Finite Group Databasebeta

← #28477760

Subgroups — #28477760

Order 15104.

420 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 87 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#21425624C211yesmaximal, characteristic
2#21425572C211yesmaximal, characteristic, complemented
3#21425567C211yesmaximal, characteristic, complemented
4#21421919C211yesmaximal, characteristic, complemented
5#21421919C211yesmaximal, characteristic, complemented
6#21421917C211yesmaximal, characteristic
7C59×#2727C211yesmaximal, characteristic
8C2×#14273660—21no—
9C2×#14273660—21no—
10C2×#14273658—21no—
11C2×#14273648—21no—
12#16837272C2211yescharacteristic
13#16837164—21no—
14#16837164—21no—
15#16837157C2211yescharacteristic
16#16837157C2211yescharacteristic
17#16837157C2211yescharacteristic
18C2×#126×C59—21no—
19C59×#399—21no—
20C59×#396—21no—
21C59×#394—21no—
22C59×#390C2211yescharacteristic
23C59×#389—21no—
24C59×#389C2211yescharacteristic
25C59×#389C2211yescharacteristic
26C59×#388—21no—
27C22×C59⋊C8—21no—
28C22×C59⋊C8—21no—
29C22×C59⋊C8—21no—
30C2×C4⋊Dic59—21no—
31C2×C4⋊Dic59—21no—
32C2×C4⋊Dic59—21no—
33#14273660—42no—
34#14273660—42no—
35#14273658—42no—
36#14273648—42no—
37C22×D4×C59D411yescharacteristic
38#126×C59—42no—
39#126×C59—42no—
40C2×C4⋊C4×C59—21no—
41C2×C4⋊C4×C59—21no—
42C2×C4⋊C4×C59—21no—
43C2×C4⋊C4×C59—21no—
44C2×C4⋊C4×C59D411yescharacteristic
45C2×C4⋊C4×C59D411yescharacteristic
46C2×C4⋊C4×C59D411yescharacteristic
47C2×C4⋊C4×C59D411yescharacteristic
48C2×C4⋊C4×C59D411yescharacteristic
49C2×C22⋊C4×C59—21no—
50C2×C22⋊C4×C59—21no—
51C2×C22⋊C4×C59—21no—
52C2×C22⋊C4×C59—21no—
53C2×C22⋊C4×C59—21no—
54C2×C22⋊C4×C59—21no—
55C2×C42×C59—21no—
56C2×C42×C59C2311yescharacteristic
57#95×C59—21no—
58#95×C59—21no—
59#95×C59—21no—
60C24×C59—21no—
61C2×D4×C59—21no—
62C2×D4×C59—21no—
63C2×D4×C59—41no—
64C2×D4×C59—41no—
65C22×C4×C59—21no—
66C22×C4×C59—21no—
67C22×C4×C59—21no—
68C22×C4×C59—21no—
69C22×C4×C59—21no—
70C22×C4×C59C2×D411yescharacteristic
71C22×C4×C59C2×D411yescharacteristic
72C22×C4×C59C2×D411yescharacteristic
73C22×Dic59—41no—
74C4⋊C4×C59—21no—
75C4⋊C4×C59—21no—
76C4⋊C4×C59—21no—
77C4⋊C4×C59—42no—
78C4⋊C4×C59—41no—
79C4⋊C4×C59—41no—
80C4⋊C4×C59—41no—
81C4⋊C4×C59—41no—
82C4⋊C4×C59—41no—
83C4⋊C4×C59—41no—
84C4⋊C4×C59—41no—
85C4⋊C4×C59—81no—
86C4⋊C4×C59D812yes—
87C22⋊C4×C59—42no—
88C22⋊C4×C59—42no—
89C22⋊C4×C59—41no—
90C22⋊C4×C59—41no—
91C22⋊C4×C59—41no—
92C22⋊C4×C59—41no—
93C22⋊C4×C59—81no—
94C22⋊C4×C59—81no—
95C42×C59—41no—
96C42×C59—42no—
97C4⋊Dic59—41no—
98C4⋊Dic59—41no—
99C4⋊Dic59—41no—
100C4⋊Dic59—41no—
101C4⋊Dic59—41no—
102C4⋊Dic59—41no—
103C2×C59⋊C8—22no—
104C2×C59⋊C8—22no—
105C2×C59⋊C8—22no—
106C2×C59⋊C8—41no—
107C2×C59⋊C8—41no—
108C2×C59⋊C8—41no—
109C2×C59⋊C8—41no—
110C2×C59⋊C8—41no—
111C2×C59⋊C8—41no—
112C23×C59—41no—
113C23×C59—41no—
114C23×C59—41no—
115C23×C59—41no—
116C23×C59—41no—
117C23×C59—41no—
118C23×C59—41no—
119C23×C59#12011yescharacteristic
120D4×C59—42no—
121D4×C59—81no—
122C2×C4×C59—21no—
123C2×C4×C59—21no—
124C2×C4×C59—21no—
125C2×C4×C59—21no—
126C2×C4×C59—21no—
127C2×C4×C59—21no—
128C2×C4×C59—22no—
129C2×C4×C59—41no—
130C2×C4×C59—41no—
131C2×C4×C59—41no—
132C2×C4×C59—41no—
133C2×C4×C59—41no—
134C2×C4×C59—41no—
135C2×C4×C59—41no—
136C2×C4×C59—41no—
137C2×C4×C59—41no—
138C2×C4×C59—41no—
139C2×C4×C59—41no—
140C2×C4×C59—41no—
141C2×C4×C59—41no—
142C2×C4×C59—41no—
143C2×C4×C59C2×D811yescharacteristic
144C2×C4×C59#13511yescharacteristic
145C2×C4×C59#13511yescharacteristic
146C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
147C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
148C2×C4×C59#13711yescharacteristic
149C2×Dic59—81no—
150C2×Dic59—81no—
151C2×Dic59—81no—
152C59⋊C8—42no—
153C59⋊C8—42no—
154C59⋊C8—42no—
155#7793—591nomaximal, Sylow
156C22×C59—81no—
157C22×C59—81no—
158C22×C59—81no—
159C22×C59—81no—
160C22×C59—81no—
161C22×C59—81no—
162C22×C59—81no—
163C22×C59#44711yescharacteristic
164C22×C59#44911yescharacteristic
165C22×C59#45211yescharacteristic
166C22×C59#45311yescharacteristic
167C22×C59#45611yescharacteristic
168C22×C59#45611yescharacteristic
169C22×C59#45811yescharacteristic
170C4×C59—21no—
171C4×C59—21no—
172C4×C59—21no—
173C4×C59—21no—
174C4×C59—21no—
175C4×C59—21no—
176C4×C59—42no—
177C4×C59—81no—
178C4×C59—81no—
179C4×C59—81no—
180C4×C59—81no—
181Dic59—161no—
182#2727D5911yescharacteristic
183#2093—591no—
184#2084—591no—
185#2083—591no—
186#1271—591no—
187#1271—591no—
188#1270—591no—
189C2×C59—161no—
190C2×C59#160911yescharacteristic
191C2×C59#161011yescharacteristic
192C2×C59#161811yescharacteristic
193C2×C59#162111yescharacteristic
194C2×C59#164711yescharacteristic
195C2×C59#165211yescharacteristic
196C2×C59#166011yescharacteristic
197C2×#126—21no—
198C2×#107—1181no—
199C2×#107—1181no—
200C2×#106—1181no—
201C2×#102—1181no—
202#399—21no—
203#396—21no—
204#394—21no—
205#390C2×D5911yescharacteristic
206#389—21no—
207#389C2×D5911yescharacteristic
208#389C2×D5911yescharacteristic
209#388—21no—
210#384—591no—
211#340—1181no—
212#340—1181no—
213#336—591no—
214#336—591no—
215#336—591no—
216C59#779311yescharacteristic, complemented, Sylow
217C22×D4C22⋊D5911yescharacteristic
218C22×C8—1181no—
219C22×C8—1181no—
220C22×C8—1181no—
221#126—42no—
222#126—42no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—1181no—
228C2×C4⋊C4—1181no—
229C2×C4⋊C4—1181no—
230C2×C4⋊C4C22⋊D5911yescharacteristic
231C2×C4⋊C4C22⋊D5911yescharacteristic
232C2×C4⋊C4C22⋊D5911yescharacteristic
233C2×C4⋊C4C22⋊D5911yescharacteristic
234C2×C4⋊C4C22⋊D5911yescharacteristic
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C42—21no—
242C2×C42C22×D5911yescharacteristic
243#107—2362no—
244#107—2362no—
245#106—2362no—
246#102—2362no—
247#95—21no—
248#95—21no—
249#95—21no—
250C24—21no—
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—41no—
254C2×D4—41no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—2361no—
261C22×C4C2×C22⋊D5911yescharacteristic
262C22×C4C2×C22⋊D5911yescharacteristic
263C22×C4C2×C22⋊D5911yescharacteristic
264C2×C8—1182no—
265C2×C8—1182no—
266C2×C8—1182no—
267C2×C8—2361no—
268C2×C8—2361no—
269C2×C8—2361no—
270C2×C8—2361no—
271C2×C8—2361no—
272C2×C8—2361no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—42no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—81no—
285C4⋊C4—2361no—
286C4⋊C4—2361no—
287C4⋊C4—2361no—
288C4⋊C4—2361no—
289C4⋊C4—2361no—
290C4⋊C4—2361no—
291C4⋊C4#1174566912yes—
292C22⋊C4—42no—
293C22⋊C4—42no—
294C22⋊C4—41no—
295C22⋊C4—41no—
296C22⋊C4—41no—
297C22⋊C4—41no—
298C22⋊C4—81no—
299C22⋊C4—81no—
300C42—41no—
301C42—42no—
302C23—41no—
303C23—41no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23#1427369611yescharacteristic
310D4—42no—
311D4—81no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—22no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—4721no—
334C2×C4—4721no—
335C2×C4—4721no—
336C2×C4C2×#1174566911yescharacteristic
337C2×C4#1427374711yescharacteristic
338C2×C4#1427374711yescharacteristic
339C2×C4#1427375211yescharacteristic
340C2×C4#1427375311yescharacteristic
341C2×C4#1427375711yescharacteristic
342C8—2362no—
343C8—2362no—
344C8—2362no—
345C22—81no—
346C22—81no—
347C22—81no—
348C22—81no—
349C22—81no—
350C22—81no—
351C22—81no—
352C22#1683753611yescharacteristic
353C22#1683754711yescharacteristic
354C22#1683756411yescharacteristic
355C22#1683757011yescharacteristic
356C22#1683758511yescharacteristic
357C22#1683758511yescharacteristic
358C22#1683759211yescharacteristic
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—42no—
366C4—81no—
367C4—81no—
368C4—81no—
369C4—81no—
370C4—9441no—
371C2—161no—
372C2#2142309311yescharacteristic
373C2#2142309911yescharacteristic
374C2#2142314411yescharacteristic
375C2#2142316111yescharacteristic
376C2#2142333111yescharacteristic
377C2#2142336611yescharacteristic
378C2#2142340711yescharacteristic
subgroupG/H
centerC23#14273696
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC59×#2727C2
socleC23×C59#120
prime radicalC22×D4×C59D4
hypercenter#2727D59
hypocenterC59#7793
LCSG ▷ C2×C42×C59 ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2727

Complements

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

NH ≅ G/Ntype
2. #21425572C2semidirect
3. #21425567C2semidirect
4. #21421919C2semidirect
5. #21421919C2semidirect
216. C59#7793semidirect

Maximal subgroups

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

subgroupindex
#214256242
#214255722
#214255672
#214219192
#214219172
C59×#27272
#779359