Finite Group Databasebeta

← #28504724

Subgroups — #28504724

Order 15104.

412 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 97 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#21432814C211yesmaximal, characteristic
2#21425781C211yesmaximal, characteristic
3#21424144C212yesmaximal, complemented
4#21422218C212yesmaximal, complemented
5C59×#2132C211yesmaximal, characteristic
6C2×#14273734C411yescharacteristic
7C2×#14273734C2211yescharacteristic
8#16837474—22no—
9#16837474—22no—
10C2×#14273662—22no—
11#16837426C411yescharacteristic
12#16837386C2212yes—
13#16837369C411yescharacteristic
14#16837369C2211yescharacteristic
15C2×#14273638C411yescharacteristic
16#16837201—21no—
17#16837199—21no—
18C59⋊#338—22no—
19C59×#532C2211yescharacteristic
20C59×#413C2212yes—
21C59×#356—21no—
22C59×#355—21no—
23#14273734—42no—
24#14273734—42no—
25C2×#11745664—21no—
26C2×#11745664—22no—
27C2×#11745664C2×C411yescharacteristic
28C2×#11745664C2×C411yescharacteristic
29C2×#11745664D412yes—
30C22×C59⋊C8—21no—
31C22×C59⋊C8C2×C411yescharacteristic
32C22×C59⋊C8C2×C411yescharacteristic
33C22×C59⋊C8D412yes—
34#14273662—44no—
35C4⋊(C59⋊C8)—22no—
36#14273646—22no—
37#14273644—22no—
38C22⋊(C59⋊C8)—22no—
39C22⋊(C59⋊C8)—22no—
40#14273638—21no—
41#14273638—21no—
42C2×#41×C59C2×C411yescharacteristic
43C2×#41×C59C2311yescharacteristic
44C2×M16×C59—22no—
45C4⋊Q8×C59D411yescharacteristic
46#127×C59Q811yescharacteristic
47#124×C59D411yescharacteristic
48#124×C59Q811yescharacteristic
49#121×C59—22no—
50C4×D4×C59—22no—
51C2×C42×C59C2×C411yescharacteristic
52#101×C59—22no—
53#100×C59—22no—
54#41×C59—41no—
55#41×C59—41no—
56#41×C59—41no—
57#41×C59—41no—
58C2×Q8×C59C4⋊C411yescharacteristic
59C2×Q8×C59C2×D411yescharacteristic
60C2×D4×C59—21no—
61C2×D4×C59—22no—
62C2×D4×C59C4⋊C411yescharacteristic
63C2×D4×C59C2×Q811yescharacteristic
64C2×D4×C59#4112yes—
65C22×C4×C59—21no—
66C22×C4×C59—21no—
67C22×C4×C59C22×C411yescharacteristic
68C22×C4×C59C2×D412yes—
69M16×C59—42no—
70M16×C59—42no—
71C2×C8×C59—42no—
72C4⋊C4×C59—22no—
73C22⋊C4×C59—22no—
74C22⋊C4×C59—22no—
75C42×C59—21no—
76C42×C59C4⋊C411yescharacteristic
77C42×C59C4⋊C411yescharacteristic
78#11745664—41no—
79#11745664—42no—
80#11745664—42no—
81#11745664—41no—
82#11745664—42no—
83#11745664—42no—
84C2×C59⋊C8—24no—
85C2×C59⋊C8—22no—
86C2×C59⋊C8—21no—
87C2×C59⋊C8—21no—
88C2×C59⋊C8—42no—
89C2×C59⋊C8—41no—
90C2×C59⋊C8—42no—
91C2×C59⋊C8C22⋊C411yescharacteristic
92C2×C59⋊C8C22⋊C412yes—
93C2×C59⋊C8C22⋊C411yescharacteristic
94C23×C59—21no—
95C23×C59#11711yescharacteristic
96C23×C59C22⋊Q812yes—
97Q8×C59—41no—
98Q8×C59—41no—
99D4×C59—41no—
100D4×C59—42no—
101D4×C59—41no—
102D4×C59—42no—
103D4×C59—82no—
104C2×C4×C59—22no—
105C2×C4×C59—21no—
106C2×C4×C59—22no—
107C2×C4×C59—21no—
108C2×C4×C59—21no—
109C2×C4×C59—21no—
110C2×C4×C59—21no—
111C2×C4×C59—41no—
112C2×C4×C59—41no—
113C2×C4×C59—81no—
114C2×C4×C59C2×C22⋊C411yescharacteristic
115C2×C4×C59C2×C4⋊C411yescharacteristic
116C2×C4×C59#12112yes—
117C8×C59—82no—
118C59⋊C8—21no—
119C59⋊C8—22no—
120C59⋊C8—41no—
121C59⋊C8—41no—
122C59⋊C8—82no—
123C59⋊C8#9912yes—
124#12810—591nomaximal, Sylow
125C22×C59—21no—
126C22×C59—22no—
127C22×C59—41no—
128C22×C59—42no—
129C22×C59—42no—
130C22×C59—41no—
131C22×C59#38011yescharacteristic
132C4×C59—22no—
133C4×C59—21no—
134C4×C59—41no—
135C4×C59—41no—
136C4×C59—41no—
137C4×C59C2×#9911yescharacteristic
138C4×C59#41011yescharacteristic
139#2932—591no—
140#2132—591no—
141#2132D5911yescharacteristic
142#1813—592no—
143#1392—592no—
144C2×C59—21no—
145C2×C59—41no—
146C2×C59—42no—
147C2×C59—81no—
148C2×C59#179911yescharacteristic
149C2×#131—591no—
150C2×#131—591no—
151#532C2×D5911yescharacteristic
152#435—1182no—
153#435—1182no—
154C2×#108—1182no—
155#423—591no—
156#413—592no—
157#413C2×D5912yes—
158#408—591no—
159#408—591no—
160C2×#97—591no—
161#356—21no—
162#356—1181no—
163#355—21no—
164#355—1181no—
165#338—1182no—
166C59#1281011yescharacteristic, complemented, Sylow
167C2×#41C4×D5911yescharacteristic
168C2×#41C22×D5911yescharacteristic
169#131—2362no—
170#131—2362no—
171C2×M16—22no—
172C2×M16—591no—
173C2×M16—592no—
174C2×M16—591no—
175C2×M16—1181no—
176C2×M16—1182no—
177C22×C8—591no—
178C22×C8—592no—
179C22×C8—591no—
180C22×C8—1181no—
181C4⋊Q8C22⋊D5911yescharacteristic
182#127Dic11811yescharacteristic
183#124Dic11811yescharacteristic
184#124C22⋊D5911yescharacteristic
185#121—22no—
186C4×D4—22no—
187C2×C42C4×D5911yescharacteristic
188#108—2364no—
189C4⋊C8—1182no—
190#101—22no—
191#101—1182no—
192#100—22no—
193#100—1182no—
194C22⋊C8—1182no—
195C22⋊C8—1182no—
196#97—1181no—
197#97—1181no—
198#41—41no—
199#41—41no—
200#41—41no—
201#41—41no—
202C2×Q8Dic59⋊C411yescharacteristic
203C2×Q8C2×C22⋊D5911yescharacteristic
204C2×D4—21no—
205C2×D4—22no—
206C2×D4Dic59⋊C411yescharacteristic
207C2×D4C2×Dic11811yescharacteristic
208C2×D4#1174568712yes—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4C2×C4×D5911yescharacteristic
212C22×C4D4×D5912yes—
213M16—42no—
214M16—42no—
215M16—2362no—
216M16—2362no—
217M16—2361no—
218M16—2362no—
219M16—2362no—
220M16—2361no—
221C2×C8—42no—
222C2×C8—591no—
223C2×C8—591no—
224C2×C8—592no—
225C2×C8—1184no—
226C2×C8—1182no—
227C2×C8—1181no—
228C2×C8—1181no—
229C2×C8—2362no—
230C2×C8—2362no—
231C2×C8—2361no—
232C4⋊C4—22no—
233C22⋊C4—22no—
234C22⋊C4—22no—
235C42—21no—
236C42Dic59⋊C411yescharacteristic
237C42Dic59⋊C411yescharacteristic
238C23—21no—
239C23#1427368311yescharacteristic
240C23#1427370412yes—
241Q8—41no—
242Q8—41no—
243D4—41no—
244D4—42no—
245D4—41no—
246D4—42no—
247D4—82no—
248C2×C4—22no—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—81no—
258C2×C4C22⋊C4×D5911yescharacteristic
259C2×C4C2×Dic59⋊C411yescharacteristic
260C2×C4#1427369912yes—
261C8—82no—
262C8—592no—
263C8—1181no—
264C8—1182no—
265C8—4722no—
266C8—2361no—
267C8—2361no—
268C22—21no—
269C22—22no—
270C22—41no—
271C22—42no—
272C22—42no—
273C22—41no—
274C22#1683726011yescharacteristic
275C4—22no—
276C4—21no—
277C4—41no—
278C4—41no—
279C4—41no—
280C4#99×D5911yescharacteristic
281C4#1683737411yescharacteristic
282C2—21no—
283C2—41no—
284C2—42no—
285C2—81no—
286C2#2142408211yescharacteristic
subgroupG/H
centerC4#16837374
commutatorC22×C4×C59C22×C4
FrattiniC2×#41C22×D59
FittingC59×#2132C2
socleC2×C59#1799
prime radicalC59×#532C22
cosocleC2×#41×C59C23
hypercenter#2132D59
hypocenterC59#12810
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #2132

Complements

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

NH ≅ G/Ntype
3. #21424144C2semidirect
4. #21422218C2semidirect
166. C59#12810semidirect

Maximal subgroups

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

subgroupindex
#214328142
#214257812
#214241442
#214222182
C59×#21322
#1281059