Finite Group Databasebeta

← #28499056

Subgroups — #28499056

Order 15104.

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

normal subgroups only

This group has 434 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#21427379C211yesmaximal, characteristic
2#21422420C211yesmaximal, characteristic, complemented
3#21422122C212yesmaximal, complemented
4#21422098C212yesmaximal
5C4×#101×C59C211yesmaximal, characteristic, complemented
6C2×C4×D236C411yescharacteristic
7#16837455—21no—
8#16837341C411yescharacteristic
9#16837341C2211yescharacteristic
10#16837328—21no—
11#16837319C411yescharacteristic
12#16837319C2211yescharacteristic
13#16837313—21no—
14C4×(C2×D59)⋊C4—21no—
15C4×(C2×D59)⋊C4—21no—
16#16837248C411yescharacteristic
17#16837128—21no—
18#16837128C2212yes—
19C2×C4×Q8×C59C2211yescharacteristic
20C2×#101×C59—21no—
21C4×M16×C59—21no—
22C59×#343C2212yes—
23C22×D236C2×C411yescharacteristic
24C22×C4×D59—21no—
25C4×D236—41no—
26C4×D236—41no—
27C2×C4⋊Dic59C2×C411yescharacteristic
28C2×(C2×D59)⋊C4—21no—
29C2×(C2×D59)⋊C4—21no—
30C2×(C2×D59)⋊C4—21no—
31C2×(C2×D59)⋊C4—21no—
32C2×(C2×D59)⋊C4—21no—
33C2×C4×Dic59—21no—
34C4⋊(C59⋊C8)—41no—
35C22⋊(C59⋊C8)—22no—
36C4×C59⋊C8—41no—
37C59⋊#95—21no—
38C59⋊#95—21no—
39C59⋊#95—21no—
40C59⋊#95—21no—
41C59⋊#95—21no—
42C59⋊#95C2×C411yescharacteristic
43C59⋊#95C2×C411yescharacteristic
44C22×Q8×C59D411yescharacteristic
45C2×M16×C59—22no—
46C4×Q8×C59—41no—
47C4×Q8×C59—41no—
48C2×C4⋊C4×C59C2×C411yescharacteristic
49C2×C4⋊C4×C59D411yescharacteristic
50C2×C4⋊C4×C59C2311yescharacteristic
51C2×C42×C59C2×C411yescharacteristic
52C2×C42×C59D411yescharacteristic
53C2×C42×C59D411yescharacteristic
54#101×C59—24no—
55#97×C59—41no—
56C4×C8×C59—41no—
57C23×D59—21no—
58C2×Q8×C59—22no—
59C2×Q8×C59—22no—
60C2×Q8×C59—41no—
61C22×C4×C59C22⋊C411yescharacteristic
62C22×C4×C59C22⋊C411yescharacteristic
63C22×C4×C59C22⋊C411yescharacteristic
64C22×C4×C59C22⋊C411yescharacteristic
65C22×C4×C59C22×C411yescharacteristic
66C22×C4×C59C2×D411yescharacteristic
67C22×C4×C59C2×D411yescharacteristic
68C22×Dic59—21no—
69C22×Dic59—21no—
70C22×Dic59—21no—
71C2×D236—41no—
72C2×D236—42no—
73C2×C4×D59—42no—
74C2×C4×D59—42no—
75C2×C4×D59—41no—
76C2×C4×D59—41no—
77M16×C59—24no—
78M16×C59—42no—
79C2×C8×C59—42no—
80C4⋊C4×C59—41no—
81C4⋊C4×C59—41no—
82C4⋊C4×C59—41no—
83C42×C59—21no—
84C42×C59—21no—
85C42×C59—21no—
86C42×C59—21no—
87C42×C59—21no—
88C42×C59—21no—
89(C2×D59)⋊C4—42no—
90(C2×D59)⋊C4—42no—
91(C2×D59)⋊C4—42no—
92(C2×D59)⋊C4—42no—
93(C2×D59)⋊C4—42no—
94C4⋊Dic59—41no—
95C4×Dic59—42no—
96C2×C59⋊C8—42no—
97C23×C59C2×C22⋊C411yescharacteristic
98C22×D59—22no—
99C22×D59—41no—
100C22×D59—41no—
101C22×D59—41no—
102C22×D59—41no—
103C22×D59—41no—
104C22×D59—41no—
105Q8×C59—44no—
106C2×C4×C59—21no—
107C2×C4×C59—21no—
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—21no—
115C2×C4×C59—21no—
116C2×C4×C59—21no—
117C2×C4×C59—41no—
118C2×C4×C59—41no—
119C2×C4×C59—41no—
120C2×C4×C59#9911yescharacteristic
121C2×C4×C59#9911yescharacteristic
122C2×C4×C59#10412yes—
123C2×C4×C59#13512yes—
124C2×Dic59—41no—
125C2×Dic59—41no—
126C2×Dic59—41no—
127C2×Dic59—41no—
128C2×Dic59—41no—
129C2×Dic59—41no—
130C2×Dic59—41no—
131C2×Dic59—41no—
132C2×Dic59—41no—
133D236—42no—
134D236—42no—
135C4×D59—82no—
136C4×D59—82no—
137C8×C59—44no—
138C59⋊C8—82no—
139#11734—591nomaximal, Sylow
140C22×C59—21no—
141C22×C59—21no—
142C22×C59C2×#9911yescharacteristic
143C22×C59#41611yescharacteristic
144C22×C59C2×#10411yescharacteristic
145C2×D59—42no—
146C2×D59—42no—
147C2×D59—42no—
148C2×D59—81no—
149C2×D59—81no—
150C2×D59—81no—
151C2×D59—81no—
152C4×C59—21no—
153C4×C59—22no—
154C4×C59—22no—
155C4×C59—41no—
156C4×C59—41no—
157C4×C59—41no—
158C4×C59—41no—
159C4×C59#35512yes—
160Dic59—81no—
161Dic59—81no—
162Dic59—81no—
163#2398—591no—
164C4×#101D5911yescharacteristic, complemented
165#1461—591no—
166#1353—592no—
167#1345—592no—
168C2×C59—21no—
169C2×C59—21no—
170C2×C59#150111yescharacteristic
171C2×C59#212011yescharacteristic
172C2×C59C2×#35511yescharacteristic
173D59—82no—
174C2×C4×Q8C2×D5911yescharacteristic
175C2×C4×D4—591no—
176#431—1181no—
177C2×#101—21no—
178C4×M16—21no—
179#400—591no—
180#400—591no—
181#397—1181no—
182#394—591no—
183#394—591no—
184#393—1181no—
185C4×C22⋊C4—1181no—
186C4×C22⋊C4—1181no—
187#376—591no—
188#343C2×D5912yes—
189#324—592no—
190#324—1181no—
191C59#1173411yescharacteristic, complemented, Sylow
192C22×Q8C22⋊D5911yescharacteristic
193C22×D4—591no—
194C23×C4—1181no—
195C2×M16—22no—
196C4×Q8—41no—
197C4×Q8—41no—
198C4×D4—2361no—
199C4×D4—2361no—
200C2×C4⋊C4—591no—
201C2×C4⋊C4C4×D5911yescharacteristic
202C2×C4⋊C4D23611yescharacteristic
203C2×C4⋊C4C22×D5911yescharacteristic
204C2×C22⋊C4—1181no—
205C2×C22⋊C4—1181no—
206C2×C22⋊C4—1181no—
207C2×C22⋊C4—1181no—
208C2×C22⋊C4—1181no—
209C2×C42—1181no—
210C2×C42C4×D5911yescharacteristic
211C2×C42D23611yescharacteristic
212C2×C42C22⋊D5911yescharacteristic
213C4⋊C8—2361no—
214#101—24no—
215C22⋊C8—1182no—
216#97—41no—
217C4×C8—41no—
218C4×C8—2361no—
219#95—591no—
220#95—591no—
221#95—1181no—
222#95—1181no—
223#95—1181no—
224#95—1181no—
225#95—1181no—
226C24—1181no—
227C2×Q8—22no—
228C2×Q8—22no—
229C2×Q8—41no—
230C2×D4—2361no—
231C2×D4—2362no—
232C22×C4—1181no—
233C22×C4—1181no—
234C22×C4—1181no—
235C22×C4—2362no—
236C22×C4—2362no—
237C22×C4—2361no—
238C22×C4—2361no—
239C22×C4(C2×D59)⋊C411yescharacteristic
240C22×C4(C2×D59)⋊C411yescharacteristic
241C22×C4(C2×D59)⋊C411yescharacteristic
242C22×C4(C2×D59)⋊C411yescharacteristic
243C22×C4C2×C4×D5911yescharacteristic
244C22×C4C2×D23611yescharacteristic
245C22×C4C2×C22⋊D5911yescharacteristic
246M16—24no—
247M16—42no—
248C2×C8—42no—
249C2×C8—2362no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—2361no—
254C22⋊C4—2362no—
255C22⋊C4—2362no—
256C22⋊C4—2362no—
257C22⋊C4—2362no—
258C22⋊C4—2362no—
259C42—21no—
260C42—21no—
261C42—21no—
262C42—21no—
263C42—21no—
264C42—21no—
265C42—2362no—
266C23—1182no—
267C23—2361no—
268C23—2361no—
269C23—2361no—
270C23—2361no—
271C23—2361no—
272C23—2361no—
273C23C2×(C2×D59)⋊C411yescharacteristic
274Q8—44no—
275D4—2362no—
276D4—2362no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—4722no—
292C2×C4—4722no—
293C2×C4—2361no—
294C2×C4—2361no—
295C2×C4—2361no—
296C2×C4—2361no—
297C2×C4—2361no—
298C2×C4—2361no—
299C2×C4—2361no—
300C2×C4—2361no—
301C2×C4—2361no—
302C2×C4#1427364111yescharacteristic
303C2×C4#1427364111yescharacteristic
304C2×C4#1427365512yes—
305C2×C4#1427374712yes—
306C8—44no—
307C8—4722no—
308C22—21no—
309C22—21no—
310C22—4721no—
311C22—4721no—
312C22—4721no—
313C22—4721no—
314C22—2362no—
315C22—2362no—
316C22—2362no—
317C22C2×#1427364111yescharacteristic
318C22#1683739611yescharacteristic
319C22C2×#1427365511yescharacteristic
320C4—21no—
321C4—22no—
322C4—22no—
323C4—41no—
324C4—41no—
325C4—41no—
326C4—41no—
327C4—4721no—
328C4—4721no—
329C4—4721no—
330C4#1683719812yes—
331C2—21no—
332C2—21no—
333C2—4722no—
334C2#2142258511yescharacteristic
335C2#2142573811yescharacteristic
336C2C2×#1683719811yescharacteristic
subgroupG/H
centerC2×C4#14273641
commutatorC22×C4×C59C22×C4
FrattiniC2×C4⋊C4C22×D59
FittingC4×#101×C59C2
socleC22×C59C2×#99
prime radicalC22×D236C2×C4
cosocleC2×C4⋊C4×C59C23
hypercenterC4×#101D59
hypocenterC59#11734
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×Q8 ◁ C4×#101

Complements

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

NH ≅ G/Ntype
2. #21422420C2semidirect
3. #21422122C2semidirect
5. C4×#101×C59C2semidirect
164. C4×#101D59semidirect
191. C59#11734semidirect

Maximal subgroups

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

subgroupindex
#214273792
#214224202
#214221222
#214220982
C4×#101×C592
#1173459