Finite Group Databasebeta

← #28508291

Subgroups — #28508291

Order 15104.

446 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14273672C211yesmaximal, characteristic
2C2×#16837207C211yesmaximal, characteristic, complemented
3C2×#16837204C211yesmaximal, characteristic
4#21424962C211yesmaximal, characteristic
5#21424957C211yesmaximal, characteristic, complemented
6#21422188C211yesmaximal, characteristic, complemented
7C2×C59×C4⋊C16C211yesmaximal, characteristic, complemented
8C22×Dic236C2211yescharacteristic
9C22×D472C2211yescharacteristic
10C2×#14273672—22no—
11C2×#14273672—22no—
12C2×#14273672—22no—
13C2×#14273661C2211yescharacteristic
14C2×#14273661C2211yescharacteristic
15C2×#14273652—21no—
16C2×#14273649—21no—
17#16837308—21no—
18#16837304—21no—
19#16837207—22no—
20#16837204—22no—
21C22×C16×C59C2211yescharacteristic
22C22×C16×C59C2211yescharacteristic
23C2×C4×C8×C59C2211yescharacteristic
24C59×C4⋊C16—22no—
25C22×Dic118—21no—
26C22×D236—21no—
27C2×Dic236—21no—
28C2×Dic236—21no—
29C2×Dic236—21no—
30C2×Dic236—21no—
31C2×Dic236—21no—
32C2×Dic236D412yes—
33C2×D472—21no—
34C2×D472—21no—
35C2×D472—21no—
36C2×D472—21no—
37C2×D472—21no—
38C2×D472D412yes—
39C2×C4⋊Dic59—21no—
40C2×C4⋊Dic59—21no—
41C2×(C2×D59)⋊C4—41no—
42C2×(C2×D59)⋊C4—41no—
43#14273672—28no—
44#14273661—21no—
45#14273661—21no—
46#14273661—21no—
47#14273661—21no—
48#14273652—42no—
49#14273649—42no—
50C59⋊#95—41no—
51C59⋊#95—41no—
52C22×C8×C59D411yescharacteristic
53C22×C8×C59C2311yescharacteristic
54C2×C42×C59D411yescharacteristic
55C2×C16×C59—21no—
56C2×C16×C59—21no—
57C2×C16×C59—21no—
58C2×C16×C59—21no—
59C2×C16×C59—21no—
60C2×C16×C59D412yes—
61C4×C8×C59—21no—
62C4×C8×C59—21no—
63C23×D59—41no—
64C22×C4×C59D811yescharacteristic
65C22×C4×C59D811yescharacteristic
66C22×C4×C59C2×D411yescharacteristic
67C22×Dic59—41no—
68C22×Dic59—41no—
69C22×Dic59—41no—
70C2×D236—22no—
71C2×D236—41no—
72C2×D236—41no—
73C2×D236—41no—
74C2×D236—41no—
75C2×D236—41no—
76C2×Dic118—22no—
77C2×Dic118—41no—
78C2×Dic118—41no—
79C2×Dic118—41no—
80C2×Dic118—41no—
81C2×Dic118—41no—
82C2×C8×C59—21no—
83C2×C8×C59—21no—
84C2×C8×C59—21no—
85C2×C8×C59C2×D411yescharacteristic
86C2×C8×C59C2×D411yescharacteristic
87C2×C8×C59C2×D411yescharacteristic
88C2×C8×C59#4111yescharacteristic
89C2×C8×C59#4111yescharacteristic
90C2×C8×C59#4111yescharacteristic
91C42×C59—21no—
92C42×C59—21no—
93(C2×D59)⋊C4—82no—
94(C2×D59)⋊C4—82no—
95C4⋊Dic59—41no—
96C4⋊Dic59—41no—
97C4⋊Dic59—41no—
98C4⋊Dic59—41no—
99Dic236—22no—
100Dic236—22no—
101Dic236—41no—
102Dic236—41no—
103D472—22no—
104D472—22no—
105D472—41no—
106D472—41no—
107C16×C59—22no—
108C16×C59—41no—
109C23×C59C2×D811yescharacteristic
110C22×D59—42no—
111C22×D59—81no—
112C22×D59—81no—
113C22×D59—81no—
114C22×D59—81no—
115C22×D59—81no—
116C22×D59—81no—
117C2×C4×C59—21no—
118C2×C4×C59—21no—
119C2×C4×C59—21no—
120C2×C4×C59—21no—
121C2×C4×C59—21no—
122C2×C4×C59SD3212yescomplemented
123C2×C4×C59#12111yescharacteristic
124C2×C4×C59#12111yescharacteristic
125C2×C4×C59#12111yescharacteristic
126C2×C4×C59#13511yescharacteristic
127C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
128C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
129C2×Dic59—42no—
130C2×Dic59—81no—
131C2×Dic59—81no—
132C2×Dic59—81no—
133C2×Dic59—81no—
134C2×Dic59—81no—
135C2×Dic59—81no—
136C2×Dic59—81no—
137C2×Dic59—81no—
138D236—42no—
139D236—42no—
140D236—81no—
141D236—81no—
142Dic118—42no—
143Dic118—42no—
144Dic118—81no—
145Dic118—81no—
146C8×C59—41no—
147C8×C59#12011yescharacteristic
148C8×C59#12311yescharacteristic
149C8×C59#12311yescharacteristic
150C8×C59#12411yescharacteristic
151#13449—591nomaximal, Sylow
152C22×C59#45911yescharacteristic
153C22×C59#45911yescharacteristic
154C22×C59#46611yescharacteristic
155C22×C59C2×SD3211yescharacteristic
156C22×C59#50811yescharacteristic
157C22×C59#50911yescharacteristic
158C22×C59#51011yescharacteristic
159C2×D59—82no—
160C2×D59—82no—
161C2×D59—82no—
162C2×D59—161no—
163C2×D59—161no—
164C2×D59—161no—
165C2×D59—161no—
166C4×C59—22no—
167C4×C59—41no—
168C4×C59#39411yescharacteristic
169C4×C59#46311yescharacteristic
170C4×C59#46311yescharacteristic
171C4×C59#46911yescharacteristic
172Dic59—82no—
173Dic59—161no—
174Dic59—161no—
175C22×SD32—591no—
176C2×C4⋊C16D5911yescharacteristic, complemented
177C2×#358—591no—
178C2×#357—591no—
179#1970—591no—
180#1969—591no—
181#1380—591no—
182C2×C59#205411yescharacteristic
183C2×C59#220711yescharacteristic
184C2×C59#220811yescharacteristic
185C2×C59#221011yescharacteristic
186C2×C59#221111yescharacteristic
187C2×C59#221711yescharacteristic
188C2×C59#222011yescharacteristic
189D59—162no—
190C22×Q16—591no—
191C22×D8—591no—
192C2×SD32—1182no—
193C2×SD32—1182no—
194C2×SD32—1182no—
195C22×C16C2×D5911yescharacteristic
196C22×C16C2×D5911yescharacteristic
197C2×#107—591no—
198C2×#107—591no—
199C2×#103—1181no—
200C2×#102—1181no—
201C2×C4×C8C2×D5911yescharacteristic
202#391—1181no—
203#390—1181no—
204C4⋊C16—22no—
205#358—1182no—
206#357—1182no—
207C59#1344911yescharacteristic, complemented, Sylow
208C22×Q8—1181no—
209C22×D4—1181no—
210C2×Q16—592no—
211C2×Q16—1181no—
212C2×Q16—1181no—
213C2×Q16—1181no—
214C2×Q16—1181no—
215C2×Q16—1181no—
216C2×D8—592no—
217C2×D8—1181no—
218C2×D8—1181no—
219C2×D8—1181no—
220C2×D8—1181no—
221C2×D8—1181no—
222C22×C8D23611yescharacteristic
223C22×C8C22×D5911yescharacteristic
224C2×C4⋊C4—1181no—
225C2×C4⋊C4—1181no—
226C2×C22⋊C4—2361no—
227C2×C22⋊C4—2361no—
228C2×C42D23611yescharacteristic
229SD32—1188no—
230C2×C16—21no—
231C2×C16—21no—
232C2×C16—21no—
233C2×C16—21no—
234C2×C16—21no—
235C2×C16C22⋊D5912yes—
236#107—1181no—
237#107—1181no—
238#107—1181no—
239#107—1181no—
240#103—2362no—
241#102—2362no—
242C4×C8—21no—
243C4×C8—21no—
244#95—2361no—
245#95—2361no—
246C24—2361no—
247C2×Q8—1182no—
248C2×Q8—2361no—
249C2×Q8—2361no—
250C2×Q8—2361no—
251C2×Q8—2361no—
252C2×Q8—2361no—
253C2×D4—1182no—
254C2×D4—2361no—
255C2×D4—2361no—
256C2×D4—2361no—
257C2×D4—2361no—
258C2×D4—2361no—
259C22×C4—2361no—
260C22×C4—2361no—
261C22×C4—2361no—
262C22×C4D47211yescharacteristic
263C22×C4D47211yescharacteristic
264C22×C4C2×D23611yescharacteristic
265Q16—1182no—
266Q16—1182no—
267Q16—2361no—
268Q16—2361no—
269D8—1182no—
270D8—1182no—
271D8—2361no—
272D8—2361no—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8#1174568511yescharacteristic
277C2×C8D4×D5911yescharacteristic
278C2×C8D4×D5911yescharacteristic
279C2×C8#1174568911yescharacteristic
280C2×C8#1174568911yescharacteristic
281C2×C8C2×C22⋊D5911yescharacteristic
282C4⋊C4—2361no—
283C4⋊C4—2361no—
284C4⋊C4—2361no—
285C4⋊C4—2361no—
286C22⋊C4—4722no—
287C22⋊C4—4722no—
288C42—21no—
289C42—21no—
290C16—22no—
291C16—41no—
292C23—4721no—
293C23—4721no—
294C23—4721no—
295C23—4721no—
296C23—4721no—
297C23—4721no—
298C23—2362no—
299C23C2×D47211yescharacteristic
300Q8—4721no—
301Q8—4721no—
302Q8—2362no—
303Q8—2362no—
304D4—4721no—
305D4—4721no—
306D4—2362no—
307D4—2362no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—4721no—
314C2×C4—4721no—
315C2×C4—4721no—
316C2×C4—4721no—
317C2×C4—4721no—
318C2×C4—4721no—
319C2×C4—4721no—
320C2×C4—4721no—
321C2×C4—2362no—
322C2×C4#1427367212yescomplemented
323C2×C4#1427370111yescharacteristic
324C2×C4#1427370211yescharacteristic
325C2×C4#1427370211yescharacteristic
326C2×C4#1427374911yescharacteristic
327C2×C4#1427375511yescharacteristic
328C2×C4#1427375511yescharacteristic
329C8—41no—
330C8#1427369511yescharacteristic
331C8#1427371211yescharacteristic
332C8#1427371211yescharacteristic
333C8#1427371611yescharacteristic
334C22—9441no—
335C22—9441no—
336C22—9441no—
337C22—9441no—
338C22—4722no—
339C22—4722no—
340C22—4722no—
341C22#1683759811yescharacteristic
342C22#1683759811yescharacteristic
343C22#1683763811yescharacteristic
344C22C2×#1427367211yescharacteristic
345C22#1683783411yescharacteristic
346C22#1683784011yescharacteristic
347C22#1683784511yescharacteristic
348C4—22no—
349C4—41no—
350C4—9441no—
351C4—9441no—
352C4—4722no—
353C4#1683731911yescharacteristic
354C4#1683761811yescharacteristic
355C4#1683761811yescharacteristic
356C4#1683765511yescharacteristic
357C2—9442no—
358C2#2142540711yescharacteristic
359C2#2142613711yescharacteristic
360C2#2142614211yescharacteristic
361C2#2142615211yescharacteristic
362C2#2142615911yescharacteristic
363C2#2142619211yescharacteristic
364C2#2142620911yescharacteristic
subgroupG/H
centerC23C2×D472
commutator, cosocleC22×C8×C59C23
FrattiniC22×C8C22×D59
FittingC2×C59×C4⋊C16C2
socleC23×C59C2×D8
prime radicalC22×D472C22
hypercenterC2×C4⋊C16D59
hypocenterC59#13449
LCSG ▷ C22×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4⋊C16

Complements

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

NH ≅ G/Ntype
2. C2×#16837207C2semidirect
5. #21424957C2semidirect
6. #21422188C2semidirect
7. C2×C59×C4⋊C16C2semidirect
122. C2×C4×C59SD32semidirect
176. C2×C4⋊C16D59semidirect
207. C59#13449semidirect
322. C2×C4#14273672semidirect

Maximal subgroups

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

subgroupindex
C22×#142736722
C2×#168372072
C2×#168372042
#214249622
#214249572
#214221882
C2×C59×C4⋊C162
#1344959