Finite Group Databasebeta

← #28574029

Subgroups — #28574029

Order 15104.

644 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21435224C211yesmaximal, characteristic
2#21433568C211yesmaximal, characteristic
3#21433177C211yesmaximal, characteristic, complemented
4#21423514C214yesmaximal, complemented
5#21423480C214yesmaximal, complemented
6#21422866C211yesmaximal, characteristic, complemented
7#21422777C211yesmaximal, characteristic, complemented
8#21422746C211yesmaximal, characteristic
9C2×C59×#477C211yesmaximal, characteristic, complemented
10C2×#14273758—21no—
11#16837952C2211yescharacteristic
12#16837798C2212yes—
13#16837750C2212yes—
14#16837720—22no—
15#16837689—22no—
16#16837624—22no—
17#16837612C2212yes—
18#16837602C2212yes—
19Q8×C59⋊C8—22no—
20Q8×C59⋊C8C2214yes—
21#16837455—22no—
22#16837455C2211yescharacteristic
23#16837455C2211yescharacteristic
24#16837426C2211yescharacteristic
25#16837401—21no—
26#16837152C2214yes—
27#16837151C2214yes—
28#16837149C2212yes—
29#16837145C2212yes—
30C2×C4⋊Q8×C59C2211yescharacteristic
31C2×C4×Q8×C59C2211yescharacteristic
32C59×#477—22no—
33C59×#477C2214yes—
34C2×#107×C59—21no—
35C2×C4⋊C8×C59C2211yescharacteristic
36C2×#103×C59—21no—
37C2×#11745685—21no—
38#14273758—44no—
39C2×#11745672—22no—
40C2×#11745671—22no—
41C2×#11745664—21no—
42#14273727C2311yescharacteristic
43#14273724C2311yescharacteristic
44#14273715—21no—
45#14273701—21no—
46#14273701—21no—
47C4×D236—21no—
48C4×D236—21no—
49#14273658—24no—
50C4⋊(C59⋊C8)—21no—
51C4⋊(C59⋊C8)—24no—
52C4⋊(C59⋊C8)D412yes—
53C4⋊(C59⋊C8)C2312yes—
54C4⋊(C59⋊C8)C2312yes—
55#14273650—22no—
56#14273647—22no—
57#14273647—24no—
58C22⋊(C59⋊C8)—21no—
59C22⋊(C59⋊C8)D412yes—
60C4×C59⋊C8—21no—
61C4×C59⋊C8—24no—
62C4×C59⋊C8C2312yes—
63C22×Q8×C59—21no—
64C22×Q8×C59D411yescharacteristic
65C22×C8×C59—21no—
66C4⋊Q8×C59—21no—
67C4⋊Q8×C59—21no—
68C4⋊Q8×C59—21no—
69C4⋊Q8×C59C2312yes—
70C4×Q8×C59—21no—
71C4×Q8×C59—21no—
72C4×Q8×C59D412yes—
73C4×Q8×C59C2312yes—
74C2×C4⋊C4×C59—21no—
75C2×C4⋊C4×C59—21no—
76C2×C4⋊C4×C59—21no—
77C2×C4⋊C4×C59D411yescharacteristic
78C2×C42×C59—21no—
79C2×C42×C59C2311yescharacteristic
80#107×C59—24no—
81C4⋊C8×C59—21no—
82C4⋊C8×C59C2312yes—
83#103×C59—24no—
84C2×Q8×C59—21no—
85C2×Q8×C59—22no—
86C2×Q8×C59—22no—
87C2×Q8×C59—41no—
88C2×Q8×C59—82no—
89C2×Q8×C59SD1614yes—
90C2×Q8×C59C2×D412yes—
91C22×C4×C59—21no—
92C22×C4×C59—21no—
93C22×C4×C59—41no—
94C22×C4×C59C2×D411yescharacteristic
95C22×C4×C59C2×D411yescharacteristic
96C22×C4×C59#4111yescharacteristic
97C2×C22⋊D59—41no—
98#11745685—42no—
99#11745685—42no—
100C2×D236—21no—
101C2×D236—21no—
102C2×D236—21no—
103C2×C4×D59—41no—
104C2×Dic118—21no—
105C2×C8×C59—22no—
106C2×C8×C59—42no—
107C4⋊C4×C59—22no—
108C4⋊C4×C59—21no—
109C4⋊C4×C59—22no—
110C4⋊C4×C59—22no—
111C4⋊C4×C59—22no—
112C4⋊C4×C59—41no—
113C4⋊C4×C59—41no—
114C4⋊C4×C59C2×D412yes—
115C42×C59—22no—
116C42×C59—22no—
117C42×C59C2×D411yescharacteristic
118C42×C59C2×D411yescharacteristic
119C42×C59#4111yescharacteristic
120C42×C59C2411yescharacteristic
121#11745672—44no—
122#11745671—44no—
123(C2×D59)⋊C4—41no—
124(C2×D59)⋊C4—41no—
125C4⋊Dic59—21no—
126C4⋊Dic59—21no—
127#11745664—42no—
128C2×C59⋊C8—22no—
129C2×C59⋊C8—22no—
130C2×C59⋊C8C2×D414yes—
131C23×C59#12111yescharacteristic
132C22×D59—41no—
133Q8×C59—22no—
134Q8×C59—24no—
135Q8×C59—82no—
136Q8×C59—82no—
137Q8×C59C2×SD1614yes—
138C2×C4×C59—22no—
139C2×C4×C59—24no—
140C2×C4×C59—22no—
141C2×C4×C59—22no—
142C2×C4×C59—41no—
143C2×C4×C59—42no—
144C2×C4×C59—81no—
145C2×C4×C59—82no—
146C2×C4×C59#12111yescharacteristic
147C2×C4×C59#12111yescharacteristic
148C2×C4×C59#12111yescharacteristic
149C2×C4×C59C2×SD1612yes—
150C2×C4×C59#13512yes—
151C2×C4×C59C22×D411yescharacteristic
152C2×C4×C59C22×D411yescharacteristic
153C2×C4×C59C2×#4111yescharacteristic
154C22⋊D59—82no—
155C2×Dic59—41no—
156D236—41no—
157D236—41no—
158D236—42no—
159D236—81no—
160C4×D59—82no—
161Dic118—41no—
162Dic118—41no—
163C8×C59—42no—
164C59⋊C8—24no—
165C59⋊C8—42no—
166C59⋊C8—42no—
167#19403—591nomaximal, Sylow
168C22×C59—21no—
169C22×C59—21no—
170C22×C59#46512yes—
171C22×C59C2×#12111yescharacteristic
172C2×D59—81no—
173C2×D59—81no—
174C2×D59—81no—
175C4×C59—21no—
176C4×C59—24no—
177C4×C59—21no—
178C4×C59—21no—
179C4×C59—42no—
180C4×C59—82no—
181C4×C59#46512yes—
182C4×C59C22×SD1611yescharacteristic
183C4×C59C2×#13511yescharacteristic
184C4×C59#57812yes—
185Dic59—81no—
186#3227—591no—
187C2×#477D5911yescharacteristic, complemented
188#3037—591no—
189#2982—591no—
190#1678—594no—
191#1672—594no—
192#1570—591no—
193#1548—591no—
194#1540—591no—
195C2×C59—22no—
196C2×C59#304411yescharacteristic
197C2×C59C2×#46511yescharacteristic
198C2×C59#306811yescharacteristic
199D59—161no—
200C2×#137—1181no—
201#532—591no—
202C2×C4⋊Q8C2×D5911yescharacteristic
203C2×C4×Q8C2×D5911yescharacteristic
204#498—592no—
205#486—592no—
206#481—1182no—
207#477—22no—
208#477C2×D5914yes—
209#476—1182no—
210#464—1182no—
211#462—592no—
212#460—592no—
213C8×Q8—594no—
214C8×Q8—1182no—
215#431—591no—
216#431—591no—
217#431—1182no—
218C2×#107—21no—
219#423—591no—
220C2×C4⋊C8C2×D5911yescharacteristic
221#417—1181no—
222C2×#103—21no—
223#334—594no—
224#333—594no—
225#332—592no—
226#331—592no—
227C59#1940311yescharacteristic, complemented, Sylow
228C2×#41—1181no—
229C22×Q8—21no—
230C22×Q8C22⋊D5911yescharacteristic
231#137—2364no—
232C2×Q16—1182no—
233C2×SD16—1182no—
234C2×M16—1181no—
235C22×C8—21no—
236C4⋊Q8—21no—
237C4⋊Q8—21no—
238C4⋊Q8—21no—
239C4⋊Q8—591no—
240C4⋊Q8C22×D5912yes—
241#127—591no—
242#124—1181no—
243#121—1181no—
244#121—1181no—
245C4×Q8—21no—
246C4×Q8—21no—
247C4×Q8C22⋊D5912yes—
248C4×Q8C22×D5912yes—
249C4×D4—1181no—
250C4×D4—1181no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4C22⋊D5911yescharacteristic
255C2×C42—21no—
256C2×C42C22×D5911yescharacteristic
257#107—24no—
258#106—1184no—
259C4⋊C8—21no—
260C4⋊C8—592no—
261C4⋊C8—592no—
262C4⋊C8—592no—
263C4⋊C8—1184no—
264C4⋊C8—1181no—
265C4⋊C8C22×D5912yes—
266#103—24no—
267#103—1182no—
268#102—1184no—
269#102—1182no—
270C22⋊C8—592no—
271C22⋊C8—1181no—
272C4×C8—592no—
273C4×C8—1184no—
274C4×C8—1181no—
275#41—2362no—
276#41—2362no—
277C2×Q8—21no—
278C2×Q8—22no—
279C2×Q8—22no—
280C2×Q8—41no—
281C2×Q8—82no—
282C2×Q8—1181no—
283C2×Q8#1174567114yes—
284C2×Q8C2×C22⋊D5912yes—
285C2×D4—1181no—
286C2×D4—1181no—
287C2×D4—1181no—
288C2×D4—2361no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—41no—
292C22×C4—2361no—
293C22×C4D4×D5911yescharacteristic
294C22×C4#1174568711yescharacteristic
295C22×C4C2×C22⋊D5911yescharacteristic
296Q16—2364no—
297SD16—2364no—
298M16—2362no—
299C2×C8—22no—
300C2×C8—42no—
301C2×C8—594no—
302C2×C8—1182no—
303C2×C8—1182no—
304C4⋊C4—22no—
305C4⋊C4—21no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—1181no—
312C4⋊C4—1181no—
313C4⋊C4C2×C22⋊D5912yes—
314C22⋊C4—2361no—
315C22⋊C4—2361no—
316C42—22no—
317C42—22no—
318C42D4×D5911yescharacteristic
319C42#1174568711yescharacteristic
320C42C2×C22⋊D5911yescharacteristic
321C42C23×D5911yescharacteristic
322C23—2361no—
323C23#1427369911yescharacteristic
324Q8—24no—
325Q8—22no—
326Q8—82no—
327Q8—82no—
328Q8—2361no—
329Q8—2361no—
330Q8C2×#1174567114yes—
331D4—4721no—
332D4—4722no—
333D4—2361no—
334D4—2361no—
335D4—2362no—
336C2×C4—24no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—81no—
343C2×C4—82no—
344C2×C4—4722no—
345C2×C4—2361no—
346C2×C4#1427369911yescharacteristic
347C2×C4#1427369911yescharacteristic
348C2×C4#1427369911yescharacteristic
349C2×C4C2×#1174567112yes—
350C2×C4#1427374712yes—
351C2×C4C2×D4×D5911yescharacteristic
352C2×C4C22×C22⋊D5911yescharacteristic
353C2×C4C2×#1174568711yescharacteristic
354C8—42no—
355C8—1184no—
356C8—2362no—
357C8—2362no—
358C22—21no—
359C22—21no—
360C22—4721no—
361C22—4721no—
362C22—4721no—
363C22#1683763312yes—
364C22C2×#1427369911yescharacteristic
365C4—21no—
366C4—24no—
367C4—21no—
368C4—21no—
369C4—42no—
370C4—82no—
371C4—4721no—
372C4#1683763312yes—
373C4C22×#1174567111yescharacteristic
374C4C2×#1427374711yescharacteristic
375C4#1683821512yes—
376C2—22no—
377C2—9441no—
378C2#2143362311yescharacteristic
379C2C2×#1683763311yescharacteristic
380C2#2143382111yescharacteristic
subgroupG/H
centerC22C2×#14273699
commutator, cosocleC42×C59C24
FrattiniC42C23×D59
FittingC2×C59×#477C2
socleC22×C59C2×#121
prime radical#16837952C22
hypercenterC2×#477D59
hypocenterC59#19403
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#477

Complements

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

NH ≅ G/Ntype
3. #21433177C2semidirect
4. #21423514C2semidirect
5. #21423480C2semidirect
6. #21422866C2semidirect
7. #21422777C2semidirect
9. C2×C59×#477C2semidirect
187. C2×#477D59semidirect
227. C59#19403semidirect

Maximal subgroups

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

subgroupindex
#214352242
#214335682
#214331772
#214235142
#214234802
#214228662
#214227772
#214227462
C2×C59×#4772
#1940359