Finite Group Databasebeta

← #28490687

Subgroups — #28490687

Order 15104.

508 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 105 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#21435044C211yesmaximal, characteristic, complemented
2#21433893C211yesmaximal, characteristic
3#21422781C211yesmaximal, characteristic, complemented
4#21422498C211yesmaximal, characteristic, complemented
5#21422417C211yesmaximal, characteristic
6#21421993C211yesmaximal, characteristic, complemented
7C2×C59×#476C211yesmaximal, characteristic
8#16837959C2211yescharacteristic
9#16837781—21no—
10#16837752—21no—
11#16837702—22no—
12#16837692—21no—
13#16837682—21no—
14#16837609—22no—
15#16837558—22no—
16#16837455C2211yescharacteristic
17#16837455C2211yescharacteristic
18#16837442—21no—
19#16837426C2211yescharacteristic
20#16837407—21no—
21#16837148—22no—
22#16837132—22no—
23#16837126—22no—
24C2×C4⋊Q8×C59C2211yescharacteristic
25C2×C4×D4×C59C2211yescharacteristic
26C59×#476—22no—
27C59×#476—22no—
28C2×#106×C59—21no—
29C2×C4⋊C8×C59C2211yescharacteristic
30C2×#102×C59—21no—
31C2×#11745672—24no—
32C2×#11745664—21no—
33#14273727D411yescharacteristic
34#14273727D411yescharacteristic
35#14273719—21no—
36#14273706—21no—
37#14273706—21no—
38C4×Dic118—21no—
39C4×Dic118—21no—
40#14273686—21no—
41#14273660—22no—
42#14273658—22no—
43C4⋊(C59⋊C8)—21no—
44C4⋊(C59⋊C8)—21no—
45C4⋊(C59⋊C8)—21no—
46C4⋊(C59⋊C8)D412yes—
47#14273651—24no—
48#14273648—22no—
49#14273648—22no—
50C22⋊(C59⋊C8)—21no—
51C22⋊(C59⋊C8)D412yes—
52C4×C59⋊C8—21no—
53C4×C59⋊C8—21no—
54C22×Q8×C59—21no—
55C22×D4×C59D411yescharacteristic
56C23×C4×C59—21no—
57C22×C8×C59—21no—
58C4⋊Q8×C59—21no—
59C4⋊Q8×C59—21no—
60C4⋊Q8×C59—21no—
61C4⋊Q8×C59D412yes—
62C4×D4×C59—21no—
63C4×D4×C59—21no—
64C4×D4×C59—21no—
65C4×D4×C59—21no—
66C2×C4⋊C4×C59—21no—
67C2×C4⋊C4×C59—21no—
68C2×C4⋊C4×C59D411yescharacteristic
69C2×C22⋊C4×C59—21no—
70C2×C42×C59C2311yescharacteristic
71#106×C59—24no—
72C4⋊C8×C59—21no—
73C4⋊C8×C59D412yes—
74#102×C59—24no—
75C24×C59—21no—
76C2×Q8×C59—22no—
77C2×Q8×C59—41no—
78C2×Q8×C59—82no—
79C2×D4×C59—21no—
80C2×D4×C59—21no—
81C2×D4×C59—21no—
82C2×D4×C59—21no—
83C2×D4×C59—21no—
84C2×D4×C59Q1612yescomplemented
85C22×C4×C59—21no—
86C22×C4×C59—41no—
87C22×C4×C59—41no—
88C22×C4×C59—41no—
89C22×C4×C59—82no—
90C22×C4×C59C2×D411yescharacteristic
91C22×C4×C59C2×D411yescharacteristic
92C22×C4×C59#4111yescharacteristic
93C2×Dic118—21no—
94C2×Dic118—21no—
95C2×C8×C59—22no—
96C2×C8×C59—41no—
97C2×C8×C59—41no—
98C4⋊C4×C59—22no—
99C4⋊C4×C59—21no—
100C4⋊C4×C59—21no—
101C4⋊C4×C59—22no—
102C4⋊C4×C59—41no—
103C4⋊C4×C59—41no—
104C22⋊C4×C59—41no—
105C22⋊C4×C59—41no—
106C42×C59C2×D411yescharacteristic
107C42×C59C2×D411yescharacteristic
108C42×C59C2×D411yescharacteristic
109C42×C59C2×D411yescharacteristic
110C22⋊Dic59—41no—
111#11745672—84no—
112C4⋊Dic59—21no—
113C4⋊Dic59—21no—
114C4⋊Dic59—21no—
115C4⋊Dic59—21no—
116Dic59⋊C4—41no—
117Dic59⋊C4—41no—
118C4×Dic59—41no—
119#11745664—41no—
120#11745664—41no—
121C2×C59⋊C8—22no—
122C2×C59⋊C8—22no—
123C2×C59⋊C8—41no—
124C23×C59—22no—
125C23×C59—41no—
126C23×C59—41no—
127C23×C59—81no—
128C23×C59—81no—
129C23×C59#12111yescharacteristic
130Q8×C59—44no—
131Q8×C59—82no—
132D4×C59—22no—
133D4×C59—22no—
134D4×C59—41no—
135D4×C59—41no—
136C2×C4×C59—22no—
137C2×C4×C59—41no—
138C2×C4×C59—41no—
139C2×C4×C59—42no—
140C2×C4×C59—82no—
141C2×C4×C59—82no—
142C2×C4×C59—82no—
143C2×C4×C59—81no—
144C2×C4×C59#12011yescharacteristic
145C2×C4×C59#12011yescharacteristic
146C2×C4×C59#12011yescharacteristic
147C2×C4×C59#12111yescharacteristic
148C2×C4×C59#12111yescharacteristic
149C2×C4×C59#12711yescharacteristic
150C2×C4×C59C2×Q1611yescharacteristic
151C2×C4×C59#13511yescharacteristic
152C2×C4×C59#13711yescharacteristic
153C2×C4×C59#13711yescharacteristic
154C2×Dic59—41no—
155C2×Dic59—41no—
156Dic118—42no—
157Dic118—42no—
158C8×C59—42no—
159C59⋊C8—42no—
160C59⋊C8—42no—
161C59⋊C8—81no—
162#10182—591nomaximal, Sylow
163C22×C59—21no—
164C22×C59—21no—
165C22×C59—42no—
166C22×C59—82no—
167C22×C59—82no—
168C22×C59—81no—
169C22×C59—81no—
170C22×C59#39211yescharacteristic
171C22×C59#46711yescharacteristic
172C22×C59#47011yescharacteristic
173C4×C59—21no—
174C4×C59—21no—
175C4×C59—21no—
176C4×C59—82no—
177C4×C59—81no—
178C4×C59#45111yescharacteristic
179C4×C59#45111yescharacteristic
180C4×C59#45211yescharacteristic
181C4×C59#45211yescharacteristic
182C4×C59#47112yes—
183Dic59—81no—
184Dic59—81no—
185#3209—591no—
186#3077—591no—
187C2×#476D5911yescharacteristic
188#1549—591no—
189#1481—591no—
190#1460—591no—
191#1301—591no—
192C2×C59—21no—
193C2×C59—21no—
194C2×C59—82no—
195C2×C59#200111yescharacteristic
196C2×C59#200811yescharacteristic
197C2×C59#201911yescharacteristic
198#533—591no—
199C2×C4⋊Q8C2×D5911yescharacteristic
200C2×C4×D4C2×D5911yescharacteristic
201#494—1181no—
202#487—1181no—
203#478—1182no—
204#476—22no—
205#476—22no—
206#476—1181no—
207#474—1181no—
208#462—1182no—
209#451—1182no—
210#431—591no—
211#431—591no—
212#428—1181no—
213C2×#106—21no—
214#423—591no—
215C2×C4⋊C8C2×D5911yescharacteristic
216#418—1181no—
217C2×#102—21no—
218#332—1182no—
219Q8⋊C8—1182no—
220#323—1182no—
221C59#1018211yescharacteristic, complemented, Sylow
222C22×Q8—21no—
223C22×D4C22⋊D5911yescharacteristic
224C23×C4—21no—
225C2×Q16—1184no—
226C2×M16—1181no—
227C22×C8—21no—
228C4⋊Q8—21no—
229C4⋊Q8—21no—
230C4⋊Q8—21no—
231C4⋊Q8—591no—
232C4⋊Q8—591no—
233C4⋊Q8C22⋊D5912yes—
234#125—1181no—
235C22⋊Q8—1181no—
236C22⋊Q8—1181no—
237C4×Q8—1181no—
238C4×Q8—1181no—
239C4×D4—21no—
240C4×D4—21no—
241C4×D4—21no—
242C4×D4—21no—
243#117—1181no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4C22⋊D5911yescharacteristic
247C2×C22⋊C4—21no—
248C2×C42C22×D5911yescharacteristic
249#107—1182no—
250#106—24no—
251#106—1182no—
252C4⋊C8—21no—
253C4⋊C8—592no—
254C4⋊C8—1181no—
255C4⋊C8—1181no—
256C4⋊C8—1181no—
257C4⋊C8C22⋊D5912yes—
258#103—1184no—
259#102—24no—
260#102—1182no—
261#102—1182no—
262C22⋊C8—592no—
263C22⋊C8—1181no—
264C4×C8—1181no—
265C4×C8—1181no—
266C24—21no—
267C2×Q8—22no—
268C2×Q8—41no—
269C2×Q8—82no—
270C2×Q8—1181no—
271C2×Q8—1181no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4—21no—
277C2×D4#1174567212yescomplemented
278C22×C4—21no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—82no—
283C22×C4D4×D5911yescharacteristic
284C22×C4#1174568711yescharacteristic
285C22×C4C2×C22⋊D5911yescharacteristic
286Q16—4724no—
287M16—2361no—
288M16—2361no—
289C2×C8—22no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—1182no—
293C2×C8—1182no—
294C2×C8—2361no—
295C4⋊C4—21no—
296C4⋊C4—22no—
297C4⋊C4—21no—
298C4⋊C4—22no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—1181no—
302C4⋊C4—1181no—
303C4⋊C4—1181no—
304C4⋊C4—1181no—
305C4⋊C4—2361no—
306C4⋊C4—2361no—
307C22⋊C4—41no—
308C22⋊C4—41no—
309C22⋊C4—2361no—
310C42—2361no—
311C42D4×D5911yescharacteristic
312C42D4×D5911yescharacteristic
313C42C2×C22⋊D5911yescharacteristic
314C42C2×C22⋊D5911yescharacteristic
315C23—22no—
316C23—41no—
317C23—41no—
318C23—81no—
319C23—81no—
320C23#1427369911yescharacteristic
321Q8—44no—
322Q8—82no—
323Q8—2362no—
324Q8—2362no—
325D4—22no—
326D4—22no—
327D4—41no—
328D4—41no—
329C2×C4—22no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4—81no—
337C2×C4—2361no—
338C2×C4—2361no—
339C2×C4#1427369511yescharacteristic
340C2×C4#1427369511yescharacteristic
341C2×C4#1427369611yescharacteristic
342C2×C4#1427369811yescharacteristic
343C2×C4#1427369911yescharacteristic
344C2×C4#1427372311yescharacteristic
345C2×C4C2×#1174567211yescharacteristic
346C2×C4#1427374711yescharacteristic
347C2×C4#1427375711yescharacteristic
348C2×C4#1427375811yescharacteristic
349C8—42no—
350C8—4721no—
351C8—2362no—
352C8—2362no—
353C22—21no—
354C22—21no—
355C22—42no—
356C22—82no—
357C22—82no—
358C22—81no—
359C22—81no—
360C22#1683731011yescharacteristic
361C22#1683764111yescharacteristic
362C22#1683765811yescharacteristic
363C4—21no—
364C4—21no—
365C4—21no—
366C4—81no—
367C4—82no—
368C4—4721no—
369C4—4721no—
370C4#1683755711yescharacteristic
371C4#1683755811yescharacteristic
372C4#1683756211yescharacteristic
373C4#1683756411yescharacteristic
374C4#1683766212yes—
375C2—21no—
376C2—21no—
377C2—82no—
378C2#2142510411yescharacteristic
379C2#2142514711yescharacteristic
380C2#2142520811yescharacteristic
subgroupG/H
centerC22#16837310
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC2×C59×#476C2
socleC22×C59#392
prime radicalC22×D4×C59D4
hypercenterC2×#476D59
hypocenterC59#10182
LCSG ▷ C2×C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#476

Complements

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

NH ≅ G/Ntype
1. #21435044C2semidirect
3. #21422781C2semidirect
4. #21422498C2semidirect
6. #21421993C2semidirect
84. C2×D4×C59Q16semidirect
221. C59#10182semidirect
277. C2×D4#11745672semidirect

Maximal subgroups

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

subgroupindex
#214350442
#214338932
#214227812
#214224982
#214224172
#214219932
C2×C59×#4762
#1018259