Finite Group Databasebeta

← #28481605

Subgroups — #28481605

Order 15104.

460 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 107 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×#16837473C211yesmaximal, characteristic
2#21424134C211yesmaximal, characteristic, complemented
3#21422432C212yesmaximal
4#21421972C212yesmaximal, complemented
5C59×#1810C211yesmaximal, characteristic, complemented
6C22×#11745659C411yescharacteristic
7C2×C4×C22⋊D59C2211yescharacteristic
8#16837473—21no—
9#16837473—21no—
10#16837473—21no—
11#16837473—21no—
12C2×#14273663—21no—
13C2×Dic59⋊C8C411yescharacteristic
14#16837366C2212yes—
15C2×C22⋊(C59⋊C8)C2211yescharacteristic
16C2×(C2×D59)⋊C8C411yescharacteristic
17C2×#14273637C411yescharacteristic
18#16837167—21no—
19#16837127—22no—
20#16837125—22no—
21C23×C8×C59C2211yescharacteristic
22C2×#108×C59—21no—
23C59×#407C2212yes—
24C59×#341—21no—
25C22×C22⋊D59Q811yescharacteristic
26C22×C4×D59C2×C411yescharacteristic
27C2×#11745664—22no—
28C2×#11745659—21no—
29C2×#11745659—21no—
30C2×#11745659—42no—
31C22×C59⋊C8C2×C411yescharacteristic
32C22×C59⋊C8C2×C411yescharacteristic
33C4×C22⋊D59—21no—
34C4×C22⋊D59—21no—
35C4×C22⋊D59—21no—
36C4×C22⋊D59—21no—
37C2×Dic59⋊C4D411yescharacteristic
38C2×C22⋊Dic59Q811yescharacteristic
39C2×(C2×D59)⋊C4D411yescharacteristic
40C2×C4×Dic59C2×C411yescharacteristic
41#14273663—42no—
42Dic59⋊C8—21no—
43Dic59⋊C8—21no—
44C22⋊(C59⋊C8)—21no—
45C22⋊(C59⋊C8)—21no—
46C22⋊(C59⋊C8)—22no—
47C22⋊(C59⋊C8)—22no—
48(C2×D59)⋊C8—21no—
49(C2×D59)⋊C8—21no—
50#14273637—21no—
51#14273637—21no—
52C23×C4×C59C2311yescharacteristic
53C2×M16×C59—22no—
54C22×C8×C59—42no—
55C22×C8×C59C2×C411yescharacteristic
56C22×C8×C59C2×C411yescharacteristic
57C22×C8×C59D411yescharacteristic
58C22×C8×C59D411yescharacteristic
59C22×C8×C59D411yescharacteristic
60C22×C8×C59D411yescharacteristic
61#108×C59—42no—
62C22⋊C8×C59—22no—
63C22⋊C8×C59—22no—
64C24×C59C2×Q811yescharacteristic
65C23×D59C4⋊C411yescharacteristic
66C22×C4×C59—41no—
67C22×C4×C59—41no—
68C22×C4×C59C22×C411yescharacteristic
69C22×C4×C59C2×D411yescharacteristic
70C22×C4×C59C2×D411yescharacteristic
71C22×C4×C59C2×D411yescharacteristic
72C22×C4×C59#4111yescharacteristic
73C22×C4×C59#4111yescharacteristic
74C2×C22⋊D59—21no—
75C2×C22⋊D59—21no—
76C2×C22⋊D59—42no—
77C22×Dic59C4⋊C411yescharacteristic
78C22×Dic59C4⋊C411yescharacteristic
79C22×Dic59C4⋊C411yescharacteristic
80C2×C4×D59—21no—
81C2×C4×D59—21no—
82C2×C4×D59—41no—
83C2×C4×D59—41no—
84M16×C59—42no—
85M16×C59—42no—
86C2×C8×C59—22no—
87C2×C8×C59—22no—
88C2×C8×C59—22no—
89C2×C8×C59—41no—
90C2×C8×C59—41no—
91C2×C8×C59—41no—
92C2×C8×C59—42no—
93C2×C8×C59C22⋊C411yescharacteristic
94C2×C8×C59C22⋊C411yescharacteristic
95C2×C8×C59C22⋊C411yescharacteristic
96C2×C8×C59C22⋊C411yescharacteristic
97C22⋊Dic59—21no—
98C22⋊Dic59—21no—
99(C2×D59)⋊C4—21no—
100(C2×D59)⋊C4—21no—
101Dic59⋊C4—21no—
102Dic59⋊C4—21no—
103C4×Dic59—21no—
104C4×Dic59—21no—
105#11745664—42no—
106#11745664—42no—
107C2×C59⋊C8—21no—
108C2×C59⋊C8—21no—
109C2×C59⋊C8—41no—
110C2×C59⋊C8—41no—
111C2×C59⋊C8—42no—
112#11745659—42no—
113#11745659—42no—
114C23×C59—22no—
115C23×C59—41no—
116C23×C59—41no—
117C23×C59C2×C4⋊C411yescharacteristic
118C23×C59C22⋊Q811yescharacteristic
119C23×C59C22⋊Q811yescharacteristic
120C22×D59—21no—
121C22×D59—22no—
122C22×D59—41no—
123C22×D59—41no—
124C2×C4×C59—21no—
125C2×C4×C59—21no—
126C2×C4×C59—21no—
127C2×C4×C59—21no—
128C2×C4×C59—21no—
129C2×C4×C59—21no—
130C2×C4×C59—41no—
131C2×C4×C59—41no—
132C2×C4×C59—41no—
133C2×C4×C59C2×C22⋊C411yescharacteristic
134C2×C4×C59#11711yescharacteristic
135C2×C4×C59#12111yescharacteristic
136C2×C4×C59#12111yescharacteristic
137C22⋊D59—42no—
138C22⋊D59—42no—
139C2×Dic59—21no—
140C2×Dic59—21no—
141C2×Dic59—21no—
142C2×Dic59—22no—
143C2×Dic59—41no—
144C2×Dic59—41no—
145C4×D59—42no—
146C4×D59—41no—
147C4×D59—41no—
148C8×C59—21no—
149C8×C59—21no—
150C8×C59—21no—
151C8×C59—82no—
152C8×C59#9912yes—
153C59⋊C8—41no—
154C59⋊C8—41no—
155C59⋊C8—82no—
156#8613—591nomaximal, Sylow
157C22×C59—21no—
158C22×C59—21no—
159C22×C59—21no—
160C22×C59—21no—
161C22×C59—21no—
162C22×C59—21no—
163C22×C59—21no—
164C22×C59—21no—
165C22×C59—44no—
166C22×C59#38011yescharacteristic
167C22×C59#43012yes—
168C2×D59—41no—
169C2×D59—41no—
170C2×D59—41no—
171C2×D59—82no—
172C4×C59—21no—
173C4×C59—21no—
174C4×C59—21no—
175C4×C59C2×#9911yescharacteristic
176C4×C59#41011yescharacteristic
177Dic59—41no—
178Dic59—41no—
179Dic59—41no—
180C2×#435—591no—
181#1810—591no—
182#1810D5911yescharacteristic, complemented
183#1465—592no—
184#1293—592no—
185C2×C59—22no—
186C2×C59—41no—
187C2×C59—41no—
188C2×C59#179911yescharacteristic
189C2×C59#181111yescharacteristic
190C2×C59#181311yescharacteristic
191D59—81no—
192C22×M16—591no—
193C23×C8C2×D5911yescharacteristic
194C2×C4×D4—591no—
195#435—1181no—
196#435—1181no—
197#435—1181no—
198#435—1181no—
199C2×#108—21no—
200C2×#108—1181no—
201C2×C4⋊C8—591no—
202#407—592no—
203#407C2×D5912yes—
204C2×C22⋊C8—591no—
205C2×C22⋊C8—591no—
206C2×#97—591no—
207#341—21no—
208#341—1181no—
209#324—1182no—
210#323—1182no—
211C59#861311yescharacteristic, complemented, Sylow
212C22×D4—591no—
213C23×C4—591no—
214C23×C4C22×D5911yescharacteristic
215C2×M16—22no—
216C2×M16—1181no—
217C2×M16—1181no—
218C2×M16—1182no—
219C2×M16—2362no—
220C22×C8—42no—
221C22×C8—591no—
222C22×C8—591no—
223C22×C8C4×D5911yescharacteristic
224C22×C8C4×D5911yescharacteristic
225C22×C8D23611yescharacteristic
226C22×C8D23611yescharacteristic
227C22×C8C22⋊D5911yescharacteristic
228C22×C8C22⋊D5911yescharacteristic
229C4×D4—1181no—
230C4×D4—1181no—
231C4×D4—1181no—
232C4×D4—1181no—
233C2×C4⋊C4—591no—
234C2×C22⋊C4—591no—
235C2×C22⋊C4—591no—
236C2×C42—591no—
237#108—42no—
238#108—2362no—
239C4⋊C8—1181no—
240C4⋊C8—1181no—
241C22⋊C8—22no—
242C22⋊C8—22no—
243C22⋊C8—1181no—
244C22⋊C8—1181no—
245C22⋊C8—1181no—
246C22⋊C8—1181no—
247C22⋊C8—1182no—
248C22⋊C8—1182no—
249#97—1181no—
250#97—1181no—
251C24—591no—
252C24Q8×D5911yescharacteristic
253C2×D4—1181no—
254C2×D4—1181no—
255C2×D4—2362no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—591no—
259C22×C4—591no—
260C22×C4—591no—
261C22×C4—1181no—
262C22×C4—1181no—
263C22×C4—2361no—
264C22×C4—2361no—
265C22×C4C2×C4×D5911yescharacteristic
266C22×C4C2×D23611yescharacteristic
267C22×C4D4×D5911yescharacteristic
268C22×C4#1174568711yescharacteristic
269C22×C4#1174568911yescharacteristic
270C22×C4C2×C22⋊D5911yescharacteristic
271M16—42no—
272M16—42no—
273M16—2362no—
274M16—2362no—
275M16—2362no—
276M16—2362no—
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—22no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—42no—
284C2×C8—1181no—
285C2×C8—1181no—
286C2×C8—2362no—
287C2×C8—2361no—
288C2×C8—2361no—
289C2×C8(C2×D59)⋊C411yescharacteristic
290C2×C8(C2×D59)⋊C411yescharacteristic
291C2×C8(C2×D59)⋊C411yescharacteristic
292C2×C8(C2×D59)⋊C411yescharacteristic
293C4⋊C4—1181no—
294C4⋊C4—1181no—
295C22⋊C4—1181no—
296C22⋊C4—1181no—
297C22⋊C4—1181no—
298C22⋊C4—1181no—
299C42—1181no—
300C42—1181no—
301C23—22no—
302C23—41no—
303C23—41no—
304C23—1182no—
305C23—1181no—
306C23—2361no—
307C23—2361no—
308C23C4⋊C4×D5911yescharacteristic
309C23#1427370311yescharacteristic
310C23#1427370711yescharacteristic
311D4—2362no—
312D4—2362no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—1182no—
323C2×C4—1181no—
324C2×C4—1181no—
325C2×C4—1181no—
326C2×C4—2362no—
327C2×C4—2361no—
328C2×C4—2361no—
329C2×C4—2361no—
330C2×C4—2361no—
331C2×C4C2×(C2×D59)⋊C411yescharacteristic
332C2×C4#1427368511yescharacteristic
333C2×C4#1427369811yescharacteristic
334C2×C4#1427370211yescharacteristic
335C8—21no—
336C8—21no—
337C8—21no—
338C8—82no—
339C8—4722no—
340C8—2361no—
341C8—2361no—
342C8#1427364112yes—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—44no—
352C22—4722no—
353C22—2361no—
354C22—2361no—
355C22—2361no—
356C22#1683725911yescharacteristic
357C22#1683744912yes—
358C4—21no—
359C4—21no—
360C4—21no—
361C4—2361no—
362C4—2361no—
363C4—2361no—
364C4C2×#1427364111yescharacteristic
365C4#1683737511yescharacteristic
366C2—22no—
367C2—41no—
368C2—41no—
369C2—4721no—
370C2#2142408111yescharacteristic
371C2#2142413511yescharacteristic
372C2#2142414311yescharacteristic
subgroupG/H
centerC22#16837259
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#1810C2
socleC22×C59#380
prime radicalC22×C22⋊D59Q8
cosocleC23×C4×C59C23
hypercenter#1810D59
hypocenterC59#8613
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1810

Complements

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

NH ≅ G/Ntype
2. #21424134C2semidirect
4. #21421972C2semidirect
5. C59×#1810C2semidirect
182. #1810D59semidirect
211. C59#8613semidirect

Maximal subgroups

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

subgroupindex
C2×#168374732
#214241342
#214224322
#214219722
C59×#18102
#861359