Finite Group Databasebeta

← #33086586

Subgroups — #33086586

Order 18688.

842 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
1C2×#17988539C211yesmaximal, characteristic
2#22612327C212yesmaximal
3#22610094C211yesmaximal, characteristic, complemented
4#22610092C211yesmaximal, characteristic, complemented
5#22610060C212yesmaximal, complemented
6#22609965C211yesmaximal, characteristic
7#22609963C212yesmaximal
8#22609961C211yesmaximal, characteristic
9#22609829C212yesmaximal, complemented
10#22609828C211yesmaximal, characteristic, complemented
11C73×#1776C211yesmaximal, characteristic, complemented
12C22×#11924011C2211yescharacteristic
13C22×#11924011C2211yescharacteristic
14C2×C42×D73C2211yescharacteristic
15C2×#14556525C2211yescharacteristic
16C2×#14556524C2212yes—
17C2×#14556523C2211yescharacteristic
18C2×#14556522C2212yes—
19#17988539C418yescomplemented
20C2×#14556502C2211yescharacteristic
21C2×C8×Dic73C2211yescharacteristic
22C2×C8×Dic73C2211yescharacteristic
23#17988454—22no—
24#17988452C2212yes—
25#17988452C2212yes—
26#17988449—22no—
27#17988444C2212yes—
28#17988444C2212yes—
29C4×C4⋊Dic73C2212yes—
30C4×#11924005—22no—
31#17988433—22no—
32#17988350C2212yes—
33#17988349C2212yes—
34#17988349C2212yes—
35C2×#107×C73C2211yescharacteristic
36C2×#106×C73C2211yescharacteristic
37C2×#97×C73C2211yescharacteristic
38C4×C4⋊C4×C73C2212yes—
39#340×C73C2212yes—
40C22×C4×D73D412yes—
41C22×C4×D73C2311yescharacteristic
42C2×#11924011—22no—
43C2×#11924011—22no—
44C2×#11924011—22no—
45C2×#11924011—22no—
46C2×#11924011C2×C414yes—
47C2×#11924011C2×C414yes—
48C22×#10596831C2311yescharacteristic
49C22×#10596831C2311yescharacteristic
50C2×C4⋊Dic73C2312yes—
51C2×C4⋊Dic73C2312yes—
52C2×#11924005—22no—
53C2×#11924005—22no—
54C2×C4×Dic73—22no—
55C2×C4×Dic73D412yes—
56C2×C4×Dic73C2311yescharacteristic
57C42×D73C2×C414yes—
58C42×D73Q814yes—
59#14556525—22no—
60#14556524—24no—
61#14556523—22no—
62#14556522—24no—
63#14556502C2×C414yes—
64C8×Dic73C2×C414yes—
65C8×Dic73C2×C414yes—
66#14556498—22no—
67#14556498—22no—
68#14556498—22no—
69#14556498—22no—
70#14556498—22no—
71#14556498—22no—
72#14556498—22no—
73C22×C8×C73C2311yescharacteristic
74C22×C8×C73C2311yescharacteristic
75C2×C4⋊C4×C73C2312yes—
76C2×C4⋊C4×C73C2312yes—
77C2×C42×C73—22no—
78C2×C42×C73C2311yescharacteristic
79#107×C73—22no—
80#106×C73—22no—
81#97×C73C2×C414yes—
82#95×C73—22no—
83C23×D73C2×D411yescharacteristic
84C22×Dic73—22no—
85C22×Dic73—22no—
86C22×Dic73C2×D411yescharacteristic
87C22×Dic73C2×D412yes—
88C2×C4×D73—21no—
89C2×C4×D73—21no—
90C2×C4×D73—21no—
91C2×C4×D73—21no—
92C2×C4×D73—42no—
93C2×C4×D73—42no—
94C2×C4×D73C4⋊C418yes—
95C2×C4×D73C22×C412yes—
96C2×C4×D73C2×Q812yes—
97#11924011—28no—
98#11924011—28no—
99C2×#10596831—21no—
100C2×#10596831—21no—
101C2×#10596831C22×C412yes—
102C2×#10596831C22×C412yes—
103C2×#10596831#4112yes—
104C2×#10596831#4112yes—
105C4⋊Dic73—22no—
106C4⋊Dic73—22no—
107C4⋊Dic73—22no—
108C4⋊Dic73—22no—
109#11924005—44no—
110#11924005—44no—
111C4×Dic73—44no—
112C4×Dic73C4⋊C418yes—
113C4×Dic73C22×C412yes—
114C4×Dic73C2×Q812yes—
115C22×C4×C73—22no—
116C22×C4×C73—22no—
117C22×C4×C73C2×D412yes—
118C22×C4×C73C2411yescharacteristic
119C2×C8×C73—21no—
120C2×C8×C73—21no—
121C2×C8×C73C22×C412yes—
122C2×C8×C73C22×C412yes—
123C2×C8×C73#4112yes—
124C2×C8×C73#4112yes—
125C4⋊C4×C73—22no—
126C4⋊C4×C73—22no—
127C4⋊C4×C73—22no—
128C4⋊C4×C73—22no—
129C42×C73—44no—
130C42×C73C22×C412yes—
131C42×C73C2×Q812yes—
132C23×C73C22×D411yescharacteristic
133C22×D73—21no—
134C22×D73—21no—
135C22×D73—22no—
136C22×D73—22no—
137C22×D73C2×C4⋊C412yes—
138C2×C4×C73—22no—
139C2×C4×C73—22no—
140C2×C4×C73—42no—
141C2×C4×C73—42no—
142C2×C4×C73—42no—
143C2×C4×C73—42no—
144C2×C4×C73—42no—
145C2×C4×C73—42no—
146C2×C4×C73C2×C4⋊C414yes—
147C2×C4×C73C23×C411yescharacteristic
148C2×C4×C73C22×Q811yescharacteristic
149C2×C4×C73C2×#4111yescharacteristic
150C2×C4×C73C2×#4111yescharacteristic
151C2×C4×C73C2×#4111yescharacteristic
152C2×C4×C73C2×#4111yescharacteristic
153C2×Dic73—21no—
154C2×Dic73—21no—
155C2×Dic73—22no—
156C2×Dic73—22no—
157C2×Dic73—42no—
158C2×Dic73—42no—
159C2×Dic73—42no—
160C2×Dic73—42no—
161C2×Dic73—42no—
162C2×Dic73—42no—
163C2×Dic73C2×C4⋊C412yes—
164C2×Dic73C2×C4⋊C414yes—
165C4×D73—22no—
166C4×D73—22no—
167C4×D73—22no—
168C4×D73—22no—
169C4×D73—48no—
170C8×C73—22no—
171C8×C73#11714yes—
172#10596831—22no—
173#10596831#11714yes—
174C22×C73C22×C4⋊C411yescharacteristic
175C22×C73#53211yescharacteristic
176C22×C73#53211yescharacteristic
177C22×C73#57512yes—
178C22×C73#57512yes—
179C2×D73—24no—
180C2×D73—22no—
181C2×D73—42no—
182C2×D73—41no—
183C2×D73—41no—
184C4×C73—24no—
185C4×C73—82no—
186C4×C73—82no—
187C4×C73C2×#11711yescharacteristic
188C4×C73C2×#11711yescharacteristic
189C4×C73#53311yescharacteristic
190C4×C73#53311yescharacteristic
191Dic73—24no—
192Dic73—22no—
193Dic73—82no—
194Dic73—82no—
195#20825—731nomaximal, Sylow
196C2×C73#228511yescharacteristic
197C2×C73#291112yes—
198C2×C73#318312yes—
199C2×C73#318311yescharacteristic
200C2×C73#318311yescharacteristic
201D73—42no—
202C2×C4×M16—731no—
203#2283—732no—
204#1846—731no—
205#1846—731no—
206#1835—732no—
207#1809—731no—
208#1809—731no—
209#1809—732no—
210#1776—731no—
211#1776—732no—
212#1776D7311yescharacteristic, complemented
213C73#2082511yescharacteristic, complemented, Sylow
214C22×M16—731no—
215C22×M16—731no—
216C22×C42—731no—
217C2×#107—731no—
218C2×#107—732no—
219C2×#107C2×D7311yescharacteristic
220C2×#106—731no—
221C2×#106—732no—
222C2×#106C2×D7311yescharacteristic
223C4×M16—738no—
224C2×#97—731no—
225C2×#97C2×D7311yescharacteristic
226C2×C4×C8—731no—
227C2×C4×C8—731no—
228#383—1462no—
229#382—732no—
230#382—732no—
231#381—1462no—
232#380—732no—
233#380—732no—
234C4×C4⋊C4—732no—
235C4×C4⋊C4C2×D7312yes—
236C4×C22⋊C4—1462no—
237#376—1462no—
238#340—732no—
239#340—732no—
240#340—732no—
241#340C2×D7312yes—
242C23×C4—732no—
243C23×C4—731no—
244C2×M16—734no—
245C2×M16—734no—
246C2×M16—1462no—
247C2×M16—1462no—
248C2×M16—1462no—
249C2×M16—1462no—
250C22×C8—731no—
251C22×C8—731no—
252C22×C8C22×D7311yescharacteristic
253C22×C8C22×D7311yescharacteristic
254C2×C4⋊C4—732no—
255C2×C4⋊C4—732no—
256C2×C4⋊C4C22×D7312yes—
257C2×C4⋊C4C22×D7312yes—
258C2×C22⋊C4—1462no—
259C2×C22⋊C4—1462no—
260C2×C42—22no—
261C2×C42—732no—
262C2×C42—731no—
263C2×C42—734no—
264C2×C42—734no—
265C2×C42—1462no—
266C2×C42C22×D7311yescharacteristic
267#107—22no—
268#107—1464no—
269#107—1462no—
270#106—22no—
271#106—1464no—
272#106—1462no—
273#97—734no—
274#97C4×D7314yes—
275C4×C8—734no—
276C4×C8—734no—
277#95—22no—
278#95—1462no—
279#95—1462no—
280#95—1462no—
281#95—1462no—
282#95—1462no—
283#95—1462no—
284#95—1462no—
285C24—731no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—2922no—
289C22×C4—2922no—
290C22×C4—738no—
291C22×C4—732no—
292C22×C4—731no—
293C22×C4—732no—
294C22×C4—732no—
295C22×C4—1462no—
296C22×C4—1462no—
297C22×C4—1461no—
298C22×C4—1461no—
299C22×C4—1461no—
300C22×C4—1461no—
301C22×C4D4×D7312yes—
302C22×C4C23×D7311yescharacteristic
303M16—1468no—
304M16—1468no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—732no—
308C2×C8—732no—
309C2×C8—732no—
310C2×C8—732no—
311C2×C8—1461no—
312C2×C8—1461no—
313C2×C8C2×C4×D7312yes—
314C2×C8C2×C4×D7312yes—
315C2×C8#1192402812yes—
316C2×C8#1192402812yes—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—1462no—
322C4⋊C4—1462no—
323C4⋊C4—1462no—
324C4⋊C4—1462no—
325C22⋊C4—2924no—
326C22⋊C4—2924no—
327C42—44no—
328C42—2924no—
329C42—738no—
330C42—732no—
331C42—732no—
332C42C2×C4×D7312yes—
333C42Q8×D7312yes—
334C23—732no—
335C23—1462no—
336C23—1462no—
337C23—1461no—
338C23—1461no—
339C23C2×D4×D7311yescharacteristic
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—2928no—
349C2×C4—2922no—
350C2×C4—2922no—
351C2×C4—2922no—
352C2×C4—2922no—
353C2×C4—2922no—
354C2×C4—2922no—
355C2×C4—732no—
356C2×C4—734no—
357C2×C4—1462no—
358C2×C4—1462no—
359C2×C4—1461no—
360C2×C4—1461no—
361C2×C4—1462no—
362C2×C4—1462no—
363C2×C4—1462no—
364C2×C4—1462no—
365C2×C4C4⋊C4×D7314yes—
366C2×C4C22×C4×D7311yescharacteristic
367C2×C4C2×Q8×D7311yescharacteristic
368C2×C4#41×D7311yescharacteristic
369C2×C4#41×D7311yescharacteristic
370C2×C4C2×#1192402811yescharacteristic
371C2×C4C2×#1192402811yescharacteristic
372C8—22no—
373C8—734no—
374C8—1462no—
375C8#1455654814yes—
376C22—2922no—
377C22—2921no—
378C22—2921no—
379C22—1462no—
380C22—1464no—
381C22C2×C4⋊C4×D7311yescharacteristic
382C22#1798913411yescharacteristic
383C22#1798913411yescharacteristic
384C22#1798938312yes—
385C22#1798938612yes—
386C4—24no—
387C4—82no—
388C4—82no—
389C4—5842no—
390C4—5842no—
391C4—1462no—
392C4—1464no—
393C4#117×D7311yescharacteristic
394C4C2×#1455654811yescharacteristic
395C4#1798914111yescharacteristic
396C4#1798914111yescharacteristic
397C2—2922no—
398C2#2261233511yescharacteristic
399C2#2261851812yes—
400C2#2262065612yes—
401C2#2262065811yescharacteristic
402C2#2262066311yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1776C2
socleC23×C73C22×D4
prime radicalC23×D73C2×D4
cosocleC22×C4×C73C24
hypercenter#1776D73
hypocenterC73#20825
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1776

Complements

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

NH ≅ G/Ntype
3. #22610094C2semidirect
4. #22610092C2semidirect
5. #22610060C2semidirect
9. #22609829C2semidirect
10. #22609828C2semidirect
11. C73×#1776C2semidirect
19. #17988539C4semidirect
212. #1776D73semidirect
213. C73#20825semidirect

Maximal subgroups

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

subgroupindex
C2×#179885392
#226123272
#226100942
#226100922
#226100602
#226099652
#226099632
#226099612
#226098292
#226098282
C73×#17762
#2082573