Finite Group Databasebeta

← #28490961

Subgroups — #28490961

Order 15104.

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

normal subgroups only

This group has 462 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#21435044C211yesmaximal, characteristic
2#21434618C211yesmaximal, characteristic
3#21425070C211yesmaximal, characteristic
4#21422661C211yesmaximal, characteristic
5#21422660C211yesmaximal, characteristic
6#21421940C211yesmaximal, characteristic
7C59×#3230C211yesmaximal, characteristic
8#16837959C2211yescharacteristic
9#16837812—21no—
10#16837812—21no—
11#16837808—21no—
12#16837808—21no—
13#16837781—21no—
14#16837752—21no—
15#16837609—22no—
16#16837558—22no—
17#16837455C2211yescharacteristic
18#16837438—22no—
19C2×#14273638C2211yescharacteristic
20#16837268C2211yescharacteristic
21#16837164—22no—
22#16837140—22no—
23#16837139—22no—
24C59×#533C2211yescharacteristic
25C2×C4⋊Q8×C59C2211yescharacteristic
26C59×#500—21no—
27C59×#487—21no—
28C59×#484—22no—
29C59×#477—22no—
30C59×#431C2211yescharacteristic
31C2×#11745672—24no—
32C22×C59⋊C8D412yes—
33#14273727D411yescharacteristic
34#14273727D411yescharacteristic
35#14273719—21no—
36#14273706—21no—
37#14273706—21no—
38C4×Dic118—21no—
39C4×Dic118—21no—
40#14273686—21no—
41C2×C4⋊Dic59—21no—
42#14273660—22no—
43#14273660—22no—
44#14273658—22no—
45#14273658—22no—
46C4⋊(C59⋊C8)—21no—
47#14273651—24no—
48C22⋊(C59⋊C8)D412yes—
49#14273638—21no—
50#14273638—21no—
51C4×C59⋊C8—21no—
52C59⋊#95—21no—
53C59⋊#95—21no—
54C22×Q8×C59—21no—
55C4⋊Q8×C59—21no—
56C4⋊Q8×C59—21no—
57C4⋊Q8×C59—21no—
58C4⋊Q8×C59D412yes—
59C4⋊Q8×C59D411yescharacteristic
60C4⋊Q8×C59D411yescharacteristic
61#125×C59—21no—
62C22⋊Q8×C59—21no—
63C22⋊Q8×C59—21no—
64C4×Q8×C59—21no—
65C4×Q8×C59—21no—
66#117×C59—21no—
67C2×C4⋊C4×C59—21no—
68C2×C4⋊C4×C59—21no—
69C2×C42×C59C2311yescharacteristic
70#107×C59—24no—
71C4⋊C8×C59—21no—
72#103×C59—24no—
73C22⋊C8×C59D412yes—
74C4×C8×C59—21no—
75C2×Q8×C59—21no—
76C2×Q8×C59—22no—
77C2×Q8×C59—21no—
78C2×Q8×C59—41no—
79C2×Q8×C59—82no—
80C22×C4×C59—41no—
81C22×C4×C59C2×D411yescharacteristic
82C22×C4×C59C2×D411yescharacteristic
83C22×C4×C59C2×D411yescharacteristic
84C22×Dic59—41no—
85C2×Dic118—21no—
86C2×Dic118—21no—
87C2×C8×C59—22no—
88C4⋊C4×C59—22no—
89C4⋊C4×C59—21no—
90C4⋊C4×C59—21no—
91C4⋊C4×C59—21no—
92C4⋊C4×C59—22no—
93C4⋊C4×C59—21no—
94C4⋊C4×C59—41no—
95C4⋊C4×C59—41no—
96C4⋊C4×C59—41no—
97C4⋊C4×C59—41no—
98C22⋊C4×C59—41no—
99C42×C59—41no—
100C42×C59C2×D411yescharacteristic
101C42×C59C2×D411yescharacteristic
102C42×C59C2×D411yescharacteristic
103C42×C59#4111yescharacteristic
104C22⋊Dic59—41no—
105#11745672—84no—
106C4⋊Dic59—21no—
107C4⋊Dic59—21no—
108C4⋊Dic59—21no—
109C4⋊Dic59—21no—
110C4⋊Dic59—42no—
111Dic59⋊C4—41no—
112Dic59⋊C4—41no—
113C4×Dic59—41no—
114C2×C59⋊C8—22no—
115C2×C59⋊C8—22no—
116C2×C59⋊C8—22no—
117C2×C59⋊C8Q1614yescomplemented
118C23×C59#12711yescharacteristic
119Q8×C59—44no—
120Q8×C59—42no—
121Q8×C59—81no—
122Q8×C59—82no—
123C2×C4×C59—41no—
124C2×C4×C59—41no—
125C2×C4×C59—41no—
126C2×C4×C59—41no—
127C2×C4×C59—42no—
128C2×C4×C59—82no—
129C2×C4×C59#12011yescharacteristic
130C2×C4×C59#12011yescharacteristic
131C2×C4×C59#12011yescharacteristic
132C2×C4×C59#12111yescharacteristic
133C2×C4×C59#12111yescharacteristic
134C2×C4×C59#12111yescharacteristic
135C2×C4×C59C2×Q1612yes—
136C2×C4×C59#13712yes—
137C2×Dic59—41no—
138C2×Dic59—41no—
139C2×Dic59—81no—
140C2×Dic59—81no—
141C2×Dic59—81no—
142Dic118—42no—
143Dic118—81no—
144C8×C59—42no—
145C59⋊C8—24no—
146C59⋊C8—42no—
147#10224—591nomaximal, Sylow
148C22×C59—21no—
149C22×C59—21no—
150C22×C59#39211yescharacteristic
151C22×C59#49411yescharacteristic
152C22×C59#49711yescharacteristic
153C4×C59—21no—
154C4×C59—21no—
155C4×C59—21no—
156C4×C59—21no—
157C4×C59—82no—
158C4×C59—81no—
159C4×C59—81no—
160C4×C59#45112yes—
161C4×C59#46212yes—
162Dic59—81no—
163Dic59—81no—
164Dic59—161no—
165#3230D5911yescharacteristic
166#3209—591no—
167#3162—591no—
168#1993—591no—
169#1518—591no—
170#1518—591no—
171#1281—591no—
172C2×C59—21no—
173C2×C59—21no—
174C2×C59#200811yescharacteristic
175C2×C59#200911yescharacteristic
176C2×C59#201611yescharacteristic
177#533—591no—
178#533C2×D5911yescharacteristic
179C2×C4⋊Q8C2×D5911yescharacteristic
180#501—1181no—
181#501—1181no—
182#501—1181no—
183#501—1181no—
184#500—21no—
185#494—1181no—
186#487—21no—
187#487—1181no—
188#484—22no—
189#477—22no—
190#462—1182no—
191#451—1182no—
192#431—591no—
193#431C2×D5911yescharacteristic
194#427—1182no—
195C2×#97—591no—
196#382—591no—
197#340—1182no—
198#328—1182no—
199#328—1182no—
200C59#1022411yescharacteristic, complemented, Sylow
201C22×Q8—21no—
202C2×Q16—1184no—
203C22×C8—592no—
204C4⋊Q8—21no—
205C4⋊Q8—21no—
206C4⋊Q8—21no—
207C4⋊Q8—591no—
208C4⋊Q8—591no—
209C4⋊Q8C22⋊D5912yes—
210C4⋊Q8C22⋊D5911yescharacteristic
211C4⋊Q8C22⋊D5911yescharacteristic
212#125—21no—
213#125—1181no—
214C22⋊Q8—21no—
215C22⋊Q8—21no—
216C22⋊Q8—1181no—
217C22⋊Q8—1181no—
218C4×Q8—21no—
219C4×Q8—21no—
220C4×Q8—1181no—
221C4×Q8—1181no—
222#117—21no—
223#117—1181no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—1181no—
227C2×C42C22×D5911yescharacteristic
228#107—24no—
229#107—1182no—
230#107—1182no—
231#106—1182no—
232#106—1182no—
233C4⋊C8—21no—
234C4⋊C8—1181no—
235#103—24no—
236#103—1184no—
237C22⋊C8—592no—
238C22⋊C8C22⋊D5912yes—
239#97—1181no—
240#97—1181no—
241C4×C8—21no—
242C4×C8—1181no—
243#95—1181no—
244#95—1181no—
245C2×Q8—22no—
246C2×Q8—21no—
247C2×Q8—21no—
248C2×Q8—41no—
249C2×Q8—82no—
250C2×Q8—1181no—
251C2×Q8—1181no—
252C22×C4—41no—
253C22×C4—2361no—
254C22×C4D4×D5911yescharacteristic
255C22×C4D4×D5911yescharacteristic
256C22×C4C2×C22⋊D5911yescharacteristic
257Q16—4724no—
258C2×C8—22no—
259C2×C8—594no—
260C2×C8—1182no—
261C2×C8—1182no—
262C2×C8—1182no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—22no—
266C4⋊C4—21no—
267C4⋊C4—22no—
268C4⋊C4—21no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—1181no—
274C4⋊C4—1181no—
275C4⋊C4—1181no—
276C4⋊C4—1181no—
277C4⋊C4—2362no—
278C4⋊C4—2361no—
279C4⋊C4—2361no—
280C22⋊C4—41no—
281C22⋊C4—2361no—
282C42—41no—
283C42—2361no—
284C42D4×D5911yescharacteristic
285C42#1174568711yescharacteristic
286C42C2×C22⋊D5911yescharacteristic
287C42C2×C22⋊D5911yescharacteristic
288C23#1427372311yescharacteristic
289Q8—44no—
290Q8—42no—
291Q8—81no—
292Q8—82no—
293Q8—4721no—
294Q8—2362no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—82no—
301C2×C4—4721no—
302C2×C4—4721no—
303C2×C4—4721no—
304C2×C4—2361no—
305C2×C4—2361no—
306C2×C4#1427369511yescharacteristic
307C2×C4#1427369511yescharacteristic
308C2×C4#1427369611yescharacteristic
309C2×C4#1427369811yescharacteristic
310C2×C4#1427369911yescharacteristic
311C2×C4#1427369911yescharacteristic
312C2×C4Q16×D5912yes—
313C2×C4#1427375912yes—
314C8—42no—
315C8—1184no—
316C8—2362no—
317C22—21no—
318C22—21no—
319C22#1683731011yescharacteristic
320C22#1683778011yescharacteristic
321C22#1683779511yescharacteristic
322C4—21no—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—81no—
327C4—82no—
328C4—81no—
329C4—9441no—
330C4—4721no—
331C4—4721no—
332C4#1683755612yes—
333C4#1683761012yes—
334C2—21no—
335C2—21no—
336C2#2142514711yescharacteristic
337C2#2142515611yescharacteristic
338C2#2142519411yescharacteristic
subgroupG/H
centerC22#16837310
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC59×#3230C2
socleC22×C59#392
prime radicalC23×C59#127
hypercenter#3230D59
hypocenterC59#10224
LCSG ▷ C2×C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3230

Complements

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

NH ≅ G/Ntype
117. C2×C59⋊C8Q16semidirect
200. C59#10224semidirect

Maximal subgroups

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

subgroupindex
#214350442
#214346182
#214250702
#214226612
#214226602
#214219402
C59×#32302
#1022459