Finite Group Databasebeta

← #356×Dic59 #28503873

Subgroups — #356×Dic59 #28503873

Order 15104.

500 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 127 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
1C4⋊Q8×Dic59C211yesmaximal, 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#16837301—22no—
13#16837272C411yescharacteristic
14C4⋊C4×Dic59—22no—
15C42×Dic59C411yescharacteristic
16#16837201C414yescomplemented
17#16837170—22no—
18C2×C4⋊Q8×C59C2211yescharacteristic
19C2×#101×C59C2212yes—
20C59×#356C414yescomplemented
21C2×#11745664—22no—
22Q8×Dic59—41no—
23Q8×Dic59—41no—
24Q8×Dic59—41no—
25Q8×Dic59—41no—
26C2×C4⋊Dic59—22no—
27C2×C4⋊Dic59—21no—
28C2×C4⋊Dic59C2×C411yescharacteristic
29C2×C4⋊Dic59C2×C411yescharacteristic
30C2×C4⋊Dic59D412yes—
31C2×C4×Dic59—21no—
32C2×C4×Dic59—21no—
33C2×C4×Dic59—21no—
34C2×C4×Dic59C2×C411yescharacteristic
35C2×C4×Dic59C2×C411yescharacteristic
36C2×C4×Dic59D412yes—
37#14273646—22no—
38#14273646C2×C414yes—
39#14273637—42no—
40C8×Dic59—42no—
41C59⋊#95—22no—
42C59⋊#95—22no—
43C22×Q8×C59C2×C411yescharacteristic
44C22×Q8×C59C2311yescharacteristic
45C2×M16×C59—22no—
46C4⋊Q8×C59—22no—
47C4⋊Q8×C59C2×C412yes—
48C4⋊Q8×C59C2×C412yes—
49C2×C4⋊C4×C59—22no—
50C2×C42×C59C2×C411yescharacteristic
51#101×C59—22no—
52#101×C59C2×C414yes—
53C2×Q8×C59—21no—
54C2×Q8×C59—41no—
55C2×Q8×C59—41no—
56C2×Q8×C59—41no—
57C2×Q8×C59—41no—
58C2×Q8×C59C4212yes—
59C2×Q8×C59C22×C411yescharacteristic
60C2×Q8×C59C22×C411yescharacteristic
61C2×Q8×C59#4112yes—
62C22×C4×C59—21no—
63C22×C4×C59—21no—
64C22×C4×C59C22×C411yescharacteristic
65C22×C4×C59C2×D412yes—
66C22×Dic59—21no—
67C22×Dic59—21no—
68C22×Dic59C22⋊C411yescharacteristic
69C22×Dic59C22⋊C412yes—
70C22×Dic59C22⋊C411yescharacteristic
71M16×C59—24no—
72M16×C59—42no—
73C2×C8×C59—42no—
74C4⋊C4×C59—24no—
75C4⋊C4×C59—24no—
76C42×C59—21no—
77C42×C59C4212yes—
78C4⋊Dic59—41no—
79C4⋊Dic59—42no—
80C4⋊Dic59—42no—
81C4⋊Dic59—44no—
82C4⋊Dic59—41no—
83C4×Dic59—24no—
84C4×Dic59—22no—
85C4×Dic59—22no—
86C4×Dic59—21no—
87C4×Dic59—21no—
88C4×Dic59—42no—
89C4×Dic59—41no—
90C4×Dic59—41no—
91C4×Dic59—41no—
92#11745664—24no—
93#11745664—42no—
94C2×C59⋊C8—42no—
95C23×C59C2×C22⋊C411yescharacteristic
96Q8×C59—41no—
97Q8×C59—41no—
98Q8×C59—42no—
99Q8×C59—42no—
100Q8×C59—42no—
101C2×C4×C59—22no—
102C2×C4×C59—22no—
103C2×C4×C59—22no—
104C2×C4×C59—22no—
105C2×C4×C59—21no—
106C2×C4×C59—22no—
107C2×C4×C59—21no—
108C2×C4×C59—22no—
109C2×C4×C59—41no—
110C2×C4×C59—41no—
111C2×C4×C59C2×C4211yescharacteristic
112C2×C4×C59#11711yescharacteristic
113C2×C4×C59C4×D412yes—
114C2×C4×C59C4×D412yes—
115C2×Dic59—22no—
116C2×Dic59—22no—
117C2×Dic59—21no—
118C2×Dic59—22no—
119C2×Dic59—41no—
120C2×Dic59—41no—
121C2×Dic59—41no—
122C2×Dic59—42no—
123C2×Dic59—42no—
124C2×Dic59—41no—
125C2×Dic59#9911yescharacteristic
126C2×Dic59#9911yescharacteristic
127C8×C59—44no—
128C59⋊C8—44no—
129C4×#356—591nomaximal, Sylow
130C22×C59—21no—
131C22×C59—21no—
132C22×C59C4×C22⋊C411yescharacteristic
133C22×C59C2×#9911yescharacteristic
134C22×C59#41011yescharacteristic
135C4×C59—24no—
136C4×C59—22no—
137C4×C59—22no—
138C4×C59—21no—
139C4×C59—21no—
140C4×C59—42no—
141Dic59—21no—
142Dic59—42no—
143Dic59—41no—
144Dic59—42no—
145Dic59—81no—
146Dic59#35612yesdirect product factor
147C4×C4⋊Q8—591no—
148C2×#356—591no—
149C2×#356D5911yescharacteristic
150C4×#101—592no—
151#1394—592no—
152C2×C59—21no—
153C2×C59—21no—
154C2×C59C4×#9911yescharacteristic
155C2×C59C2×#35611yescharacteristic
156C2×C59#213211yescharacteristic
157C2×C4⋊Q8C2×D5911yescharacteristic
158C2×C4×Q8—591no—
159C2×C4×Q8—591no—
160C2×#101—592no—
161C2×#101C2×D5912yes—
162C4×M16—1182no—
163#391—591no—
164#391—591no—
165#389—1182no—
166#384—591no—
167C4×C4⋊C4—1182no—
168C43—591no—
169#356—594no—
170#356Dic5914yesdirect product factor
171#343—1182no—
172C59C4×#35611yescharacteristic, complemented, Sylow
173C22×Q8C4×D5911yescharacteristic
174C22×Q8C22×D5911yescharacteristic
175C2×M16—22no—
176C2×M16—1182no—
177C4⋊Q8—22no—
178C4⋊Q8C4×D5912yes—
179C4⋊Q8C2×Dic5912yes—
180C4×Q8—2361no—
181C4×Q8—2361no—
182C4×Q8—2361no—
183C4×Q8—2361no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—591no—
186C2×C4⋊C4—592no—
187C2×C4⋊C4—591no—
188C2×C4⋊C4—1181no—
189C2×C4⋊C4—1182no—
190C2×C42—591no—
191C2×C42—592no—
192C2×C42—591no—
193C2×C42—1181no—
194C2×C42—1181no—
195C2×C42—1181no—
196C2×C42C4×D5911yescharacteristic
197#101—22no—
198#101—594no—
199#101—1182no—
200#101C2×Dic5914yes—
201#97—2362no—
202C4×C8—2362no—
203#95—1182no—
204#95—1182no—
205C2×Q8—21no—
206C2×Q8—41no—
207C2×Q8—41no—
208C2×Q8—41no—
209C2×Q8—41no—
210C2×Q8C4×Dic5912yes—
211C2×Q8C2×C4×D5911yescharacteristic
212C2×Q8#1174568712yes—
213C2×Q8C22×Dic5911yescharacteristic
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—591no—
217C22×C4—592no—
218C22×C4—591no—
219C22×C4—1181no—
220C22×C4—1181no—
221C22×C4C2×C4×D5911yescharacteristic
222C22×C4D4×D5912yes—
223M16—24no—
224M16—42no—
225M16—1184no—
226M16—2362no—
227C2×C8—42no—
228C2×C8—2362no—
229C4⋊C4—24no—
230C4⋊C4—24no—
231C4⋊C4—2361no—
232C4⋊C4—2362no—
233C4⋊C4—2361no—
234C4⋊C4—2364no—
235C4⋊C4—2362no—
236C42—21no—
237C42—1182no—
238C42—1182no—
239C42—1181no—
240C42—1181no—
241C42—1184no—
242C42—2361no—
243C42—2362no—
244C42—2361no—
245C42—2361no—
246C42C4×Dic5912yes—
247C23C22⋊C4×D5911yescharacteristic
248Q8—41no—
249Q8—42no—
250Q8—41no—
251Q8—42no—
252Q8—42no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—591no—
264C2×C4—591no—
265C2×C4—1182no—
266C2×C4—1182no—
267C2×C4—1181no—
268C2×C4—1182no—
269C2×C4—2362no—
270C2×C4—2361no—
271C2×C4—2361no—
272C2×C4—2361no—
273C2×C4—2361no—
274C2×C4—2362no—
275C2×C4C2×C4×Dic5911yescharacteristic
276C2×C4#1427368311yescharacteristic
277C2×C4(C22⋊D59)⋊C412yes—
278C2×C4D4×Dic5912yes—
279C8—44no—
280C8—2364no—
281C22—21no—
282C22—21no—
283C22C22⋊C4×Dic5911yescharacteristic
284C22#99×D5911yescharacteristic
285C22#1683737411yescharacteristic
286C4—24no—
287C4—22no—
288C4—22no—
289C4—21no—
290C4—21no—
291C4—42no—
292C4—592no—
293C4—1181no—
294C4—4721no—
295C4—2362no—
296C4—2361no—
297C4—2362no—
298C2—21no—
299C2—21no—
300C2#99×Dic5911yescharacteristic
301C2#356×D5911yescharacteristic
302C2#2142578011yescharacteristic
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
hypocenterC59C4×#356
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
146. Dic59#356direct (N × H)
170. #356Dic59direct (N × H)
172. C59C4×#356semidirect

Maximal subgroups

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

subgroupindex
C4⋊Q8×Dic592
C2×#168372012
#101×Dic592
#214222222
C2×C59×#3562
C4×#35659