Finite Group Databasebeta

← #28501158

Subgroups — #28501158

Order 15104.

360 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 67 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#21426029C211yesmaximal, characteristic, complemented
2#21426012C211yesmaximal, characteristic
3#21425419C211yesmaximal, characteristic, complemented
4#21422108C211yesmaximal, characteristic, complemented
5#21422105C211yesmaximal, characteristic
6#21422036C211yesmaximal, characteristic, complemented
7C59×#3250C211yesmaximal, characteristic
8C22×Dic236C2211yescharacteristic
9C2×#14273673—21no—
10C2×#14273671—21no—
11C2×#14273652—21no—
12#16837308—21no—
13#16837205—21no—
14#16837205—21no—
15C22⋊(C59⋊C16)C2211yescharacteristic
16C22⋊(C59⋊C16)C2211yescharacteristic
17#16837165C2211yescharacteristic
18C59×#529—21no—
19C59×#479—21no—
20C59×#478—21no—
21C59×#467C2211yescharacteristic
22C59×#466C2211yescharacteristic
23C59×#465—21no—
24C59×#464—21no—
25C59×#463—21no—
26C2×C4⋊C8×C59C2211yescharacteristic
27C22×Dic118—21no—
28C2×Dic236—21no—
29C2×Dic236—21no—
30C2×Dic236—42no—
31C2×C4⋊Dic59—21no—
32#14273673—42no—
33#14273671—42no—
34C2×C59⋊C16—21no—
35C2×C59⋊C16—21no—
36#14273652—42no—
37C59⋊#95—41no—
38C59⋊#95—41no—
39C2×Q16×C59D411yescharacteristic
40C2×SD16×C59—21no—
41C2×D8×C59D411yescharacteristic
42C22×C8×C59D411yescharacteristic
43C22×C8×C59C2311yescharacteristic
44#126×C59—41no—
45#125×C59—21no—
46#124×C59—21no—
47#123×C59—41no—
48C22⋊Q8×C59—21no—
49#121×C59—21no—
50C4×Q8×C59—21no—
51C4×D4×C59—21no—
52C4×D4×C59—41no—
53#117×C59—41no—
54C2×C42×C59D411yescharacteristic
55#107×C59D411yescharacteristic
56#107×C59D411yescharacteristic
57#106×C59—21no—
58C4⋊C8×C59—21no—
59C4⋊C8×C59—21no—
60#103×C59—21no—
61#103×C59—21no—
62#102×C59—21no—
63#102×C59—21no—
64C2×Q8×C59—21no—
65C2×D4×C59—21no—
66C2×D4×C59—41no—
67C22×C4×C59—41no—
68C22×C4×C59D811yescharacteristic
69C22×C4×C59D811yescharacteristic
70C22×C4×C59C2×D411yescharacteristic
71C22×Dic59—41no—
72C22×Dic59—41no—
73C2×Dic118—22no—
74C2×Dic118—41no—
75C2×Dic118—41no—
76C2×Dic118—41no—
77C2×Dic118—41no—
78C2×Dic118—41no—
79Q16×C59—41no—
80SD16×C59—81no—
81D8×C59—41no—
82C2×C8×C59—21no—
83C2×C8×C59—41no—
84C2×C8×C59—41no—
85C2×C8×C59C2×D411yescharacteristic
86C2×C8×C59C2×D411yescharacteristic
87C4⋊C4×C59—21no—
88C4⋊C4×C59—21no—
89C4⋊C4×C59—41no—
90C4⋊C4×C59—41no—
91C4⋊C4×C59—41no—
92C4⋊C4×C59—41no—
93C22⋊C4×C59—41no—
94C22⋊C4×C59—41no—
95C22⋊C4×C59—41no—
96C22⋊C4×C59—41no—
97C22⋊C4×C59—41no—
98C42×C59—21no—
99C42×C59—21no—
100C42×C59—41no—
101C4⋊Dic59—41no—
102C4⋊Dic59—41no—
103Dic236—22no—
104Dic236—22no—
105Dic236—41no—
106Dic236—41no—
107C59⋊C16—41no—
108C59⋊C16—41no—
109C23×C59—41no—
110C23×C59C2×D811yescharacteristic
111Q8×C59—41no—
112Q8×C59—41no—
113D4×C59—41no—
114D4×C59—41no—
115D4×C59—81no—
116D4×C59—81no—
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—41no—
124C2×C4×C59—41no—
125C2×C4×C59—41no—
126C2×C4×C59—41no—
127C2×C4×C59—82no—
128C2×C4×C59#12011yescharacteristic
129C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
130C2×Dic59—42no—
131C2×Dic59—81no—
132C2×Dic59—81no—
133C2×Dic59—81no—
134C2×Dic59—81no—
135C2×Dic59—81no—
136Dic118—42no—
137Dic118—42no—
138Dic118—81no—
139Dic118—81no—
140C8×C59—21no—
141C8×C59—21no—
142C8×C59—41no—
143#12163—591nomaximal, Sylow
144C22×C59—21no—
145C22×C59—21no—
146C22×C59—81no—
147C22×C59—81no—
148C22×C59—81no—
149C22×C59#44711yescharacteristic
150C22×C59#50811yescharacteristic
151C22×C59#51011yescharacteristic
152C4×C59—21no—
153C4×C59—41no—
154C4×C59—41no—
155C4×C59—81no—
156C4×C59—81no—
157C4×C59—81no—
158C4×C59#45411yescharacteristic
159C4×C59B(4,𝔽2)11yescharacteristic
160Dic59—82no—
161Dic59—161no—
162#3250D5911yescharacteristic
163#2188—591no—
164#2185—591no—
165#2056—591no—
166#1348—591no—
167#1347—591no—
168#1322—591no—
169C2×C59—21no—
170C2×C59—21no—
171C2×C59—161no—
172C2×C59#159811yescharacteristic
173C2×C59#218911yescharacteristic
174C2×C59#219111yescharacteristic
175C22×Q16—591no—
176#529—21no—
177C2×Q32—1181no—
178C2×SD32—1181no—
179#479—21no—
180#478—21no—
181#467C2×D5911yescharacteristic
182#466C2×D5911yescharacteristic
183#465—21no—
184#464—21no—
185#463—21no—
186C2×C4⋊C8C2×D5911yescharacteristic
187C2×#103—1181no—
188#391—1181no—
189#358—1181no—
190#358—1181no—
191C22⋊C16—591no—
192C22⋊C16—591no—
193#340—591no—
194C59#1216311yescharacteristic, complemented, Sylow
195C22×Q8—1181no—
196C2×Q16—1181no—
197C2×Q16—1181no—
198C2×Q16—2362no—
199C2×Q16C22⋊D5911yescharacteristic
200C2×SD16—21no—
201C2×D8C22⋊D5911yescharacteristic
202C22×C8C22⋊D5911yescharacteristic
203C22×C8C22×D5911yescharacteristic
204#126—41no—
205#125—21no—
206#124—21no—
207#123—41no—
208C22⋊Q8—21no—
209#121—21no—
210C4×Q8—21no—
211C4×D4—21no—
212C4×D4—41no—
213#117—41no—
214C2×C4⋊C4—1181no—
215C2×C42C22⋊D5911yescharacteristic
216Q32—2362no—
217SD32—2362no—
218C2×C16—1181no—
219C2×C16—1181no—
220#107C22⋊D5911yescharacteristic
221#107C22⋊D5911yescharacteristic
222#106—21no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225#103—21no—
226#103—21no—
227#103—2362no—
228#102—21no—
229#102—21no—
230#95—2361no—
231#95—2361no—
232C2×Q8—21no—
233C2×Q8—1182no—
234C2×Q8—2361no—
235C2×Q8—2361no—
236C2×Q8—2361no—
237C2×Q8—2361no—
238C2×Q8—2361no—
239C2×D4—21no—
240C2×D4—41no—
241C22×C4—41no—
242C22×C4—2361no—
243C22×C4—2361no—
244C22×C4#1174566911yescharacteristic
245C22×C4#1174566911yescharacteristic
246C22×C4C2×C22⋊D5911yescharacteristic
247Q16—41no—
248Q16—1182no—
249Q16—1182no—
250Q16—2361no—
251Q16—2361no—
252SD16—81no—
253D8—41no—
254C2×C8—21no—
255C2×C8—41no—
256C2×C8—41no—
257C2×C8C2×C22⋊D5911yescharacteristic
258C2×C8C2×C22⋊D5911yescharacteristic
259C4⋊C4—21no—
260C4⋊C4—21no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—2361no—
266C4⋊C4—2361no—
267C22⋊C4—41no—
268C22⋊C4—41no—
269C22⋊C4—41no—
270C22⋊C4—41no—
271C22⋊C4—41no—
272C42—21no—
273C42—21no—
274C42—41no—
275C16—2361no—
276C16—2361no—
277C23—41no—
278C23C2×#1174566911yescharacteristic
279Q8—41no—
280Q8—41no—
281Q8—4721no—
282Q8—4721no—
283Q8—2362no—
284Q8—2362no—
285D4—41no—
286D4—41no—
287D4—81no—
288D4—81no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—82no—
300C2×C4—4721no—
301C2×C4—4721no—
302C2×C4—4721no—
303C2×C4—4721no—
304C2×C4—4721no—
305C2×C4—2362no—
306C2×C4#1427369611yescharacteristic
307C2×C4#1427375311yescharacteristic
308C8—21no—
309C8—21no—
310C8—41no—
311C22—21no—
312C22—21no—
313C22—81no—
314C22—81no—
315C22—81no—
316C22#1683753611yescharacteristic
317C22#1683783211yescharacteristic
318C22#1683784211yescharacteristic
319C4—21no—
320C4—41no—
321C4—41no—
322C4—81no—
323C4—81no—
324C4—81no—
325C4—9441no—
326C4—4722no—
327C4#1683757511yescharacteristic
328C4#1683758911yescharacteristic
329C2—21no—
330C2—21no—
331C2—161no—
332C2#2142303111yescharacteristic
333C2#2142603611yescharacteristic
334C2#2142604711yescharacteristic
subgroupG/H
centerC22#16837536
commutator, cosocleC22×C8×C59C23
FrattiniC22×C8C22×D59
FittingC59×#3250C2
socleC22×C59#447
prime radicalC59×#466C22
hypercenter#3250D59
hypocenterC59#12163
LCSG ▷ C22×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C8 ◁ #3250

Complements

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

NH ≅ G/Ntype
1. #21426029C2semidirect
3. #21425419C2semidirect
4. #21422108C2semidirect
6. #21422036C2semidirect
194. C59#12163semidirect

Maximal subgroups

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

subgroupindex
#214260292
#214260122
#214254192
#214221082
#214221052
#214220362
C59×#32502
#1216359