Finite Group Databasebeta

← #28493659

Subgroups — #28493659

Order 15104.

356 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 93 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#21424894C211yesmaximal, characteristic
2#21424237C211yesmaximal, characteristic, complemented
3#21422871C211yesmaximal, characteristic, complemented
4#21422867C211yesmaximal, characteristic, complemented
5#21422522C211yesmaximal, characteristic
6#21422522C211yesmaximal, characteristic
7C59×#1275C211yesmaximal, characteristic, complemented
8C2×C4×D236C2211yescharacteristic
9#16837426C2211yescharacteristic
10#16837426C2211yescharacteristic
11C2×C4⋊(C59⋊C8)C2211yescharacteristic
12C2×(C2×D59)⋊C8—22no—
13#16837167—22no—
14#16837155—22no—
15#16837153—22no—
16#16837131—22no—
17#16837131—22no—
18C59×#431C2211yescharacteristic
19C59×#431C2211yescharacteristic
20C2×#97×C59C2211yescharacteristic
21C22×D236Q811yescharacteristic
22C22×C4×D59—21no—
23C2×#11745664—21no—
24C2×#11745664—21no—
25C22×C59⋊C8—21no—
26C4×D236—21no—
27C4×D236—21no—
28C4×D236—21no—
29C4×D236—21no—
30C2×C4⋊Dic59Q811yescharacteristic
31C2×(C2×D59)⋊C4—21no—
32C4⋊(C59⋊C8)—21no—
33C4⋊(C59⋊C8)—21no—
34C4⋊(C59⋊C8)—21no—
35C4⋊(C59⋊C8)—21no—
36C4⋊(C59⋊C8)—21no—
37C4⋊(C59⋊C8)—21no—
38(C2×D59)⋊C8—44no—
39C22×C8×C59D412yes—
40C2×C42×C59C2311yescharacteristic
41C4⋊C8×C59—21no—
42C4⋊C8×C59—21no—
43C22⋊C8×C59—21no—
44C22⋊C8×C59—21no—
45#97×C59—21no—
46#97×C59—21no—
47C4×C8×C59D412yes—
48C4×C8×C59D412yes—
49C23×D59—21no—
50C22×C4×C59C2×D411yescharacteristic
51C22×C4×C59C2×Q811yescharacteristic
52C22×C4×C59#4111yescharacteristic
53C22×Dic59—21no—
54C2×D236—21no—
55C2×D236—21no—
56C2×D236—42no—
57C2×C4×D59—41no—
58C2×C4×D59—41no—
59C2×C4×D59—42no—
60C2×C4×D59—42no—
61C2×C8×C59—22no—
62C2×C8×C59—22no—
63C2×C8×C59—22no—
64C2×C8×C59—22no—
65C2×C8×C59—22no—
66C42×C59C2×D411yescharacteristic
67C42×C59C2×D411yescharacteristic
68C42×C59#4111yescharacteristic
69C42×C59#4111yescharacteristic
70(C2×D59)⋊C4—41no—
71(C2×D59)⋊C4—41no—
72C4⋊Dic59—21no—
73C4⋊Dic59—21no—
74#11745664—42no—
75#11745664—42no—
76C2×C59⋊C8—41no—
77C2×C59⋊C8—41no—
78C2×C59⋊C8—41no—
79C2×C59⋊C8—42no—
80C23×C59C22⋊Q811yescharacteristic
81C22×D59—22no—
82C22×D59—41no—
83C22×D59—42no—
84C22×D59—41no—
85C22×D59—42no—
86C2×C4×C59—41no—
87C2×C4×C59—41no—
88C2×C4×C59C22⋊Q811yescharacteristic
89C2×C4×C59C22⋊Q811yescharacteristic
90C2×C4×C59#12311yescharacteristic
91C2×C4×C59#12311yescharacteristic
92C2×C4×C59#12311yescharacteristic
93C2×C4×C59#12711yescharacteristic
94C2×C4×C59#13512yes—
95C2×C4×C59AGL(1,ℤ/8)12yes—
96C2×Dic59—41no—
97C2×Dic59—42no—
98D236—22no—
99D236—22no—
100D236—41no—
101D236—41no—
102C4×D59—82no—
103C4×D59—81no—
104C4×D59—81no—
105C8×C59—42no—
106C8×C59—42no—
107C8×C59—42no—
108C59⋊C8—81no—
109C59⋊C8—81no—
110C59⋊C8—81no—
111#10659—591nomaximal, Sylow
112C22×C59—21no—
113C22×C59—21no—
114C22×C59#39911yescharacteristic
115C22×C59#47812yes—
116C2×D59—44no—
117C2×D59—42no—
118C2×D59—82no—
119C2×D59—81no—
120C2×D59—81no—
121C4×C59—21no—
122C4×C59—21no—
123C4×C59—21no—
124C4×C59—21no—
125C4×C59#48212yes—
126C4×C59#49511yescharacteristic
127C4×C59#49611yescharacteristic
128Dic59—81no—
129#1955—591no—
130#1838—591no—
131#1571—591no—
132#1570—591no—
133#1486—591no—
134#1486—591no—
135#1275D5911yescharacteristic, complemented
136C2×C59—22no—
137C2×C59#209111yescharacteristic
138C2×C59#209211yescharacteristic
139C2×C59#209811yescharacteristic
140D59—82no—
141C2×C4×D4—591no—
142#431C2×D5911yescharacteristic
143#431C2×D5911yescharacteristic
144#423—591no—
145#423—591no—
146C2×C4⋊C8—591no—
147C2×C22⋊C8—1182no—
148C2×#97C2×D5911yescharacteristic
149#341—1182no—
150#335—1182no—
151#334—1182no—
152D4⋊C8—1182no—
153D4⋊C8—1182no—
154C59#1065911yescharacteristic, complemented, Sylow
155C22×D4—591no—
156C23×C4—1181no—
157C2×M16—1181no—
158C2×M16—1181no—
159C22×C8—1181no—
160C22×C8D23612yes—
161C4×D4—1181no—
162C4×D4—1181no—
163C4×D4—1181no—
164C4×D4—1181no—
165C2×C4⋊C4—591no—
166C2×C22⋊C4—1181no—
167C2×C42C22×D5911yescharacteristic
168C4⋊C8—21no—
169C4⋊C8—21no—
170C4⋊C8—1181no—
171C4⋊C8—1181no—
172C4⋊C8—1181no—
173C4⋊C8—1181no—
174C4⋊C8—1181no—
175C4⋊C8—1181no—
176C22⋊C8—21no—
177C22⋊C8—21no—
178C22⋊C8—2364no—
179#97—21no—
180#97—21no—
181C4×C8D23612yes—
182C4×C8D23612yes—
183C24—1181no—
184C2×D4—1181no—
185C2×D4—1181no—
186C2×D4—2362no—
187C22×C4—1181no—
188C22×C4—2361no—
189C22×C4—2361no—
190C22×C4—2362no—
191C22×C4—2362no—
192C22×C4C2×D23611yescharacteristic
193C22×C4#1174568711yescharacteristic
194C22×C4Q8×D5911yescharacteristic
195M16—2362no—
196M16—2362no—
197C2×C8—22no—
198C2×C8—22no—
199C2×C8—22no—
200C2×C8—22no—
201C2×C8—22no—
202C2×C8—2361no—
203C2×C8—2361no—
204C2×C8—2361no—
205C2×C8—2362no—
206C4⋊C4—1181no—
207C4⋊C4—1181no—
208C22⋊C4—2361no—
209C22⋊C4—2361no—
210C42C2×D23611yescharacteristic
211C42C2×D23611yescharacteristic
212C42#1174568711yescharacteristic
213C42#1174568711yescharacteristic
214C23—1182no—
215C23—2361no—
216C23—2362no—
217C23—2362no—
218C23—2361no—
219C23#1427370711yescharacteristic
220D4—1182no—
221D4—1182no—
222D4—2361no—
223D4—2361no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—4722no—
227C2×C4—4721no—
228C2×C4—4721no—
229C2×C4—2362no—
230C2×C4—2361no—
231C2×C4#1427370711yescharacteristic
232C2×C4#1427370711yescharacteristic
233C2×C4#1427371111yescharacteristic
234C2×C4#1427371111yescharacteristic
235C2×C4#1427371111yescharacteristic
236C2×C4#1427372411yescharacteristic
237C2×C4#1427374912yes—
238C2×C4#1427375512yes—
239C8—42no—
240C8—42no—
241C8—42no—
242C8—4721no—
243C8—4721no—
244C8—4721no—
245C22—21no—
246C22—21no—
247C22—4722no—
248C22—4721no—
249C22—4721no—
250C22—2364no—
251C22—2362no—
252C22#1683733911yescharacteristic
253C22#1683770312yes—
254C4—21no—
255C4—21no—
256C4—21no—
257C4—21no—
258C4—4721no—
259C4#1683772712yes—
260C4#1683778611yescharacteristic
261C4#1683779111yescharacteristic
262C2—22no—
263C2—4722no—
264C2#2142561311yescharacteristic
265C2#2142561611yescharacteristic
266C2#2142565011yescharacteristic
subgroupG/H
centerC2×C4#14273724
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC59×#1275C2
socleC22×C59#399
prime radicalC22×D236Q8
hypercenter#1275D59
hypocenterC59#10659
LCSG ▷ C2×C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1275

Complements

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

NH ≅ G/Ntype
2. #21424237C2semidirect
3. #21422871C2semidirect
4. #21422867C2semidirect
7. C59×#1275C2semidirect
135. #1275D59semidirect
154. C59#10659semidirect

Maximal subgroups

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

subgroupindex
#214248942
#214242372
#214228712
#214228672
#214225222
C59×#12752
#1065959