Finite Group Databasebeta

← #28501129

Subgroups — #28501129

Order 15104.

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

normal subgroups only

This group has 434 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#21435406C211yesmaximal, characteristic, complemented
2#21426008C212yesmaximal, complemented
3#21422104C212yesmaximal, complemented
4#21422036C211yesmaximal, characteristic, complemented
5C59×#2054C211yesmaximal, characteristic, complemented
6#16837952—21no—
7C2×#14273668—22no—
8#16837683—22no—
9#16837638—21no—
10#16837638C2212yes—
11#16837598—22no—
12#16837203—22no—
13C22⋊(C59⋊C16)C2212yes—
14C22×D8×C59C2211yescharacteristic
15C2×C4⋊C8×C59C2211yescharacteristic
16C2×#102×C59—21no—
17C59×#390—21no—
18C59×#340C2211yescharacteristic
19C2×#11745685—41no—
20C2×D472—21no—
21C2×D472D412yes—
22#14273727—21no—
23#14273724—21no—
24#14273715—41no—
25#14273701—22no—
26#14273701—41no—
27C4×D236—22no—
28C4×D236—41no—
29#14273668—44no—
30C2×C59⋊C16—22no—
31#14273661—21no—
32#14273661D412yes—
33#14273649—22no—
34#14273649—22no—
35C22×D4×C59—21no—
36C2×D8×C59—22no—
37C2×D8×C59—42no—
38C22×C8×C59D411yescharacteristic
39C22×C8×C59C2311yescharacteristic
40C2×C4⋊C4×C59—21no—
41C2×C22⋊C4×C59—41no—
42C2×C22⋊C4×C59—41no—
43C2×C42×C59D411yescharacteristic
44C4⋊C8×C59—22no—
45#102×C59—42no—
46C24×C59—41no—
47C2×D4×C59—22no—
48C2×D4×C59—41no—
49C2×D4×C59—41no—
50C2×D4×C59—41no—
51C2×D4×C59—41no—
52C2×D4×C59—41no—
53C22×C4×C59—41no—
54C22×C4×C59D811yescharacteristic
55C22×C4×C59D811yescharacteristic
56C22×C4×C59C2×D411yescharacteristic
57C2×C22⋊D59—42no—
58#11745685—82no—
59#11745685—82no—
60C2×D236—22no—
61C2×D236—41no—
62C2×D236—41no—
63C2×C4×D59—42no—
64C2×Dic118—41no—
65D8×C59—24no—
66D8×C59—42no—
67C2×C8×C59—21no—
68C2×C8×C59—41no—
69C2×C8×C59—41no—
70C2×C8×C59C2×D412yes—
71C4⋊C4×C59—41no—
72C4⋊C4×C59—41no—
73C22⋊C4×C59—82no—
74C22⋊C4×C59—82no—
75C42×C59—21no—
76C42×C59—21no—
77(C2×D59)⋊C4—42no—
78(C2×D59)⋊C4—42no—
79C4⋊Dic59—22no—
80C4⋊Dic59—41no—
81D472—42no—
82D472—81no—
83C59⋊C16—42no—
84C23×C59—42no—
85C23×C59—81no—
86C23×C59—81no—
87C23×C59—81no—
88C23×C59—81no—
89C23×C59—81no—
90C23×C59—81no—
91C23×C59C2×D811yescharacteristic
92C22×D59—42no—
93D4×C59—42no—
94D4×C59—42no—
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—21no—
102C2×C4×C59—41no—
103C2×C4×C59—81no—
104C2×C4×C59—81no—
105C2×C4×C59—81no—
106C2×C4×C59D1612yescomplemented
107C2×C4×C59#12011yescharacteristic
108C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
109C22⋊D59—82no—
110C22⋊D59—82no—
111C2×Dic59—42no—
112D236—42no—
113D236—42no—
114D236—82no—
115D236—82no—
116C4×D59—84no—
117Dic118—82no—
118C8×C59—22no—
119C8×C59—41no—
120#12158—591nomaximal, Sylow
121C22×C59—21no—
122C22×C59—21no—
123C22×C59—82no—
124C22×C59—82no—
125C22×C59—82no—
126C22×C59—161no—
127C22×C59—161no—
128C22×C59—161no—
129C22×C59—161no—
130C22×C59#44711yescharacteristic
131C22×C59C2×D1611yescharacteristic
132C22×C59#50911yescharacteristic
133C2×D59—82no—
134C2×D59—82no—
135C2×D59—82no—
136C4×C59—21no—
137C4×C59—22no—
138C4×C59—41no—
139C4×C59—161no—
140C4×C59#45411yescharacteristic
141C4×C59B(4,𝔽2)11yescharacteristic
142Dic59—82no—
143#3245—591no—
144#2184—592no—
145#2054D5911yescharacteristic, complemented
146#1347—592no—
147#1322—591no—
148C2×C59—21no—
149C2×C59—21no—
150C2×C59—162no—
151C2×C59#159811yescharacteristic
152C2×C59#218411yescharacteristic
153C2×C59#219011yescharacteristic
154D59—162no—
155C22×D8C2×D5911yescharacteristic
156#532—1181no—
157C2×D16—1182no—
158#474—1182no—
159#466—592no—
160#466—1181no—
161#459—1182no—
162C2×C4⋊C8C2×D5911yescharacteristic
163C2×#102—21no—
164#390—21no—
165#357—1182no—
166C22⋊C16—592no—
167#340C2×D5911yescharacteristic
168C59#1215811yescharacteristic, complemented, Sylow
169C2×#41—2361no—
170C22×D4—21no—
171C2×D8—22no—
172C2×D8—42no—
173C2×D8—592no—
174C2×D8—1181no—
175C22×C8C22⋊D5911yescharacteristic
176C22×C8C22×D5911yescharacteristic
177C4⋊Q8—1181no—
178#127—1181no—
179#124—2361no—
180#121—1182no—
181#121—2361no—
182C4×D4—1182no—
183C4×D4—2361no—
184C2×C4⋊C4—21no—
185C2×C22⋊C4—41no—
186C2×C22⋊C4—41no—
187C2×C42C22⋊D5911yescharacteristic
188D16—2364no—
189C2×C16—1182no—
190#107—592no—
191#107—1181no—
192C4⋊C8—22no—
193#102—42no—
194#102—1182no—
195#102—1182no—
196C24—41no—
197#41—4722no—
198#41—4722no—
199C2×Q8—2361no—
200C2×D4—22no—
201C2×D4—41no—
202C2×D4—41no—
203C2×D4—41no—
204C2×D4—41no—
205C2×D4—41no—
206C2×D4—1182no—
207C2×D4—2361no—
208C2×D4—2361no—
209C2×D4—2362no—
210C22×C4—41no—
211C22×C4—2362no—
212C22×C4#1174566911yescharacteristic
213C22×C4#1174566911yescharacteristic
214C22×C4C2×C22⋊D5911yescharacteristic
215D8—24no—
216D8—42no—
217D8—4721no—
218D8—2362no—
219C2×C8—21no—
220C2×C8—41no—
221C2×C8—41no—
222C2×C8D4×D5912yes—
223C4⋊C4—41no—
224C4⋊C4—41no—
225C4⋊C4—1182no—
226C4⋊C4—2361no—
227C22⋊C4—82no—
228C22⋊C4—82no—
229C22⋊C4—2362no—
230C22⋊C4—2362no—
231C42—21no—
232C42—21no—
233C16—2362no—
234C23—42no—
235C23—81no—
236C23—81no—
237C23—81no—
238C23—81no—
239C23—81no—
240C23—81no—
241C23—2362no—
242C23C2×#1174566911yescharacteristic
243Q8—4722no—
244D4—42no—
245D4—42no—
246D4—81no—
247D4—81no—
248D4—4722no—
249D4—4722no—
250D4—4722no—
251D4—4722no—
252D4—2362no—
253D4—2362no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—41no—
260C2×C4—81no—
261C2×C4—81no—
262C2×C4—81no—
263C2×C4—4724no—
264C2×C4—2362no—
265C2×C4#1427366812yescomplemented
266C2×C4#1427369511yescharacteristic
267C2×C4#1427375211yescharacteristic
268C8—22no—
269C8—41no—
270C22—21no—
271C22—21no—
272C22—82no—
273C22—82no—
274C22—82no—
275C22—161no—
276C22—161no—
277C22—161no—
278C22—161no—
279C22—4722no—
280C22—4722no—
281C22—4722no—
282C22#1683753511yescharacteristic
283C22C2×#1427366811yescharacteristic
284C22#1683783711yescharacteristic
285C4—22no—
286C4—21no—
287C4—41no—
288C4—161no—
289C4—4722no—
290C4#1683757411yescharacteristic
291C4#1683758811yescharacteristic
292C2—21no—
293C2—21no—
294C2—162no—
295C2—9442no—
296C2#2142303011yescharacteristic
297C2#2142600711yescharacteristic
298C2#2142604111yescharacteristic
subgroupG/H
centerC22#16837535
commutator, cosocleC22×C8×C59C23
FrattiniC22×C8C22×D59
FittingC59×#2054C2
socleC22×C59#447
hypercenter#2054D59
hypocenterC59#12158
LCSG ▷ C22×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C8 ◁ #2054

Complements

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

NH ≅ G/Ntype
1. #21435406C2semidirect
2. #21426008C2semidirect
3. #21422104C2semidirect
4. #21422036C2semidirect
5. C59×#2054C2semidirect
106. C2×C4×C59D16semidirect
145. #2054D59semidirect
168. C59#12158semidirect
265. C2×C4#14273668semidirect

Maximal subgroups

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

subgroupindex
#214354062
#214260082
#214221042
#214220362
C59×#20542
#1215859