Finite Group Databasebeta

← #15778550

Subgroups — #15778550

Order 3328.

714 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 714 conjugacy classes of subgroups — too many to draw. Only its 127 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#13116505C212yesmaximal
2C2×#11718133C212yesmaximal, complemented
3#13114944C212yesmaximal, complemented
4C13×#3510C211yesmaximal, characteristic
5C13×#559—22no—
6C13×#556—22no—
7C13×#551—24no—
8C13×#542—22no—
9C13×#536C411yescharacteristic, complemented
10C13×#535C2211yescharacteristic
11C13×#533C411yescharacteristic, complemented
12C13×#529—22no—
13C2×C13×#124C2211yescharacteristic
14C13×#519C411yescharacteristic, complemented
15C13×#518C2211yescharacteristic
16C4×C13×#41C411yescharacteristic, complemented
17C2×#88074C2214yes—
18#11718137—22no—
19#11718133—22no—
20#11718133—22no—
21#11718133—22no—
22#11718119—24no—
23C2×C13×#41C2×C411yescharacteristic
24C22×Q8×C13C2×C411yescharacteristic
25C22×D4×C13C2311yescharacteristic
26C13×C4⋊Q8—21no—
27C13×C4⋊Q8—21no—
28C13×#127D412yes—
29C13×#126—24no—
30C13×#126—24no—
31C13×#125—21no—
32C13×#124—24no—
33C13×#124—21no—
34C13×#124—22no—
35C13×#124D412yes—
36C13×#124D412yes—
37C13×#124D412yes—
38C13×#123—24no—
39C13×#123—24no—
40C13×#123—22no—
41C13×C22⋊Q8—24no—
42C13×C22⋊Q8—22no—
43C13×C22⋊Q8—22no—
44C13×#121—22no—
45C13×#120—22no—
46C4×Q8×C13—21no—
47C4×Q8×C13—22no—
48C4×Q8×C13—22no—
49C4×Q8×C13—22no—
50C4×D4×C13—24no—
51C4×D4×C13—21no—
52C4×D4×C13—22no—
53C4×D4×C13—22no—
54C4×D4×C13—22no—
55C13×#117—24no—
56C13×#117—22no—
57C13×#117—21no—
58C13×#117C2×C411yescharacteristic
59C13×#117C2×C411yescharacteristic
60C13×#117C2×C411yescharacteristic
61C2×C13×C22⋊C4—22no—
62C2×C13×C22⋊C4D414yes—
63C2×C42×C13—21no—
64C2×C42×C13C2×C411yescharacteristic
65C2×C22⋊Dic13—24no—
66#88074—24no—
67#88074—24no—
68#12728—131nomaximal, Sylow
69C24×C13C2×D412yes—
70C13×#41—42no—
71C2×Q8×C13—22no—
72C2×Q8×C13—21no—
73C2×Q8×C13—42no—
74C2×Q8×C13C22⋊C412yes—
75C2×Q8×C13C22⋊C411yescharacteristic
76C2×D4×C13—21no—
77C2×D4×C13—24no—
78C2×D4×C13—42no—
79C2×D4×C13C22⋊C411yescharacteristic
80C2×D4×C13C2×D412yes—
81C2×D4×C13C2×D412yes—
82C22×C4×C13—21no—
83C22×C4×C13—22no—
84C22×C4×C13—21no—
85C22×C4×C13—24no—
86C22×C4×C13—21no—
87C22×C4×C13C22×C411yescharacteristic
88C22×Dic13—44no—
89C13×C4⋊C4—22no—
90C13×C4⋊C4—22no—
91C13×C4⋊C4—24no—
92C13×C4⋊C4—24no—
93C13×C4⋊C4—22no—
94C13×C4⋊C4—22no—
95C13×C4⋊C4—21no—
96C13×C4⋊C4—21no—
97C13×C4⋊C4—21no—
98C13×C22⋊C4—22no—
99C13×C22⋊C4—24no—
100C13×C22⋊C4—21no—
101C13×C22⋊C4—24no—
102C13×C22⋊C4—24no—
103C13×C22⋊C4—24no—
104C13×C22⋊C4—24no—
105C13×C22⋊C4C22⋊C412yes—
106C13×C22⋊C4C22⋊C412yes—
107C42×C13—22no—
108C42×C13—24no—
109C42×C13—21no—
110C42×C13—21no—
111C42×C13—21no—
112C42×C13C22⋊C411yescharacteristic
113C42×C13C22⋊C411yescharacteristic
114C42×C13C22⋊C412yes—
115C22⋊Dic13—44no—
116C22⋊Dic13—44no—
117#3510D1311yescharacteristic
118#2127—132no—
119C2×#354—132no—
120#1828—132no—
121C23×C13—24no—
122C23×C13—21no—
123C23×C13—82no—
124C23×C13C2×C22⋊C411yescharacteristic
125C23×C13#12014yes—
126Q8×C13—42no—
127Q8×C13—42no—
128Q8×C13—41no—
129Q8×C13—82no—
130D4×C13—42no—
131D4×C13—42no—
132D4×C13—41no—
133D4×C13—42no—
134D4×C13—84no—
135C2×C4×C13—22no—
136C2×C4×C13—22no—
137C2×C4×C13—21no—
138C2×C4×C13—22no—
139C2×C4×C13—22no—
140C2×C4×C13—21no—
141C2×C4×C13—21no—
142C2×C4×C13—22no—
143C2×C4×C13—42no—
144C2×C4×C13—42no—
145C2×C4×C13—82no—
146C2×C4×C13—84no—
147C2×C4×C13—81no—
148C2×C4×C13C2×C22⋊C411yescharacteristic
149C2×C4×C13C2×C22⋊C411yescharacteristic
150C2×Dic13—84no—
151C2×Dic13—164no—
152#559—22no—
153#556—22no—
154#551—24no—
155#542—22no—
156#536Dic1311yescharacteristic, complemented
157#535C2×D1311yescharacteristic
158#533Dic1311yescharacteristic, complemented
159#529—22no—
160C2×#124C2×D1311yescharacteristic
161#519Dic1311yescharacteristic, complemented
162#518C2×D1311yescharacteristic
163C4×#41Dic1311yescharacteristic, complemented
164C2×#99—134no—
165#354—262no—
166#354—262no—
167#354—262no—
168#353—262no—
169#342—264no—
170C22×C13—24no—
171C22×C13—41no—
172C22×C13—44no—
173C22×C13—42no—
174C22×C13—41no—
175C22×C13—84no—
176C22×C13#37911yescharacteristic
177C22×C13#41012yes—
178C4×C13—21no—
179C4×C13—21no—
180C4×C13—42no—
181C4×C13—41no—
182C4×C13—42no—
183C4×C13—41no—
184C4×C13—42no—
185C4×C13—41no—
186C4×C13—42no—
187Dic13—164no—
188C2×#41C2×Dic1311yescharacteristic
189C22×Q8C2×Dic1311yescharacteristic
190C22×D4C22×D1311yescharacteristic
191C4⋊Q8—21no—
192C4⋊Q8—21no—
193#127C22⋊D1312yes—
194#126—24no—
195#126—24no—
196#125—21no—
197#124—24no—
198#124—21no—
199#124—22no—
200#124C22⋊D1312yes—
201#124C22⋊D1312yes—
202#124C22⋊D1312yes—
203#123—24no—
204#123—24no—
205#123—22no—
206C22⋊Q8—24no—
207C22⋊Q8—22no—
208C22⋊Q8—22no—
209#121—22no—
210#120—22no—
211C4×Q8—21no—
212C4×Q8—22no—
213C4×Q8—22no—
214C4×Q8—22no—
215C4×D4—24no—
216C4×D4—21no—
217C4×D4—22no—
218C4×D4—22no—
219C4×D4—22no—
220#117—24no—
221#117—22no—
222#117—21no—
223#117C2×Dic1311yescharacteristic
224#117C2×Dic1311yescharacteristic
225#117C2×Dic1311yescharacteristic
226C2×C22⋊C4—22no—
227C2×C22⋊C4—264no—
228C2×C22⋊C4C22⋊D1314yes—
229C2×C42—21no—
230C2×C42C2×Dic1311yescharacteristic
231#99—264no—
232#99—264no—
233C2×C13—21no—
234C2×C13—22no—
235C2×C13—44no—
236C2×C13—81no—
237C2×C13#178211yescharacteristic
238C24C2×C22⋊D1312yes—
239#41—42no—
240C2×Q8—22no—
241C2×Q8—21no—
242C2×Q8—42no—
243C2×Q8C22⋊Dic1312yes—
244C2×Q8C22⋊Dic1311yescharacteristic
245C2×D4—21no—
246C2×D4—24no—
247C2×D4—42no—
248C2×D4C22⋊Dic1311yescharacteristic
249C2×D4C2×C22⋊D1312yes—
250C2×D4C2×C22⋊D1312yes—
251C22×C4—21no—
252C22×C4—22no—
253C22×C4—21no—
254C22×C4—24no—
255C22×C4—21no—
256C22×C4—524no—
257C22×C4C22×Dic1311yescharacteristic
258C4⋊C4—22no—
259C4⋊C4—22no—
260C4⋊C4—24no—
261C4⋊C4—24no—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C22⋊C4—22no—
268C22⋊C4—24no—
269C22⋊C4—21no—
270C22⋊C4—24no—
271C22⋊C4—24no—
272C22⋊C4—24no—
273C22⋊C4—24no—
274C22⋊C4—524no—
275C22⋊C4—524no—
276C22⋊C4C22⋊Dic1312yes—
277C22⋊C4C22⋊Dic1312yes—
278C42—22no—
279C42—24no—
280C42—21no—
281C42—21no—
282C42—21no—
283C42C22⋊Dic1311yescharacteristic
284C42C22⋊Dic1311yescharacteristic
285C42C22⋊Dic1312yes—
286C13#1272811yescharacteristic, complemented, Sylow
287C23—24no—
288C23—21no—
289C23—82no—
290C23C2×C22⋊Dic1311yescharacteristic
291C23#8820714yes—
292Q8—42no—
293Q8—41no—
294Q8—42no—
295Q8—82no—
296D4—42no—
297D4—42no—
298D4—41no—
299D4—42no—
300D4—84no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—21no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—82no—
312C2×C4—84no—
313C2×C4—81no—
314C2×C4—1044no—
315C2×C4—2084no—
316C2×C4C2×C22⋊Dic1311yescharacteristic
317C2×C4C2×C22⋊Dic1311yescharacteristic
318C22—24no—
319C22—41no—
320C22—44no—
321C22—42no—
322C22—41no—
323C22—84no—
324C22#1171890112yes—
325C22#1171890811yescharacteristic
326C4—21no—
327C4—21no—
328C4—42no—
329C4—41no—
330C4—42no—
331C4—41no—
332C4—42no—
333C4—41no—
334C4—42no—
335C4—2084no—
336C2—21no—
337C2—22no—
338C2—44no—
339C2—81no—
340C2#1311475411yescharacteristic
subgroupG/H
centerC2#13114754
commutatorC22×C4×C13C22×C4
FrattiniC22×D4C22×D13
FittingC13×#3510C2
socleC2×C13#1782
prime radicalC13×#535C22
cosocleC22×D4×C13C23
hypercenter#3510D13
hypocenterC13#12728
LCSG ▷ C22×C4×C13 ▷ C22×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C2 ◁ C23 ◁ #518 ◁ #3510

Complements

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

NH ≅ G/Ntype
2. C2×#11718133C2semidirect
3. #13114944C2semidirect
9. C13×#536C4semidirect
11. C13×#533C4semidirect
14. C13×#519C4semidirect
16. C4×C13×#41C4semidirect
156. #536Dic13semidirect
158. #533Dic13semidirect
161. #519Dic13semidirect
163. C4×#41Dic13semidirect
286. C13#12728semidirect

Maximal subgroups

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

subgroupindex
#131165052
C2×#117181332
#131149442
C13×#35102
#1272813