Finite Group Databasebeta

← #28484757

Subgroups — #28484757

Order 15104.

622 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 622 conjugacy classes of subgroups — too many to draw. Only its 131 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#21429838C211yesmaximal, characteristic
2#21422663C212yesmaximal
3C2×#16837128C211yesmaximal, characteristic, complemented
4#21421963C212yesmaximal, complemented
5C59×#2316C211yesmaximal, characteristic, complemented
6C2×#14273710—21no—
7C2×#14273701C2211yescharacteristic
8C2×C22⋊(C59⋊C8)C2212yes—
9#16837332C412yescomplemented
10#16837324—22no—
11#16837319C412yescomplemented
12#16837281—21no—
13#16837262C2211yescharacteristic
14#16837135—24no—
15#16837128—22no—
16C2×#117×C59—21no—
17C22×C4⋊C4×C59C2211yescharacteristic
18C59×#389—24no—
19C59×#380—21no—
20C59×#380C2212yes—
21C22×D236C2×C411yescharacteristic
22C22×C22⋊D59—21no—
23C22×C59⋊C8—22no—
24#14273710—42no—
25#14273710—42no—
26#14273701—22no—
27#14273701—22no—
28C2×C4⋊Dic59—21no—
29C2×C4⋊Dic59C2×C411yescharacteristic
30C2×Dic59⋊C4—22no—
31C2×C22⋊Dic59—21no—
32C2×(C2×D59)⋊C4—22no—
33C2×(C2×D59)⋊C4—21no—
34C2×(C2×D59)⋊C4—22no—
35C22⋊(C59⋊C8)—22no—
36C22⋊(C59⋊C8)—22no—
37C59⋊#95—22no—
38C59⋊#95C2×C412yes—
39C23×C4×C59D411yescharacteristic
40C23×C4×C59D411yescharacteristic
41C23×C4×C59C2311yescharacteristic
42#117×C59—42no—
43#117×C59—42no—
44C2×C4⋊C4×C59—22no—
45C2×C4⋊C4×C59—22no—
46C2×C4⋊C4×C59—22no—
47C2×C4⋊C4×C59—22no—
48C2×C4⋊C4×C59—22no—
49C2×C4⋊C4×C59C2×C412yes—
50C2×C4⋊C4×C59D412yes—
51C2×C22⋊C4×C59—22no—
52C2×C42×C59—22no—
53#95×C59—22no—
54#95×C59D414yes—
55C24×C59C2×D411yescharacteristic
56C23×D59—21no—
57C22×C4×C59—21no—
58C22×C4×C59—21no—
59C22×C4×C59—21no—
60C22×C4×C59—21no—
61C22×C4×C59—21no—
62C22×C4×C59—21no—
63C22×C4×C59—24no—
64C22×C4×C59—81no—
65C22×C4×C59—81no—
66C22×C4×C59C22⋊C412yes—
67C22×C4×C59C22⋊C412yes—
68C22×C4×C59D812yes—
69C22×C4×C59SD1612yes—
70C22×C4×C59C22×C411yescharacteristic
71C22×C4×C59C2×D412yes—
72C22×C4×C59C2×D411yescharacteristic
73C22×C4×C59#4112yes—
74C2×C22⋊D59—41no—
75C2×C22⋊D59—41no—
76C2×C22⋊D59—41no—
77C2×C22⋊D59—41no—
78C2×C22⋊D59—41no—
79C2×C22⋊D59—41no—
80C22×Dic59—21no—
81C22×Dic59—22no—
82C2×D236—22no—
83C2×D236—81no—
84C4⋊C4×C59—24no—
85C4⋊C4×C59—24no—
86C4⋊C4×C59—44no—
87C4⋊C4×C59—82no—
88C4⋊C4×C59—82no—
89C22⋊C4×C59—42no—
90C22⋊C4×C59—42no—
91C42×C59—44no—
92C22⋊Dic59—41no—
93C22⋊Dic59—41no—
94(C2×D59)⋊C4—44no—
95(C2×D59)⋊C4—42no—
96(C2×D59)⋊C4—82no—
97C4⋊Dic59—22no—
98C4⋊Dic59—81no—
99Dic59⋊C4—44no—
100C2×C59⋊C8—42no—
101C2×C59⋊C8—82no—
102C23×C59—41no—
103C23×C59—41no—
104C23×C59C2×C22⋊C411yescharacteristic
105C23×C59#12011yescharacteristic
106C23×C59#12311yescharacteristic
107C23×C59C2×D811yescharacteristic
108C23×C59C2×SD1611yescharacteristic
109C23×C59AGL(1,ℤ/8)11yescharacteristic
110C23×C59#13711yescharacteristic
111C22×D59—42no—
112C22×D59—42no—
113C22×D59—41no—
114C22×D59—41no—
115C22×D59—41no—
116C2×C4×C59—22no—
117C2×C4×C59—22no—
118C2×C4×C59—22no—
119C2×C4×C59—22no—
120C2×C4×C59—22no—
121C2×C4×C59—22no—
122C2×C4×C59—22no—
123C2×C4×C59—42no—
124C2×C4×C59—41no—
125C2×C4×C59—42no—
126C2×C4×C59—41no—
127C2×C4×C59—42no—
128C2×C4×C59—42no—
129C2×C4×C59—44no—
130C2×C4×C59—44no—
131C2×C4×C59—44no—
132C2×C4×C59—81no—
133C2×C4×C59#10214yes—
134C2×C4×C59C4×D412yes—
135C2×C4×C59#12112yes—
136C22⋊D59—81no—
137C22⋊D59—81no—
138C22⋊D59—81no—
139C22⋊D59—81no—
140C2×Dic59—42no—
141C2×Dic59—42no—
142C2×Dic59—41no—
143C2×Dic59—41no—
144C2×Dic59—42no—
145C2×Dic59—41no—
146D236—82no—
147C59⋊C8—82no—
148#9204—591nomaximal, Sylow
149C22×C59—21no—
150C22×C59—21no—
151C22×C59—21no—
152C22×C59—21no—
153C22×C59—21no—
154C22×C59—21no—
155C22×C59—41no—
156C22×C59—41no—
157C22×C59—41no—
158C22×C59—41no—
159C22×C59#38611yescharacteristic
160C22×C59C2×#10211yescharacteristic
161C22×C59#41711yescharacteristic
162C22×C59#44711yescharacteristic
163C22×C59#44811yescharacteristic
164C22×C59#48011yescharacteristic
165C22×C59#48111yescharacteristic
166C2×D59—82no—
167C2×D59—82no—
168C2×D59—81no—
169C2×D59—81no—
170C2×D59—81no—
171C4×C59—24no—
172C4×C59—22no—
173C4×C59—22no—
174C4×C59—42no—
175C4×C59—84no—
176Dic59—82no—
177Dic59—81no—
178#2618—591no—
179#2316D5911yescharacteristic, complemented
180#1519—592no—
181C2×#324—591no—
182#1289—592no—
183C2×C59—21no—
184C2×C59—21no—
185C2×C59—21no—
186C2×C59—21no—
187C2×C59—21no—
188C2×C59—21no—
189C2×C59#187411yescharacteristic
190C2×C59#191211yescharacteristic
191C2×C59#191311yescharacteristic
192D59—161no—
193C2×#123—1181no—
194C2×#121—591no—
195C2×#117—21no—
196C22×C4⋊C4C2×D5911yescharacteristic
197C2×C22⋊C8—592no—
198#398—592no—
199#396—1182no—
200#394—592no—
201#389—24no—
202#386—1181no—
203#380—21no—
204#380—591no—
205#380C2×D5912yes—
206#327—1184no—
207#324—1182no—
208C59#920411yescharacteristic, complemented, Sylow
209C22×D4—591no—
210C22×D4—1181no—
211C23×C4C22⋊D5911yescharacteristic
212C23×C4C22⋊D5911yescharacteristic
213C23×C4C22×D5911yescharacteristic
214C22×C8—1182no—
215#123—2362no—
216#123—2362no—
217#121—1182no—
218#121—1182no—
219#117—42no—
220#117—42no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—591no—
227C2×C4⋊C4—1181no—
228C2×C4⋊C4—1182no—
229C2×C4⋊C4C4×D5912yes—
230C2×C4⋊C4D23612yes—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—1182no—
233C2×C22⋊C4—1181no—
234C2×C22⋊C4—1182no—
235C2×C22⋊C4—1181no—
236C2×C42—22no—
237C22⋊C8—1182no—
238C22⋊C8—1182no—
239#95—22no—
240#95—592no—
241#95—1182no—
242#95C22⋊D5914yes—
243C24—1181no—
244C24C2×C22⋊D5911yescharacteristic
245C2×D4—1182no—
246C2×D4—4721no—
247C2×D4—2361no—
248C2×D4—2361no—
249C2×D4—2361no—
250C2×D4—2361no—
251C2×D4—2361no—
252C2×D4—2361no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—24no—
260C22×C4—81no—
261C22×C4—81no—
262C22×C4—1182no—
263C22×C4—1181no—
264C22×C4(C2×D59)⋊C412yes—
265C22×C4(C2×D59)⋊C412yes—
266C22×C4#1174566912yes—
267C22×C4#1174567112yes—
268C22×C4C2×C4×D5911yescharacteristic
269C22×C4C2×D23611yescharacteristic
270C22×C4#1174568512yes—
271C22×C4C2×C22⋊D5912yes—
272C2×C8—4722no—
273C2×C8—2362no—
274C4⋊C4—24no—
275C4⋊C4—24no—
276C4⋊C4—44no—
277C4⋊C4—82no—
278C4⋊C4—82no—
279C4⋊C4—1182no—
280C4⋊C4—4721no—
281C4⋊C4—2364no—
282C22⋊C4—42no—
283C22⋊C4—42no—
284C22⋊C4—4722no—
285C22⋊C4—2361no—
286C22⋊C4—2361no—
287C22⋊C4—2364no—
288C22⋊C4—2362no—
289C42—44no—
290C23—41no—
291C23—41no—
292C23—2362no—
293C23—2362no—
294C23—2361no—
295C23—2361no—
296C23—2361no—
297C23C2×(C2×D59)⋊C411yescharacteristic
298C23#1427369611yescharacteristic
299C23#1427371011yescharacteristic
300C23C2×#1174566911yescharacteristic
301C23C2×#1174567111yescharacteristic
302C23#1427375311yescharacteristic
303C23#1427375811yescharacteristic
304D4—4722no—
305D4—4721no—
306D4—4721no—
307D4—4721no—
308D4—4721no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—42no—
319C2×C4—41no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—44no—
323C2×C4—44no—
324C2×C4—44no—
325C2×C4—81no—
326C2×C4—2362no—
327C2×C4—2362no—
328C2×C4—2362no—
329C2×C4—2361no—
330C2×C4—2361no—
331C2×C4—2361no—
332C2×C4#1427364714yes—
333C2×C4C4×C22⋊D5912yes—
334C2×C4#1427370112yes—
335C8—4722no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—4722no—
347C22—4722no—
348C22—4721no—
349C22—4721no—
350C22—4721no—
351C22#1683728111yescharacteristic
352C22C2×#1427364711yescharacteristic
353C22#1683740111yescharacteristic
354C22#1683753611yescharacteristic
355C22#1683754111yescharacteristic
356C22#1683771511yescharacteristic
357C22#1683772011yescharacteristic
358C4—24no—
359C4—22no—
360C4—22no—
361C4—42no—
362C4—84no—
363C4—4722no—
364C4—4721no—
365C2—21no—
366C2—21no—
367C2—21no—
368C2—21no—
369C2—21no—
370C2—21no—
371C2—9441no—
372C2#2142441311yescharacteristic
373C2#2142463311yescharacteristic
374C2#2142463811yescharacteristic
subgroupG/H
centerC22#16837281
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#2316C2
socleC22×C59#386
prime radicalC2×#14273701C22
cosocleC23×C4×C59C23
hypercenter#2316D59
hypocenterC59#9204
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2316

Complements

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

NH ≅ G/Ntype
3. C2×#16837128C2semidirect
4. #21421963C2semidirect
5. C59×#2316C2semidirect
9. #16837332C4semidirect
11. #16837319C4semidirect
179. #2316D59semidirect
208. C59#9204semidirect

Maximal subgroups

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

subgroupindex
#214298382
#214226632
C2×#168371282
#214219632
C59×#23162
#920459