Finite Group Databasebeta

← #28477983

Subgroups — #28477983

Order 15104.

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

normal subgroups only

This group has 404 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#21425659C211yesmaximal, characteristic, complemented
2#21425659C211yesmaximal, characteristic, complemented
3#21425629C211yesmaximal, characteristic
4#21421919C211yesmaximal, characteristic, complemented
5#21421919C211yesmaximal, characteristic, complemented
6#21421917C211yesmaximal, characteristic
7C59×#2727C211yesmaximal, characteristic
8C2×#14273658—21no—
9C2×#14273648—21no—
10#16837272C2211yescharacteristic
11#16837164—21no—
12#16837164—21no—
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#14273658—42no—
34#14273648—42no—
35C22×D4×C59D411yescharacteristic
36#126×C59—42no—
37#126×C59—42no—
38C2×C4⋊C4×C59—21no—
39C2×C4⋊C4×C59—21no—
40C2×C4⋊C4×C59—21no—
41C2×C4⋊C4×C59—21no—
42C2×C4⋊C4×C59D411yescharacteristic
43C2×C4⋊C4×C59D411yescharacteristic
44C2×C4⋊C4×C59D411yescharacteristic
45C2×C4⋊C4×C59D411yescharacteristic
46C2×C4⋊C4×C59D411yescharacteristic
47C2×C22⋊C4×C59—21no—
48C2×C22⋊C4×C59—21no—
49C2×C22⋊C4×C59—21no—
50C2×C22⋊C4×C59—21no—
51C2×C22⋊C4×C59—21no—
52C2×C22⋊C4×C59—21no—
53C2×C42×C59—21no—
54C2×C42×C59C2311yescharacteristic
55#95×C59—21no—
56#95×C59—21no—
57#95×C59—21no—
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×C59—21no—
65C22×C4×C59—21no—
66C22×C4×C59—21no—
67C22×C4×C59—21no—
68C22×C4×C59C2×D411yescharacteristic
69C22×C4×C59C2×D411yescharacteristic
70C22×C4×C59C2×D411yescharacteristic
71C22×Dic59—41no—
72C4⋊C4×C59—21no—
73C4⋊C4×C59—21no—
74C4⋊C4×C59—21no—
75C4⋊C4×C59—42no—
76C4⋊C4×C59—41no—
77C4⋊C4×C59—41no—
78C4⋊C4×C59—41no—
79C4⋊C4×C59—41no—
80C4⋊C4×C59—41no—
81C4⋊C4×C59—41no—
82C4⋊C4×C59—41no—
83C4⋊C4×C59—81no—
84C4⋊C4×C59SD1612yes—
85C22⋊C4×C59—42no—
86C22⋊C4×C59—42no—
87C22⋊C4×C59—41no—
88C22⋊C4×C59—41no—
89C22⋊C4×C59—41no—
90C22⋊C4×C59—41no—
91C22⋊C4×C59—81no—
92C22⋊C4×C59—81no—
93C42×C59—41no—
94C42×C59—42no—
95C4⋊Dic59—41no—
96C4⋊Dic59—41no—
97C4⋊Dic59—42no—
98C4⋊Dic59—42no—
99C2×C59⋊C8—22no—
100C2×C59⋊C8—41no—
101C2×C59⋊C8—41no—
102C2×C59⋊C8—41no—
103C2×C59⋊C8—41no—
104C2×C59⋊C8—41no—
105C2×C59⋊C8—41no—
106C2×C59⋊C8—41no—
107C2×C59⋊C8—41no—
108C23×C59—41no—
109C23×C59—41no—
110C23×C59—41no—
111C23×C59—41no—
112C23×C59—41no—
113C23×C59—41no—
114C23×C59—41no—
115C23×C59#12011yescharacteristic
116D4×C59—42no—
117D4×C59—81no—
118C2×C4×C59—21no—
119C2×C4×C59—21no—
120C2×C4×C59—21no—
121C2×C4×C59—21no—
122C2×C4×C59—21no—
123C2×C4×C59—21no—
124C2×C4×C59—22no—
125C2×C4×C59—41no—
126C2×C4×C59—41no—
127C2×C4×C59—41no—
128C2×C4×C59—41no—
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×C59C2×SD1611yescharacteristic
140C2×C4×C59#13511yescharacteristic
141C2×C4×C59#13511yescharacteristic
142C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
143C2×C4×C59#13711yescharacteristic
144C2×C4×C59#13711yescharacteristic
145C2×Dic59—81no—
146C2×Dic59—81no—
147C2×Dic59—81no—
148C59⋊C8—42no—
149C59⋊C8—81no—
150C59⋊C8—81no—
151#7828—591nomaximal, Sylow
152C22×C59—81no—
153C22×C59—81no—
154C22×C59—81no—
155C22×C59—81no—
156C22×C59—81no—
157C22×C59—81no—
158C22×C59—81no—
159C22×C59#45011yescharacteristic
160C22×C59#45211yescharacteristic
161C22×C59#45211yescharacteristic
162C22×C59#45311yescharacteristic
163C22×C59#45411yescharacteristic
164C22×C59#45411yescharacteristic
165C22×C59#45811yescharacteristic
166C4×C59—21no—
167C4×C59—21no—
168C4×C59—21no—
169C4×C59—21no—
170C4×C59—21no—
171C4×C59—21no—
172C4×C59—42no—
173C4×C59—81no—
174C4×C59—81no—
175C4×C59—81no—
176C4×C59—81no—
177Dic59—161no—
178#2727D5911yescharacteristic
179#2100—591no—
180#2100—591no—
181#2094—591no—
182#1271—591no—
183#1271—591no—
184#1270—591no—
185C2×C59—161no—
186C2×C59#161511yescharacteristic
187C2×C59#161611yescharacteristic
188C2×C59#161811yescharacteristic
189C2×C59#163111yescharacteristic
190C2×C59#163511yescharacteristic
191C2×C59#163811yescharacteristic
192C2×C59#165711yescharacteristic
193C2×#126—21no—
194C2×#106—1181no—
195C2×#102—1181no—
196#399—21no—
197#396—21no—
198#394—21no—
199#390C2×D5911yescharacteristic
200#389—21no—
201#389C2×D5911yescharacteristic
202#389C2×D5911yescharacteristic
203#388—21no—
204#384—591no—
205#340—1181no—
206#340—1181no—
207#340—1181no—
208#340—1181no—
209#336—591no—
210#336—591no—
211#336—591no—
212C59#782811yescharacteristic, complemented, Sylow
213C22×D4C22⋊D5911yescharacteristic
214C22×C8—1181no—
215C22×C8—1181no—
216C22×C8—1181no—
217#126—42no—
218#126—42no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—1181no—
224C2×C4⋊C4—1181no—
225C2×C4⋊C4—1181no—
226C2×C4⋊C4C22⋊D5911yescharacteristic
227C2×C4⋊C4C22⋊D5911yescharacteristic
228C2×C4⋊C4C22⋊D5911yescharacteristic
229C2×C4⋊C4C22⋊D5911yescharacteristic
230C2×C4⋊C4C22⋊D5911yescharacteristic
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C42—21no—
238C2×C42C22×D5911yescharacteristic
239#106—2362no—
240#102—2362no—
241#95—21no—
242#95—21no—
243#95—21no—
244C24—21no—
245C2×D4—21no—
246C2×D4—21no—
247C2×D4—41no—
248C2×D4—41no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—2361no—
255C22×C4C2×C22⋊D5911yescharacteristic
256C22×C4C2×C22⋊D5911yescharacteristic
257C22×C4C2×C22⋊D5911yescharacteristic
258C2×C8—1182no—
259C2×C8—2361no—
260C2×C8—2361no—
261C2×C8—2361no—
262C2×C8—2361no—
263C2×C8—2361no—
264C2×C8—2361no—
265C2×C8—2361no—
266C2×C8—2361no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—42no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—81no—
279C4⋊C4—2361no—
280C4⋊C4—2361no—
281C4⋊C4—2362no—
282C4⋊C4—2362no—
283C4⋊C4#1174567012yes—
284C22⋊C4—42no—
285C22⋊C4—42no—
286C22⋊C4—41no—
287C22⋊C4—41no—
288C22⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—81no—
291C22⋊C4—81no—
292C42—41no—
293C42—42no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23—41no—
301C23#1427369611yescharacteristic
302D4—42no—
303D4—81no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—22no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—4721no—
326C2×C4—4721no—
327C2×C4—4721no—
328C2×C4C2×#1174567011yescharacteristic
329C2×C4#1427374711yescharacteristic
330C2×C4#1427374711yescharacteristic
331C2×C4#1427375311yescharacteristic
332C2×C4#1427375711yescharacteristic
333C2×C4#1427375711yescharacteristic
334C8—4721no—
335C8—4721no—
336C8—2362no—
337C22—81no—
338C22—81no—
339C22—81no—
340C22—81no—
341C22—81no—
342C22—81no—
343C22—81no—
344C22#1683755311yescharacteristic
345C22#1683756411yescharacteristic
346C22#1683756411yescharacteristic
347C22#1683757011yescharacteristic
348C22#1683757511yescharacteristic
349C22#1683757511yescharacteristic
350C22#1683759211yescharacteristic
351C4—21no—
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—42no—
358C4—81no—
359C4—81no—
360C4—81no—
361C4—81no—
362C4—9441no—
363C2—161no—
364C2#2142312711yescharacteristic
365C2#2142313311yescharacteristic
366C2#2142314411yescharacteristic
367C2#2142322311yescharacteristic
368C2#2142324711yescharacteristic
369C2#2142326811yescharacteristic
370C2#2142339411yescharacteristic
subgroupG/H
centerC23#14273696
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC59×#2727C2
socleC23×C59#120
prime radicalC22×D4×C59D4
hypercenter#2727D59
hypocenterC59#7828
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
1. #21425659C2semidirect
2. #21425659C2semidirect
4. #21421919C2semidirect
5. #21421919C2semidirect
212. C59#7828semidirect

Maximal subgroups

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

subgroupindex
#214256592
#214256292
#214219192
#214219172
C59×#27272
#782859