Finite Group Databasebeta

← #28509103

Subgroups — #28509103

Order 15104.

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

normal subgroups only

This group has 316 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#21435157C211yesmaximal, characteristic
2#21425791C211yesmaximal, characteristic, complemented
3#21422268C212yesmaximal, complemented
4#21422262C212yesmaximal, complemented
5C59×#2133C211yesmaximal, characteristic, complemented
6#16838007C411yescharacteristic
7#16837792C411yescharacteristic
8#16837776C2211yescharacteristic
9#16837770C2211yescharacteristic
10#16837756C411yescharacteristic
11#16837614—21no—
12#16837604—21no—
13#16837547—21no—
14#16837541—21no—
15#16837463C411yescharacteristic
16#16837386—21no—
17#16837201—21no—
18#16837201C2212yes—
19C59×#536C2211yescharacteristic
20C2×#101×C59—21no—
21C59×#356C2212yes—
22C59×#355—21no—
23C2×#11745671—21no—
24C2×#11745671—21no—
25C2×#11745671—21no—
26C2×#11745669—21no—
27C2×#11745664—41no—
28#14273724C2×C411yescharacteristic
29#14273719C2×C411yescharacteristic
30#14273712—41no—
31#14273701—21no—
32#14273700—41no—
33C4×D236—21no—
34C4⋊(C59⋊C8)—21no—
35#14273651—21no—
36#14273651—21no—
37#14273651—21no—
38#14273648—21no—
39#14273646—22no—
40#14273644—41no—
41C22⋊(C59⋊C8)—21no—
42#14273638C2×C411yescharacteristic
43C4×C59⋊C8C2×C411yescharacteristic
44C2×#41×C59D411yescharacteristic
45C22×Q8×C59D411yescharacteristic
46C2×M16×C59—41no—
47C4⋊Q8×C59D411yescharacteristic
48C4⋊Q8×C59C2311yescharacteristic
49#124×C59C2×C411yescharacteristic
50#124×C59D411yescharacteristic
51#123×C59—41no—
52C22⋊Q8×C59—41no—
53#117×C59C2×C411yescharacteristic
54#101×C59—22no—
55#101×C59—41no—
56#41×C59—81no—
57C2×Q8×C59—21no—
58C2×Q8×C59—81no—
59C2×Q8×C59C22⋊C411yescharacteristic
60C2×Q8×C59C2×D411yescharacteristic
61C2×Q8×C59C2×D411yescharacteristic
62C2×D4×C59—21no—
63C2×D4×C59C22⋊C411yescharacteristic
64C22×C4×C59—21no—
65C22×C4×C59—21no—
66C22×C4×C59C22⋊C411yescharacteristic
67C2×C22⋊D59—41no—
68C2×D236—21no—
69C2×D236—41no—
70C2×C4×D59—41no—
71M16×C59—42no—
72M16×C59—81no—
73C2×C8×C59—81no—
74C4⋊C4×C59—21no—
75C4⋊C4×C59—41no—
76C4⋊C4×C59—41no—
77C22⋊C4×C59—21no—
78C22⋊C4×C59—41no—
79C22⋊C4×C59—41no—
80C42×C59C22⋊C411yescharacteristic
81C42×C59C22×C411yescharacteristic
82#11745671—81no—
83#11745671—81no—
84#11745671—81no—
85#11745669—81no—
86(C2×D59)⋊C4—41no—
87(C2×D59)⋊C4—41no—
88C4⋊Dic59—21no—
89Dic59⋊C4—41no—
90#11745664—42no—
91#11745664—81no—
92C2×C59⋊C8—21no—
93C2×C59⋊C8—21no—
94C2×C59⋊C8—81no—
95C23×C59—21no—
96C23×C59#9911yescharacteristic
97C22×D59—41no—
98Q8×C59—41no—
99Q8×C59—41no—
100Q8×C59—41no—
101Q8×C59—81no—
102D4×C59—41no—
103D4×C59—81no—
104C2×C4×C59—21no—
105C2×C4×C59—21no—
106C2×C4×C59—21no—
107C2×C4×C59—21no—
108C2×C4×C59—21no—
109C2×C4×C59—41no—
110C2×C4×C59—41no—
111C2×C4×C59—41no—
112C2×C4×C59—41no—
113C2×C4×C59—41no—
114C2×C4×C59#9911yescharacteristic
115C2×C4×C59C2×C22⋊C411yescharacteristic
116C22⋊D59—161no—
117C2×Dic59—41no—
118D236—41no—
119D236—41no—
120D236—161no—
121C4×D59—161no—
122C8×C59—82no—
123C59⋊C8—41no—
124C59⋊C8—41no—
125C59⋊C8—82no—
126#13611—591nomaximal, Sylow
127C22×C59—21no—
128C22×C59—41no—
129C22×C59—41no—
130C22×C59—81no—
131C22×C59C2×#9911yescharacteristic
132C2×D59—81no—
133C2×D59—81no—
134C2×D59—81no—
135C4×C59—21no—
136C4×C59—21no—
137C4×C59—41no—
138C4×C59—41no—
139C4×C59—41no—
140C4×C59—41no—
141C4×C59—41no—
142Dic59—81no—
143#3221—591no—
144#2135—591no—
145#2133D5911yescharacteristic, complemented
146#1413—592no—
147#1410—592no—
148C2×C59—21no—
149C2×C59—41no—
150C2×C59—81no—
151C2×C59#212711yescharacteristic
152D59—161no—
153#540—591no—
154#536C2×D5911yescharacteristic
155#496—591no—
156#492—591no—
157#491—591no—
158#488—591no—
159#463—1181no—
160#461—1181no—
161#449—1181no—
162#448—1181no—
163#433—591no—
164#413—1181no—
165C2×#101—21no—
166#356—592no—
167#356—1181no—
168#356C2×D5912yes—
169#355—21no—
170C59#1361111yescharacteristic, complemented, Sylow
171C2×#41D23611yescharacteristic
172C22×Q8C22⋊D5911yescharacteristic
173C2×SD16—1181no—
174C2×SD16—1181no—
175C2×SD16—1181no—
176C2×D8—1181no—
177C2×M16—41no—
178C2×M16—2361no—
179C4⋊Q8D23611yescharacteristic
180C4⋊Q8C22×D5911yescharacteristic
181#127—591no—
182#125—591no—
183#124C4×D5911yescharacteristic
184#124C22⋊D5911yescharacteristic
185#123—41no—
186#123—2361no—
187C22⋊Q8—41no—
188#121—1181no—
189#121—2361no—
190C4×D4—1181no—
191#117C4×D5911yescharacteristic
192C4⋊C8—1181no—
193#103—1181no—
194#103—1181no—
195#103—1181no—
196#102—1181no—
197#101—22no—
198#101—41no—
199#101—1182no—
200#100—2361no—
201C22⋊C8—1181no—
202#97—591no—
203C4×C8—591no—
204#41—81no—
205C2×Q8—21no—
206C2×Q8—81no—
207C2×Q8(C2×D59)⋊C411yescharacteristic
208C2×Q8C2×D23611yescharacteristic
209C2×Q8C2×C22⋊D5911yescharacteristic
210C2×D4—21no—
211C2×D4—1181no—
212C2×D4—2361no—
213C2×D4—2361no—
214C2×D4(C2×D59)⋊C411yescharacteristic
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—2361no—
218C22×C4(C2×D59)⋊C411yescharacteristic
219SD16—4721no—
220SD16—4721no—
221SD16—4721no—
222D8—4721no—
223M16—42no—
224M16—81no—
225M16—4721no—
226M16—2362no—
227C2×C8—81no—
228C2×C8—1181no—
229C2×C8—1181no—
230C2×C8—4721no—
231C4⋊C4—21no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4—1181no—
235C4⋊C4—2361no—
236C22⋊C4—21no—
237C22⋊C4—41no—
238C22⋊C4—41no—
239C22⋊C4—2361no—
240C22⋊C4—2361no—
241C42(C2×D59)⋊C411yescharacteristic
242C42C2×C4×D5911yescharacteristic
243C23—21no—
244C23—2361no—
245C23#1427364111yescharacteristic
246Q8—41no—
247Q8—41no—
248Q8—41no—
249Q8—81no—
250D4—41no—
251D4—81no—
252D4—9441no—
253D4—9441no—
254D4—2361no—
255D4—2361no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—9441no—
267C2×C4—2361no—
268C2×C4#1427364111yescharacteristic
269C2×C4C2×(C2×D59)⋊C411yescharacteristic
270C8—82no—
271C8—4722no—
272C8—2361no—
273C8—2361no—
274C22—21no—
275C22—41no—
276C22—41no—
277C22—81no—
278C22—4721no—
279C22—4721no—
280C22—4721no—
281C22C2×#1427364111yescharacteristic
282C4—21no—
283C4—21no—
284C4—41no—
285C4—41no—
286C4—41no—
287C4—41no—
288C4—41no—
289C4—4721no—
290C2—21no—
291C2—41no—
292C2—81no—
293C2—9441no—
294C2#2142576311yescharacteristic
subgroupG/H
centerC2#21425763
commutatorC42×C59C22×C4
FrattiniC4⋊Q8C22×D59
FittingC59×#2133C2
socleC2×C59#2127
prime radical#21435157C2
cosocleC4⋊Q8×C59C23
hypercenter#2133D59
hypocenterC59#13611
LCSG ▷ C42×C59 ▷ C2×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C22 ◁ C22×C4 ◁ #536 ◁ #2133

Complements

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

NH ≅ G/Ntype
2. #21425791C2semidirect
3. #21422268C2semidirect
4. #21422262C2semidirect
5. C59×#2133C2semidirect
145. #2133D59semidirect
170. C59#13611semidirect

Maximal subgroups

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

subgroupindex
#214351572
#214257912
#214222682
#214222622
C59×#21332
#1361159