Finite Group Databasebeta

← #28501130

Subgroups — #28501130

Order 15104.

434 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 71 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#21426007C212yesmaximal, complemented
2#21425407C211yesmaximal, characteristic, complemented
3#21422105C212yesmaximal, complemented
4#21422036C211yesmaximal, characteristic, complemented
5C59×#3245C211yesmaximal, characteristic, complemented
6C22×D472C2211yescharacteristic
7C2×#14273668—22no—
8C2×#14273649—21no—
9#16837304—21no—
10#16837202—22no—
11C22⋊(C59⋊C16)C2212yes—
12#16837165C2211yescharacteristic
13C59×#532—21no—
14C59×#474—22no—
15C59×#466—21no—
16C59×#466C2212yes—
17C59×#459—22no—
18C2×C4⋊C8×C59C2211yescharacteristic
19C22×D236—21no—
20C2×D472—22no—
21C2×D472—42no—
22C2×C4⋊Dic59—21no—
23C2×(C2×D59)⋊C4—41no—
24C2×(C2×D59)⋊C4—41no—
25#14273668—44no—
26C2×C59⋊C16—22no—
27#14273649—42no—
28C2×#41×C59—41no—
29C2×D8×C59—21no—
30C2×D8×C59D412yes—
31C22×C8×C59D411yescharacteristic
32C22×C8×C59C2311yescharacteristic
33C4⋊Q8×C59—21no—
34#127×C59—21no—
35#124×C59—41no—
36#121×C59—22no—
37#121×C59—41no—
38C4×D4×C59—22no—
39C4×D4×C59—41no—
40C2×C42×C59D411yescharacteristic
41#107×C59—21no—
42#107×C59D412yes—
43C4⋊C8×C59—22no—
44#102×C59—22no—
45#102×C59—22no—
46C23×D59—41no—
47#41×C59—82no—
48#41×C59—82no—
49C2×Q8×C59—41no—
50C2×D4×C59—22no—
51C2×D4×C59—41no—
52C2×D4×C59—41no—
53C2×D4×C59—42no—
54C22×C4×C59—42no—
55C22×C4×C59D811yescharacteristic
56C22×C4×C59D811yescharacteristic
57C22×C4×C59C2×D411yescharacteristic
58C22×Dic59—41no—
59C2×D236—22no—
60C2×D236—41no—
61C2×D236—41no—
62C2×D236—41no—
63C2×D236—41no—
64C2×D236—41no—
65D8×C59—42no—
66D8×C59—81no—
67C2×C8×C59—21no—
68C2×C8×C59—41no—
69C2×C8×C59—41no—
70C2×C8×C59C2×D412yes—
71C4⋊C4×C59—22no—
72C4⋊C4×C59—41no—
73C22⋊C4×C59—42no—
74C22⋊C4×C59—42no—
75C42×C59—21no—
76C42×C59—21no—
77(C2×D59)⋊C4—82no—
78(C2×D59)⋊C4—82no—
79C4⋊Dic59—41no—
80C4⋊Dic59—41no—
81D472—24no—
82D472—42no—
83C59⋊C16—42no—
84C23×C59—42no—
85C23×C59C2×D811yescharacteristic
86C22×D59—42no—
87C22×D59—81no—
88C22×D59—81no—
89C22×D59—81no—
90C22×D59—81no—
91C22×D59—81no—
92C22×D59—81no—
93Q8×C59—82no—
94D4×C59—42no—
95D4×C59—42no—
96D4×C59—82no—
97D4×C59—82no—
98D4×C59—82no—
99D4×C59—82no—
100C2×C4×C59—21no—
101C2×C4×C59—21no—
102C2×C4×C59—21no—
103C2×C4×C59—21no—
104C2×C4×C59—21no—
105C2×C4×C59—41no—
106C2×C4×C59—42no—
107C2×C4×C59—84no—
108C2×C4×C59D1612yescomplemented
109C2×C4×C59#12011yescharacteristic
110C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
111C2×Dic59—81no—
112C2×Dic59—81no—
113C2×Dic59—81no—
114D236—42no—
115D236—42no—
116D236—81no—
117D236—81no—
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#44711yescharacteristic
127C22×C59C2×D1611yescharacteristic
128C22×C59#50911yescharacteristic
129C2×D59—82no—
130C2×D59—82no—
131C2×D59—82no—
132C2×D59—161no—
133C2×D59—161no—
134C2×D59—161no—
135C2×D59—161no—
136C4×C59—21no—
137C4×C59—22no—
138C4×C59—41no—
139C4×C59—82no—
140C4×C59#45411yescharacteristic
141C4×C59B(4,𝔽2)11yescharacteristic
142Dic59—161no—
143#3245D5911yescharacteristic, complemented
144#2184—592no—
145#2054—591no—
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×D8—591no—
156#532—21no—
157C2×D16—1182no—
158#474—22no—
159#466—21no—
160#466C2×D5912yes—
161#459—22no—
162C2×C4⋊C8C2×D5911yescharacteristic
163C2×#102—1181no—
164#390—1181no—
165#357—1182no—
166C22⋊C16—592no—
167#340—591no—
168C59#1215811yescharacteristic, complemented, Sylow
169C2×#41—41no—
170C22×D4—1181no—
171C2×D8—21no—
172C2×D8—1182no—
173C2×D8—2362no—
174C2×D8C22⋊D5912yes—
175C22×C8C22⋊D5911yescharacteristic
176C22×C8C22×D5911yescharacteristic
177C4⋊Q8—21no—
178#127—21no—
179#124—41no—
180#121—22no—
181#121—41no—
182C4×D4—22no—
183C4×D4—41no—
184C2×C4⋊C4—1181no—
185C2×C22⋊C4—2361no—
186C2×C22⋊C4—2361no—
187C2×C42C22⋊D5911yescharacteristic
188D16—2364no—
189C2×C16—1182no—
190#107—21no—
191#107C22⋊D5912yes—
192C4⋊C8—22no—
193#102—22no—
194#102—22no—
195#102—2362no—
196C24—2361no—
197#41—82no—
198#41—82no—
199C2×Q8—41no—
200C2×D4—22no—
201C2×D4—41no—
202C2×D4—41no—
203C2×D4—42no—
204C2×D4—1182no—
205C2×D4—2361no—
206C2×D4—2361no—
207C2×D4—2361no—
208C2×D4—2361no—
209C2×D4—2361no—
210C22×C4—42no—
211C22×C4—2361no—
212C22×C4#1174566911yescharacteristic
213C22×C4#1174566911yescharacteristic
214C22×C4C2×C22⋊D5911yescharacteristic
215D8—42no—
216D8—81no—
217D8—1184no—
218D8—2362no—
219C2×C8—21no—
220C2×C8—41no—
221C2×C8—41no—
222C2×C8C2×C22⋊D5912yes—
223C4⋊C4—22no—
224C4⋊C4—41no—
225C4⋊C4—2361no—
226C4⋊C4—2361no—
227C22⋊C4—42no—
228C22⋊C4—42no—
229C22⋊C4—4722no—
230C22⋊C4—4722no—
231C42—21no—
232C42—21no—
233C16—2362no—
234C23—42no—
235C23—4721no—
236C23—4721no—
237C23—4721no—
238C23—4721no—
239C23—4721no—
240C23—4721no—
241C23—2362no—
242C23C2×#1174566911yescharacteristic
243Q8—82no—
244D4—42no—
245D4—42no—
246D4—82no—
247D4—82no—
248D4—82no—
249D4—82no—
250D4—4721no—
251D4—4721no—
252D4—2362no—
253D4—2362no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—41no—
260C2×C4—42no—
261C2×C4—84no—
262C2×C4—4721no—
263C2×C4—4721no—
264C2×C4—4721no—
265C2×C4#1427366812yescomplemented
266C2×C4#1427369611yescharacteristic
267C2×C4#1427375311yescharacteristic
268C8—22no—
269C8—41no—
270C22—21no—
271C22—21no—
272C22—82no—
273C22—82no—
274C22—82no—
275C22—9441no—
276C22—9441no—
277C22—9441no—
278C22—9441no—
279C22—4722no—
280C22—4722no—
281C22—4722no—
282C22#1683753611yescharacteristic
283C22C2×#1427366811yescharacteristic
284C22#1683783811yescharacteristic
285C4—22no—
286C4—21no—
287C4—41no—
288C4—82no—
289C4—9441no—
290C4#1683757511yescharacteristic
291C4#1683758911yescharacteristic
292C2—21no—
293C2—21no—
294C2—162no—
295C2—9442no—
296C2#2142303111yescharacteristic
297C2#2142600811yescharacteristic
298C2#2142604211yescharacteristic
subgroupG/H
centerC22#16837536
commutator, cosocleC22×C8×C59C23
FrattiniC22×C8C22×D59
FittingC59×#3245C2
socleC22×C59#447
hypercenter#3245D59
hypocenterC59#12158
LCSG ▷ C22×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C8 ◁ #3245

Complements

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

NH ≅ G/Ntype
1. #21426007C2semidirect
2. #21425407C2semidirect
3. #21422105C2semidirect
4. #21422036C2semidirect
5. C59×#3245C2semidirect
108. C2×C4×C59D16semidirect
143. #3245D59semidirect
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
#214260072
#214254072
#214221052
#214220362
C59×#32452
#1215859