Finite Group Databasebeta

← #28505742

Subgroups — #28505742

Order 15104.

476 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837832C211yesmaximal, characteristic
2C2×#16837520C211yesmaximal, characteristic, complemented
3C2×#16837209C211yesmaximal, characteristic, complemented
4C2×#16837208C211yesmaximal, characteristic, complemented
5#21422189C211yesmaximal, characteristic, complemented
6#21422174C211yesmaximal, characteristic, complemented
7C59×#1974C211yesmaximal, characteristic, complemented
8C2×#14273749C2211yescharacteristic
9C2×#14273733—21no—
10#16837832—22no—
11#16837832—22no—
12C2×#14273673C411yescharacteristic, complemented
13C2×#14273671C411yescharacteristic, complemented
14C2×#14273670C411yescharacteristic, complemented
15C2×#14273668C411yescharacteristic, complemented
16C22×C59⋊C16C2211yescharacteristic
17C22×C59⋊C16C2211yescharacteristic
18#16837520—22no—
19#16837520—22no—
20C2×#14273663C2211yescharacteristic
21C2×#14273655—21no—
22#16837209—22no—
23#16837208—22no—
24C2×#135×C59C2211yescharacteristic
25C2×#108×C59C2211yescharacteristic
26C2×#104×C59—21no—
27C59×#408—21no—
28C2×C4×C8×C59C2211yescharacteristic
29C2×#11745685—21no—
30#14273749—21no—
31#14273749—21no—
32#14273749D412yes—
33#14273749D412yes—
34C2×Dic236C2×C411yescharacteristic
35C2×#11745660—21no—
36C2×D472C2×C411yescharacteristic
37#14273733—24no—
38#14273733—42no—
39C2×#11745664—21no—
40C2×#11745659—41no—
41C2×C8×D59—41no—
42#14273673—22no—
43#14273671—22no—
44#14273670—22no—
45#14273668—22no—
46C2×C59⋊C16—21no—
47C2×C59⋊C16—21no—
48C2×C59⋊C16—21no—
49C2×C59⋊C16—21no—
50C2×C59⋊C16—21no—
51C2×C59⋊C16C2×C411yescharacteristic
52C2×C59⋊C16C2×C411yescharacteristic
53#14273663—21no—
54#14273663D412yes—
55#14273655—24no—
56C2×#41×C59—21no—
57#135×C59—21no—
58#135×C59—21no—
59#135×C59—21no—
60#135×C59—21no—
61C2×Q16×C59C2×C411yescharacteristic
62C2×SD16×C59—21no—
63C2×D8×C59C2×C411yescharacteristic
64C2×M16×C59—21no—
65C22×C8×C59D411yescharacteristic
66C22×C8×C59C2311yescharacteristic
67C2×C42×C59D411yescharacteristic
68#108×C59—21no—
69#108×C59—21no—
70#104×C59—42no—
71C22⋊C8×C59—41no—
72C22⋊C8×C59—41no—
73C4×C8×C59—21no—
74C4×C8×C59—21no—
75#41×C59—41no—
76#41×C59—41no—
77#41×C59—41no—
78#41×C59—41no—
79C2×Q8×C59—21no—
80C2×D4×C59—21no—
81C2×D4×C59—41no—
82C22×C4×C59—21no—
83C22×C4×C59—41no—
84C22×C4×C59C2×D411yescharacteristic
85C2×C22⋊D59—41no—
86#11745685—22no—
87#11745685—22no—
88#11745685—41no—
89#11745685—41no—
90C2×D236—21no—
91C2×C4×D59—41no—
92C2×Dic118—21no—
93Q16×C59—21no—
94Q16×C59—21no—
95SD16×C59—41no—
96SD16×C59—41no—
97D8×C59—21no—
98D8×C59—21no—
99M16×C59—41no—
100M16×C59—41no—
101C2×C8×C59—21no—
102C2×C8×C59—21no—
103C2×C8×C59—41no—
104C2×C8×C59D811yescharacteristic
105C2×C8×C59D811yescharacteristic
106C2×C8×C59C22×C411yescharacteristic
107C2×C8×C59C2×D411yescharacteristic
108C2×C8×C59C2×D411yescharacteristic
109C2×C8×C59C2×D411yescharacteristic
110C2×C8×C59#4111yescharacteristic
111C2×C8×C59#4111yescharacteristic
112C42×C59—21no—
113C42×C59—21no—
114#11745664—22no—
115#11745664—41no—
116C2×C59⋊C8—41no—
117Dic236—21no—
118Dic236C22⋊C412yes—
119D472—21no—
120D472C22⋊C412yes—
121#11745660—22no—
122#11745660—22no—
123#11745659—44no—
124C8×D59—44no—
125C59⋊C16—21no—
126C59⋊C16—21no—
127C59⋊C16—41no—
128C23×C59—41no—
129C23×C59#13511yescharacteristic
130C22×D59—41no—
131Q8×C59—41no—
132Q8×C59—41no—
133D4×C59—41no—
134D4×C59—41no—
135D4×C59—81no—
136D4×C59—81no—
137C2×C4×C59—22no—
138C2×C4×C59—41no—
139C2×C4×C59—41no—
140C2×C4×C59—41no—
141C2×C4×C59—81no—
142C2×C4×C59—81no—
143C2×C4×C59C4×D411yescharacteristic
144C2×C4×C59#12011yescharacteristic
145C2×C4×C59#12111yescharacteristic
146C2×C4×C59C2×D811yescharacteristic
147C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
148C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
149C22⋊D59—42no—
150C22⋊D59—42no—
151C2×Dic59—41no—
152D236—22no—
153D236—41no—
154C4×D59—42no—
155C4×D59—42no—
156Dic118—22no—
157Dic118—41no—
158C8×C59—21no—
159C8×C59—21no—
160C8×C59—81no—
161C8×C59C2×C22⋊C411yescharacteristic
162C8×C59C4×D411yescharacteristic
163C8×C59#12111yescharacteristic
164C8×C59#12311yescharacteristic
165C59⋊C8—42no—
166#13011—591nomaximal, Sylow
167C22×C59—21no—
168C22×C59—21no—
169C22×C59—81no—
170C22×C59—81no—
171C22×C59—81no—
172C22×C59C4×D811yescharacteristic
173C22×C59#44911yescharacteristic
174C22×C59#46311yescharacteristic
175C2×D59—42no—
176C2×D59—81no—
177C2×D59—81no—
178C4×C59—42no—
179C4×C59—81no—
180C4×C59#38611yescharacteristic
181C4×C59#44211yescharacteristic
182C4×C59#44711yescharacteristic
183C4×C59#45911yescharacteristic
184Dic59—42no—
185C2×#508—591no—
186C2×#443—591no—
187C2×#359—591no—
188C2×#359—591no—
189#1974D5911yescharacteristic, complemented
190#1381—591no—
191#1374—591no—
192C2×C59—21no—
193C2×C59—21no—
194C2×C59—161no—
195C2×C59#187911yescharacteristic
196C2×C59#217811yescharacteristic
197C2×C59#217911yescharacteristic
198D59—82no—
199C2×#135—591no—
200C2×#135C2×D5911yescharacteristic
201C2×#131—1181no—
202#508—1182no—
203#508—1182no—
204C2×Q32—591no—
205C2×SD32—591no—
206C2×SD32—591no—
207C2×D16—591no—
208C22×C16—591no—
209C22×C16—591no—
210#443—1182no—
211#443—1182no—
212C2×#108—591no—
213C2×#108C2×D5911yescharacteristic
214C2×#104—21no—
215C2×#104—1181no—
216#408—21no—
217C2×C4×C8C2×D5911yescharacteristic
218#359—1182no—
219#359—1182no—
220C59#1301111yescharacteristic, complemented, Sylow
221C2×#41—21no—
222C2×#41—1181no—
223#135—21no—
224#135—21no—
225#135—21no—
226#135—21no—
227#135—592no—
228#135—592no—
229#135—1181no—
230#135—1181no—
231C2×Q16—591no—
232C2×Q16C4×D5911yescharacteristic
233C2×SD16—21no—
234C2×SD16—1181no—
235C2×D8—591no—
236C2×D8C4×D5911yescharacteristic
237#131—1184no—
238#131—2362no—
239C2×M16—21no—
240C2×M16—1181no—
241C2×M16—2361no—
242C22×C8—2361no—
243C22×C8C22⋊D5911yescharacteristic
244C22×C8C22×D5911yescharacteristic
245C2×C42C22⋊D5911yescharacteristic
246Q32—1182no—
247SD32—1182no—
248SD32—1182no—
249D16—1182no—
250C2×C16—591no—
251C2×C16—591no—
252C2×C16—1181no—
253C2×C16—1181no—
254C2×C16—1181no—
255C2×C16—1181no—
256C2×C16—1181no—
257#108—21no—
258#108—21no—
259#108—592no—
260#108—1181no—
261#104—42no—
262#104—1184no—
263C22⋊C8—41no—
264C22⋊C8—41no—
265C4×C8—21no—
266C4×C8—21no—
267#41—41no—
268#41—41no—
269#41—41no—
270#41—41no—
271#41—1182no—
272#41—1182no—
273#41—2361no—
274#41—2361no—
275C2×Q8—21no—
276C2×Q8—1181no—
277C2×D4—21no—
278C2×D4—41no—
279C2×D4—1181no—
280C2×D4—2361no—
281C22×C4—21no—
282C22×C4—41no—
283C22×C4—2361no—
284C22×C4C2×C22⋊D5911yescharacteristic
285Q16—21no—
286Q16—21no—
287Q16—592no—
288Q16—1181no—
289SD16—41no—
290SD16—41no—
291SD16—1182no—
292SD16—1182no—
293D8—21no—
294D8—21no—
295D8—592no—
296D8—1181no—
297M16—41no—
298M16—41no—
299M16—1182no—
300M16—2364no—
301M16—2361no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—41no—
305C2×C8—2364no—
306C2×C8—2361no—
307C2×C8#1174566911yescharacteristic
308C2×C8#1174566911yescharacteristic
309C2×C8C2×C4×D5911yescharacteristic
310C2×C8#1174568511yescharacteristic
311C2×C8D4×D5911yescharacteristic
312C2×C8D4×D5911yescharacteristic
313C2×C8D4×D5911yescharacteristic
314C2×C8#1174568711yescharacteristic
315C42—21no—
316C42—21no—
317C16—1181no—
318C16—1181no—
319C16—2361no—
320C23—41no—
321C23—2361no—
322C23#1427374711yescharacteristic
323Q8—41no—
324Q8—41no—
325Q8—1182no—
326Q8—2361no—
327D4—41no—
328D4—41no—
329D4—81no—
330D4—81no—
331D4—1182no—
332D4—2362no—
333D4—2362no—
334D4—2361no—
335C2×C4—22no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—2361no—
342C2×C4—2362no—
343C2×C4—2362no—
344C2×C4C4×C22⋊D5911yescharacteristic
345C2×C4#1427369511yescharacteristic
346C2×C4#1427369811yescharacteristic
347C2×C4C2×#1174566911yescharacteristic
348C2×C4#1427375211yescharacteristic
349C2×C4#1427375311yescharacteristic
350C8—21no—
351C8—21no—
352C8—81no—
353C8—2362no—
354C8C22⋊C4×D5911yescharacteristic
355C8(C22⋊D59)⋊C411yescharacteristic
356C8#1427370011yescharacteristic
357C8#1427370911yescharacteristic
358C22—21no—
359C22—21no—
360C22—81no—
361C22—81no—
362C22—81no—
363C22—4721no—
364C22—4721no—
365C22—2362no—
366C22C4×#1174566911yescharacteristic
367C22#1683754511yescharacteristic
368C22#1683761411yescharacteristic
369C4—42no—
370C4—81no—
371C4—2362no—
372C4#1683727811yescharacteristic
373C4#1683751211yescharacteristic
374C4#1683753511yescharacteristic
375C4#1683759411yescharacteristic
376C2—21no—
377C2—21no—
378C2—161no—
379C2—4722no—
380C2#2142444711yescharacteristic
381C2#2142597511yescharacteristic
382C2#2142598011yescharacteristic
subgroupG/H
centerC2×C4C2×#11745669
commutatorC2×C8×C59C22×C4
FrattiniC22×C8C22×D59
FittingC59×#1974C2
socleC22×C59C4×D8
prime radicalC2×#16837832C2
cosocleC22×C8×C59C23
hypercenter#1974D59
hypocenterC59#13011
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C4×C8 ◁ #1974

Complements

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

NH ≅ G/Ntype
2. C2×#16837520C2semidirect
3. C2×#16837209C2semidirect
4. C2×#16837208C2semidirect
5. #21422189C2semidirect
6. #21422174C2semidirect
7. C59×#1974C2semidirect
12. C2×#14273673C4semidirect
13. C2×#14273671C4semidirect
14. C2×#14273670C4semidirect
15. C2×#14273668C4semidirect
189. #1974D59semidirect
220. C59#13011semidirect

Maximal subgroups

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

subgroupindex
C2×#168378322
C2×#168375202
C2×#168372092
C2×#168372082
#214221892
#214221742
C59×#19742
#1301159