Finite Group Databasebeta

← #28492289

Subgroups — #28492289

Order 15104.

440 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 97 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×#16837604C211yesmaximal, characteristic
2#21424085C211yesmaximal, characteristic
3#21422867C211yesmaximal, characteristic, complemented
4#21422768C211yesmaximal, characteristic, complemented
5#21422486C211yesmaximal, characteristic
6#21421991C211yesmaximal, characteristic, complemented
7C59×#1546C211yesmaximal, characteristic, complemented
8C22×#11745671—21no—
9C2×C4×D236C2211yescharacteristic
10#16837604—22no—
11#16837604—22no—
12#16837455C2211yescharacteristic
13#16837426C2211yescharacteristic
14C2×C4⋊(C59⋊C8)C2211yescharacteristic
15C2×#14273651—21no—
16C2×#14273643—21no—
17#16837170—21no—
18#16837153—22no—
19#16837148—22no—
20#16837129—22no—
21C2×C4⋊Q8×C59C2211yescharacteristic
22C59×#431C2211yescharacteristic
23C59×#423C2211yescharacteristic
24C59×#332—22no—
25C22×D236D411yescharacteristic
26C22×C4×D59—21no—
27C2×#11745671—24no—
28C2×#11745671—42no—
29C2×#11745671—42no—
30C2×#11745664—21no—
31C22×C59⋊C8—21no—
32C4×D236—21no—
33C4×D236—21no—
34C4×D236—21no—
35C4×D236—21no—
36C2×C4⋊Dic59D411yescharacteristic
37C2×(C2×D59)⋊C4—21no—
38C4⋊(C59⋊C8)—21no—
39C4⋊(C59⋊C8)—21no—
40C4⋊(C59⋊C8)—21no—
41C4⋊(C59⋊C8)—21no—
42C4⋊(C59⋊C8)D412yes—
43#14273651—24no—
44#14273643—42no—
45C22⋊(C59⋊C8)—21no—
46C4×C59⋊C8—21no—
47C22×Q8×C59—21no—
48C2×M16×C59—21no—
49C4⋊Q8×C59—21no—
50C4⋊Q8×C59—21no—
51C4⋊Q8×C59—21no—
52C4⋊Q8×C59D412yes—
53C2×C4⋊C4×C59—21no—
54C2×C4⋊C4×C59—21no—
55C2×C42×C59C2311yescharacteristic
56C4⋊C8×C59—21no—
57C4⋊C8×C59—21no—
58C4⋊C8×C59D412yes—
59C22⋊C8×C59—21no—
60C4×C8×C59—21no—
61C23×D59—21no—
62C2×Q8×C59—22no—
63C2×Q8×C59—41no—
64C2×Q8×C59—41no—
65C2×Q8×C59—41no—
66C2×Q8×C59—41no—
67C2×Q8×C59—41no—
68C22×C4×C59—41no—
69C22×C4×C59C2×D411yescharacteristic
70C22×C4×C59#4111yescharacteristic
71C22×C4×C59#4111yescharacteristic
72C22×Dic59—21no—
73C2×D236—21no—
74C2×D236—21no—
75C2×D236—21no—
76C2×D236—21no—
77C2×D236—21no—
78C2×D236SD1612yes—
79C2×C4×D59—41no—
80C2×C4×D59—41no—
81C2×C4×D59—82no—
82M16×C59—41no—
83M16×C59—41no—
84C2×C8×C59—22no—
85C2×C8×C59—41no—
86C4⋊C4×C59—22no—
87C4⋊C4×C59—22no—
88C4⋊C4×C59—41no—
89C4⋊C4×C59—41no—
90C42×C59C2×D411yescharacteristic
91C42×C59C2×D411yescharacteristic
92C42×C59C2×D411yescharacteristic
93C42×C59#4111yescharacteristic
94#11745671—48no—
95(C2×D59)⋊C4—41no—
96(C2×D59)⋊C4—41no—
97C4⋊Dic59—21no—
98C4⋊Dic59—21no—
99#11745664—41no—
100#11745664—41no—
101C2×C59⋊C8—22no—
102C2×C59⋊C8—41no—
103C2×C59⋊C8—41no—
104C2×C59⋊C8—41no—
105C2×C59⋊C8—41no—
106C23×C59#12311yescharacteristic
107C22×D59—22no—
108C22×D59—41no—
109C22×D59—41no—
110C22×D59—81no—
111C22×D59—81no—
112Q8×C59—42no—
113Q8×C59—42no—
114Q8×C59—81no—
115Q8×C59—81no—
116C2×C4×C59—22no—
117C2×C4×C59—41no—
118C2×C4×C59—42no—
119C2×C4×C59—81no—
120C2×C4×C59—81no—
121C2×C4×C59#12011yescharacteristic
122C2×C4×C59#12111yescharacteristic
123C2×C4×C59#12111yescharacteristic
124C2×C4×C59#12111yescharacteristic
125C2×C4×C59#12311yescharacteristic
126C2×C4×C59#12411yescharacteristic
127C2×C4×C59C2×SD1611yescharacteristic
128C2×C4×C59#13511yescharacteristic
129C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
130C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
131C2×Dic59—41no—
132C2×Dic59—81no—
133D236—22no—
134D236—22no—
135D236—41no—
136D236—41no—
137C4×D59—82no—
138C4×D59—82no—
139C8×C59—42no—
140C8×C59—81no—
141C59⋊C8—42no—
142C59⋊C8—81no—
143C59⋊C8—81no—
144#10445—591nomaximal, Sylow
145C22×C59—21no—
146C22×C59—21no—
147C22×C59#39411yescharacteristic
148C22×C59#48111yescharacteristic
149C22×C59#48211yescharacteristic
150C2×D59—42no—
151C2×D59—82no—
152C2×D59—81no—
153C2×D59—81no—
154C2×D59—82no—
155C4×C59—21no—
156C4×C59—21no—
157C4×C59—21no—
158C4×C59—82no—
159C4×C59#44911yescharacteristic
160C4×C59#45011yescharacteristic
161C4×C59#46511yescharacteristic
162C4×C59#46911yescharacteristic
163C4×C59#47112yes—
164Dic59—81no—
165C2×#461—591no—
166#1800—591no—
167#1570—591no—
168#1546—591no—
169#1546D5911yescharacteristic, complemented
170#1478—591no—
171#1301—591no—
172C2×C59—21no—
173C2×C59—21no—
174C2×C59#203411yescharacteristic
175C2×C59#204611yescharacteristic
176C2×C59#204911yescharacteristic
177D59—82no—
178C22×SD16—1181no—
179C2×C4⋊Q8C2×D5911yescharacteristic
180C2×C4×D4—591no—
181#461—1182no—
182#461—1182no—
183#431—591no—
184#431C2×D5911yescharacteristic
185#423—591no—
186#423C2×D5911yescharacteristic
187C2×C4⋊C8—591no—
188C2×#103—1181no—
189C2×#100—1181no—
190#343—1181no—
191#334—1182no—
192#332—22no—
193#332—1182no—
194D4⋊C8—1182no—
195C59#1044511yescharacteristic, complemented, Sylow
196C22×Q8—21no—
197C22×D4—591no—
198C23×C4—1181no—
199C2×SD16—1184no—
200C2×SD16—2362no—
201C2×SD16—2362no—
202C2×M16—21no—
203C2×M16—1181no—
204C22×C8—1181no—
205C4⋊Q8—21no—
206C4⋊Q8—21no—
207C4⋊Q8—21no—
208C4⋊Q8D23612yes—
209C4×D4—1181no—
210C4×D4—1181no—
211C4×D4—1181no—
212C4×D4—1181no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—591no—
216C2×C22⋊C4—1181no—
217C2×C42C22×D5911yescharacteristic
218C4⋊C8—21no—
219C4⋊C8—21no—
220C4⋊C8—592no—
221C4⋊C8—1181no—
222C4⋊C8—1181no—
223C4⋊C8—1181no—
224C4⋊C8—1181no—
225C4⋊C8C22⋊D5912yes—
226#103—1184no—
227#100—2362no—
228C22⋊C8—21no—
229C22⋊C8—1181no—
230C4×C8—21no—
231C4×C8—1181no—
232C24—1181no—
233C2×Q8—22no—
234C2×Q8—41no—
235C2×Q8—41no—
236C2×Q8—41no—
237C2×Q8—41no—
238C2×Q8—41no—
239C2×D4—592no—
240C2×D4—1181no—
241C2×D4—1181no—
242C2×D4—1181no—
243C2×D4—1181no—
244C2×D4—1181no—
245C22×C4—41no—
246C22×C4—1181no—
247C22×C4—4722no—
248C22×C4—2361no—
249C22×C4—2361no—
250C22×C4#1174568511yescharacteristic
251C22×C4D4×D5911yescharacteristic
252C22×C4#1174568711yescharacteristic
253SD16—2368no—
254M16—41no—
255M16—41no—
256M16—2361no—
257M16—2361no—
258C2×C8—22no—
259C2×C8—41no—
260C2×C8—1182no—
261C2×C8—2361no—
262C2×C8—2361no—
263C2×C8—2361no—
264C2×C8—2361no—
265C4⋊C4—22no—
266C4⋊C4—22no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—1181no—
270C4⋊C4—1181no—
271C22⋊C4—2361no—
272C22⋊C4—2361no—
273C42C2×D23611yescharacteristic
274C42D4×D5911yescharacteristic
275C42#1174568711yescharacteristic
276C42C2×C22⋊D5911yescharacteristic
277C23—1182no—
278C23—4721no—
279C23—4721no—
280C23—2361no—
281C23—2361no—
282C23#1427370911yescharacteristic
283Q8—42no—
284Q8—42no—
285Q8—81no—
286Q8—81no—
287D4—1182no—
288D4—1182no—
289D4—2361no—
290D4—2361no—
291C2×C4—22no—
292C2×C4—41no—
293C2×C4—42no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—4722no—
297C2×C4—4722no—
298C2×C4—4721no—
299C2×C4—2361no—
300C2×C4#1427369711yescharacteristic
301C2×C4#1427369811yescharacteristic
302C2×C4#1427369911yescharacteristic
303C2×C4#1427370111yescharacteristic
304C2×C4#1427371111yescharacteristic
305C2×C4#1427371311yescharacteristic
306C2×C4SD16×D5911yescharacteristic
307C2×C4#1427374811yescharacteristic
308C2×C4#1427375111yescharacteristic
309C2×C4#1427375111yescharacteristic
310C8—42no—
311C8—81no—
312C8—4721no—
313C8—4721no—
314C8—2362no—
315C22—21no—
316C22—21no—
317C22—4722no—
318C22—4722no—
319C22—4721no—
320C22—4721no—
321C22—2362no—
322C22#1683731811yescharacteristic
323C22#1683771911yescharacteristic
324C22#1683772511yescharacteristic
325C4—21no—
326C4—21no—
327C4—21no—
328C4—82no—
329C4—4721no—
330C4#1683754811yescharacteristic
331C4#1683755411yescharacteristic
332C4#1683762811yescharacteristic
333C4#1683765211yescharacteristic
334C4#1683766712yes—
335C2—21no—
336C2—21no—
337C2—4722no—
338C2#2142528411yescharacteristic
339C2#2142535611yescharacteristic
340C2#2142537511yescharacteristic
subgroupG/H
centerC22#16837318
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC59×#1546C2
socleC22×C59#394
prime radicalC22×D236D4
hypercenter#1546D59
hypocenterC59#10445
LCSG ▷ C2×C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1546

Complements

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

NH ≅ G/Ntype
3. #21422867C2semidirect
4. #21422768C2semidirect
6. #21421991C2semidirect
7. C59×#1546C2semidirect
169. #1546D59semidirect
195. C59#10445semidirect

Maximal subgroups

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

subgroupindex
C2×#168376042
#214240852
#214228672
#214227682
#214224862
#214219912
C59×#15462
#1044559