Finite Group Databasebeta

← #28507015

Subgroups — #28507015

Order 15104.

472 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 107 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#21429081C211yesmaximal, characteristic
2C2×#16837200C214yesmaximal
3#21424092C211yesmaximal, characteristic
4C4×#101×C59C211yesmaximal, characteristic
5C2×#14273727C411yescharacteristic
6C2×#14273725C2212yes—
7#16837426—21no—
8C2×#14273646C2212yes—
9#16837332—21no—
10#16837332—21no—
11#16837307C412yes—
12#16837300—21no—
13#16837300—21no—
14C4×C4⋊Dic59C411yescharacteristic
15#16837200—28no—
16#16837171—21no—
17C2×C4×Q8×C59C2211yescharacteristic
18C2×#101×C59C2212yes—
19C4×M16×C59—21no—
20C59×#343—21no—
21C22×Dic118C2×C412yes—
22C2×#11745664—22no—
23#14273727—41no—
24#14273727—41no—
25#14273725—24no—
26#14273725—24no—
27C2×C4⋊Dic59—21no—
28C2×C4⋊Dic59—21no—
29C2×C4⋊Dic59—21no—
30C2×Dic59⋊C4—22no—
31C2×Dic59⋊C4—22no—
32C2×C4×Dic59C2×C412yes—
33C4⋊(C59⋊C8)—41no—
34C4⋊(C59⋊C8)—41no—
35#14273646—24no—
36C59⋊#95—21no—
37C59⋊#95—22no—
38C59⋊#95—22no—
39C22×Q8×C59C2311yescharacteristic
40C2×M16×C59—22no—
41C4×Q8×C59—41no—
42C4×Q8×C59—41no—
43C2×C4⋊C4×C59C2×C411yescharacteristic
44C2×C4⋊C4×C59D411yescharacteristic
45C2×C4⋊C4×C59D411yescharacteristic
46C2×C42×C59C2×C411yescharacteristic
47C2×C42×C59D411yescharacteristic
48C2×C42×C59D411yescharacteristic
49#101×C59—22no—
50#101×C59D414yes—
51#97×C59—41no—
52C4×C8×C59—41no—
53C2×Q8×C59—41no—
54C2×Q8×C59—41no—
55C2×Q8×C59C2×D412yes—
56C2×Q8×C59#4112yes—
57C22×C4×C59C22⋊C411yescharacteristic
58C22×C4×C59C22⋊C411yescharacteristic
59C22×C4×C59C22⋊C411yescharacteristic
60C22×C4×C59C22⋊C411yescharacteristic
61C22×C4×C59C22×C411yescharacteristic
62C22×C4×C59C2×D411yescharacteristic
63C22×C4×C59C2×D411yescharacteristic
64C22×Dic59—22no—
65C22×Dic59—22no—
66C2×Dic118—24no—
67C2×Dic118—42no—
68C2×Dic118—42no—
69M16×C59—24no—
70M16×C59—42no—
71C2×C8×C59—42no—
72C4⋊C4×C59—41no—
73C4⋊C4×C59—41no—
74C4⋊C4×C59—41no—
75C42×C59—21no—
76C42×C59—21no—
77C42×C59—21no—
78C42×C59—21no—
79C42×C59—21no—
80C42×C59—21no—
81C4⋊Dic59—24no—
82C4⋊Dic59—42no—
83C4⋊Dic59—41no—
84C4⋊Dic59—41no—
85Dic59⋊C4—44no—
86Dic59⋊C4—42no—
87Dic59⋊C4—42no—
88C4×Dic59—24no—
89#11745664—42no—
90#11745664—42no—
91C2×C59⋊C8—42no—
92C23×C59C2×C22⋊C411yescharacteristic
93Q8×C59—42no—
94Q8×C59—42no—
95C2×C4×C59—21no—
96C2×C4×C59—21no—
97C2×C4×C59—21no—
98C2×C4×C59—21no—
99C2×C4×C59—21no—
100C2×C4×C59—21no—
101C2×C4×C59—22no—
102C2×C4×C59—21no—
103C2×C4×C59—21no—
104C2×C4×C59—21no—
105C2×C4×C59—41no—
106C2×C4×C59—41no—
107C2×C4×C59—41no—
108C2×C4×C59#9911yescharacteristic
109C2×C4×C59#9911yescharacteristic
110C2×C4×C59C4×D412yes—
111C2×C4×C59#12112yes—
112C2×C4×C59#12411yescharacteristic
113C2×C4×C59#12711yescharacteristic
114C2×Dic59—24no—
115C2×Dic59—42no—
116C2×Dic59—42no—
117C2×Dic59—42no—
118C2×Dic59—42no—
119C2×Dic59—42no—
120Dic118—84no—
121C8×C59—44no—
122C59⋊C8—82no—
123#13227—591nomaximal, Sylow
124C22×C59—21no—
125C22×C59—21no—
126C22×C59#39011yescharacteristic
127C22×C59C2×#9911yescharacteristic
128C22×C59#41011yescharacteristic
129C4×C59—21no—
130C4×C59—21no—
131C4×C59—22no—
132C4×C59—22no—
133C4×C59—21no—
134C4×C59—41no—
135C4×C59—41no—
136C4×C59—41no—
137C4×C59#35612yescomplemented
138Dic59—44no—
139Dic59—82no—
140#2546—591no—
141C2×#356—594no—
142#1801—591no—
143C4×#101D5911yescharacteristic
144C2×C59—21no—
145C2×C59—21no—
146C2×C59#196411yescharacteristic
147C2×C59C2×#35611yescharacteristic
148C2×C59#213511yescharacteristic
149C2×C4⋊Q8—592no—
150C2×C4⋊Q8—591no—
151C2×C4×Q8C2×D5911yescharacteristic
152#423—1181no—
153C2×#101—592no—
154C2×#101C2×D5912yes—
155C4×M16—21no—
156#398—1181no—
157#398—1181no—
158#391—592no—
159#389—1181no—
160#389—1181no—
161C4×C4⋊C4—591no—
162#356—1188no—
163#343—21no—
164#343—1181no—
165C59#1322711yescharacteristic, complemented, Sylow
166C22×Q8—592no—
167C22×Q8C22×D5911yescharacteristic
168C2×M16—22no—
169C2×M16—1182no—
170C4⋊Q8—1184no—
171C4⋊Q8—1184no—
172C4⋊Q8—2361no—
173C4⋊Q8—2361no—
174C4×Q8—41no—
175C4×Q8—41no—
176C2×C4⋊C4—1181no—
177C2×C4⋊C4—1181no—
178C2×C4⋊C4—1181no—
179C2×C4⋊C4—1182no—
180C2×C4⋊C4—1182no—
181C2×C4⋊C4C4×D5911yescharacteristic
182C2×C4⋊C4D23611yescharacteristic
183C2×C4⋊C4C22⋊D5911yescharacteristic
184C2×C42—592no—
185C2×C42C4×D5911yescharacteristic
186C2×C42D23611yescharacteristic
187C2×C42C22⋊D5911yescharacteristic
188C4⋊C8—2361no—
189C4⋊C8—2361no—
190#101—22no—
191#101—1184no—
192#101D23614yes—
193#97—41no—
194C4×C8—41no—
195#95—1181no—
196#95—1182no—
197#95—1182no—
198C2×Q8—41no—
199C2×Q8—41no—
200C2×Q8—1184no—
201C2×Q8—2362no—
202C2×Q8—2362no—
203C2×Q8C2×D23612yes—
204C2×Q8#1174568512yes—
205C22×C4—1182no—
206C22×C4—1182no—
207C22×C4(C2×D59)⋊C411yescharacteristic
208C22×C4(C2×D59)⋊C411yescharacteristic
209C22×C4(C2×D59)⋊C411yescharacteristic
210C22×C4(C2×D59)⋊C411yescharacteristic
211C22×C4C2×C4×D5911yescharacteristic
212C22×C4C2×D23611yescharacteristic
213C22×C4C2×C22⋊D5911yescharacteristic
214M16—24no—
215M16—42no—
216M16—2362no—
217M16—2362no—
218C2×C8—42no—
219C2×C8—2362no—
220C4⋊C4—41no—
221C4⋊C4—41no—
222C4⋊C4—41no—
223C4⋊C4—1184no—
224C4⋊C4—2362no—
225C4⋊C4—2361no—
226C4⋊C4—2361no—
227C4⋊C4—2364no—
228C4⋊C4—2362no—
229C4⋊C4—2362no—
230C42—21no—
231C42—21no—
232C42—21no—
233C42—21no—
234C42—21no—
235C42—21no—
236C42—1184no—
237C23C2×(C2×D59)⋊C411yescharacteristic
238Q8—42no—
239Q8—42no—
240Q8—4724no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—1184no—
255C2×C4—2362no—
256C2×C4—2362no—
257C2×C4—2362no—
258C2×C4—2362no—
259C2×C4—2362no—
260C2×C4#1427364111yescharacteristic
261C2×C4#1427364111yescharacteristic
262C2×C4C4×D23612yes—
263C2×C4#1427370112yes—
264C2×C4#1427371511yescharacteristic
265C2×C4#1427372411yescharacteristic
266C8—44no—
267C8—4722no—
268C22—21no—
269C22—21no—
270C22#1683730411yescharacteristic
271C22C2×#1427364111yescharacteristic
272C22#1683737511yescharacteristic
273C4—22no—
274C4—21no—
275C4—22no—
276C4—21no—
277C4—21no—
278C4—41no—
279C4—41no—
280C4—41no—
281C4—4722no—
282C4—2364no—
283C4#1683720012yescomplemented
284C2—21no—
285C2—21no—
286C2#2142493211yescharacteristic
287C2C2×#1683720011yescharacteristic
288C2#2142579011yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC22×Q8C22×D59
FittingC4×#101×C59C2
socleC22×C59C2×#99
prime radicalC23×C59C2×C22⋊C4
cosocleC22×Q8×C59C23
hypercenterC4×#101D59
hypocenterC59#13227
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ C4×#101

Complements

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

NH ≅ G/Ntype
137. C4×C59#356semidirect
165. C59#13227semidirect
283. C4#16837200semidirect

Maximal subgroups

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

subgroupindex
#214290812
C2×#168372002
#214240922
C4×#101×C592
#1322759