Finite Group Databasebeta

← #28508032

Subgroups — #28508032

Order 15104.

566 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837777C211yesmaximal, characteristic
2C2×#16837204C211yesmaximal, characteristic, complemented
3C2×#16837203C211yesmaximal, characteristic, complemented
4C2×#16837202C211yesmaximal, characteristic, complemented
5#21424830C211yesmaximal, characteristic, complemented
6#21422173C211yesmaximal, characteristic, complemented
7C2×C59×#357C211yesmaximal, characteristic, complemented
8C22×D472C2211yescharacteristic
9C22×#11745669—21no—
10C2×#14273723—21no—
11C22×C59⋊C16C2211yescharacteristic
12#16837777—22no—
13#16837777—22no—
14C2×#14273661C2211yescharacteristic
15C2×C8×Dic59C2211yescharacteristic
16#16837301—21no—
17#16837204—22no—
18#16837203—22no—
19#16837202—22no—
20#16837164—21no—
21C22×D8×C59C2211yescharacteristic
22C22×C16×C59C2211yescharacteristic
23C2×#107×C59C2211yescharacteristic
24C59×#357—22no—
25C22×D236—21no—
26C22×C22⋊D59—41no—
27C22×C22⋊D59—41no—
28C2×D472—21no—
29C2×D472—21no—
30C2×D472—21no—
31C2×D472—21no—
32C2×D472—21no—
33C2×D472D412yes—
34C2×#11745669—24no—
35C2×#11745669—42no—
36C2×#11745669—42no—
37C22×C59⋊C8D411yescharacteristic
38#14273723—24no—
39#14273723—42no—
40C2×C4⋊Dic59—21no—
41C2×C4⋊Dic59—41no—
42C2×C4×Dic59D411yescharacteristic
43C2×C59⋊C16—21no—
44C2×C59⋊C16—21no—
45C2×C59⋊C16—21no—
46#14273661—21no—
47#14273661—21no—
48C8×Dic59—21no—
49C8×Dic59—21no—
50C59⋊#95—41no—
51C22×D4×C59—21no—
52C2×D8×C59—21no—
53C2×D8×C59—21no—
54C2×D8×C59—21no—
55C2×D8×C59—21no—
56C2×D8×C59—21no—
57C2×D8×C59D412yes—
58C22×C8×C59C2311yescharacteristic
59C2×C4⋊C4×C59—21no—
60C2×C16×C59—21no—
61C2×C16×C59—21no—
62C2×C16×C59—21no—
63#107×C59—21no—
64#107×C59D412yescomplemented
65C24×C59—41no—
66C23×D59—41no—
67C2×D4×C59—22no—
68C2×D4×C59—41no—
69C2×D4×C59—41no—
70C2×D4×C59—41no—
71C2×D4×C59—41no—
72C2×D4×C59—41no—
73C22×C4×C59—41no—
74C22×C4×C59C2×D411yescharacteristic
75C2×C22⋊D59—44no—
76C2×C22⋊D59—44no—
77C2×C22⋊D59—82no—
78C2×C22⋊D59—82no—
79C2×C22⋊D59—82no—
80C2×C22⋊D59—82no—
81C22×Dic59—41no—
82C22×Dic59D811yescharacteristic
83C22×Dic59D811yescharacteristic
84C2×D236—22no—
85C2×D236—41no—
86C2×D236—41no—
87C2×D236—41no—
88C2×D236—41no—
89C2×D236—41no—
90D8×C59—22no—
91D8×C59—22no—
92D8×C59—41no—
93D8×C59—41no—
94C2×C8×C59C2×D411yescharacteristic
95C2×C8×C59C2×D411yescharacteristic
96C2×C8×C59C2×D411yescharacteristic
97C2×C8×C59#4111yescharacteristic
98C2×C8×C59#4111yescharacteristic
99C2×C8×C59#4111yescharacteristic
100C4⋊C4×C59—22no—
101C4⋊C4×C59—41no—
102#11745669—48no—
103C4⋊Dic59—41no—
104C4⋊Dic59—41no—
105C4⋊Dic59—82no—
106C4×Dic59—21no—
107C4×Dic59—21no—
108C2×C59⋊C8—21no—
109C2×C59⋊C8—21no—
110C2×C59⋊C8SD1612yes—
111D472—22no—
112D472—22no—
113D472—41no—
114D472—41no—
115C16×C59—41no—
116C59⋊C16—41no—
117C23×C59—42no—
118C23×C59—81no—
119C23×C59—81no—
120C23×C59—81no—
121C23×C59—81no—
122C23×C59—81no—
123C23×C59—81no—
124C23×C59C2×D811yescharacteristic
125C22×D59—42no—
126C22×D59—81no—
127C22×D59—81no—
128C22×D59—81no—
129C22×D59—81no—
130C22×D59—81no—
131C22×D59—81no—
132D4×C59—42no—
133D4×C59—42no—
134D4×C59—81no—
135D4×C59—81no—
136C2×C4×C59—42no—
137C2×C4×C59—81no—
138C2×C4×C59—81no—
139C2×C4×C59#12011yescharacteristic
140C2×C4×C59#12111yescharacteristic
141C2×C4×C59#12411yescharacteristic
142C2×C4×C59C2×SD1611yescharacteristic
143C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
144C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
145C22⋊D59—88no—
146C22⋊D59—88no—
147C2×Dic59—21no—
148C2×Dic59—21no—
149C2×Dic59—21no—
150C2×Dic59—21no—
151C2×Dic59—81no—
152C2×Dic59—81no—
153C2×Dic59—81no—
154C2×Dic59D1612yes—
155C2×Dic59SD3212yes—
156D236—42no—
157D236—42no—
158D236—81no—
159D236—81no—
160C8×C59#12111yescharacteristic
161C8×C59#12111yescharacteristic
162C8×C59#12311yescharacteristic
163C8×C59#12311yescharacteristic
164C59⋊C8—22no—
165#13407—591nomaximal, Sylow
166C22×C59—82no—
167C22×C59—82no—
168C22×C59—82no—
169C22×C59—161no—
170C22×C59—161no—
171C22×C59—161no—
172C22×C59—161no—
173C22×C59#44711yescharacteristic
174C22×C59#45911yescharacteristic
175C22×C59#48611yescharacteristic
176C22×C59C2×D1611yescharacteristic
177C22×C59C2×SD3211yescharacteristic
178C22×C59#50911yescharacteristic
179C22×C59#50911yescharacteristic
180C2×D59—82no—
181C2×D59—82no—
182C2×D59—82no—
183C2×D59—161no—
184C2×D59—161no—
185C2×D59—161no—
186C2×D59—161no—
187C4×C59—82no—
188C4×C59#39411yescharacteristic
189C4×C59#45011yescharacteristic
190C4×C59#46011yescharacteristic
191C4×C59#49011yescharacteristic
192Dic59—22no—
193Dic59—22no—
194Dic59—161no—
195C2×#493—591no—
196C2×#357—591no—
197C2×#357—591no—
198C2×#357—591no—
199C2×#357D5911yescharacteristic, complemented
200#1942—591no—
201#1374—591no—
202C2×C59—162no—
203C2×C59#204111yescharacteristic
204C2×C59#218411yescharacteristic
205C2×C59#218711yescharacteristic
206C2×C59#220611yescharacteristic
207C2×C59#220711yescharacteristic
208C2×C59#224011yescharacteristic
209C2×C59#224411yescharacteristic
210D59—162no—
211C22×D8—591no—
212C22×D8—1181no—
213C22×D8C2×D5911yescharacteristic
214C2×#127—1181no—
215C22×C16—591no—
216C22×C16C2×D5911yescharacteristic
217#493—1182no—
218#493—1182no—
219C2×#107—591no—
220C2×#107C2×D5911yescharacteristic
221C2×C4×C8—591no—
222#389—1181no—
223#357—22no—
224#357—1182no—
225#357—1182no—
226#357—1182no—
227#340—1181no—
228C59#1340711yescharacteristic, complemented, Sylow
229C22×D4—21no—
230C22×D4—1181no—
231C22×D4—2361no—
232C22×D4—2361no—
233C2×D8—21no—
234C2×D8—21no—
235C2×D8—21no—
236C2×D8—21no—
237C2×D8—21no—
238C2×D8—592no—
239C2×D8—1184no—
240C2×D8—1181no—
241C2×D8—1181no—
242C2×D8—1181no—
243C2×D8—1181no—
244C2×D8—1181no—
245C2×D8—2362no—
246C2×D8—2362no—
247C2×D8D23612yes—
248C22×C8—591no—
249C22×C8C22×D5911yescharacteristic
250#127—1184no—
251#127—2362no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—1181no—
254C2×C4⋊C4—2361no—
255C2×C42—591no—
256C2×C16—21no—
257C2×C16—21no—
258C2×C16—21no—
259C2×C16—1181no—
260C2×C16—1181no—
261C2×C16—1181no—
262#107—21no—
263#107—1181no—
264#107—1181no—
265#107C22⋊D5912yescomplemented
266C4×C8—1181no—
267C4×C8—1181no—
268#95—2361no—
269C24—41no—
270C24—2361no—
271C2×D4—22no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—1182no—
278C2×D4—4722no—
279C2×D4—4722no—
280C2×D4—4722no—
281C2×D4—4722no—
282C2×D4—2364no—
283C2×D4—2364no—
284C2×D4—2361no—
285C2×D4—2361no—
286C2×D4—2361no—
287C2×D4—2361no—
288C2×D4—2361no—
289C22×C4—41no—
290C22×C4—591no—
291C22×C4—591no—
292C22×C4—2361no—
293C22×C4D4×D5911yescharacteristic
294D8—22no—
295D8—22no—
296D8—41no—
297D8—41no—
298D8—1182no—
299D8—1182no—
300D8—2368no—
301D8—2361no—
302D8—2361no—
303C2×C8—592no—
304C2×C8—1181no—
305C2×C8—1181no—
306C2×C8C2×D23611yescharacteristic
307C2×C8#1174568511yescharacteristic
308C2×C8D4×D5911yescharacteristic
309C2×C8#1174568711yescharacteristic
310C2×C8#1174568711yescharacteristic
311C2×C8C2×C22⋊D5911yescharacteristic
312C4⋊C4—22no—
313C4⋊C4—41no—
314C4⋊C4—4722no—
315C4⋊C4—2361no—
316C4⋊C4—2361no—
317C42—1181no—
318C42—1181no—
319C16—41no—
320C16—2361no—
321C23—42no—
322C23—81no—
323C23—81no—
324C23—81no—
325C23—81no—
326C23—81no—
327C23—81no—
328C23—4721no—
329C23—4721no—
330C23—4721no—
331C23—4721no—
332C23—4721no—
333C23—4721no—
334C23—2362no—
335C23D8×D5911yescharacteristic
336D4—42no—
337D4—42no—
338D4—81no—
339D4—81no—
340D4—4728no—
341D4—4728no—
342D4—4721no—
343D4—4721no—
344D4—2362no—
345D4—2362no—
346C2×C4—42no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—592no—
350C2×C4—592no—
351C2×C4—1181no—
352C2×C4—1181no—
353C2×C4—1181no—
354C2×C4—1181no—
355C2×C4—4721no—
356C2×C4—4721no—
357C2×C4—4721no—
358C2×C4#1427369711yescharacteristic
359C2×C4#1427369911yescharacteristic
360C2×C4#1427371311yescharacteristic
361C2×C4SD16×D5911yescharacteristic
362C2×C4#1427375111yescharacteristic
363C2×C4#1427375411yescharacteristic
364C8—1182no—
365C8#1427369811yescharacteristic
366C8#1427370111yescharacteristic
367C8#1427370911yescharacteristic
368C8#1427371111yescharacteristic
369C22—82no—
370C22—82no—
371C22—82no—
372C22—161no—
373C22—161no—
374C22—161no—
375C22—161no—
376C22—9441no—
377C22—9441no—
378C22—9441no—
379C22—9441no—
380C22—4722no—
381C22—4722no—
382C22—4722no—
383C22#1683753711yescharacteristic
384C22#1683759511yescharacteristic
385C22#1683774711yescharacteristic
386C22D16×D5911yescharacteristic
387C22SD32×D5911yescharacteristic
388C22#1683783611yescharacteristic
389C22#1683783911yescharacteristic
390C4—82no—
391C4—1182no—
392C4—1182no—
393C4—9441no—
394C4#1683731811yescharacteristic
395C4#1683755411yescharacteristic
396C4#1683760011yescharacteristic
397C4#1683776511yescharacteristic
398C2—162no—
399C2—9442no—
400C2#2142532911yescharacteristic
401C2#2142600911yescharacteristic
402C2#2142602611yescharacteristic
403C2#2142612911yescharacteristic
404C2#2142613411yescharacteristic
405C2#2142631811yescharacteristic
406C2#2142633611yescharacteristic
subgroupG/H
centerC23D8×D59
commutator, cosocleC22×C8×C59C23
FrattiniC22×C8C22×D59
FittingC2×C59×#357C2
socleC23×C59C2×D8
prime radicalC2×#16837777C2
hypercenterC2×#357D59
hypocenterC59#13407
LCSG ▷ C22×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ C2×#357

Complements

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

NH ≅ G/Ntype
2. C2×#16837204C2semidirect
3. C2×#16837203C2semidirect
4. C2×#16837202C2semidirect
5. #21424830C2semidirect
6. #21422173C2semidirect
7. C2×C59×#357C2semidirect
64. #107×C59D4semidirect
199. C2×#357D59semidirect
228. C59#13407semidirect
265. #107C22⋊D59semidirect

Maximal subgroups

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

subgroupindex
C2×#168377772
C2×#168372042
C2×#168372032
C2×#168372022
#214248302
#214221732
C2×C59×#3572
#1340759