Finite Group Databasebeta

← #28486108

Subgroups — #28486108

Order 15104.

450 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 109 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#21433179C211yesmaximal, characteristic
2#21429311C211yesmaximal, characteristic
3#21424089C211yesmaximal, characteristic
4#21422782C211yesmaximal, characteristic
5#21422780C211yesmaximal, characteristic
6#21421999C211yesmaximal, characteristic
7C59×#1303C211yesmaximal, characteristic
8C2×#14273727C2211yescharacteristic
9C2×C4×Dic118C2211yescharacteristic
10#16837532—22no—
11#16837529—22no—
12#16837456—22no—
13#16837455C2211yescharacteristic
14#16837426C2211yescharacteristic
15C2×#14273645—21no—
16#16837342C414yes—
17#16837308—21no—
18#16837307—22no—
19#16837268—21no—
20#16837170—21no—
21#16837149—22no—
22#16837147—22no—
23C59×#431C2211yescharacteristic
24C59×#423C2211yescharacteristic
25C4×C4⋊C4×C59C2211yescharacteristic
26C22×Dic118—21no—
27C22×Dic118D411yescharacteristic
28C2×#11745664—21no—
29#14273727—21no—
30#14273727—21no—
31#14273727—21no—
32#14273727—21no—
33C4×Dic118—21no—
34C4×Dic118—21no—
35C4×Dic118—21no—
36C4×Dic118—21no—
37C2×C4⋊Dic59—21no—
38C2×C4⋊Dic59C2×C412yes—
39C2×C4⋊Dic59D411yescharacteristic
40C2×Dic59⋊C4C2×C412yes—
41C2×C4×Dic59D412yes—
42C4⋊(C59⋊C8)—21no—
43C4⋊(C59⋊C8)—21no—
44C4⋊(C59⋊C8)D412yes—
45Dic59⋊C8—24no—
46#14273645—42no—
47C22⋊(C59⋊C8)—21no—
48#14273637—24no—
49C4×C59⋊C8—21no—
50C59⋊#95—24no—
51C59⋊#95—24no—
52C2×M16×C59—21no—
53C2×C4⋊C4×C59—21no—
54C2×C42×C59—21no—
55C2×C42×C59C2311yescharacteristic
56C4⋊C8×C59—21no—
57C4⋊C8×C59—21no—
58C4⋊C8×C59D412yes—
59C22⋊C8×C59—21no—
60C4×C8×C59—21no—
61#95×C59C2×C412yes—
62C22×C4×C59—22no—
63C22×C4×C59C22×C411yescharacteristic
64C22×C4×C59C2×D411yescharacteristic
65C22×C4×C59C2×D411yescharacteristic
66C22×Dic59—22no—
67C22×Dic59C22⋊C412yes—
68C22×Dic59C22⋊C412yes—
69C2×Dic118—21no—
70C2×Dic118—21no—
71C2×Dic118—21no—
72C2×Dic118—21no—
73C2×Dic118—21no—
74C2×Dic118—21no—
75C2×Dic118—44no—
76C2×Dic118—41no—
77C2×Dic118—41no—
78M16×C59—41no—
79M16×C59—41no—
80C2×C8×C59—22no—
81C2×C8×C59—41no—
82C4⋊C4×C59—42no—
83C42×C59—41no—
84C42×C59—41no—
85C42×C59C2×D411yescharacteristic
86C42×C59C2×D411yescharacteristic
87C42×C59#4111yescharacteristic
88C42×C59#4111yescharacteristic
89C4⋊Dic59—21no—
90C4⋊Dic59—21no—
91C4⋊Dic59—22no—
92C4⋊Dic59—22no—
93C4⋊Dic59—41no—
94C4⋊Dic59—41no—
95Dic59⋊C4—22no—
96Dic59⋊C4—22no—
97C4×Dic59—22no—
98C4×Dic59—22no—
99#11745664—41no—
100#11745664—41no—
101C2×C59⋊C8—22no—
102C2×C59⋊C8—41no—
103C23×C59C2×C22⋊C411yescharacteristic
104C2×C4×C59—22no—
105C2×C4×C59—41no—
106C2×C4×C59—42no—
107C2×C4×C59—42no—
108C2×C4×C59—42no—
109C2×C4×C59C4×D411yescharacteristic
110C2×C4×C59C4×D411yescharacteristic
111C2×C4×C59#12011yescharacteristic
112C2×C4×C59#12111yescharacteristic
113C2×C4×C59#12111yescharacteristic
114C2×C4×C59#12311yescharacteristic
115C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
116C2×C4×C59#13711yescharacteristic
117C2×C4×C59#13711yescharacteristic
118C2×C4×C59#13711yescharacteristic
119C2×Dic59—22no—
120C2×Dic59—22no—
121C2×Dic59—22no—
122C2×Dic59—22no—
123C2×Dic59—42no—
124C2×Dic59—44no—
125C2×Dic59—42no—
126Dic118—41no—
127Dic118—41no—
128Dic118—41no—
129Dic118—41no—
130Dic118—82no—
131Dic118—82no—
132C8×C59—42no—
133C8×C59—81no—
134C59⋊C8—42no—
135C59⋊C8—81no—
136#9404—591nomaximal, Sylow
137C22×C59—21no—
138C22×C59—21no—
139C22×C59#38611yescharacteristic
140C22×C59#41711yescharacteristic
141C22×C59#41911yescharacteristic
142C4×C59—21no—
143C4×C59—21no—
144C4×C59—21no—
145C4×C59—82no—
146C4×C59#45412yes—
147C4×C59#46811yescharacteristic
148C4×C59#46811yescharacteristic
149C4×C59#47011yescharacteristic
150C4×C59#47011yescharacteristic
151Dic59—42no—
152Dic59—42no—
153Dic59—82no—
154#2983—591no—
155#2568—591no—
156#1801—591no—
157#1549—591no—
158#1549—591no—
159#1303—591no—
160#1303D5911yescharacteristic
161C2×C59—21no—
162C2×C59—21no—
163C2×C59#188811yescharacteristic
164C2×C59#189411yescharacteristic
165C2×C59#191511yescharacteristic
166C2×C4⋊Q8—591no—
167C2×C4×Q8—591no—
168#446—1182no—
169#446—1182no—
170#432—1182no—
171#431—591no—
172#431C2×D5911yescharacteristic
173#423—591no—
174#423C2×D5911yescharacteristic
175C2×#101—1181no—
176#400—594no—
177#391—1181no—
178#391—1182no—
179#382—1181no—
180C4×C4⋊C4C2×D5911yescharacteristic
181#343—1181no—
182#332—1182no—
183#332—1182no—
184C59#940411yescharacteristic, complemented, Sylow
185C22×Q8—591no—
186C22×Q8—1181no—
187C2×M16—21no—
188C2×M16—1181no—
189C4⋊Q8—1181no—
190C4⋊Q8—1181no—
191C4⋊Q8—1181no—
192C4⋊Q8—1181no—
193C4×Q8—1181no—
194C4×Q8—1181no—
195C4×Q8—1181no—
196C4×Q8—1181no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—592no—
199C2×C4⋊C4—591no—
200C2×C4⋊C4—592no—
201C2×C4⋊C4—1181no—
202C2×C42—21no—
203C2×C42—592no—
204C2×C42C22×D5911yescharacteristic
205C4⋊C8—21no—
206C4⋊C8—21no—
207C4⋊C8—592no—
208C4⋊C8—1184no—
209C4⋊C8—1181no—
210C4⋊C8—1181no—
211C4⋊C8D23612yes—
212#101—2362no—
213C22⋊C8—21no—
214C22⋊C8—1181no—
215#97—1184no—
216C4×C8—21no—
217C4×C8—1181no—
218#95—1184no—
219#95—1184no—
220#95C4×D5912yes—
221C2×Q8—1181no—
222C2×Q8—1181no—
223C2×Q8—1181no—
224C2×Q8—1181no—
225C2×Q8—1181no—
226C2×Q8—1181no—
227C2×Q8—2364no—
228C2×Q8—2361no—
229C2×Q8—2361no—
230C22×C4—22no—
231C22×C4—592no—
232C22×C4—592no—
233C22×C4—1182no—
234C22×C4C2×C4×D5911yescharacteristic
235C22×C4D4×D5911yescharacteristic
236C22×C4D4×D5911yescharacteristic
237M16—41no—
238M16—41no—
239M16—2361no—
240M16—2361no—
241C2×C8—22no—
242C2×C8—41no—
243C2×C8—1182no—
244C2×C8—2361no—
245C4⋊C4—42no—
246C4⋊C4—1182no—
247C4⋊C4—1182no—
248C4⋊C4—1181no—
249C4⋊C4—1181no—
250C4⋊C4—1182no—
251C4⋊C4—1182no—
252C4⋊C4—2361no—
253C4⋊C4—2361no—
254C42—41no—
255C42—41no—
256C42—1182no—
257C42—1182no—
258C42C2×D23611yescharacteristic
259C42#1174568511yescharacteristic
260C42D4×D5911yescharacteristic
261C42#1174568911yescharacteristic
262C23C22⋊C4×D5911yescharacteristic
263Q8—4722no—
264Q8—4722no—
265Q8—2361no—
266Q8—2361no—
267Q8—2361no—
268Q8—2361no—
269C2×C4—22no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—1182no—
275C2×C4—1182no—
276C2×C4—1182no—
277C2×C4—1182no—
278C2×C4—2364no—
279C2×C4—2362no—
280C2×C4—2362no—
281C2×C4#1427368911yescharacteristic
282C2×C4C4×D23611yescharacteristic
283C2×C4#1427369711yescharacteristic
284C2×C4#1427370011yescharacteristic
285C2×C4#1427370211yescharacteristic
286C2×C4#1427371211yescharacteristic
287C2×C4#1427375111yescharacteristic
288C2×C4#1427375611yescharacteristic
289C2×C4#1427375611yescharacteristic
290C2×C4#1427375911yescharacteristic
291C8—42no—
292C8—81no—
293C8—4721no—
294C8—2362no—
295C22—21no—
296C22—21no—
297C22#1683728011yescharacteristic
298C22#1683740011yescharacteristic
299C22#1683740911yescharacteristic
300C4—21no—
301C4—21no—
302C4—21no—
303C4—82no—
304C4—4722no—
305C4—2362no—
306C4—2362no—
307C4#1683757612yes—
308C4#1683764711yescharacteristic
309C4#1683764911yescharacteristic
310C4#1683765911yescharacteristic
311C4#1683766111yescharacteristic
312C2—21no—
313C2—21no—
314C2#2142449811yescharacteristic
315C2#2142452811yescharacteristic
316C2#2142464711yescharacteristic
subgroupG/H
centerC22#16837280
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1303C2
socleC22×C59#386
prime radicalC23×C59C2×C22⋊C4
cosocleC2×C42×C59C23
hypercenter#1303D59
hypocenterC59#9404
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1303

Complements

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

NH ≅ G/Ntype
184. C59#9404semidirect

Maximal subgroups

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

subgroupindex
#214331792
#214293112
#214240892
#214227822
#214227802
#214219992
C59×#13032
#940459