Finite Group Databasebeta

← #28573974

Subgroups — #28573974

Order 15104.

656 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21435023C211yesmaximal, characteristic
2C2×#16837687C211yesmaximal, characteristic, complemented
3#21433568C211yesmaximal, characteristic
4#21423526C214yesmaximal, complemented
5#21423444C214yesmaximal, complemented
6#21422867C211yesmaximal, characteristic, complemented
7#21422780C211yesmaximal, characteristic, complemented
8#21422743C211yesmaximal, characteristic
9C59×#2982C211yesmaximal, characteristic, complemented
10C2×#14273758—21no—
11#16837952C2211yescharacteristic
12C2×#14273727C2211yescharacteristic
13#16837775C2212yes—
14#16837753C2212yes—
15#16837687—22no—
16#16837687C2214yes—
17#16837624—22no—
18#16837612C2212yes—
19#16837608—22no—
20#16837602C2212yes—
21#16837543—22no—
22C2×#14273658—21no—
23#16837426C2211yescharacteristic
24C2×C4⋊(C59⋊C8)C2211yescharacteristic
25#16837401—21no—
26C2×#14273651—21no—
27#16837153C2214yes—
28#16837149C2214yes—
29#16837147C2212yes—
30#16837145C2212yes—
31C2×C4×Q8×C59C2211yescharacteristic
32C8×Q8×C59—22no—
33C8×Q8×C59C2214yes—
34C59×#431—22no—
35C59×#431C2211yescharacteristic
36C59×#431C2211yescharacteristic
37C2×#11745685—21no—
38C22×Dic118—21no—
39#14273758—44no—
40C2×#11745672—22no—
41C2×#11745660—24no—
42C2×#11745671—22no—
43C2×#11745664—21no—
44C22×C59⋊C8—21no—
45#14273727—21no—
46#14273727—21no—
47#14273727—21no—
48#14273727C2311yescharacteristic
49#14273727C2312yes—
50#14273724C2311yescharacteristic
51#14273715—21no—
52#14273701—21no—
53#14273701—21no—
54C4×D236—21no—
55C4×D236—21no—
56C2×C4⋊Dic59—21no—
57C2×C4⋊Dic59—21no—
58#14273658—24no—
59C4⋊(C59⋊C8)—21no—
60C4⋊(C59⋊C8)—21no—
61C4⋊(C59⋊C8)C2312yes—
62C4⋊(C59⋊C8)C2312yes—
63#14273652—24no—
64#14273651—24no—
65#14273650—22no—
66#14273647—22no—
67C22×Q8×C59D411yescharacteristic
68C4×Q8×C59—21no—
69C4×Q8×C59—21no—
70C4×Q8×C59D412yes—
71C4×Q8×C59C2312yes—
72C2×C4⋊C4×C59—21no—
73C2×C4⋊C4×C59D411yescharacteristic
74C2×C42×C59—21no—
75C2×C42×C59C2311yescharacteristic
76C4⋊C8×C59—24no—
77C4⋊C8×C59D412yes—
78C4⋊C8×C59C2312yes—
79C22⋊C8×C59—21no—
80C22⋊C8×C59D412yes—
81C4×C8×C59—24no—
82C4×C8×C59—21no—
83C4×C8×C59C2312yes—
84C2×Q8×C59—21no—
85C2×Q8×C59—22no—
86C2×Q8×C59SD1614yescomplemented
87C2×Q8×C59C2×D412yes—
88C22×C4×C59—21no—
89C22×C4×C59—21no—
90C22×C4×C59C2×D411yescharacteristic
91C22×C4×C59C2×D411yescharacteristic
92C22×C4×C59#4111yescharacteristic
93C2×C22⋊D59—41no—
94C22×Dic59—41no—
95#11745685—42no—
96#11745685—42no—
97C2×D236—21no—
98C2×D236—21no—
99C2×D236—21no—
100C2×C4×D59—41no—
101C2×Dic118—21no—
102C2×Dic118—22no—
103C2×Dic118—41no—
104C2×Dic118—82no—
105C2×C8×C59—22no—
106C2×C8×C59C2×D414yes—
107C4⋊C4×C59—22no—
108C4⋊C4×C59—21no—
109C4⋊C4×C59—22no—
110C4⋊C4×C59C2×D412yes—
111C42×C59—22no—
112C42×C59—22no—
113C42×C59C2×D411yescharacteristic
114C42×C59C2×D411yescharacteristic
115C42×C59#4111yescharacteristic
116C42×C59C2411yescharacteristic
117#11745672—44no—
118#11745671—44no—
119(C2×D59)⋊C4—41no—
120(C2×D59)⋊C4—41no—
121C4⋊Dic59—22no—
122C4⋊Dic59—21no—
123C4⋊Dic59—22no—
124C4⋊Dic59—21no—
125C4⋊Dic59—41no—
126C4⋊Dic59—41no—
127#11745664—42no—
128C2×C59⋊C8—22no—
129C2×C59⋊C8—22no—
130C2×C59⋊C8—42no—
131#11745660—84no—
132C23×C59#12111yescharacteristic
133C22×D59—41no—
134Q8×C59—22no—
135Q8×C59—24no—
136Q8×C59C2×SD1614yes—
137C2×C4×C59—22no—
138C2×C4×C59—24no—
139C2×C4×C59—22no—
140C2×C4×C59—22no—
141C2×C4×C59—41no—
142C2×C4×C59—81no—
143C2×C4×C59#12111yescharacteristic
144C2×C4×C59#12111yescharacteristic
145C2×C4×C59#12111yescharacteristic
146C2×C4×C59C2×SD1612yes—
147C2×C4×C59#13512yes—
148C2×C4×C59C22×D411yescharacteristic
149C2×C4×C59C22×D411yescharacteristic
150C2×C4×C59C2×#4111yescharacteristic
151C22⋊D59—82no—
152C2×Dic59—42no—
153C2×Dic59—41no—
154C2×Dic59—82no—
155D236—41no—
156D236—41no—
157D236—42no—
158D236—81no—
159C4×D59—82no—
160Dic118—41no—
161Dic118—41no—
162Dic118—44no—
163Dic118—82no—
164C8×C59—24no—
165C8×C59—42no—
166C59⋊C8—42no—
167C59⋊C8—42no—
168#19397—591nomaximal, Sylow
169C22×C59—21no—
170C22×C59—21no—
171C22×C59#46512yes—
172C22×C59C2×#12111yescharacteristic
173C2×D59—81no—
174C2×D59—81no—
175C2×D59—81no—
176C4×C59—21no—
177C4×C59—24no—
178C4×C59—21no—
179C4×C59—21no—
180C4×C59—42no—
181C4×C59#46512yes—
182C4×C59C22×SD1611yescharacteristic
183C4×C59C2×#13511yescharacteristic
184C4×C59#57712yes—
185Dic59—82no—
186Dic59—81no—
187#3206—591no—
188C2×#475—591no—
189#3037—591no—
190#2982D5911yescharacteristic, complemented
191#1680—594no—
192#1666—594no—
193#1570—591no—
194#1549—591no—
195#1540—591no—
196C2×C59—22no—
197C2×C59#304311yescharacteristic
198C2×C59C2×#46511yescharacteristic
199C2×C59#306911yescharacteristic
200D59—161no—
201C2×#137—1181no—
202#532—591no—
203C2×C4⋊Q8—591no—
204C2×C4×Q8C2×D5911yescharacteristic
205#492—592no—
206#487—592no—
207#475—594no—
208#475—1182no—
209#464—1182no—
210#462—592no—
211#461—1182no—
212#460—592no—
213#448—1182no—
214C8×Q8—22no—
215C8×Q8C2×D5914yes—
216#431—22no—
217#431C2×D5911yescharacteristic
218#431C2×D5911yescharacteristic
219C2×#106—1181no—
220#423—591no—
221C2×C4⋊C8—591no—
222#417—1181no—
223C2×#103—1181no—
224#334—594no—
225#332—592no—
226#332—594no—
227#331—592no—
228C59#1939711yescharacteristic, complemented, Sylow
229C2×#41—1181no—
230C22×Q8—1181no—
231C22×Q8D23611yescharacteristic
232#137—2364no—
233C2×Q16—1182no—
234C2×SD16—1182no—
235C2×SD16—1184no—
236C2×M16—1181no—
237C22×C8—1181no—
238C4⋊Q8—592no—
239C4⋊Q8—591no—
240C4⋊Q8—1181no—
241C4⋊Q8—1181no—
242C4⋊Q8—1181no—
243#127—591no—
244#124—1181no—
245#121—1181no—
246#121—1181no—
247C4×Q8—21no—
248C4×Q8—21no—
249C4×Q8D23612yes—
250C4×Q8C22×D5912yes—
251C4×D4—1181no—
252C4×D4—1181no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—1181no—
255C2×C4⋊C4—1181no—
256C2×C4⋊C4D23611yescharacteristic
257C2×C42—21no—
258C2×C42C22×D5911yescharacteristic
259#106—1184no—
260C4⋊C8—24no—
261C4⋊C8—592no—
262C4⋊C8—592no—
263C4⋊C8—1181no—
264C4⋊C8—1181no—
265C4⋊C8C22⋊D5912yes—
266C4⋊C8C22×D5912yes—
267#103—1184no—
268#103—1184no—
269#103—1182no—
270#102—1182no—
271C22⋊C8—21no—
272C22⋊C8C22⋊D5912yes—
273C4×C8—24no—
274C4×C8—21no—
275C4×C8C22×D5912yes—
276#41—2362no—
277#41—2362no—
278C2×Q8—21no—
279C2×Q8—22no—
280C2×Q8—1182no—
281C2×Q8—1181no—
282C2×Q8—4722no—
283C2×Q8—2361no—
284C2×Q8#1174566014yescomplemented
285C2×Q8C2×D23612yes—
286C2×D4—1181no—
287C2×D4—1181no—
288C2×D4—1181no—
289C2×D4—2361no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—2361no—
293C22×C4—2361no—
294C22×C4C2×D23611yescharacteristic
295C22×C4#1174568511yescharacteristic
296C22×C4C2×C22⋊D5911yescharacteristic
297Q16—2364no—
298SD16—4724no—
299SD16—2364no—
300M16—2362no—
301C2×C8—22no—
302C2×C8—1182no—
303C2×C8—1182no—
304C2×C8—2362no—
305C2×C8C2×C22⋊D5914yes—
306C4⋊C4—22no—
307C4⋊C4—21no—
308C4⋊C4—22no—
309C4⋊C4—1181no—
310C4⋊C4—1182no—
311C4⋊C4—1182no—
312C4⋊C4—1181no—
313C4⋊C4—2361no—
314C4⋊C4—2361no—
315C4⋊C4C2×D23612yes—
316C22⋊C4—2361no—
317C22⋊C4—2361no—
318C42—22no—
319C42—22no—
320C42C2×D23611yescharacteristic
321C42#1174568511yescharacteristic
322C42C2×C22⋊D5911yescharacteristic
323C42C23×D5911yescharacteristic
324C23—2361no—
325C23#1427370111yescharacteristic
326Q8—22no—
327Q8—24no—
328Q8—4722no—
329Q8—2361no—
330Q8—2361no—
331Q8—2364no—
332Q8C2×#1174566014yes—
333D4—4721no—
334D4—4722no—
335D4—2361no—
336D4—2361no—
337D4—2362no—
338C2×C4—22no—
339C2×C4—24no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—41no—
343C2×C4—81no—
344C2×C4—4722no—
345C2×C4—4722no—
346C2×C4—2361no—
347C2×C4—2362no—
348C2×C4#1427370111yescharacteristic
349C2×C4#1427370111yescharacteristic
350C2×C4#1427370111yescharacteristic
351C2×C4C2×#1174566012yes—
352C2×C4#1427374912yes—
353C2×C4C22×C22⋊D5911yescharacteristic
354C2×C4C22×D23611yescharacteristic
355C2×C4C2×#1174568511yescharacteristic
356C8—24no—
357C8—42no—
358C8—2362no—
359C8—2362no—
360C22—21no—
361C22—21no—
362C22—4721no—
363C22—4721no—
364C22—4721no—
365C22#1683763112yes—
366C22C2×#1427370111yescharacteristic
367C4—21no—
368C4—24no—
369C4—21no—
370C4—21no—
371C4—42no—
372C4—4722no—
373C4—4721no—
374C4#1683763112yes—
375C4C22×#1174566011yescharacteristic
376C4C2×#1427374911yescharacteristic
377C4#1683821012yes—
378C2—22no—
379C2—9441no—
380C2#2143361711yescharacteristic
381C2C2×#1683763111yescharacteristic
382C2#2143383311yescharacteristic
subgroupG/H
centerC22C2×#14273701
commutator, cosocleC42×C59C24
FrattiniC42C23×D59
FittingC59×#2982C2
socleC22×C59C2×#121
prime radical#16837952C22
hypercenter#2982D59
hypocenterC59#19397
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2982

Complements

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

NH ≅ G/Ntype
2. C2×#16837687C2semidirect
4. #21423526C2semidirect
5. #21423444C2semidirect
6. #21422867C2semidirect
7. #21422780C2semidirect
9. C59×#2982C2semidirect
86. C2×Q8×C59SD16semidirect
190. #2982D59semidirect
228. C59#19397semidirect
284. C2×Q8#11745660semidirect

Maximal subgroups

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

subgroupindex
#214350232
C2×#168376872
#214335682
#214235262
#214234442
#214228672
#214227802
#214227432
C59×#29822
#1939759