Finite Group Databasebeta

← #28596557

Subgroups — #28596557

Order 15104.

974 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837526C211yesmaximal, characteristic, complemented
2C2×D4×C59⋊C8C211yesmaximal, characteristic
3C4×#14273714C211yesmaximal, characteristic
4#21425032C211yesmaximal, characteristic, complemented
5#21424699C214yesmaximal, complemented
6#21424212C211yesmaximal, characteristic, complemented
7#21423870C214yesmaximal, complemented
8C4×C8×Dic59C211yesmaximal, characteristic, complemented
9C59×#1954C211yesmaximal, characteristic
10C23×C59⋊C8C2212yes—
11C2×#14273714C411yescharacteristic
12C2×C4×Dic118C2211yescharacteristic
13#16837526—24no—
14D4×C59⋊C8—24no—
15C2×C4⋊(C59⋊C8)C2211yescharacteristic
16C2×Dic59⋊C8C2212yes—
17C2×C22⋊(C59⋊C8)C2212yes—
18C2×C4×C59⋊C8C2211yescharacteristic
19C2×C8×Dic59C2212yes—
20C2×C8×Dic59C2212yes—
21C2×C8×Dic59C2212yes—
22#16837307C411yescharacteristic
23#16837305C411yescharacteristic
24#16837296C414yes—
25#16837267C411yescharacteristic
26C4×C22⋊Dic59C2214yes—
27C42×Dic59C2211yescharacteristic
28#16837156C2214yes—
29#16837156C2214yes—
30C2×C4×D4×C59C2211yescharacteristic
31C2×C4⋊C8×C59C2211yescharacteristic
32C2×C22⋊C8×C59C2214yes—
33C2×C4×C8×C59C2211yescharacteristic
34C22×Dic118C2×C411yescharacteristic
35C22×C59⋊C8—24no—
36C22×C59⋊C8—28no—
37C22×C59⋊C8C2312yes—
38C22×C59⋊C8C2311yescharacteristic
39C22×C59⋊C8C2311yescharacteristic
40#14273714C818yescomplemented
41C4×Dic118—22no—
42C4×Dic118—22no—
43C2×C4⋊Dic59C2×C411yescharacteristic
44C2×Dic59⋊C4C2×C412yes—
45C2×C22⋊Dic59C2×C414yes—
46C2×C22⋊Dic59C2×C414yes—
47C2×C4×Dic59C2×C412yes—
48C2×C4×Dic59C2×C411yescharacteristic
49C2×C4×Dic59C2×C411yescharacteristic
50C2×C4×Dic59C2312yes—
51C2×C4×Dic59C2312yes—
52C4⋊(C59⋊C8)—22no—
53Dic59⋊C8—24no—
54C22⋊(C59⋊C8)—24no—
55C4×C59⋊C8D414yes—
56C8×Dic59—24no—
57C8×Dic59—24no—
58C8×Dic59—24no—
59C59⋊#95C2×C414yes—
60C59⋊#95C2×C414yes—
61C22×D4×C59C2×C411yescharacteristic
62C23×C4×C59C2312yes—
63C22×C8×C59C2312yes—
64C22×C8×C59C2312yes—
65C4×D4×C59—22no—
66C4×D4×C59—22no—
67C2×C4⋊C4×C59C2×C411yescharacteristic
68C2×C22⋊C4×C59C2×C412yes—
69C2×C42×C59C2311yescharacteristic
70C4⋊C8×C59—22no—
71C22⋊C8×C59—28no—
72C4×C8×C59—22no—
73C24×C59C22×C412yes—
74C2×D4×C59—22no—
75C2×D4×C59—22no—
76C2×D4×C59C2×C814yes—
77C22×C4×C59—22no—
78C22×C4×C59—24no—
79C22×C4×C59—22no—
80C22×C4×C59—24no—
81C22×C4×C59C22×C412yes—
82C22×C4×C59C22×C411yescharacteristic
83C22×C4×C59C22×C411yescharacteristic
84C22×C4×C59C2411yescharacteristic
85C22×Dic59C22×C412yes—
86C22×Dic59C22×C412yes—
87C22×Dic59C22×C412yes—
88C22×Dic59C22×C412yes—
89C2×Dic118—22no—
90C2×Dic118—22no—
91C2×Dic118C2×C814yescomplemented
92C2×C8×C59—22no—
93C2×C8×C59—22no—
94C2×C8×C59—22no—
95C2×C8×C59—24no—
96C2×C8×C59#4114yes—
97C4⋊C4×C59—22no—
98C22⋊C4×C59—24no—
99C42×C59C2×D412yes—
100C42×C59#4112yes—
101C22⋊Dic59—28no—
102C22⋊Dic59C2×C8116yescomplemented
103C4⋊Dic59—22no—
104Dic59⋊C4—24no—
105C4×Dic59—22no—
106C4×Dic59—22no—
107C4×Dic59—22no—
108C4×Dic59—22no—
109C4×Dic59—22no—
110C4×Dic59—24no—
111C4×Dic59C2×C814yescomplemented
112C2×C59⋊C8—22no—
113C2×C59⋊C8—22no—
114C2×C59⋊C8—24no—
115C2×C59⋊C8—48no—
116C2×C59⋊C8C2×D412yes—
117C2×C59⋊C8C2×D412yes—
118C2×C59⋊C8#4114yes—
119C23×C59—22no—
120C23×C59—24no—
121C23×C59—24no—
122C23×C59—22no—
123C23×C59C22×C814yes—
124C23×C59C23×C411yescharacteristic
125D4×C59—28no—
126C2×C4×C59—22no—
127C2×C4×C59—22no—
128C2×C4×C59—24no—
129C2×C4×C59—42no—
130C2×C4×C59—44no—
131C2×C4×C59—42no—
132C2×C4×C59C4×D412yes—
133C2×C4×C59C4×D412yes—
134C2×C4×C59C22×C812yes—
135C2×C4×C59#13112yes—
136C2×C4×C59C22×D411yescharacteristic
137C2×C4×C59C2×#4112yes—
138C2×C4×C59C2×#4111yescharacteristic
139C2×C4×C59C2×#4112yes—
140C2×Dic59—22no—
141C2×Dic59—22no—
142C2×Dic59—22no—
143C2×Dic59—22no—
144C2×Dic59—22no—
145C2×Dic59—22no—
146C2×Dic59—22no—
147C2×Dic59—24no—
148C2×Dic59C22×C814yes—
149C2×Dic59C22×C814yes—
150C2×Dic59#13114yes—
151Dic118—28no—
152C8×C59—24no—
153C8×C59—42no—
154C59⋊C8—22no—
155C59⋊C8—42no—
156C59⋊C8#12414yescomplemented
157C8×#124—591nomaximal, Sylow
158C22×C59—24no—
159C22×C59—28no—
160C22×C59—42no—
161C22×C59—44no—
162C22×C59—42no—
163C22×C59C2×C4×D411yescharacteristic
164C22×C59C4×#4112yes—
165C22×C59C23×C811yescharacteristic
166C22×C59C2×#13112yes—
167C22×C59C2×#13111yescharacteristic
168C4×C59—22no—
169C4×C59—42no—
170C4×C59C8×D414yes—
171C4×C59C2×#12411yescharacteristic
172C4×C59#52912yes—
173C4×C59#53211yescharacteristic
174Dic59—24no—
175Dic59—24no—
176Dic59—24no—
177Dic59—42no—
178C2×C8×Q8—591no—
179C2×C8×D4—591no—
180C4×#124—591no—
181#1984—591no—
182#1954D5911yescharacteristic
183#1923—594no—
184#1831—591no—
185C8×C22⋊C4—594no—
186C42×C8—591no—
187C2×C59—44no—
188C2×C59C4×#12411yescharacteristic
189C2×C59C2×C8×D411yescharacteristic
190C2×C59#293111yescharacteristic
191C2×C59C8×#4112yes—
192C2×C59#296612yes—
193C23×C8—592no—
194C2×#124—591no—
195C2×C4×Q8—591no—
196C2×C4×D4C2×D5911yescharacteristic
197C8×Q8—1184no—
198C8×D4—1184no—
199C2×C4⋊C8—592no—
200C2×C4⋊C8—591no—
201C2×C4⋊C8C2×D5911yescharacteristic
202C2×C22⋊C8—592no—
203C2×C22⋊C8C2×D5914yes—
204C2×C4×C8—591no—
205C2×C4×C8—592no—
206C2×C4×C8—592no—
207C2×C4×C8—592no—
208C2×C4×C8C2×D5911yescharacteristic
209#391—591no—
210#390—591no—
211#388—594no—
212#382—591no—
213C4×C22⋊C4—594no—
214C43—591no—
215#336—594no—
216#336—594no—
217C59C8×#12411yescharacteristic, complemented, Sylow
218C22×Q8—591no—
219C22×D4C4×D5911yescharacteristic
220C23×C4C22×D5912yes—
221C22×C8—591no—
222C22×C8—591no—
223C22×C8—592no—
224C22×C8—1184no—
225C22×C8—1188no—
226C22×C8C22×D5912yes—
227C22×C8C22×D5912yes—
228#124—598no—
229C4×Q8—1182no—
230C4×Q8—1182no—
231C4×D4—22no—
232C4×D4—22no—
233C2×C4⋊C4—591no—
234C2×C4⋊C4—592no—
235C2×C4⋊C4C4×D5911yescharacteristic
236C2×C22⋊C4—594no—
237C2×C22⋊C4—594no—
238C2×C22⋊C4C4×D5912yes—
239C2×C42—591no—
240C2×C42—592no—
241C2×C42—591no—
242C2×C42—592no—
243C2×C42—592no—
244C2×C42C22×D5911yescharacteristic
245C4⋊C8—22no—
246C4⋊C8—1184no—
247C4⋊C8—1182no—
248C22⋊C8—28no—
249C22⋊C8—1184no—
250C4×C8—22no—
251C4×C8—594no—
252C4×C8—1184no—
253C4×C8—1184no—
254C4×C8—1184no—
255#95—594no—
256#95—594no—
257C24C2×C4×D5912yes—
258C2×Q8—594no—
259C2×Q8—1182no—
260C2×Q8—1182no—
261C2×D4—22no—
262C2×D4—22no—
263C2×D4C8×D5914yes—
264C22×C4—22no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—22no—
268C22×C4—592no—
269C22×C4—592no—
270C22×C4—592no—
271C22×C4—592no—
272C22×C4C2×C4×D5912yes—
273C22×C4C2×C4×D5911yescharacteristic
274C22×C4C2×C4×D5911yescharacteristic
275C22×C4C23×D5911yescharacteristic
276C2×C8—22no—
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—24no—
280C2×C8—592no—
281C2×C8—594no—
282C2×C8—592no—
283C2×C8—1182no—
284C2×C8—1182no—
285C2×C8—1184no—
286C2×C8—2368no—
287C2×C8#1174568714yes—
288C4⋊C4—22no—
289C4⋊C4—1184no—
290C4⋊C4—1182no—
291C22⋊C4—24no—
292C22⋊C4—5916no—
293C22⋊C4—1188no—
294C42—594no—
295C42—1184no—
296C42—1182no—
297C42—1182no—
298C42—1182no—
299C42—1182no—
300C42—1182no—
301C42D4×D5912yes—
302C42#1174568712yes—
303C23—24no—
304C23—22no—
305C23—22no—
306C23—24no—
307C23C2×C8×D5914yes—
308C23C22×C4×D5911yescharacteristic
309Q8—1188no—
310D4—28no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—24no—
314C2×C4—42no—
315C2×C4—44no—
316C2×C4—42no—
317C2×C4—594no—
318C2×C4—594no—
319C2×C4—594no—
320C2×C4—1182no—
321C2×C4—1182no—
322C2×C4—1182no—
323C2×C4—1182no—
324C2×C4—1182no—
325C2×C4—1182no—
326C2×C4—1184no—
327C2×C4—1182no—
328C2×C4(C22⋊D59)⋊C412yes—
329C2×C4(C22⋊D59)⋊C412yes—
330C2×C4C2×C8×D5912yes—
331C2×C4#1427373512yes—
332C2×C4C2×D4×D5911yescharacteristic
333C2×C4#41×D5912yes—
334C2×C4C2×#1174568711yescharacteristic
335C2×C4C2×#1174568712yes—
336C8—24no—
337C8—42no—
338C8—594no—
339C8—1182no—
340C8—2362no—
341C22—24no—
342C22—28no—
343C22—42no—
344C22—42no—
345C22—44no—
346C22C2×(C22⋊D59)⋊C411yescharacteristic
347C22C4×#1174568712yes—
348C22C22×C8×D5911yescharacteristic
349C22#131×D5912yes—
350C22C2×#1427373511yescharacteristic
351C4—22no—
352C4—42no—
353C4—1184no—
354C4—1184no—
355C4—1184no—
356C4—2362no—
357C4(C22⋊D59)⋊C814yes—
358C4#124×D5911yescharacteristic
359C4#1683793312yes—
360C4#1683795011yescharacteristic
361C2—44no—
362C2#2142664311yescharacteristic
363C2C2×(C22⋊D59)⋊C811yescharacteristic
364C2#2143280411yescharacteristic
365C2C8×#1174568712yes—
366C2#2143307412yes—
subgroupG/H
center, FrattiniC22×C4C23×D59
commutatorC22×C59C23×C8
FittingC59×#1954C2
socleC23×C59C23×C4
prime radicalC22×D4×C59C2×C4
cosocleC22×C4×C59C24
hypercenter#1954D59
hypocenterC59C8×#124
LCSG ▷ C22×C59 ▷ C59
UCSC1 ◁ C22×C4 ◁ #1954

Complements

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

NH ≅ G/Ntype
1. C2×#16837526C2semidirect
4. #21425032C2semidirect
5. #21424699C2semidirect
6. #21424212C2semidirect
7. #21423870C2semidirect
8. C4×C8×Dic59C2semidirect
40. #14273714C8semidirect
91. C2×Dic118C2×C8semidirect
102. C22⋊Dic59C2×C8semidirect
111. C4×Dic59C2×C8semidirect
156. C59⋊C8#124semidirect
217. C59C8×#124semidirect

Maximal subgroups

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

subgroupindex
C2×#168375262
C2×D4×C59⋊C82
C4×#142737142
#214250322
#214246992
#214242122
#214238702
C4×C8×Dic592
C59×#19542
C8×#12459