Finite Group Databasebeta

← #32982208

Subgroups — #32982208

Order 18688.

562 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988868C211yesmaximal, characteristic
2C2×#17988664C211yesmaximal, characteristic
3#22610732C211yesmaximal, characteristic, complemented
4#22609928C211yesmaximal, characteristic
5C2×#17988334C211yesmaximal, characteristic, complemented
6#22607765C211yesmaximal, characteristic, complemented
7C73×#1296C211yesmaximal, characteristic, complemented
8C22×#11924011—21no—
9C2×#14556591C2211yescharacteristic
10C2×C4×D292C2211yescharacteristic
11#17988868—22no—
12#17988868C414yescomplemented
13#17988664—22no—
14#17988664—22no—
15C2×#14556525—21no—
16C2×C4⋊#10596831C2211yescharacteristic
17C2×#14556514—21no—
18C2×#14556514—21no—
19C2×#14556513—21no—
20C2×#14556511—21no—
21C2×#14556503—21no—
22C2×C4×#10596831C2211yescharacteristic
23#17988349—21no—
24#17988334—22no—
25C2×C4⋊C8×C73C2211yescharacteristic
26C2×C4⋊C8×C73C2211yescharacteristic
27#384×C73C2211yescharacteristic
28C22×Dic146—21no—
29C22×D292D411yescharacteristic
30C22×C4×D73—21no—
31C2×#11924011—24no—
32C2×#11924011—42no—
33C2×#11924011—42no—
34C22×#10596831—21no—
35C22×#10596831D411yescharacteristic
36C22×#10596831D411yescharacteristic
37#14556591—21no—
38#14556591—21no—
39#14556591—21no—
40#14556591C2×C412yes—
41C4×D292—21no—
42C4×D292—21no—
43C4×D292C2×C412yes—
44C4×D292D412yes—
45C2×C4⋊Dic73—21no—
46C2×C4⋊Dic73—21no—
47C2×C4⋊Dic73D411yescharacteristic
48C2×#11924005—21no—
49#14556525—24no—
50C4⋊#10596831—21no—
51C4⋊#10596831—21no—
52#14556514—42no—
53#14556514—42no—
54#14556513—24no—
55#14556511—42no—
56#14556503—24no—
57C4×#10596831—21no—
58C4×#10596831—21no—
59C22×C8×C73—21no—
60C22×C8×C73—21no—
61C2×C4⋊C4×C73—21no—
62C2×C4⋊C4×C73—21no—
63C2×C4⋊C4×C73—21no—
64C2×C42×C73C2311yescharacteristic
65C4⋊C8×C73—21no—
66C4⋊C8×C73—21no—
67C4⋊C8×C73C2×C412yes—
68C4⋊C8×C73D412yes—
69C23×D73—21no—
70C2×Dic146—22no—
71C2×Dic146—41no—
72C2×Dic146—41no—
73C2×Dic146—41no—
74C2×Dic146—41no—
75C2×Dic146—41no—
76C2×D292—21no—
77C2×D292—21no—
78C2×D292—21no—
79C2×D292—21no—
80C2×D292—21no—
81C2×D292C22⋊C412yes—
82C22×Dic73—21no—
83C22×Dic73—41no—
84C2×C4×D73—22no—
85C2×C4×D73—22no—
86C2×C4×D73—41no—
87C2×C4×D73—41no—
88C2×C4×D73—41no—
89C2×C4×D73—41no—
90#11924011—48no—
91C2×#10596831—21no—
92C2×#10596831—21no—
93C2×#10596831—21no—
94C2×#10596831—21no—
95C2×#10596831—21no—
96C2×#10596831—41no—
97C2×#10596831—41no—
98C2×#10596831—41no—
99C2×#10596831D812yes—
100C4⋊Dic73—21no—
101C4⋊Dic73—22no—
102C4⋊Dic73—22no—
103C4⋊Dic73—41no—
104C4⋊Dic73—41no—
105C4⋊Dic73C22⋊C412yes—
106#11924005—22no—
107#11924005—22no—
108C22×C4×C73—41no—
109C22×C4×C73C2×D411yescharacteristic
110C22×C4×C73C2×D411yescharacteristic
111C22×C4×C73#4111yescharacteristic
112C2×C8×C73—22no—
113C2×C8×C73—22no—
114C2×C8×C73—41no—
115C2×C8×C73—41no—
116C2×C8×C73—41no—
117C2×C8×C73—41no—
118C4⋊C4×C73—41no—
119C4⋊C4×C73—41no—
120C4⋊C4×C73—41no—
121C4⋊C4×C73—41no—
122C4⋊C4×C73—81no—
123C42×C73C22×C411yescharacteristic
124C42×C73C2×D411yescharacteristic
125C42×C73C2×D411yescharacteristic
126C42×C73#4111yescharacteristic
127C23×C73#12111yescharacteristic
128C22×D73—22no—
129C22×D73—41no—
130C22×D73—41no—
131C22×D73—41no—
132C22×D73—41no—
133C22×D73—41no—
134C22×D73—41no—
135C2×C4×C73—21no—
136C2×C4×C73—21no—
137C2×C4×C73—81no—
138C2×C4×C73—81no—
139C2×C4×C73—81no—
140C2×C4×C73C2×C22⋊C411yescharacteristic
141C2×C4×C73C4×D411yescharacteristic
142C2×C4×C73C4×D411yescharacteristic
143C2×C4×C73#12011yescharacteristic
144C2×C4×C73#12111yescharacteristic
145C2×C4×C73#12311yescharacteristic
146C2×C4×C73C2×D811yescharacteristic
147C2×C4×C73#13511yescharacteristic
148C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
149C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
150C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
151C2×C4×C73#13711yescharacteristic
152C2×C4×C73#13711yescharacteristic
153C2×C4×C73#13711yescharacteristic
154C2×Dic73—22no—
155C2×Dic73—41no—
156C2×Dic73—41no—
157C2×Dic73—42no—
158C2×Dic73—81no—
159C2×Dic73—81no—
160D292—22no—
161D292—22no—
162D292—41no—
163D292—41no—
164C4×D73—42no—
165C4×D73—42no—
166C4×D73—42no—
167C4×D73—42no—
168Dic146—42no—
169Dic146—42no—
170Dic146—81no—
171Dic146—81no—
172C8×C73—42no—
173C8×C73—42no—
174#10596831—22no—
175#10596831—41no—
176#10596831—81no—
177C22×C73#38611yescharacteristic
178C22×C73#45911yescharacteristic
179C22×C73#46311yescharacteristic
180C22×C73#46811yescharacteristic
181C22×C73#47011yescharacteristic
182C22×C73#47111yescharacteristic
183C22×C73#47311yescharacteristic
184C2×D73—42no—
185C2×D73—42no—
186C2×D73—42no—
187C2×D73—81no—
188C2×D73—81no—
189C2×D73—81no—
190C2×D73—81no—
191C4×C73—21no—
192C4×C73—21no—
193C4×C73—161no—
194C4×C73#41711yescharacteristic
195C4×C73#41911yescharacteristic
196C4×C73C4×D811yescharacteristic
197C4×C73#44211yescharacteristic
198C4×C73#44711yescharacteristic
199C4×C73#44911yescharacteristic
200C4×C73#46811yescharacteristic
201C4×C73#47011yescharacteristic
202Dic73—42no—
203Dic73—82no—
204#9227—731nomaximal, Sylow
205C2×C73#168511yescharacteristic
206C2×C73#168911yescharacteristic
207C2×C73#169911yescharacteristic
208C2×C73#170311yescharacteristic
209C2×C73#187911yescharacteristic
210C2×C73#189411yescharacteristic
211C2×C73#190811yescharacteristic
212D73—82no—
213C2×#474—731no—
214C2×#436—731no—
215#1956—731no—
216#1802—731no—
217C2×#332—731no—
218#1296D7311yescharacteristic, complemented
219#1277—731no—
220C73#922711yescharacteristic, complemented, Sylow
221C22×M16—1461no—
222C2×C4⋊Q8—731no—
223C2×C4×D4—731no—
224#474—734no—
225#474—1462no—
226#436—1462no—
227#436—1462no—
228C2×#107—1461no—
229C2×C4⋊C8—731no—
230C2×C4⋊C8C2×D7311yescharacteristic
231C2×C4⋊C8C2×D7311yescharacteristic
232C2×#103—1461no—
233C2×#103—1461no—
234C2×#102—1461no—
235C2×#102—1461no—
236C2×C22⋊C8—1461no—
237C2×C4×C8—731no—
238#384C2×D7311yescharacteristic
239#340—1461no—
240#332—1462no—
241C22×Q8—1461no—
242C22×D4—731no—
243C23×C4—1461no—
244C2×M16—2922no—
245C2×M16—2922no—
246C2×M16—1464no—
247C22×C8—21no—
248C22×C8—21no—
249C22×C8—731no—
250C22×C8—731no—
251C22×C8—1461no—
252C4⋊Q8—732no—
253C4⋊Q8—1461no—
254C4⋊Q8—1461no—
255C4⋊Q8—1461no—
256C4×D4—732no—
257C4×D4—732no—
258C4×D4—1461no—
259C4×D4—1461no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—731no—
264C2×C4⋊C4—1461no—
265C2×C4⋊C4—1461no—
266C2×C22⋊C4—1461no—
267C2×C42C22×D7311yescharacteristic
268#107—1464no—
269C4⋊C8—21no—
270C4⋊C8—21no—
271C4⋊C8—1461no—
272C4⋊C8—1461no—
273C4⋊C8C4×D7312yes—
274C4⋊C8D29212yes—
275#103—2922no—
276#103—2922no—
277#102—2922no—
278#102—1464no—
279C22⋊C8—1464no—
280C4×C8—1461no—
281C4×C8—1461no—
282C24—1461no—
283C2×Q8—2921no—
284C2×Q8—2921no—
285C2×Q8—2921no—
286C2×Q8—2921no—
287C2×Q8—2921no—
288C2×Q8—1462no—
289C2×D4—732no—
290C2×D4—1461no—
291C2×D4—1461no—
292C2×D4—1461no—
293C2×D4—1461no—
294C2×D4—1461no—
295C22×C4—41no—
296C22×C4—2921no—
297C22×C4—2921no—
298C22×C4—2921no—
299C22×C4—2921no—
300C22×C4—2921no—
301C22×C4—1461no—
302C22×C4—1462no—
303C22×C4—1462no—
304C22×C4D4×D7311yescharacteristic
305C22×C4D4×D7311yescharacteristic
306C22×C4#1192402811yescharacteristic
307M16—2928no—
308C2×C8—22no—
309C2×C8—22no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—2921no—
315C2×C8—2921no—
316C2×C8—2921no—
317C2×C8—732no—
318C2×C8—1461no—
319C2×C8—1461no—
320C2×C8—1461no—
321C2×C8—1461no—
322C2×C8—1461no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—81no—
328C4⋊C4—2921no—
329C4⋊C4—2921no—
330C4⋊C4—732no—
331C4⋊C4—1462no—
332C4⋊C4—1462no—
333C4⋊C4—1461no—
334C22⋊C4—1462no—
335C22⋊C4—1462no—
336C42C2×C4×D7311yescharacteristic
337C42D4×D7311yescharacteristic
338C42C2×D29211yescharacteristic
339C42#1192402911yescharacteristic
340C23—2921no—
341C23—2921no—
342C23—2921no—
343C23—2921no—
344C23—2921no—
345C23—2921no—
346C23—1462no—
347C23#1455656411yescharacteristic
348Q8—2922no—
349Q8—2922no—
350Q8—5841no—
351Q8—5841no—
352D4—2921no—
353D4—2921no—
354D4—1462no—
355D4—1462no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—2921no—
362C2×C4—2921no—
363C2×C4—2922no—
364C2×C4—2922no—
365C2×C4—2922no—
366C2×C4—2922no—
367C2×C4—2922no—
368C2×C4—5841no—
369C2×C4—5841no—
370C2×C4—1462no—
371C2×C4C22⋊C4×D7311yescharacteristic
372C2×C4#1455655311yescharacteristic
373C2×C4C4×D29211yescharacteristic
374C2×C4#1455656111yescharacteristic
375C2×C4#1455656611yescharacteristic
376C2×C4#1455657611yescharacteristic
377C2×C4D8×D7311yescharacteristic
378C2×C4#1455661411yescharacteristic
379C2×C4#1455661511yescharacteristic
380C2×C4#1455661811yescharacteristic
381C2×C4#1455661811yescharacteristic
382C2×C4#1455662011yescharacteristic
383C2×C4#1455662011yescharacteristic
384C2×C4#1455662311yescharacteristic
385C8—42no—
386C8—42no—
387C8—2921no—
388C8—5841no—
389C8—1462no—
390C22—2922no—
391C22—2922no—
392C22—2922no—
393C22—5841no—
394C22—5841no—
395C22—5841no—
396C22—5841no—
397C22#1798846511yescharacteristic
398C22#1798878111yescharacteristic
399C22#1798880511yescharacteristic
400C22#1798883211yescharacteristic
401C22#1798884411yescharacteristic
402C22#1798884911yescharacteristic
403C22#1798886111yescharacteristic
404C4—21no—
405C4—21no—
406C4—161no—
407C4—2922no—
408C4—5842no—
409C4#1798858511yescharacteristic
410C4#1798859411yescharacteristic
411C4#1798866911yescharacteristic
412C4#1798869811yescharacteristic
413C4#1798872211yescharacteristic
414C4#1798873511yescharacteristic
415C4#1798883411yescharacteristic
416C4#1798884611yescharacteristic
417C2—5842no—
418C2#2260939511yescharacteristic
419C2#2260941911yescharacteristic
420C2#2260948111yescharacteristic
421C2#2260950911yescharacteristic
422C2#2261028011yescharacteristic
423C2#2261036311yescharacteristic
424C2#2261044711yescharacteristic
subgroupG/H
centerC23#14556564
commutatorC42×C73C22×C4
FrattiniC2×C42C22×D73
FittingC73×#1296C2
socleC23×C73#121
prime radicalC22×D292D4
cosocleC2×C42×C73C23
hypercenter#1296D73
hypocenterC73#9227
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1296

Complements

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

NH ≅ G/Ntype
3. #22610732C2semidirect
5. C2×#17988334C2semidirect
6. #22607765C2semidirect
7. C73×#1296C2semidirect
12. #17988868C4semidirect
218. #1296D73semidirect
220. C73#9227semidirect

Maximal subgroups

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

subgroupindex
C2×#179888682
C2×#179886642
#226107322
#226099282
C2×#179883342
#226077652
C73×#12962
#922773