Finite Group Databasebeta

← #28490979

Subgroups — #28490979

Order 15104.

466 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 101 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#21435240C211yesmaximal, characteristic
2#21434618C211yesmaximal, characteristic
3#21424917C211yesmaximal, characteristic
4#21422661C211yesmaximal, characteristic
5#21422660C211yesmaximal, characteristic
6#21421940C211yesmaximal, characteristic
7C59×#3229C211yesmaximal, characteristic
8#16837959C2211yescharacteristic
9#16837812—21no—
10#16837812—21no—
11#16837808—21no—
12#16837808—21no—
13#16837798—21no—
14#16837754—21no—
15#16837732—22no—
16#16837684—22no—
17#16837455C2211yescharacteristic
18#16837438—22no—
19C2×#14273651—22no—
20C2×#14273638C2211yescharacteristic
21#16837268C2211yescharacteristic
22#16837140—22no—
23#16837139—22no—
24C59×#533C2211yescharacteristic
25C2×C4⋊Q8×C59C2211yescharacteristic
26C59×#498—21no—
27C59×#487—21no—
28C59×#483—22no—
29C59×#475—22no—
30C59×#431C2211yescharacteristic
31C22×C59⋊C8D412yes—
32#14273727D411yescharacteristic
33#14273727D411yescharacteristic
34#14273719—21no—
35#14273706—21no—
36#14273706—21no—
37C4×Dic118—21no—
38C4×Dic118—21no—
39#14273686—21no—
40C2×C4⋊Dic59—21no—
41#14273660—22no—
42#14273660—22no—
43#14273658—24no—
44#14273658—22no—
45#14273658—22no—
46C4⋊(C59⋊C8)—21no—
47#14273651—44no—
48#14273650—24no—
49C22⋊(C59⋊C8)D412yes—
50#14273638—21no—
51#14273638—21no—
52C4×C59⋊C8—21no—
53C59⋊#95—21no—
54C59⋊#95—21no—
55C22×Q8×C59—21no—
56C4⋊Q8×C59—21no—
57C4⋊Q8×C59—21no—
58C4⋊Q8×C59—21no—
59C4⋊Q8×C59D412yes—
60C4⋊Q8×C59D411yescharacteristic
61C4⋊Q8×C59D411yescharacteristic
62#125×C59—21no—
63C22⋊Q8×C59—21no—
64C22⋊Q8×C59—21no—
65C4×Q8×C59—21no—
66C4×Q8×C59—21no—
67#117×C59—21no—
68C2×C4⋊C4×C59—21no—
69C2×C4⋊C4×C59—21no—
70C2×C42×C59C2311yescharacteristic
71#106×C59—24no—
72C4⋊C8×C59—21no—
73#103×C59—24no—
74C22⋊C8×C59D412yes—
75C4×C8×C59—21no—
76C2×Q8×C59—21no—
77C2×Q8×C59—22no—
78C2×Q8×C59—21no—
79C2×Q8×C59—41no—
80C2×Q8×C59—42no—
81C2×Q8×C59—42no—
82C22×C4×C59—41no—
83C22×C4×C59C2×D411yescharacteristic
84C22×C4×C59C2×D411yescharacteristic
85C22×C4×C59C2×D411yescharacteristic
86C22×Dic59—41no—
87C2×Dic118—21no—
88C2×Dic118—21no—
89C2×C8×C59—22no—
90C4⋊C4×C59—22no—
91C4⋊C4×C59—21no—
92C4⋊C4×C59—22no—
93C4⋊C4×C59—21no—
94C4⋊C4×C59—21no—
95C4⋊C4×C59—21no—
96C4⋊C4×C59—41no—
97C4⋊C4×C59—41no—
98C4⋊C4×C59—41no—
99C4⋊C4×C59—41no—
100C22⋊C4×C59—41no—
101C42×C59—41no—
102C42×C59C2×D411yescharacteristic
103C42×C59C2×D411yescharacteristic
104C42×C59C2×D411yescharacteristic
105C42×C59#4111yescharacteristic
106C22⋊Dic59—41no—
107C4⋊Dic59—21no—
108C4⋊Dic59—21no—
109C4⋊Dic59—21no—
110C4⋊Dic59—21no—
111C4⋊Dic59—42no—
112Dic59⋊C4—41no—
113Dic59⋊C4—41no—
114C4×Dic59—41no—
115C2×C59⋊C8—22no—
116C2×C59⋊C8—22no—
117C2×C59⋊C8—22no—
118C2×C59⋊C8SD1614yes—
119C23×C59#12711yescharacteristic
120Q8×C59—42no—
121Q8×C59—44no—
122Q8×C59—81no—
123Q8×C59—82no—
124C2×C4×C59—41no—
125C2×C4×C59—41no—
126C2×C4×C59—41no—
127C2×C4×C59—41no—
128C2×C4×C59—42no—
129C2×C4×C59—81no—
130C2×C4×C59—81no—
131C2×C4×C59#12011yescharacteristic
132C2×C4×C59#12011yescharacteristic
133C2×C4×C59#12011yescharacteristic
134C2×C4×C59#12111yescharacteristic
135C2×C4×C59#12111yescharacteristic
136C2×C4×C59#12111yescharacteristic
137C2×C4×C59C2×SD1612yes—
138C2×C4×C59#13712yes—
139C2×Dic59—41no—
140C2×Dic59—41no—
141C2×Dic59—81no—
142C2×Dic59—81no—
143C2×Dic59—81no—
144Dic118—42no—
145Dic118—81no—
146C8×C59—42no—
147C59⋊C8—24no—
148C59⋊C8—42no—
149#10227—591nomaximal, Sylow
150C22×C59—21no—
151C22×C59—21no—
152C22×C59#39211yescharacteristic
153C22×C59#49211yescharacteristic
154C22×C59#49711yescharacteristic
155C4×C59—21no—
156C4×C59—21no—
157C4×C59—21no—
158C4×C59—21no—
159C4×C59—81no—
160C4×C59—81no—
161C4×C59—82no—
162C4×C59#44812yes—
163C4×C59#46112yes—
164Dic59—81no—
165Dic59—81no—
166Dic59—161no—
167#3229—591no—
168#3229D5911yescharacteristic
169#3162—591no—
170#1960—591no—
171#1518—591no—
172#1518—591no—
173#1281—591no—
174C2×C59—21no—
175C2×C59—21no—
176C2×C59#200911yescharacteristic
177C2×C59#200911yescharacteristic
178C2×C59#201311yescharacteristic
179#533—591no—
180#533C2×D5911yescharacteristic
181C2×C4⋊Q8C2×D5911yescharacteristic
182#501—1181no—
183#501—1181no—
184#501—1181no—
185#501—1181no—
186#498—21no—
187#498—1181no—
188#487—21no—
189#487—1181no—
190#483—22no—
191#483—1182no—
192#475—22no—
193#475—1182no—
194#431—591no—
195#431C2×D5911yescharacteristic
196#427—1182no—
197C2×#103—1182no—
198C2×#97—591no—
199#382—591no—
200#328—1182no—
201#328—1182no—
202C59#1022711yescharacteristic, complemented, Sylow
203C22×Q8—21no—
204C22×C8—592no—
205C4⋊Q8—21no—
206C4⋊Q8—21no—
207C4⋊Q8—21no—
208C4⋊Q8—591no—
209C4⋊Q8—591no—
210C4⋊Q8C22⋊D5912yes—
211C4⋊Q8C22⋊D5911yescharacteristic
212C4⋊Q8C22⋊D5911yescharacteristic
213#125—21no—
214#125—1181no—
215C22⋊Q8—21no—
216C22⋊Q8—21no—
217C22⋊Q8—1181no—
218C22⋊Q8—1181no—
219C4×Q8—21no—
220C4×Q8—21no—
221C4×Q8—1181no—
222C4×Q8—1181no—
223#117—21no—
224#117—1181no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—1181no—
228C2×C42C22×D5911yescharacteristic
229#107—1182no—
230#107—1182no—
231#106—24no—
232#106—1182no—
233#106—1182no—
234#106—1184no—
235C4⋊C8—21no—
236C4⋊C8—1181no—
237#103—24no—
238#103—1184no—
239#103—2364no—
240C22⋊C8—592no—
241C22⋊C8C22⋊D5912yes—
242#97—1181no—
243#97—1181no—
244C4×C8—21no—
245C4×C8—1181no—
246#95—1181no—
247#95—1181no—
248C2×Q8—21no—
249C2×Q8—22no—
250C2×Q8—21no—
251C2×Q8—41no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×Q8—1181no—
255C2×Q8—1181no—
256C22×C4—41no—
257C22×C4—2361no—
258C22×C4D4×D5911yescharacteristic
259C22×C4D4×D5911yescharacteristic
260C22×C4C2×C22⋊D5911yescharacteristic
261C2×C8—22no—
262C2×C8—594no—
263C2×C8—1182no—
264C2×C8—1182no—
265C2×C8—1182no—
266C4⋊C4—22no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—22no—
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—1181no—
277C4⋊C4—1181no—
278C4⋊C4—1181no—
279C4⋊C4—1181no—
280C4⋊C4—2362no—
281C4⋊C4—2361no—
282C4⋊C4—2361no—
283C22⋊C4—41no—
284C22⋊C4—2361no—
285C42—41no—
286C42—2361no—
287C42D4×D5911yescharacteristic
288C42#1174568711yescharacteristic
289C42C2×C22⋊D5911yescharacteristic
290C42C2×C22⋊D5911yescharacteristic
291C23#1427372311yescharacteristic
292Q8—42no—
293Q8—44no—
294Q8—81no—
295Q8—82no—
296Q8—4721no—
297Q8—2362no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—42no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—4721no—
306C2×C4—4721no—
307C2×C4—4721no—
308C2×C4—2361no—
309C2×C4—2361no—
310C2×C4#1427369511yescharacteristic
311C2×C4#1427369511yescharacteristic
312C2×C4#1427369611yescharacteristic
313C2×C4#1427369811yescharacteristic
314C2×C4#1427369911yescharacteristic
315C2×C4#1427369911yescharacteristic
316C2×C4SD16×D5912yes—
317C2×C4#1427375612yes—
318C8—42no—
319C8—1184no—
320C8—2362no—
321C22—21no—
322C22—21no—
323C22#1683731011yescharacteristic
324C22#1683777311yescharacteristic
325C22#1683779311yescharacteristic
326C4—21no—
327C4—21no—
328C4—21no—
329C4—21no—
330C4—81no—
331C4—81no—
332C4—82no—
333C4—9441no—
334C4—4721no—
335C4—4721no—
336C4#1683753912yes—
337C4#1683760512yes—
338C2—21no—
339C2—21no—
340C2#2142515411yescharacteristic
341C2#2142515611yescharacteristic
342C2#2142517711yescharacteristic
subgroupG/H
centerC22#16837310
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC59×#3229C2
socleC22×C59#392
prime radicalC23×C59#127
hypercenter#3229D59
hypocenterC59#10227
LCSG ▷ C2×C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3229

Complements

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

NH ≅ G/Ntype
202. C59#10227semidirect

Maximal subgroups

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

subgroupindex
#214352402
#214346182
#214249172
#214226612
#214226602
#214219402
C59×#32292
#1022759