Finite Group Databasebeta

← #28502260

Subgroups — #28502260

Order 15104.

324 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 67 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#21426363C211yesmaximal, characteristic
2#21426136C211yesmaximal, characteristic, complemented
3#21423580C211yesmaximal, characteristic, complemented
4#21422139C211yesmaximal, characteristic, complemented
5#21422078C211yesmaximal, characteristic, complemented
6#21422066C211yesmaximal, characteristic, complemented
7C59×#3402C211yesmaximal, characteristic, complemented
8C2×#14273670—21no—
9C2×#14273670—21no—
10C2×#14273668—21no—
11#16837779C2211yescharacteristic
12#16837602—21no—
13C4⋊(C59⋊C16)C2211yescharacteristic
14#16837202—21no—
15#16837178C2211yescharacteristic
16#16837153C2211yescharacteristic
17#16837145—21no—
18C59×#556—21no—
19C59×#497—21no—
20C59×#495C2211yescharacteristic
21C59×#464—21no—
22C59×#463—21no—
23C59×#462—21no—
24C59×#446C2211yescharacteristic
25C59×#440—21no—
26C4×Q16×C59C2211yescharacteristic
27C2×#11745671—41no—
28C2×D472—21no—
29C2×D472—21no—
30#14273724—21no—
31#14273670—42no—
32#14273670—42no—
33#14273668—42no—
34C2×C59⋊C16—21no—
35C2×C59⋊C16—21no—
36C4⋊(C59⋊C8)—21no—
37#14273647—41no—
38C2×Q16×C59—21no—
39C2×Q16×C59D411yescharacteristic
40C2×Q16×C59D411yescharacteristic
41C2×SD16×C59—21no—
42C2×SD16×C59—21no—
43C2×D8×C59D411yescharacteristic
44C4⋊Q8×C59—21no—
45#126×C59—41no—
46#126×C59—41no—
47#124×C59—21no—
48#124×C59—21no—
49C22⋊Q8×C59—41no—
50C4×Q8×C59—21no—
51C4×Q8×C59—21no—
52C4×Q8×C59D411yescharacteristic
53C4×D4×C59—21no—
54#117×C59—41no—
55#107×C59D411yescharacteristic
56C4⋊C8×C59—21no—
57C4⋊C8×C59D411yescharacteristic
58#103×C59—21no—
59#103×C59—21no—
60#102×C59—21no—
61#97×C59—21no—
62C4×C8×C59C2311yescharacteristic
63C2×Q8×C59—21no—
64C2×Q8×C59—21no—
65C2×Q8×C59D811yescharacteristic
66C2×D4×C59—21no—
67C22×C4×C59—41no—
68C2×D236—41no—
69C2×D236—41no—
70C2×D236—41no—
71Q16×C59—21no—
72Q16×C59—21no—
73Q16×C59—41no—
74Q16×C59—81no—
75SD16×C59—41no—
76SD16×C59—41no—
77SD16×C59—81no—
78D8×C59—41no—
79C2×C8×C59—21no—
80C2×C8×C59C2×D411yescharacteristic
81C2×C8×C59C2×D411yescharacteristic
82C4⋊C4×C59—21no—
83C4⋊C4×C59—21no—
84C4⋊C4×C59—41no—
85C4⋊C4×C59—41no—
86C4⋊C4×C59—41no—
87C4⋊C4×C59—41no—
88C4⋊C4×C59D811yescharacteristic
89C22⋊C4×C59—41no—
90C22⋊C4×C59—41no—
91C22⋊C4×C59—41no—
92C22⋊C4×C59—41no—
93C22⋊C4×C59—41no—
94C42×C59—21no—
95C42×C59—41no—
96C42×C59—41no—
97C42×C59C2×D411yescharacteristic
98#11745671—82no—
99C2×C59⋊C8—41no—
100D472—41no—
101D472—41no—
102D472—41no—
103D472—41no—
104C59⋊C16—22no—
105C59⋊C16—41no—
106C23×C59—41no—
107C22×D59—81no—
108Q8×C59—41no—
109Q8×C59—41no—
110Q8×C59—41no—
111Q8×C59—41no—
112Q8×C59—41no—
113D4×C59—41no—
114D4×C59—41no—
115C2×C4×C59—21no—
116C2×C4×C59—21no—
117C2×C4×C59—41no—
118C2×C4×C59—41no—
119C2×C4×C59—41no—
120C2×C4×C59—82no—
121C2×C4×C59#12011yescharacteristic
122C2×C4×C59C2×D811yescharacteristic
123C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
124D236—81no—
125D236—81no—
126D236—81no—
127D236—81no—
128D236—161no—
129C8×C59—21no—
130C8×C59—41no—
131C8×C59#13511yescharacteristic
132C8×C59AGL(1,ℤ/8)11yescharacteristic
133C59⋊C8—81no—
134#12337—591nomaximal, Sylow
135C22×C59—81no—
136C22×C59—81no—
137C22×C59—81no—
138C22×C59#44711yescharacteristic
139C2×D59—161no—
140C2×D59—161no—
141C2×D59—161no—
142C4×C59—21no—
143C4×C59—21no—
144C4×C59—41no—
145C4×C59—41no—
146C4×C59—81no—
147C4×C59—81no—
148C4×C59—81no—
149C4×C59#44911yescharacteristic
150C4×C59#45311yescharacteristic
151#3402D5911yescharacteristic, complemented
152#2250—591no—
153#2207—591no—
154#1688—591no—
155#1360—591no—
156#1338—591no—
157#1334—591no—
158C2×C59—161no—
159C2×C59#162111yescharacteristic
160C2×C59#219011yescharacteristic
161C2×C59#219311yescharacteristic
162D59—321no—
163#556—21no—
164C2×SD32—1181no—
165C2×SD32—1181no—
166C2×D16—1181no—
167#497—21no—
168#495C2×D5911yescharacteristic
169#493—591no—
170#464—21no—
171#463—21no—
172#462—21no—
173#460—1181no—
174#446C2×D5911yescharacteristic
175#440—21no—
176C4×Q16C2×D5911yescharacteristic
177C4⋊C16—591no—
178#357—1181no—
179#346—591no—
180#334—591no—
181#331—1181no—
182C59#1233711yescharacteristic, complemented, Sylow
183C2×Q16—21no—
184C2×Q16C22⋊D5911yescharacteristic
185C2×Q16C22⋊D5911yescharacteristic
186C2×SD16—21no—
187C2×SD16—21no—
188C2×SD16—2361no—
189C2×D8—1181no—
190C2×D8—1181no—
191C2×D8C22⋊D5911yescharacteristic
192C4⋊Q8—21no—
193#127—1181no—
194#126—41no—
195#126—41no—
196#124—21no—
197#124—21no—
198C22⋊Q8—41no—
199C4×Q8—21no—
200C4×Q8—21no—
201C4×Q8C22⋊D5911yescharacteristic
202C4×D4—21no—
203#117—41no—
204SD32—2362no—
205SD32—2362no—
206D16—2362no—
207C2×C16—1181no—
208C2×C16—1181no—
209#107C22⋊D5911yescharacteristic
210C4⋊C8—21no—
211C4⋊C8—1181no—
212C4⋊C8C22⋊D5911yescharacteristic
213#103—21no—
214#103—21no—
215#102—21no—
216#102—2361no—
217#97—21no—
218C4×C8C22×D5911yescharacteristic
219C2×Q8—21no—
220C2×Q8—21no—
221C2×Q8#1174566911yescharacteristic
222C2×D4—21no—
223C2×D4—2361no—
224C2×D4—2361no—
225C2×D4—2361no—
226C22×C4—41no—
227Q16—21no—
228Q16—21no—
229Q16—41no—
230Q16—81no—
231SD16—41no—
232SD16—41no—
233SD16—81no—
234SD16—4722no—
235D8—41no—
236D8—2361no—
237D8—2361no—
238D8—2361no—
239D8—2361no—
240C2×C8—21no—
241C2×C8—2361no—
242C2×C8C2×C22⋊D5911yescharacteristic
243C2×C8C2×C22⋊D5911yescharacteristic
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4#1174566911yescharacteristic
251C22⋊C4—41no—
252C22⋊C4—41no—
253C22⋊C4—41no—
254C22⋊C4—41no—
255C22⋊C4—41no—
256C42—21no—
257C42—41no—
258C42—41no—
259C42C2×C22⋊D5911yescharacteristic
260C16—1182no—
261C16—2361no—
262C23—41no—
263C23—4721no—
264Q8—41no—
265Q8—41no—
266Q8—41no—
267Q8—41no—
268Q8—41no—
269D4—41no—
270D4—41no—
271D4—9441no—
272D4—4721no—
273D4—4721no—
274D4—4721no—
275D4—4721no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—82no—
282C2×C4#1427369611yescharacteristic
283C2×C4C2×#1174566911yescharacteristic
284C2×C4#1427375311yescharacteristic
285C8—21no—
286C8—41no—
287C8—4721no—
288C8#1427374711yescharacteristic
289C8#1427375211yescharacteristic
290C22—81no—
291C22—81no—
292C22—81no—
293C22—9441no—
294C22—9441no—
295C22—9441no—
296C22#1683753611yescharacteristic
297C4—21no—
298C4—21no—
299C4—41no—
300C4—41no—
301C4—81no—
302C4—81no—
303C4—81no—
304C4#1683754711yescharacteristic
305C4#1683757011yescharacteristic
306C2—161no—
307C2—18881no—
308C2#2142316111yescharacteristic
309C2#2142604211yescharacteristic
310C2#2142606011yescharacteristic
subgroupG/H
centerC22#16837536
commutator, cosocleC4×C8×C59C23
FrattiniC4×C8C22×D59
FittingC59×#3402C2
socleC22×C59#447
prime radical#21426363C2
hypercenter#3402D59
hypocenterC59#12337
LCSG ▷ C4×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ #446 ◁ #3402

Complements

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

NH ≅ G/Ntype
2. #21426136C2semidirect
3. #21423580C2semidirect
4. #21422139C2semidirect
5. #21422078C2semidirect
6. #21422066C2semidirect
7. C59×#3402C2semidirect
151. #3402D59semidirect
182. C59#12337semidirect

Maximal subgroups

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

subgroupindex
#214263632
#214261362
#214235802
#214221392
#214220782
#214220662
C59×#34022
#1233759