Finite Group Databasebeta

← #28490686

Subgroups — #28490686

Order 15104.

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

normal subgroups only

This group has 508 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×#16837689C211yesmaximal, characteristic
3#21422781C211yesmaximal, characteristic, complemented
4#21422499C211yesmaximal, characteristic, complemented
5#21422416C211yesmaximal, characteristic
6#21421992C211yesmaximal, characteristic, complemented
7C59×#3209C211yesmaximal, characteristic, complemented
8C2×C4×D236C2211yescharacteristic
9#16837699—22no—
10#16837689—22no—
11#16837689—21no—
12#16837689—22no—
13#16837679—21no—
14#16837455C2211yescharacteristic
15#16837445—21no—
16C2×#14273658—21no—
17#16837426C2211yescharacteristic
18C2×C4⋊(C59⋊C8)C2211yescharacteristic
19#16837406—21no—
20C2×#14273647—21no—
21#16837148—22no—
22Q8⋊(C59⋊C8)—22no—
23#16837125—22no—
24C59×#533C2211yescharacteristic
25C2×C4⋊Q8×C59C2211yescharacteristic
26C59×#494—21no—
27C59×#487—21no—
28C59×#462—22no—
29C59×#451—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—22no—
42#14273658—24no—
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—
55C2×Q16×C59—24no—
56C4⋊Q8×C59—21no—
57C4⋊Q8×C59—21no—
58C4⋊Q8×C59—21no—
59C4⋊Q8×C59D411yescharacteristic
60C4⋊Q8×C59D412yes—
61C4⋊Q8×C59D411yescharacteristic
62#125×C59—21no—
63C22⋊Q8×C59—21no—
64C22⋊Q8×C59—21no—
65C4×Q8×C59—21no—
66C4×Q8×C59—21no—
67#117×C59—21no—
68C2×C4⋊C4×C59—21no—
69C2×C4⋊C4×C59—21no—
70C2×C42×C59C2311yescharacteristic
71C4⋊C8×C59—21no—
72#103×C59—24no—
73C22⋊C8×C59D412yes—
74C4×C8×C59—21no—
75C23×D59—21no—
76C2×Q8×C59—22no—
77C2×Q8×C59—21no—
78C2×Q8×C59—21no—
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×D236Q1612yescomplemented
92C2×C4×D59—41no—
93C2×C4×D59—41no—
94C2×C4×D59—82no—
95Q16×C59—84no—
96C2×C8×C59—22no—
97C4⋊C4×C59—21no—
98C4⋊C4×C59—22no—
99C4⋊C4×C59—21no—
100C4⋊C4×C59—21no—
101C4⋊C4×C59—22no—
102C4⋊C4×C59—21no—
103C4⋊C4×C59—41no—
104C4⋊C4×C59—41no—
105C4⋊C4×C59—41no—
106C4⋊C4×C59—41no—
107C22⋊C4×C59—41no—
108C42×C59—41no—
109C42×C59C2×D411yescharacteristic
110C42×C59C2×D411yescharacteristic
111C42×C59C2×D411yescharacteristic
112C42×C59C2×D411yescharacteristic
113(C2×D59)⋊C4—41no—
114(C2×D59)⋊C4—41no—
115C4⋊Dic59—21no—
116C4⋊Dic59—21no—
117#11745664—41no—
118#11745664—41no—
119C2×C59⋊C8—22no—
120C2×C59⋊C8—22no—
121C2×C59⋊C8—41no—
122C2×C59⋊C8—41no—
123C2×C59⋊C8—41no—
124C23×C59#12111yescharacteristic
125C22×D59—22no—
126C22×D59—41no—
127C22×D59—41no—
128C22×D59—81no—
129C22×D59—81no—
130Q8×C59—42no—
131Q8×C59—44no—
132Q8×C59—42no—
133Q8×C59—82no—
134C2×C4×C59—22no—
135C2×C4×C59—41no—
136C2×C4×C59—41no—
137C2×C4×C59—41no—
138C2×C4×C59—42no—
139C2×C4×C59—82no—
140C2×C4×C59#12011yescharacteristic
141C2×C4×C59#12011yescharacteristic
142C2×C4×C59#12011yescharacteristic
143C2×C4×C59#12111yescharacteristic
144C2×C4×C59#12111yescharacteristic
145C2×C4×C59#12711yescharacteristic
146C2×C4×C59C2×Q1611yescharacteristic
147C2×C4×C59#13511yescharacteristic
148C2×C4×C59#13711yescharacteristic
149C2×C4×C59#13711yescharacteristic
150C2×Dic59—41no—
151C2×Dic59—81no—
152D236—22no—
153D236—22no—
154D236—41no—
155D236—41no—
156C4×D59—82no—
157C4×D59—82no—
158C8×C59—42no—
159C59⋊C8—42no—
160C59⋊C8—42no—
161C59⋊C8—81no—
162#10182—591nomaximal, Sylow
163C22×C59—21no—
164C22×C59—21no—
165C22×C59#39211yescharacteristic
166C22×C59#46711yescharacteristic
167C22×C59#47011yescharacteristic
168C2×D59—42no—
169C2×D59—82no—
170C2×D59—81no—
171C2×D59—81no—
172C2×D59—82no—
173C4×C59—21no—
174C4×C59—21no—
175C4×C59—21no—
176C4×C59—82no—
177C4×C59—81no—
178C4×C59—81no—
179C4×C59#45111yescharacteristic
180C4×C59#45111yescharacteristic
181C4×C59#45211yescharacteristic
182C4×C59#45211yescharacteristic
183C4×C59#47112yes—
184Dic59—81no—
185#3209D5911yescharacteristic, complemented
186#3077—591no—
187C2×#476—591no—
188#1549—591no—
189#1481—591no—
190#1460—591no—
191#1301—591no—
192C2×C59—21no—
193C2×C59—21no—
194C2×C59#200111yescharacteristic
195C2×C59#200811yescharacteristic
196C2×C59#201911yescharacteristic
197D59—82no—
198#533C2×D5911yescharacteristic
199C2×C4⋊Q8C2×D5911yescharacteristic
200C2×C4×D4—591no—
201#494—21no—
202#487—21no—
203#478—1182no—
204#476—1182no—
205#476—1181no—
206#476—1182no—
207#474—1181no—
208#462—22no—
209#451—22no—
210#431—591no—
211#431C2×D5911yescharacteristic
212#428—1181no—
213C2×#106—1181no—
214#423—591no—
215C2×C4⋊C8—591no—
216#418—1181no—
217C2×#102—1181no—
218#332—1182no—
219Q8⋊C8—1182no—
220#323—1182no—
221C59#1018211yescharacteristic, complemented, Sylow
222C22×Q8—21no—
223C22×D4—591no—
224C23×C4—1181no—
225C2×Q16—24no—
226C2×M16—1181no—
227C22×C8—1181no—
228C4⋊Q8—21no—
229C4⋊Q8—21no—
230C4⋊Q8—21no—
231C4⋊Q8C22⋊D5912yes—
232C4⋊Q8C22⋊D5911yescharacteristic
233C4⋊Q8C22⋊D5911yescharacteristic
234#125—21no—
235C22⋊Q8—21no—
236C22⋊Q8—21no—
237C4×Q8—21no—
238C4×Q8—21no—
239C4×D4—1181no—
240C4×D4—1181no—
241C4×D4—1181no—
242C4×D4—1181no—
243#117—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—591no—
247C2×C22⋊C4—1181no—
248C2×C42C22×D5911yescharacteristic
249#107—1182no—
250#106—1184no—
251#106—1182no—
252C4⋊C8—21no—
253C4⋊C8—592no—
254C4⋊C8—592no—
255C4⋊C8—1181no—
256C4⋊C8—1181no—
257C4⋊C8—1181no—
258#103—24no—
259#102—1184no—
260#102—1182no—
261#102—1182no—
262C22⋊C8—1181no—
263C22⋊C8C22⋊D5912yes—
264C4×C8—21no—
265C4×C8—1181no—
266C24—1181no—
267C2×Q8—22no—
268C2×Q8—21no—
269C2×Q8—21no—
270C2×Q8—41no—
271C2×Q8—82no—
272C2×D4—592no—
273C2×D4—1181no—
274C2×D4—1181no—
275C2×D4—1181no—
276C2×D4—1181no—
277C2×D4—1181no—
278C22×C4—41no—
279C22×C4—1181no—
280C22×C4—4722no—
281C22×C4—2361no—
282C22×C4—2361no—
283C22×C4D4×D5911yescharacteristic
284C22×C4#1174568711yescharacteristic
285C22×C4C2×C22⋊D5911yescharacteristic
286Q16—84no—
287M16—2361no—
288M16—2361no—
289C2×C8—22no—
290C2×C8—1182no—
291C2×C8—1182no—
292C2×C8—2361no—
293C2×C8—2361no—
294C2×C8—2361no—
295C4⋊C4—22no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—22no—
300C4⋊C4—21no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—1181no—
306C4⋊C4—1181no—
307C22⋊C4—41no—
308C22⋊C4—2361no—
309C22⋊C4—2361no—
310C42—41no—
311C42D4×D5911yescharacteristic
312C42D4×D5911yescharacteristic
313C42C2×C22⋊D5911yescharacteristic
314C42C2×C22⋊D5911yescharacteristic
315C23—1182no—
316C23—4721no—
317C23—4721no—
318C23—2361no—
319C23—2361no—
320C23#1427369811yescharacteristic
321Q8—42no—
322Q8—44no—
323Q8—42no—
324Q8—82no—
325D4—1182no—
326D4—1182no—
327D4—2361no—
328D4—2361no—
329C2×C4—22no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—82no—
335C2×C4—4722no—
336C2×C4—4722no—
337C2×C4—4721no—
338C2×C4—2361no—
339C2×C4#1427369511yescharacteristic
340C2×C4#1427369511yescharacteristic
341C2×C4#1427369611yescharacteristic
342C2×C4#1427369911yescharacteristic
343C2×C4#1427369911yescharacteristic
344C2×C4#1427372311yescharacteristic
345C2×C4Q16×D5911yescharacteristic
346C2×C4#1427375011yescharacteristic
347C2×C4#1427375911yescharacteristic
348C2×C4#1427375911yescharacteristic
349C8—42no—
350C8—4721no—
351C8—2362no—
352C8—2362no—
353C22—21no—
354C22—21no—
355C22—4722no—
356C22—4722no—
357C22—4721no—
358C22—4721no—
359C22—2362no—
360C22#1683731011yescharacteristic
361C22#1683764011yescharacteristic
362C22#1683765711yescharacteristic
363C4—21no—
364C4—21no—
365C4—21no—
366C4—81no—
367C4—82no—
368C4—81no—
369C4—4721no—
370C4#1683755611yescharacteristic
371C4#1683755611yescharacteristic
372C4#1683756311yescharacteristic
373C4#1683756311yescharacteristic
374C4#1683766712yes—
375C2—21no—
376C2—21no—
377C2—4722no—
378C2#2142510311yescharacteristic
379C2#2142514911yescharacteristic
380C2#2142520711yescharacteristic
subgroupG/H
centerC22#16837310
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC59×#3209C2
socleC22×C59#392
prime radicalC22×D236D4
hypercenter#3209D59
hypocenterC59#10182
LCSG ▷ C2×C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3209

Complements

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

NH ≅ G/Ntype
3. #21422781C2semidirect
4. #21422499C2semidirect
6. #21421992C2semidirect
7. C59×#3209C2semidirect
91. C2×D236Q16semidirect
185. #3209D59semidirect
221. C59#10182semidirect

Maximal subgroups

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

subgroupindex
#214338892
C2×#168376892
#214227812
#214224992
#214224162
#214219922
C59×#32092
#1018259