Finite Group Databasebeta

← C59×#7394 #28419503

Subgroups — C59×#7394 #28419503

Order 15104.

418 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 42 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
1C59×#2021C211yesmaximal, characteristic
2C59×#1316C211yesmaximal, characteristic, complemented
3C59×#1304C211yesmaximal, characteristic, complemented
4C59×#534C411yescharacteristic, complemented
5C2×#120×C59—21no—
6C2×#99×C59—21no—
7C59×#379—21no—
8C59×#379C2211yescharacteristic
9C59×#342C411yescharacteristic, complemented
10C59×#323—21no—
11C25×C59D411yescharacteristic
12C22×D4×C59—21no—
13C22×D4×C59—41no—
14#123×C59—41no—
15#121×C59—41no—
16#120×C59—42no—
17#120×C59—41no—
18#120×C59—41no—
19#120×C59—41no—
20C2×C22⋊C4×C59—21no—
21C2×C22⋊C4×C59—22no—
22C2×C22⋊C4×C59—41no—
23C2×C22⋊C4×C59—41no—
24C2×C22⋊C4×C59—41no—
25C2×C22⋊C4×C59—41no—
26C2×C22⋊C4×C59C2×C411yescharacteristic
27C2×C22⋊C4×C59D411yescharacteristic
28#99×C59—42no—
29C22⋊C8×C59—41no—
30C24×C59—21no—
31C24×C59—21no—
32C24×C59—21no—
33C24×C59—41no—
34C24×C59—41no—
35C24×C59—82no—
36C24×C59C22⋊C411yescharacteristic
37C24×C59D811yescharacteristic
38C24×C59SD1611yescharacteristic
39C2×D4×C59—41no—
40C2×D4×C59—41no—
41C2×D4×C59—41no—
42C2×D4×C59—42no—
43C2×D4×C59—42no—
44C2×D4×C59—41no—
45C2×D4×C59—41no—
46C2×D4×C59—41no—
47C2×D4×C59—41no—
48C2×D4×C59—84no—
49C22×C4×C59—21no—
50C22×C4×C59—21no—
51C22×C4×C59—21no—
52C22×C4×C59—41no—
53C22×C4×C59—42no—
54C2×C8×C59—81no—
55C4⋊C4×C59—41no—
56C22⋊C4×C59—22no—
57C22⋊C4×C59—42no—
58C22⋊C4×C59—42no—
59C22⋊C4×C59—42no—
60C22⋊C4×C59—41no—
61C22⋊C4×C59—41no—
62C22⋊C4×C59—41no—
63C22⋊C4×C59—82no—
64C22⋊C4×C59—82no—
65C22⋊C4×C59—81no—
66C22⋊C4×C59—81no—
67C22⋊C4×C59—82no—
68C23×C59—21no—
69C23×C59—21no—
70C23×C59—21no—
71C23×C59—21no—
72C23×C59—41no—
73C23×C59—41no—
74C23×C59—42no—
75C23×C59—42no—
76C23×C59—42no—
77C23×C59—41no—
78C23×C59—41no—
79C23×C59—44no—
80C23×C59—41no—
81C23×C59—41no—
82C23×C59—41no—
83C23×C59—81no—
84C23×C59—81no—
85C23×C59—81no—
86C23×C59—81no—
87C23×C59—81no—
88C23×C59—82no—
89C23×C59—82no—
90C23×C59—82no—
91C23×C59—82no—
92C23×C59#9911yescharacteristic
93C23×C59#10211yescharacteristic
94C23×C59#10411yescharacteristic
95D4×C59—81no—
96D4×C59—81no—
97D4×C59—84no—
98D4×C59—84no—
99D4×C59—81no—
100D4×C59—81no—
101D4×C59—161no—
102C2×C4×C59—42no—
103C2×C4×C59—41no—
104C2×C4×C59—41no—
105C2×C4×C59—41no—
106C2×C4×C59—41no—
107C2×C4×C59—41no—
108C2×C4×C59—82no—
109C2×C4×C59—81no—
110C2×C4×C59—81no—
111C2×C4×C59—82no—
112C2×C4×C59—82no—
113C2×C4×C59—82no—
114C8×C59—161no—
115#7394C5911yesmaximal, characteristic, direct product factor, Sylow
116C22×C59—22no—
117C22×C59—21no—
118C22×C59—21no—
119C22×C59—41no—
120C22×C59—41no—
121C22×C59—41no—
122C22×C59—41no—
123C22×C59—42no—
124C22×C59—41no—
125C22×C59—41no—
126C22×C59—42no—
127C22×C59—41no—
128C22×C59—42no—
129C22×C59—41no—
130C22×C59—81no—
131C22×C59—82no—
132C22×C59—82no—
133C22×C59—82no—
134C22×C59—81no—
135C22×C59—84no—
136C22×C59—81no—
137C22×C59—82no—
138C22×C59—81no—
139C22×C59—161no—
140C22×C59#32711yescharacteristic
141C22×C59AΣL(1,𝔽16)11yescharacteristic
142C22×C59#35311yescharacteristic
143C4×C59—81no—
144C4×C59—81no—
145C4×C59—82no—
146C4×C59—81no—
147C4×C59—162no—
148#2021C2×C5911yescharacteristic
149#1316C2×C5911yescharacteristic, complemented
150#1304C2×C5911yescharacteristic, complemented
151C2×C59—21no—
152C2×C59—21no—
153C2×C59—21no—
154C2×C59—41no—
155C2×C59—42no—
156C2×C59—41no—
157C2×C59—81no—
158C2×C59—161no—
159C2×C59#134311yescharacteristic
160#534C4×C5911yescharacteristic, complemented
161C2×#120—21no—
162C2×#99—21no—
163#379—21no—
164#379C22×C5911yescharacteristic
165#342C4×C5911yescharacteristic, complemented
166#323—21no—
167C59#739411yescharacteristic, direct product factor, Sylow
168C25D4×C5911yescharacteristic
169C22×D4—21no—
170C22×D4—41no—
171#123—41no—
172#121—41no—
173#120—42no—
174#120—41no—
175#120—41no—
176#120—41no—
177C2×C22⋊C4—21no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—41no—
180C2×C22⋊C4—41no—
181C2×C22⋊C4—41no—
182C2×C22⋊C4—41no—
183C2×C22⋊C4C2×C4×C5911yescharacteristic
184C2×C22⋊C4D4×C5911yescharacteristic
185#99—42no—
186C22⋊C8—41no—
187C24—21no—
188C24—21no—
189C24—21no—
190C24—41no—
191C24—41no—
192C24—82no—
193C24C22⋊C4×C5911yescharacteristic
194C24D8×C5911yescharacteristic
195C24SD16×C5911yescharacteristic
196C2×D4—41no—
197C2×D4—41no—
198C2×D4—41no—
199C2×D4—42no—
200C2×D4—42no—
201C2×D4—41no—
202C2×D4—41no—
203C2×D4—41no—
204C2×D4—41no—
205C2×D4—84no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—41no—
210C22×C4—42no—
211C2×C8—81no—
212C4⋊C4—41no—
213C22⋊C4—22no—
214C22⋊C4—42no—
215C22⋊C4—42no—
216C22⋊C4—42no—
217C22⋊C4—41no—
218C22⋊C4—41no—
219C22⋊C4—41no—
220C22⋊C4—82no—
221C22⋊C4—82no—
222C22⋊C4—81no—
223C22⋊C4—81no—
224C22⋊C4—82no—
225C23—21no—
226C23—21no—
227C23—21no—
228C23—21no—
229C23—41no—
230C23—41no—
231C23—42no—
232C23—42no—
233C23—42no—
234C23—41no—
235C23—41no—
236C23—44no—
237C23—41no—
238C23—41no—
239C23—41no—
240C23—81no—
241C23—81no—
242C23—81no—
243C23—81no—
244C23—81no—
245C23—82no—
246C23—82no—
247C23—82no—
248C23—82no—
249C23#99×C5911yescharacteristic
250C23#102×C5911yescharacteristic
251C23#104×C5911yescharacteristic
252D4—81no—
253D4—81no—
254D4—84no—
255D4—84no—
256D4—81no—
257D4—81no—
258D4—161no—
259C2×C4—42no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—82no—
266C2×C4—81no—
267C2×C4—81no—
268C2×C4—82no—
269C2×C4—82no—
270C2×C4—82no—
271C8—161no—
272C22—22no—
273C22—21no—
274C22—21no—
275C22—41no—
276C22—41no—
277C22—41no—
278C22—41no—
279C22—42no—
280C22—41no—
281C22—41no—
282C22—42no—
283C22—41no—
284C22—42no—
285C22—41no—
286C22—81no—
287C22—82no—
288C22—82no—
289C22—82no—
290C22—81no—
291C22—84no—
292C22—81no—
293C22—82no—
294C22—81no—
295C22—161no—
296C22C59×#32711yescharacteristic
297C22C59×AΣL(1,𝔽16)11yescharacteristic
298C22C59×#35311yescharacteristic
299C4—81no—
300C4—81no—
301C4—82no—
302C4—81no—
303C4—162no—
304C2—21no—
305C2—21no—
306C2—21no—
307C2—41no—
308C2—42no—
309C2—41no—
310C2—81no—
311C2—161no—
312C2C59×#134311yescharacteristic
subgroupG/H
center, socleC2×C59#1343
commutatorC2×C22⋊C4C2×C4×C59
Frattini, cosocle#379C22×C59
prime radicalC59×#2021C2
LCSG ▷ C2×C22⋊C4 ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C59 ◁ C23×C59 ◁ C24×C59 ◁ C59×#379 ◁ G

Complements

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

NH ≅ G/Ntype
2. C59×#1316C2semidirect
3. C59×#1304C2semidirect
4. C59×#534C4semidirect
9. C59×#342C4semidirect
115. #7394C59direct (N × H)
149. #1316C2×C59semidirect
150. #1304C2×C59semidirect
160. #534C4×C59semidirect
165. #342C4×C59semidirect
167. C59#7394direct (N × H)

Maximal subgroups

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

subgroupindex
C59×#20212
C59×#13162
C59×#13042
#739459