Finite Group Databasebeta

← #28492355

Subgroups — #28492355

Order 15104.

376 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 97 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
1C2×#16837698C211yesmaximal, characteristic
2#21424089C211yesmaximal, characteristic
3#21422782C211yesmaximal, characteristic
4#21422776C211yesmaximal, characteristic
5#21422490C211yesmaximal, characteristic
6#21421993C211yesmaximal, characteristic
7C59×#1570C211yesmaximal, characteristic
8C2×#14273727C2211yescharacteristic
9C2×C4×Dic118C2211yescharacteristic
10#16837698—22no—
11#16837698—22no—
12#16837455C2211yescharacteristic
13C2×#14273661—21no—
14#16837426C2211yescharacteristic
15C2×#14273652—21no—
16C2×#14273645—21no—
17#16837170—21no—
18#16837150—22no—
19#16837147—22no—
20#16837132—22no—
21C59×#431C2211yescharacteristic
22C59×#423C2211yescharacteristic
23C2×C4⋊C8×C59C2211yescharacteristic
24C59×#334—22no—
25C22×Dic118—21no—
26C22×Dic118D411yescharacteristic
27C2×#11745664—21no—
28#14273727—21no—
29#14273727—21no—
30#14273727—21no—
31#14273727D412yes—
32C4×Dic118—21no—
33C4×Dic118—21no—
34C4×Dic118—21no—
35C4×Dic118—21no—
36C2×C4⋊Dic59—21no—
37C2×C4⋊Dic59—21no—
38C2×C4⋊Dic59D411yescharacteristic
39C2×Dic59⋊C4—21no—
40C2×C4×Dic59—21no—
41#14273661—24no—
42C4⋊(C59⋊C8)—21no—
43C4⋊(C59⋊C8)—21no—
44C4⋊(C59⋊C8)—21no—
45#14273652—24no—
46#14273645—42no—
47C22⋊(C59⋊C8)—21no—
48C4×C59⋊C8—21no—
49C2×M16×C59—21no—
50C22×C8×C59—21no—
51C2×C42×C59C2311yescharacteristic
52C4⋊C8×C59—21no—
53C4⋊C8×C59—21no—
54C4⋊C8×C59—21no—
55C4⋊C8×C59D412yes—
56C4⋊C8×C59D412yes—
57C22⋊C8×C59—21no—
58C4×C8×C59—21no—
59C22×C4×C59C2×D411yescharacteristic
60C22×C4×C59#4111yescharacteristic
61C22×C4×C59#4111yescharacteristic
62C22×Dic59—21no—
63C22×Dic59—21no—
64C22×Dic59—41no—
65C2×Dic118—21no—
66C2×Dic118—21no—
67C2×Dic118—21no—
68C2×Dic118—21no—
69C2×Dic118—21no—
70C2×Dic118—22no—
71C2×Dic118—41no—
72C2×Dic118—82no—
73C2×Dic118SD1612yes—
74M16×C59—41no—
75M16×C59—41no—
76C2×C8×C59—22no—
77C2×C8×C59—22no—
78C2×C8×C59—41no—
79C2×C8×C59—41no—
80C2×C8×C59—41no—
81C42×C59C2×D411yescharacteristic
82C42×C59C2×D411yescharacteristic
83C42×C59C2×D411yescharacteristic
84C42×C59#4111yescharacteristic
85C4⋊Dic59—22no—
86C4⋊Dic59—21no—
87C4⋊Dic59—21no—
88C4⋊Dic59—22no—
89C4⋊Dic59—41no—
90C4⋊Dic59—41no—
91Dic59⋊C4—41no—
92Dic59⋊C4—41no—
93C4×Dic59—41no—
94C4×Dic59—41no—
95#11745664—41no—
96#11745664—41no—
97C2×C59⋊C8—41no—
98C2×C59⋊C8—41no—
99C23×C59#12311yescharacteristic
100C2×C4×C59—22no—
101C2×C4×C59—41no—
102C2×C4×C59#12011yescharacteristic
103C2×C4×C59#12111yescharacteristic
104C2×C4×C59#12111yescharacteristic
105C2×C4×C59#12111yescharacteristic
106C2×C4×C59#12311yescharacteristic
107C2×C4×C59#12411yescharacteristic
108C2×C4×C59C2×SD1611yescharacteristic
109C2×C4×C59#13511yescharacteristic
110C2×C4×C59#13711yescharacteristic
111C2×C4×C59#13711yescharacteristic
112C2×Dic59—22no—
113C2×Dic59—41no—
114C2×Dic59—41no—
115C2×Dic59—41no—
116C2×Dic59—42no—
117C2×Dic59—82no—
118C2×Dic59—81no—
119Dic118—22no—
120Dic118—22no—
121Dic118—41no—
122Dic118—41no—
123Dic118—82no—
124Dic118—82no—
125C8×C59—42no—
126C8×C59—42no—
127C8×C59—81no—
128C59⋊C8—81no—
129C59⋊C8—81no—
130#10454—591nomaximal, Sylow
131C22×C59—21no—
132C22×C59—21no—
133C22×C59#39411yescharacteristic
134C22×C59#48311yescharacteristic
135C22×C59#48511yescharacteristic
136C4×C59—21no—
137C4×C59—21no—
138C4×C59—21no—
139C4×C59#44811yescharacteristic
140C4×C59#45211yescharacteristic
141C4×C59#46511yescharacteristic
142C4×C59#47011yescharacteristic
143C4×C59#47312yes—
144Dic59—42no—
145Dic59—82no—
146Dic59—81no—
147C2×#477—591no—
148#1801—591no—
149#1570D5911yescharacteristic
150#1549—591no—
151#1548—591no—
152#1479—591no—
153#1301—591no—
154C2×C59—21no—
155C2×C59—21no—
156C2×C59#203511yescharacteristic
157C2×C59#204711yescharacteristic
158C2×C59#204811yescharacteristic
159C2×C4⋊Q8—591no—
160C2×C4×Q8—591no—
161#477—1182no—
162#477—1182no—
163#431—591no—
164#431C2×D5911yescharacteristic
165C2×#107—1181no—
166#423—591no—
167#423C2×D5911yescharacteristic
168C2×C4⋊C8C2×D5911yescharacteristic
169C2×#103—1181no—
170C2×#101—1181no—
171#343—1181no—
172#334—22no—
173#333—1182no—
174#332—1182no—
175Q8⋊C8—1182no—
176C59#1045411yescharacteristic, complemented, Sylow
177C22×Q8—591no—
178C22×Q8—1181no—
179C2×M16—21no—
180C2×M16—1181no—
181C22×C8—21no—
182C4⋊Q8—592no—
183C4⋊Q8—1181no—
184C4⋊Q8—1181no—
185C4⋊Q8—1181no—
186C4×Q8—1181no—
187C4×Q8—1181no—
188C4×Q8—1181no—
189C4×Q8—1181no—
190C2×C4⋊C4—591no—
191C2×C4⋊C4—1181no—
192C2×C4⋊C4—1181no—
193C2×C4⋊C4—1181no—
194C2×C42—1181no—
195C2×C42C22×D5911yescharacteristic
196#107—1184no—
197C4⋊C8—21no—
198C4⋊C8—21no—
199C4⋊C8—21no—
200C4⋊C8—1181no—
201C4⋊C8—1181no—
202C4⋊C8—1181no—
203C4⋊C8D23612yes—
204C4⋊C8C22⋊D5912yes—
205#103—1184no—
206#101—2362no—
207C22⋊C8—21no—
208C22⋊C8—1181no—
209C4×C8—21no—
210C4×C8—1181no—
211C2×Q8—592no—
212C2×Q8—1181no—
213C2×Q8—1181no—
214C2×Q8—1181no—
215C2×Q8—1182no—
216C2×Q8—1181no—
217C2×Q8—1181no—
218C2×Q8—4722no—
219C2×Q8—2361no—
220C22×C4—1181no—
221C22×C4—1181no—
222C22×C4—2361no—
223C22×C4#1174568511yescharacteristic
224C22×C4D4×D5911yescharacteristic
225C22×C4#1174568911yescharacteristic
226M16—41no—
227M16—41no—
228M16—2361no—
229M16—2361no—
230C2×C8—22no—
231C2×C8—22no—
232C2×C8—41no—
233C2×C8—41no—
234C2×C8—41no—
235C2×C8—2361no—
236C2×C8—2361no—
237C4⋊C4—1182no—
238C4⋊C4—1181no—
239C4⋊C4—1181no—
240C4⋊C4—1182no—
241C4⋊C4—2361no—
242C4⋊C4—2361no—
243C4⋊C4—2361no—
244C4⋊C4—2361no—
245C42—2361no—
246C42—2361no—
247C42C2×D23611yescharacteristic
248C42D4×D5911yescharacteristic
249C42#1174568911yescharacteristic
250C42C2×C22⋊D5911yescharacteristic
251C23#1427371211yescharacteristic
252Q8—1182no—
253Q8—1182no—
254Q8—4722no—
255Q8—4722no—
256Q8—2361no—
257Q8—2361no—
258C2×C4—22no—
259C2×C4—41no—
260C2×C4—1182no—
261C2×C4—4722no—
262C2×C4—4721no—
263C2×C4—2361no—
264C2×C4—2362no—
265C2×C4—2361no—
266C2×C4—2361no—
267C2×C4#1427369511yescharacteristic
268C2×C4#1427370111yescharacteristic
269C2×C4#1427370211yescharacteristic
270C2×C4#1427370211yescharacteristic
271C2×C4#1427371211yescharacteristic
272C2×C4#1427371611yescharacteristic
273C2×C4SD16×D5911yescharacteristic
274C2×C4#1427374811yescharacteristic
275C2×C4#1427375611yescharacteristic
276C2×C4#1427375611yescharacteristic
277C8—42no—
278C8—42no—
279C8—81no—
280C8—4721no—
281C8—4721no—
282C22—21no—
283C22—21no—
284C22#1683731911yescharacteristic
285C22#1683773411yescharacteristic
286C22#1683774611yescharacteristic
287C4—21no—
288C4—21no—
289C4—21no—
290C4—4722no—
291C4—4721no—
292C4—2362no—
293C4#1683753911yescharacteristic
294C4#1683756111yescharacteristic
295C4#1683763211yescharacteristic
296C4#1683766111yescharacteristic
297C4#1683767712yes—
298C2—21no—
299C2—21no—
300C2#2142529411yescharacteristic
301C2#2142536411yescharacteristic
302C2#2142537111yescharacteristic
subgroupG/H
centerC22#16837319
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC59×#1570C2
socleC22×C59#394
prime radicalC23×C59#123
hypercenter#1570D59
hypocenterC59#10454
LCSG ▷ C2×C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1570

Complements

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

NH ≅ G/Ntype
176. C59#10454semidirect

Maximal subgroups

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

subgroupindex
C2×#168376982
#214240892
#214227822
#214227762
#214224902
#214219932
C59×#15702
#1045459