Finite Group Databasebeta

← #28504232

Subgroups — #28504232

Order 15104.

952 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 952 conjugacy classes of subgroups — too many to draw. Only its 135 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#21425785C214yesmaximal, complemented
2#21424020C212yesmaximal, complemented
3C59×#3486C211yesmaximal, characteristic
4C2×#14273646—22no—
5C2×#14273646C2214yes—
6#16837366—22no—
7#16837201—24no—
8#16837199—24no—
9C2×2−1+4×C59C411yescharacteristic
10C23×Q8×C59C2211yescharacteristic
11D4×Q8×C59—24no—
12C59×#548—24no—
13C59×#536—22no—
14C59×#536C2212yes—
15C59×#525—22no—
16C59×#525C412yes—
17C59×#519C411yescharacteristic
18C2×#11745664—24no—
19#14273646—28no—
20#14273646—44no—
21C22⋊(C59⋊C8)—44no—
222−1+4×C59—44no—
23C2×#41×C59—22no—
24C2×#41×C59—22no—
25C2×#41×C59C2×C412yes—
26C22×Q8×C59—22no—
27C22×Q8×C59—24no—
28C22×Q8×C59—44no—
29C22×Q8×C59C2×C411yescharacteristic
30C22×Q8×C59D412yes—
31C22×Q8×C59C2311yescharacteristic
32C23×C4×C59C2×C411yescharacteristic
33C23×C4×C59D412yes—
34C4⋊Q8×C59—24no—
35C4⋊Q8×C59D414yes—
36#124×C59—24no—
37#124×C59D414yes—
38#123×C59—22no—
39#123×C59—24no—
40C22⋊Q8×C59—24no—
41C22⋊Q8×C59—28no—
42C22⋊Q8×C59—22no—
43C22⋊Q8×C59—24no—
44#121×C59—24no—
45#121×C59—22no—
46#120×C59—22no—
47C4×Q8×C59—24no—
48C4×D4×C59—28no—
49C4×D4×C59—24no—
50#117×C59—22no—
51#117×C59C2×C412yes—
52C24×C59C22⋊C411yescharacteristic
53#41×C59—44no—
54#41×C59—44no—
55#41×C59—48no—
56#41×C59—44no—
57C2×Q8×C59—22no—
58C2×Q8×C59—24no—
59C2×Q8×C59—24no—
60C2×Q8×C59—22no—
61C2×Q8×C59—24no—
62C2×Q8×C59—42no—
63C2×Q8×C59—42no—
64C2×Q8×C59—416no—
65C2×Q8×C59—48no—
66C2×Q8×C59C22⋊C412yes—
67C2×Q8×C59C2×D414yes—
68C2×D4×C59—24no—
69C2×D4×C59—22no—
70C2×D4×C59—22no—
71C2×D4×C59C22⋊C412yes—
72C22×C4×C59—24no—
73C22×C4×C59—24no—
74C22×C4×C59—22no—
75C22×C4×C59—22no—
76C22×C4×C59—24no—
77C22×C4×C59—42no—
78C22×C4×C59—42no—
79C22×C4×C59C22⋊C411yescharacteristic
80C22×C4×C59C22⋊C412yes—
81C22×C4×C59C22×C411yescharacteristic
82C22×C4×C59C2×D412yes—
83C4⋊C4×C59—24no—
84C4⋊C4×C59—24no—
85C4⋊C4×C59—22no—
86C4⋊C4×C59—28no—
87C22⋊C4×C59—24no—
88C22⋊C4×C59—24no—
89C22⋊C4×C59—22no—
90C22⋊C4×C59—28no—
91C42×C59—24no—
92C42×C59C22⋊C414yes—
93#11745664—44no—
94#11745664—44no—
95C2×C59⋊C8—44no—
96C23×C59—21no—
97C23×C59—22no—
98C23×C59—22no—
99C23×C59—42no—
100C23×C59C2×C22⋊C411yescharacteristic
101Q8×C59—28no—
102Q8×C59—44no—
103Q8×C59—48no—
104Q8×C59—42no—
105Q8×C59—44no—
106D4×C59—42no—
107D4×C59—44no—
108D4×C59—48no—
109D4×C59—44no—
110C2×C4×C59—22no—
111C2×C4×C59—22no—
112C2×C4×C59—22no—
113C2×C4×C59—22no—
114C2×C4×C59—22no—
115C2×C4×C59—24no—
116C2×C4×C59—21no—
117C2×C4×C59—22no—
118C2×C4×C59—22no—
119C2×C4×C59—48no—
120C2×C4×C59—44no—
121C2×C4×C59—48no—
122C2×C4×C59—44no—
123C2×C4×C59#9914yes—
124C2×C4×C59C2×C22⋊C412yes—
125C2×C4×C59#12014yes—
126C59⋊C8—84no—
127#12710—591nomaximal, Sylow
128C22×C59—22no—
129C22×C59—24no—
130C22×C59—41no—
131C22×C59—41no—
132C22×C59—42no—
133C22×C59—44no—
134C22×C59—41no—
135C22×C59—42no—
136C22×C59#37911yescharacteristic
137C22×C59C2×#9912yes—
138C4×C59—24no—
139C4×C59—24no—
140C4×C59—22no—
141C4×C59—44no—
142C4×C59—41no—
143C4×C59—42no—
144#3486D5911yescharacteristic
145#2133—594no—
146#1784—592no—
147C2×C59—21no—
148C2×C59—22no—
149C2×C59—42no—
150C2×C59—82no—
151C2×C59#178111yescharacteristic
152C2×2−1+4Dic5911yescharacteristic
153C23×Q8C2×D5911yescharacteristic
154D4×Q8—24no—
155#548—24no—
156#536—22no—
157#536C2×D5912yes—
158#525—22no—
159#525Dic5912yes—
160#519Dic5911yescharacteristic
161C2×#101—594no—
162C2×#101—1182no—
163#407—1182no—
164#356—1184no—
165#355—1184no—
166C59#1271011yescharacteristic, complemented, Sylow
1672−1+4—44no—
168C2×#41—22no—
169C2×#41—22no—
170C2×#41C2×Dic5912yes—
171C22×Q8—22no—
172C22×Q8—24no—
173C22×Q8—44no—
174C22×Q8C2×Dic5911yescharacteristic
175C22×Q8C22⋊D5912yes—
176C22×Q8C22×D5911yescharacteristic
177C23×C4C2×Dic5911yescharacteristic
178C23×C4C22⋊D5912yes—
179C2×M16—1184no—
180C4⋊Q8—24no—
181C4⋊Q8C22⋊D5914yes—
182#124—24no—
183#124C22⋊D5914yes—
184#123—22no—
185#123—24no—
186C22⋊Q8—24no—
187C22⋊Q8—28no—
188C22⋊Q8—22no—
189C22⋊Q8—24no—
190#121—24no—
191#121—22no—
192#120—22no—
193C4×Q8—24no—
194C4×D4—28no—
195C4×D4—24no—
196#117—22no—
197#117C2×Dic5912yes—
198#101—1188no—
199#101—2364no—
200C22⋊C8—2364no—
201C24C22⋊Dic5911yescharacteristic
202#41—44no—
203#41—44no—
204#41—48no—
205#41—44no—
206C2×Q8—22no—
207C2×Q8—24no—
208C2×Q8—24no—
209C2×Q8—22no—
210C2×Q8—24no—
211C2×Q8—42no—
212C2×Q8—42no—
213C2×Q8—416no—
214C2×Q8—48no—
215C2×Q8C22⋊Dic5912yes—
216C2×Q8C2×C22⋊D5914yes—
217C2×D4—24no—
218C2×D4—22no—
219C2×D4—22no—
220C2×D4C22⋊Dic5912yes—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—24no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4C22⋊Dic5911yescharacteristic
229C22×C4C22⋊Dic5912yes—
230C22×C4C22×Dic5911yescharacteristic
231C22×C4C2×C22⋊D5912yes—
232M16—2364no—
233M16—2364no—
234C2×C8—2364no—
235C4⋊C4—24no—
236C4⋊C4—24no—
237C4⋊C4—22no—
238C4⋊C4—28no—
239C22⋊C4—24no—
240C22⋊C4—24no—
241C22⋊C4—22no—
242C22⋊C4—28no—
243C42—24no—
244C42C22⋊Dic5914yes—
245C23—21no—
246C23—22no—
247C23—22no—
248C23—42no—
249C23C2×C22⋊Dic5911yescharacteristic
250Q8—28no—
251Q8—44no—
252Q8—48no—
253Q8—42no—
254Q8—44no—
255D4—42no—
256D4—44no—
257D4—48no—
258D4—44no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—24no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—48no—
269C2×C4—44no—
270C2×C4—48no—
271C2×C4—44no—
272C2×C4#1427364214yes—
273C2×C4C2×C22⋊Dic5912yes—
274C2×C4#1427369614yes—
275C8—4724no—
276C22—22no—
277C22—24no—
278C22—41no—
279C22—41no—
280C22—42no—
281C22—44no—
282C22—41no—
283C22—42no—
284C22#1683725811yescharacteristic
285C22C2×#1427364212yes—
286C4—24no—
287C4—24no—
288C4—22no—
289C4—44no—
290C4—41no—
291C4—42no—
292C2—21no—
293C2—22no—
294C2—42no—
295C2—82no—
296C2#2142401111yescharacteristic
subgroupG/H
centerC2#21424011
commutatorC22×C4×C59C22×C4
FrattiniC22×Q8C22×D59
Fitting, prime radicalC59×#3486C2
socleC2×C59#1781
cosocleC22×Q8×C59C23
hypercenter#3486D59
hypocenterC59#12710
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C23 ◁ C23×Q8 ◁ #3486

Complements

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

NH ≅ G/Ntype
1. #21425785C2semidirect
2. #21424020C2semidirect
166. C59#12710semidirect

Maximal subgroups

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

subgroupindex
#214257852
#214240202
C59×#34862
#1271059