Finite Group Databasebeta

← #33002288

Subgroups — #33002288

Order 18688.

420 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#427×D73C211yesmaximal, characteristic
2#22611541C211yesmaximal, characteristic
3#22609953C211yesmaximal, characteristic
4#22608038C211yesmaximal, characteristic, complemented
5#22608037C211yesmaximal, characteristic, complemented
6#22608036C211yesmaximal, characteristic, complemented
7C73×#1386C211yesmaximal, characteristic, complemented
8C22×C8×D73C2211yescharacteristic
9#117×D73—21no—
10C2×#14556531C2211yescharacteristic
11#17988625C2211yescharacteristic
12#17988624C411yescharacteristic, complemented
13#17988624C411yescharacteristic, complemented
14#17988623—21no—
15#17988622—21no—
16#107×D73C411yescharacteristic, complemented
17#107×D73C411yescharacteristic, complemented
18C2×#14556523C2211yescharacteristic
19#106×D73—21no—
20#17988444—21no—
21#17988366—21no—
22#17988366—21no—
23#17988349—21no—
24C2×M32×C73C2211yescharacteristic
25#427×C73C2211yescharacteristic
26C2×#106×C73C2211yescharacteristic
27C22×C4×D73D411yescharacteristic
28C22×#10596831D411yescharacteristic
29C2×C8×D73—21no—
30C2×C8×D73—21no—
31C2×C8×D73—21no—
32C2×C8×D73—21no—
33C2×C8×D73C2×C411yescharacteristic
34C2×C8×D73C2×C411yescharacteristic
35C2×C8×D73D411yescharacteristic
36C2×C8×D73D411yescharacteristic
37#14556550—21no—
38#14556549—21no—
39#14556549—21no—
40#14556548—41no—
41#14556547—41no—
42C2×C4⋊Dic73—21no—
43C4⋊C4×D73—21no—
44C4⋊C4×D73—21no—
45C2×#11924005—41no—
46C22⋊C4×D73—41no—
47C42×D73—41no—
48#14556531—21no—
49#14556531—21no—
50C2×#11924003—21no—
51#14556525C2×C411yescharacteristic
52#14556525C2×C411yescharacteristic
53#14556524—21no—
54#14556524—21no—
55#14556523—21no—
56#14556523—21no—
57#14556523—21no—
58#14556522—21no—
59#14556522—21no—
60#14556498—41no—
61C22×C8×C73C2311yescharacteristic
62#117×C73—21no—
63C2×C4⋊C4×C73—21no—
64M32×C73—21no—
65M32×C73—21no—
66C2×C16×C73—21no—
67#107×C73C2×C411yescharacteristic
68#107×C73C2×C411yescharacteristic
69#106×C73—21no—
70#106×C73—21no—
71#106×C73—21no—
72C23×D73D811yescharacteristic
73C22×Dic73—41no—
74C22×Dic73D811yescharacteristic
75C2×C4×D73—21no—
76C2×C4×D73—21no—
77C2×C4×D73—21no—
78C2×C4×D73—21no—
79C2×C4×D73—41no—
80C2×C4×D73—41no—
81C2×C4×D73C22⋊C411yescharacteristic
82C2×C4×D73C22⋊C411yescharacteristic
83C2×C4×D73SD1611yescharacteristic
84C2×C4×D73SD1611yescharacteristic
85C2×#10596831—21no—
86C2×#10596831—21no—
87C2×#10596831C22⋊C411yescharacteristic
88C2×#10596831C22⋊C411yescharacteristic
89C8×D73—21no—
90C8×D73—21no—
91C8×D73—21no—
92C8×D73—21no—
93C8×D73—21no—
94C8×D73—21no—
95C8×D73—21no—
96C8×D73—21no—
97C4⋊Dic73—21no—
98C4⋊Dic73—21no—
99C4⋊Dic73—41no—
100C4⋊Dic73—41no—
101#11924007—41no—
102#11924007—41no—
103C22⋊Dic73—41no—
104#11924005—41no—
105#11924005—41no—
106#11924005—82no—
107C4×Dic73—41no—
108C4×Dic73—41no—
109C4×Dic73—41no—
110#11924003—41no—
111C22×C4×C73—41no—
112C22×C4×C73C2×D411yescharacteristic
113C2×C8×C73C22×C411yescharacteristic
114C2×C8×C73C2×D411yescharacteristic
115C2×C8×C73#4111yescharacteristic
116C2×C8×C73#4111yescharacteristic
117C2×C8×C73#4111yescharacteristic
118C2×C8×C73#4111yescharacteristic
119C4⋊C4×C73—21no—
120C4⋊C4×C73—21no—
121C4⋊C4×C73—41no—
122C4⋊C4×C73—41no—
123C22⋊C4×C73—41no—
124C42×C73—41no—
125C16×C73—41no—
126C23×C73C2×D811yescharacteristic
127C22×D73—21no—
128C22×D73—21no—
129C22×D73—22no—
130C22×D73—41no—
131C22×D73#10211yescharacteristic
132C22×D73#10211yescharacteristic
133C2×C4×C73—41no—
134C2×C4×C73—41no—
135C2×C4×C73—81no—
136C2×C4×C73—81no—
137C2×C4×C73—81no—
138C2×C4×C73C2×C22⋊C411yescharacteristic
139C2×C4×C73#12311yescharacteristic
140C2×C4×C73#12311yescharacteristic
141C2×C4×C73C2×SD1611yescharacteristic
142C2×C4×C73#13711yescharacteristic
143C2×C4×C73#13711yescharacteristic
144C2×Dic73—21no—
145C2×Dic73—21no—
146C2×Dic73—41no—
147C2×Dic73—41no—
148C2×Dic73—81no—
149C2×Dic73—81no—
150C2×Dic73—81no—
151C2×Dic73#10211yescharacteristic
152C2×Dic73#10211yescharacteristic
153C4×D73—21no—
154C4×D73—21no—
155C4×D73—21no—
156C4×D73—21no—
157C4×D73—21no—
158C4×D73—21no—
159C4×D73—21no—
160C4×D73—21no—
161C4×D73—82no—
162C4×D73—82no—
163C8×C73#11711yescharacteristic
164C8×C73#11711yescharacteristic
165C8×C73#12311yescharacteristic
166C8×C73#12311yescharacteristic
167#10596831—21no—
168#10596831—21no—
169C22×C73—21no—
170C22×C73—21no—
171C22×C73C2×#10211yescharacteristic
172C22×C73#48011yescharacteristic
173C22×C73#48011yescharacteristic
174C2×D73—21no—
175C2×D73—21no—
176C2×D73—21no—
177C2×D73—21no—
178C2×D73—21no—
179C2×D73—42no—
180C2×D73—42no—
181C2×D73#36012yes—
182C4×C73—81no—
183C4×C73—81no—
184C4×C73—161no—
185C4×C73#38111yescharacteristic
186C4×C73#41911yescharacteristic
187C4×C73#48311yescharacteristic
188C4×C73#48311yescharacteristic
189Dic73—21no—
190Dic73—21no—
191Dic73—81no—
192Dic73—81no—
193Dic73—161no—
194#12870—731nomaximal, Sylow
195C2×C73—21no—
196C2×C73—21no—
197C2×C73#182311yescharacteristic
198C2×C73C2×#36011yescharacteristic
199C2×C73#214511yescharacteristic
200D73—22no—
201D73—41no—
202C2×#427—731no—
203#2112—731no—
204#1807—731no—
205#1386—731no—
206#1386—731no—
207#1386—731no—
208#1386D7311yescharacteristic, complemented
209C73#1287011yescharacteristic, complemented, Sylow
210C23×C8—731no—
211C2×#117—1461no—
212C2×M32—731no—
213C2×M32C2×D7311yescharacteristic
214#427—731no—
215#427—731no—
216#427—731no—
217#427—1461no—
218#427—1461no—
219#427C2×D7311yescharacteristic
220C2×#107—731no—
221C2×#107—731no—
222C2×#106—731no—
223C2×#106—1461no—
224C2×#106C2×D7311yescharacteristic
225#380—1461no—
226C22⋊C16—1461no—
227C22⋊C16—1461no—
228#340—1461no—
229C23×C4—731no—
230C22×C8—731no—
231C22×C8—731no—
232C22×C8—731no—
233C22×C8—731no—
234C22×C8—731no—
235C22×C8—1461no—
236C22×C8—1461no—
237C22×C8—1461no—
238C22×C8—1461no—
239C22×C8C22×D7311yescharacteristic
240#117—21no—
241#117—2921no—
242#117—2921no—
243#117—1461no—
244#117—1461no—
245#117—1461no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—1461no—
248C2×C4⋊C4—1461no—
249C2×C4⋊C4—1461no—
250C2×C22⋊C4—2921no—
251C2×C22⋊C4—2921no—
252C2×C42—2921no—
253M32—21no—
254M32—21no—
255M32—1461no—
256M32—1461no—
257C2×C16—21no—
258C2×C16—1461no—
259#107—731no—
260#107—731no—
261#107—1461no—
262#107—1461no—
263#107C4×D7311yescharacteristic
264#107C4×D7311yescharacteristic
265#106—21no—
266#106—21no—
267#106—21no—
268#106—1461no—
269#106—1461no—
270#106—1461no—
271#106—1461no—
272#106—1461no—
273#95—2921no—
274C24—731no—
275C22×C4—41no—
276C22×C4—2921no—
277C22×C4—2921no—
278C22×C4—2921no—
279C22×C4—731no—
280C22×C4—731no—
281C22×C4—731no—
282C22×C4—731no—
283C22×C4—731no—
284C22×C4—1461no—
285C22×C4—1461no—
286C22×C4—1461no—
287C22×C4—1461no—
288C22×C4D4×D7311yescharacteristic
289C2×C8—731no—
290C2×C8—731no—
291C2×C8—1461no—
292C2×C8—1461no—
293C2×C8—1461no—
294C2×C8—1461no—
295C2×C8—1461no—
296C2×C8—1461no—
297C2×C8—1461no—
298C2×C8—1461no—
299C2×C8—1461no—
300C2×C8—1461no—
301C2×C8C2×C4×D7311yescharacteristic
302C2×C8D4×D7311yescharacteristic
303C2×C8#1192402711yescharacteristic
304C2×C8#1192402811yescharacteristic
305C2×C8#1192402811yescharacteristic
306C2×C8#1192402911yescharacteristic
307C4⋊C4—21no—
308C4⋊C4—21no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—2921no—
312C4⋊C4—2921no—
313C4⋊C4—2921no—
314C4⋊C4—2921no—
315C4⋊C4—1461no—
316C4⋊C4—1461no—
317C22⋊C4—41no—
318C22⋊C4—2921no—
319C22⋊C4—2921no—
320C22⋊C4—2921no—
321C22⋊C4—5842no—
322C42—41no—
323C42—2921no—
324C42—2921no—
325C42—2921no—
326C16—41no—
327C16—2921no—
328C23—2921no—
329C23—731no—
330C23—731no—
331C23—1462no—
332C23—1461no—
333C23—1461no—
334C23D8×D7311yescharacteristic
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—2921no—
341C2×C4—2921no—
342C2×C4—5842no—
343C2×C4—5842no—
344C2×C4—5841no—
345C2×C4—5841no—
346C2×C4—5841no—
347C2×C4—731no—
348C2×C4—731no—
349C2×C4—1461no—
350C2×C4—1461no—
351C2×C4—1461no—
352C2×C4—1461no—
353C2×C4—1461no—
354C2×C4—1461no—
355C2×C4—1461no—
356C2×C4—1461no—
357C2×C4—1461no—
358C2×C4—1461no—
359C2×C4C22⋊C4×D7311yescharacteristic
360C2×C4#1455657311yescharacteristic
361C2×C4#1455657611yescharacteristic
362C2×C4SD16×D7311yescharacteristic
363C2×C4#1455662011yescharacteristic
364C2×C4#1455662311yescharacteristic
365C8—1461no—
366C8—1461no—
367C8#1455654811yescharacteristic
368C8#1455654911yescharacteristic
369C8#1455657311yescharacteristic
370C8#1455657611yescharacteristic
371C22—21no—
372C22—21no—
373C22—2922no—
374C22—2922no—
375C22—732no—
376C22—1461no—
377C22—1461no—
378C22—1461no—
379C22—1461no—
380C22—1461no—
381C22#102×D7311yescharacteristic
382C22#1798889911yescharacteristic
383C22#1798890211yescharacteristic
384C4—81no—
385C4—81no—
386C4—161no—
387C4—5841no—
388C4—5841no—
389C4—11681no—
390C4—1461no—
391C4—1461no—
392C4#1798844911yescharacteristic
393C4#1798859411yescharacteristic
394C4#1798891611yescharacteristic
395C4#1798891911yescharacteristic
396C2—21no—
397C2—21no—
398C2—2921no—
399C2—1462no—
400C2#2261001611yescharacteristic
401C2#360×D7311yescharacteristic
402C2#2261166511yescharacteristic
subgroupG/H
centerC22#102×D73
commutatorC2×C8×C73C22×C4
FrattiniC22×C8C22×D73
FittingC73×#1386C2
socleC22×C73C2×#102
prime radicalC23×D73D8
cosocleC22×C8×C73C23
hypercenter#1386D73
hypocenterC73#12870
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #1386

Complements

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

NH ≅ G/Ntype
4. #22608038C2semidirect
5. #22608037C2semidirect
6. #22608036C2semidirect
7. C73×#1386C2semidirect
12. #17988624C4semidirect
13. #17988624C4semidirect
16. #107×D73C4semidirect
17. #107×D73C4semidirect
208. #1386D73semidirect
209. C73#12870semidirect

Maximal subgroups

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

subgroupindex
#427×D732
#226115412
#226099532
#226080382
#226080372
#226080362
C73×#13862
#1287073