Finite Group Databasebeta

← #28499383

Subgroups — #28499383

Order 15104.

448 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 79 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#21429838C211yesmaximal, characteristic
2#21422426C211yesmaximal, characteristic, complemented
3#21422101C212yesmaximal, complemented
4#21422098C212yesmaximal, complemented
5C59×#1821C211yesmaximal, characteristic, complemented
6C2×#14273710—21no—
7C2×#14273701C411yescharacteristic
8#16837366—21no—
9#16837332C411yescharacteristic
10#16837332C2211yescharacteristic
11#16837324—21no—
12#16837324—21no—
13#16837319C411yescharacteristic
14#16837319C2211yescharacteristic
15#16837281—21no—
16#16837262C411yescharacteristic
17#16837128—21no—
18#16837128C2212yes—
19C22×C4⋊C4×C59C2211yescharacteristic
20C59×#407—21no—
21C59×#343—21no—
22C59×#343C2212yes—
23C22×D236C2×C411yescharacteristic
24C22×C22⋊D59—21no—
25C2×#11745664—41no—
26#14273710—44no—
27#14273701—41no—
28#14273701—41no—
29C2×C4⋊Dic59—21no—
30C2×C4⋊Dic59C2×C411yescharacteristic
31C2×Dic59⋊C4—21no—
32C2×Dic59⋊C4—21no—
33C2×C22⋊Dic59—21no—
34C2×(C2×D59)⋊C4—21no—
35C2×(C2×D59)⋊C4—21no—
36C2×(C2×D59)⋊C4—21no—
37C2×(C2×D59)⋊C4—21no—
38C2×(C2×D59)⋊C4—21no—
39C22⋊(C59⋊C8)—22no—
40C22⋊(C59⋊C8)—41no—
41C59⋊#95—21no—
42C59⋊#95—21no—
43C59⋊#95C2×C411yescharacteristic
44C59⋊#95C2×C411yescharacteristic
45C23×C4×C59C2×C411yescharacteristic
46C23×C4×C59D411yescharacteristic
47C23×C4×C59D411yescharacteristic
48C2×M16×C59—22no—
49C2×C4⋊C4×C59—41no—
50C2×C4⋊C4×C59—41no—
51C2×C4⋊C4×C59C2×C411yescharacteristic
52C2×C4⋊C4×C59D411yescharacteristic
53C2×C4⋊C4×C59D411yescharacteristic
54C2×C4⋊C4×C59C2311yescharacteristic
55C22⋊C8×C59—41no—
56C22⋊C8×C59—41no—
57C24×C59C22⋊C411yescharacteristic
58C23×D59—21no—
59C22×C4×C59—21no—
60C22×C4×C59—21no—
61C22×C4×C59—21no—
62C22×C4×C59—21no—
63C22×C4×C59—21no—
64C22×C4×C59—21no—
65C22×C4×C59—22no—
66C22×C4×C59—41no—
67C22×C4×C59—81no—
68C22×C4×C59—81no—
69C22×C4×C59C22⋊C411yescharacteristic
70C22×C4×C59C22⋊C411yescharacteristic
71C22×C4×C59C22⋊C411yescharacteristic
72C22×C4×C59C22×C411yescharacteristic
73C22×C4×C59C2×D411yescharacteristic
74C22×C4×C59C2×D411yescharacteristic
75C2×C22⋊D59—42no—
76C2×C22⋊D59—42no—
77C2×C22⋊D59—41no—
78C2×C22⋊D59—41no—
79C22×Dic59—21no—
80C22×Dic59—21no—
81C22×Dic59—21no—
82C2×D236—41no—
83C2×D236—81no—
84M16×C59—42no—
85M16×C59—42no—
86C2×C8×C59—42no—
87C4⋊C4×C59—41no—
88C4⋊C4×C59—41no—
89C4⋊C4×C59—41no—
90C4⋊C4×C59—41no—
91C22⋊Dic59—42no—
92(C2×D59)⋊C4—42no—
93(C2×D59)⋊C4—42no—
94(C2×D59)⋊C4—42no—
95(C2×D59)⋊C4—81no—
96(C2×D59)⋊C4—81no—
97C4⋊Dic59—41no—
98C4⋊Dic59—81no—
99Dic59⋊C4—42no—
100Dic59⋊C4—42no—
101#11745664—82no—
102C2×C59⋊C8—42no—
103C23×C59—21no—
104C23×C59—21no—
105C23×C59—41no—
106C23×C59—41no—
107C23×C59#9911yescharacteristic
108C23×C59#9911yescharacteristic
109C23×C59C2×C22⋊C411yescharacteristic
110C22×D59—41no—
111C22×D59—41no—
112C22×D59—41no—
113C22×D59—41no—
114C22×D59—41no—
115C22×D59—41no—
116C22×D59—41no—
117C2×C4×C59—21no—
118C2×C4×C59—21no—
119C2×C4×C59—21no—
120C2×C4×C59—21no—
121C2×C4×C59—21no—
122C2×C4×C59—21no—
123C2×C4×C59—21no—
124C2×C4×C59—21no—
125C2×C4×C59—21no—
126C2×C4×C59—42no—
127C2×C4×C59—42no—
128C2×C4×C59—42no—
129C2×C4×C59—41no—
130C2×C4×C59—41no—
131C2×C4×C59—41no—
132C2×C4×C59—41no—
133C2×C4×C59—81no—
134C2×C4×C59—81no—
135C2×C4×C59—81no—
136C2×C4×C59AGL(1,ℤ/8)12yes—
137C22⋊D59—82no—
138C22⋊D59—82no—
139C2×Dic59—41no—
140C2×Dic59—41no—
141C2×Dic59—41no—
142C2×Dic59—41no—
143C2×Dic59—41no—
144C2×Dic59—41no—
145C2×Dic59—41no—
146C2×Dic59—41no—
147C2×Dic59—41no—
148D236—82no—
149C8×C59—82no—
150C59⋊C8—82no—
151#11805—591nomaximal, Sylow
152C22×C59—21no—
153C22×C59—21no—
154C22×C59—21no—
155C22×C59—21no—
156C22×C59—21no—
157C22×C59—21no—
158C22×C59—21no—
159C22×C59—21no—
160C22×C59—42no—
161C22×C59—42no—
162C22×C59C2×#9911yescharacteristic
163C22×C59#41811yescharacteristic
164C22×C59#42111yescharacteristic
165C2×D59—81no—
166C2×D59—81no—
167C2×D59—81no—
168C2×D59—81no—
169C2×D59—81no—
170C2×D59—81no—
171C2×D59—81no—
172C4×C59—22no—
173C4×C59—41no—
174C4×C59—41no—
175C4×C59—41no—
176C4×C59—41no—
177C4×C59—41no—
178Dic59—81no—
179Dic59—81no—
180Dic59—81no—
181#2618—591no—
182#1821D5911yescharacteristic, complemented
183#1463—591no—
184#1346—592no—
185#1345—592no—
186C2×C59—21no—
187C2×C59—21no—
188C2×C59—41no—
189C2×C59—41no—
190C2×C59#151011yescharacteristic
191C2×C59#212111yescharacteristic
192C2×C59#213311yescharacteristic
193D59—161no—
194C2×#123—1181no—
195C2×#121—591no—
196C22×C4⋊C4C2×D5911yescharacteristic
197#407—21no—
198#407—1181no—
199#398—591no—
200#398—591no—
201#396—1181no—
202#396—1181no—
203#394—591no—
204#394—591no—
205#386—1181no—
206#380—591no—
207#343—21no—
208#343C2×D5912yes—
209#324—592no—
210#324—1181no—
211C59#1180511yescharacteristic, complemented, Sylow
212C22×D4—591no—
213C22×D4—1181no—
214C23×C4C4×D5911yescharacteristic
215C23×C4D23611yescharacteristic
216C23×C4C22⋊D5911yescharacteristic
217C2×M16—22no—
218C2×M16—2361no—
219#123—2364no—
220#121—2361no—
221#121—2361no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—591no—
225C2×C4⋊C4—1181no—
226C2×C4⋊C4—1181no—
227C2×C4⋊C4—1181no—
228C2×C4⋊C4C4×D5911yescharacteristic
229C2×C4⋊C4D23611yescharacteristic
230C2×C4⋊C4C22⋊D5911yescharacteristic
231C2×C4⋊C4C22×D5911yescharacteristic
232C2×C22⋊C4—1181no—
233C2×C22⋊C4—1181no—
234C2×C22⋊C4—1181no—
235C2×C22⋊C4—1181no—
236C2×C22⋊C4—1181no—
237C2×C22⋊C4—1181no—
238C22⋊C8—41no—
239C22⋊C8—41no—
240C22⋊C8—1182no—
241C22⋊C8—2361no—
242#95—591no—
243#95—591no—
244#95—1181no—
245#95—1181no—
246C24—1181no—
247C24(C2×D59)⋊C411yescharacteristic
248C2×D4—4721no—
249C2×D4—2361no—
250C2×D4—2362no—
251C2×D4—2362no—
252C2×D4—2361no—
253C2×D4—2361no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—22no—
261C22×C4—41no—
262C22×C4—81no—
263C22×C4—81no—
264C22×C4—1181no—
265C22×C4—1181no—
266C22×C4—1181no—
267C22×C4(C2×D59)⋊C411yescharacteristic
268C22×C4(C2×D59)⋊C411yescharacteristic
269C22×C4(C2×D59)⋊C411yescharacteristic
270C22×C4C2×C4×D5911yescharacteristic
271C22×C4C2×D23611yescharacteristic
272C22×C4C2×C22⋊D5911yescharacteristic
273M16—42no—
274M16—42no—
275M16—4722no—
276C2×C8—42no—
277C2×C8—2362no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—4721no—
283C4⋊C4—2361no—
284C4⋊C4—2362no—
285C4⋊C4—2362no—
286C22⋊C4—4721no—
287C22⋊C4—4721no—
288C22⋊C4—2362no—
289C22⋊C4—2362no—
290C22⋊C4—2362no—
291C22⋊C4—2362no—
292C23—21no—
293C23—21no—
294C23—41no—
295C23—41no—
296C23—2361no—
297C23—2361no—
298C23—2361no—
299C23—2361no—
300C23—2361no—
301C23—2361no—
302C23—2361no—
303C23#1427364111yescharacteristic
304C23#1427364111yescharacteristic
305C23C2×(C2×D59)⋊C411yescharacteristic
306D4—4722no—
307D4—4722no—
308D4—4722no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—2361no—
329C2×C4—2361no—
330C2×C4—2361no—
331C2×C4—2361no—
332C2×C4—2361no—
333C2×C4—2361no—
334C2×C4—2361no—
335C2×C4—2361no—
336C2×C4—2361no—
337C2×C4#1427375212yes—
338C8—82no—
339C8—4722no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—42no—
349C22—42no—
350C22—4721no—
351C22—4721no—
352C22—4721no—
353C22—4721no—
354C22—4721no—
355C22—4721no—
356C22—4721no—
357C22C2×#1427364111yescharacteristic
358C22#1683740611yescharacteristic
359C22#1683742011yescharacteristic
360C4—22no—
361C4—41no—
362C4—41no—
363C4—41no—
364C4—41no—
365C4—41no—
366C4—4721no—
367C4—4721no—
368C4—4721no—
369C2—21no—
370C2—21no—
371C2—41no—
372C2—41no—
373C2—9441no—
374C2#2142262411yescharacteristic
375C2#2142574311yescharacteristic
376C2#2142578311yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC2×C4⋊C4C22×D59
FittingC59×#1821C2
socleC22×C59C2×#99
prime radicalC2×#14273701C4
cosocleC2×C4⋊C4×C59C23
hypercenter#1821D59
hypocenterC59#11805
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #1821

Complements

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

NH ≅ G/Ntype
2. #21422426C2semidirect
3. #21422101C2semidirect
4. #21422098C2semidirect
5. C59×#1821C2semidirect
182. #1821D59semidirect
211. C59#11805semidirect

Maximal subgroups

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

subgroupindex
#214298382
#214224262
#214221012
#214220982
C59×#18212
#1180559