Finite Group Databasebeta

← #33240655

Subgroups — #33240655

Order 18688.

574 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988991C211yesmaximal, characteristic, complemented
2#22618702C211yesmaximal, characteristic
3#22612082C212yesmaximal, complemented
4#22612077C212yesmaximal, complemented
5#22612072C212yesmaximal, complemented
6#22612067C212yesmaximal, complemented
7#22611720C211yesmaximal, characteristic, complemented
8#22611686C211yesmaximal, characteristic, complemented
9#22611644C211yesmaximal, characteristic
10#22611641C211yesmaximal, characteristic
11C73×#2159C211yesmaximal, characteristic, complemented
12C2×#14556613C2211yescharacteristic
13C2×#14556590—21no—
14#17989055—21no—
15C2×#14556531C2211yescharacteristic
16#17988991—21no—
17#17988991—21no—
18#17988991C2212yes—
19#17988991C2212yes—
20#17988681C2211yescharacteristic
21#17988681C2211yescharacteristic
22#17988674—21no—
23#17988674—21no—
24#17988669C2211yescharacteristic
25#17988669C2211yescharacteristic
26C2×#14556525C2211yescharacteristic
27C2×#14556524—21no—
28#17988581—21no—
29C2×C8×Dic73C2211yescharacteristic
30#17988414C2212yes—
31#17988412C2212yes—
32#17988405C2212yes—
33#17988405C2212yes—
34#17988392C2212yes—
35#17988390C2212yes—
36#17988389C2212yes—
37#17988387C2212yes—
38C2×M32×C73C2211yescharacteristic
39#427×C73C2211yescharacteristic
40C2×#107×C73C2211yescharacteristic
41#367×C73C2212yes—
42#366×C73C2212yes—
43C2×#11924028—21no—
44C22×Dic146—41no—
45#14556613—22no—
46#14556613Q814yescomplemented
47C2×Dic292C2311yescharacteristic
48C2×#11924017—21no—
49C2×D584C2311yescharacteristic
50C22×#10596831D411yescharacteristic
51#14556590—22no—
52#14556590—22no—
53#14556590—41no—
54#14556590—41no—
55#14556556—21no—
56#14556556—21no—
57#14556553—21no—
58#14556553—21no—
59#14556551—41no—
60#14556551—41no—
61#14556548—41no—
62C2×C4⋊Dic73—21no—
63C2×#11924007—41no—
64C2×C4×Dic73D411yescharacteristic
65C42×D73—41no—
66#14556531—22no—
67C2×#11924003C2312yes—
68#14556525—21no—
69#14556525C2312yes—
70#14556524—22no—
71#14556524—22no—
72#14556514—21no—
73#14556514—21no—
74#14556511—21no—
75#14556511—21no—
76C8×Dic73D411yescharacteristic
77C8×Dic73D411yescharacteristic
78C8×Dic73C2311yescharacteristic
79C8×Dic73C2311yescharacteristic
80C22×C8×C73C2311yescharacteristic
81#117×C73—21no—
82C2×C4⋊C4×C73—21no—
83M32×C73—22no—
84C2×C16×C73C2312yes—
85#107×C73—21no—
86#107×C73C2312yes—
87#107×C73C2311yescharacteristic
88#107×C73C2311yescharacteristic
89#106×C73—21no—
90#11924028—24no—
91#11924028—42no—
92C2×Dic146—21no—
93C2×Dic146—42no—
94C2×Dic146—42no—
95C2×Dic146—84no—
96C2×D292—21no—
97C2×C22⋊D73—41no—
98C22×Dic73—41no—
99C22×Dic73D811yescharacteristic
100C22×Dic73D811yescharacteristic
101C2×C4×D73—41no—
102C2×C4×D73—41no—
103C2×C4×D73—41no—
104Dic292—21no—
105Dic292C2×Q812yescomplemented
106#11924017—24no—
107D584—21no—
108D584C2×Q812yescomplemented
109C2×#10596831—22no—
110C2×#10596831C2×D411yescharacteristic
111C2×#10596831C2×D411yescharacteristic
112C4⋊Dic73—22no—
113C4⋊Dic73—41no—
114#11924007—42no—
115#11924007—42no—
116#11924007—41no—
117#11924007—41no—
118#11924005—41no—
119#11924005—41no—
120C4×Dic73—41no—
121C4×Dic73—41no—
122C4×Dic73D811yescharacteristic
123C4×Dic73D811yescharacteristic
124C4×Dic73C2×D411yescharacteristic
125C4×Dic73C2×D411yescharacteristic
126#11924003—22no—
127C22×C4×C73—41no—
128C22×C4×C73C2×D411yescharacteristic
129C2×C8×C73C2×D411yescharacteristic
130C2×C8×C73C2×Q812yes—
131C2×C8×C73#4112yes—
132C2×C8×C73C2411yescharacteristic
133C4⋊C4×C73—21no—
134C4⋊C4×C73—21no—
135C4⋊C4×C73—22no—
136C4⋊C4×C73—41no—
137C22⋊C4×C73—41no—
138C42×C73—41no—
139C16×C73—22no—
140C23×C73C2×D811yescharacteristic
141C22×D73—41no—
142C2×C4×C73—41no—
143C2×C4×C73—41no—
144C2×C4×C73—42no—
145C2×C4×C73—82no—
146C2×C4×C73C22⋊Q812yes—
147C2×C4×C73C2×D811yescharacteristic
148C2×C4×C73#13712yes—
149C2×C4×C73C22×D411yescharacteristic
150C22⋊D73—44no—
151C2×Dic73—22no—
152C2×Dic73—22no—
153C2×Dic73—41no—
154C2×Dic73—42no—
155C2×Dic73—82no—
156C2×Dic73C2×D811yescharacteristic
157C2×Dic73C2×D811yescharacteristic
158C2×Dic73C2×D811yescharacteristic
159C2×Dic73C2×D811yescharacteristic
160D292—22no—
161D292—41no—
162C4×D73—44no—
163C4×D73—82no—
164C4×D73—82no—
165Dic146—22no—
166Dic146—41no—
167Dic146—84no—
168Dic146—84no—
169C8×C73C22⋊Q812yes—
170C8×C73C22×Q811yescharacteristic
171C8×C73C2×#4111yescharacteristic
172#10596831—21no—
173#10596831—21no—
174C22×C73—21no—
175C22×C73—21no—
176C22×C73#47412yes—
177C22×C73C22×D811yescharacteristic
178C2×D73—42no—
179C2×D73—81no—
180C2×D73—81no—
181C4×C73—82no—
182C4×C73—81no—
183C4×C73—81no—
184C4×C73#47412yes—
185C4×C73C2×C22⋊Q811yescharacteristic
186C4×C73C2×#13711yescharacteristic
187Dic73—21no—
188Dic73—21no—
189Dic73—42no—
190Dic73—82no—
191Dic73#50912yes—
192Dic73#51012yes—
193#33629—731nomaximal, Sylow
194C2×C73—22no—
195C2×C73C2×#47411yescharacteristic
196C2×C73C2×#50911yescharacteristic
197C2×C73C2×#51011yescharacteristic
198D73—82no—
199C2×#500—731no—
200C4×#135—731no—
201#2227—732no—
202#2226—732no—
203#2225—732no—
204#2224—732no—
205#2159—731no—
206#2159D7311yescharacteristic, complemented
207#2150—731no—
208#2139—731no—
209#2139—731no—
210C73#3362911yescharacteristic, complemented, Sylow
211C2×#135—731no—
212C2×C4⋊Q8—1461no—
213C4×#41—1461no—
214C2×M32—731no—
215C2×M32C2×D7311yescharacteristic
216#500—732no—
217#500—732no—
218#500—1461no—
219#500—1461no—
220C4×Q16—731no—
221C4×Q16—731no—
222C4×SD16—1461no—
223C4×SD16—1461no—
224C4×D8—731no—
225C4×D8—731no—
226#427C2×D7311yescharacteristic
227C2×#107—731no—
228C2×#107—1461no—
229C2×#107C2×D7311yescharacteristic
230#416—1461no—
231C2×C4×C8—731no—
232#367—732no—
233#367C2×D7312yes—
234#366—732no—
235#366C2×D7312yes—
236C4⋊C16—732no—
237C4⋊C16—732no—
238#358—732no—
239#358—732no—
240#357—732no—
241#357—732no—
242C2×#41—1461no—
243C22×Q8—2921no—
244#135—734no—
245#135—1462no—
246C2×Q16—731no—
247C2×SD16—1461no—
248C2×D8—731no—
249C22×C8—731no—
250C22×C8C22×D7311yescharacteristic
251C4⋊Q8—2921no—
252C4⋊Q8—2921no—
253C4⋊Q8—1462no—
254C4⋊Q8—1462no—
255C4×Q8—1461no—
256C4×Q8—1461no—
257C4×D4—2921no—
258C4×D4—2921no—
259C4×D4—1461no—
260C4×D4—1461no—
261#117—21no—
262#117—2921no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—2921no—
265C2×C4⋊C4—1461no—
266C2×C42—2921no—
267C2×C42—731no—
268M32—22no—
269M32—1462no—
270C2×C16—732no—
271C2×C16C22×D7312yes—
272#107—21no—
273#107—732no—
274#107—1462no—
275#107—1462no—
276#107—1461no—
277#107C22×D7312yes—
278#107C22×D7311yescharacteristic
279#107C22×D7311yescharacteristic
280#106—21no—
281#103—1461no—
282#103—1461no—
283#102—1461no—
284#102—1461no—
285C4×C8—731no—
286C4×C8—731no—
287C4×C8—731no—
288C4×C8—731no—
289#41—2922no—
290#41—1464no—
291C2×Q8—2922no—
292C2×Q8—2922no—
293C2×Q8—5844no—
294C2×Q8—1461no—
295C2×D4—2921no—
296C2×D4—1461no—
297C22×C4—41no—
298C22×C4—2921no—
299C22×C4—2921no—
300C22×C4—2921no—
301C22×C4—2921no—
302C22×C4—731no—
303C22×C4—731no—
304C22×C4D4×D7311yescharacteristic
305Q16—732no—
306Q16—1461no—
307SD16—1464no—
308D8—732no—
309D8—1461no—
310C2×C8—731no—
311C2×C8—731no—
312C2×C8—1462no—
313C2×C8D4×D7311yescharacteristic
314C2×C8Q8×D7312yes—
315C2×C8#1192402812yes—
316C2×C8C23×D7311yescharacteristic
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—22no—
320C4⋊C4—41no—
321C4⋊C4—2922no—
322C4⋊C4—2922no—
323C4⋊C4—2921no—
324C4⋊C4—2921no—
325C4⋊C4—2921no—
326C4⋊C4—1462no—
327C22⋊C4—41no—
328C22⋊C4—2921no—
329C22⋊C4—2921no—
330C42—41no—
331C42—2921no—
332C42—2921no—
333C42—731no—
334C42—731no—
335C42—731no—
336C42—731no—
337C16—22no—
338C16—1462no—
339C23—2921no—
340C23D8×D7311yescharacteristic
341Q8—2921no—
342Q8—5844no—
343Q8—5844no—
344Q8—1462no—
345D4—2924no—
346D4—2921no—
347D4—1462no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—82no—
352C2×C4—2921no—
353C2×C4—2924no—
354C2×C4—2922no—
355C2×C4—5842no—
356C2×C4—5842no—
357C2×C4—5842no—
358C2×C4—731no—
359C2×C4—731no—
360C2×C4—731no—
361C2×C4—731no—
362C2×C4—1462no—
363C2×C4—1462no—
364C2×C4#1455656912yes—
365C2×C4D8×D7311yescharacteristic
366C2×C4#1455662312yes—
367C2×C4C2×D4×D7311yescharacteristic
368C8—1461no—
369C8—1461no—
370C8#1455656912yes—
371C8C2×Q8×D7311yescharacteristic
372C8C2×#1192402811yescharacteristic
373C22—21no—
374C22—21no—
375C22—2922no—
376C22—5841no—
377C22—5841no—
378C22#1798886612yes—
379C22C2×D8×D7311yescharacteristic
380C4—82no—
381C4—81no—
382C4—81no—
383C4—2922no—
384C4—5842no—
385C4—732no—
386C4—732no—
387C4—1461no—
388C4—1461no—
389C4#1798886612yes—
390C4C2×#1455656911yescharacteristic
391C4C2×#1455662311yescharacteristic
392C2—22no—
393C2—5842no—
394C2C2×#1798886611yescharacteristic
395C2#509×D7311yescharacteristic
396C2#510×D7311yescharacteristic
subgroupG/H
centerC22C2×D8×D73
commutator, cosocleC2×C8×C73C24
FrattiniC2×C8C23×D73
FittingC73×#2159C2
socleC22×C73C22×D8
prime radicalC2×#14556613C22
hypercenter#2159D73
hypocenterC73#33629
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2159

Complements

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

NH ≅ G/Ntype
1. C2×#17988991C2semidirect
3. #22612082C2semidirect
4. #22612077C2semidirect
5. #22612072C2semidirect
6. #22612067C2semidirect
7. #22611720C2semidirect
8. #22611686C2semidirect
11. C73×#2159C2semidirect
46. #14556613Q8semidirect
105. Dic292C2×Q8semidirect
108. D584C2×Q8semidirect
206. #2159D73semidirect
210. C73#33629semidirect

Maximal subgroups

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

subgroupindex
C2×#179889912
#226187022
#226120822
#226120772
#226120722
#226120672
#226117202
#226116862
#226116442
#226116412
C73×#21592
#3362973