Finite Group Databasebeta

← #28483876

Subgroups — #28483876

Order 15104.

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

normal subgroups only

This group has 424 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
1#21434136C211yesmaximal, characteristic
2#21433033C211yesmaximal, characteristic
3#21425035C211yesmaximal, characteristic
4#21422754C211yesmaximal, characteristic
5#21422524C211yesmaximal, characteristic
6#21421940C211yesmaximal, characteristic
7C59×#1301C211yesmaximal, characteristic
8#16837959C2211yescharacteristic
9C2×C4×Dic118C2211yescharacteristic
10#16837738C412yes—
11#16837698—21no—
12#16837613—21no—
13#16837560C412yes—
14#16837531—22no—
15#16837531—22no—
16C4×Dic236—22no—
17#16837455C2211yescharacteristic
18#16837430—21no—
19#16837360—21no—
20C2×#14273638C2211yescharacteristic
21#16837156—21no—
22#16837156—21no—
23#16837150—21no—
24#16837147—21no—
25#16837132—22no—
26C59×#431C2211yescharacteristic
27C59×#423C2211yescharacteristic
28C2×C4⋊C8×C59C2211yescharacteristic
29C22×Dic118C2×C411yescharacteristic
30C2×Dic236—22no—
31C22×C59⋊C8D411yescharacteristic
32C22×C59⋊C8D411yescharacteristic
33#14273727D411yescharacteristic
34#14273727D411yescharacteristic
35#14273719—21no—
36#14273706—21no—
37#14273706C2×C412yes—
38C4×Dic118—21no—
39C4×Dic118—21no—
40C4×Dic118—21no—
41C4×Dic118—21no—
42C4×Dic118—21no—
43C4×Dic118D412yes—
44#14273686—21no—
45C2×C4⋊Dic59C2×C411yescharacteristic
46C2×Dic59⋊C4—21no—
47C2×C4×Dic59—21no—
48#14273661—22no—
49C4⋊(C59⋊C8)—21no—
50Dic59⋊C8—24no—
51#14273652—22no—
52#14273652—22no—
53C22⋊(C59⋊C8)—21no—
54#14273638—21no—
55#14273638—21no—
56#14273637—22no—
57C4×C59⋊C8—21no—
58C8×Dic59—22no—
59C2×M16×C59—21no—
60C22×C8×C59—21no—
61C2×C42×C59C2311yescharacteristic
62C4⋊C8×C59—21no—
63C4⋊C8×C59—21no—
64C4⋊C8×C59—21no—
65C4⋊C8×C59—21no—
66C4⋊C8×C59D412yes—
67C22⋊C8×C59C2×C412yes—
68C4×C8×C59—21no—
69C22×C4×C59C22×C411yescharacteristic
70C22×C4×C59C2×D411yescharacteristic
71C22×C4×C59#4111yescharacteristic
72C22×Dic59—21no—
73C22×Dic59—21no—
74C2×Dic118—21no—
75C2×Dic118—21no—
76C2×Dic118—21no—
77C2×Dic118—42no—
78C2×Dic118C22⋊C412yes—
79M16×C59—41no—
80M16×C59—41no—
81C2×C8×C59—22no—
82C2×C8×C59—22no—
83C2×C8×C59—41no—
84C2×C8×C59—41no—
85C2×C8×C59—41no—
86C42×C59C2×D411yescharacteristic
87C42×C59C2×D411yescharacteristic
88C42×C59C2×D411yescharacteristic
89C42×C59#4111yescharacteristic
90C22⋊Dic59—22no—
91C4⋊Dic59—21no—
92C4⋊Dic59—21no—
93C4⋊Dic59—21no—
94C4⋊Dic59—21no—
95C4⋊Dic59—21no—
96C4⋊Dic59C22⋊C412yes—
97Dic59⋊C4—22no—
98Dic59⋊C4—22no—
99Dic59⋊C4—41no—
100Dic59⋊C4—41no—
101C4×Dic59—22no—
102C4×Dic59—41no—
103C4×Dic59—41no—
104C2×C59⋊C8—21no—
105C2×C59⋊C8—21no—
106C2×C59⋊C8—21no—
107C2×C59⋊C8—21no—
108C2×C59⋊C8—21no—
109C2×C59⋊C8—41no—
110C2×C59⋊C8Q1612yescomplemented
111Dic236—44no—
112C23×C59C4×D411yescharacteristic
113C2×C4×C59—41no—
114C2×C4×C59—41no—
115C2×C4×C59C2×C22⋊C411yescharacteristic
116C2×C4×C59C4×D411yescharacteristic
117C2×C4×C59#12011yescharacteristic
118C2×C4×C59#12111yescharacteristic
119C2×C4×C59#12111yescharacteristic
120C2×C4×C59#12311yescharacteristic
121C2×C4×C59C2×Q1611yescharacteristic
122C2×C4×C59#13511yescharacteristic
123C2×C4×C59#13711yescharacteristic
124C2×C4×C59#13711yescharacteristic
125C2×Dic59—22no—
126C2×Dic59—22no—
127C2×Dic59—22no—
128C2×Dic59—41no—
129C2×Dic59—41no—
130C2×Dic59—42no—
131C2×Dic59—41no—
132Dic118—22no—
133Dic118—22no—
134Dic118—41no—
135Dic118—41no—
136Dic118—42no—
137Dic118—81no—
138C8×C59—42no—
139C8×C59—42no—
140C8×C59—81no—
141C59⋊C8—22no—
142C59⋊C8—41no—
143C59⋊C8—81no—
144#9048—591nomaximal, Sylow
145C22×C59—21no—
146C22×C59—21no—
147C22×C59#38611yescharacteristic
148C22×C59C4×Q1611yescharacteristic
149C22×C59#44111yescharacteristic
150C4×C59—21no—
151C4×C59—21no—
152C4×C59—21no—
153C4×C59—21no—
154C4×C59#45111yescharacteristic
155C4×C59#45211yescharacteristic
156C4×C59#46211yescharacteristic
157C4×C59#46411yescharacteristic
158Dic59—42no—
159Dic59—42no—
160Dic59—42no—
161Dic59—81no—
162#3104—591no—
163#2961—591no—
164#1985—591no—
165#1542—591no—
166#1487—591no—
167#1301D5911yescharacteristic
168#1281—591no—
169C2×C59—21no—
170C2×C59—21no—
171C2×C59#187811yescharacteristic
172C2×C59#188111yescharacteristic
173C2×C59#189111yescharacteristic
174#533—591no—
175C2×C4×Q8—591no—
176#484—592no—
177#477—1181no—
178#462—1181no—
179#451—592no—
180#446—1182no—
181#446—1182no—
182C4×Q16—1182no—
183#431—591no—
184#431C2×D5911yescharacteristic
185#424—1181no—
186#423C2×D5911yescharacteristic
187C2×C4⋊C8C2×D5911yescharacteristic
188#405—1181no—
189C2×#97—591no—
190#336—1181no—
191#336—1181no—
192#333—1181no—
193#332—1181no—
194Q8⋊C8—1182no—
195C59#904811yescharacteristic, complemented, Sylow
196C22×Q8—591no—
197C2×Q16—1182no—
198C2×M16—21no—
199C22×C8—21no—
200C22×C8—591no—
201C22×C8—591no—
202C4⋊Q8—591no—
203C4⋊Q8—591no—
204#125—1181no—
205C22⋊Q8—592no—
206C22⋊Q8—1181no—
207C4×Q8—592no—
208C4×Q8—1181no—
209C4×Q8—1181no—
210C4×Q8—1181no—
211C4×Q8—1181no—
212C4×Q8—1181no—
213#117—1181no—
214C2×C4⋊C4—591no—
215C2×C4⋊C4—1181no—
216C2×C42—1181no—
217C2×C42C22×D5911yescharacteristic
218#107—1182no—
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8—21no—
222C4⋊C8—21no—
223C4⋊C8—1184no—
224C4⋊C8—1181no—
225C4⋊C8D23612yes—
226#103—1182no—
227#103—1182no—
228C22⋊C8—1181no—
229C22⋊C8C4×D5912yes—
230#97—1182no—
231#97—1181no—
232#97—1181no—
233C4×C8—21no—
234C4×C8—1182no—
235C4×C8—1181no—
236C2×Q8—592no—
237C2×Q8—1181no—
238C2×Q8—1181no—
239C2×Q8—1181no—
240C2×Q8—2362no—
241C22×C4—1181no—
242C22×C4—1181no—
243C22×C4C2×C4×D5911yescharacteristic
244C22×C4D4×D5911yescharacteristic
245C22×C4#1174568911yescharacteristic
246Q16—2364no—
247M16—41no—
248M16—41no—
249C2×C8—22no—
250C2×C8—22no—
251C2×C8—41no—
252C2×C8—41no—
253C2×C8—41no—
254C2×C8—592no—
255C2×C8—1181no—
256C2×C8—1181no—
257C2×C8—1181no—
258C2×C8—1181no—
259C2×C8—1181no—
260C2×C8—2361no—
261C4⋊C4—592no—
262C4⋊C4—1182no—
263C4⋊C4—1181no—
264C4⋊C4—1181no—
265C4⋊C4—1181no—
266C4⋊C4—1182no—
267C4⋊C4—1181no—
268C4⋊C4—1181no—
269C4⋊C4—2361no—
270C4⋊C4—2361no—
271C22⋊C4—1182no—
272C42—1182no—
273C42—2361no—
274C42—2361no—
275C42C2×D23611yescharacteristic
276C42#1174568511yescharacteristic
277C42D4×D5911yescharacteristic
278C42D4×D5911yescharacteristic
279C23#1427368911yescharacteristic
280Q8—1182no—
281Q8—1182no—
282Q8—4721no—
283Q8—2362no—
284Q8—2361no—
285Q8—2361no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—1182no—
289C2×C4—1182no—
290C2×C4—1182no—
291C2×C4—2362no—
292C2×C4—2361no—
293C2×C4—2361no—
294C2×C4—2361no—
295C2×C4C22⋊C4×D5911yescharacteristic
296C2×C4C4×D23611yescharacteristic
297C2×C4#1427369711yescharacteristic
298C2×C4#1427370011yescharacteristic
299C2×C4#1427370211yescharacteristic
300C2×C4#1427371211yescharacteristic
301C2×C4Q16×D5911yescharacteristic
302C2×C4#1427374611yescharacteristic
303C2×C4#1427375611yescharacteristic
304C2×C4#1427375611yescharacteristic
305C8—42no—
306C8—42no—
307C8—81no—
308C8—1182no—
309C8—4721no—
310C8—2361no—
311C22—21no—
312C22—21no—
313C22#1683728011yescharacteristic
314C22#1683749611yescharacteristic
315C22#1683750811yescharacteristic
316C4—21no—
317C4—21no—
318C4—21no—
319C4—21no—
320C4—4721no—
321C4—2362no—
322C4—2362no—
323C4—2362no—
324C4#1683755911yescharacteristic
325C4#1683756511yescharacteristic
326C4#1683761111yescharacteristic
327C4#1683762311yescharacteristic
328C2—21no—
329C2—21no—
330C2#2142443811yescharacteristic
331C2#2142445911yescharacteristic
332C2#2142451311yescharacteristic
subgroupG/H
centerC2×C4#14273697
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1301C2
socleC22×C59#386
prime radicalC23×C59C4×D4
cosocleC2×C42×C59C23
hypercenter#1301D59
hypocenterC59#9048
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1301

Complements

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

NH ≅ G/Ntype
110. C2×C59⋊C8Q16semidirect
195. C59#9048semidirect

Maximal subgroups

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

subgroupindex
#214341362
#214330332
#214250352
#214227542
#214225242
#214219402
C59×#13012
#904859