Finite Group Databasebeta

← #28489253

Subgroups — #28489253

Order 15104.

772 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 772 conjugacy classes of subgroups — too many to draw. Only its 121 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
1C2×#16837718C212yesmaximal, complemented
2C2×#16837136C212yesmaximal, complemented
3#21422365C211yesmaximal, characteristic, complemented
4#21421939C211yesmaximal, characteristic, complemented
5C59×#2656C211yesmaximal, characteristic
6C22×C4⋊Dic59C2211yescharacteristic
7#16837718—24no—
8#16837718—24no—
9C2×#14273658—22no—
10C2×#14273648—22no—
11C2×C22⋊(C59⋊C8)C2212yes—
12C2×C59⋊#95—22no—
13C2×C59⋊#95C2211yescharacteristic
14#16837136—24no—
15C2×#121×C59—22no—
16C2×#121×C59C2212yes—
17C59×#399—22no—
18C59×#392—22no—
19C2×#95×C59C2211yescharacteristic
20C23×Dic59—21no—
21C23×Dic59—21no—
22C22×C59⋊C8—22no—
23C2×C4⋊Dic59—22no—
24C2×C4⋊Dic59—22no—
25C2×C4⋊Dic59—21no—
26C2×C4⋊Dic59D412yescomplemented
27#14273658—28no—
28#14273648—28no—
29C22⋊(C59⋊C8)—22no—
30C22⋊(C59⋊C8)D414yescomplemented
31C59⋊#95—22no—
32C59⋊#95—44no—
33C22×D4×C59—22no—
34C22×D4×C59—22no—
35C22×D4×C59D412yes—
36C23×C4×C59D412yes—
37C23×C4×C59C2311yescharacteristic
38#121×C59—22no—
39#121×C59—22no—
40#121×C59—22no—
41#121×C59—22no—
42#121×C59—44no—
43#121×C59—44no—
44C2×C4⋊C4×C59—22no—
45C2×C4⋊C4×C59D412yes—
46C2×C22⋊C4×C59—24no—
47C2×C22⋊C4×C59—22no—
48#95×C59—22no—
49C24×C59—22no—
50C24×C59C2×D411yescharacteristic
51C2×D4×C59—22no—
52C2×D4×C59—22no—
53C2×D4×C59—44no—
54C2×D4×C59—44no—
55C2×D4×C59—42no—
56C2×D4×C59—42no—
57C2×D4×C59—44no—
58C2×D4×C59—44no—
59C2×D4×C59—82no—
60C2×D4×C59—82no—
61C22×C4×C59—22no—
62C22×C4×C59—22no—
63C22×C4×C59—22no—
64C22×C4×C59—42no—
65C22×C4×C59—42no—
66C22×C4×C59—42no—
67C22×C4×C59SD1614yes—
68C22×C4×C59C2×D412yes—
69C22×C4×C59C2×D411yescharacteristic
70C22×C4×C59C2×D412yes—
71C22×C4×C59#4111yescharacteristic
72C22×Dic59—22no—
73C22×Dic59—22no—
74C22×Dic59—42no—
75C22×Dic59—41no—
76C22×Dic59—42no—
77C22×Dic59—42no—
78C22×Dic59—41no—
79C22×Dic59—41no—
80C22×Dic59—41no—
81C22×Dic59—42no—
82C4⋊C4×C59—22no—
83C4⋊C4×C59—22no—
84C4⋊C4×C59—44no—
85C22⋊C4×C59—48no—
86C22⋊C4×C59—42no—
87C22⋊C4×C59—42no—
88C4⋊Dic59—24no—
89C4⋊Dic59—42no—
90C2×C59⋊C8—24no—
91C2×C59⋊C8—44no—
92C23×C59—21no—
93C23×C59—21no—
94C23×C59—22no—
95C23×C59—42no—
96C23×C59—44no—
97C23×C59—42no—
98C23×C59—42no—
99C23×C59—44no—
100C23×C59#12011yescharacteristic
101C23×C59#12111yescharacteristic
102C23×C59#12711yescharacteristic
103C23×C59C2×SD1612yes—
104C23×C59AGL(1,ℤ/8)12yes—
105D4×C59—44no—
106D4×C59—82no—
107D4×C59—88no—
108D4×C59—84no—
109D4×C59—82no—
110D4×C59—82no—
111C2×C4×C59—24no—
112C2×C4×C59—24no—
113C2×C4×C59—41no—
114C2×C4×C59—42no—
115C2×C4×C59—41no—
116C2×C4×C59—42no—
117C2×C4×C59—42no—
118C2×C4×C59—44no—
119C2×C4×C59—42no—
120C2×C4×C59—44no—
121C2×C4×C59—82no—
122C2×C4×C59#12012yes—
123C2×C4×C59#12112yes—
124C2×Dic59—44no—
125C2×Dic59—42no—
126C2×Dic59—44no—
127C2×Dic59—42no—
128C2×Dic59—82no—
129C2×Dic59—81no—
130C2×Dic59—82no—
131C2×Dic59—81no—
132C2×Dic59—81no—
133C2×Dic59—81no—
134C59⋊C8—44no—
135#9968—591nomaximal, Sylow
136C22×C59—22no—
137C22×C59—22no—
138C22×C59—22no—
139C22×C59—22no—
140C22×C59—21no—
141C22×C59—21no—
142C22×C59—21no—
143C22×C59—21no—
144C22×C59—84no—
145C22×C59—84no—
146C22×C59—82no—
147C22×C59—82no—
148C22×C59—82no—
149C22×C59#39211yescharacteristic
150C22×C59#45012yes—
151C22×C59#45311yescharacteristic
152C22×C59#45411yescharacteristic
153C22×C59#45812yes—
154C22×C59#46012yes—
155C22×C59#49211yescharacteristic
156C22×C59#49611yescharacteristic
157C4×C59—22no—
158C4×C59—22no—
159C4×C59—44no—
160C4×C59—82no—
161Dic59—82no—
162Dic59—82no—
163C2×#481—592no—
164#2656D5911yescharacteristic
165C2×#327—592no—
166#1445—591no—
167#1280—591no—
168C2×C59—21no—
169C2×C59—22no—
170C2×C59—21no—
171C2×C59—162no—
172C2×C59#161312yes—
173C2×C59#161612yes—
174C2×C59#200611yescharacteristic
175C2×C59#201411yescharacteristic
176C2×C59#202211yescharacteristic
177C2×#121—22no—
178C2×#121C2×D5912yes—
179C22×C4⋊C4—591no—
180#481—1184no—
181#481—1184no—
182C2×#106—1182no—
183C2×#102—1182no—
184C2×C22⋊C8—592no—
185#399—22no—
186#392—22no—
187C2×#95—591no—
188C2×#95—1182no—
189C2×#95C2×D5911yescharacteristic
190#327—1184no—
191C59#996811yescharacteristic, complemented, Sylow
192C22×D4—22no—
193C22×D4—22no—
194C22×D4C22⋊D5912yes—
195C23×C4—1181no—
196C23×C4—1181no—
197C23×C4C22⋊D5912yes—
198C23×C4C22×D5911yescharacteristic
199C22×C8—1182no—
200#121—22no—
201#121—22no—
202#121—22no—
203#121—22no—
204#121—44no—
205#121—44no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—592no—
208C2×C4⋊C4—1182no—
209C2×C4⋊C4—1182no—
210C2×C4⋊C4—1181no—
211C2×C4⋊C4C22⋊D5912yes—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—22no—
214#106—1188no—
215#102—1188no—
216C22⋊C8—594no—
217C22⋊C8—1182no—
218#95—22no—
219#95—1182no—
220#95—2364no—
221C24—22no—
222C24C2×C22⋊D5911yescharacteristic
223C2×D4—22no—
224C2×D4—22no—
225C2×D4—44no—
226C2×D4—44no—
227C2×D4—42no—
228C2×D4—42no—
229C2×D4—44no—
230C2×D4—44no—
231C2×D4—82no—
232C2×D4—82no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—1182no—
240C22×C4—1182no—
241C22×C4—2362no—
242C22×C4—2361no—
243C22×C4—2362no—
244C22×C4—2362no—
245C22×C4—2361no—
246C22×C4—2361no—
247C22×C4—2361no—
248C22×C4—2362no—
249C22×C4#1174567014yes—
250C22×C4D4×D5911yescharacteristic
251C22×C4D4×D5912yes—
252C22×C4#1174568711yescharacteristic
253C22×C4C2×C22⋊D5912yes—
254C2×C8—1184no—
255C2×C8—2364no—
256C4⋊C4—22no—
257C4⋊C4—22no—
258C4⋊C4—44no—
259C4⋊C4—1184no—
260C4⋊C4—2362no—
261C22⋊C4—48no—
262C22⋊C4—42no—
263C22⋊C4—42no—
264C23—21no—
265C23—21no—
266C23—22no—
267C23—44no—
268C23—42no—
269C23—42no—
270C23—42no—
271C23—44no—
272C23#1427369611yescharacteristic
273C23#1427369811yescharacteristic
274C23#1427372311yescharacteristic
275C23C2×#1174567012yes—
276C23#1427375312yes—
277D4—44no—
278D4—82no—
279D4—88no—
280D4—82no—
281D4—82no—
282D4—84no—
283C2×C4—24no—
284C2×C4—24no—
285C2×C4—41no—
286C2×C4—42no—
287C2×C4—41no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—44no—
291C2×C4—44no—
292C2×C4—42no—
293C2×C4—82no—
294C2×C4—4722no—
295C2×C4—4721no—
296C2×C4—4722no—
297C2×C4—4721no—
298C2×C4—4721no—
299C2×C4—4721no—
300C2×C4—2364no—
301C2×C4—2362no—
302C2×C4—2364no—
303C2×C4—2362no—
304C2×C4#1427369512yes—
305C2×C4#1427369912yes—
306C8—2364no—
307C22—22no—
308C22—22no—
309C22—22no—
310C22—22no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—84no—
316C22—84no—
317C22—82no—
318C22—82no—
319C22—82no—
320C22#1683731011yescharacteristic
321C22#1683755312yes—
322C22#1683757011yescharacteristic
323C22#1683757511yescharacteristic
324C22#1683759212yes—
325C22#1683759912yes—
326C22#1683777411yescharacteristic
327C22#1683778911yescharacteristic
328C4—22no—
329C4—22no—
330C4—44no—
331C4—82no—
332C4—4722no—
333C4—4722no—
334C2—21no—
335C2—21no—
336C2—22no—
337C2—162no—
338C2#2142311612yes—
339C2#2142313312yes—
340C2#2142513811yescharacteristic
341C2#2142518411yescharacteristic
342C2#2142522511yescharacteristic
subgroupG/H
centerC23#14273696
commutator, cosocleC23×C4×C59C23
FrattiniC23×C4C22×D59
Fitting, prime radicalC59×#2656C2
socleC23×C59#120
hypercenter#2656D59
hypocenterC59#9968
LCSG ▷ C23×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2656

Complements

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

NH ≅ G/Ntype
1. C2×#16837718C2semidirect
2. C2×#16837136C2semidirect
3. #21422365C2semidirect
4. #21421939C2semidirect
26. C2×C4⋊Dic59D4semidirect
30. C22⋊(C59⋊C8)D4semidirect
191. C59#9968semidirect

Maximal subgroups

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

subgroupindex
C2×#168377182
C2×#168371362
#214223652
#214219392
C59×#26562
#996859