Finite Group Databasebeta

← #28586725

Subgroups — #28586725

Order 15104.

1010 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
1C2×#16837440C211yesmaximal, characteristic
2#21432514C212yesmaximal
3C2×#16837357C211yesmaximal, characteristic
4#21423983C214yesmaximal, complemented
5#21423978C214yesmaximal, complemented
6#21423807C212yesmaximal
7C22×#108×C59C211yesmaximal, characteristic
8C22×#11745664C2211yescharacteristic
9C2×#14273686C2212yes—
10C2×#14273686C2211yescharacteristic
11#16837440C412yescomplemented
12#16837440C412yescomplemented
13#16837440C2212yes—
14#16837440C2212yes—
15C2×#14273661C412yescomplemented
16C2×#14273659C412yescomplemented
17#16837360C2218yes—
18#16837357—24no—
19#16837357C418yescomplemented
20M16×Dic59—24no—
21C2×#14273637C2212yes—
22C2×C8×Dic59C2212yes—
23#16837163C2218yes—
24C22×M16×C59C2212yes—
25C23×C8×C59C2211yescharacteristic
26C2×#108×C59C418yescomplemented
27C2×#108×C59C2212yes—
28C2×#108×C59C2212yes—
29C2×#11745664C2×C414yes—
30C2×#11745664D412yes—
31C2×#11745664D412yes—
32C2×#11745664Q814yes—
33C22×C59⋊C8C2312yes—
34#14273686—22no—
35#14273686—22no—
36#14273686—22no—
37#14273686C2×C414yes—
38#14273686C2×C414yes—
39#14273686C2314yes—
40C2×C4⋊Dic59—21no—
41C2×C4⋊Dic59C2×C414yes—
42C2×C22⋊Dic59—21no—
43C2×C22⋊Dic59—22no—
44C2×C4×Dic59—22no—
45C2×C4×Dic59C2312yes—
46#14273661C2×C414yes—
47#14273661C2×C414yes—
48#14273659C2×C414yes—
49#14273659C2×C414yes—
50#14273637—28no—
51#14273637C2×C418yescomplemented
52C8×Dic59—28no—
53C8×Dic59C2×C418yescomplemented
54C23×C4×C59C2311yescharacteristic
55C2×M16×C59—22no—
56C2×M16×C59—24no—
57C2×M16×C59C2×C418yes—
58C2×M16×C59C2314yes—
59C22×C8×C59—22no—
60C22×C8×C59C2×C411yescharacteristic
61C22×C8×C59C2×C411yescharacteristic
62C22×C8×C59C2×C411yescharacteristic
63C22×C8×C59C2×C411yescharacteristic
64C22×C8×C59C2×C414yes—
65C22×C8×C59C2×C414yes—
66C22×C8×C59C2311yescharacteristic
67C22×C8×C59C2311yescharacteristic
68#108×C59C2×C418yescomplemented
69#108×C59C2×C418yescomplemented
70C24×C59#4111yescharacteristic
71C22×C4×C59C22×C412yes—
72C22×C4×C59C22×C412yes—
73C22×C4×C59C22×C411yescharacteristic
74C22×C4×C59C22×C411yescharacteristic
75C22×C4×C59C2×D411yescharacteristic
76C22×C4×C59C2×D411yescharacteristic
77C22×C4×C59C2×Q812yes—
78C22×C4×C59#4111yescharacteristic
79C22×C4×C59#4112yes—
80C22×C4×C59C2411yescharacteristic
81C22×Dic59—22no—
82C22×Dic59—24no—
83M16×C59—28no—
84M16×C59—24no—
85M16×C59C22×C418yes—
86C2×C8×C59—28no—
87C2×C8×C59—24no—
88C2×C8×C59C4214yes—
89C2×C8×C59C4214yes—
90C2×C8×C59C4214yes—
91C2×C8×C59C4214yes—
92C2×C8×C59C22×C412yes—
93C2×C8×C59C22×C412yes—
94C2×C8×C59C22×C411yescharacteristic
95C2×C8×C59C22×C411yescharacteristic
96C2×C8×C59C22×C411yescharacteristic
97C2×C8×C59C22×C411yescharacteristic
98C2×C8×C59C22×C412yes—
99C2×C8×C59C22×C412yes—
100C22⋊Dic59—24no—
101C22⋊Dic59—24no—
102C22⋊Dic59—24no—
103C4⋊Dic59—24no—
104C4⋊Dic59—24no—
105C4⋊Dic59C22×C418yes—
106C4×Dic59—24no—
107C4×Dic59—24no—
108C4×Dic59—22no—
109C4×Dic59C22×C414yes—
110#11745664C4⋊C418yes—
111#11745664C4⋊C418yes—
112C2×C59⋊C8C22×C414yes—
113C2×C59⋊C8C2×D414yes—
114C2×C59⋊C8C2×Q814yes—
115C23×C59—42no—
116C23×C59#11712yes—
117C23×C59C4×D411yescharacteristic
118C23×C59C4×D411yescharacteristic
119C23×C59C4×Q812yes—
120C23×C59C2×#4111yescharacteristic
121C2×C4×C59C2×C4212yes—
122C2×C4×C59C2×C4212yes—
123C2×C4×C59C2×C4⋊C412yes—
124C2×C4×C59C2×C4⋊C412yes—
125C2×C4×C59#11712yes—
126C2×C4×C59C4×D411yescharacteristic
127C2×C4×C59C4×D411yescharacteristic
128C2×C4×C59C4×D412yes—
129C2×C4×C59C4×D412yes—
130C2×C4×C59C4×Q812yes—
131C2×C4×C59C4×Q812yes—
132C2×C4×C59C4×Q812yes—
133C2×C4×C59C23×C411yescharacteristic
134C2×C4×C59C23×C411yescharacteristic
135C2×C4×C59C23×C411yescharacteristic
136C2×C4×C59C22×D411yescharacteristic
137C2×C4×C59C22×Q811yescharacteristic
138C2×C4×C59C2×#4111yescharacteristic
139C2×Dic59—24no—
140C2×Dic59—28no—
141C2×Dic59—44no—
142C2×Dic59—84no—
143C8×C59—28no—
144C8×C59C2×C4212yes—
145C8×C59C2×C4212yes—
146C8×C59C2×C4212yes—
147C8×C59C2×C4212yes—
148C59⋊C8C2×C4⋊C418yes—
149#20446—591nomaximal, Sylow
150C22×C59—22no—
151C22×C59—22no—
152C22×C59—22no—
153C22×C59—42no—
154C22×C59—42no—
155C22×C59C4×C4⋊C412yes—
156C22×C59C4×C4⋊C412yes—
157C22×C59C2×#11711yescharacteristic
158C22×C59C2×C4×D411yescharacteristic
159C22×C59C2×C4×Q811yescharacteristic
160C4×C59C4×C4⋊C412yes—
161C4×C59C4×C4⋊C412yes—
162C4×C59C22×C4211yescharacteristic
163C4×C59C22×C4⋊C411yescharacteristic
164C4×C59C2×C4×D411yescharacteristic
165C4×C59C2×C4×Q811yescharacteristic
166Dic59—44no—
167Dic59—48no—
168C22×#108D5911yescharacteristic
169C2×#427—591no—
170#2873—592no—
171C2×#405—591no—
172#1773—594no—
173#1772—594no—
174#1733—592no—
175C2×C59—21no—
176C2×C59—21no—
177C2×C59—22no—
178C2×C59—22no—
179C2×C59C2×C4×C4⋊C411yescharacteristic
180C2×C59#295712yes—
181C22×M16—591no—
182C22×M16C2×D5912yes—
183C23×C8C2×D5911yescharacteristic
184C2×#117—591no—
185C2×#117—592no—
186C2×#108Dic5918yescomplemented
187C2×#108C2×D5912yes—
188C2×#108C2×D5912yes—
189#427—592no—
190#427—592no—
191#427—592no—
192#427—592no—
193C2×#107—592no—
194C2×#106—592no—
195#405—598no—
196#405—598no—
197#405—1184no—
198C4×M16—1184no—
199C2×#97—592no—
200C2×C4×C8—592no—
201#339—598no—
202C59#2044611yescharacteristic, complemented, Sylow
203C23×C4C22×D5911yescharacteristic
204C2×M16—22no—
205C2×M16—24no—
206C2×M16—594no—
207C2×M16—592no—
208C2×M16—592no—
209C2×M16—594no—
210C2×M16C2×Dic5918yes—
211C2×M16C22×D5914yes—
212C22×C8—22no—
213C22×C8—592no—
214C22×C8C4×D5914yes—
215C22×C8C4×D5911yescharacteristic
216C22×C8C4×D5911yescharacteristic
217C22×C8C4×D5911yescharacteristic
218C22×C8C4×D5911yescharacteristic
219C22×C8C2×Dic5914yes—
220C22×C8C22×D5911yescharacteristic
221C22×C8C22×D5911yescharacteristic
222#117—594no—
223#117—594no—
224#117—594no—
225#117—1182no—
226#117—1182no—
227#117—1182no—
228C2×C4⋊C4—594no—
229C2×C4⋊C4—1181no—
230C2×C22⋊C4—1181no—
231C2×C22⋊C4—1182no—
232C2×C42—592no—
233C2×C42—1182no—
234#108C2×Dic5918yescomplemented
235#108C2×Dic5918yescomplemented
236#107—594no—
237#107—594no—
238#106—594no—
239#106—594no—
240#97—598no—
241#97—1188no—
242C4×C8—598no—
243C4×C8—1188no—
244C24#1174568911yescharacteristic
245C22×C4—1184no—
246C22×C4—1182no—
247C22×C4C2×C4×D5912yes—
248C22×C4C2×C4×D5911yescharacteristic
249C22×C4C2×C4×D5911yescharacteristic
250C22×C4D4×D5911yescharacteristic
251C22×C4D4×D5911yescharacteristic
252C22×C4#1174568712yes—
253C22×C4Q8×D5912yes—
254C22×C4#1174568911yescharacteristic
255C22×C4C22×Dic5912yes—
256C22×C4C23×D5911yescharacteristic
257M16—28no—
258M16—24no—
259M16—598no—
260M16—598no—
261M16C22×Dic5918yes—
262C2×C8—28no—
263C2×C8—24no—
264C2×C8—594no—
265C2×C8—594no—
266C2×C8—594no—
267C2×C8C4×Dic5914yes—
268C2×C8C4×Dic5914yes—
269C2×C8C4×Dic5914yes—
270C2×C8C4×Dic5914yes—
271C2×C8C2×C4×D5912yes—
272C2×C8C2×C4×D5912yes—
273C2×C8C2×C4×D5911yescharacteristic
274C2×C8C2×C4×D5911yescharacteristic
275C2×C8C2×C4×D5911yescharacteristic
276C2×C8C2×C4×D5911yescharacteristic
277C2×C8C22×Dic5912yes—
278C2×C8C22×Dic5912yes—
279C4⋊C4—598no—
280C4⋊C4—1184no—
281C4⋊C4—1184no—
282C22⋊C4—1184no—
283C22⋊C4—1184no—
284C22⋊C4—1184no—
285C42—594no—
286C42—1184no—
287C42—1184no—
288C42—1182no—
289C23—42no—
290C23#1427368512yes—
291C23#1427368911yescharacteristic
292C23#1427368911yescharacteristic
293C23Q8×Dic5912yes—
294C23C2×#1174568911yescharacteristic
295C2×C4—1184no—
296C2×C4—1188no—
297C2×C4—4724no—
298C2×C4—2364no—
299C2×C4C2×C4×Dic5912yes—
300C2×C4C2×C4×Dic5912yes—
301C2×C4C4⋊C4×D5912yes—
302C2×C4C4⋊C4×D5912yes—
303C2×C4#1427368512yes—
304C2×C4#1427368911yescharacteristic
305C2×C4#1427368911yescharacteristic
306C2×C4D4×Dic5912yes—
307C2×C4D4×Dic5912yes—
308C2×C4#1427369212yes—
309C2×C4#1427369212yes—
310C2×C4Q8×Dic5912yes—
311C2×C4C22×C4×D5911yescharacteristic
312C2×C4C22×C4×D5911yescharacteristic
313C2×C4C23×Dic5911yescharacteristic
314C2×C4C2×D4×D5911yescharacteristic
315C2×C4C2×Q8×D5911yescharacteristic
316C2×C4C2×#1174568711yescharacteristic
317C8—28no—
318C8—598no—
319C8C2×C4×Dic5912yes—
320C8C2×C4×Dic5912yes—
321C8C2×C4×Dic5912yes—
322C8C2×C4×Dic5912yes—
323C22—22no—
324C22—22no—
325C22—22no—
326C22—42no—
327C22—42no—
328C22C4⋊C4×Dic5912yes—
329C22C4⋊C4×Dic5912yes—
330C22C2×#1427368511yescharacteristic
331C22C2×#1427368911yescharacteristic
332C22C2×Q8×Dic5911yescharacteristic
333C4—2364no—
334C4—2368no—
335C4C4⋊C4×Dic5912yes—
336C4C4⋊C4×Dic5912yes—
337C4C22×C4×Dic5911yescharacteristic
338C4C2×C4⋊C4×D5911yescharacteristic
339C4C2×D4×Dic5911yescharacteristic
340C4C2×#1427369211yescharacteristic
341C2—21no—
342C2—22no—
343C2—22no—
344C2—21no—
345C2C2×C4⋊C4×Dic5911yescharacteristic
346C2#2143301212yes—
subgroupG/H
centerC2×C4C2×D4×D59
commutatorC4×C59C22×C42
FrattiniC22×C4C23×D59
FittingC22×#108×C59C2
socleC22×C59C2×C4×D4
prime radicalC24×C59#41
cosocleC22×C4×C59C24
hypercenterC22×#108D59
hypocenterC59#20446
LCSG ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#108

Complements

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

NH ≅ G/Ntype
4. #21423983C2semidirect
5. #21423978C2semidirect
11. #16837440C4semidirect
12. #16837440C4semidirect
15. C2×#14273661C4semidirect
16. C2×#14273659C4semidirect
19. #16837357C4semidirect
26. C2×#108×C59C4semidirect
51. #14273637C2×C4semidirect
53. C8×Dic59C2×C4semidirect
68. #108×C59C2×C4semidirect
69. #108×C59C2×C4semidirect
186. C2×#108Dic59semidirect
202. C59#20446semidirect
234. #108C2×Dic59semidirect
235. #108C2×Dic59semidirect

Maximal subgroups

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

subgroupindex
C2×#168374402
#214325142
C2×#168373572
#214239832
#214239782
#214238072
C22×#108×C592
#2044659