Finite Group Databasebeta

← #28506108

Subgroups — #28506108

Order 15104.

456 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21437199C211yesmaximal, characteristic, complemented
2#21433816C211yesmaximal, characteristic
3C2×#16837211C211yesmaximal, characteristic, complemented
4#21425823C211yesmaximal, characteristic, complemented
5#21422209C211yesmaximal, characteristic, complemented
6#21422194C211yesmaximal, characteristic, complemented
7C59×#1815C211yesmaximal, characteristic, complemented
8#16838200—21no—
9C2×#14273749C2211yescharacteristic
10#16837819—22no—
11#16837819—22no—
12C2×#14273667—22no—
13C2×#14273667C2211yescharacteristic
14C2×#14273667C2211yescharacteristic
15#16837677C412yescomplemented
16#16837672C412yescomplemented
17#16837665—22no—
18C2×#14273663C2211yescharacteristic
19#16837384—21no—
20#16837211—22no—
21#16837208—22no—
22C2×#131×C59C2211yescharacteristic
23C2×#108×C59C2211yescharacteristic
24C2×#106×C59C2211yescharacteristic
25C59×#417—21no—
26C59×#343—21no—
27C2×#11745685—21no—
28#14273758—41no—
29#14273757—22no—
30#14273753—41no—
31#14273752—22no—
32#14273749—21no—
33#14273749—21no—
34#14273749—21no—
35#14273749—21no—
36#14273747—42no—
37C2×Dic236C2×C411yescharacteristic
38C2×#11745660—21no—
39C2×D472C2×C411yescharacteristic
40C2×#11745664—21no—
41#14273667—24no—
42#14273667—21no—
43#14273667—21no—
44#14273667—21no—
45#14273667D412yes—
46C2×C59⋊C16—24no—
47C2×C59⋊C16—21no—
48C2×C59⋊C16D412yes—
49#14273663—21no—
50#14273663C2×C412yes—
51#14273645—22no—
52#14273643—22no—
53C2×#41×C59D411yescharacteristic
54#131×C59—21no—
55#131×C59—21no—
56#131×C59C2×C412yes—
57#131×C59D412yes—
58C2×M16×C59—21no—
59C2×M16×C59—21no—
60C2×M16×C59D411yescharacteristic
61C22×C8×C59—21no—
62C22×C8×C59C2311yescharacteristic
63C2×C4⋊C4×C59—21no—
64#108×C59—21no—
65#108×C59—21no—
66#106×C59—21no—
67#106×C59—21no—
68#103×C59—41no—
69#102×C59—41no—
70#41×C59—21no—
71#41×C59—21no—
72#41×C59C22⋊C412yes—
73#41×C59SD1612yes—
74C2×Q8×C59D811yescharacteristic
75C2×D4×C59—21no—
76C2×D4×C59D811yescharacteristic
77C22×C4×C59—21no—
78C22×C4×C59—41no—
79C22×C4×C59C2×D411yescharacteristic
80C2×C22⋊D59—41no—
81#11745685—41no—
82#11745685—41no—
83#11745685—41no—
84#11745685—41no—
85C2×D236—21no—
86C2×C4×D59—41no—
87C2×Dic118—21no—
88M16×C59—21no—
89M16×C59—22no—
90M16×C59—22no—
91M16×C59—41no—
92M16×C59—41no—
93M16×C59C22⋊C412yes—
94C2×C8×C59—21no—
95C2×C8×C59—22no—
96C2×C8×C59—22no—
97C2×C8×C59—41no—
98C2×C8×C59C22×C411yescharacteristic
99C2×C8×C59C2×D411yescharacteristic
100C2×C8×C59C2×D411yescharacteristic
101C2×C8×C59C2×D411yescharacteristic
102C2×C8×C59#4111yescharacteristic
103C2×C8×C59#4111yescharacteristic
104C4⋊C4×C59—41no—
105C4⋊C4×C59—41no—
106#11745672—42no—
107#11745671—42no—
108#11745670—42no—
109#11745669—42no—
110#11745664—22no—
111#11745664—41no—
112C2×C59⋊C8—41no—
113Dic236—21no—
114Dic236—21no—
115D472—21no—
116D472—21no—
117#11745660—41no—
118#11745660—41no—
119C59⋊C16—22no—
120C59⋊C16—22no—
121C23×C59—21no—
122C23×C59#13511yescharacteristic
123C22×D59—41no—
124Q8×C59—21no—
125Q8×C59#10212yes—
126D4×C59—22no—
127D4×C59—22no—
128D4×C59—21no—
129D4×C59#10212yes—
130C2×C4×C59—21no—
131C2×C4×C59—22no—
132C2×C4×C59—22no—
133C2×C4×C59—81no—
134C2×C4×C59—161no—
135C2×C4×C59C2×C22⋊C411yescharacteristic
136C2×C4×C59#12111yescharacteristic
137C2×C4×C59#12111yescharacteristic
138C2×C4×C59C2×D811yescharacteristic
139C2×C4×C59C2×SD1611yescharacteristic
140C2×C4×C59#13511yescharacteristic
141C22⋊D59—81no—
142C22⋊D59—81no—
143C2×Dic59—41no—
144D236—41no—
145D236—41no—
146C4×D59—81no—
147C4×D59—81no—
148Dic118—41no—
149Dic118—41no—
150C8×C59—22no—
151C8×C59—81no—
152C8×C59C4×D411yescharacteristic
153C8×C59C4×D411yescharacteristic
154C8×C59#12011yescharacteristic
155C8×C59#12311yescharacteristic
156C59⋊C8—42no—
157#13070—591nomaximal, Sylow
158C22×C59—22no—
159C22×C59—41no—
160C22×C59—81no—
161C22×C59#41611yescharacteristic
162C22×C59#46511yescharacteristic
163C22×C59#46611yescharacteristic
164C2×D59—81no—
165C2×D59—161no—
166C4×C59—22no—
167C4×C59—161no—
168C4×C59#38611yescharacteristic
169C4×C59C2×#10211yescharacteristic
170C4×C59#46511yescharacteristic
171C4×C59#46611yescharacteristic
172Dic59—81no—
173#3407—591no—
174#3067—591no—
175C2×#360—591no—
176#2143—591no—
177#1815D5911yescharacteristic, complemented
178#1388—591no—
179#1383—591no—
180C2×C59—21no—
181C2×C59—21no—
182C2×C59—21no—
183C2×C59—42no—
184C2×C59#189311yescharacteristic
185D59—161no—
186#575—1181no—
187C2×#135—591no—
188C2×#131C2×D5911yescharacteristic
189#504—1182no—
190#504—1182no—
191C2×M32—591no—
192C2×M32—591no—
193C2×M32—1182no—
194#473—592no—
195#472—592no—
196#471—1182no—
197C2×#108—591no—
198C2×#108C2×D5911yescharacteristic
199C2×#106C2×D5911yescharacteristic
200#417—21no—
201#413—1181no—
202#360—1182no—
203#359—1182no—
204#343—21no—
205C59#1307011yescharacteristic, complemented, Sylow
206C2×#41—1181no—
207C2×#41C22⋊D5911yescharacteristic
208#137—1182no—
209#137—2361no—
210AGL(1,ℤ/8)—1182no—
211AGL(1,ℤ/8)—2361no—
212#135—1181no—
213#135—1181no—
214#135—1181no—
215#135—1181no—
216#135—2362no—
217C2×Q16—591no—
218C2×SD16—1181no—
219C2×D8—591no—
220#131—21no—
221#131—21no—
222#131C4×D5912yes—
223#131D23612yes—
224C2×M16—21no—
225C2×M16—21no—
226C2×M16—1181no—
227C2×M16C22⋊D5911yescharacteristic
228C22×C8—21no—
229C22×C8C22×D5911yescharacteristic
230C2×C4⋊C4—21no—
231M32—592no—
232M32—1184no—
233M32—1181no—
234M32—1181no—
235M32—1181no—
236C2×C16—592no—
237C2×C16—1184no—
238C2×C16—1181no—
239#108—21no—
240#108—21no—
241#108—592no—
242#108—1181no—
243#106—21no—
244#106—21no—
245#103—41no—
246#102—41no—
247#101—1182no—
248#100—1182no—
249#41—21no—
250#41—21no—
251#41—2361no—
252#41—2361no—
253#41—2361no—
254#41—2361no—
255#41(C2×D59)⋊C412yes—
256#41#1174567112yes—
257C2×Q8—1181no—
258C2×Q8#1174566911yescharacteristic
259C2×D4—21no—
260C2×D4—1181no—
261C2×D4—2361no—
262C2×D4#1174566911yescharacteristic
263C22×C4—21no—
264C22×C4—41no—
265C22×C4—2361no—
266C22×C4C2×C22⋊D5911yescharacteristic
267Q16—1181no—
268Q16—1181no—
269Q16—2362no—
270SD16—2362no—
271SD16—2362no—
272SD16—2361no—
273SD16—2361no—
274D8—1181no—
275D8—1181no—
276D8—2362no—
277M16—21no—
278M16—22no—
279M16—22no—
280M16—41no—
281M16—41no—
282M16—1182no—
283M16—2361no—
284M16(C2×D59)⋊C412yes—
285C2×C8—21no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—41no—
289C2×C8—2361no—
290C2×C8C2×C4×D5911yescharacteristic
291C2×C8C2×D23611yescharacteristic
292C2×C8D4×D5911yescharacteristic
293C2×C8D4×D5911yescharacteristic
294C2×C8#1174568711yescharacteristic
295C2×C8#1174568711yescharacteristic
296C4⋊C4—41no—
297C4⋊C4—41no—
298C16—1182no—
299C16—1182no—
300C23—21no—
301C23—2361no—
302C23#1427374711yescharacteristic
303Q8—21no—
304Q8—2361no—
305Q8—2361no—
306Q8#1427364712yes—
307D4—22no—
308D4—22no—
309D4—21no—
310D4—4721no—
311D4—4721no—
312D4—2361no—
313D4—2361no—
314D4#1427364712yes—
315C2×C4—21no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—81no—
319C2×C4—161no—
320C2×C4—4721no—
321C2×C4—4721no—
322C2×C4—2361no—
323C2×C4C2×(C2×D59)⋊C411yescharacteristic
324C2×C4#1427369911yescharacteristic
325C2×C4#1427369911yescharacteristic
326C2×C4C2×#1174566911yescharacteristic
327C2×C4C2×#1174567111yescharacteristic
328C2×C4#1427374711yescharacteristic
329C8—22no—
330C8—81no—
331C8—2362no—
332C8(C22⋊D59)⋊C411yescharacteristic
333C8(C22⋊D59)⋊C411yescharacteristic
334C8#1427369711yescharacteristic
335C8#1427371111yescharacteristic
336C22—22no—
337C22—41no—
338C22—81no—
339C22—9441no—
340C22—4721no—
341C22#1683739611yescharacteristic
342C22#1683763311yescharacteristic
343C22#1683763611yescharacteristic
344C4—22no—
345C4—161no—
346C4—4721no—
347C4#1683727911yescharacteristic
348C4C2×#1427364711yescharacteristic
349C4#1683763311yescharacteristic
350C4#1683763611yescharacteristic
351C2—21no—
352C2—21no—
353C2—21no—
354C2—42no—
355C2—9441no—
356C2#2142452211yescharacteristic
subgroupG/H
centerC2#21424522
commutatorC2×C8×C59C22×C4
FrattiniC22×C8C22×D59
FittingC59×#1815C2
socleC2×C59#1893
prime radical#21433816C2
cosocleC22×C8×C59C23
hypercenter#1815D59
hypocenterC59#13070
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ #1815

Complements

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

NH ≅ G/Ntype
1. #21437199C2semidirect
3. C2×#16837211C2semidirect
4. #21425823C2semidirect
5. #21422209C2semidirect
6. #21422194C2semidirect
7. C59×#1815C2semidirect
15. #16837677C4semidirect
16. #16837672C4semidirect
177. #1815D59semidirect
205. C59#13070semidirect

Maximal subgroups

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

subgroupindex
#214371992
#214338162
C2×#168372112
#214258232
#214222092
#214221942
C59×#18152
#1307059