Finite Group Databasebeta

← #28506110

Subgroups — #28506110

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#21433816C211yesmaximal, characteristic
2C2×#16837213C211yesmaximal, characteristic, complemented
3#21425825C211yesmaximal, characteristic, complemented
4#21424155C211yesmaximal, characteristic, complemented
5#21422208C211yesmaximal, characteristic, complemented
6#21422195C211yesmaximal, characteristic, complemented
7C59×#3407C211yesmaximal, characteristic, complemented
8#16838200—21no—
9C2×#14273749C2211yescharacteristic
10#16837677C412yescomplemented
11#16837672C412yescomplemented
12#16837665—22no—
13C2×#14273663C2211yescharacteristic
14C2×#14273663C2211yescharacteristic
15C2×#14273659C2211yescharacteristic
16#16837403—21no—
17#16837384—21no—
18#16837213—22no—
19#16837210—22no—
20#16837170—21no—
21C2×#131×C59C2211yescharacteristic
22C59×#504—22no—
23C59×#504—22no—
24C2×M32×C59—22no—
25C2×M32×C59C2211yescharacteristic
26C2×M32×C59C2211yescharacteristic
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—
41C2×#11745664—21no—
42C2×C4⋊Dic59—21no—
43#14273663—21no—
44#14273663—21no—
45#14273663—21no—
46#14273663C2×C412yes—
47#14273659—21no—
48#14273659—21no—
49#14273651—41no—
50#14273648—41no—
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×C59D411yescharacteristic
60C22×C8×C59—21no—
61C22×C8×C59C2311yescharacteristic
62M32×C59—24no—
63M32×C59—21no—
64M32×C59—21no—
65M32×C59—21no—
66M32×C59D412yes—
67C2×C16×C59—24no—
68C2×C16×C59—21no—
69C2×C16×C59D412yes—
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×C59C2×D411yescharacteristic
79C2×C22⋊D59—41no—
80C22×Dic59—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×C59C22⋊C412yes—
92C2×C8×C59—21no—
93C2×C8×C59—22no—
94C2×C8×C59—22no—
95C2×C8×C59C22×C411yescharacteristic
96C2×C8×C59C2×D411yescharacteristic
97C2×C8×C59C2×D411yescharacteristic
98C2×C8×C59C2×D411yescharacteristic
99C2×C8×C59#4111yescharacteristic
100C2×C8×C59#4111yescharacteristic
101#11745672—42no—
102#11745671—42no—
103#11745670—42no—
104#11745669—42no—
105C4⋊Dic59—41no—
106C4⋊Dic59—41no—
107#11745664—22no—
108#11745664—41no—
109#11745664—41no—
110#11745664—41no—
111C2×C59⋊C8—41no—
112C2×C59⋊C8—41no—
113Dic236—21no—
114Dic236—21no—
115D472—21no—
116D472—21no—
117#11745660—41no—
118#11745660—41no—
119C16×C59—22no—
120C16×C59—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×C59C2×C22⋊C411yescharacteristic
134C2×C4×C59#12111yescharacteristic
135C2×C4×C59#12111yescharacteristic
136C2×C4×C59C2×D811yescharacteristic
137C2×C4×C59C2×SD1611yescharacteristic
138C2×C4×C59#13511yescharacteristic
139C22⋊D59—81no—
140C22⋊D59—81no—
141C2×Dic59—41no—
142C2×Dic59—81no—
143C2×Dic59—161no—
144D236—41no—
145D236—41no—
146C4×D59—81no—
147C4×D59—81no—
148Dic118—41no—
149Dic118—41no—
150C8×C59—22no—
151C8×C59C4×D411yescharacteristic
152C8×C59C4×D411yescharacteristic
153C8×C59#12011yescharacteristic
154C8×C59#12311yescharacteristic
155C59⋊C8—42no—
156C59⋊C8—81no—
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#38611yescharacteristic
168C4×C59C2×#10211yescharacteristic
169C4×C59#46511yescharacteristic
170C4×C59#46611yescharacteristic
171Dic59—81no—
172Dic59—161no—
173#3407D5911yescharacteristic, complemented
174#3067—591no—
175C2×#360—591no—
176#2143—591no—
177#1815—591no—
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—22no—
190#504—22no—
191C2×M32—22no—
192C2×M32C2×D5911yescharacteristic
193C2×M32C2×D5911yescharacteristic
194#473—592no—
195#472—592no—
196#471—1182no—
197C2×#108—591no—
198C2×#108—591no—
199C2×#106—591no—
200#417—1181no—
201#413—1181no—
202#360—1182no—
203#359—1182no—
204#343—1181no—
205C59#1307011yescharacteristic, complemented, Sylow
206C2×#41—1181no—
207C2×#41D23611yescharacteristic
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#131C22⋊D5912yes—
224C2×M16—21no—
225C2×M16—1181no—
226C2×M16—1181no—
227C2×M16D23611yescharacteristic
228C22×C8—21no—
229C22×C8C22×D5911yescharacteristic
230C2×C4⋊C4—1181no—
231M32—24no—
232M32—21no—
233M32—21no—
234M32—21no—
235M32C22⋊D5912yes—
236C2×C16—24no—
237C2×C16—21no—
238C2×C16C22⋊D5912yes—
239#108—592no—
240#108—1181no—
241#108—1181no—
242#108—1181no—
243#106—1181no—
244#106—1181no—
245#103—2361no—
246#102—2361no—
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#1174566012yes—
256#41(C2×D59)⋊C412yes—
257C2×Q8—1181no—
258C2×Q8D47211yescharacteristic
259C2×D4—21no—
260C2×D4—1181no—
261C2×D4—2361no—
262C2×D4D47211yescharacteristic
263C22×C4—21no—
264C22×C4—2361no—
265C22×C4—2361no—
266C22×C4C2×D23611yescharacteristic
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—1182no—
281M16—2361no—
282M16—2361no—
283M16—2361no—
284M16(C2×D59)⋊C412yes—
285C2×C8—21no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—2361no—
289C2×C8—2361no—
290C2×C8C2×C4×D5911yescharacteristic
291C2×C8#1174568511yescharacteristic
292C2×C8#1174568511yescharacteristic
293C2×C8C2×C22⋊D5911yescharacteristic
294C2×C8C2×C22⋊D5911yescharacteristic
295C2×C8C2×C22⋊D5911yescharacteristic
296C4⋊C4—2361no—
297C4⋊C4—2361no—
298C16—22no—
299C16—22no—
300C23—21no—
301C23—2361no—
302C23#1427374911yescharacteristic
303Q8—21no—
304Q8—2361no—
305Q8—2361no—
306Q8#1427364912yes—
307D4—22no—
308D4—22no—
309D4—21no—
310D4—4721no—
311D4—4721no—
312D4—2361no—
313D4—2361no—
314D4#1427364912yes—
315C2×C4—21no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—9441no—
319C2×C4—4721no—
320C2×C4—4721no—
321C2×C4—4721no—
322C2×C4—2361no—
323C2×C4C2×(C2×D59)⋊C411yescharacteristic
324C2×C4#1427370111yescharacteristic
325C2×C4#1427370111yescharacteristic
326C2×C4C2×D47211yescharacteristic
327C2×C4C2×#1174566011yescharacteristic
328C2×C4#1427374911yescharacteristic
329C8—22no—
330C8—4721no—
331C8—2362no—
332C8C4×C22⋊D5911yescharacteristic
333C8C4×C22⋊D5911yescharacteristic
334C8#1427369611yescharacteristic
335C8#1427371011yescharacteristic
336C22—22no—
337C22—41no—
338C22—81no—
339C22—9441no—
340C22—4721no—
341C22#1683739911yescharacteristic
342C22#1683763111yescharacteristic
343C22#1683763811yescharacteristic
344C4—22no—
345C4—9441no—
346C4—4721no—
347C4#1683728111yescharacteristic
348C4C2×#1427364911yescharacteristic
349C4#1683763111yescharacteristic
350C4#1683763811yescharacteristic
351C2—21no—
352C2—21no—
353C2—21no—
354C2—42no—
355C2—9441no—
356C2#2142452411yescharacteristic
subgroupG/H
centerC2#21424524
commutatorC2×C8×C59C22×C4
FrattiniC22×C8C22×D59
FittingC59×#3407C2
socleC2×C59#1893
prime radical#21433816C2
cosocleC22×C8×C59C23
hypercenter#3407D59
hypocenterC59#13070
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ #3407

Complements

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

NH ≅ G/Ntype
2. C2×#16837213C2semidirect
3. #21425825C2semidirect
4. #21424155C2semidirect
5. #21422208C2semidirect
6. #21422195C2semidirect
7. C59×#3407C2semidirect
10. #16837677C4semidirect
11. #16837672C4semidirect
173. #3407D59semidirect
205. C59#13070semidirect

Maximal subgroups

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

subgroupindex
#214338162
C2×#168372132
#214258252
#214241552
#214222082
#214221952
C59×#34072
#1307059