Finite Group Databasebeta

← #28486091

Subgroups — #28486091

Order 15104.

534 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 109 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#21433136C211yesmaximal, characteristic
2#21429092C211yesmaximal, characteristic
3#21424085C211yesmaximal, characteristic
4#21422768C211yesmaximal, characteristic, complemented
5#21421998C211yesmaximal, characteristic, complemented
6#21421997C211yesmaximal, characteristic, complemented
7C59×#1546C211yesmaximal, characteristic, complemented
8C2×#14273716—21no—
9C2×C4×D236C2211yescharacteristic
10#16837481—22no—
11#16837479—22no—
12#16837455C2211yescharacteristic
13#16837426C2211yescharacteristic
14C2×#14273643—21no—
15#16837364—22no—
16#16837328C414yes—
17#16837309—21no—
18#16837259—22no—
19C4×C4⋊Dic59C2211yescharacteristic
20#16837170—21no—
21#16837148—22no—
22C2×C4⋊Q8×C59C2211yescharacteristic
23C59×#431C2211yescharacteristic
24C59×#423C2211yescharacteristic
25C59×#332—22no—
26C22×D236D411yescharacteristic
27C22×C4×D59D412yes—
28C2×#11745664—21no—
29C2×#11745659—24no—
30#14273716—42no—
31#14273716—42no—
32C4×D236—21no—
33C4×D236—21no—
34C4×D236—21no—
35C4×D236—21no—
36C2×C4⋊Dic59—21no—
37C2×C4⋊Dic59D411yescharacteristic
38C2×(C2×D59)⋊C4—24no—
39C2×(C2×D59)⋊C4C2×C412yes—
40C2×C4×Dic59—21no—
41C4⋊(C59⋊C8)—21no—
42C4⋊(C59⋊C8)—21no—
43C4⋊(C59⋊C8)D412yes—
44#14273643—42no—
45C22⋊(C59⋊C8)—21no—
46(C2×D59)⋊C8—24no—
47C4×C59⋊C8—21no—
48C59⋊#95—24no—
49C59⋊#95C2×C412yes—
50C22×Q8×C59—21no—
51C2×M16×C59—21no—
52C4⋊Q8×C59—21no—
53C4⋊Q8×C59—21no—
54C4⋊Q8×C59—21no—
55C4⋊Q8×C59—21no—
56C2×C4⋊C4×C59—21no—
57C2×C4⋊C4×C59C2×C412yes—
58C2×C42×C59C2311yescharacteristic
59C4⋊C8×C59—21no—
60C4⋊C8×C59—21no—
61C4⋊C8×C59D412yes—
62C22⋊C8×C59—21no—
63C4×C8×C59—21no—
64C23×D59C22⋊C412yes—
65C2×Q8×C59—41no—
66C2×Q8×C59—41no—
67C2×Q8×C59—41no—
68C2×Q8×C59—41no—
69C2×Q8×C59—41no—
70C2×Q8×C59—41no—
71C22×C4×C59—22no—
72C22×C4×C59C22×C411yescharacteristic
73C22×C4×C59C2×D411yescharacteristic
74C22×C4×C59C2×D411yescharacteristic
75C22×Dic59—22no—
76C22×Dic59C22⋊C412yes—
77C2×D236—21no—
78C2×D236—21no—
79C2×D236—21no—
80C2×D236—21no—
81C2×D236—21no—
82C2×D236—21no—
83C2×C4×D59—22no—
84C2×C4×D59—22no—
85C2×C4×D59—44no—
86M16×C59—41no—
87M16×C59—41no—
88C2×C8×C59—22no—
89C2×C8×C59—41no—
90C4⋊C4×C59—22no—
91C4⋊C4×C59—22no—
92C4⋊C4×C59—41no—
93C4⋊C4×C59—41no—
94C42×C59C2×D411yescharacteristic
95C42×C59C2×D411yescharacteristic
96C42×C59#4111yescharacteristic
97C42×C59#4111yescharacteristic
98(C2×D59)⋊C4—22no—
99(C2×D59)⋊C4—22no—
100(C2×D59)⋊C4—48no—
101C4⋊Dic59—21no—
102C4⋊Dic59—21no—
103C4⋊Dic59—42no—
104C4×Dic59—41no—
105C4×Dic59—41no—
106#11745664—41no—
107#11745664—41no—
108C2×C59⋊C8—22no—
109C2×C59⋊C8—41no—
110#11745659—84no—
111C23×C59C2×C22⋊C411yescharacteristic
112C22×D59—22no—
113C22×D59—22no—
114C22×D59—22no—
115C22×D59—42no—
116C22×D59—42no—
117Q8×C59—81no—
118Q8×C59—81no—
119Q8×C59—81no—
120Q8×C59—81no—
121C2×C4×C59—22no—
122C2×C4×C59—41no—
123C2×C4×C59—42no—
124C2×C4×C59—42no—
125C2×C4×C59—42no—
126C2×C4×C59C4×D411yescharacteristic
127C2×C4×C59C4×D411yescharacteristic
128C2×C4×C59#12011yescharacteristic
129C2×C4×C59#12111yescharacteristic
130C2×C4×C59#12111yescharacteristic
131C2×C4×C59#12311yescharacteristic
132C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
133C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
134C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
135C2×C4×C59#13711yescharacteristic
136C2×Dic59—22no—
137C2×Dic59—42no—
138C2×Dic59—42no—
139C2×Dic59—42no—
140C2×Dic59—42no—
141D236—41no—
142D236—41no—
143D236—41no—
144D236—41no—
145C4×D59—44no—
146C4×D59—82no—
147C8×C59—42no—
148C8×C59—81no—
149C59⋊C8—42no—
150C59⋊C8—81no—
151#9401—591nomaximal, Sylow
152C22×C59—21no—
153C22×C59—21no—
154C22×C59#38611yescharacteristic
155C22×C59#41711yescharacteristic
156C22×C59#41811yescharacteristic
157C2×D59—44no—
158C2×D59—42no—
159C2×D59—42no—
160C2×D59—42no—
161C2×D59—82no—
162C4×C59—21no—
163C4×C59—21no—
164C4×C59—21no—
165C4×C59—82no—
166C4×C59#45512yes—
167C4×C59#46811yescharacteristic
168C4×C59#46811yescharacteristic
169C4×C59#46911yescharacteristic
170C4×C59#46911yescharacteristic
171Dic59—42no—
172Dic59—82no—
173#2975—591no—
174#2547—591no—
175#1800—591no—
176#1546—591no—
177#1546D5911yescharacteristic, complemented
178#1303—591no—
179#1303—591no—
180C2×C59—21no—
181C2×C59—21no—
182C2×C59#188511yescharacteristic
183C2×C59#189511yescharacteristic
184C2×C59#191511yescharacteristic
185D59—82no—
186C2×C4⋊Q8C2×D5911yescharacteristic
187C2×#124—1181no—
188C2×C4×D4—591no—
189#436—1182no—
190#436—1182no—
191#431—591no—
192#431C2×D5911yescharacteristic
193#423—591no—
194#423C2×D5911yescharacteristic
195C2×#100—1181no—
196#407—1182no—
197#397—594no—
198#391—1181no—
199#380—1182no—
200C4×C4⋊C4—591no—
201#343—1181no—
202#332—22no—
203#332—1182no—
204C59#940111yescharacteristic, complemented, Sylow
205C22×Q8—21no—
206C22×D4—591no—
207C23×C4—592no—
208C2×M16—21no—
209C2×M16—1184no—
210C2×M16—1181no—
211C4⋊Q8—21no—
212C4⋊Q8—21no—
213C4⋊Q8—21no—
214C4⋊Q8—21no—
215#124—2362no—
216#124—2362no—
217C4×D4—1181no—
218C4×D4—1181no—
219C4×D4—1181no—
220C4×D4—1181no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—591no—
223C2×C4⋊C4—1181no—
224C2×C4⋊C4C4×D5912yes—
225C2×C22⋊C4—592no—
226C2×C22⋊C4—1184no—
227C2×C42—1181no—
228C2×C42C22×D5911yescharacteristic
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—592no—
232C4⋊C8—1181no—
233C4⋊C8—1181no—
234C4⋊C8C22⋊D5912yes—
235#100—2362no—
236C22⋊C8—21no—
237C22⋊C8—1184no—
238C22⋊C8—1181no—
239C4×C8—21no—
240C4×C8—1181no—
241#95—592no—
242#95—1184no—
243C24—592no—
244C2×Q8—41no—
245C2×Q8—41no—
246C2×Q8—41no—
247C2×Q8—41no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×D4—1181no—
251C2×D4—1181no—
252C2×D4—1181no—
253C2×D4—1181no—
254C2×D4—1181no—
255C2×D4—1181no—
256C22×C4—22no—
257C22×C4—592no—
258C22×C4—1182no—
259C22×C4—1182no—
260C22×C4—1182no—
261C22×C4—2364no—
262C22×C4C2×C4×D5911yescharacteristic
263C22×C4D4×D5911yescharacteristic
264C22×C4D4×D5911yescharacteristic
265M16—41no—
266M16—41no—
267M16—4724no—
268M16—2361no—
269M16—2361no—
270C2×C8—22no—
271C2×C8—41no—
272C2×C8—1182no—
273C2×C8—2361no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—1181no—
279C4⋊C4—1181no—
280C4⋊C4—2362no—
281C22⋊C4—1182no—
282C22⋊C4—1182no—
283C22⋊C4—2368no—
284C42—2361no—
285C42—2361no—
286C42#1174568511yescharacteristic
287C42D4×D5911yescharacteristic
288C42#1174568711yescharacteristic
289C42C2×C22⋊D5911yescharacteristic
290C23—1182no—
291C23—1182no—
292C23—1182no—
293C23—2362no—
294C23—2362no—
295C23C22⋊C4×D5911yescharacteristic
296Q8—81no—
297Q8—81no—
298Q8—81no—
299Q8—81no—
300D4—2361no—
301D4—2361no—
302D4—2361no—
303D4—2361no—
304C2×C4—22no—
305C2×C4—41no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—1182no—
310C2×C4—4722no—
311C2×C4—2362no—
312C2×C4—2362no—
313C2×C4—2362no—
314C2×C4—2364no—
315C2×C4—2362no—
316C2×C4(C22⋊D59)⋊C411yescharacteristic
317C2×C4C4×C22⋊D5911yescharacteristic
318C2×C4#1427369511yescharacteristic
319C2×C4#1427369811yescharacteristic
320C2×C4#1427370011yescharacteristic
321C2×C4#1427370911yescharacteristic
322C2×C4#1427375111yescharacteristic
323C2×C4#1427375111yescharacteristic
324C2×C4#1427375411yescharacteristic
325C2×C4#1427375611yescharacteristic
326C8—42no—
327C8—81no—
328C8—4721no—
329C8—2362no—
330C22—21no—
331C22—21no—
332C22—4722no—
333C22—2364no—
334C22—2362no—
335C22—2362no—
336C22—2362no—
337C22#1683727811yescharacteristic
338C22#1683740011yescharacteristic
339C22#1683740511yescharacteristic
340C4—21no—
341C4—21no—
342C4—21no—
343C4—82no—
344C4—4722no—
345C4—2362no—
346C4#1683757812yes—
347C4#1683764511yescharacteristic
348C4#1683764711yescharacteristic
349C4#1683765211yescharacteristic
350C4#1683765411yescharacteristic
351C2—21no—
352C2—21no—
353C2—4722no—
354C2#2142448111yescharacteristic
355C2#2142453111yescharacteristic
356C2#2142464511yescharacteristic
subgroupG/H
centerC22#16837278
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1546C2
socleC22×C59#386
prime radicalC22×D236D4
cosocleC2×C42×C59C23
hypercenter#1546D59
hypocenterC59#9401
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1546

Complements

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

NH ≅ G/Ntype
4. #21422768C2semidirect
5. #21421998C2semidirect
6. #21421997C2semidirect
7. C59×#1546C2semidirect
177. #1546D59semidirect
204. C59#9401semidirect

Maximal subgroups

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

subgroupindex
#214331362
#214290922
#214240852
#214227682
#214219982
#214219972
C59×#15462
#940159