Finite Group Databasebeta

← #28486066

Subgroups — #28486066

Order 15104.

470 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 117 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#21433176C211yesmaximal, characteristic
2#21429312C211yesmaximal, characteristic
3#21424089C211yesmaximal, characteristic
4#21422781C211yesmaximal, characteristic
5#21421998C211yesmaximal, characteristic
6#21421997C211yesmaximal, characteristic
7C59×#1549C211yesmaximal, characteristic
8C2×C4×Dic118C2211yescharacteristic
9C8×Dic118—22no—
10#16837526—22no—
11#16837455C2211yescharacteristic
12#16837452—22no—
13#16837426C2211yescharacteristic
14C2×#14273645—21no—
15#16837340C414yes—
16#16837309—22no—
17#16837307—21no—
18#16837267—21no—
19C4×C4⋊Dic59C2211yescharacteristic
20#16837170—21no—
21#16837148—22no—
22C2×C4⋊Q8×C59C2211yescharacteristic
23C59×#431C2211yescharacteristic
24C59×#423C2211yescharacteristic
25C59×#332—22no—
26C22×Dic118D411yescharacteristic
27C2×#11745664—21no—
28C4×Dic118—21no—
29C4×Dic118—21no—
30C4×Dic118—21no—
31C4×Dic118—21no—
32C2×C4⋊Dic59—21no—
33C2×C4⋊Dic59D411yescharacteristic
34C2×Dic59⋊C4C2×C412yes—
35C2×C4×Dic59—21no—
36C2×C4×Dic59D412yes—
37C4⋊(C59⋊C8)—21no—
38C4⋊(C59⋊C8)—21no—
39C4⋊(C59⋊C8)D412yes—
40Dic59⋊C8—24no—
41#14273645—42no—
42C22⋊(C59⋊C8)—21no—
43C4×C59⋊C8—21no—
44C8×Dic59—24no—
45C59⋊#95—24no—
46C59⋊#95—24no—
47C59⋊#95C2×C412yes—
48C22×Q8×C59—21no—
49C2×M16×C59—21no—
50C4⋊Q8×C59—21no—
51C4⋊Q8×C59—21no—
52C4⋊Q8×C59—21no—
53C4⋊Q8×C59—21no—
54C2×C4⋊C4×C59—21no—
55C2×C4⋊C4×C59C2×C412yes—
56C2×C42×C59C2311yescharacteristic
57C4⋊C8×C59—21no—
58C4⋊C8×C59—21no—
59C4⋊C8×C59D412yes—
60C22⋊C8×C59—21no—
61C4×C8×C59—21no—
62C2×Q8×C59—44no—
63C2×Q8×C59—41no—
64C2×Q8×C59—41no—
65C22×C4×C59—22no—
66C22×C4×C59C22×C411yescharacteristic
67C22×C4×C59C2×D411yescharacteristic
68C22×C4×C59C2×D411yescharacteristic
69C22×Dic59—22no—
70C22×Dic59C22⋊C412yes—
71C22×Dic59C22⋊C412yes—
72C2×Dic118—21no—
73C2×Dic118—21no—
74C2×Dic118—21no—
75C2×Dic118—21no—
76C2×Dic118SD1612yes—
77C2×Dic118Q1612yes—
78M16×C59—41no—
79M16×C59—41no—
80C2×C8×C59—22no—
81C2×C8×C59—41no—
82C4⋊C4×C59—22no—
83C4⋊C4×C59—22no—
84C4⋊C4×C59—41no—
85C4⋊C4×C59—41no—
86C42×C59C2×D411yescharacteristic
87C42×C59C2×D411yescharacteristic
88C42×C59#4111yescharacteristic
89C42×C59#4111yescharacteristic
90C4⋊Dic59—21no—
91C4⋊Dic59—21no—
92C4⋊Dic59—42no—
93Dic59⋊C4—22no—
94Dic59⋊C4—22no—
95C4×Dic59—22no—
96C4×Dic59—22no—
97C4×Dic59—41no—
98C4×Dic59—41no—
99#11745664—41no—
100#11745664—41no—
101C2×C59⋊C8—22no—
102C2×C59⋊C8—41no—
103C23×C59C2×C22⋊C411yescharacteristic
104Q8×C59—82no—
105Q8×C59—82no—
106C2×C4×C59—22no—
107C2×C4×C59—41no—
108C2×C4×C59—44no—
109C2×C4×C59—42no—
110C2×C4×C59C4×D411yescharacteristic
111C2×C4×C59C4×D411yescharacteristic
112C2×C4×C59#12011yescharacteristic
113C2×C4×C59#12111yescharacteristic
114C2×C4×C59#12111yescharacteristic
115C2×C4×C59#12311yescharacteristic
116C2×C4×C59C2×SD1611yescharacteristic
117C2×C4×C59C2×Q1611yescharacteristic
118C2×C4×C59#13511yescharacteristic
119C2×C4×C59#13511yescharacteristic
120C2×Dic59—22no—
121C2×Dic59—22no—
122C2×Dic59—22no—
123C2×Dic59—42no—
124C2×Dic59—42no—
125C2×Dic59—42no—
126C2×Dic59—42no—
127C2×Dic59#10314yes—
128Dic118—22no—
129Dic118—22no—
130Dic118—22no—
131Dic118—22no—
132C8×C59—42no—
133C8×C59—81no—
134C59⋊C8—42no—
135C59⋊C8—81no—
136#9396—591nomaximal, Sylow
137C22×C59—21no—
138C22×C59—21no—
139C22×C59#38611yescharacteristic
140C22×C59C2×#10311yescharacteristic
141C22×C59#41611yescharacteristic
142C4×C59—21no—
143C4×C59—21no—
144C4×C59—21no—
145C4×C59—82no—
146C4×C59#45412yes—
147C4×C59#46511yescharacteristic
148C4×C59#46511yescharacteristic
149C4×C59#46711yescharacteristic
150C4×C59#46711yescharacteristic
151Dic59—24no—
152Dic59—42no—
153Dic59—82no—
154#2982—591no—
155#2568—591no—
156#1801—591no—
157#1549—591no—
158#1549D5911yescharacteristic
159#1303—591no—
160#1303—591no—
161C2×C59—21no—
162C2×C59—21no—
163C2×C59#188811yescharacteristic
164C2×C59#189211yescharacteristic
165C2×C59#191511yescharacteristic
166C2×C4⋊Q8C2×D5911yescharacteristic
167C2×C4×Q8—591no—
168C8×Q8—1182no—
169C8×Q8—1182no—
170#431—591no—
171#431—1182no—
172#431C2×D5911yescharacteristic
173#423—591no—
174#423C2×D5911yescharacteristic
175C2×#101—1181no—
176#400—594no—
177#391—1182no—
178#391—1181no—
179#382—1181no—
180C4×C4⋊C4—591no—
181#343—1181no—
182#332—22no—
183#332—1182no—
184C59#939611yescharacteristic, complemented, Sylow
185C22×Q8—21no—
186C22×Q8—591no—
187C2×M16—21no—
188C2×M16—1181no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8—21no—
192C4⋊Q8—21no—
193C4×Q8—1181no—
194C4×Q8—1181no—
195C4×Q8—1181no—
196C4×Q8—1181no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—592no—
199C2×C4⋊C4—591no—
200C2×C4⋊C4—1181no—
201C2×C4⋊C4C4×D5912yes—
202C2×C42—592no—
203C2×C42—1181no—
204C2×C42C22×D5911yescharacteristic
205C4⋊C8—21no—
206C4⋊C8—21no—
207C4⋊C8—592no—
208C4⋊C8—1184no—
209C4⋊C8—1181no—
210C4⋊C8—1181no—
211C4⋊C8C22⋊D5912yes—
212#101—2362no—
213C22⋊C8—21no—
214C22⋊C8—1181no—
215C4×C8—21no—
216C4×C8—1184no—
217C4×C8—1181no—
218#95—592no—
219#95—1184no—
220#95—1184no—
221C2×Q8—44no—
222C2×Q8—41no—
223C2×Q8—41no—
224C2×Q8—592no—
225C2×Q8—592no—
226C2×Q8—1181no—
227C2×Q8—1181no—
228C2×Q8—1181no—
229C2×Q8—1181no—
230C22×C4—22no—
231C22×C4—592no—
232C22×C4—592no—
233C22×C4—1182no—
234C22×C4C2×C4×D5911yescharacteristic
235C22×C4D4×D5911yescharacteristic
236C22×C4D4×D5911yescharacteristic
237M16—41no—
238M16—41no—
239M16—2361no—
240M16—2361no—
241C2×C8—22no—
242C2×C8—41no—
243C2×C8—1182no—
244C2×C8—2361no—
245C4⋊C4—22no—
246C4⋊C4—22no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—1182no—
250C4⋊C4—1182no—
251C4⋊C4—1181no—
252C4⋊C4—1181no—
253C4⋊C4—2362no—
254C42—1182no—
255C42—1182no—
256C42—2361no—
257C42—2361no—
258C42#1174568511yescharacteristic
259C42D4×D5911yescharacteristic
260C42#1174568711yescharacteristic
261C42C2×C22⋊D5911yescharacteristic
262C23C22⋊C4×D5911yescharacteristic
263Q8—82no—
264Q8—82no—
265Q8—1182no—
266Q8—1182no—
267Q8—1182no—
268Q8—1182no—
269C2×C4—22no—
270C2×C4—41no—
271C2×C4—44no—
272C2×C4—42no—
273C2×C4—594no—
274C2×C4—1182no—
275C2×C4—1182no—
276C2×C4—1182no—
277C2×C4—2362no—
278C2×C4—2362no—
279C2×C4—2362no—
280C2×C4—2362no—
281C2×C4(C22⋊D59)⋊C411yescharacteristic
282C2×C4C4×C22⋊D5911yescharacteristic
283C2×C4#1427369511yescharacteristic
284C2×C4#1427369811yescharacteristic
285C2×C4#1427370011yescharacteristic
286C2×C4#1427370911yescharacteristic
287C2×C4SD16×D5911yescharacteristic
288C2×C4Q16×D5911yescharacteristic
289C2×C4#1427374811yescharacteristic
290C2×C4#1427375011yescharacteristic
291C8—42no—
292C8—81no—
293C8—4721no—
294C8—2362no—
295C22—21no—
296C22—21no—
297C22#1683727811yescharacteristic
298C22#103×D5911yescharacteristic
299C22#1683739711yescharacteristic
300C4—21no—
301C4—21no—
302C4—21no—
303C4—82no—
304C4—1184no—
305C4—4722no—
306C4—2362no—
307C4#1683757312yes—
308C4#1683762811yescharacteristic
309C4#1683763411yescharacteristic
310C4#1683764011yescharacteristic
311C4#1683764211yescharacteristic
312C2—21no—
313C2—21no—
314C2#2142449611yescharacteristic
315C2#2142451611yescharacteristic
316C2#2142464511yescharacteristic
subgroupG/H
centerC22#16837278
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1549C2
socleC22×C59#386
prime radicalC23×C59C2×C22⋊C4
cosocleC2×C42×C59C23
hypercenter#1549D59
hypocenterC59#9396
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1549

Complements

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

NH ≅ G/Ntype
184. C59#9396semidirect

Maximal subgroups

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

subgroupindex
#214331762
#214293122
#214240892
#214227812
#214219982
#214219972
C59×#15492
#939659