Finite Group Databasebeta

← #28505463

Subgroups — #28505463

Order 15104.

504 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 99 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837192C211yesmaximal, characteristic
2#21424510C211yesmaximal, characteristic
3#21424376C211yesmaximal, characteristic
4#21424305C211yesmaximal, characteristic
5#21422221C211yesmaximal, characteristic
6#21422214C211yesmaximal, characteristic
7C59×#2376C211yesmaximal, characteristic
8#16837877C2211yescharacteristic
9#16837442—21no—
10C2×#14273644C2211yescharacteristic
11C2×#14273642C2211yescharacteristic
12C2×#14273642C2211yescharacteristic
13#16837275—21no—
14#16837265—21no—
15#16837192C414yescomplemented
16#16837168—21no—
17#16837168—21no—
18#16837162—21no—
19C2×#123×C59C2211yescharacteristic
20C59×#390C2211yescharacteristic
21C59×#389—21no—
22C59×#388—21no—
23C59×#388—21no—
24C59×#387—21no—
25C59×#386C2211yescharacteristic
26C59×#385—21no—
27C59×#380—21no—
28C2×#11745664—21no—
29D4×Dic59—41no—
30D4×Dic59—41no—
31#14273686—21no—
32C2×Dic59⋊C4—41no—
33C2×C22⋊Dic59—21no—
34C2×C22⋊Dic59—21no—
35C2×C22⋊Dic59—21no—
36C2×C22⋊Dic59—21no—
37#14273660—41no—
38#14273658—41no—
39#14273644—21no—
40#14273644C2×C412yes—
41#14273642—21no—
42#14273642—21no—
43#14273642—21no—
44#14273642C2×C412yes—
45C59⋊#95—41no—
46C59⋊#95—41no—
47C59⋊#95—41no—
48C22×D4×C59C2311yescharacteristic
49C23×C4×C59D411yescharacteristic
50C23×C4×C59D411yescharacteristic
51#123×C59—21no—
52#123×C59—21no—
53#123×C59C2×C412yes—
54#123×C59Q812yes—
55C2×C4⋊C4×C59—21no—
56C2×C4⋊C4×C59—21no—
57C2×C4⋊C4×C59—21no—
58C2×C4⋊C4×C59—21no—
59C2×C4⋊C4×C59D411yescharacteristic
60C2×C22⋊C4×C59—21no—
61C2×C22⋊C4×C59—21no—
62C2×C22⋊C4×C59—21no—
63C2×C22⋊C4×C59—21no—
64C2×C22⋊C4×C59D411yescharacteristic
65C2×C22⋊C4×C59D411yescharacteristic
66C2×C42×C59—21no—
67C2×C42×C59D411yescharacteristic
68#95×C59—21no—
69#95×C59—21no—
70#95×C59—21no—
71#95×C59—21no—
72#95×C59—21no—
73C24×C59C2×D411yescharacteristic
74C24×C59C2×D411yescharacteristic
75C2×D4×C59—41no—
76C2×D4×C59—41no—
77C2×D4×C59C22×C411yescharacteristic
78C2×D4×C59C2×Q811yescharacteristic
79C2×D4×C59#4111yescharacteristic
80C2×D4×C59#4111yescharacteristic
81C22×C4×C59—21no—
82C22×C4×C59—21no—
83C22×C4×C59—21no—
84C22×C4×C59—21no—
85C22×C4×C59—21no—
86C22×C4×C59—21no—
87C22×C4×C59—21no—
88C22×C4×C59—41no—
89C22×C4×C59—41no—
90C22×C4×C59—41no—
91C22×C4×C59—81no—
92C22×C4×C59C4⋊C412yescomplemented
93C22×C4×C59C2×D411yescharacteristic
94C22×Dic59—21no—
95C22×Dic59—21no—
96C22×Dic59—21no—
97C22×Dic59—21no—
98C22×Dic59—41no—
99C22×Dic59—41no—
100C4⋊C4×C59—22no—
101C4⋊C4×C59—22no—
102C4⋊C4×C59—41no—
103C4⋊C4×C59—81no—
104C4⋊C4×C59—81no—
105C4⋊C4×C59—81no—
106C22⋊C4×C59—22no—
107C22⋊C4×C59—22no—
108C22⋊C4×C59—21no—
109C22⋊C4×C59—41no—
110C22⋊C4×C59—81no—
111C22⋊C4×C59—81no—
112C22⋊C4×C59—81no—
113C22⋊C4×C59C4⋊C412yescomplemented
114C42×C59—41no—
115C42×C59—81no—
116C22⋊Dic59—22no—
117C22⋊Dic59—41no—
118C22⋊Dic59—41no—
119C22⋊Dic59—41no—
120C22⋊Dic59—41no—
121C22⋊Dic59—41no—
122C22⋊Dic59—41no—
123C4⋊Dic59—21no—
124C4⋊Dic59—21no—
125Dic59⋊C4—82no—
126C4×Dic59—41no—
127#11745664—22no—
128#11745664—41no—
129C2×C59⋊C8—41no—
130C23×C59—21no—
131C23×C59—21no—
132C23×C59—21no—
133C23×C59—21no—
134C23×C59—41no—
135C23×C59—41no—
136C23×C59—41no—
137C23×C59—41no—
138C23×C59C2×C4⋊C411yescharacteristic
139C23×C59C4×D411yescharacteristic
140C23×C59#12011yescharacteristic
141C23×C59C22⋊Q811yescharacteristic
142C23×C59#12311yescharacteristic
143D4×C59—41no—
144D4×C59—41no—
145D4×C59—41no—
146D4×C59—41no—
147C2×C4×C59—21no—
148C2×C4×C59—21no—
149C2×C4×C59—22no—
150C2×C4×C59—22no—
151C2×C4×C59—41no—
152C2×C4×C59—41no—
153C2×C4×C59—41no—
154C2×C4×C59—41no—
155C2×C4×C59—42no—
156C2×C4×C59—41no—
157C2×C4×C59—41no—
158C2×C4×C59—41no—
159C2×C4×C59—41no—
160C2×C4×C59—41no—
161C2×C4×C59—41no—
162C2×C4×C59—41no—
163C2×C4×C59—41no—
164C2×C4×C59—42no—
165C2×C4×C59—41no—
166C2×C4×C59—41no—
167C2×C4×C59—41no—
168C2×C4×C59—41no—
169C2×C4×C59—81no—
170C2×C4×C59—81no—
171C2×C4×C59C4×D411yescharacteristic
172C2×C4×C59C22⋊Q811yescharacteristic
173C2×Dic59—42no—
174C2×Dic59—41no—
175C2×Dic59—41no—
176C2×Dic59—81no—
177C2×Dic59—81no—
178C2×Dic59—81no—
179C2×Dic59—81no—
180C2×Dic59—81no—
181C2×Dic59—81no—
182C2×Dic59—81no—
183C2×Dic59—81no—
184C2×Dic59—81no—
185C59⋊C8—42no—
186#12961—591nomaximal, Sylow
187C22×C59—21no—
188C22×C59—21no—
189C22×C59—21no—
190C22×C59—21no—
191C22×C59—21no—
192C22×C59—21no—
193C22×C59—41no—
194C22×C59—41no—
195C22×C59—41no—
196C22×C59—41no—
197C22×C59—41no—
198C22×C59—41no—
199C22×C59—41no—
200C22×C59—41no—
201C22×C59—81no—
202C22×C59—81no—
203C22×C59#38511yescharacteristic
204C22×C59B(4,𝔽2)11yescharacteristic
205C22×C59#45811yescharacteristic
206C4×C59—21no—
207C4×C59—21no—
208C4×C59—42no—
209C4×C59—42no—
210C4×C59—81no—
211C4×C59—81no—
212C4×C59—81no—
213C4×C59—81no—
214Dic59—82no—
215Dic59—81no—
216Dic59—81no—
217Dic59—161no—
218#2376D5911yescharacteristic
219C2×#352—591no—
220#1890—591no—
221#1867—591no—
222#1856—591no—
223#1393—591no—
224#1390—591no—
225C2×C59—21no—
226C2×C59—21no—
227C2×C59—41no—
228C2×C59—41no—
229C2×C59—41no—
230C2×C59—41no—
231C2×C59#186711yescharacteristic
232C2×C59#220011yescharacteristic
233C2×C59#220111yescharacteristic
234C2×#123C2×D5911yescharacteristic
235#518—591no—
236#428—1181no—
237C2×#100—591no—
238C2×#99—591no—
239C2×#99—591no—
240#390C2×D5911yescharacteristic
241#389—21no—
242#388—21no—
243#388—21no—
244#387—21no—
245#386C2×D5911yescharacteristic
246#385—21no—
247#385—1181no—
248#381—1181no—
249#380—21no—
250#352—594no—
251#342—1181no—
252#342—1181no—
253#339—1181no—
254C59#1296111yescharacteristic, complemented, Sylow
255C22×D4C22×D5911yescharacteristic
256C23×C4C22⋊D5911yescharacteristic
257C23×C4C22⋊D5911yescharacteristic
258C2×M16—1181no—
259#123—21no—
260#123—21no—
261#123Dic11812yes—
262#123C4×D5912yes—
263C4×D4—2361no—
264C4×D4—2361no—
265#117—1181no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—2361no—
271C2×C4⋊C4C22⋊D5911yescharacteristic
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—1181no—
277C2×C22⋊C4—1181no—
278C2×C22⋊C4—1181no—
279C2×C22⋊C4—1181no—
280C2×C22⋊C4C22⋊D5911yescharacteristic
281C2×C22⋊C4C22⋊D5911yescharacteristic
282C2×C42—21no—
283C2×C42C22⋊D5911yescharacteristic
284#107—2361no—
285#106—2361no—
286#100—592no—
287#100—1181no—
288#99—592no—
289#99—1181no—
290#99—1181no—
291#99—1181no—
292#95—21no—
293#95—21no—
294#95—21no—
295#95—21no—
296#95—21no—
297#95—2361no—
298#95—2361no—
299#95—2361no—
300C24C2×C22⋊D5911yescharacteristic
301C24C2×C22⋊D5911yescharacteristic
302C2×D4—41no—
303C2×D4—41no—
304C2×D4C2×Dic11811yescharacteristic
305C2×D4C2×C4×D5911yescharacteristic
306C2×D4#1174568511yescharacteristic
307C2×D4#1174568511yescharacteristic
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—81no—
319C22×C4—1181no—
320C22×C4—1181no—
321C22×C4—1181no—
322C22×C4—1181no—
323C22×C4—2361no—
324C22×C4—2361no—
325C22×C4Dic59⋊C412yescomplemented
326C22×C4C2×C22⋊D5911yescharacteristic
327M16—1182no—
328M16—2361no—
329C2×C8—2361no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—41no—
333C4⋊C4—81no—
334C4⋊C4—81no—
335C4⋊C4—81no—
336C4⋊C4—1181no—
337C4⋊C4—1181no—
338C4⋊C4—4722no—
339C22⋊C4—22no—
340C22⋊C4—22no—
341C22⋊C4—21no—
342C22⋊C4—41no—
343C22⋊C4—81no—
344C22⋊C4—81no—
345C22⋊C4—81no—
346C22⋊C4—1182no—
347C22⋊C4—2361no—
348C22⋊C4—2361no—
349C22⋊C4—2361no—
350C22⋊C4—2361no—
351C22⋊C4—2361no—
352C22⋊C4—2361no—
353C22⋊C4Dic59⋊C412yescomplemented
354C42—41no—
355C42—81no—
356C42—2361no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23C2×Dic59⋊C411yescharacteristic
366C23C4×C22⋊D5911yescharacteristic
367C23#1427369611yescharacteristic
368C23#1427370611yescharacteristic
369C23#1427371011yescharacteristic
370D4—41no—
371D4—41no—
372D4—41no—
373D4—41no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—42no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—42no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—4721no—
399C2×C4—4721no—
400C2×C4—4721no—
401C2×C4—4721no—
402C2×C4—4721no—
403C2×C4—4721no—
404C2×C4—4721no—
405C2×C4—4721no—
406C2×C4—4721no—
407C2×C4—2362no—
408C2×C4—2361no—
409C2×C4—2361no—
410C2×C4C4×C22⋊D5911yescharacteristic
411C2×C4#1427370611yescharacteristic
412C8—2362no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—41no—
420C22—41no—
421C22—41no—
422C22—41no—
423C22—41no—
424C22—41no—
425C22—41no—
426C22—41no—
427C22—81no—
428C22—81no—
429C22#1683727511yescharacteristic
430C22#1683758911yescharacteristic
431C22#1683759211yescharacteristic
432C4—21no—
433C4—21no—
434C4—42no—
435C4—42no—
436C4—81no—
437C4—81no—
438C4—81no—
439C4—81no—
440C4—9441no—
441C4—4722no—
442C4—4721no—
443C4—4721no—
444C2—21no—
445C2—21no—
446C2—41no—
447C2—41no—
448C2—41no—
449C2—41no—
450C2#2142437711yescharacteristic
451C2#2142610311yescharacteristic
452C2#2142610911yescharacteristic
subgroupG/H
centerC22#16837589
commutatorC2×D4×C59C22×C4
FrattiniC22×D4C22×D59
FittingC59×#2376C2
socleC22×C59B(4,𝔽2)
prime radical, cosocleC22×D4×C59C23
hypercenter#2376D59
hypocenterC59#12961
LCSG ▷ C2×D4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ #2376

Complements

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

NH ≅ G/Ntype
15. #16837192C4semidirect
92. C22×C4×C59C4⋊C4semidirect
113. C22⋊C4×C59C4⋊C4semidirect
254. C59#12961semidirect
325. C22×C4Dic59⋊C4semidirect
353. C22⋊C4Dic59⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#168371922
#214245102
#214243762
#214243052
#214222212
#214222142
C59×#23762
#1296159