Finite Group Databasebeta

← #28477802

Subgroups — #28477802

Order 15104.

404 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 81 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#21431055C211yesmaximal, characteristic
2#21425655C211yesmaximal, characteristic, complemented
3#21425569C211yesmaximal, characteristic, complemented
4#21425499C211yesmaximal, characteristic
5#21421920C211yesmaximal, characteristic, complemented
6#21421918C211yesmaximal, characteristic, complemented
7C59×#1270C211yesmaximal, characteristic
8C2×#14273720—21no—
9C2×#14273661—21no—
10C2×#14273648—21no—
11#16837338—21no—
12#16837325—21no—
13#16837316—21no—
14#16837300C2211yescharacteristic
15#16837300C2211yescharacteristic
16#16837297—21no—
17#16837295—21no—
18#16837272C2211yescharacteristic
19#16837165—21no—
20#16837165—21no—
21#16837165—21no—
22#16837164—21no—
23#16837157C2211yescharacteristic
24C59×#390C2211yescharacteristic
25C59×#336C2211yescharacteristic
26C59×#336C2211yescharacteristic
27C22×C59⋊C8—21no—
28#14273720—42no—
29#14273720—42no—
30C2×C4⋊Dic59—21no—
31C2×C4⋊Dic59—21no—
32C2×C4⋊Dic59—21no—
33C2×C4⋊Dic59D411yescharacteristic
34C2×C4⋊Dic59D411yescharacteristic
35C2×C4⋊Dic59D411yescharacteristic
36C2×C4⋊Dic59D411yescharacteristic
37C2×Dic59⋊C4—21no—
38C2×Dic59⋊C4—21no—
39C2×Dic59⋊C4—21no—
40C2×Dic59⋊C4—21no—
41C2×C22⋊Dic59—21no—
42C2×C22⋊Dic59—21no—
43C2×C22⋊Dic59—21no—
44C2×C22⋊Dic59—21no—
45C2×C4×Dic59—21no—
46#14273661—42no—
47#14273648—42no—
48C59⋊#95—21no—
49C59⋊#95—21no—
50C59⋊#95—21no—
51C22×D4×C59D411yescharacteristic
52C22×C8×C59—21no—
53C22×C8×C59—21no—
54C2×C4⋊C4×C59D411yescharacteristic
55C2×C22⋊C4×C59—21no—
56C2×C22⋊C4×C59—21no—
57C2×C42×C59C2311yescharacteristic
58C24×C59—21no—
59C2×D4×C59—21no—
60C2×D4×C59—21no—
61C2×D4×C59—41no—
62C2×D4×C59—41no—
63C22×C4×C59—21no—
64C22×C4×C59C2×D411yescharacteristic
65C22×C4×C59C2×D411yescharacteristic
66C22×C4×C59C2×D411yescharacteristic
67C22×Dic59—21no—
68C22×Dic59—21no—
69C22×Dic59—21no—
70C22×Dic59—21no—
71C22×Dic59—41no—
72C2×C8×C59—22no—
73C2×C8×C59—41no—
74C2×C8×C59—41no—
75C2×C8×C59—41no—
76C2×C8×C59—41no—
77C2×C8×C59—41no—
78C4⋊C4×C59—41no—
79C22⋊C4×C59—41no—
80C22⋊C4×C59—41no—
81C22⋊C4×C59—41no—
82C22⋊C4×C59—41no—
83C42×C59—41no—
84C22⋊Dic59—42no—
85C22⋊Dic59—42no—
86C22⋊Dic59—81no—
87C22⋊Dic59—81no—
88C4⋊Dic59—21no—
89C4⋊Dic59—21no—
90C4⋊Dic59—21no—
91C4⋊Dic59—41no—
92C4⋊Dic59—41no—
93C4⋊Dic59—42no—
94C4⋊Dic59—41no—
95C4⋊Dic59—41no—
96C4⋊Dic59—41no—
97C4⋊Dic59—41no—
98C4⋊Dic59D812yes—
99Dic59⋊C4—42no—
100Dic59⋊C4—41no—
101Dic59⋊C4—41no—
102Dic59⋊C4—41no—
103Dic59⋊C4—41no—
104Dic59⋊C4—81no—
105C4×Dic59—42no—
106C2×C59⋊C8—41no—
107C2×C59⋊C8—41no—
108C2×C59⋊C8—41no—
109C23×C59—41no—
110C23×C59—41no—
111C23×C59—41no—
112C23×C59—41no—
113C23×C59—41no—
114C23×C59—41no—
115C23×C59—41no—
116C23×C59#12011yescharacteristic
117D4×C59—42no—
118D4×C59—81no—
119C2×C4×C59—21no—
120C2×C4×C59—21no—
121C2×C4×C59—21no—
122C2×C4×C59—21no—
123C2×C4×C59—21no—
124C2×C4×C59—21no—
125C2×C4×C59—41no—
126C2×C4×C59—41no—
127C2×C4×C59—41no—
128C2×C4×C59C2×D811yescharacteristic
129C2×C4×C59#13511yescharacteristic
130C2×C4×C59#13511yescharacteristic
131C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
132C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
133C2×C4×C59#13711yescharacteristic
134C2×Dic59—22no—
135C2×Dic59—41no—
136C2×Dic59—41no—
137C2×Dic59—41no—
138C2×Dic59—41no—
139C2×Dic59—41no—
140C2×Dic59—41no—
141C2×Dic59—41no—
142C2×Dic59—41no—
143C2×Dic59—41no—
144C2×Dic59—41no—
145C2×Dic59—41no—
146C2×Dic59—81no—
147C2×Dic59—81no—
148C2×Dic59—81no—
149C8×C59—42no—
150C8×C59—81no—
151C59⋊C8—81no—
152#7799—591nomaximal, Sylow
153C22×C59—81no—
154C22×C59—81no—
155C22×C59—81no—
156C22×C59—81no—
157C22×C59—81no—
158C22×C59—81no—
159C22×C59—81no—
160C22×C59#44711yescharacteristic
161C22×C59#44911yescharacteristic
162C22×C59#45211yescharacteristic
163C22×C59#45411yescharacteristic
164C22×C59#45511yescharacteristic
165C22×C59#45611yescharacteristic
166C22×C59#45811yescharacteristic
167C4×C59—21no—
168C4×C59—21no—
169C4×C59—21no—
170C4×C59—21no—
171C4×C59—21no—
172C4×C59—21no—
173C4×C59—81no—
174Dic59—42no—
175Dic59—81no—
176Dic59—81no—
177Dic59—81no—
178Dic59—161no—
179#2727—591no—
180#2099—591no—
181#2083—591no—
182#2071—591no—
183#1271—591no—
184#1271—591no—
185#1270D5911yescharacteristic
186C2×C59—161no—
187C2×C59#161011yescharacteristic
188C2×C59#161211yescharacteristic
189C2×C59#161511yescharacteristic
190C2×C59#162511yescharacteristic
191C2×C59#164111yescharacteristic
192C2×C59#165011yescharacteristic
193C2×C59#166411yescharacteristic
194C2×#126—1181no—
195C2×#107—1181no—
196C2×#102—1181no—
197#399—1181no—
198#396—1181no—
199#394—1181no—
200#390C2×D5911yescharacteristic
201#389—591no—
202#389—591no—
203#389—1181no—
204#388—1181no—
205#384—591no—
206#340—1181no—
207#340—1181no—
208#340—1181no—
209#340—1181no—
210#336—591no—
211#336C2×D5911yescharacteristic
212#336C2×D5911yescharacteristic
213C59#779911yescharacteristic, complemented, Sylow
214C22×D4D23611yescharacteristic
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—1181no—
218#126—2362no—
219#126—2362no—
220C2×C4⋊C4—591no—
221C2×C4⋊C4—591no—
222C2×C4⋊C4—591no—
223C2×C4⋊C4—591no—
224C2×C4⋊C4—1181no—
225C2×C4⋊C4—1181no—
226C2×C4⋊C4—1181no—
227C2×C4⋊C4—1181no—
228C2×C4⋊C4—1181no—
229C2×C4⋊C4—1181no—
230C2×C4⋊C4—1181no—
231C2×C4⋊C4D23611yescharacteristic
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—1181no—
235C2×C22⋊C4—1181no—
236C2×C22⋊C4—1181no—
237C2×C22⋊C4—1181no—
238C2×C42—1181no—
239C2×C42C22×D5911yescharacteristic
240#107—2362no—
241#102—2362no—
242#95—1181no—
243#95—1181no—
244#95—1181no—
245C24—21no—
246C2×D4—21no—
247C2×D4—21no—
248C2×D4—41no—
249C2×D4—41no—
250C22×C4—21no—
251C22×C4—1181no—
252C22×C4—1181no—
253C22×C4—1181no—
254C22×C4—1181no—
255C22×C4—2361no—
256C22×C4C2×D23611yescharacteristic
257C22×C4D4×D5911yescharacteristic
258C22×C4D4×D5911yescharacteristic
259C2×C8—22no—
260C2×C8—41no—
261C2×C8—41no—
262C2×C8—41no—
263C2×C8—41no—
264C2×C8—41no—
265C2×C8—2361no—
266C2×C8—2361no—
267C2×C8—2361no—
268C4⋊C4—41no—
269C4⋊C4—592no—
270C4⋊C4—1181no—
271C4⋊C4—1181no—
272C4⋊C4—1181no—
273C4⋊C4—4721no—
274C4⋊C4—2362no—
275C4⋊C4—2361no—
276C4⋊C4—2361no—
277C4⋊C4—2362no—
278C4⋊C4—2361no—
279C4⋊C4—2361no—
280C4⋊C4—2361no—
281C4⋊C4—2361no—
282C4⋊C4—2361no—
283C4⋊C4—2361no—
284C4⋊C4—2361no—
285C4⋊C4—2361no—
286C22⋊C4—41no—
287C22⋊C4—41no—
288C22⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—4721no—
291C22⋊C4—4721no—
292C22⋊C4—2362no—
293C22⋊C4—2362no—
294C42—41no—
295C42—2362no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23—41no—
301C23—41no—
302C23—41no—
303C23#1427369711yescharacteristic
304D4—42no—
305D4—81no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—1182no—
316C2×C4—4721no—
317C2×C4—4721no—
318C2×C4—4721no—
319C2×C4—2361no—
320C2×C4—2361no—
321C2×C4—2361no—
322C2×C4—2361no—
323C2×C4—2361no—
324C2×C4—2361no—
325C2×C4—2361no—
326C2×C4—2361no—
327C2×C4—2361no—
328C2×C4—2361no—
329C2×C4—2361no—
330C2×C4D8×D5911yescharacteristic
331C2×C4#1427374611yescharacteristic
332C2×C4#1427374911yescharacteristic
333C2×C4#1427375411yescharacteristic
334C2×C4#1427375511yescharacteristic
335C2×C4#1427375611yescharacteristic
336C8—42no—
337C8—81no—
338C8—4721no—
339C22—81no—
340C22—81no—
341C22—81no—
342C22—81no—
343C22—81no—
344C22—81no—
345C22—81no—
346C22#1683753711yescharacteristic
347C22#1683754911yescharacteristic
348C22#1683756511yescharacteristic
349C22#1683757611yescharacteristic
350C22#1683758211yescharacteristic
351C22#1683758611yescharacteristic
352C22#1683759311yescharacteristic
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—21no—
359C4—81no—
360C4—9441no—
361C4—4721no—
362C4—4721no—
363C4—4721no—
364C4—2362no—
365C2—161no—
366C2#2142310011yescharacteristic
367C2#2142311211yescharacteristic
368C2#2142312811yescharacteristic
369C2#2142318811yescharacteristic
370C2#2142329011yescharacteristic
371C2#2142335411yescharacteristic
372C2#2142343311yescharacteristic
subgroupG/H
centerC23#14273697
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC59×#1270C2
socleC23×C59#120
prime radicalC22×D4×C59D4
hypercenter#1270D59
hypocenterC59#7799
LCSG ▷ C2×C42×C59 ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1270

Complements

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

NH ≅ G/Ntype
2. #21425655C2semidirect
3. #21425569C2semidirect
5. #21421920C2semidirect
6. #21421918C2semidirect
213. C59#7799semidirect

Maximal subgroups

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

subgroupindex
#214310552
#214256552
#214255692
#214254992
#214219202
#214219182
C59×#12702
#779959