Finite Group Databasebeta

← #28505754

Subgroups — #28505754

Order 15104.

526 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14273672C211yesmaximal, characteristic
2C2×#16837496C211yesmaximal, characteristic, complemented
3C2×#16837205C211yesmaximal, characteristic, complemented
4C2×#16837202C211yesmaximal, characteristic
5#21424960C211yesmaximal, characteristic
6#21422173C211yesmaximal, characteristic, complemented
7C59×#1380C211yesmaximal, characteristic, complemented
8C22×Dic236C2211yescharacteristic
9C22×D472C2211yescharacteristic
10C2×#14273692—21no—
11C2×#14273672—22no—
12C2×#14273672—22no—
13C2×#14273672C414yescomplemented
14C22×C59⋊C16C2211yescharacteristic
15#16837496—22no—
16#16837496—22no—
17C2×#14273650—21no—
18C2×#14273647—21no—
19C2×C8×Dic59C2211yescharacteristic
20#16837303—21no—
21#16837205—22no—
22#16837202—22no—
23C22×C16×C59C2211yescharacteristic
24C2×#107×C59C2211yescharacteristic
25C2×#107×C59C2211yescharacteristic
26C22×Dic118—21no—
27C22×D236—21no—
28C2×Dic236—21no—
29C2×Dic236—21no—
30C2×Dic236—21no—
31C2×Dic236C2×C412yes—
32C2×Dic236D412yes—
33C2×Dic236D412yes—
34C2×D472—21no—
35C2×D472—21no—
36C2×D472—21no—
37C2×D472—21no—
38C2×D472—21no—
39C2×D472C2×C412yes—
40C22×C59⋊C8D411yescharacteristic
41#14273692—24no—
42#14273692—42no—
43C2×Dic59⋊C4—41no—
44C2×(C2×D59)⋊C4—41no—
45C2×(C2×D59)⋊C4—41no—
46C2×C4×Dic59—41no—
47C2×C4×Dic59D411yescharacteristic
48#14273672—28no—
49C2×C59⋊C16—21no—
50C2×C59⋊C16—21no—
51C2×C59⋊C16—21no—
52#14273650—24no—
53#14273647—42no—
54C8×Dic59—21no—
55C8×Dic59—21no—
56C22×C8×C59C2311yescharacteristic
57C2×C4⋊C4×C59—21no—
58C2×C4⋊C4×C59—21no—
59C2×C16×C59—21no—
60C2×C16×C59—21no—
61C2×C16×C59C2×C412yes—
62#107×C59—21no—
63#107×C59—21no—
64#107×C59—21no—
65#107×C59D412yes—
66C23×D59—41no—
67C22×C4×C59—41no—
68C22×C4×C59—41no—
69C22×C4×C59C2×D411yescharacteristic
70C22×Dic59—41no—
71C22×Dic59D811yescharacteristic
72C22×Dic59D811yescharacteristic
73C2×D236—22no—
74C2×D236—41no—
75C2×D236—41no—
76C2×D236—41no—
77C2×D236—41no—
78C2×D236—41no—
79C2×Dic118—22no—
80C2×Dic118—22no—
81C2×Dic118—22no—
82C2×Dic118—41no—
83C2×Dic118—41no—
84C2×Dic118—41no—
85C2×C8×C59C22×C411yescharacteristic
86C2×C8×C59C2×D411yescharacteristic
87C2×C8×C59C2×D411yescharacteristic
88C2×C8×C59C2×D411yescharacteristic
89C2×C8×C59#4111yescharacteristic
90C2×C8×C59#4111yescharacteristic
91C4⋊C4×C59—22no—
92C4⋊C4×C59—41no—
93C4⋊C4×C59—41no—
94C4⋊C4×C59—41no—
95(C2×D59)⋊C4—82no—
96(C2×D59)⋊C4—82no—
97Dic59⋊C4—44no—
98C4×Dic59—21no—
99C4×Dic59—21no—
100C4×Dic59—44no—
101C2×C59⋊C8—21no—
102C2×C59⋊C8—21no—
103C2×C59⋊C8—21no—
104Dic236—22no—
105Dic236—22no—
106Dic236—22no—
107Dic236C22⋊C414yescomplemented
108D472—22no—
109D472—22no—
110D472—41no—
111D472—41no—
112C16×C59—22no—
113C59⋊C16—41no—
114C23×C59C2×D811yescharacteristic
115C22×D59—42no—
116C22×D59—81no—
117C22×D59—81no—
118C22×D59—81no—
119C22×D59—81no—
120C22×D59—81no—
121C22×D59—81no—
122C2×C4×C59—42no—
123C2×C4×C59—81no—
124C2×C4×C59—81no—
125C2×C4×C59—81no—
126C2×C4×C59—81no—
127C2×C4×C59—81no—
128C2×C4×C59C4×D411yescharacteristic
129C2×C4×C59#12011yescharacteristic
130C2×C4×C59#12111yescharacteristic
131C2×C4×C59#13511yescharacteristic
132C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
133C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
134C2×Dic59—21no—
135C2×Dic59—21no—
136C2×Dic59—21no—
137C2×Dic59—21no—
138C2×Dic59—21no—
139C2×Dic59—42no—
140C2×Dic59—42no—
141C2×Dic59—42no—
142C2×Dic59SD3212yescomplemented
143D236—42no—
144D236—42no—
145D236—81no—
146D236—81no—
147Dic118—24no—
148Dic118—42no—
149Dic118—42no—
150Dic118—42no—
151C8×C59C2×C22⋊C411yescharacteristic
152C8×C59C4×D411yescharacteristic
153C8×C59#12111yescharacteristic
154C8×C59#12311yescharacteristic
155C59⋊C8—41no—
156#13013—591nomaximal, Sylow
157C22×C59C4×D811yescharacteristic
158C22×C59#44711yescharacteristic
159C22×C59#45911yescharacteristic
160C22×C59C2×SD3211yescharacteristic
161C22×C59#50811yescharacteristic
162C22×C59#50911yescharacteristic
163C22×C59#51011yescharacteristic
164C2×D59—82no—
165C2×D59—82no—
166C2×D59—82no—
167C2×D59—161no—
168C2×D59—161no—
169C2×D59—161no—
170C2×D59—161no—
171C4×C59—82no—
172C4×C59—161no—
173C4×C59#38611yescharacteristic
174C4×C59#44211yescharacteristic
175C4×C59#44911yescharacteristic
176C4×C59#46311yescharacteristic
177Dic59—22no—
178Dic59—44no—
179Dic59—41no—
180C22×SD32—591no—
181C2×C4×Q16—591no—
182C2×#358—591no—
183C2×#357—591no—
184#1970—591no—
185#1380D5911yescharacteristic, complemented
186#1374—591no—
187C2×C59#187911yescharacteristic
188C2×C59C4×SD3211yescharacteristic
189C2×C59#217511yescharacteristic
190C2×C59#218511yescharacteristic
191C2×C59#218611yescharacteristic
192C2×C59#220711yescharacteristic
193C2×C59#221111yescharacteristic
194D59—162no—
195C22×Q16—591no—
196C22×D8—591no—
197C2×C4×Q8—1181no—
198C2×SD32—594no—
199C2×SD32—1182no—
200C2×SD32—1182no—
201C22×C16—591no—
202C22×C16C2×D5911yescharacteristic
203C4×Q16—1182no—
204C4×Q16—1182no—
205C2×#107C2×D5911yescharacteristic
206C2×#107C2×D5911yescharacteristic
207C2×#103—1181no—
208C2×#102—1181no—
209C2×C4×C8—591no—
210#390—1181no—
211#358—1182no—
212#357—1182no—
213C59#1301311yescharacteristic, complemented, Sylow
214C22×Q8—1181no—
215C22×D4—1181no—
216C2×Q16—592no—
217C2×Q16—592no—
218C2×Q16—592no—
219C2×Q16—1181no—
220C2×Q16—1181no—
221C2×Q16—1181no—
222C2×D8—592no—
223C2×D8—1181no—
224C2×D8—1181no—
225C2×D8—1181no—
226C2×D8—1181no—
227C2×D8—1181no—
228C22×C8—591no—
229C22×C8C22×D5911yescharacteristic
230C4×Q8—1184no—
231C4×Q8—2362no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—2361no—
235C2×C22⋊C4—2361no—
236C2×C22⋊C4—2361no—
237C2×C42—591no—
238C2×C42—2361no—
239SD32—1188no—
240C2×C16—21no—
241C2×C16—21no—
242C2×C16—1181no—
243C2×C16—1181no—
244C2×C16—1181no—
245C2×C16C4×D5912yes—
246#107—21no—
247#107—21no—
248#107—21no—
249#107D23612yes—
250#103—1184no—
251#102—2362no—
252C4×C8—1181no—
253C4×C8—1181no—
254C24—2361no—
255C2×Q8—1182no—
256C2×Q8—1182no—
257C2×Q8—1182no—
258C2×Q8—2361no—
259C2×Q8—2361no—
260C2×Q8—2361no—
261C2×D4—1182no—
262C2×D4—2361no—
263C2×D4—2361no—
264C2×D4—2361no—
265C2×D4—2361no—
266C2×D4—2361no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—591no—
270C22×C4—591no—
271C22×C4—2361no—
272C22×C4D4×D5911yescharacteristic
273Q16—594no—
274Q16—1182no—
275Q16—1182no—
276Q16—1182no—
277D8—1182no—
278D8—1182no—
279D8—2361no—
280D8—2361no—
281C2×C8—1181no—
282C2×C8—1181no—
283C2×C8—1181no—
284C2×C8C2×C4×D5911yescharacteristic
285C2×C8C2×D23611yescharacteristic
286C2×C8#1174568511yescharacteristic
287C2×C8D4×D5911yescharacteristic
288C2×C8D4×D5911yescharacteristic
289C2×C8#1174568911yescharacteristic
290C4⋊C4—22no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—2364no—
295C22⋊C4—4722no—
296C22⋊C4—4722no—
297C42—1181no—
298C42—1181no—
299C42—2364no—
300C16—22no—
301C16—2361no—
302C23—4721no—
303C23—4721no—
304C23—4721no—
305C23—4721no—
306C23—4721no—
307C23—4721no—
308C23—2362no—
309C23D8×D5911yescharacteristic
310Q8—1184no—
311Q8—2362no—
312Q8—2362no—
313Q8—2362no—
314D4—4721no—
315D4—4721no—
316D4—2362no—
317D4—2362no—
318C2×C4—42no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—592no—
325C2×C4—1181no—
326C2×C4—1181no—
327C2×C4—1181no—
328C2×C4—1181no—
329C2×C4—1181no—
330C2×C4—2362no—
331C2×C4—2362no—
332C2×C4—2362no—
333C2×C4#1427368911yescharacteristic
334C2×C4#1427369711yescharacteristic
335C2×C4#1427370011yescharacteristic
336C2×C4#1427375011yescharacteristic
337C2×C4#1427375411yescharacteristic
338C2×C4#1427375411yescharacteristic
339C8—2361no—
340C8C22⋊C4×D5911yescharacteristic
341C8C4×D23611yescharacteristic
342C8#1427370211yescharacteristic
343C8#1427371211yescharacteristic
344C22—9441no—
345C22—9441no—
346C22—9441no—
347C22—9441no—
348C22—4722no—
349C22—4722no—
350C22—4722no—
351C22#1683748411yescharacteristic
352C22#1683753711yescharacteristic
353C22#1683759611yescharacteristic
354C22SD32×D5911yescharacteristic
355C22#1683783311yescharacteristic
356C22#1683783611yescharacteristic
357C22#1683784411yescharacteristic
358C4—82no—
359C4—161no—
360C4—1182no—
361C4—2364no—
362C4—2361no—
363C4#1683728011yescharacteristic
364C4#1683751311yescharacteristic
365C4#1683755011yescharacteristic
366C4#1683762011yescharacteristic
367C2—9442no—
368C2#2142444511yescharacteristic
369C2#2142594711yescharacteristic
370C2#2142595911yescharacteristic
371C2#2142601411yescharacteristic
372C2#2142602011yescharacteristic
373C2#2142613511yescharacteristic
374C2#2142616111yescharacteristic
subgroupG/H
centerC23D8×D59
commutatorC2×C8×C59C22×C4
FrattiniC22×C8C22×D59
FittingC59×#1380C2
socleC23×C59C2×D8
prime radicalC22×D472C22
cosocleC22×C8×C59C23
hypercenter#1380D59
hypocenterC59#13013
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ #1380

Complements

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

NH ≅ G/Ntype
2. C2×#16837496C2semidirect
3. C2×#16837205C2semidirect
6. #21422173C2semidirect
7. C59×#1380C2semidirect
13. C2×#14273672C4semidirect
107. Dic236C22⋊C4semidirect
142. C2×Dic59SD32semidirect
185. #1380D59semidirect
213. C59#13013semidirect

Maximal subgroups

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

subgroupindex
C22×#142736722
C2×#168374962
C2×#168372052
C2×#168372022
#214249602
#214221732
C59×#13802
#1301359