Finite Group Databasebeta

← #28617814

Subgroups — #28617814

Order 15104.

766 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#21432686C211yesmaximal, characteristic
2#21432685C211yesmaximal, characteristic
3C2×#16837438C211yesmaximal, characteristic
4C2×#16837430C211yesmaximal, characteristic
5#21425610C214yesmaximal, complemented
6#21425605C214yesmaximal, complemented
7#21423834C211yesmaximal, characteristic
8C2×#16837166C211yesmaximal, characteristic
9C59×#1742C211yesmaximal, characteristic
10C22×#11745664C2211yescharacteristic
11C23×C59⋊C8C2211yescharacteristic
12C2×#14273686C2211yescharacteristic
13C2×#14273686C2211yescharacteristic
14#16837445—22no—
15#16837444—22no—
16#16837444C2214yes—
17#16837442C2214yes—
18#16837440—22no—
19#16837438—22no—
20#16837438—22no—
21#16837438—22no—
22C2×#14273660C2212yes—
23C2×#14273658C2212yes—
24#16837430—22no—
25#16837430C2212yes—
26#16837430C2212yes—
27C2×Dic59⋊C8—22no—
28#16837170C2214yes—
29#16837166C2212yes—
30#16837166C2212yes—
31C22×M16×C59C2211yescharacteristic
32C22×M16×C59C2211yescharacteristic
33C2×#117×C59C2211yescharacteristic
34C59×#343C2214yes—
35C2×#11745664—21no—
36C2×#11745664—22no—
37C2×#11745664Q814yes—
38C2×#11745664C2312yes—
39C22×C59⋊C8—21no—
40C22×C59⋊C8—22no—
41C22×C59⋊C8—22no—
42C22×C59⋊C8D412yes—
43C22×C59⋊C8D412yes—
44C22×C59⋊C8C2311yescharacteristic
45C22×C59⋊C8C2311yescharacteristic
46#14273686—21no—
47#14273686—21no—
48#14273686—21no—
49#14273686—21no—
50#14273686—21no—
51#14273686—21no—
52#14273686D412yes—
53#14273686C2312yes—
54C2×C4⋊Dic59D412yes—
55C2×C4⋊Dic59C2312yes—
56C2×C22⋊Dic59—21no—
57C2×C22⋊Dic59—21no—
58C2×C4×Dic59—21no—
59C2×C4×Dic59—21no—
60#14273661—24no—
61#14273660—24no—
62#14273660—24no—
63#14273659—24no—
64#14273658—24no—
65#14273658—24no—
66Dic59⋊C8—24no—
67Dic59⋊C8—24no—
68C23×C4×C59C2311yescharacteristic
69C2×M16×C59—21no—
70C2×M16×C59—22no—
71C2×M16×C59—21no—
72C2×M16×C59—22no—
73C2×M16×C59—22no—
74C2×M16×C59Q814yes—
75C2×M16×C59C2312yes—
76C2×M16×C59C2312yes—
77C22×C8×C59—21no—
78C22×C8×C59—21no—
79#117×C59—21no—
80#117×C59—21no—
81#117×C59—21no—
82#117×C59—21no—
83C2×C4⋊C4×C59C2312yes—
84C2×C22⋊C4×C59—21no—
85C2×C42×C59—21no—
86C24×C59#4111yescharacteristic
87C22×C4×C59—22no—
88C22×C4×C59C2×D411yescharacteristic
89C22×C4×C59C2×D411yescharacteristic
90C22×C4×C59C2×D411yescharacteristic
91C22×C4×C59C2×D411yescharacteristic
92C22×C4×C59C2×Q812yes—
93C22×C4×C59C2×Q812yes—
94C22×C4×C59#4111yescharacteristic
95C22×C4×C59#4112yes—
96C22×C4×C59#4112yes—
97C22×C4×C59C2411yescharacteristic
98C22×Dic59—22no—
99C22×Dic59—22no—
100M16×C59—24no—
101M16×C59—22no—
102M16×C59—22no—
103M16×C59—22no—
104M16×C59—42no—
105M16×C59C2×Q814yes—
106C2×C8×C59—24no—
107C2×C8×C59—22no—
108C2×C8×C59—22no—
109C2×C8×C59—42no—
110C4⋊C4×C59—22no—
111C4⋊C4×C59—22no—
112C22⋊C4×C59—41no—
113C22⋊C4×C59—41no—
114C42×C59—41no—
115C42×C59—41no—
116C22⋊Dic59—22no—
117C22⋊Dic59—22no—
118C22⋊Dic59—41no—
119C22⋊Dic59—41no—
120C4⋊Dic59—22no—
121C4⋊Dic59—22no—
122C4⋊Dic59—22no—
123C4⋊Dic59C2×D414yes—
124C4×Dic59—22no—
125C4×Dic59—22no—
126C4×Dic59—41no—
127C4×Dic59—41no—
128#11745664—24no—
129#11745664—22no—
130#11745664C2×Q814yes—
131C2×C59⋊C8—24no—
132C2×C59⋊C8—24no—
133C2×C59⋊C8—24no—
134C2×C59⋊C8—22no—
135C2×C59⋊C8—21no—
136C2×C59⋊C8—21no—
137C2×C59⋊C8—21no—
138C2×C59⋊C8—21no—
139C2×C59⋊C8C2×D412yes—
140C2×C59⋊C8C2×D412yes—
141C23×C59—42no—
142C23×C59#12111yescharacteristic
143C23×C59#12111yescharacteristic
144C23×C59#12512yes—
145C23×C59#12512yes—
146C23×C59C2×#4111yescharacteristic
147C2×C4×C59—42no—
148C2×C4×C59—82no—
149C2×C4×C59#12111yescharacteristic
150C2×C4×C59#12111yescharacteristic
151C2×C4×C59C22⋊Q812yes—
152C2×C4×C59C22⋊Q812yes—
153C2×C4×C59C22⋊Q812yes—
154C2×C4×C59C22⋊Q812yes—
155C2×C4×C59#12312yes—
156C2×C4×C59#12312yes—
157C2×C4×C59#12512yes—
158C2×C4×C59#12512yes—
159C2×C4×C59C4⋊Q812yes—
160C2×C4×C59C4⋊Q812yes—
161C2×C4×C59C22×D411yescharacteristic
162C2×C4×C59C22×D411yescharacteristic
163C2×C4×C59C22×Q811yescharacteristic
164C2×C4×C59C22×Q811yescharacteristic
165C2×C4×C59C2×#4111yescharacteristic
166C2×C4×C59C2×#4111yescharacteristic
167C2×Dic59—24no—
168C2×Dic59—44no—
169C2×Dic59—42no—
170C2×Dic59—82no—
171C8×C59—24no—
172C8×C59—42no—
173C59⋊C8—24no—
174C59⋊C8—22no—
175C59⋊C8—22no—
176#23674—591nomaximal, Sylow
177C22×C59—22no—
178C22×C59—22no—
179C22×C59—22no—
180C22×C59—42no—
181C22×C59—42no—
182C22×C59#39812yes—
183C22×C59#39812yes—
184C22×C59C2×#12111yescharacteristic
185C22×C59C2×#12511yescharacteristic
186C22×C59C2×#12511yescharacteristic
187C4×C59—82no—
188C4×C59#39812yes—
189C4×C59#39812yes—
190C4×C59C2×C22⋊Q811yescharacteristic
191C4×C59C2×C22⋊Q811yescharacteristic
192C4×C59C2×#12311yescharacteristic
193C4×C59C2×C4⋊Q811yescharacteristic
194Dic59—44no—
195Dic59—82no—
196#2911—591no—
197#2911—591no—
198C2×#427—591no—
199C2×#424—591no—
200#2090—594no—
201#2089—594no—
202#1742—591no—
203#1742D5911yescharacteristic
204C2×#341—591no—
205C2×C59—21no—
206C2×C59—21no—
207C2×C59—22no—
208C2×C59—22no—
209C2×C59C2×#39811yescharacteristic
210C2×C59#306712yes—
211C22×M16—591no—
212C22×M16C2×D5911yescharacteristic
213C22×M16C2×D5911yescharacteristic
214C23×C8—591no—
215C2×#117—591no—
216C2×#117—591no—
217C2×#117C2×D5911yescharacteristic
218#428—594no—
219#428—594no—
220#428—1182no—
221#428—1182no—
222#427—1182no—
223#427—1182no—
224#427—1182no—
225#427—1182no—
226C2×#107—592no—
227C2×#106—592no—
228#424—592no—
229#424—592no—
230#424—1182no—
231C2×C4⋊C8—1182no—
232#343—594no—
233#343C2×D5914yes—
234#341—592no—
235#341—592no—
236C59#2367411yescharacteristic, complemented, Sylow
237C23×C4C22×D5911yescharacteristic
238C2×M16—21no—
239C2×M16—22no—
240C2×M16—21no—
241C2×M16—22no—
242C2×M16—22no—
243C2×M16—594no—
244C2×M16—592no—
245C2×M16—1182no—
246C2×M16—1181no—
247C2×M16Dic11814yes—
248C2×M16C22×D5912yes—
249C2×M16C22×D5912yes—
250C22×C8—21no—
251C22×C8—21no—
252C22×C8—592no—
253C22×C8—592no—
254C22×C8—591no—
255C22×C8—591no—
256C22×C8—1182no—
257C22×C8—1181no—
258C22×C8—1182no—
259#117—21no—
260#117—21no—
261#117—21no—
262#117—21no—
263#117—592no—
264#117—592no—
265#117—1181no—
266#117—1181no—
267#117—1181no—
268#117—1181no—
269#117—1181no—
270#117—1181no—
271C2×C4⋊C4—592no—
272C2×C4⋊C4—592no—
273C2×C4⋊C4C22×D5912yes—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—1181no—
276C2×C22⋊C4—1181no—
277C2×C42—21no—
278C2×C42—1181no—
279C2×C42—1181no—
280#107—1184no—
281#107—1184no—
282#107—1184no—
283#106—1184no—
284#106—1184no—
285#106—1184no—
286C4⋊C8—1184no—
287C4⋊C8—1184no—
288C24#1174568511yescharacteristic
289C22×C4—22no—
290C22×C4—1182no—
291C22×C4—1182no—
292C22×C4C2×Dic11812yes—
293C22×C4#1174568511yescharacteristic
294C22×C4#1174568512yes—
295C22×C4D4×D5911yescharacteristic
296C22×C4D4×D5911yescharacteristic
297C22×C4D4×D5911yescharacteristic
298C22×C4D4×D5911yescharacteristic
299C22×C4#1174568712yes—
300C22×C4Q8×D5912yes—
301C22×C4C23×D5911yescharacteristic
302M16—24no—
303M16—22no—
304M16—22no—
305M16—22no—
306M16—42no—
307M16—594no—
308M16—1184no—
309M16—1182no—
310M16C2×Dic11814yes—
311C2×C8—24no—
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—42no—
315C2×C8—592no—
316C2×C8—592no—
317C2×C8—1184no—
318C2×C8—1184no—
319C2×C8—1184no—
320C2×C8—1182no—
321C2×C8—1181no—
322C2×C8—1181no—
323C2×C8—1181no—
324C2×C8—1181no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—594no—
328C4⋊C4—1182no—
329C4⋊C4—1182no—
330C4⋊C4—1182no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—1182no—
334C22⋊C4—1182no—
335C22⋊C4—2361no—
336C22⋊C4—2361no—
337C42—41no—
338C42—41no—
339C42—1182no—
340C42—1182no—
341C42—2361no—
342C42—2361no—
343C23—42no—
344C23#1427370011yescharacteristic
345C23#1427370011yescharacteristic
346C23#1427371712yes—
347C23#1427371912yes—
348C23C2×#1174568511yescharacteristic
349C2×C4—42no—
350C2×C4—82no—
351C2×C4—1184no—
352C2×C4—4722no—
353C2×C4—2362no—
354C2×C4—2364no—
355C2×C4#1427370011yescharacteristic
356C2×C4#1427370011yescharacteristic
357C2×C4#1427370412yes—
358C2×C4#1427370412yes—
359C2×C4#1427370512yes—
360C2×C4#1427370512yes—
361C2×C4#1427370912yes—
362C2×C4#1427370912yes—
363C2×C4#1427371712yes—
364C2×C4#1427371912yes—
365C2×C4#1427372612yes—
366C2×C4#1427372612yes—
367C2×C4C2×D4×D5911yescharacteristic
368C2×C4C2×D4×D5911yescharacteristic
369C2×C4C2×Q8×D5911yescharacteristic
370C2×C4C22×Dic11811yescharacteristic
371C2×C4C2×#1174568711yescharacteristic
372C2×C4C2×#1174568511yescharacteristic
373C8—24no—
374C8—42no—
375C8—1184no—
376C8—1182no—
377C8—1182no—
378C22—22no—
379C22—22no—
380C22—22no—
381C22—42no—
382C22—42no—
383C22#1683733612yes—
384C22#1683733612yes—
385C22C2×#1427370011yescharacteristic
386C22C2×#1427371711yescharacteristic
387C22C2×#1427371911yescharacteristic
388C4—82no—
389C4—4722no—
390C4—2364no—
391C4#1683733612yes—
392C4#1683733612yes—
393C4C2×#1427370411yescharacteristic
394C4C2×#1427370511yescharacteristic
395C4C2×#1427370911yescharacteristic
396C4C2×#1427372611yescharacteristic
397C2—21no—
398C2—21no—
399C2—22no—
400C2—22no—
401C2C2×#1683733611yescharacteristic
402C2#2143381512yes—
subgroupG/H
centerC2×C4#14273700
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#1742C2
socleC22×C59C2×#121
prime radicalC24×C59#41
hypercenter#1742D59
hypocenterC59#23674
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1742

Complements

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

NH ≅ G/Ntype
5. #21425610C2semidirect
6. #21425605C2semidirect
236. C59#23674semidirect

Maximal subgroups

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

subgroupindex
#214326862
#214326852
C2×#168374382
C2×#168374302
#214256102
#214256052
#214238342
C2×#168371662
C59×#17422
#2367459