Finite Group Databasebeta

← #28486061

Subgroups — #28486061

Order 15104.

594 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 117 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#21433119C211yesmaximal, characteristic
2#21429050C211yesmaximal, characteristic
3#21424085C211yesmaximal, characteristic
4#21422781C211yesmaximal, characteristic, complemented
5#21421998C211yesmaximal, characteristic, complemented
6#21421997C211yesmaximal, characteristic, complemented
7C59×#1549C211yesmaximal, characteristic, complemented
8C2×#14273703—22no—
9C2×C4×D236C2211yescharacteristic
10C8×D236—22no—
11#16837467—22no—
12#16837455C2211yescharacteristic
13#16837426C2211yescharacteristic
14C2×#14273643—21no—
15C22⋊C8×D59—22no—
16#16837337C414yes—
17#16837303—21no—
18#16837271—21no—
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×C8×D59—24no—
30#14273703—48no—
31C4×D236—21no—
32C4×D236—21no—
33C4×D236—21no—
34C4×D236—21no—
35C2×C4⋊Dic59—21no—
36C2×C4⋊Dic59D411yescharacteristic
37C2×Dic59⋊C4—24no—
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⋊#95C2×C412yes—
49C22×Q8×C59—21no—
50C2×M16×C59—21no—
51C4⋊Q8×C59—21no—
52C4⋊Q8×C59—21no—
53C4⋊Q8×C59—21no—
54C4⋊Q8×C59—21no—
55C2×C4⋊C4×C59—21no—
56C2×C4⋊C4×C59C2×C412yes—
57C2×C42×C59C2311yescharacteristic
58C4⋊C8×C59—21no—
59C4⋊C8×C59—21no—
60C4⋊C8×C59D412yes—
61C22⋊C8×C59—21no—
62C4×C8×C59—21no—
63C23×D59C22⋊C412yes—
64C2×Q8×C59—44no—
65C2×Q8×C59—41no—
66C2×Q8×C59—41no—
67C22×C4×C59—22no—
68C22×C4×C59C22×C411yescharacteristic
69C22×C4×C59C2×D411yescharacteristic
70C22×C4×C59C2×D411yescharacteristic
71C22×Dic59—22no—
72C22×Dic59C22⋊C412yes—
73C2×D236—21no—
74C2×D236—21no—
75C2×D236—21no—
76C2×D236—21no—
77C2×D236SD1612yes—
78C2×D236Q1612yes—
79C2×C4×D59—22no—
80C2×C4×D59—22no—
81C2×C4×D59—44no—
82M16×C59—41no—
83M16×C59—41no—
84C2×C8×C59—22no—
85C2×C8×C59—41no—
86C4⋊C4×C59—22no—
87C4⋊C4×C59—22no—
88C4⋊C4×C59—41no—
89C4⋊C4×C59—41no—
90C42×C59C2×D411yescharacteristic
91C42×C59C2×D411yescharacteristic
92C42×C59#4111yescharacteristic
93C42×C59#4111yescharacteristic
94(C2×D59)⋊C4—22no—
95(C2×D59)⋊C4—22no—
96(C2×D59)⋊C4—48no—
97C4⋊Dic59—24no—
98C4⋊Dic59—21no—
99C4⋊Dic59—21no—
100Dic59⋊C4—48no—
101C4×Dic59—41no—
102C4×Dic59—41no—
103#11745664—41no—
104#11745664—41no—
105C2×C59⋊C8—22no—
106C2×C59⋊C8—41no—
107C8×D59—84no—
108C23×C59C2×C22⋊C411yescharacteristic
109C22×D59—22no—
110C22×D59—22no—
111C22×D59—42no—
112C22×D59—42no—
113C22×D59#10314yes—
114Q8×C59—82no—
115Q8×C59—82no—
116C2×C4×C59—22no—
117C2×C4×C59—41no—
118C2×C4×C59—44no—
119C2×C4×C59—42no—
120C2×C4×C59C4×D411yescharacteristic
121C2×C4×C59C4×D411yescharacteristic
122C2×C4×C59#12011yescharacteristic
123C2×C4×C59#12111yescharacteristic
124C2×C4×C59#12111yescharacteristic
125C2×C4×C59#12311yescharacteristic
126C2×C4×C59C2×SD1611yescharacteristic
127C2×C4×C59C2×Q1611yescharacteristic
128C2×C4×C59#13511yescharacteristic
129C2×C4×C59#13511yescharacteristic
130C2×Dic59—24no—
131C2×Dic59—22no—
132C2×Dic59—42no—
133C2×Dic59—42no—
134C2×Dic59—42no—
135D236—22no—
136D236—22no—
137D236—22no—
138D236—22no—
139C4×D59—44no—
140C4×D59—82no—
141C8×C59—42no—
142C8×C59—81no—
143C59⋊C8—42no—
144C59⋊C8—81no—
145#9395—591nomaximal, Sylow
146C22×C59—21no—
147C22×C59—21no—
148C22×C59#38611yescharacteristic
149C22×C59C2×#10311yescharacteristic
150C22×C59#41611yescharacteristic
151C2×D59—24no—
152C2×D59—42no—
153C2×D59—42no—
154C2×D59—44no—
155C2×D59—82no—
156C4×C59—21no—
157C4×C59—21no—
158C4×C59—21no—
159C4×C59—82no—
160C4×C59#45312yes—
161C4×C59#46511yescharacteristic
162C4×C59#46511yescharacteristic
163C4×C59#46711yescharacteristic
164C4×C59#46711yescharacteristic
165Dic59—44no—
166Dic59—42no—
167#2972—591no—
168#2542—591no—
169#1800—591no—
170#1549—591no—
171#1549D5911yescharacteristic, complemented
172#1303—591no—
173#1303—591no—
174C2×C59—21no—
175C2×C59—21no—
176C2×C59#188711yescharacteristic
177C2×C59#189211yescharacteristic
178C2×C59#191511yescharacteristic
179D59—44no—
180C2×C4⋊Q8C2×D5911yescharacteristic
181C2×C22⋊Q8—1182no—
182C2×C4×D4—591no—
183C8×D4—1182no—
184C8×D4—1182no—
185#431—591no—
186#431C2×D5911yescharacteristic
187#423—591no—
188#423C2×D5911yescharacteristic
189C2×#100—1181no—
190C2×C22⋊C8—1182no—
191#399—594no—
192#390—1181no—
193#384—1181no—
194C4×C4⋊C4—591no—
195#343—1181no—
196#332—22no—
197#332—1182no—
198C59#939511yescharacteristic, complemented, Sylow
199C22×Q8—21no—
200C22×D4—591no—
201C23×C4—592no—
202C2×M16—21no—
203C2×M16—1181no—
204C22×C8—1184no—
205C4⋊Q8—21no—
206C4⋊Q8—21no—
207C4⋊Q8—21no—
208C4⋊Q8—21no—
209C22⋊Q8—2368no—
210C4×D4—1181no—
211C4×D4—1181no—
212C4×D4—1181no—
213C4×D4—1181no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—591no—
216C2×C4⋊C4—1181no—
217C2×C4⋊C4—1184no—
218C2×C4⋊C4C4×D5912yes—
219C2×C22⋊C4—592no—
220C2×C22⋊C4—1184no—
221C2×C42—1181no—
222C2×C42C22×D5911yescharacteristic
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—592no—
226C4⋊C8—1181no—
227C4⋊C8—1181no—
228C4⋊C8C22⋊D5912yes—
229#100—2362no—
230C22⋊C8—21no—
231C22⋊C8—1184no—
232C22⋊C8—1181no—
233C4×C8—21no—
234C4×C8—1181no—
235#95—592no—
236C24—592no—
237C2×Q8—44no—
238C2×Q8—41no—
239C2×Q8—41no—
240C2×D4—592no—
241C2×D4—592no—
242C2×D4—1181no—
243C2×D4—1181no—
244C2×D4—1181no—
245C2×D4—1181no—
246C22×C4—22no—
247C22×C4—592no—
248C22×C4—1182no—
249C22×C4—1182no—
250C22×C4—1182no—
251C22×C4—2364no—
252C22×C4C2×C4×D5911yescharacteristic
253C22×C4D4×D5911yescharacteristic
254C22×C4D4×D5911yescharacteristic
255M16—41no—
256M16—41no—
257M16—2361no—
258M16—2361no—
259C2×C8—22no—
260C2×C8—41no—
261C2×C8—1182no—
262C2×C8—4724no—
263C2×C8—2361no—
264C4⋊C4—22no—
265C4⋊C4—22no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—1184no—
269C4⋊C4—1181no—
270C4⋊C4—1181no—
271C4⋊C4—2368no—
272C22⋊C4—1182no—
273C22⋊C4—1182no—
274C22⋊C4—2368no—
275C42—2361no—
276C42—2361no—
277C42#1174568511yescharacteristic
278C42D4×D5911yescharacteristic
279C42#1174568711yescharacteristic
280C42C2×C22⋊D5911yescharacteristic
281C23—594no—
282C23—1182no—
283C23—1182no—
284C23—2362no—
285C23—2362no—
286C23C22⋊C4×D5911yescharacteristic
287Q8—82no—
288Q8—82no—
289D4—1182no—
290D4—1182no—
291D4—1182no—
292D4—1182no—
293C2×C4—22no—
294C2×C4—41no—
295C2×C4—44no—
296C2×C4—42no—
297C2×C4—1184no—
298C2×C4—1182no—
299C2×C4—4722no—
300C2×C4—2362no—
301C2×C4—2362no—
302C2×C4—2364no—
303C2×C4—2362no—
304C2×C4(C22⋊D59)⋊C411yescharacteristic
305C2×C4C4×C22⋊D5911yescharacteristic
306C2×C4#1427369511yescharacteristic
307C2×C4#1427369811yescharacteristic
308C2×C4#1427370011yescharacteristic
309C2×C4#1427370911yescharacteristic
310C2×C4SD16×D5911yescharacteristic
311C2×C4Q16×D5911yescharacteristic
312C2×C4#1427374811yescharacteristic
313C2×C4#1427375011yescharacteristic
314C8—42no—
315C8—81no—
316C8—4721no—
317C8—2362no—
318C22—21no—
319C22—21no—
320C22—1184no—
321C22—4722no—
322C22—2364no—
323C22—2362no—
324C22—2362no—
325C22#1683727811yescharacteristic
326C22#103×D5911yescharacteristic
327C22#1683739711yescharacteristic
328C4—21no—
329C4—21no—
330C4—21no—
331C4—82no—
332C4—2364no—
333C4—2362no—
334C4#1683756812yes—
335C4#1683762811yescharacteristic
336C4#1683763411yescharacteristic
337C4#1683764011yescharacteristic
338C4#1683764211yescharacteristic
339C2—21no—
340C2—21no—
341C2—2364no—
342C2#2142449111yescharacteristic
343C2#2142451611yescharacteristic
344C2#2142464511yescharacteristic
subgroupG/H
centerC22#16837278
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1549C2
socleC22×C59#386
prime radicalC22×D236D4
cosocleC2×C42×C59C23
hypercenter#1549D59
hypocenterC59#9395
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1549

Complements

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

NH ≅ G/Ntype
4. #21422781C2semidirect
5. #21421998C2semidirect
6. #21421997C2semidirect
7. C59×#1549C2semidirect
171. #1549D59semidirect
198. C59#9395semidirect

Maximal subgroups

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

subgroupindex
#214331192
#214290502
#214240852
#214227812
#214219982
#214219972
C59×#15492
#939559