Finite Group Databasebeta

← #28490742

Subgroups — #28490742

Order 15104.

496 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 103 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
1#21433889C211yesmaximal, characteristic
2C2×#16837679C211yesmaximal, characteristic
3#21422777C211yesmaximal, characteristic, complemented
4#21422499C211yesmaximal, characteristic, complemented
5#21422416C211yesmaximal, characteristic
6#21421992C211yesmaximal, characteristic, complemented
7C59×#3230C211yesmaximal, characteristic, complemented
8C2×C4×D236C2211yescharacteristic
9#16837699—22no—
10#16837689—21no—
11#16837679—22no—
12#16837679—21no—
13#16837679—22no—
14#16837455C2211yescharacteristic
15#16837445—21no—
16C2×#14273660—21no—
17#16837426C2211yescharacteristic
18C2×C4⋊(C59⋊C8)C2211yescharacteristic
19#16837406—21no—
20C2×#14273647—21no—
21#16837151—22no—
22Q8⋊(C59⋊C8)—22no—
23#16837125—22no—
24C59×#533C2211yescharacteristic
25C2×C4⋊Q8×C59C2211yescharacteristic
26C59×#500—21no—
27C59×#487—21no—
28C59×#484—22no—
29C59×#477—22no—
30C59×#431C2211yescharacteristic
31C22×D236D411yescharacteristic
32C22×C4×D59—21no—
33C2×#11745664—21no—
34C22×C59⋊C8—21no—
35C4×D236—21no—
36C4×D236—21no—
37C4×D236—21no—
38C4×D236—21no—
39C2×C4⋊Dic59D411yescharacteristic
40C2×(C2×D59)⋊C4—21no—
41#14273660—24no—
42#14273660—22no—
43#14273658—22no—
44C4⋊(C59⋊C8)—21no—
45C4⋊(C59⋊C8)—21no—
46C4⋊(C59⋊C8)—21no—
47C4⋊(C59⋊C8)D412yes—
48C4⋊(C59⋊C8)D412yes—
49#14273647—24no—
50#14273647—22no—
51#14273647—22no—
52C22⋊(C59⋊C8)—21no—
53C4×C59⋊C8—21no—
54C22×Q8×C59—21no—
55C4⋊Q8×C59—21no—
56C4⋊Q8×C59—21no—
57C4⋊Q8×C59—21no—
58C4⋊Q8×C59D412yes—
59C4⋊Q8×C59D411yescharacteristic
60C4⋊Q8×C59D411yescharacteristic
61#125×C59—21no—
62C22⋊Q8×C59—21no—
63C22⋊Q8×C59—21no—
64C4×Q8×C59—21no—
65C4×Q8×C59—21no—
66#117×C59—21no—
67C2×C4⋊C4×C59—21no—
68C2×C4⋊C4×C59—21no—
69C2×C42×C59C2311yescharacteristic
70#107×C59—24no—
71C4⋊C8×C59—21no—
72#103×C59—24no—
73C22⋊C8×C59D412yes—
74C4×C8×C59—21no—
75C23×D59—21no—
76C2×Q8×C59—21no—
77C2×Q8×C59—21no—
78C2×Q8×C59—22no—
79C2×Q8×C59—41no—
80C2×Q8×C59—82no—
81C22×C4×C59—41no—
82C22×C4×C59C2×D411yescharacteristic
83C22×C4×C59C2×D411yescharacteristic
84C22×C4×C59#4111yescharacteristic
85C22×Dic59—21no—
86C2×D236—21no—
87C2×D236—21no—
88C2×D236—21no—
89C2×D236—21no—
90C2×D236—21no—
91C2×D236Q1612yes—
92C2×C4×D59—41no—
93C2×C4×D59—41no—
94C2×C4×D59—82no—
95C2×C8×C59—22no—
96C4⋊C4×C59—22no—
97C4⋊C4×C59—21no—
98C4⋊C4×C59—22no—
99C4⋊C4×C59—21no—
100C4⋊C4×C59—21no—
101C4⋊C4×C59—21no—
102C4⋊C4×C59—41no—
103C4⋊C4×C59—41no—
104C4⋊C4×C59—41no—
105C4⋊C4×C59—41no—
106C22⋊C4×C59—41no—
107C42×C59—41no—
108C42×C59C2×D411yescharacteristic
109C42×C59C2×D411yescharacteristic
110C42×C59C2×D411yescharacteristic
111C42×C59C2×D411yescharacteristic
112(C2×D59)⋊C4—41no—
113(C2×D59)⋊C4—41no—
114C4⋊Dic59—21no—
115C4⋊Dic59—21no—
116#11745664—41no—
117#11745664—41no—
118C2×C59⋊C8—22no—
119C2×C59⋊C8—22no—
120C2×C59⋊C8—41no—
121C2×C59⋊C8—41no—
122C2×C59⋊C8—41no—
123C23×C59#12111yescharacteristic
124C22×D59—22no—
125C22×D59—41no—
126C22×D59—41no—
127C22×D59—81no—
128C22×D59—81no—
129Q8×C59—42no—
130Q8×C59—42no—
131Q8×C59—82no—
132Q8×C59—82no—
133C2×C4×C59—22no—
134C2×C4×C59—41no—
135C2×C4×C59—41no—
136C2×C4×C59—41no—
137C2×C4×C59—42no—
138C2×C4×C59—82no—
139C2×C4×C59#12011yescharacteristic
140C2×C4×C59#12011yescharacteristic
141C2×C4×C59#12011yescharacteristic
142C2×C4×C59#12111yescharacteristic
143C2×C4×C59#12111yescharacteristic
144C2×C4×C59#12711yescharacteristic
145C2×C4×C59C2×Q1611yescharacteristic
146C2×C4×C59#13511yescharacteristic
147C2×C4×C59#13711yescharacteristic
148C2×C4×C59#13711yescharacteristic
149C2×Dic59—41no—
150C2×Dic59—81no—
151D236—22no—
152D236—22no—
153D236—41no—
154D236—41no—
155C4×D59—82no—
156C4×D59—82no—
157C8×C59—42no—
158C59⋊C8—42no—
159C59⋊C8—42no—
160C59⋊C8—81no—
161#10190—591nomaximal, Sylow
162C22×C59—21no—
163C22×C59—21no—
164C22×C59#39211yescharacteristic
165C22×C59#46711yescharacteristic
166C22×C59#47011yescharacteristic
167C2×D59—42no—
168C2×D59—82no—
169C2×D59—81no—
170C2×D59—81no—
171C2×D59—82no—
172C4×C59—21no—
173C4×C59—21no—
174C4×C59—21no—
175C4×C59—81no—
176C4×C59—82no—
177C4×C59—81no—
178C4×C59#45111yescharacteristic
179C4×C59#45111yescharacteristic
180C4×C59#45211yescharacteristic
181C4×C59#45211yescharacteristic
182C4×C59#47212yes—
183Dic59—81no—
184#3230D5911yescharacteristic, complemented
185#3077—591no—
186C2×#474—591no—
187#1548—591no—
188#1481—591no—
189#1460—591no—
190#1301—591no—
191C2×C59—21no—
192C2×C59—21no—
193C2×C59#200111yescharacteristic
194C2×C59#200911yescharacteristic
195C2×C59#201811yescharacteristic
196D59—82no—
197#533C2×D5911yescharacteristic
198C2×C4⋊Q8C2×D5911yescharacteristic
199C2×C4×D4—591no—
200#500—21no—
201#487—21no—
202#484—22no—
203#478—1182no—
204#477—22no—
205#476—1181no—
206#474—1182no—
207#474—1181no—
208#474—1182no—
209#431—591no—
210#431C2×D5911yescharacteristic
211#428—1181no—
212C2×#107—1181no—
213#423—591no—
214C2×C4⋊C8—591no—
215#418—1181no—
216C2×#102—1181no—
217#333—1182no—
218Q8⋊C8—1182no—
219#323—1182no—
220C59#1019011yescharacteristic, complemented, Sylow
221C22×Q8—21no—
222C22×D4—591no—
223C23×C4—1181no—
224C2×M16—1181no—
225C22×C8—1181no—
226C4⋊Q8—21no—
227C4⋊Q8—21no—
228C4⋊Q8—21no—
229C4⋊Q8C22⋊D5912yes—
230C4⋊Q8C22⋊D5911yescharacteristic
231C4⋊Q8C22⋊D5911yescharacteristic
232#125—21no—
233C22⋊Q8—21no—
234C22⋊Q8—21no—
235C4×Q8—21no—
236C4×Q8—21no—
237C4×D4—1181no—
238C4×D4—1181no—
239C4×D4—1181no—
240C4×D4—1181no—
241#117—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—591no—
245C2×C22⋊C4—1181no—
246C2×C42C22×D5911yescharacteristic
247#107—24no—
248#107—1184no—
249#107—1182no—
250#106—1182no—
251C4⋊C8—21no—
252C4⋊C8—592no—
253C4⋊C8—592no—
254C4⋊C8—1181no—
255C4⋊C8—1181no—
256C4⋊C8—1181no—
257#103—24no—
258#102—1184no—
259#102—1182no—
260#102—1182no—
261C22⋊C8—1181no—
262C22⋊C8C22⋊D5912yes—
263C4×C8—21no—
264C4×C8—1181no—
265C24—1181no—
266C2×Q8—21no—
267C2×Q8—21no—
268C2×Q8—22no—
269C2×Q8—41no—
270C2×Q8—82no—
271C2×D4—592no—
272C2×D4—1181no—
273C2×D4—1181no—
274C2×D4—1181no—
275C2×D4—1181no—
276C2×D4—1181no—
277C22×C4—41no—
278C22×C4—1181no—
279C22×C4—4722no—
280C22×C4—2361no—
281C22×C4—2361no—
282C22×C4D4×D5911yescharacteristic
283C22×C4#1174568711yescharacteristic
284C22×C4C2×C22⋊D5911yescharacteristic
285M16—2361no—
286M16—2361no—
287C2×C8—22no—
288C2×C8—1182no—
289C2×C8—1182no—
290C2×C8—2361no—
291C2×C8—2361no—
292C2×C8—2361no—
293C4⋊C4—21no—
294C4⋊C4—22no—
295C4⋊C4—21no—
296C4⋊C4—22no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—1181no—
304C4⋊C4—1181no—
305C22⋊C4—41no—
306C22⋊C4—2361no—
307C22⋊C4—2361no—
308C42—41no—
309C42D4×D5911yescharacteristic
310C42D4×D5911yescharacteristic
311C42C2×C22⋊D5911yescharacteristic
312C42C2×C22⋊D5911yescharacteristic
313C23—1182no—
314C23—4721no—
315C23—4721no—
316C23—2361no—
317C23—2361no—
318C23#1427369811yescharacteristic
319Q8—42no—
320Q8—42no—
321Q8—82no—
322Q8—82no—
323D4—1182no—
324D4—1182no—
325D4—2361no—
326D4—2361no—
327C2×C4—22no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—42no—
332C2×C4—82no—
333C2×C4—4722no—
334C2×C4—4722no—
335C2×C4—4721no—
336C2×C4—2361no—
337C2×C4#1427369511yescharacteristic
338C2×C4#1427369511yescharacteristic
339C2×C4#1427369611yescharacteristic
340C2×C4#1427369911yescharacteristic
341C2×C4#1427369911yescharacteristic
342C2×C4#1427372311yescharacteristic
343C2×C4Q16×D5911yescharacteristic
344C2×C4#1427375011yescharacteristic
345C2×C4#1427375911yescharacteristic
346C2×C4#1427375911yescharacteristic
347C8—42no—
348C8—4721no—
349C8—2362no—
350C8—2362no—
351C22—21no—
352C22—21no—
353C22—4722no—
354C22—4722no—
355C22—4721no—
356C22—4721no—
357C22—2362no—
358C22#1683731011yescharacteristic
359C22#1683764011yescharacteristic
360C22#1683765711yescharacteristic
361C4—21no—
362C4—21no—
363C4—21no—
364C4—81no—
365C4—82no—
366C4—81no—
367C4—4721no—
368C4#1683755611yescharacteristic
369C4#1683755611yescharacteristic
370C4#1683756311yescharacteristic
371C4#1683756311yescharacteristic
372C4#1683767012yes—
373C2—21no—
374C2—21no—
375C2—4722no—
376C2#2142510311yescharacteristic
377C2#2142515611yescharacteristic
378C2#2142520211yescharacteristic
subgroupG/H
centerC22#16837310
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC59×#3230C2
socleC22×C59#392
prime radicalC22×D236D4
hypercenter#3230D59
hypocenterC59#10190
LCSG ▷ C2×C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3230

Complements

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

NH ≅ G/Ntype
3. #21422777C2semidirect
4. #21422499C2semidirect
6. #21421992C2semidirect
7. C59×#3230C2semidirect
184. #3230D59semidirect
220. C59#10190semidirect

Maximal subgroups

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

subgroupindex
#214338892
C2×#168376792
#214227772
#214224992
#214224162
#214219922
C59×#32302
#1019059