Finite Group Databasebeta

← #28477410

Subgroups — #28477410

Order 15104.

604 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 95 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#21425330C212yesmaximal, complemented
2#21425166C211yesmaximal, characteristic
3#21421919C212yesmaximal, complemented
4#21421917C211yesmaximal, characteristic, complemented
5C59×#2683C211yesmaximal, characteristic, complemented
6C22×#11745669—21no—
7C2×#14273724C2211yescharacteristic
8C2×#14273647—21no—
9C2×#14273647—22no—
10C2×#14273647—22no—
11#16837157C2211yescharacteristic
12#16837157C2212yes—
13C2×C22⋊Q8×C59—22no—
14C59×#396—22no—
15C59×#395—21no—
16C59×#390C2211yescharacteristic
17C59×#389C2212yes—
18C59×#386—21no—
19C22×D236—22no—
20C22×D236—21no—
21C2×#11745669—42no—
22C2×#11745669—42no—
23C2×#11745669—42no—
24C22×C59⋊C8—21no—
25C22×C59⋊C8—22no—
26#14273724—42no—
27#14273647—42no—
28#14273647—44no—
29#14273647—44no—
30C22×Q8×C59—21no—
31C22×D4×C59D411yescharacteristic
32C23×C4×C59—21no—
33C22⋊Q8×C59—44no—
34C22⋊Q8×C59—44no—
35C2×C4⋊C4×C59—22no—
36C2×C4⋊C4×C59—22no—
37C2×C4⋊C4×C59D412yes—
38C2×C4⋊C4×C59D412yes—
39C2×C4⋊C4×C59D411yescharacteristic
40C2×C22⋊C4×C59—22no—
41C2×C22⋊C4×C59—22no—
42C2×C22⋊C4×C59—21no—
43C2×C42×C59C2311yescharacteristic
44#95×C59—21no—
45#95×C59—22no—
46C24×C59—21no—
47C23×D59—41no—
48C2×Q8×C59—44no—
49C2×Q8×C59—81no—
50C2×D4×C59—21no—
51C2×D4×C59—41no—
52C2×D4×C59—41no—
53C2×D4×C59SD1612yes—
54C22×C4×C59—22no—
55C22×C4×C59—22no—
56C22×C4×C59—21no—
57C22×C4×C59—44no—
58C22×C4×C59—81no—
59C22×C4×C59C2×D412yes—
60C22×C4×C59C2×D411yescharacteristic
61C2×D236—24no—
62C2×D236—22no—
63C2×D236—41no—
64C2×D236—41no—
65C2×D236—42no—
66C2×D236—42no—
67C2×D236—42no—
68C2×D236—41no—
69C2×D236—42no—
70C2×D236—41no—
71C2×D236—42no—
72C2×D236—41no—
73C4⋊C4×C59—22no—
74C4⋊C4×C59—44no—
75C4⋊C4×C59—41no—
76C4⋊C4×C59—42no—
77C4⋊C4×C59—42no—
78C4⋊C4×C59—42no—
79C4⋊C4×C59D814yescomplemented
80C22⋊C4×C59—44no—
81C22⋊C4×C59—42no—
82C22⋊C4×C59—42no—
83C22⋊C4×C59—81no—
84C42×C59—41no—
85#11745669—48no—
86C2×C59⋊C8—22no—
87C2×C59⋊C8—41no—
88C2×C59⋊C8—41no—
89C2×C59⋊C8—42no—
90C2×C59⋊C8—42no—
91C2×C59⋊C8—42no—
92C23×C59—22no—
93C23×C59—42no—
94C23×C59—41no—
95C23×C59—42no—
96C23×C59—41no—
97C23×C59#12011yescharacteristic
98C22×D59—42no—
99C22×D59—82no—
100C22×D59—81no—
101C22×D59—81no—
102C22×D59—82no—
103Q8×C59—84no—
104D4×C59—42no—
105D4×C59—81no—
106C2×C4×C59—21no—
107C2×C4×C59—22no—
108C2×C4×C59—22no—
109C2×C4×C59—21no—
110C2×C4×C59—24no—
111C2×C4×C59—42no—
112C2×C4×C59—42no—
113C2×C4×C59—42no—
114C2×C4×C59—42no—
115C2×C4×C59—41no—
116C2×C4×C59—42no—
117C2×C4×C59—42no—
118C2×C4×C59—84no—
119C2×C4×C59C2×D812yes—
120C2×C4×C59C2×SD1611yescharacteristic
121C2×C4×C59AGL(1,ℤ/8)12yes—
122C2×C4×C59#13711yescharacteristic
123D236—42no—
124D236—42no—
125D236—44no—
126D236—44no—
127D236—81no—
128D236—81no—
129D236—82no—
130D236—82no—
131C59⋊C8—42no—
132C59⋊C8—82no—
133#7738—591nomaximal, Sylow
134C22×C59—42no—
135C22×C59—44no—
136C22×C59—82no—
137C22×C59—81no—
138C22×C59—81no—
139C22×C59#44712yes—
140C22×C59#44811yescharacteristic
141C22×C59#45311yescharacteristic
142C22×C59#45412yes—
143C22×C59B(4,𝔽2)11yescharacteristic
144C2×D59—82no—
145C2×D59—84no—
146C2×D59—161no—
147C2×D59—162no—
148C2×D59—161no—
149C4×C59—21no—
150C4×C59—21no—
151C4×C59—22no—
152C4×C59—22no—
153C4×C59—44no—
154C4×C59—82no—
155C4×C59—81no—
156#2683D5911yescharacteristic, complemented
157#2041—592no—
158#2011—591no—
159#1271—592no—
160#1270—591no—
161C2×C59—82no—
162C2×C59#159611yescharacteristic
163C2×C59#159812yes—
164C2×C59#161911yescharacteristic
165C2×C59#162712yes—
166C2×C59#165711yescharacteristic
167D59—162no—
168C22×D8—1181no—
169C2×#127—591no—
170C2×C22⋊Q8—22no—
171C2×#102—1181no—
172C2×#102—1182no—
173C2×#102—1182no—
174#396—22no—
175#395—21no—
176#390C2×D5911yescharacteristic
177#389C2×D5912yes—
178#386—21no—
179#336—591no—
180#336—592no—
181C59#773811yescharacteristic, complemented, Sylow
182C22×Q8—21no—
183C22×D4—1181no—
184C22×D4—1182no—
185C22×D4C22⋊D5911yescharacteristic
186C23×C4—21no—
187C2×D8—2362no—
188C2×D8—2362no—
189C2×D8—2362no—
190C22×C8—1181no—
191C22×C8—1182no—
192#127—2362no—
193C22⋊Q8—44no—
194C22⋊Q8—44no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4C22⋊D5912yes—
198C2×C4⋊C4C22⋊D5912yes—
199C2×C4⋊C4C22⋊D5911yescharacteristic
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—21no—
203C2×C42C22×D5911yescharacteristic
204#102—2362no—
205#102—2364no—
206#102—2364no—
207#95—21no—
208#95—22no—
209C24—21no—
210C24—2361no—
211C2×Q8—44no—
212C2×Q8—81no—
213C2×D4—21no—
214C2×D4—41no—
215C2×D4—41no—
216C2×D4—1184no—
217C2×D4—1182no—
218C2×D4—2362no—
219C2×D4—2361no—
220C2×D4—2361no—
221C2×D4—2361no—
222C2×D4—2362no—
223C2×D4—2362no—
224C2×D4—2362no—
225C2×D4—2361no—
226C2×D4—2362no—
227C2×D4—2361no—
228C2×D4#1174567112yes—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—21no—
232C22×C4—44no—
233C22×C4—81no—
234C22×C4C2×C22⋊D5912yes—
235C22×C4C2×C22⋊D5911yescharacteristic
236D8—2368no—
237C2×C8—1182no—
238C2×C8—2362no—
239C2×C8—2362no—
240C2×C8—2362no—
241C2×C8—2361no—
242C2×C8—2361no—
243C4⋊C4—22no—
244C4⋊C4—41no—
245C4⋊C4—42no—
246C4⋊C4—44no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4#1174566914yescomplemented
250C22⋊C4—44no—
251C22⋊C4—42no—
252C22⋊C4—42no—
253C22⋊C4—81no—
254C42—41no—
255C23—22no—
256C23—42no—
257C23—41no—
258C23—42no—
259C23—41no—
260C23—4722no—
261C23—4721no—
262C23—4721no—
263C23—4722no—
264C23—2362no—
265C23#1427369611yescharacteristic
266Q8—84no—
267D4—42no—
268D4—81no—
269D4—4721no—
270D4—4721no—
271D4—4722no—
272D4—4722no—
273D4—2362no—
274D4—2362no—
275D4—2364no—
276D4—2364no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—21no—
281C2×C4—24no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—84no—
290C2×C4C2×#1174566912yes—
291C2×C4C2×#1174567111yescharacteristic
292C2×C4#1427375312yes—
293C2×C4#1427375811yescharacteristic
294C8—4722no—
295C8—2362no—
296C22—44no—
297C22—42no—
298C22—81no—
299C22—81no—
300C22—82no—
301C22—9441no—
302C22—9441no—
303C22—9442no—
304C22—4722no—
305C22—4724no—
306C22#1683753612yes—
307C22#1683754111yescharacteristic
308C22#1683757011yescharacteristic
309C22#1683757512yes—
310C22#1683758911yescharacteristic
311C4—21no—
312C4—21no—
313C4—22no—
314C4—22no—
315C4—44no—
316C4—81no—
317C4—82no—
318C2—82no—
319C2—9442no—
320C2#2142301911yescharacteristic
321C2#2142303112yes—
322C2#2142315011yescharacteristic
323C2#2142319912yes—
324C2#2142339411yescharacteristic
subgroupG/H
centerC23#14273696
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC59×#2683C2
socleC23×C59#120
prime radical#21425166C2
hypercenter#2683D59
hypocenterC59#7738
LCSG ▷ C2×C42×C59 ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2683

Complements

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

NH ≅ G/Ntype
1. #21425330C2semidirect
3. #21421919C2semidirect
4. #21421917C2semidirect
5. C59×#2683C2semidirect
79. C4⋊C4×C59D8semidirect
156. #2683D59semidirect
181. C59#7738semidirect
249. C4⋊C4#11745669semidirect

Maximal subgroups

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

subgroupindex
#214253302
#214251662
#214219192
#214219172
C59×#26832
#773859