Finite Group Databasebeta

← #28483091

Subgroups — #28483091

Order 15104.

392 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 95 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#21433167C211yesmaximal, characteristic
2#21422861C212yesmaximal, complemented
3#21422688C211yesmaximal, characteristic
4#21421989C212yesmaximal, complemented
5C59×#1963C211yesmaximal, characteristic
6#16837474—22no—
7#16837474—22no—
8#16837459C2212yes—
9#16837369—21no—
10#16837369—21no—
11C4×#11745664C2212yes—
12#16837154—22no—
13#16837152—22no—
14#16837142C411yescharacteristic, complemented
15#16837142C412yescomplemented
16#16837142C411yescharacteristic, complemented
17C59×#532C2211yescharacteristic
18C59×#423C2211yescharacteristic
19C2×#104×C59—21no—
20C2×#104×C59—21no—
21C59×#376C2211yescharacteristic
22C2×#11745664—22no—
23C2×#11745664—21no—
24C2×#11745664—21no—
25C22×C59⋊C8—21no—
26C22×C59⋊C8—21no—
27C4⋊(C59⋊C8)D412yes—
28C4⋊(C59⋊C8)D412yes—
29C22⋊(C59⋊C8)—22no—
30C22⋊(C59⋊C8)—22no—
31C22⋊(C59⋊C8)—22no—
32#14273638—22no—
33#14273638C2×C412yes—
34#14273638C2×C412yes—
35C4×C59⋊C8—22no—
36C2×#41×C59D411yescharacteristic
37C2×#41×C59D411yescharacteristic
38C2×M16×C59—21no—
39C4⋊Q8×C59Q811yescharacteristic
40#127×C59Q811yescharacteristic
41#124×C59C2×C411yescharacteristic
42#124×C59C2×C411yescharacteristic
43#121×C59—22no—
44C4×D4×C59—22no—
45C2×C42×C59—21no—
46C2×C42×C59C2311yescharacteristic
47C4⋊C8×C59—22no—
48#104×C59—42no—
49#104×C59—42no—
50#95×C59—22no—
51#41×C59—41no—
52#41×C59—41no—
53#41×C59—41no—
54#41×C59—41no—
55C2×Q8×C59C4⋊C411yescharacteristic
56C2×Q8×C59C4⋊C411yescharacteristic
57C2×D4×C59—21no—
58C2×D4×C59—21no—
59C2×D4×C59—22no—
60C2×D4×C59C4⋊C411yescharacteristic
61C2×D4×C59C4⋊C411yescharacteristic
62C22×C4×C59—21no—
63C22×C4×C59—21no—
64C22×C4×C59—41no—
65C22×C4×C59C2×D411yescharacteristic
66C22×C4×C59#4112yes—
67M16×C59—41no—
68M16×C59—41no—
69C2×C8×C59—41no—
70C4⋊C4×C59—22no—
71C22⋊C4×C59—22no—
72C22⋊C4×C59—22no—
73C42×C59—22no—
74C42×C59—41no—
75C42×C59C22×C411yescharacteristic
76C42×C59C2×D412yes—
77C42×C59C2×Q811yescharacteristic
78#11745664—44no—
79#11745664—81no—
80#11745664—81no—
81C2×C59⋊C8—22no—
82C2×C59⋊C8—22no—
83C2×C59⋊C8—22no—
84C2×C59⋊C8—22no—
85C2×C59⋊C8—81no—
86C2×C59⋊C8—81no—
87C23×C59—21no—
88C23×C59—21no—
89C23×C59#12311yescharacteristic
90Q8×C59—41no—
91Q8×C59—41no—
92D4×C59—41no—
93D4×C59—41no—
94D4×C59—82no—
95D4×C59—81no—
96D4×C59—81no—
97C2×C4×C59—21no—
98C2×C4×C59—21no—
99C2×C4×C59—21no—
100C2×C4×C59—21no—
101C2×C4×C59—42no—
102C2×C4×C59—41no—
103C2×C4×C59—41no—
104C2×C4×C59—41no—
105C2×C4×C59—41no—
106C2×C4×C59—42no—
107C2×C4×C59—82no—
108C2×C4×C59C2×C4⋊C411yescharacteristic
109C2×C4×C59C4×D412yes—
110C2×C4×C59#12011yescharacteristic
111C2×C4×C59C22⋊Q812yes—
112C8×C59—81no—
113C59⋊C8—24no—
114C59⋊C8—42no—
115C59⋊C8—42no—
116C59⋊C8—42no—
117#8918—591nomaximal, Sylow
118C22×C59—21no—
119C22×C59—22no—
120C22×C59—41no—
121C22×C59—41no—
122C22×C59—42no—
123C22×C59—42no—
124C22×C59#38511yescharacteristic
125C4×C59—21no—
126C4×C59—22no—
127C4×C59—22no—
128C4×C59—41no—
129C4×C59—41no—
130C4×C59—82no—
131C4×C59#45311yescharacteristic
132C4×C59#45511yescharacteristic
133#2980—591no—
134#1963D5911yescharacteristic
135#1569—592no—
136#1525—591no—
137#1300—592no—
138C2×C59—41no—
139C2×C59—81no—
140C2×C59—81no—
141C2×C59#185611yescharacteristic
142C2×C59#186112yes—
143#532C2×D5911yescharacteristic
144#435—1182no—
145#435—1182no—
146#432—592no—
147#423C2×D5911yescharacteristic
148C2×#104—21no—
149C2×#104—21no—
150#408—1181no—
151#408—1181no—
152C4×M16—592no—
153#376C2×D5911yescharacteristic
154#335—1182no—
155#334—1182no—
156#329—591no—
157#329—592no—
158#329—591no—
159C59#891811yescharacteristic, complemented, Sylow
160C2×#41C22⋊D5911yescharacteristic
161C2×#41C22⋊D5911yescharacteristic
162C2×M16—21no—
163C2×M16—1181no—
164C2×M16—1181no—
165C2×M16—1182no—
166C22×C8—1181no—
167C22×C8—1181no—
168C4⋊Q8Dic11811yescharacteristic
169#127Dic11811yescharacteristic
170#124C4×D5911yescharacteristic
171#124C4×D5911yescharacteristic
172#121—22no—
173C4×D4—22no—
174C2×C42—21no—
175C2×C42C22×D5911yescharacteristic
176C4⋊C8—22no—
177C4⋊C8—592no—
178C4⋊C8—592no—
179#104—42no—
180#104—42no—
181C22⋊C8—1182no—
182C22⋊C8—1182no—
183C22⋊C8—1182no—
184#97—592no—
185#97—592no—
186#97—1182no—
187C4×C8—1182no—
188#95—22no—
189#41—41no—
190#41—41no—
191#41—41no—
192#41—41no—
193C2×Q8Dic59⋊C411yescharacteristic
194C2×Q8Dic59⋊C411yescharacteristic
195C2×D4—21no—
196C2×D4—21no—
197C2×D4—22no—
198C2×D4Dic59⋊C411yescharacteristic
199C2×D4Dic59⋊C411yescharacteristic
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—41no—
203C22×C4#1174568712yes—
204C22×C4C2×C22⋊D5911yescharacteristic
205M16—41no—
206M16—41no—
207M16—4721no—
208M16—4721no—
209M16—2364no—
210C2×C8—41no—
211C2×C8—1182no—
212C2×C8—1182no—
213C2×C8—1182no—
214C2×C8—1182no—
215C2×C8—4721no—
216C2×C8—4721no—
217C4⋊C4—22no—
218C22⋊C4—22no—
219C22⋊C4—22no—
220C42—22no—
221C42—41no—
222C42C2×Dic11811yescharacteristic
223C42C2×C4×D5911yescharacteristic
224C42D4×D5912yes—
225C23—21no—
226C23—21no—
227C23#1427370811yescharacteristic
228Q8—41no—
229Q8—41no—
230D4—41no—
231D4—41no—
232D4—82no—
233D4—81no—
234D4—81no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—42no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—42no—
245C2×C4—82no—
246C2×C4C2×Dic59⋊C411yescharacteristic
247C2×C4(C22⋊D59)⋊C412yes—
248C2×C4#1427369511yescharacteristic
249C2×C4#1427370412yes—
250C8—81no—
251C8—1184no—
252C8—2362no—
253C8—2362no—
254C8—2362no—
255C22—21no—
256C22—22no—
257C22—41no—
258C22—41no—
259C22—42no—
260C22—42no—
261C22#1683727411yescharacteristic
262C4—21no—
263C4—22no—
264C4—22no—
265C4—41no—
266C4—41no—
267C4—82no—
268C4#1683756911yescharacteristic
269C4#1683757911yescharacteristic
270C2—41no—
271C2—81no—
272C2—81no—
273C2#2142430511yescharacteristic
274C2#2142433412yes—
subgroupG/H
centerC2×C4#14273695
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1963C2
socleC22×C59#385
prime radicalC59×#532C22
cosocleC2×C42×C59C23
hypercenter#1963D59
hypocenterC59#8918
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1963

Complements

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

NH ≅ G/Ntype
2. #21422861C2semidirect
4. #21421989C2semidirect
14. #16837142C4semidirect
15. #16837142C4semidirect
16. #16837142C4semidirect
159. C59#8918semidirect

Maximal subgroups

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

subgroupindex
#214331672
#214228612
#214226882
#214219892
C59×#19632
#891859