no—
162C2×C4×C73—42no—
163C2×C4×C73—41no—
164C2×C4×C73—41no—
165C2×C4×C73—41no—
166C2×C4×C73—41no—
167C2×C4×C73—44no—
168C2×C4×C73—44no—
169C2×C4×C73—42no—
170C2×C4×C73—41no—
171C2×C4×C73—41no—
172C2×C4×C73C2×C22⋊C411yescharacteristic
173C2×C4×C73C2×C22⋊C411yescharacteristic
174C2×C4×C73#12012yes—
175C2×Dic73—82no—
176C2×Dic73—82no—
177C2×Dic73—82no—
178#10596831—82no—
179C22×C73—22no—
180C22×C73—21no—
181C22×C73—22no—
182C22×C73—21no—
183C22×C73—42no—
184C22×C73—41no—
185C22×C73—41no—
186C22×C73—41no—
187C22×C73—41no—
188C22×C73—41no—
189C22×C73—81no—
190C22×C73—81no—
191C22×C73#37911yescharacteristic
192C22×C73#41611yescharacteristic
193C22×C73#41711yescharacteristic
194C22×C73#44912yes—
195C22×C73#45212yes—
196C4×C73—22no—
197C4×C73—21no—
198C4×C73—41no—
199C4×C73—42no—
200C4×C73—41no—
201C4×C73—42no—
202C4×C73—41no—
203C4×C73—42no—
204C4×C73—42no—
205C4×C73#41012yes—
206Dic73—162no—
207#8382—731nomaximal, Sylow
208C2×C73—22no—
209C2×C73—21no—
210C2×C73—21no—
211C2×C73—22no—
212C2×C73—81no—
213C2×C73#178211yescharacteristic
214C2×C73#178811yescharacteristic
215C2×C73#179611yescharacteristic
216C2×#552D7311yescharacteristic
217#1823—732no—
218#1799—731no—
219#1520—732no—
220#1459—731no—
221C73#838211yescharacteristic, complemented, Sylow
222#552—24no—
223#552—24no—
224C2×#126—21no—
225C2×#125—21no—
226C2×#125Dic7311yescharacteristic, complemented
227C2×#123—21no—
228C2×C22⋊Q8Dic7311yescharacteristic, complemented
229C2×C22⋊Q8C2×D7311yescharacteristic
230C2×C4×Q8Dic7311yescharacteristic, complemented
231C2×C4×D4C2×D7311yescharacteristic
232C2×#117—21no—
233C2×#117Dic7311yescharacteristic, complemented
234C22×C4⋊C4C2×D7311yescharacteristic
235C2×#99—1461no—
236C2×C22⋊C8—732no—
237#380—732no—
238#342—1461no—
239#340—1464no—
240#328—1464no—
241#324—1461no—
242#323—1461no—
243C22×Q8C2×Dic7311yescharacteristic
244C22×D4C22⋊D7311yescharacteristic
245C23×C4—21no—
246C23×C4C2×Dic7311yescharacteristic
247C23×C4C22×D7311yescharacteristic
248C22×C8—1462no—
249#126—28no—
250#125—28no—
251#125—22no—
252#125—22no—
253#123—28no—
254C22⋊Q8—22no—
255C22⋊Q8—22no—
256C22⋊Q8—22no—
257C22⋊Q8C22⋊D7314yes—
258C4×Q8—22no—
259C4×Q8—22no—
260C4×D4—22no—
261C4×D4—22no—
262#117—28no—
263#117—22no—
264#117—22no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—1462no—
273C2×C4⋊C4C2×Dic7311yescharacteristic
274C2×C4⋊C4C2×Dic7311yescharacteristic
275C2×C4⋊C4C2×Dic7311yescharacteristic
276C2×C4⋊C4C22⋊D7311yescharacteristic
277C2×C4⋊C4C22⋊D7314yes—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—1462no—
282C2×C22⋊C4C22⋊D7311yescharacteristic
283C2×C22⋊C4C22⋊D7311yescharacteristic
284C2×C42—21no—
285C2×C42—21no—
286C2×C42C2×Dic7311yescharacteristic
287#99—2922no—
288C22⋊C8—1462no—
289C22⋊C8—1462no—
290#95—1462no—
291C24C22⋊Dic7311yescharacteristic
292C24C2×C22⋊D7311yescharacteristic
293C2×Q8—21no—
294C2×Q8—42no—
295C2×Q8C22⋊Dic7312yes—
296C2×D4—22no—
297C2×D4—22no—
298C2×D4—21no—
299C2×D4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—24no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—2922no—
313C22×C4C22⋊Dic7311yescharacteristic
314C22×C4C22⋊Dic7311yescharacteristic
315C22×C4C22⋊Dic7311yescharacteristic
316C22×C4C22×Dic7311yescharacteristic
317C22×C4C2×C22⋊D7312yes—
318C22×C4C2×C22⋊D7312yes—
319C22×C4C2×C22⋊D7311yescharacteristic
320C2×C8—2924no—
321C2×C8—2922no—
322C4⋊C4—24no—
323C4⋊C4—24no—
324C4⋊C4—24no—
325C4⋊C4—24no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—24no—
330C4⋊C4—24no—
331C4⋊C4—22no—
332C4⋊C4—2924no—
333C4⋊C4C22⋊Dic7312yes—
334C4⋊C4C22⋊Dic7312yes—
335C4⋊C4C22⋊Dic7312yes—
336C22⋊C4—24no—
337C22⋊C4—24no—
338C22⋊C4—24no—
339C22⋊C4—22no—
340C22⋊C4—22no—
341C22⋊C4—2922no—
342C22⋊C4—2922no—
343C42—24no—
344C42—24no—
345C42—21no—
346C42—21no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23C2×C22⋊Dic7311yescharacteristic
355C23#1455656012yes—
356C23#1455661112yes—
357C23#1455661611yescharacteristic
358C23#1455662111yescharacteristic
359Q8—42no—
360Q8—42no—
361D4—42no—
362D4—42no—
363C2×C4—24no—
364C2×C4—22no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—21no—
371C2×C4—22no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—41no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—42no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—5842no—
389C2×C4—5842no—
390C2×C4—5842no—
391C2×C4C2×C22⋊Dic7311yescharacteristic
392C2×C4C2×C22⋊Dic7311yescharacteristic
393C2×C4#1455656012yes—
394C8—5842no—
395C22—21no—
396C22—22no—
397C22—21no—
398C22—22no—
399C22—42no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—81no—
406C22—81no—
407C22#1798844311yescharacteristic
408C22#1798858311yescharacteristic
409C22#1798858811yescharacteristic
410C22#1798873212yes—
411C22#1798874912yes—
412C4—22no—
413C4—21no—
414C4—41no—
415C4—42no—
416C4—41no—
417C4—42no—
418C4—42no—
419C4—42no—
420C4—41no—
421C4—11682no—
422C4#1798856112yes—
423C2—22no—
424C2—21no—
425C2—21no—
426C2—22no—
427C2—81no—
428C2#2260984911yescharacteristic
429C2#2260987111yescharacteristic
430C2#2260990611yescharacteristic
subgroupG/H
centerC22#17988443
commutatorC22×C4×C73C22×C4
FrattiniC23×C4C22×D73
FittingC2×#552×C73C2
socleC22×C73#379
prime radicalC22×D4×C73D4
cosocleC23×C4×C73C23
hypercenterC2×#552D73
hypocenterC73#8382
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#552

Complements

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

NH ≅ G/Ntype
1. #22610017C2semidirect
3. #22608503C2semidirect
18. C2×#125×C73C4semidirect
20. C2×C22⋊Q8×C73C4semidirect
22. C2×C4×Q8×C73C4semidirect
25. C2×#117×C73C4semidirect
221. C73#8382semidirect
226. C2×#125Dic73semidirect
228. C2×C22⋊Q8Dic73semidirect
230. C2×C4×Q8Dic73semidirect
233. C2×#117Dic73semidirect

Maximal subgroups

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

subgroupindex
#226100172
#226099192
#226085032
#226082492
C2×#552×C732
#838273