Finite Group Databasebeta

← #28506972

Subgroups — #28506972

Order 15104.

536 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 107 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#21428404C211yesmaximal, characteristic
2C2×#16837193C214yesmaximal
3#21424084C211yesmaximal, characteristic
4C4×#100×C59C211yesmaximal, characteristic
5C2×#14273708C2212yes—
6C22×C4⋊Dic59C411yescharacteristic, complemented
7C2×#14273642C2212yes—
8#16837332—21no—
9#16837327—21no—
10#16837300—21no—
11#16837263C412yescomplemented
12#16837262—21no—
13#16837262—21no—
14#16837262C411yescharacteristic, complemented
15#16837193—28no—
16#16837169—21no—
17C2×C4×D4×C59C2211yescharacteristic
18C2×#100×C59C2212yes—
19C4×M16×C59—21no—
20C59×#343—21no—
21C23×Dic59C2×C412yes—
22#14273708—24no—
23#14273708—24no—
24C2×C4⋊Dic59—21no—
25C2×C4⋊Dic59—21no—
26C2×C4⋊Dic59—21no—
27C2×C4⋊Dic59—41no—
28C2×C4⋊Dic59—41no—
29C2×C4⋊Dic59—41no—
30C2×Dic59⋊C4—22no—
31C2×C22⋊Dic59—22no—
32C2×C22⋊Dic59—22no—
33C2×C22⋊Dic59C2×C412yes—
34#14273642—24no—
35C59⋊#95—22no—
36C59⋊#95—21no—
37C59⋊#95—22no—
38C59⋊#95—41no—
39C22×D4×C59C2311yescharacteristic
40C23×C4×C59C2×C411yescharacteristic
41C23×C4×C59D411yescharacteristic
42C2×M16×C59—22no—
43C4×D4×C59—41no—
44C4×D4×C59—41no—
45C2×C4⋊C4×C59D411yescharacteristic
46C2×C22⋊C4×C59C2×C411yescharacteristic
47C2×C22⋊C4×C59D411yescharacteristic
48C2×C42×C59D411yescharacteristic
49#100×C59—22no—
50#100×C59D414yes—
51#97×C59—41no—
52C4×C8×C59—41no—
53C24×C59C22×C411yescharacteristic
54C24×C59C2×D411yescharacteristic
55C2×D4×C59—41no—
56C2×D4×C59—41no—
57C2×D4×C59C2×D412yes—
58C2×D4×C59#4112yes—
59C22×C4×C59—21no—
60C22×C4×C59—21no—
61C22×C4×C59—21no—
62C22×C4×C59—21no—
63C22×C4×C59—42no—
64C22×C4×C59—42no—
65C22×C4×C59C22⋊C411yescharacteristic
66C22×C4×C59C22⋊C411yescharacteristic
67C22×C4×C59C22⋊C411yescharacteristic
68C22×C4×C59C22⋊C411yescharacteristic
69C22×C4×C59C2×D411yescharacteristic
70C22×Dic59—22no—
71C22×Dic59—22no—
72C22×Dic59—24no—
73C22×Dic59—42no—
74C22×Dic59—42no—
75C22×Dic59—42no—
76M16×C59—24no—
77M16×C59—42no—
78C2×C8×C59—42no—
79C4⋊C4×C59—41no—
80C22⋊C4×C59—41no—
81C22⋊C4×C59—41no—
82C42×C59—21no—
83C42×C59—21no—
84C22⋊Dic59—24no—
85C22⋊Dic59—42no—
86C22⋊Dic59—42no—
87C22⋊Dic59—42no—
88C22⋊Dic59—42no—
89C4⋊Dic59—42no—
90C4⋊Dic59—42no—
91C4⋊Dic59—41no—
92C4⋊Dic59—41no—
93C4⋊Dic59—82no—
94Dic59⋊C4—44no—
95C23×C59—21no—
96C23×C59—21no—
97C23×C59—21no—
98C23×C59—21no—
99C23×C59—41no—
100C23×C59—41no—
101C23×C59—41no—
102C23×C59—41no—
103C23×C59C2×C22⋊C411yescharacteristic
104C23×C59C4×D412yes—
105C23×C59#12411yescharacteristic
106C23×C59#12711yescharacteristic
107D4×C59—42no—
108D4×C59—42no—
109C2×C4×C59—21no—
110C2×C4×C59—21no—
111C2×C4×C59—22no—
112C2×C4×C59—21no—
113C2×C4×C59—21no—
114C2×C4×C59—21no—
115C2×C4×C59—42no—
116C2×C4×C59—42no—
117C2×C4×C59—42no—
118C2×C4×C59—41no—
119C2×C4×C59—41no—
120C2×C4×C59—41no—
121C2×C4×C59—81no—
122C2×C4×C59#9911yescharacteristic
123C2×C4×C59#9911yescharacteristic
124C2×C4×C59#12112yes—
125C2×Dic59—42no—
126C2×Dic59—42no—
127C2×Dic59—42no—
128C2×Dic59—42no—
129C2×Dic59—42no—
130C2×Dic59—42no—
131C2×Dic59—84no—
132C2×Dic59—82no—
133C2×Dic59—82no—
134C2×Dic59—82no—
135C8×C59—44no—
136#13217—591nomaximal, Sylow
137C22×C59—21no—
138C22×C59—21no—
139C22×C59—22no—
140C22×C59—21no—
141C22×C59—21no—
142C22×C59—42no—
143C22×C59—41no—
144C22×C59—41no—
145C22×C59—42no—
146C22×C59—41no—
147C22×C59—41no—
148C22×C59—41no—
149C22×C59—41no—
150C22×C59—81no—
151C22×C59#39011yescharacteristic
152C22×C59C2×#9911yescharacteristic
153C22×C59#41011yescharacteristic
154C4×C59—21no—
155C4×C59—22no—
156C4×C59—41no—
157C4×C59—41no—
158C4×C59—41no—
159C4×C59#35212yescomplemented
160Dic59—82no—
161Dic59—82no—
162Dic59—162no—
163#2486—591no—
164C2×#352—594no—
165#1799—591no—
166C4×#100D5911yescharacteristic
167C2×C59—21no—
168C2×C59—21no—
169C2×C59—41no—
170C2×C59—42no—
171C2×C59—41no—
172C2×C59#196411yescharacteristic
173C2×C59C2×#35211yescharacteristic
174C2×C59#212311yescharacteristic
175C2×#123—592no—
176C2×C4×D4C2×D5911yescharacteristic
177C22×C4⋊C4—591no—
178C2×#100C2×D5912yes—
179C2×#99—592no—
180C4×M16—21no—
181#398—1181no—
182#397—1181no—
183#389—1181no—
184#381—592no—
185#380—591no—
186#380—1181no—
187#380—1181no—
188#352—1188no—
189#343—21no—
190#342—1181no—
191C59#1321711yescharacteristic, complemented, Sylow
192C22×D4C22×D5911yescharacteristic
193C23×C4—592no—
194C23×C4C4×D5911yescharacteristic
195C23×C4D23611yescharacteristic
196C2×M16—22no—
197#123—1184no—
198#123—1184no—
199C4×D4—41no—
200C4×D4—41no—
201C2×C4⋊C4—1181no—
202C2×C4⋊C4—1181no—
203C2×C4⋊C4—1181no—
204C2×C4⋊C4—1182no—
205C2×C4⋊C4—2361no—
206C2×C4⋊C4—2361no—
207C2×C4⋊C4—2361no—
208C2×C4⋊C4C22⋊D5911yescharacteristic
209C2×C22⋊C4—592no—
210C2×C22⋊C4—1182no—
211C2×C22⋊C4—1182no—
212C2×C22⋊C4C4×D5911yescharacteristic
213C2×C22⋊C4D23611yescharacteristic
214C2×C42C22⋊D5911yescharacteristic
215#100—22no—
216#100D23614yes—
217#99—1184no—
218#97—41no—
219C4×C8—41no—
220#95—1181no—
221#95—1182no—
222#95—1182no—
223#95—2361no—
224C24C2×C4×D5911yescharacteristic
225C24C2×D23611yescharacteristic
226C2×D4—41no—
227C2×D4—41no—
228C2×D4C2×D23612yes—
229C2×D4#1174568512yes—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—1184no—
237C22×C4—1182no—
238C22×C4—1182no—
239C22×C4—2362no—
240C22×C4—2362no—
241C22×C4—2362no—
242C22×C4(C2×D59)⋊C411yescharacteristic
243C22×C4(C2×D59)⋊C411yescharacteristic
244C22×C4(C2×D59)⋊C411yescharacteristic
245C22×C4(C2×D59)⋊C411yescharacteristic
246C22×C4C2×C22⋊D5911yescharacteristic
247M16—24no—
248M16—42no—
249C2×C8—42no—
250C4⋊C4—41no—
251C4⋊C4—4722no—
252C4⋊C4—2362no—
253C4⋊C4—2362no—
254C4⋊C4—2361no—
255C4⋊C4—2361no—
256C4⋊C4—2364no—
257C22⋊C4—41no—
258C22⋊C4—41no—
259C22⋊C4—1184no—
260C22⋊C4—2362no—
261C22⋊C4—2362no—
262C22⋊C4—2362no—
263C22⋊C4—2362no—
264C42—21no—
265C42—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—41no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23C2×(C2×D59)⋊C411yescharacteristic
275C23C4×D23612yes—
276C23#1427371511yescharacteristic
277C23#1427372411yescharacteristic
278D4—42no—
279D4—42no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—81no—
293C2×C4—4722no—
294C2×C4—4722no—
295C2×C4—4722no—
296C2×C4—4724no—
297C2×C4—2362no—
298C2×C4—2362no—
299C2×C4—2362no—
300C2×C4—2362no—
301C2×C4—2362no—
302C2×C4—2362no—
303C2×C4#1427364111yescharacteristic
304C2×C4#1427364111yescharacteristic
305C2×C4#1427370112yes—
306C8—44no—
307C22—21no—
308C22—21no—
309C22—22no—
310C22—21no—
311C22—21no—
312C22—42no—
313C22—41no—
314C22—41no—
315C22—42no—
316C22—41no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22—81no—
321C22#1683730411yescharacteristic
322C22C2×#1427364111yescharacteristic
323C22#1683737511yescharacteristic
324C4—22no—
325C4—21no—
326C4—41no—
327C4—41no—
328C4—41no—
329C4—9442no—
330C4—4722no—
331C4—4722no—
332C4#1683719312yescomplemented
333C2—21no—
334C2—21no—
335C2—41no—
336C2—42no—
337C2—41no—
338C2#2142493211yescharacteristic
339C2C2×#1683719311yescharacteristic
340C2#2142575111yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC24×C59C22×C4
FrattiniC22×D4C22×D59
FittingC4×#100×C59C2
socleC22×C59C2×#99
prime radical, cosocleC22×D4×C59C23
hypercenterC4×#100D59
hypocenterC59#13217
LCSG ▷ C24×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#100

Complements

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

NH ≅ G/Ntype
6. C22×C4⋊Dic59C4semidirect
11. #16837263C4semidirect
14. #16837262C4semidirect
159. C4×C59#352semidirect
191. C59#13217semidirect
332. C4#16837193semidirect

Maximal subgroups

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

subgroupindex
#214284042
C2×#168371932
#214240842
C4×#100×C592
#1321759