Finite Group Databasebeta

← #28503877

Subgroups — #28503877

Order 15104.

464 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 125 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#21426816C211yesmaximal, characteristic
2C2×#16837201C211yesmaximal, characteristic
3#101×Dic59C212yesmaximal
4#21422222C212yesmaximal
5C2×C59×#356C211yesmaximal, characteristic
6C2×Q8×Dic59C411yescharacteristic
7C2×Q8×Dic59C2211yescharacteristic
8C2×#14273646C2212yes—
9M16×Dic59—22no—
10#16837309C411yescharacteristic
11#16837309C2211yescharacteristic
12#16837288—22no—
13#16837288—22no—
14#16837268C411yescharacteristic
15#16837248C411yescharacteristic
16#16837201C414yescomplemented
17#16837170—22no—
18C2×C4⋊Q8×C59C2211yescharacteristic
19C2×#101×C59C2212yes—
20C59×#356C414yescomplemented
21C2×#11745664—22no—
22Q8×Dic59—42no—
23Q8×Dic59—42no—
24C2×C4⋊Dic59—21no—
25C2×C4⋊Dic59C2×C411yescharacteristic
26C2×C4⋊Dic59C2×C411yescharacteristic
27C2×C4⋊Dic59D412yes—
28C2×C4×Dic59—21no—
29C2×C4×Dic59C2×C411yescharacteristic
30C2×C4×Dic59C2×C411yescharacteristic
31C2×C4×Dic59D412yes—
32#14273646—22no—
33#14273646C2×C414yes—
34#14273637—42no—
35C8×Dic59—42no—
36C59⋊#95—22no—
37C59⋊#95—21no—
38C59⋊#95—21no—
39C59⋊#95—22no—
40C59⋊#95—22no—
41C22×Q8×C59C2×C411yescharacteristic
42C22×Q8×C59C2311yescharacteristic
43C2×M16×C59—22no—
44C4⋊Q8×C59—22no—
45C4⋊Q8×C59C2×C412yes—
46C4⋊Q8×C59C2×C412yes—
47C2×C4⋊C4×C59—22no—
48C2×C42×C59C2×C411yescharacteristic
49#101×C59—22no—
50#101×C59C2×C414yes—
51C2×Q8×C59—21no—
52C2×Q8×C59—41no—
53C2×Q8×C59—41no—
54C2×Q8×C59—41no—
55C2×Q8×C59—41no—
56C2×Q8×C59C4212yes—
57C2×Q8×C59C22×C411yescharacteristic
58C2×Q8×C59C22×C411yescharacteristic
59C2×Q8×C59#4112yes—
60C22×C4×C59—21no—
61C22×C4×C59—21no—
62C22×C4×C59C22×C411yescharacteristic
63C22×C4×C59C2×D412yes—
64C22×Dic59—21no—
65C22×Dic59—21no—
66C22×Dic59C22⋊C412yes—
67C22×Dic59C22⋊C411yescharacteristic
68C22×Dic59C22⋊C411yescharacteristic
69M16×C59—24no—
70M16×C59—42no—
71C2×C8×C59—42no—
72C4⋊C4×C59—24no—
73C4⋊C4×C59—24no—
74C42×C59—21no—
75C42×C59C4212yes—
76C4⋊Dic59—41no—
77C4⋊Dic59—42no—
78C4⋊Dic59—42no—
79C4⋊Dic59—41no—
80C4×Dic59—22no—
81C4×Dic59—22no—
82C4×Dic59—22no—
83C4×Dic59—42no—
84C4×Dic59—41no—
85#11745664—24no—
86#11745664—42no—
87C2×C59⋊C8—42no—
88C23×C59C2×C22⋊C411yescharacteristic
89Q8×C59—41no—
90Q8×C59—41no—
91Q8×C59—42no—
92Q8×C59—42no—
93Q8×C59—42no—
94C2×C4×C59—22no—
95C2×C4×C59—21no—
96C2×C4×C59—22no—
97C2×C4×C59—21no—
98C2×C4×C59—22no—
99C2×C4×C59—22no—
100C2×C4×C59—41no—
101C2×C4×C59—41no—
102C2×C4×C59—41no—
103C2×C4×C59—41no—
104C2×C4×C59C2×C4211yescharacteristic
105C2×C4×C59#11711yescharacteristic
106C2×C4×C59C4×D412yes—
107C2×C4×C59C4×D412yes—
108C2×Dic59—21no—
109C2×Dic59—22no—
110C2×Dic59—41no—
111C2×Dic59—42no—
112C2×Dic59—41no—
113C2×Dic59—41no—
114C2×Dic59—41no—
115C2×Dic59—42no—
116C2×Dic59—41no—
117C2×Dic59—41no—
118C2×Dic59#9912yes—
119C8×C59—44no—
120C59⋊C8—44no—
121#12631—591nomaximal, Sylow
122C22×C59—21no—
123C22×C59—21no—
124C22×C59C4×C22⋊C411yescharacteristic
125C22×C59C2×#9911yescharacteristic
126C22×C59#41011yescharacteristic
127C4×C59—22no—
128C4×C59—22no—
129C4×C59—21no—
130C4×C59—21no—
131C4×C59—42no—
132C4×C59—42no—
133Dic59—22no—
134Dic59—41no—
135Dic59—42no—
136Dic59—81no—
137Dic59—81no—
138#2327—591no—
139C2×#356—591no—
140C2×#356D5911yescharacteristic
141C4×#101—592no—
142#1394—592no—
143C2×C59—21no—
144C2×C59—21no—
145C2×C59C4×#9911yescharacteristic
146C2×C59#213311yescharacteristic
147C2×C59#213511yescharacteristic
148C2×C4⋊Q8C2×D5911yescharacteristic
149C2×C4×Q8—591no—
150C2×C4×Q8—591no—
151C2×#101—592no—
152C2×#101C2×D5912yes—
153C4×M16—1182no—
154#391—591no—
155#391—591no—
156#387—1182no—
157#387—1182no—
158#382—591no—
159#376—591no—
160#356—594no—
161#356Dic5914yescomplemented
162#343—1182no—
163C59#1263111yescharacteristic, complemented, Sylow
164C22×Q8C4×D5911yescharacteristic
165C22×Q8C22×D5911yescharacteristic
166C2×M16—22no—
167C2×M16—1182no—
168C4⋊Q8—22no—
169C4⋊Q8C4×D5912yes—
170C4⋊Q8C2×Dic5912yes—
171C4×Q8—2362no—
172C4×Q8—2362no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—591no—
175C2×C4⋊C4—592no—
176C2×C4⋊C4—591no—
177C2×C4⋊C4—1181no—
178C2×C42—591no—
179C2×C42—592no—
180C2×C42—591no—
181C2×C42—1181no—
182C2×C42C4×D5911yescharacteristic
183#101—22no—
184#101—594no—
185#101—1182no—
186#101C2×Dic5914yes—
187#97—2362no—
188C4×C8—2362no—
189#95—1181no—
190#95—1181no—
191#95—1182no—
192#95—1182no—
193#95—1182no—
194C2×Q8—21no—
195C2×Q8—41no—
196C2×Q8—41no—
197C2×Q8—41no—
198C2×Q8—41no—
199C2×Q8C4×Dic5912yes—
200C2×Q8C2×C4×D5911yescharacteristic
201C2×Q8#1174568712yes—
202C2×Q8C22×Dic5911yescharacteristic
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—591no—
206C22×C4—592no—
207C22×C4—591no—
208C22×C4—1181no—
209C22×C4—1181no—
210C22×C4C2×C4×D5911yescharacteristic
211C22×C4D4×D5912yes—
212M16—24no—
213M16—42no—
214M16—1184no—
215M16—2362no—
216C2×C8—42no—
217C2×C8—2362no—
218C4⋊C4—24no—
219C4⋊C4—24no—
220C4⋊C4—2361no—
221C4⋊C4—2362no—
222C4⋊C4—2361no—
223C4⋊C4—2362no—
224C42—21no—
225C42—1182no—
226C42—1182no—
227C42—1182no—
228C42—2361no—
229C42—2362no—
230C42C4×Dic5912yes—
231C23C22⋊C4×D5911yescharacteristic
232Q8—41no—
233Q8—42no—
234Q8—41no—
235Q8—42no—
236Q8—42no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—592no—
248C2×C4—1181no—
249C2×C4—1182no—
250C2×C4—2361no—
251C2×C4—2361no—
252C2×C4—2361no—
253C2×C4—2362no—
254C2×C4—2361no—
255C2×C4—2361no—
256C2×C4—2362no—
257C2×C4—2361no—
258C2×C4C2×C4×Dic5911yescharacteristic
259C2×C4#1427368311yescharacteristic
260C2×C4(C22⋊D59)⋊C412yes—
261C2×C4D4×Dic5912yes—
262C8—44no—
263C8—2364no—
264C22—21no—
265C22—21no—
266C22C22⋊C4×Dic5911yescharacteristic
267C22#99×D5911yescharacteristic
268C22#1683737411yescharacteristic
269C4—22no—
270C4—22no—
271C4—21no—
272C4—21no—
273C4—42no—
274C4—42no—
275C4—1182no—
276C4—4721no—
277C4—4721no—
278C4—2361no—
279C4—2362no—
280C2—21no—
281C2—21no—
282C2#99×Dic5911yescharacteristic
283C2#2142578411yescharacteristic
284C2#2142578911yescharacteristic
subgroupG/H
centerC22#99×D59
commutatorC2×C4×C59C2×C42
FrattiniC22×Q8C22×D59
FittingC2×C59×#356C2
socleC22×C59C2×#99
prime radicalC23×C59C2×C22⋊C4
cosocleC22×Q8×C59C23
hypercenterC2×#356D59
hypocenterC59#12631
LCSG ▷ C2×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ C2×#356

Complements

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

NH ≅ G/Ntype
16. #16837201C4semidirect
20. C59×#356C4semidirect
161. #356Dic59semidirect
163. C59#12631semidirect

Maximal subgroups

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

subgroupindex
#214268162
C2×#168372012
#101×Dic592
#214222222
C2×C59×#3562
#1263159