Finite Group Databasebeta

← #28499515

Subgroups — #28499515

Order 15104.

382 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 87 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#21424092C211yesmaximal, characteristic, complemented
2#21422440C211yesmaximal, characteristic, complemented
3#21422126C212yesmaximal, complemented
4#21422099C212yesmaximal
5C59×#2632C211yesmaximal, characteristic
6#16837459—21no—
7#16837426—21no—
8C2×#14273646—21no—
9#16837171C2212yes—
10#16837128—21no—
11#16837128C2212yes—
12C2×#124×C59C411yescharacteristic
13C2×C4×Q8×C59C2211yescharacteristic
14C59×#399—21no—
15C59×#395C411yescharacteristic
16C59×#395C2211yescharacteristic
17C59×#394C411yescharacteristic
18C59×#394C2211yescharacteristic
19C59×#393—21no—
20C59×#388—21no—
21C59×#388—21no—
22C59×#383C411yescharacteristic
23C2×#11745664—22no—
24C4⋊(C59⋊C8)—41no—
25C4⋊(C59⋊C8)—41no—
26C4⋊(C59⋊C8)—41no—
27#14273646—42no—
28C22⋊(C59⋊C8)—22no—
29#14273638—41no—
30C22×Q8×C59C2×C411yescharacteristic
31C22×Q8×C59D411yescharacteristic
32C22×D4×C59C2×C411yescharacteristic
33#124×C59—41no—
34#124×C59—41no—
35C4×Q8×C59—41no—
36C4×Q8×C59—41no—
37C2×C4⋊C4×C59—21no—
38C2×C4⋊C4×C59—21no—
39C2×C4⋊C4×C59C2×C411yescharacteristic
40C2×C4⋊C4×C59D411yescharacteristic
41C2×C4⋊C4×C59C2311yescharacteristic
42C2×C22⋊C4×C59—21no—
43C2×C22⋊C4×C59—21no—
44C2×C22⋊C4×C59—21no—
45C2×C22⋊C4×C59—21no—
46C2×C22⋊C4×C59—21no—
47C2×C22⋊C4×C59—21no—
48C2×C42×C59C2×C411yescharacteristic
49C2×C42×C59D411yescharacteristic
50C2×C42×C59D411yescharacteristic
51#95×C59—21no—
52#95×C59—21no—
53#95×C59—21no—
54#95×C59—21no—
55#95×C59C2×C411yescharacteristic
56#95×C59C2×C411yescharacteristic
57C24×C59—21no—
58C2×Q8×C59—21no—
59C2×Q8×C59—21no—
60C2×Q8×C59—41no—
61C2×Q8×C59—41no—
62C2×Q8×C59—41no—
63C2×Q8×C59—81no—
64C2×D4×C59—41no—
65C2×D4×C59—81no—
66C22×C4×C59—21no—
67C22×C4×C59—21no—
68C22×C4×C59—21no—
69C22×C4×C59C22⋊C411yescharacteristic
70C22×C4×C59C22⋊C411yescharacteristic
71C22×C4×C59C22⋊C411yescharacteristic
72C22×C4×C59C22⋊C411yescharacteristic
73C22×C4×C59C22×C411yescharacteristic
74C22×C4×C59C2×D411yescharacteristic
75C22×C4×C59C2×D411yescharacteristic
76C4⋊C4×C59—41no—
77C4⋊C4×C59—41no—
78C4⋊C4×C59—41no—
79C4⋊C4×C59—81no—
80C4⋊C4×C59—81no—
81C22⋊C4×C59—42no—
82C22⋊C4×C59—42no—
83C22⋊C4×C59—41no—
84C22⋊C4×C59—41no—
85C22⋊C4×C59—41no—
86C22⋊C4×C59—41no—
87C22⋊C4×C59—81no—
88C22⋊C4×C59—81no—
89C42×C59—21no—
90C42×C59—21no—
91C42×C59—21no—
92C42×C59—21no—
93C42×C59—21no—
94C42×C59—21no—
95#11745664—42no—
96#11745664—42no—
97C2×C59⋊C8—42no—
98C2×C59⋊C8—42no—
99C23×C59—41no—
100C23×C59—41no—
101C23×C59—41no—
102C23×C59—41no—
103C23×C59—41no—
104C23×C59—41no—
105C23×C59—41no—
106C23×C59C2×C22⋊C411yescharacteristic
107Q8×C59—42no—
108Q8×C59—42no—
109Q8×C59—82no—
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—21no—
118C2×C4×C59—21no—
119C2×C4×C59—21no—
120C2×C4×C59—21no—
121C2×C4×C59—21no—
122C2×C4×C59—41no—
123C2×C4×C59—41no—
124C2×C4×C59—41no—
125C2×C4×C59—41no—
126C2×C4×C59—41no—
127C2×C4×C59—41no—
128C2×C4×C59—41no—
129C2×C4×C59—41no—
130C2×C4×C59—41no—
131C2×C4×C59—41no—
132C2×C4×C59—41no—
133C2×C4×C59—41no—
134C2×C4×C59—41no—
135C2×C4×C59#9911yescharacteristic
136C2×C4×C59#9911yescharacteristic
137C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
138C2×C4×C59#13711yescharacteristic
139C59⋊C8—82no—
140C59⋊C8—82no—
141#11833—591nomaximal, Sylow
142C22×C59—21no—
143C22×C59—21no—
144C22×C59—81no—
145C22×C59—81no—
146C22×C59—81no—
147C22×C59—81no—
148C22×C59—81no—
149C22×C59—81no—
150C22×C59—81no—
151C22×C59C2×#9911yescharacteristic
152C22×C59#41711yescharacteristic
153C22×C59#42111yescharacteristic
154C4×C59—21no—
155C4×C59—21no—
156C4×C59—21no—
157C4×C59—21no—
158C4×C59—41no—
159C4×C59—41no—
160C4×C59—41no—
161C4×C59—41no—
162C4×C59—41no—
163C4×C59—81no—
164C4×C59—81no—
165C4×C59—81no—
166#2632D5911yescharacteristic
167#1801—591no—
168#1467—591no—
169#1354—592no—
170#1345—592no—
171C2×C59—21no—
172C2×C59—21no—
173C2×C59—161no—
174C2×C59#151311yescharacteristic
175C2×C59#212211yescharacteristic
176C2×C59#213311yescharacteristic
177C2×#124Dic5911yescharacteristic
178C2×C4×Q8C2×D5911yescharacteristic
179#432—1181no—
180#423—1181no—
181C2×#101—1181no—
182#399—21no—
183#395Dic5911yescharacteristic
184#395C2×D5911yescharacteristic
185#394Dic5911yescharacteristic
186#394C2×D5911yescharacteristic
187#393—21no—
188#388—21no—
189#388—21no—
190#383Dic5911yescharacteristic
191#343—592no—
192#324—592no—
193#324—1181no—
194C59#1183311yescharacteristic, complemented, Sylow
195C22×Q8C2×Dic5911yescharacteristic
196C22×Q8C22⋊D5911yescharacteristic
197C22×D4C2×Dic5911yescharacteristic
198C2×M16—1182no—
199#124—41no—
200#124—41no—
201C4×Q8—41no—
202C4×Q8—41no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4C2×Dic5911yescharacteristic
206C2×C4⋊C4C22⋊D5911yescharacteristic
207C2×C4⋊C4C22×D5911yescharacteristic
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C42C2×Dic5911yescharacteristic
215C2×C42C22⋊D5911yescharacteristic
216C2×C42C22⋊D5911yescharacteristic
217C4⋊C8—2361no—
218C4⋊C8—2361no—
219C4⋊C8—2361no—
220#101—2362no—
221C22⋊C8—1182no—
222#97—2361no—
223#95—21no—
224#95—21no—
225#95—21no—
226#95—21no—
227#95C2×Dic5911yescharacteristic
228#95C2×Dic5911yescharacteristic
229C24—21no—
230C2×Q8—21no—
231C2×Q8—21no—
232C2×Q8—41no—
233C2×Q8—41no—
234C2×Q8—41no—
235C2×Q8—81no—
236C2×D4—41no—
237C2×D4—81no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4C22⋊Dic5911yescharacteristic
242C22×C4C22⋊Dic5911yescharacteristic
243C22×C4C22⋊Dic5911yescharacteristic
244C22×C4C22⋊Dic5911yescharacteristic
245C22×C4C22×Dic5911yescharacteristic
246C22×C4C2×C22⋊D5911yescharacteristic
247C22×C4C2×C22⋊D5911yescharacteristic
248M16—2362no—
249M16—2362no—
250C2×C8—2362no—
251C2×C8—2362no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—81no—
256C4⋊C4—81no—
257C22⋊C4—42no—
258C22⋊C4—42no—
259C22⋊C4—41no—
260C22⋊C4—41no—
261C22⋊C4—41no—
262C22⋊C4—41no—
263C22⋊C4—81no—
264C22⋊C4—81no—
265C42—21no—
266C42—21no—
267C42—21no—
268C42—21no—
269C42—21no—
270C42—21no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23C2×C22⋊Dic5911yescharacteristic
279Q8—42no—
280Q8—42no—
281Q8—82no—
282D4—82no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4#1427364211yescharacteristic
308C2×C4#1427364211yescharacteristic
309C2×C4#1427375211yescharacteristic
310C2×C4#1427375711yescharacteristic
311C8—4722no—
312C8—4722no—
313C22—21no—
314C22—21no—
315C22—81no—
316C22—81no—
317C22—81no—
318C22—81no—
319C22—81no—
320C22—81no—
321C22—81no—
322C22C2×#1427364211yescharacteristic
323C22#1683740311yescharacteristic
324C22#1683741911yescharacteristic
325C4—21no—
326C4—21no—
327C4—21no—
328C4—21no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—41no—
334C4—81no—
335C4—81no—
336C4—81no—
337C2—21no—
338C2—21no—
339C2—161no—
340C2#2142263911yescharacteristic
341C2#2142574611yescharacteristic
342C2#2142578511yescharacteristic
subgroupG/H
centerC22C2×#14273642
commutatorC22×C4×C59C22×C4
FrattiniC2×C4⋊C4C22×D59
FittingC59×#2632C2
socleC22×C59C2×#99
prime radicalC22×D4×C59C2×C4
cosocleC2×C4⋊C4×C59C23
hypercenter#2632D59
hypocenterC59#11833
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2632

Complements

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

NH ≅ G/Ntype
1. #21424092C2semidirect
2. #21422440C2semidirect
3. #21422126C2semidirect
194. C59#11833semidirect

Maximal subgroups

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

subgroupindex
#214240922
#214224402
#214221262
#214220992
C59×#26322
#1183359