Finite Group Databasebeta

← #28506984

Subgroups — #28506984

Order 15104.

540 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 103 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
3C4×#14273642C211yesmaximal, characteristic
4C59×#1800C211yesmaximal, characteristic
5C2×#14273708C2212yes—
6C22×C4⋊Dic59C411yescharacteristic, complemented
7C2×#14273642C2212yes—
8#16837332—21no—
9#16837327—21no—
10#16837300—21no—
11#16837263C412yescomplemented
12#16837262—21no—
13#16837262C411yescharacteristic, complemented
14C4×C22⋊Dic59—21no—
15#16837193—28no—
16#16837169—21no—
17C2×C4×D4×C59C2211yescharacteristic
18C59×#423—21no—
19C2×#100×C59C2212yes—
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×Dic59⋊C4—22no—
30C2×C22⋊Dic59—22no—
31C2×C22⋊Dic59—22no—
32C2×C22⋊Dic59C2×C412yes—
33C2×C4×Dic59—41no—
34#14273642—22no—
35#14273642D414yes—
36C59⋊#95—22no—
37C59⋊#95—21no—
38C59⋊#95—22no—
39C59⋊#95—41no—
40C22×D4×C59C2311yescharacteristic
41C23×C4×C59C2×C411yescharacteristic
42C23×C4×C59D411yescharacteristic
43C2×M16×C59—22no—
44C4×D4×C59—41no—
45C4×D4×C59—41no—
46C2×C4⋊C4×C59D411yescharacteristic
47C2×C22⋊C4×C59C2×C411yescharacteristic
48C2×C22⋊C4×C59D411yescharacteristic
49C2×C42×C59D411yescharacteristic
50C4⋊C8×C59—41no—
51C4⋊C8×C59—41no—
52#100×C59—24no—
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—42no—
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—24no—
86C22⋊Dic59—42no—
87C22⋊Dic59—44no—
88C4⋊Dic59—42no—
89C4⋊Dic59—42no—
90C4⋊Dic59—41no—
91C4⋊Dic59—41no—
92Dic59⋊C4—42no—
93Dic59⋊C4—42no—
94C4×Dic59—82no—
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#12112yes—
106D4×C59—42no—
107D4×C59—42no—
108C2×C4×C59—21no—
109C2×C4×C59—21no—
110C2×C4×C59—22no—
111C2×C4×C59—21no—
112C2×C4×C59—21no—
113C2×C4×C59—21no—
114C2×C4×C59—42no—
115C2×C4×C59—42no—
116C2×C4×C59—42no—
117C2×C4×C59—41no—
118C2×C4×C59—41no—
119C2×C4×C59—41no—
120C2×C4×C59—81no—
121C2×C4×C59#9911yescharacteristic
122C2×C4×C59#9911yescharacteristic
123C2×C4×C59#12411yescharacteristic
124C2×C4×C59#12711yescharacteristic
125C2×Dic59—42no—
126C2×Dic59—44no—
127C2×Dic59—42no—
128C2×Dic59—42no—
129C2×Dic59—42no—
130C2×Dic59—42no—
131C2×Dic59—42no—
132C2×Dic59—82no—
133C2×Dic59—82no—
134C2×Dic59—84no—
135C8×C59—82no—
136#13220—591nomaximal, Sylow
137C22×C59—21no—
138C22×C59—21no—
139C22×C59—22no—
140C22×C59—22no—
141C22×C59—42no—
142C22×C59—42no—
143C22×C59—42no—
144C22×C59—41no—
145C22×C59—41no—
146C22×C59—41no—
147C22×C59—41no—
148C22×C59—81no—
149C22×C59#39011yescharacteristic
150C22×C59C2×#9911yescharacteristic
151C22×C59#41011yescharacteristic
152C4×C59—21no—
153C4×C59—21no—
154C4×C59—21no—
155C4×C59—41no—
156C4×C59—41no—
157C4×C59—41no—
158C4×C59#35212yescomplemented
159Dic59—82no—
160Dic59—84no—
161Dic59—82no—
162#2486—591no—
163C2×#352—594no—
164#1800D5911yescharacteristic
165C4×#99—591no—
166C2×C59—21no—
167C2×C59—21no—
168C2×C59—42no—
169C2×C59—42no—
170C2×C59#196411yescharacteristic
171C2×C59C2×#35211yescharacteristic
172C2×C59#212311yescharacteristic
173C2×#123—592no—
174C2×C4×D4C2×D5911yescharacteristic
175C22×C4⋊C4—591no—
176#423—21no—
177C2×#100C2×D5912yes—
178C2×#99—592no—
179#398—1181no—
180#397—1181no—
181#389—1181no—
182#381—592no—
183#380—591no—
184#380—1181no—
185C4×C22⋊C4—1181no—
186#352—1188no—
187#343—21no—
188#342—1181no—
189C59#1322011yescharacteristic, complemented, Sylow
190C22×D4C22×D5911yescharacteristic
191C23×C4—592no—
192C23×C4C4×D5911yescharacteristic
193C23×C4D23611yescharacteristic
194C2×M16—22no—
195#123—1184no—
196#123—1184no—
197C4×D4—41no—
198C4×D4—41no—
199C2×C4⋊C4—1181no—
200C2×C4⋊C4—1181no—
201C2×C4⋊C4—1181no—
202C2×C4⋊C4—1182no—
203C2×C4⋊C4—2361no—
204C2×C4⋊C4—2361no—
205C2×C4⋊C4C22⋊D5911yescharacteristic
206C2×C22⋊C4—592no—
207C2×C22⋊C4—1182no—
208C2×C22⋊C4—1182no—
209C2×C22⋊C4C4×D5911yescharacteristic
210C2×C22⋊C4D23611yescharacteristic
211C2×C42—2361no—
212C2×C42C22⋊D5911yescharacteristic
213C4⋊C8—41no—
214C4⋊C8—41no—
215#100—24no—
216#99—594no—
217#99—1182no—
218#95—1181no—
219#95—1182no—
220#95—1182no—
221#95—2361no—
222C24C2×C4×D5911yescharacteristic
223C24C2×D23611yescharacteristic
224C2×D4—41no—
225C2×D4—41no—
226C2×D4D4×D5912yes—
227C2×D4#1174568912yes—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—1184no—
235C22×C4—1182no—
236C22×C4—1182no—
237C22×C4—2362no—
238C22×C4—2362no—
239C22×C4—2362no—
240C22×C4(C2×D59)⋊C411yescharacteristic
241C22×C4(C2×D59)⋊C411yescharacteristic
242C22×C4(C2×D59)⋊C411yescharacteristic
243C22×C4(C2×D59)⋊C411yescharacteristic
244C22×C4C2×C22⋊D5911yescharacteristic
245M16—42no—
246M16—42no—
247C2×C8—42no—
248C4⋊C4—41no—
249C4⋊C4—2362no—
250C4⋊C4—2362no—
251C4⋊C4—2361no—
252C4⋊C4—2361no—
253C4⋊C4—2362no—
254C4⋊C4—2362no—
255C22⋊C4—41no—
256C22⋊C4—41no—
257C22⋊C4—1184no—
258C22⋊C4—1184no—
259C22⋊C4—2362no—
260C22⋊C4—2364no—
261C42—21no—
262C42—21no—
263C42—4722no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23C2×(C2×D59)⋊C411yescharacteristic
273C23#1427368912yes—
274C23#1427370212yes—
275D4—42no—
276D4—42no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—81no—
290C2×C4—4722no—
291C2×C4—4722no—
292C2×C4—4724no—
293C2×C4—2364no—
294C2×C4—2362no—
295C2×C4—2362no—
296C2×C4—2362no—
297C2×C4—2362no—
298C2×C4—2362no—
299C2×C4—2362no—
300C2×C4#1427364111yescharacteristic
301C2×C4#1427364111yescharacteristic
302C2×C4#1427371611yescharacteristic
303C2×C4#1427372311yescharacteristic
304C8—82no—
305C22—21no—
306C22—21no—
307C22—22no—
308C22—22no—
309C22—42no—
310C22—42no—
311C22—42no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—41no—
316C22—81no—
317C22#1683730311yescharacteristic
318C22C2×#1427364111yescharacteristic
319C22#1683737511yescharacteristic
320C4—21no—
321C4—21no—
322C4—21no—
323C4—41no—
324C4—41no—
325C4—41no—
326C4—4722no—
327C4—4724no—
328C4—4722no—
329C4#1683719312yescomplemented
330C2—21no—
331C2—21no—
332C2—42no—
333C2—42no—
334C2#2142493111yescharacteristic
335C2C2×#1683719311yescharacteristic
336C2#2142575111yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC24×C59C22×C4
FrattiniC22×D4C22×D59
FittingC59×#1800C2
socleC22×C59C2×#99
prime radical, cosocleC22×D4×C59C23
hypercenter#1800D59
hypocenterC59#13220
LCSG ▷ C24×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1800

Complements

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

NH ≅ G/Ntype
6. C22×C4⋊Dic59C4semidirect
11. #16837263C4semidirect
13. #16837262C4semidirect
158. C4×C59#352semidirect
189. C59#13220semidirect
329. 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
C4×#142736422
C59×#18002
#1322059