Finite Group Databasebeta

← #33240654

Subgroups — #33240654

Order 18688.

574 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#500×D73C211yesmaximal, characteristic, complemented
2C4×#14556610C211yesmaximal, characteristic
3#22612084C211yesmaximal, characteristic, complemented
4#22612080C211yesmaximal, characteristic, complemented
5#22612078C211yesmaximal, characteristic, complemented
6#22612072C212yesmaximal, complemented
7#22612066C212yesmaximal, complemented
8#22611718C212yesmaximal, complemented
9#22611685C211yesmaximal, characteristic, complemented
10#22611640C212yesmaximal
11C73×#2226C211yesmaximal, characteristic, complemented
12C2×#14556610C2211yescharacteristic
13C4⋊Q8×D73—21no—
14C4×#11924027—21no—
15C2×#14556530C2212yes—
16#17988992C2211yescharacteristic
17#17988991—21no—
18#17988991C2212yes—
19#17988990—21no—
20C4×Dic292C2211yescharacteristic
21#17988681C2211yescharacteristic
22#17988678—21no—
23C4×#11924016—21no—
24D8×Dic73C2211yescharacteristic
25#17988624C2211yescharacteristic
26#107×D73—21no—
27#107×D73C2212yes—
28#17988580—21no—
29C4×C8×D73C2211yescharacteristic
30#17988414C2212yes—
31#17988413C2212yes—
32#17988412C2212yes—
33C4⋊#11924003C2211yescharacteristic
34#17988405C2212yes—
35#17988392C2212yes—
36#17988390C2212yes—
37#17988388C2212yes—
38#500×C73C2211yescharacteristic
39C4×D8×C73C2211yescharacteristic
40#367×C73C2212yes—
41C4⋊C16×C73C2211yescharacteristic
42#357×C73C2212yes—
43C2×#11924027—21no—
44C2×Q8×D73—41no—
45#14556610—22no—
46#14556610Q814yescomplemented
47C2×Dic292C2311yescharacteristic
48C2×#11924016—21no—
49C2×C8×D73D411yescharacteristic
50C2×C8×D73C2311yescharacteristic
51#14556591—21no—
52#14556590—22no—
53#14556590—41no—
54#14556589—41no—
55C4×Dic146—21no—
56#14556556—21no—
57D4×Dic73—21no—
58C4×C22⋊D73—41no—
59#14556551—41no—
60#14556549—21no—
61#14556547—41no—
62C4⋊C4×D73—22no—
63C4⋊C4×D73—41no—
64C2×C4×Dic73—41no—
65C42×D73D411yescharacteristic
66#14556530—24no—
67C2×#11924003C2312yes—
68#14556525—21no—
69#14556525C2312yes—
70#14556525C2311yescharacteristic
71#14556524—22no—
72#14556524—22no—
73#14556522—21no—
74#14556516—21no—
75#14556514—21no—
76#14556512—21no—
77C4×#10596831D411yescharacteristic
78C8×Dic73D411yescharacteristic
79C8×Dic73C2311yescharacteristic
80C2×D8×C73C2311yescharacteristic
81C4⋊Q8×C73—21no—
82C4×D4×C73—21no—
83C2×C16×C73C2312yes—
84#107×C73—21no—
85#107×C73C2312yes—
86#107×C73C2311yescharacteristic
87#102×C73—21no—
88C4×C8×C73C2311yescharacteristic
89#11924027—24no—
90#11924027—42no—
91C2×Dic146—21no—
92C2×Dic146—41no—
93C2×Dic146—42no—
94Q8×D73—84no—
95C2×C22⋊D73—41no—
96C22×Dic73—41no—
97C22×Dic73—41no—
98C2×C4×D73—42no—
99C2×C4×D73D811yescharacteristic
100C2×C4×D73D811yescharacteristic
101C2×C4×D73C2×D411yescharacteristic
102Dic292—21no—
103Dic292C2×Q812yescomplemented
104#11924016—24no—
105C2×#10596831C2×D411yescharacteristic
106C2×#10596831C2×D411yescharacteristic
107C8×D73—22no—
108C8×D73C2×Q812yes—
109C8×D73#4112yes—
110C4⋊Dic73—21no—
111C4⋊Dic73—22no—
112C4⋊Dic73—41no—
113#11924007—41no—
114#11924007—42no—
115#11924007—42no—
116#11924007—41no—
117C22⋊Dic73—41no—
118#11924005—41no—
119C4×Dic73—41no—
120C4×Dic73—41no—
121C4×Dic73—41no—
122C4×Dic73D811yescharacteristic
123C4×Dic73D811yescharacteristic
124C4×Dic73C2×D411yescharacteristic
125#11924003—22no—
126C2×Q8×C73—41no—
127C2×D4×C73—21no—
128C22×C4×C73—41no—
129D8×C73—21no—
130D8×C73C2×Q812yescomplemented
131C2×C8×C73C2×D411yescharacteristic
132C2×C8×C73C2411yescharacteristic
133C4⋊C4×C73—21no—
134C4⋊C4×C73—22no—
135C4⋊C4×C73—41no—
136C22⋊C4×C73—41no—
137C42×C73C2×D411yescharacteristic
138C16×C73—22no—
139C23×C73—41no—
140C22×D73C2×D811yescharacteristic
141Q8×C73—82no—
142D4×C73—22no—
143D4×C73—41no—
144C2×C4×C73—41no—
145C2×C4×C73—42no—
146C2×C4×C73—82no—
147C2×C4×C73C2×D811yescharacteristic
148C2×C4×C73C2×D811yescharacteristic
149C2×C4×C73C22×D411yescharacteristic
150C22⋊D73—44no—
151C2×Dic73—41no—
152C2×Dic73—44no—
153C2×Dic73—41no—
154C2×Dic73—42no—
155C2×Dic73—82no—
156C2×Dic73C2×D811yescharacteristic
157C2×Dic73C2×D811yescharacteristic
158C2×Dic73C2×D811yescharacteristic
159C4×D73—22no—
160C4×D73—22no—
161C4×D73—84no—
162C4×D73C22⋊Q812yes—
163C4×D73#13712yes—
164Dic146—22no—
165Dic146—41no—
166Dic146—82no—
167Dic146—84no—
168C8×C73—21no—
169C8×C73C22×Q811yescharacteristic
170C8×C73C2×#4111yescharacteristic
171#10596831—21no—
172#10596831C22⋊Q812yes—
173C22×C73—42no—
174C22×C73—81no—
175C22×C73—81no—
176C22×C73C22×D811yescharacteristic
177C2×D73—21no—
178C2×D73—21no—
179C2×D73#47412yes—
180C4×C73—21no—
181C4×C73—82no—
182C4×C73—81no—
183C4×C73#51012yes—
184C4×C73C2×C22⋊Q811yescharacteristic
185C4×C73C2×#13711yescharacteristic
186Dic73—21no—
187Dic73—42no—
188Dic73—82no—
189Dic73—81no—
190Dic73#47412yes—
191Dic73#50912yes—
192#33629—731nomaximal, Sylow
193C2×C73—82no—
194C2×C73C2×#47411yescharacteristic
195C2×C73C2×#50911yescharacteristic
196C2×C73C2×#51011yescharacteristic
197D73—22no—
198C2×#500—731no—
199C4×#135—731no—
200#2227—731no—
201#2227—731no—
202#2226—731no—
203#2226D7311yescharacteristic, complemented
204#2225—732no—
205#2224—732no—
206#2159—732no—
207#2150—731no—
208#2139—732no—
209C73#3362911yescharacteristic, complemented, Sylow
210C2×#135—731no—
211C2×C4⋊Q8—1461no—
212C4×#41—1461no—
213C2×M32—732no—
214#500—732no—
215#500—731no—
216#500—1461no—
217#500—1461no—
218#500C2×D7311yescharacteristic
219C4×Q16—731no—
220C4×Q16—731no—
221C4×SD16—1461no—
222C4×SD16—1461no—
223C4×D8—731no—
224C4×D8C2×D7311yescharacteristic
225#427—731no—
226C2×#107—732no—
227C2×#107—1461no—
228#416—1461no—
229C2×C4×C8—731no—
230#367—732no—
231#367C2×D7312yes—
232#366—732no—
233#366—732no—
234C4⋊C16—732no—
235C4⋊C16—731no—
236C4⋊C16C2×D7311yescharacteristic
237#358—732no—
238#358—732no—
239#357—732no—
240#357C2×D7312yes—
241C2×#41—1461no—
242C22×Q8—2921no—
243#135—734no—
244#135—1462no—
245C2×Q16—731no—
246C2×SD16—1461no—
247C2×D8C22×D7311yescharacteristic
248C22×C8—731no—
249C22×C8—731no—
250C4⋊Q8—21no—
251C4⋊Q8—2921no—
252C4⋊Q8—2921no—
253C4⋊Q8—1462no—
254C4⋊Q8—1461no—
255C4×Q8—1461no—
256C4×Q8—1461no—
257C4×D4—21no—
258C4×D4—2921no—
259C4×D4—2921no—
260C4×D4—1461no—
261#117—2921no—
262#117—1461no—
263C2×C4⋊C4—2921no—
264C2×C4⋊C4—1462no—
265C2×C42—2921no—
266C2×C42—731no—
267M32—1464no—
268C2×C16—732no—
269C2×C16C22×D7312yes—
270#107—21no—
271#107—732no—
272#107—731no—
273#107—1462no—
274#107—1462no—
275#107—1461no—
276#107C22×D7312yes—
277#107C22×D7311yescharacteristic
278#106—1461no—
279#103—1461no—
280#103—1461no—
281#102—21no—
282#102—1461no—
283C4×C8—731no—
284C4×C8—731no—
285C4×C8—731no—
286C4×C8C22×D7311yescharacteristic
287#41—2922no—
288#41—1464no—
289C2×Q8—41no—
290C2×Q8—2922no—
291C2×Q8—2921no—
292C2×Q8—5844no—
293C2×Q8—1461no—
294C2×D4—21no—
295C2×D4—2921no—
296C22×C4—41no—
297C22×C4—2922no—
298C22×C4—2921no—
299C22×C4—2921no—
300C22×C4—731no—
301C22×C4—731no—
302C22×C4—731no—
303Q16—732no—
304Q16—1461no—
305SD16—1464no—
306D8—21no—
307D8Q8×D7312yescomplemented
308C2×C8—731no—
309C2×C8—732no—
310C2×C8—732no—
311C2×C8—731no—
312C2×C8—1462no—
313C2×C8D4×D7311yescharacteristic
314C2×C8C23×D7311yescharacteristic
315C4⋊C4—21no—
316C4⋊C4—22no—
317C4⋊C4—41no—
318C4⋊C4—2922no—
319C4⋊C4—2922no—
320C4⋊C4—2921no—
321C4⋊C4—2921no—
322C4⋊C4—2921no—
323C4⋊C4—1461no—
324C4⋊C4—1462no—
325C22⋊C4—41no—
326C22⋊C4—2921no—
327C22⋊C4—2921no—
328C42—2921no—
329C42—2921no—
330C42—2921no—
331C42—731no—
332C42—731no—
333C42—731no—
334C42D4×D7311yescharacteristic
335C16—22no—
336C16—1462no—
337C23—41no—
338C23—731no—
339Q8—82no—
340Q8—2921no—
341Q8—5842no—
342Q8—5844no—
343Q8—1462no—
344D4—22no—
345D4—41no—
346D4—2924no—
347C2×C4—41no—
348C2×C4—42no—
349C2×C4—82no—
350C2×C4—2921no—
351C2×C4—2924no—
352C2×C4—2921no—
353C2×C4—2922no—
354C2×C4—5844no—
355C2×C4—5842no—
356C2×C4—731no—
357C2×C4—731no—
358C2×C4—732no—
359C2×C4—732no—
360C2×C4—731no—
361C2×C4—1462no—
362C2×C4—1462no—
363C2×C4D8×D7311yescharacteristic
364C2×C4D8×D7311yescharacteristic
365C2×C4C2×D4×D7311yescharacteristic
366C8—21no—
367C8—732no—
368C8—1461no—
369C8C2×Q8×D7311yescharacteristic
370C8#41×D7311yescharacteristic
371C22—42no—
372C22—81no—
373C22—81no—
374C22—732no—
375C22—1461no—
376C22—1461no—
377C22C2×D8×D7311yescharacteristic
378C4—21no—
379C4—82no—
380C4—81no—
381C4—2922no—
382C4—5842no—
383C4—5841no—
384C4—732no—
385C4—732no—
386C4—1461no—
387C4#1798902612yes—
388C4C22⋊Q8×D7311yescharacteristic
389C4#137×D7311yescharacteristic
390C2—82no—
391C2—1462no—
392C2#474×D7311yescharacteristic
393C2#509×D7311yescharacteristic
394C2C2×#1798902611yescharacteristic
subgroupG/H
centerC22C2×D8×D73
commutator, cosocleC2×C8×C73C24
FrattiniC2×C8C23×D73
FittingC73×#2226C2
socleC22×C73C22×D8
prime radicalC2×#14556610C22
hypercenter#2226D73
hypocenterC73#33629
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2226

Complements

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

NH ≅ G/Ntype
1. #500×D73C2semidirect
3. #22612084C2semidirect
4. #22612080C2semidirect
5. #22612078C2semidirect
6. #22612072C2semidirect
7. #22612066C2semidirect
8. #22611718C2semidirect
9. #22611685C2semidirect
11. C73×#2226C2semidirect
46. #14556610Q8semidirect
103. Dic292C2×Q8semidirect
130. D8×C73C2×Q8semidirect
203. #2226D73semidirect
209. C73#33629semidirect
307. D8Q8×D73semidirect

Maximal subgroups

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

subgroupindex
#500×D732
C4×#145566102
#226120842
#226120802
#226120782
#226120722
#226120662
#226117182
#226116852
#226116402
C73×#22262
#3362973