Finite Group Databasebeta

← #28743407

Subgroups — #28743407

Order 15104.

534 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21434636C211yesmaximal, characteristic, complemented
2#21432972C211yesmaximal, characteristic
3#21426258C212yesmaximal, complemented
4#21426255C212yesmaximal, complemented
5#21426241C212yesmaximal, complemented
6#21426236C212yesmaximal, complemented
7#21425885C211yesmaximal, characteristic, complemented
8C2×#16837230C211yesmaximal, characteristic, complemented
9#21425854C211yesmaximal, characteristic, complemented
10#21425808C211yesmaximal, characteristic
11C59×#2138C211yesmaximal, characteristic
12#16838030—21no—
13#16837885—21no—
14C2×#14273667C2211yescharacteristic
15#16837809—22no—
16#16837806C2212yes—
17#16837803C2212yes—
18#16837503—22no—
19Q16×Dic59C2212yes—
20D8×Dic59C2212yes—
21#16837445—21no—
22C2×#14273661C2211yescharacteristic
23C2×#14273661C2211yescharacteristic
24#16837403—21no—
25#16837357C2211yescharacteristic
26#16837230C2212yes—
27#16837230C2212yes—
28#16837229C2212yes—
29#16837227C2212yes—
30Dic59⋊C16C2212yes—
31Dic59⋊C16C2212yes—
32#16837206C2212yes—
33#16837203C2212yes—
34C2×#135×C59C2211yescharacteristic
35C22×C16×C59C2211yescharacteristic
36C59×#427C2211yescharacteristic
37C59×#358C2211yescharacteristic
38C59×#358C2211yescharacteristic
39C59×#357C2211yescharacteristic
40C59×#357C2211yescharacteristic
41C2×#11745664D411yescharacteristic
42#14273727—41no—
43#14273726—22no—
44#14273719—41no—
45#14273718—22no—
46#14273704—42no—
47Q8×Dic59—22no—
48D4×Dic59—22no—
49D4×Dic59—42no—
50#14273686—41no—
51#14273686D411yescharacteristic
52C2×C4⋊Dic59—21no—
53C2×C4⋊Dic59—21no—
54C2×C4⋊Dic59—41no—
55#14273667—22no—
56C2×C59⋊C16C2312yes—
57#14273661—21no—
58#14273661—21no—
59#14273661C2312yes—
60#14273661C2312yes—
61#14273660—22no—
62#14273658—22no—
63#14273651—22no—
64#14273648—22no—
65#14273637D412yes—
66C8×Dic59C2312yes—
67C2×#41×C59—21no—
68#135×C59—22no—
69#135×C59Q814yescomplemented
70C2×Q16×C59C2311yescharacteristic
71C2×SD16×C59—21no—
72C2×D8×C59C2311yescharacteristic
73C22×C8×C59C2311yescharacteristic
74#117×C59—21no—
75C2×C16×C59—22no—
76C2×C16×C59C2311yescharacteristic
77C2×C16×C59C2311yescharacteristic
78#107×C59C2311yescharacteristic
79#107×C59C2311yescharacteristic
80#106×C59—21no—
81#41×C59—24no—
82#41×C59—42no—
83C2×Q8×C59—21no—
84C2×D4×C59—21no—
85C2×D4×C59—41no—
86C22×C4×C59—41no—
87C22×C4×C59C2×D411yescharacteristic
88C22×Dic59—41no—
89C22×Dic59—41no—
90C22×Dic59—42no—
91C2×Dic118—42no—
92Q16×C59—21no—
93Q16×C59C2×Q812yes—
94SD16×C59—24no—
95D8×C59—21no—
96D8×C59C2×Q812yes—
97C2×C8×C59C2×D411yescharacteristic
98C2×C8×C59C2×Q812yes—
99C2×C8×C59#4112yes—
100C2×C8×C59C2411yescharacteristic
101C4⋊C4×C59—21no—
102C4⋊C4×C59—21no—
103C22⋊C4×C59—41no—
104C42×C59—41no—
105C22⋊Dic59—42no—
106C22⋊Dic59D812yes—
107C4⋊Dic59—22no—
108C4⋊Dic59—22no—
109C4⋊Dic59—41no—
110C4⋊Dic59—41no—
111C4⋊Dic59—42no—
112C4⋊Dic59—42no—
113C4⋊Dic59D811yescharacteristic
114C4⋊Dic59D811yescharacteristic
115Dic59⋊C4—42no—
116Dic59⋊C4—42no—
117C4×Dic59—42no—
118C4×Dic59C2×D412yes—
119#11745664—22no—
120C2×C59⋊C8C2×D412yes—
121C16×C59—21no—
122C16×C59—21no—
123C59⋊C16—22no—
124C23×C59—41no—
125C23×C59C2×D811yescharacteristic
126Q8×C59—22no—
127Q8×C59—41no—
128D4×C59—22no—
129D4×C59—44no—
130D4×C59—41no—
131C2×C4×C59—41no—
132C2×C4×C59—44no—
133C2×C4×C59—41no—
134C2×C4×C59—41no—
135C2×C4×C59C22⋊Q812yes—
136C2×C4×C59C2×D811yescharacteristic
137C2×C4×C59#13712yes—
138C2×C4×C59C22×D411yescharacteristic
139C2×Dic59—42no—
140C2×Dic59—42no—
141C2×Dic59—162no—
142C2×Dic59—161no—
143C2×Dic59—161no—
144C2×Dic59C2×D812yes—
145C2×Dic59C2×D812yes—
146Dic118—162no—
147C8×C59C22⋊Q812yes—
148C8×C59C22×Q811yescharacteristic
149C8×C59C2×#4111yescharacteristic
150C59⋊C8—22no—
151#33634—591nomaximal, Sylow
152C22×C59—21no—
153C22×C59—21no—
154C22×C59—42no—
155C22×C59—81no—
156C22×C59—81no—
157C22×C59#47412yes—
158C22×C59C22×D811yescharacteristic
159C4×C59—42no—
160C4×C59—81no—
161C4×C59—81no—
162C4×C59#47412yes—
163C4×C59C2×C22⋊Q811yescharacteristic
164C4×C59C2×#13711yescharacteristic
165Dic59—22no—
166Dic59—22no—
167Dic59—82no—
168Dic59—82no—
169#3164—591no—
170#2951—591no—
171#2229—592no—
172#2228—592no—
173#2226—592no—
174#2225—592no—
175#2159—591no—
176C2×#367—591no—
177#2151—591no—
178#2139—591no—
179#2138D5911yescharacteristic
180C2×C59—22no—
181C2×C59—82no—
182C2×C59C2×#47411yescharacteristic
183C2×C59#341611yescharacteristic
184C2×C59#341611yescharacteristic
185C2×#135C2×D5911yescharacteristic
186#544—1181no—
187#520—1181no—
188C2×M32—591no—
189C22×C16C2×D5911yescharacteristic
190#501—1182no—
191#500—592no—
192#499—592no—
193#440—1182no—
194C4×Q16—592no—
195C4×D8—592no—
196#428—1181no—
197#427C2×D5911yescharacteristic
198C2×#107—591no—
199C2×#107—591no—
200#417—1181no—
201#405—591no—
202#367—592no—
203#367—592no—
204#367—592no—
205#366—592no—
206C4⋊C16—592no—
207C4⋊C16—592no—
208#358—592no—
209#358C2×D5911yescharacteristic
210#358C2×D5911yescharacteristic
211#357—592no—
212#357C2×D5911yescharacteristic
213#357C2×D5911yescharacteristic
214C59#3363411yescharacteristic, complemented, Sylow
215C2×#41—21no—
216#135—22no—
217#135Dic11814yescomplemented
218C2×Q16C22×D5911yescharacteristic
219C2×SD16—21no—
220C2×D8C22×D5911yescharacteristic
221C2×M16—591no—
222C22×C8C22×D5911yescharacteristic
223C4⋊Q8—1182no—
224C4⋊Q8—2361no—
225#125—1182no—
226#125—2361no—
227C22⋊Q8—2362no—
228C4×Q8—1182no—
229C4×D4—1182no—
230C4×D4—2362no—
231#117—21no—
232#117—591no—
233#117—2361no—
234C2×C4⋊C4—1181no—
235C2×C4⋊C4—1181no—
236C2×C4⋊C4—2361no—
237M32—1182no—
238C2×C16—22no—
239C2×C16—592no—
240C2×C16C22×D5911yescharacteristic
241C2×C16C22×D5911yescharacteristic
242#107—592no—
243#107—592no—
244#107—1182no—
245#107—1181no—
246#107—1181no—
247#107C22×D5911yescharacteristic
248#107C22×D5911yescharacteristic
249#106—21no—
250#106—1182no—
251#103—1182no—
252#102—1182no—
253#97—592no—
254C4×C8—592no—
255#41—24no—
256#41—42no—
257C2×Q8—21no—
258C2×Q8—2362no—
259C2×D4—21no—
260C2×D4—41no—
261C22×C4—41no—
262C22×C4—2362no—
263C22×C4—2361no—
264C22×C4—2361no—
265C22×C4D4×D5911yescharacteristic
266Q16—21no—
267Q16C2×Dic11812yes—
268SD16—24no—
269D8—21no—
270D8C2×Dic11812yes—
271M16—1182no—
272C2×C8—592no—
273C2×C8C2×Dic11812yes—
274C2×C8D4×D5911yescharacteristic
275C2×C8#1174568712yes—
276C2×C8C23×D5911yescharacteristic
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—591no—
280C4⋊C4—591no—
281C4⋊C4—1182no—
282C4⋊C4—1182no—
283C4⋊C4—2362no—
284C4⋊C4—2362no—
285C4⋊C4—2362no—
286C4⋊C4—2362no—
287C4⋊C4—2361no—
288C4⋊C4—2361no—
289C22⋊C4—41no—
290C22⋊C4—592no—
291C22⋊C4—2362no—
292C42—41no—
293C42—592no—
294C42—2362no—
295C16—21no—
296C16—21no—
297C16—1182no—
298C23—41no—
299C23D8×D5911yescharacteristic
300Q8—22no—
301Q8—41no—
302Q8—9442no—
303D4—22no—
304D4—44no—
305D4—41no—
306C2×C4—41no—
307C2×C4—44no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—592no—
311C2×C4—592no—
312C2×C4—9442no—
313C2×C4—9441no—
314C2×C4—9441no—
315C2×C4—2362no—
316C2×C4—2362no—
317C2×C4#1427370412yes—
318C2×C4D8×D5911yescharacteristic
319C2×C4#1427375612yes—
320C2×C4C2×D4×D5911yescharacteristic
321C8—1182no—
322C8#1427370412yes—
323C8C22×Dic11811yescharacteristic
324C8C2×#1174568711yescharacteristic
325C22—21no—
326C22—21no—
327C22—42no—
328C22—81no—
329C22—81no—
330C22#1683768012yes—
331C22C2×D8×D5911yescharacteristic
332C4—42no—
333C4—81no—
334C4—81no—
335C4—1182no—
336C4—1182no—
337C4—4722no—
338C4—4722no—
339C4#1683768012yes—
340C4C2×#1427370411yescharacteristic
341C4C2×#1427375611yescharacteristic
342C2—22no—
343C2—82no—
344C2C2×#1683768011yescharacteristic
345C2#2143723911yescharacteristic
346C2#2143723911yescharacteristic
subgroupG/H
centerC22C2×D8×D59
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC59×#2138C2
socleC22×C59C22×D8
prime radicalC2×#135×C59C22
hypercenter#2138D59
hypocenterC59#33634
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2138

Complements

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

NH ≅ G/Ntype
1. #21434636C2semidirect
3. #21426258C2semidirect
4. #21426255C2semidirect
5. #21426241C2semidirect
6. #21426236C2semidirect
7. #21425885C2semidirect
8. C2×#16837230C2semidirect
9. #21425854C2semidirect
69. #135×C59Q8semidirect
214. C59#33634semidirect
217. #135Dic118semidirect

Maximal subgroups

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

subgroupindex
#214346362
#214329722
#214262582
#214262552
#214262412
#214262362
#214258852
C2×#168372302
#214258542
#214258082
C59×#21382
#3363459