Finite Group Databasebeta

← #32980741

Subgroups — #32980741

Order 18688.

592 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988874C211yesmaximal, characteristic
2#22612878C211yesmaximal, characteristic
3#22610734C211yesmaximal, characteristic, complemented
4C2×#17988314C211yesmaximal, characteristic
5#22607815C211yesmaximal, characteristic, complemented
6#22607772C211yesmaximal, characteristic, complemented
7C73×#1987C211yesmaximal, characteristic, complemented
8C2×C4×D292C2211yescharacteristic
9C2×#14556549—21no—
10#17988874—22no—
11#17988874C414yescomplemented
12C2×#14556522—21no—
13C2×C4⋊#10596831C2211yescharacteristic
14C2×#14556515—22no—
15C2×#14556511—21no—
16C2×C4×#10596831C2211yescharacteristic
17#17988489—21no—
18#17988479—22no—
19#17988457C2211yescharacteristic
20#17988444—21no—
21C4×C4⋊Dic73—21no—
22#17988314—22no—
23C2×C4⋊Q8×C73C2211yescharacteristic
24C2×C4⋊C8×C73C2211yescharacteristic
25C4×C4⋊C4×C73C2211yescharacteristic
26#340×C73—21no—
27#340×C73—21no—
28C22×D292D411yescharacteristic
29C22×C4×D73—21no—
30C22×#10596831—21no—
31C22×#10596831—21no—
32C4×D292—21no—
33C4×D292—21no—
34C4×D292C2×C412yes—
35C4×D292D412yes—
36#14556549—42no—
37#14556549—42no—
38C2×C4⋊Dic73—21no—
39C2×C4⋊Dic73—21no—
40C2×C4⋊Dic73D411yescharacteristic
41C2×C4⋊Dic73D412yes—
42C2×#11924005—22no—
43C2×#11924005—22no—
44C2×#11924005—21no—
45C2×C4×Dic73—22no—
46#14556522—24no—
47C4⋊#10596831—21no—
48C4⋊#10596831C2×C412yes—
49#14556515—44no—
50#14556511—24no—
51C4×#10596831—21no—
52C4×#10596831—21no—
53#14556498—22no—
54C22×Q8×C73—21no—
55C22×C8×C73—21no—
56C4⋊Q8×C73—21no—
57C4⋊Q8×C73—21no—
58C4⋊Q8×C73—21no—
59C4⋊Q8×C73C2×C412yes—
60C2×C4⋊C4×C73—21no—
61C2×C4⋊C4×C73—21no—
62C2×C4⋊C4×C73D411yescharacteristic
63C2×C4⋊C4×C73D411yescharacteristic
64C2×C42×C73—21no—
65C2×C42×C73C2311yescharacteristic
66C4⋊C8×C73—21no—
67C4⋊C8×C73—21no—
68#95×C73—21no—
69C23×D73—21no—
70C2×D292—21no—
71C2×D292—21no—
72C2×D292—21no—
73C2×D292—21no—
74C2×D292—21no—
75C2×D292C22⋊C412yes—
76C22×Dic73—21no—
77C22×Dic73—22no—
78C2×C4×D73—22no—
79C2×C4×D73—22no—
80C2×C4×D73—41no—
81C2×C4×D73—41no—
82C2×C4×D73—41no—
83C2×C4×D73—41no—
84C2×#10596831—22no—
85C2×#10596831—22no—
86C2×#10596831—41no—
87C2×#10596831—41no—
88C2×#10596831—41no—
89C2×#10596831—41no—
90C4⋊Dic73—21no—
91C4⋊Dic73—41no—
92C4⋊Dic73—41no—
93C4⋊Dic73—41no—
94C4⋊Dic73—41no—
95C4⋊Dic73—42no—
96C4⋊Dic73C22⋊C412yes—
97#11924005—22no—
98#11924005—22no—
99#11924005—44no—
100#11924005—82no—
101C4×Dic73—44no—
102C2×Q8×C73—22no—
103C2×Q8×C73—42no—
104C2×Q8×C73—42no—
105C2×Q8×C73—41no—
106C22×C4×C73—21no—
107C22×C4×C73—21no—
108C22×C4×C73—41no—
109C22×C4×C73C2×D411yescharacteristic
110C22×C4×C73C2×D411yescharacteristic
111C22×C4×C73C2×D411yescharacteristic
112C2×C8×C73—41no—
113C2×C8×C73—41no—
114C2×C8×C73—41no—
115C4⋊C4×C73—21no—
116C4⋊C4×C73—22no—
117C4⋊C4×C73—22no—
118C4⋊C4×C73—21no—
119C4⋊C4×C73—21no—
120C4⋊C4×C73—41no—
121C4⋊C4×C73—41no—
122C4⋊C4×C73SD1612yes—
123C42×C73—41no—
124C42×C73—41no—
125C42×C73C22×C411yescharacteristic
126C42×C73C2×D411yescharacteristic
127C42×C73#4111yescharacteristic
128C42×C73#4111yescharacteristic
129C23×C73#12011yescharacteristic
130C22×D73—22no—
131C22×D73—41no—
132C22×D73—41no—
133C22×D73—41no—
134C22×D73—41no—
135C22×D73—41no—
136C22×D73—41no—
137Q8×C73—44no—
138Q8×C73—82no—
139C2×C4×C73—21no—
140C2×C4×C73—21no—
141C2×C4×C73—22no—
142C2×C4×C73—41no—
143C2×C4×C73—41no—
144C2×C4×C73—42no—
145C2×C4×C73—41no—
146C2×C4×C73—41no—
147C2×C4×C73—41no—
148C2×C4×C73—81no—
149C2×C4×C73—81no—
150C2×C4×C73C2×C22⋊C411yescharacteristic
151C2×C4×C73C4×D411yescharacteristic
152C2×C4×C73C4×D411yescharacteristic
153C2×C4×C73#12111yescharacteristic
154C2×C4×C73#12111yescharacteristic
155C2×C4×C73#12311yescharacteristic
156C2×C4×C73C2×SD1611yescharacteristic
157C2×C4×C73#13511yescharacteristic
158C2×C4×C73#13511yescharacteristic
159C2×C4×C73#13511yescharacteristic
160C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
161C2×C4×C73#13711yescharacteristic
162C2×C4×C73#13711yescharacteristic
163C2×C4×C73#13711yescharacteristic
164C2×Dic73—22no—
165C2×Dic73—41no—
166C2×Dic73—41no—
167C2×Dic73—42no—
168C2×Dic73—42no—
169C2×Dic73—42no—
170D292—22no—
171D292—22no—
172D292—41no—
173D292—41no—
174C4×D73—42no—
175C4×D73—42no—
176C4×D73—42no—
177C4×D73—42no—
178C8×C73—81no—
179#10596831—42no—
180#10596831—42no—
181C22×C73#38611yescharacteristic
182C22×C73#45011yescharacteristic
183C22×C73#45211yescharacteristic
184C22×C73#45211yescharacteristic
185C22×C73#45211yescharacteristic
186C22×C73#45311yescharacteristic
187C22×C73#45411yescharacteristic
188C2×D73—42no—
189C2×D73—42no—
190C2×D73—42no—
191C2×D73—81no—
192C2×D73—81no—
193C2×D73—81no—
194C2×D73—81no—
195C4×C73—21no—
196C4×C73—21no—
197C4×C73—42no—
198C4×C73—82no—
199C4×C73—81no—
200C4×C73#41611yescharacteristic
201C4×C73#41911yescharacteristic
202C4×C73C4×SD1611yescharacteristic
203C4×C73#44011yescharacteristic
204C4×C73#46111yescharacteristic
205C4×C73#46311yescharacteristic
206C4×C73#48511yescharacteristic
207C4×C73#48511yescharacteristic
208Dic73—42no—
209Dic73—82no—
210#8976—731nomaximal, Sylow
211C2×C73#163111yescharacteristic
212C2×C73#163411yescharacteristic
213C2×C73#163511yescharacteristic
214C2×C73#163811yescharacteristic
215C2×C73#187311yescharacteristic
216C2×C73#187711yescharacteristic
217C2×C73#188311yescharacteristic
218D73—82no—
219C2×#476—731no—
220#2354—731no—
221#1987D7311yescharacteristic, complemented
222#1956—731no—
223C2×D4⋊C8—731no—
224#1296—731no—
225#1279—731no—
226C73#897611yescharacteristic, complemented, Sylow
227C2×C4⋊Q8C2×D7311yescharacteristic
228C2×C4×D4—731no—
229C2×#117—1461no—
230#476—734no—
231#476—1462no—
232C2×#106—1461no—
233C2×C4⋊C8—731no—
234C2×C4⋊C8C2×D7311yescharacteristic
235C2×#103—1462no—
236C2×#102—1461no—
237C2×C4×C8—731no—
238#390—1461no—
239#388—1462no—
240#384—731no—
241#380—1461no—
242C4×C4⋊C4—1461no—
243C4×C4⋊C4C2×D7311yescharacteristic
244#340—21no—
245#340—21no—
246D4⋊C8—1462no—
247C22×Q8—21no—
248C22×D4—731no—
249C23×C4—1461no—
250C22×C8—21no—
251C22×C8—1461no—
252C22×C8—1461no—
253C4⋊Q8—21no—
254C4⋊Q8—21no—
255C4⋊Q8—21no—
256C4⋊Q8C4×D7312yes—
257C4×D4—732no—
258C4×D4—732no—
259C4×D4—1461no—
260C4×D4—1461no—
261#117—2922no—
262#117—2922no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—732no—
266C2×C4⋊C4—731no—
267C2×C4⋊C4—1461no—
268C2×C4⋊C4—1461no—
269C2×C4⋊C4C22⋊D7311yescharacteristic
270C2×C4⋊C4C22⋊D7311yescharacteristic
271C2×C22⋊C4—1462no—
272C2×C22⋊C4—1462no—
273C2×C22⋊C4—1461no—
274C2×C42—21no—
275C2×C42—1462no—
276C2×C42C22×D7311yescharacteristic
277#106—1464no—
278C4⋊C8—21no—
279C4⋊C8—21no—
280C4⋊C8—732no—
281C4⋊C8—1461no—
282#103—2924no—
283#102—1464no—
284C4×C8—1461no—
285C4×C8—1461no—
286#95—21no—
287#95—1462no—
288C24—1461no—
289C2×Q8—22no—
290C2×Q8—42no—
291C2×Q8—42no—
292C2×Q8—41no—
293C2×D4—732no—
294C2×D4—1461no—
295C2×D4—1461no—
296C2×D4—1461no—
297C2×D4—1461no—
298C2×D4—1461no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—41no—
302C22×C4—2921no—
303C22×C4—2921no—
304C22×C4—2921no—
305C22×C4—2921no—
306C22×C4—1462no—
307C22×C4—1461no—
308C22×C4—1462no—
309C22×C4—1462no—
310C22×C4D4×D7311yescharacteristic
311C22×C4D4×D7311yescharacteristic
312C22×C4C2×C22⋊D7311yescharacteristic
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—2921no—
317C2×C8—2921no—
318C2×C8—2921no—
319C2×C8—2921no—
320C2×C8—1462no—
321C2×C8—1462no—
322C4⋊C4—21no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—2922no—
330C4⋊C4—2921no—
331C4⋊C4—2921no—
332C4⋊C4—2921no—
333C4⋊C4—2921no—
334C4⋊C4—732no—
335C4⋊C4—1461no—
336C4⋊C4#1192401512yes—
337C22⋊C4—2924no—
338C22⋊C4—5842no—
339C22⋊C4—1462no—
340C22⋊C4—1462no—
341C42—41no—
342C42—41no—
343C42—2924no—
344C42C2×C4×D7311yescharacteristic
345C42D4×D7311yescharacteristic
346C42#1192402711yescharacteristic
347C42#1192402811yescharacteristic
348C23—2921no—
349C23—2921no—
350C23—2921no—
351C23—2921no—
352C23—2921no—
353C23—2921no—
354C23—1462no—
355C23#1455655911yescharacteristic
356Q8—44no—
357Q8—82no—
358D4—2921no—
359D4—2921no—
360D4—1462no—
361D4—1462no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—22no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—2921no—
374C2×C4—2921no—
375C2×C4—2922no—
376C2×C4—2922no—
377C2×C4—2922no—
378C2×C4—2922no—
379C2×C4—2922no—
380C2×C4—2922no—
381C2×C4—2922no—
382C2×C4—1462no—
383C2×C4C22⋊C4×D7311yescharacteristic
384C2×C4#1455655111yescharacteristic
385C2×C4C4×C22⋊D7311yescharacteristic
386C2×C4#1455656211yescharacteristic
387C2×C4#1455656411yescharacteristic
388C2×C4#1455657311yescharacteristic
389C2×C4C2×#1192401511yescharacteristic
390C2×C4#1455661111yescharacteristic
391C2×C4#1455661211yescharacteristic
392C2×C4#1455661411yescharacteristic
393C2×C4#1455661611yescharacteristic
394C2×C4#1455662011yescharacteristic
395C2×C4#1455662211yescharacteristic
396C2×C4#1455662311yescharacteristic
397C8—81no—
398C8—2922no—
399C8—2922no—
400C22—2922no—
401C22—2922no—
402C22—2922no—
403C22—5841no—
404C22—5841no—
405C22—5841no—
406C22—5841no—
407C22#1798846311yescharacteristic
408C22#1798873711yescharacteristic
409C22#1798874611yescharacteristic
410C22#1798874711yescharacteristic
411C22#1798874811yescharacteristic
412C22#1798875311yescharacteristic
413C22#1798875811yescharacteristic
414C4—21no—
415C4—21no—
416C4—42no—
417C4—82no—
418C4—81no—
419C4—2922no—
420C4—5842no—
421C4#1798858211yescharacteristic
422C4#1798859411yescharacteristic
423C4C4×#1192401511yescharacteristic
424C4#1798868511yescharacteristic
425C4#1798878911yescharacteristic
426C4#1798879911yescharacteristic
427C4#1798892611yescharacteristic
428C4#1798892711yescharacteristic
429C2—5842no—
430C2#2260905611yescharacteristic
431C2#2260907511yescharacteristic
432C2#2260908011yescharacteristic
433C2#2260910211yescharacteristic
434C2#2261024011yescharacteristic
435C2#2261026811yescharacteristic
436C2#2261030411yescharacteristic
subgroupG/H
centerC23#14556559
commutatorC42×C73C22×C4
FrattiniC2×C42C22×D73
FittingC73×#1987C2
socleC23×C73#120
prime radicalC22×D292D4
cosocleC2×C42×C73C23
hypercenter#1987D73
hypocenterC73#8976
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1987

Complements

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

NH ≅ G/Ntype
3. #22610734C2semidirect
5. #22607815C2semidirect
6. #22607772C2semidirect
7. C73×#1987C2semidirect
11. #17988874C4semidirect
221. #1987D73semidirect
226. C73#8976semidirect

Maximal subgroups

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

subgroupindex
C2×#179888742
#226128782
#226107342
C2×#179883142
#226078152
#226077722
C73×#19872
#897673