Finite Group Databasebeta

← #28507012

Subgroups — #28507012

Order 15104.

540 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 111 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
1C2×#16837199C214yesmaximal, complemented
2#21424092C211yesmaximal, characteristic, complemented
3C4×#14273646C211yesmaximal, characteristic, complemented
4C59×#2548C211yesmaximal, characteristic
5#16837426—21no—
6C2×#14273646C2212yes—
7C2×#14273646C2212yes—
8C4×#11745664—21no—
9#16837199—28no—
10#16837171—21no—
11#16837171—21no—
12C2×#127×C59C411yescharacteristic
13C2×#124×C59C2212yes—
14C2×C4×Q8×C59C2211yescharacteristic
15C59×#394—21no—
16C59×#394—21no—
17C59×#390—21no—
18C59×#390—21no—
19C59×#390C412yes—
20C4×C4⋊C4×C59C411yescharacteristic
21C2×#11745664—22no—
22C2×#11745664—22no—
23C4⋊(C59⋊C8)—41no—
24C4⋊(C59⋊C8)—41no—
25#14273646—22no—
26#14273646—24no—
27#14273646D414yescomplemented
28#14273638—41no—
29C4×C59⋊C8—41no—
30C22×Q8×C59C2311yescharacteristic
31C22×D4×C59—21no—
32C22×D4×C59—21no—
33C22×D4×C59C2×C412yes—
34#127×C59—41no—
35#127×C59—41no—
36#124×C59—24no—
37#124×C59—24no—
38C4×Q8×C59—41no—
39C4×Q8×C59—41no—
40C2×C4⋊C4×C59—21no—
41C2×C4⋊C4×C59C2×C411yescharacteristic
42C2×C4⋊C4×C59D411yescharacteristic
43C2×C4⋊C4×C59D411yescharacteristic
44C2×C22⋊C4×C59—22no—
45C2×C22⋊C4×C59—22no—
46C2×C22⋊C4×C59—22no—
47C2×C22⋊C4×C59—22no—
48C2×C42×C59C2×C412yes—
49C2×C42×C59C2×C411yescharacteristic
50C2×C42×C59D411yescharacteristic
51C2×C42×C59D411yescharacteristic
52#95×C59—21no—
53C24×C59—22no—
54C2×Q8×C59—41no—
55C2×Q8×C59—41no—
56C2×Q8×C59C2×D412yes—
57C2×Q8×C59#4112yes—
58C2×D4×C59—24no—
59C2×D4×C59—42no—
60C2×D4×C59—42no—
61C2×D4×C59—42no—
62C2×D4×C59—42no—
63C2×D4×C59—42no—
64C2×D4×C59—41no—
65C2×D4×C59—41no—
66C2×D4×C59—82no—
67C22×C4×C59—22no—
68C22×C4×C59C22⋊C411yescharacteristic
69C22×C4×C59C22⋊C411yescharacteristic
70C22×C4×C59C22⋊C411yescharacteristic
71C22×C4×C59C22⋊C411yescharacteristic
72C22×C4×C59C22×C411yescharacteristic
73C22×C4×C59C2×D411yescharacteristic
74C22×C4×C59C2×D411yescharacteristic
75C4⋊C4×C59—41no—
76C4⋊C4×C59—41no—
77C4⋊C4×C59—41no—
78C4⋊C4×C59—42no—
79C22⋊C4×C59—44no—
80C22⋊C4×C59—42no—
81C22⋊C4×C59—42no—
82C22⋊C4×C59—82no—
83C22⋊C4×C59—82no—
84C42×C59—21no—
85C42×C59—21no—
86C42×C59—21no—
87C42×C59—21no—
88C42×C59—21no—
89C42×C59—21no—
90C42×C59—24no—
91#11745664—24no—
92#11745664—42no—
93#11745664—42no—
94#11745664—42no—
95C2×C59⋊C8—42no—
96C2×C59⋊C8—42no—
97C23×C59—42no—
98C23×C59—42no—
99C23×C59—42no—
100C23×C59—42no—
101C23×C59—42no—
102C23×C59—42no—
103C23×C59—42no—
104C23×C59C2×C22⋊C411yescharacteristic
105Q8×C59—42no—
106Q8×C59—42no—
107D4×C59—84no—
108D4×C59—82no—
109D4×C59—84no—
110D4×C59—82no—
111C2×C4×C59—21no—
112C2×C4×C59—21no—
113C2×C4×C59—21no—
114C2×C4×C59—21no—
115C2×C4×C59—21no—
116C2×C4×C59—21no—
117C2×C4×C59—22no—
118C2×C4×C59—21no—
119C2×C4×C59—21no—
120C2×C4×C59—21no—
121C2×C4×C59—41no—
122C2×C4×C59—41no—
123C2×C4×C59—41no—
124C2×C4×C59—42no—
125C2×C4×C59—42no—
126C2×C4×C59—42no—
127C2×C4×C59#9911yescharacteristic
128C2×C4×C59#9911yescharacteristic
129C2×C4×C59C4×D412yes—
130C2×C4×C59#12112yes—
131C2×C4×C59#12411yescharacteristic
132C2×C4×C59#12711yescharacteristic
133C59⋊C8—44no—
134C59⋊C8—82no—
135#13226—591nomaximal, Sylow
136C22×C59—21no—
137C22×C59—21no—
138C22×C59—82no—
139C22×C59—82no—
140C22×C59—82no—
141C22×C59—82no—
142C22×C59—82no—
143C22×C59—82no—
144C22×C59—82no—
145C22×C59#39011yescharacteristic
146C22×C59C2×#9911yescharacteristic
147C22×C59#41011yescharacteristic
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—82no—
157C4×C59#35512yescomplemented
158#2548D5911yescharacteristic
159C2×#355—594no—
160#1801—591no—
161C4×#101—591no—
162C2×C59—21no—
163C2×C59—21no—
164C2×C59—162no—
165C2×C59#196411yescharacteristic
166C2×C59C2×#35511yescharacteristic
167C2×C59#213411yescharacteristic
168C2×#127Dic5911yescharacteristic
169C2×#124C2×D5912yes—
170C2×C4×Q8C2×D5911yescharacteristic
171#423—1181no—
172C2×#101—592no—
173C2×#101—592no—
174C4×M16—1181no—
175#394—21no—
176#394—21no—
177#390—21no—
178#390—21no—
179#390Dic5912yes—
180C4×C4⋊C4Dic5911yescharacteristic
181#355—1188no—
182#343—1181no—
183#343—1181no—
184C59#1322611yescharacteristic, complemented, Sylow
185C22×Q8C22×D5911yescharacteristic
186C22×D4—21no—
187C22×D4—21no—
188C22×D4C2×Dic5912yes—
189C2×M16—1182no—
190C2×M16—1182no—
191#127—41no—
192#127—41no—
193#124—24no—
194#124—24no—
195C4×Q8—41no—
196C4×Q8—41no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4C2×Dic5911yescharacteristic
199C2×C4⋊C4C22⋊D5911yescharacteristic
200C2×C4⋊C4C22⋊D5911yescharacteristic
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C42C2×Dic5912yes—
206C2×C42C2×Dic5911yescharacteristic
207C2×C42C22⋊D5911yescharacteristic
208C2×C42C22⋊D5911yescharacteristic
209C4⋊C8—2361no—
210C4⋊C8—2361no—
211#101—594no—
212#101—1182no—
213#101—1184no—
214#97—2361no—
215C4×C8—2361no—
216#95—21no—
217C24—22no—
218C2×Q8—41no—
219C2×Q8—41no—
220C2×Q8D4×D5912yes—
221C2×Q8#1174568712yes—
222C2×D4—24no—
223C2×D4—42no—
224C2×D4—42no—
225C2×D4—42no—
226C2×D4—42no—
227C2×D4—42no—
228C2×D4—41no—
229C2×D4—41no—
230C2×D4—82no—
231C22×C4—22no—
232C22×C4C22⋊Dic5911yescharacteristic
233C22×C4C22⋊Dic5911yescharacteristic
234C22×C4C22⋊Dic5911yescharacteristic
235C22×C4C22⋊Dic5911yescharacteristic
236C22×C4C22×Dic5911yescharacteristic
237C22×C4C2×C22⋊D5911yescharacteristic
238C22×C4C2×C22⋊D5911yescharacteristic
239M16—1184no—
240M16—2362no—
241M16—2362no—
242M16—2362no—
243C2×C8—2362no—
244C2×C8—2362no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—42no—
249C22⋊C4—44no—
250C22⋊C4—42no—
251C22⋊C4—42no—
252C22⋊C4—82no—
253C22⋊C4—82no—
254C42—21no—
255C42—21no—
256C42—21no—
257C42—21no—
258C42—21no—
259C42—21no—
260C42—24no—
261C23—42no—
262C23—42no—
263C23—42no—
264C23—42no—
265C23—42no—
266C23—42no—
267C23—42no—
268C23C2×C22⋊Dic5911yescharacteristic
269Q8—42no—
270Q8—42no—
271D4—84no—
272D4—82no—
273D4—84no—
274D4—82no—
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—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4#1427364211yescharacteristic
292C2×C4#1427364211yescharacteristic
293C2×C4D4×Dic5912yes—
294C2×C4#1427369812yes—
295C2×C4#1427371411yescharacteristic
296C2×C4#1427372311yescharacteristic
297C8—4722no—
298C8—2364no—
299C22—21no—
300C22—21no—
301C22—82no—
302C22—82no—
303C22—82no—
304C22—82no—
305C22—82no—
306C22—82no—
307C22—82no—
308C22#1683730511yescharacteristic
309C22C2×#1427364211yescharacteristic
310C22#1683737611yescharacteristic
311C4—21no—
312C4—22no—
313C4—22no—
314C4—21no—
315C4—21no—
316C4—41no—
317C4—41no—
318C4—41no—
319C4—82no—
320C4#1683719912yescomplemented
321C2—21no—
322C2—21no—
323C2—162no—
324C2#2142493311yescharacteristic
325C2C2×#1683719911yescharacteristic
326C2#2142578811yescharacteristic
subgroupG/H
centerC22C2×#14273642
commutatorC22×C4×C59C22×C4
FrattiniC22×Q8C22×D59
FittingC59×#2548C2
socleC22×C59C2×#99
prime radicalC2×#127×C59C4
cosocleC22×Q8×C59C23
hypercenter#2548D59
hypocenterC59#13226
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2548

Complements

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

NH ≅ G/Ntype
1. C2×#16837199C2semidirect
2. #21424092C2semidirect
3. C4×#14273646C2semidirect
27. #14273646D4semidirect
157. C4×C59#355semidirect
184. C59#13226semidirect
320. C4#16837199semidirect

Maximal subgroups

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

subgroupindex
C2×#168371992
#214240922
C4×#142736462
C59×#25482
#1322659