Finite Group Databasebeta

← #28506129

Subgroups — #28506129

Order 15104.

478 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837819C211yesmaximal, characteristic, complemented
2C2×#16837672C211yesmaximal, characteristic
3C2×#16837214C211yesmaximal, characteristic, complemented
4C2×#16837202C211yesmaximal, characteristic, complemented
5#21422204C211yesmaximal, characteristic, complemented
6#21422189C211yesmaximal, characteristic, complemented
7C59×#1816C211yesmaximal, characteristic, complemented
8C2×#14273752—21no—
9C22×D472C2211yescharacteristic
10#16837819—22no—
11#16837819—22no—
12C2×#14273667—21no—
13C2×#14273667C2211yescharacteristic
14C22×C59⋊C16—21no—
15C22×C59⋊C16C2211yescharacteristic
16#16837672—22no—
17#16837672C414yescomplemented
18C2×#14273663C2211yescharacteristic
19C2×#14273643—21no—
20#16837214—22no—
21#16837202—21no—
22#16837202—21no—
23C2×#131×C59C2211yescharacteristic
24C2×#108×C59C2211yescharacteristic
25C2×#107×C59C2211yescharacteristic
26C59×#417—21no—
27C59×#343—21no—
28C22×D236—21no—
29#14273752—24no—
30#14273752—42no—
31C2×#11745671—41no—
32C2×D472—21no—
33C2×D472—21no—
34C2×D472—21no—
35C2×D472—21no—
36C2×D472—21no—
37C2×D472C2×C412yes—
38C2×#11745669—41no—
39C2×#11745664—21no—
40#14273667—24no—
41#14273667—21no—
42#14273667D412yes—
43C2×C59⋊C16—24no—
44C2×C59⋊C16—21no—
45C2×C59⋊C16—21no—
46C2×C59⋊C16—21no—
47C2×C59⋊C16D411yescharacteristic
48C2×C59⋊C16D411yescharacteristic
49#14273663—21no—
50#14273663C2×C412yes—
51#14273643—24no—
52C2×#41×C59D411yescharacteristic
53#131×C59—21no—
54#131×C59—21no—
55#131×C59C2×C412yes—
56#131×C59D412yes—
57C2×M16×C59—21no—
58C2×M16×C59—21no—
59C2×M16×C59D411yescharacteristic
60C22×C8×C59—21no—
61C22×C8×C59C2311yescharacteristic
62C2×C4⋊C4×C59—21no—
63#108×C59—21no—
64#108×C59—21no—
65#107×C59—21no—
66#107×C59—21no—
67#103×C59—41no—
68#102×C59—41no—
69C23×D59—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×D236—22no—
81C2×D236—41no—
82C2×D236—41no—
83C2×D236—41no—
84C2×D236—41no—
85C2×D236—41no—
86M16×C59—21no—
87M16×C59—22no—
88M16×C59—22no—
89M16×C59—41no—
90M16×C59—41no—
91M16×C59C22⋊C412yes—
92C2×C8×C59—21no—
93C2×C8×C59—22no—
94C2×C8×C59—22no—
95C2×C8×C59—41no—
96C2×C8×C59C22×C411yescharacteristic
97C2×C8×C59C2×D411yescharacteristic
98C2×C8×C59C2×D411yescharacteristic
99C2×C8×C59C2×D411yescharacteristic
100C2×C8×C59#4111yescharacteristic
101C2×C8×C59#4111yescharacteristic
102C4⋊C4×C59—41no—
103C4⋊C4×C59—41no—
104#11745671—44no—
105#11745669—44no—
106#11745664—22no—
107#11745664—41no—
108C2×C59⋊C8—41no—
109D472—22no—
110D472—22no—
111D472—41no—
112D472—41no—
113C59⋊C16—21no—
114C59⋊C16—22no—
115C59⋊C16—21no—
116C23×C59—21no—
117C23×C59#13511yescharacteristic
118C22×D59—42no—
119C22×D59—81no—
120C22×D59—81no—
121C22×D59—161no—
122C22×D59—161no—
123Q8×C59—21no—
124Q8×C59#10212yes—
125D4×C59—22no—
126D4×C59—22no—
127D4×C59—21no—
128D4×C59#10212yes—
129C2×C4×C59—21no—
130C2×C4×C59—22no—
131C2×C4×C59—22no—
132C2×C4×C59—81no—
133C2×C4×C59—161no—
134C2×C4×C59C2×C22⋊C411yescharacteristic
135C2×C4×C59#12111yescharacteristic
136C2×C4×C59#12111yescharacteristic
137C2×C4×C59C2×D811yescharacteristic
138C2×C4×C59C2×SD1611yescharacteristic
139C2×C4×C59#13511yescharacteristic
140D236—42no—
141D236—42no—
142D236—81no—
143D236—81no—
144C8×C59—22no—
145C8×C59—81no—
146C8×C59C4×D411yescharacteristic
147C8×C59C4×D411yescharacteristic
148C8×C59#12011yescharacteristic
149C8×C59#12311yescharacteristic
150C59⋊C8—42no—
151#13073—591nomaximal, Sylow
152C22×C59—21no—
153C22×C59—21no—
154C22×C59—22no—
155C22×C59—41no—
156C22×C59—41no—
157C22×C59#41611yescharacteristic
158C22×C59#46511yescharacteristic
159C22×C59#46611yescharacteristic
160C2×D59—82no—
161C2×D59—162no—
162C2×D59—161no—
163C2×D59—161no—
164C2×D59—162no—
165C4×C59—22no—
166C4×C59—161no—
167C4×C59#38611yescharacteristic
168C4×C59C2×#10211yescharacteristic
169C4×C59#46511yescharacteristic
170C4×C59#46611yescharacteristic
171C2×#504—591no—
172C2×#472—591no—
173C2×#361—591no—
174C2×#357—591no—
175#1816D5911yescharacteristic, complemented
176#1387—591no—
177#1381—591no—
178C2×C59—21no—
179C2×C59—21no—
180C2×C59—42no—
181C2×C59#189311yescharacteristic
182C2×C59#222111yescharacteristic
183C2×C59#222311yescharacteristic
184D59—162no—
185C2×AGL(1,ℤ/8)—1181no—
186C22×D8—591no—
187C2×#131C2×D5911yescharacteristic
188#504—1182no—
189#504—1182no—
190C2×M32—591no—
191C2×M32—1181no—
192C22×C16—591no—
193C22×C16—1181no—
194#472—594no—
195#472—1182no—
196C2×#108—591no—
197C2×#108C2×D5911yescharacteristic
198C2×#107C2×D5911yescharacteristic
199#417—21no—
200C2×#100—1181no—
201#361—1182no—
202#357—1181no—
203#357—1181no—
204#343—21no—
205C59#1307311yescharacteristic, complemented, Sylow
206C2×#41C22⋊D5911yescharacteristic
207C22×D4—1181no—
208AGL(1,ℤ/8)—1184no—
209AGL(1,ℤ/8)—2362no—
210C2×SD16—2361no—
211C2×D8—592no—
212C2×D8—1181no—
213C2×D8—1181no—
214C2×D8—1181no—
215C2×D8—1181no—
216C2×D8—1181no—
217C2×D8—2361no—
218#131—21no—
219#131—21no—
220#131C4×D5912yes—
221#131D23612yes—
222C2×M16—21no—
223C2×M16—21no—
224C2×M16—1181no—
225C2×M16C22⋊D5911yescharacteristic
226C22×C8—21no—
227C22×C8C22×D5911yescharacteristic
228C2×C4⋊C4—21no—
229M32—592no—
230M32—1184no—
231M32—1181no—
232C2×C16—591no—
233C2×C16—591no—
234C2×C16—1184no—
235C2×C16—1181no—
236C2×C16—1181no—
237C2×C16—1181no—
238#108—21no—
239#108—21no—
240#108—592no—
241#108—1181no—
242#107—21no—
243#107—21no—
244#103—41no—
245#102—41no—
246#100—1184no—
247C24—2361no—
248#41—21no—
249#41—21no—
250#41(C2×D59)⋊C412yes—
251#41#1174567112yes—
252C2×Q8#1174566911yescharacteristic
253C2×D4—21no—
254C2×D4—1182no—
255C2×D4—2361no—
256C2×D4—2361no—
257C2×D4—2361no—
258C2×D4—2361no—
259C2×D4—2361no—
260C2×D4#1174566911yescharacteristic
261C22×C4—21no—
262C22×C4—41no—
263C22×C4C2×C22⋊D5911yescharacteristic
264SD16—2364no—
265D8—1182no—
266D8—1182no—
267D8—2364no—
268D8—2361no—
269D8—2361no—
270M16—21no—
271M16—22no—
272M16—22no—
273M16—41no—
274M16—41no—
275M16—1182no—
276M16—2361no—
277M16(C2×D59)⋊C412yes—
278C2×C8—21no—
279C2×C8—22no—
280C2×C8—22no—
281C2×C8—41no—
282C2×C8—2361no—
283C2×C8C2×C4×D5911yescharacteristic
284C2×C8C2×D23611yescharacteristic
285C2×C8D4×D5911yescharacteristic
286C2×C8D4×D5911yescharacteristic
287C2×C8#1174568711yescharacteristic
288C2×C8#1174568711yescharacteristic
289C4⋊C4—41no—
290C4⋊C4—41no—
291C16—1181no—
292C16—1181no—
293C16—1182no—
294C23—21no—
295C23—9441no—
296C23—9441no—
297C23—4721no—
298C23—4721no—
299C23—2362no—
300C23#1427374711yescharacteristic
301Q8—21no—
302Q8#1427364712yes—
303D4—22no—
304D4—22no—
305D4—21no—
306D4—4721no—
307D4—4721no—
308D4—2362no—
309D4—2362no—
310D4#1427364712yes—
311C2×C4—21no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—81no—
315C2×C4—161no—
316C2×C4C2×(C2×D59)⋊C411yescharacteristic
317C2×C4#1427369911yescharacteristic
318C2×C4#1427369911yescharacteristic
319C2×C4C2×#1174566911yescharacteristic
320C2×C4C2×#1174567111yescharacteristic
321C2×C4#1427374711yescharacteristic
322C8—22no—
323C8—81no—
324C8—2362no—
325C8(C22⋊D59)⋊C411yescharacteristic
326C8(C22⋊D59)⋊C411yescharacteristic
327C8#1427369711yescharacteristic
328C8#1427371111yescharacteristic
329C22—21no—
330C22—21no—
331C22—22no—
332C22—41no—
333C22—41no—
334C22—9442no—
335C22—9442no—
336C22—9441no—
337C22—9441no—
338C22—4722no—
339C22#1683739611yescharacteristic
340C22#1683763311yescharacteristic
341C22#1683763611yescharacteristic
342C4—22no—
343C4—161no—
344C4#1683727911yescharacteristic
345C4C2×#1427364711yescharacteristic
346C4#1683763311yescharacteristic
347C4#1683763611yescharacteristic
348C2—21no—
349C2—21no—
350C2—42no—
351C2—9442no—
352C2#2142452211yescharacteristic
353C2#2142621411yescharacteristic
354C2#2142622811yescharacteristic
subgroupG/H
centerC22#16837636
commutatorC2×C8×C59C22×C4
FrattiniC22×C8C22×D59
FittingC59×#1816C2
socleC22×C59#466
prime radicalC2×#16837672C2
cosocleC22×C8×C59C23
hypercenter#1816D59
hypocenterC59#13073
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#131 ◁ #1816

Complements

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

NH ≅ G/Ntype
1. C2×#16837819C2semidirect
3. C2×#16837214C2semidirect
4. C2×#16837202C2semidirect
5. #21422204C2semidirect
6. #21422189C2semidirect
7. C59×#1816C2semidirect
17. #16837672C4semidirect
175. #1816D59semidirect
205. C59#13073semidirect

Maximal subgroups

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

subgroupindex
C2×#168378192
C2×#168376722
C2×#168372142
C2×#168372022
#214222042
#214221892
C59×#18162
#1307359