Finite Group Databasebeta

← #28598171

Subgroups — #28598171

Order 15104.

1302 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837455C211yesmaximal, characteristic
2#21426503C211yesmaximal, characteristic
3#21425058C214yesmaximal, complemented
4#21425049C214yesmaximal, complemented
5#21424182C214yesmaximal
6C22×C4⋊Q8×C59C211yesmaximal, characteristic
7C2×#14273686—21no—
8#16837455—24no—
9C2×C4⋊(C59⋊C8)C2212yes—
10C2×C22⋊(C59⋊C8)C2212yes—
11C2×C4×C59⋊C8C2212yes—
12#16837272C2212yes—
13#16837268—21no—
14#16837262—21no—
15#16837262C2212yes—
16C4×C4⋊Dic59—21no—
17C4×C4⋊Dic59C2212yes—
18#16837164C22116yes—
19C23×Q8×C59—21no—
20C2×C4⋊Q8×C59—21no—
21C2×C4⋊Q8×C59—21no—
22C2×C4⋊Q8×C59—24no—
23C2×C4⋊Q8×C59C418yescomplemented
24C2×C4⋊Q8×C59C2212yes—
25C2×C4⋊Q8×C59C2212yes—
26C22×C4⋊C4×C59—21no—
27C22×C4⋊C4×C59C2212yes—
28C22×C42×C59C2211yescharacteristic
29C22×C59⋊C8C2314yes—
30#14273686—44no—
31C2×C4⋊Dic59—22no—
32C2×C4⋊Dic59—22no—
33C2×C4⋊Dic59C2314yes—
34C2×C22⋊Dic59—22no—
35C2×C4×Dic59—22no—
36C4⋊(C59⋊C8)—24no—
37C22⋊(C59⋊C8)—24no—
38C4×C59⋊C8—24no—
39C59⋊#95—22no—
40C59⋊#95—22no—
41C22×Q8×C59—28no—
42C22×Q8×C59—22no—
43C22×Q8×C59—22no—
44C22×Q8×C59—44no—
45C22×Q8×C59—44no—
46C23×C4×C59—22no—
47C23×C4×C59D411yescharacteristic
48C23×C4×C59D411yescharacteristic
49C23×C4×C59C2311yescharacteristic
50C4⋊Q8×C59—28no—
51C4⋊Q8×C59—28no—
52C4⋊Q8×C59—24no—
53C4⋊Q8×C59—24no—
54C4⋊Q8×C59C2×C418yes—
55C4⋊Q8×C59C2×C418yes—
56C2×C4⋊C4×C59—28no—
57C2×C4⋊C4×C59—22no—
58C2×C4⋊C4×C59—22no—
59C2×C4⋊C4×C59—22no—
60C2×C4⋊C4×C59—24no—
61C2×C4⋊C4×C59C2×C418yes—
62C2×C4⋊C4×C59C2314yes—
63C2×C42×C59C2×C414yes—
64C2×C42×C59D411yescharacteristic
65C2×C42×C59D411yescharacteristic
66C2×C42×C59D414yes—
67C2×C42×C59C2311yescharacteristic
68C2×C42×C59C2311yescharacteristic
69C24×C59C2×D411yescharacteristic
70C2×Q8×C59—28no—
71C2×Q8×C59—28no—
72C2×Q8×C59—432no—
73C2×Q8×C59—44no—
74C2×Q8×C59—44no—
75C2×Q8×C59—44no—
76C2×Q8×C59—44no—
77C22×C4×C59—24no—
78C22×C4×C59—24no—
79C22×C4×C59—28no—
80C22×C4×C59—24no—
81C22×C4×C59—44no—
82C22×C4×C59—44no—
83C22×C4×C59C22⋊C414yes—
84C22×C4×C59C22⋊C414yes—
85C22×C4×C59D814yes—
86C22×C4×C59SD1614yes—
87C22×C4×C59C22×C412yes—
88C22×C4×C59C2×D411yescharacteristic
89C22×C4×C59C2×D411yescharacteristic
90C22×C4×C59C2×D412yes—
91C22×C4×C59C2×D411yescharacteristic
92C22×C4×C59C2×D411yescharacteristic
93C22×C4×C59C2×D411yescharacteristic
94C22×C4×C59C2411yescharacteristic
95C22×Dic59—24no—
96C4⋊C4×C59—28no—
97C4⋊C4×C59—28no—
98C4⋊C4×C59—28no—
99C4⋊C4×C59—24no—
100C4⋊C4×C59C22×C418yes—
101C42×C59C22⋊C414yes—
102C42×C59C22⋊C414yes—
103C42×C59C22×C412yes—
104C42×C59C22×C412yes—
105C42×C59C2×D412yes—
106C42×C59C2×D412yes—
107C22⋊Dic59—44no—
108C4⋊Dic59—28no—
109C4⋊Dic59—44no—
110C4⋊Dic59—44no—
111C4×Dic59—44no—
112C2×C59⋊C8—24no—
113C2×C59⋊C8—28no—
114C23×C59—22no—
115C23×C59—22no—
116C23×C59C2×C22⋊C412yes—
117C23×C59C2×D812yes—
118C23×C59C2×SD1612yes—
119C23×C59C22×D411yescharacteristic
120Q8×C59—416no—
121Q8×C59—416no—
122C2×C4×C59—22no—
123C2×C4×C59—22no—
124C2×C4×C59—24no—
125C2×C4×C59—24no—
126C2×C4×C59—28no—
127C2×C4×C59—28no—
128C2×C4×C59—44no—
129C2×C4×C59—44no—
130C2×C4×C59—416no—
131C2×C4×C59#102116yes—
132C2×C4×C59C2×C22⋊C412yes—
133C2×C4×C59C2×C22⋊C412yes—
134C2×C4×C59C2×C22⋊C412yes—
135C2×C4×C59C2×C22⋊C412yes—
136C2×C4×C59C2×C22⋊C412yes—
137C2×C4×C59C2×D812yes—
138C2×C4×C59C2×D812yes—
139C2×C4×C59C2×SD1612yes—
140C2×C4×C59C2×SD1612yes—
141C2×C4×C59#13714yes—
142C2×C4×C59C23×C411yescharacteristic
143C2×C4×C59C22×D411yescharacteristic
144C2×C4×C592−1+414yes—
145C2×Dic59—44no—
146C2×Dic59—44no—
147C2×Dic59—44no—
148C59⋊C8—44no—
149#21833—591nomaximal, Sylow
150C22×C59—22no—
151C22×C59—22no—
152C22×C59—24no—
153C22×C59—24no—
154C22×C59C2×#10214yes—
155C22×C59C22×C22⋊C411yescharacteristic
156C22×C59#53612yes—
157C22×C59C22×D811yescharacteristic
158C22×C59C22×SD1611yescharacteristic
159C22×C59C2×#13712yes—
160C4×C59—22no—
161C4×C59—22no—
162C4×C59—48no—
163C4×C59C2×#10214yes—
164C4×C59C2×#10214yes—
165C4×C59#41914yes—
166C4×C59#51912yes—
167C4×C59#53612yes—
168Dic59—84no—
169C22×C4⋊Q8D5911yescharacteristic
170C2×#431—591no—
171#2285—591no—
172#1990—594no—
173#1988—594no—
174#1823—594no—
175C2×C59—24no—
176C2×C59#231011yescharacteristic
177C2×C59C22×#10211yescharacteristic
178C2×C59C2×#41911yescharacteristic
179C2×C59#322612yes—
180C2×C59#322912yes—
181C23×Q8—21no—
182C2×C4⋊Q8—21no—
183C2×C4⋊Q8—21no—
184C2×C4⋊Q8—24no—
185C2×C4⋊Q8Dic5918yescomplemented
186C2×C4⋊Q8C2×D5912yes—
187C2×C4⋊Q8C2×D5912yes—
188C2×#117—1181no—
189C22×C4⋊C4—21no—
190C22×C4⋊C4C2×D5912yes—
191C22×C42C2×D5911yescharacteristic
192#431—1184no—
193C2×C4⋊C8—592no—
194C2×C22⋊C8—592no—
195C2×C4×C8—592no—
196#384—592no—
197#382—1181no—
198#380—592no—
199#380—1181no—
200C4×C4⋊C4—592no—
201C4×C4⋊C4—1181no—
202#340—5916no—
203C59#2183311yescharacteristic, complemented, Sylow
204C22×Q8—28no—
205C22×Q8—22no—
206C22×Q8—22no—
207C22×Q8—44no—
208C22×Q8—44no—
209C23×C4—22no—
210C23×C4C22⋊D5911yescharacteristic
211C23×C4C22⋊D5911yescharacteristic
212C23×C4C22×D5911yescharacteristic
213C22×C8—594no—
214C4⋊Q8—28no—
215C4⋊Q8—28no—
216C4⋊Q8—24no—
217C4⋊Q8—24no—
218C4⋊Q8C2×Dic5918yes—
219C4⋊Q8C2×Dic5918yes—
220#117—2364no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—594no—
227C2×C4⋊C4—1182no—
228C2×C4⋊C4—1182no—
229C2×C4⋊C4C2×Dic5918yes—
230C2×C4⋊C4C22×D5914yes—
231C2×C22⋊C4—1182no—
232C2×C42—1182no—
233C2×C42C2×Dic5914yes—
234C2×C42C22⋊D5911yescharacteristic
235C2×C42C22⋊D5911yescharacteristic
236C2×C42C22⋊D5914yes—
237C2×C42C22×D5911yescharacteristic
238C2×C42C22×D5911yescharacteristic
239C4⋊C8—1184no—
240C22⋊C8—1184no—
241C4×C8—1184no—
242#95—1182no—
243#95—1182no—
244C24C2×C22⋊D5911yescharacteristic
245C2×Q8—28no—
246C2×Q8—28no—
247C2×Q8—432no—
248C2×Q8—44no—
249C2×Q8—44no—
250C2×Q8—44no—
251C2×Q8—44no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—28no—
255C22×C4—24no—
256C22×C4—44no—
257C22×C4—44no—
258C22×C4—1184no—
259C22×C4#1174566914yes—
260C22×C4#1174567014yes—
261C22×C4C22⋊Dic5914yes—
262C22×C4C22⋊Dic5914yes—
263C22×C4C22×Dic5912yes—
264C22×C4C2×C22⋊D5911yescharacteristic
265C22×C4C2×C22⋊D5911yescharacteristic
266C22×C4C2×C22⋊D5912yes—
267C22×C4C2×C22⋊D5911yescharacteristic
268C22×C4C2×C22⋊D5911yescharacteristic
269C22×C4C2×C22⋊D5911yescharacteristic
270C22×C4C23×D5911yescharacteristic
271C2×C8—1184no—
272C2×C8—1188no—
273C4⋊C4—28no—
274C4⋊C4—28no—
275C4⋊C4—28no—
276C4⋊C4—24no—
277C4⋊C4—1188no—
278C4⋊C4—2364no—
279C4⋊C4—2364no—
280C4⋊C4C22×Dic5918yes—
281C22⋊C4—2364no—
282C42—2364no—
283C42C22⋊Dic5914yes—
284C42C22⋊Dic5914yes—
285C42C22×Dic5912yes—
286C42C22×Dic5912yes—
287C42C2×C22⋊D5912yes—
288C42C2×C22⋊D5912yes—
289C23—22no—
290C23—22no—
291C23C2×C22⋊Dic5912yes—
292C23C2×#1174566912yes—
293C23C2×#1174567012yes—
294C23C22×C22⋊D5911yescharacteristic
295Q8—416no—
296Q8—416no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—28no—
300C2×C4—24no—
301C2×C4—24no—
302C2×C4—28no—
303C2×C4—416no—
304C2×C4—44no—
305C2×C4—44no—
306C2×C4—2364no—
307C2×C4—2364no—
308C2×C4—2364no—
309C2×C4#14273648116yes—
310C2×C4C2×C22⋊Dic5912yes—
311C2×C4C2×C22⋊Dic5912yes—
312C2×C4C2×C22⋊Dic5912yes—
313C2×C4C2×C22⋊Dic5912yes—
314C2×C4C2×C22⋊Dic5912yes—
315C2×C4C2×#1174566912yes—
316C2×C4C2×#1174566912yes—
317C2×C4C2×#1174567012yes—
318C2×C4C2×#1174567012yes—
319C2×C4#1427375814yes—
320C2×C4C23×Dic5911yescharacteristic
321C2×C4C22×C22⋊D5911yescharacteristic
322C2×C4#1427377514yes—
323C8—2364no—
324C22—22no—
325C22—24no—
326C22—24no—
327C22—22no—
328C22C2×#1427364814yes—
329C22C22×C22⋊Dic5911yescharacteristic
330C22#1683797612yes—
331C22C22×#1174566911yescharacteristic
332C22C22×#1174567011yescharacteristic
333C22C2×#1427375812yes—
334C4—22no—
335C4—22no—
336C4—48no—
337C4—4724no—
338C4C2×#1427364814yes—
339C4C2×#1427364814yes—
340C4#1683741114yes—
341C4#1683788012yes—
342C4#1683797612yes—
343C2—24no—
344C2#2142668211yescharacteristic
345C2C22×#1427364811yescharacteristic
346C2C2×#1683741111yescharacteristic
347C2#2143521712yes—
348C2#2143524012yes—
subgroupG/H
centerC23C22×C22⋊D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC22×C4⋊Q8×C59C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
cosocleC22×C4×C59C24
hypercenterC22×C4⋊Q8D59
hypocenterC59#21833
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4⋊Q8

Complements

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

NH ≅ G/Ntype
3. #21425058C2semidirect
4. #21425049C2semidirect
23. C2×C4⋊Q8×C59C4semidirect
185. C2×C4⋊Q8Dic59semidirect
203. C59#21833semidirect

Maximal subgroups

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

subgroupindex
C2×#168374552
#214265032
#214250582
#214250492
#214241822
C22×C4⋊Q8×C592
#2183359