Finite Group Databasebeta

← #28507010

Subgroups — #28507010

Order 15104.

540 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 103 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21429095C211yesmaximal, characteristic
2C2×#16837198C214yesmaximal, complemented
3C4×#14273646C211yesmaximal, characteristic, complemented
4C59×#1801C211yesmaximal, characteristic, complemented
5C2×#14273724C411yescharacteristic
6C2×#14273716C2212yes—
7C2×#14273646C2212yes—
8C4×#11745664—21no—
9#16837319—21no—
10#16837319—21no—
11#16837304—21no—
12#16837304—21no—
13#16837303C412yes—
14C4×C4⋊Dic59C411yescharacteristic
15#16837198—28no—
16#16837171—21no—
17C2×C4×Q8×C59C2211yescharacteristic
18C59×#423—21no—
19C2×#101×C59C2212yes—
20C59×#343—21no—
21C22×D236—21no—
22C22×D236—21no—
23C22×D236C2×C412yes—
24C2×#11745664—22no—
25#14273724—41no—
26#14273724—41no—
27#14273716—24no—
28#14273716—24no—
29C2×C4⋊Dic59—21no—
30C2×(C2×D59)⋊C4—22no—
31C2×(C2×D59)⋊C4—22no—
32C2×(C2×D59)⋊C4—22no—
33C2×(C2×D59)⋊C4—22no—
34C2×C4×Dic59C2×C412yes—
35#14273646—22no—
36#14273646D414yescomplemented
37#14273638—41no—
38C4×C59⋊C8—41no—
39C59⋊#95—21no—
40C22×Q8×C59C2311yescharacteristic
41C2×M16×C59—22no—
42C4×Q8×C59—41no—
43C4×Q8×C59—41no—
44C2×C4⋊C4×C59C2×C411yescharacteristic
45C2×C4⋊C4×C59D411yescharacteristic
46C2×C4⋊C4×C59D411yescharacteristic
47C2×C42×C59C2×C411yescharacteristic
48C2×C42×C59D411yescharacteristic
49C2×C42×C59D411yescharacteristic
50C4⋊C8×C59—41no—
51C4⋊C8×C59—41no—
52#101×C59—24no—
53C23×D59—22no—
54C2×Q8×C59—41no—
55C2×Q8×C59—41no—
56C2×Q8×C59C2×D412yes—
57C2×Q8×C59#4112yes—
58C22×C4×C59C22⋊C411yescharacteristic
59C22×C4×C59C22⋊C411yescharacteristic
60C22×C4×C59C22⋊C411yescharacteristic
61C22×C4×C59C22⋊C411yescharacteristic
62C22×C4×C59C22×C411yescharacteristic
63C22×C4×C59C2×D411yescharacteristic
64C22×C4×C59C2×D411yescharacteristic
65C22×Dic59—22no—
66C2×D236—24no—
67C2×D236—42no—
68C2×D236—42no—
69C2×D236—42no—
70C2×D236—42no—
71C2×D236—42no—
72C2×D236—41no—
73C2×D236—41no—
74C2×D236—82no—
75M16×C59—42no—
76M16×C59—42no—
77C2×C8×C59—42no—
78C4⋊C4×C59—41no—
79C4⋊C4×C59—41no—
80C4⋊C4×C59—41no—
81C42×C59—21no—
82C42×C59—21no—
83C42×C59—21no—
84C42×C59—21no—
85C42×C59—21no—
86C42×C59—21no—
87(C2×D59)⋊C4—44no—
88(C2×D59)⋊C4—42no—
89(C2×D59)⋊C4—42no—
90(C2×D59)⋊C4—82no—
91(C2×D59)⋊C4—82no—
92C4⋊Dic59—42no—
93C4×Dic59—24no—
94#11745664—24no—
95#11745664—42no—
96C2×C59⋊C8—42no—
97C23×C59C2×C22⋊C411yescharacteristic
98C22×D59—42no—
99C22×D59—42no—
100C22×D59—42no—
101C22×D59—42no—
102C22×D59—42no—
103C22×D59—42no—
104C22×D59—42no—
105Q8×C59—42no—
106Q8×C59—42no—
107C2×C4×C59—21no—
108C2×C4×C59—21no—
109C2×C4×C59—21no—
110C2×C4×C59—21no—
111C2×C4×C59—21no—
112C2×C4×C59—21no—
113C2×C4×C59—22no—
114C2×C4×C59—21no—
115C2×C4×C59—21no—
116C2×C4×C59—21no—
117C2×C4×C59—41no—
118C2×C4×C59—41no—
119C2×C4×C59—41no—
120C2×C4×C59#9911yescharacteristic
121C2×C4×C59#9911yescharacteristic
122C2×C4×C59C4×D412yes—
123C2×C4×C59#12112yes—
124C2×C4×C59#12411yescharacteristic
125C2×C4×C59#12711yescharacteristic
126C2×Dic59—42no—
127C2×Dic59—42no—
128C2×Dic59—42no—
129D236—84no—
130D236—82no—
131D236—84no—
132D236—82no—
133C8×C59—82no—
134C59⋊C8—44no—
135#13226—591nomaximal, Sylow
136C22×C59—21no—
137C22×C59—21no—
138C22×C59#39011yescharacteristic
139C22×C59C2×#9911yescharacteristic
140C22×C59#41011yescharacteristic
141C2×D59—82no—
142C2×D59—82no—
143C2×D59—82no—
144C2×D59—82no—
145C2×D59—82no—
146C2×D59—82no—
147C2×D59—82no—
148C4×C59—21no—
149C4×C59—22no—
150C4×C59—21no—
151C4×C59—22no—
152C4×C59—21no—
153C4×C59—41no—
154C4×C59—41no—
155C4×C59—41no—
156C4×C59#35512yescomplemented
157Dic59—82no—
158#2548—591no—
159C2×#355—594no—
160#1801D5911yescharacteristic, complemented
161C4×#101—591no—
162C2×C59—21no—
163C2×C59—21no—
164C2×C59#196411yescharacteristic
165C2×C59C2×#35511yescharacteristic
166C2×C59#213411yescharacteristic
167D59—162no—
168C2×#127—591no—
169C2×#124—592no—
170C2×C4×Q8C2×D5911yescharacteristic
171#423—21no—
172C2×#101—592no—
173C2×#101C2×D5912yes—
174C4×M16—1181no—
175#394—1181no—
176#394—1181no—
177#390—592no—
178#390—1181no—
179#390—1181no—
180C4×C4⋊C4—591no—
181#355—1188no—
182#343—21no—
183#343—1181no—
184C59#1322611yescharacteristic, complemented, Sylow
185C22×Q8C22×D5911yescharacteristic
186C22×D4—592no—
187C22×D4—1181no—
188C22×D4—1181no—
189C2×M16—22no—
190C2×M16—1182no—
191#127—2361no—
192#127—2361no—
193#124—1184no—
194#124—1184no—
195C4×Q8—41no—
196C4×Q8—41no—
197C2×C4⋊C4—1181no—
198C2×C4⋊C4C4×D5911yescharacteristic
199C2×C4⋊C4D23611yescharacteristic
200C2×C4⋊C4C22⋊D5911yescharacteristic
201C2×C22⋊C4—1182no—
202C2×C22⋊C4—1182no—
203C2×C22⋊C4—1182no—
204C2×C22⋊C4—1182no—
205C2×C42—592no—
206C2×C42C4×D5911yescharacteristic
207C2×C42D23611yescharacteristic
208C2×C42C22⋊D5911yescharacteristic
209C4⋊C8—41no—
210C4⋊C8—41no—
211#101—24no—
212#101—594no—
213#101—1182no—
214#97—2361no—
215C4×C8—2361no—
216#95—1181no—
217C24—1182no—
218C2×Q8—41no—
219C2×Q8—41no—
220C2×Q8D4×D5912yes—
221C2×Q8#1174568912yes—
222C2×D4—1184no—
223C2×D4—4722no—
224C2×D4—2362no—
225C2×D4—2362no—
226C2×D4—2362no—
227C2×D4—2362no—
228C2×D4—2362no—
229C2×D4—2361no—
230C2×D4—2361no—
231C22×C4—1182no—
232C22×C4(C2×D59)⋊C411yescharacteristic
233C22×C4(C2×D59)⋊C411yescharacteristic
234C22×C4(C2×D59)⋊C411yescharacteristic
235C22×C4(C2×D59)⋊C411yescharacteristic
236C22×C4C2×C4×D5911yescharacteristic
237C22×C4C2×D23611yescharacteristic
238C22×C4C2×C22⋊D5911yescharacteristic
239M16—42no—
240M16—42no—
241M16—1184no—
242M16—2362no—
243C2×C8—42no—
244C2×C8—2362no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—2362no—
249C22⋊C4—4722no—
250C22⋊C4—4722no—
251C22⋊C4—2364no—
252C22⋊C4—2362no—
253C22⋊C4—2362no—
254C42—21no—
255C42—21no—
256C42—21no—
257C42—21no—
258C42—21no—
259C42—21no—
260C42—1184no—
261C23—2362no—
262C23—2362no—
263C23—2362no—
264C23—2362no—
265C23—2362no—
266C23—2362no—
267C23—2362no—
268C23C2×(C2×D59)⋊C411yescharacteristic
269Q8—42no—
270Q8—42no—
271D4—4724no—
272D4—4722no—
273D4—4724no—
274D4—4722no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—2362no—
289C2×C4—2362no—
290C2×C4—2362no—
291C2×C4#1427364111yescharacteristic
292C2×C4#1427364111yescharacteristic
293C2×C4#1427368912yes—
294C2×C4#1427370212yes—
295C2×C4#1427371611yescharacteristic
296C2×C4#1427372311yescharacteristic
297C8—82no—
298C8—2364no—
299C22—21no—
300C22—21no—
301C22—4722no—
302C22—4722no—
303C22—4722no—
304C22—4722no—
305C22—4722no—
306C22—4722no—
307C22—4722no—
308C22#1683730311yescharacteristic
309C22C2×#1427364111yescharacteristic
310C22#1683737511yescharacteristic
311C4—21no—
312C4—22no—
313C4—22no—
314C4—21no—
315C4—21no—
316C4—41no—
317C4—41no—
318C4—41no—
319C4—4722no—
320C4#1683719812yescomplemented
321C2—21no—
322C2—21no—
323C2—9442no—
324C2#2142493111yescharacteristic
325C2C2×#1683719811yescharacteristic
326C2#2142578711yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC22×Q8C22×D59
FittingC59×#1801C2
socleC22×C59C2×#99
prime radicalC2×#14273724C4
cosocleC22×Q8×C59C23
hypercenter#1801D59
hypocenterC59#13226
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1801

Complements

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

NH ≅ G/Ntype
2. C2×#16837198C2semidirect
3. C4×#14273646C2semidirect
4. C59×#1801C2semidirect
36. #14273646D4semidirect
156. C4×C59#355semidirect
160. #1801D59semidirect
184. C59#13226semidirect
320. C4#16837198semidirect

Maximal subgroups

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

subgroupindex
#214290952
C2×#168371982
C4×#142736462
C59×#18012
#1322659