Finite Group Databasebeta

← #15778453

Subgroups — #15778453

Order 3328.

1216 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 1216 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#13116505C214yesmaximal, complemented
2#13114751C212yesmaximal, complemented
3C13×#3484C211yesmaximal, characteristic
4C2×C13×H(5,𝔽2)C411yescharacteristic, complemented
5C23×D4×C13C2211yescharacteristic
6C13×#546—24no—
7D42×C13—24no—
8C13×#535C2212yes—
9C13×#534—22no—
10C13×#525C412yescomplemented
11C2×C13×#120—22no—
12C13×#518C411yescharacteristic, complemented
13#11718908—22no—
14C2×#88074—22no—
15C2×#88074C2214yescomplemented
16#11718137—24no—
17#11718133—24no—
18C25×C13D412yes—
19C13×H(5,𝔽2)—44no—
20C2×C13×#41—22no—
21C2×C13×#41C2×C412yes—
22C22×D4×C13—22no—
23C22×D4×C13—24no—
24C22×D4×C13—22no—
25C22×D4×C13—24no—
26C22×D4×C13—44no—
27C22×D4×C13C2×C411yescharacteristic
28C22×D4×C13D412yes—
29C22×D4×C13C2311yescharacteristic
30C23×C4×C13C2×C411yescharacteristic
31C13×#127D414yes—
32C13×#124D414yes—
33C13×#123—24no—
34C13×#123—22no—
35C13×C22⋊Q8—22no—
36C13×#121—28no—
37C13×#121—22no—
38C13×#121—24no—
39C13×#120—28no—
40C13×#120—24no—
41C13×#120—22no—
42C13×#120—24no—
43C4×D4×C13—24no—
44C4×D4×C13—24no—
45C13×#117C2×C412yes—
46C2×C13×C22⋊C4—24no—
47C2×C13×C22⋊C4—22no—
48C2×C22⋊Dic13—24no—
49C2×C22⋊Dic13—44no—
50#88074—28no—
51#88074—44no—
52#12707—131nomaximal, Sylow
53C24×C13—22no—
54C24×C13—24no—
55C24×C13—24no—
56C24×C13—42no—
57C24×C13—84no—
58C24×C13C22⋊C411yescharacteristic
59C24×C13C22⋊C412yes—
60C24×C13C2×D412yes—
61C13×#41—44no—
62C13×#41—48no—
63C2×Q8×C13C22⋊C412yescomplemented
64C2×D4×C13—24no—
65C2×D4×C13—24no—
66C2×D4×C13—24no—
67C2×D4×C13—24no—
68C2×D4×C13—22no—
69C2×D4×C13—24no—
70C2×D4×C13—28no—
71C2×D4×C13—22no—
72C2×D4×C13—42no—
73C2×D4×C13—42no—
74C2×D4×C13—416no—
75C2×D4×C13—48no—
76C2×D4×C13—44no—
77C2×D4×C13—44no—
78C2×D4×C13—88no—
79C2×D4×C13C22⋊C412yescomplemented
80C2×D4×C13C2×D414yes—
81C22×C4×C13—22no—
82C22×C4×C13—24no—
83C22×C4×C13—22no—
84C22×C4×C13—22no—
85C22×C4×C13—42no—
86C22×C4×C13C22⋊C411yescharacteristic
87C22×C4×C13C22×C411yescharacteristic
88C22×Dic13—44no—
89C13×C4⋊C4—24no—
90C13×C4⋊C4—22no—
91C13×C22⋊C4—24no—
92C13×C22⋊C4—24no—
93C13×C22⋊C4—24no—
94C13×C22⋊C4—22no—
95C13×C22⋊C4—28no—
96C42×C13C22⋊C414yescomplemented
97C22⋊Dic13—44no—
98C22⋊Dic13—44no—
99C22⋊Dic13—88no—
100#3484D1311yescharacteristic
101#2127—134no—
102#1781—132no—
103C23×C13—21no—
104C23×C13—22no—
105C23×C13—22no—
106C23×C13—22no—
107C23×C13—22no—
108C23×C13—22no—
109C23×C13—22no—
110C23×C13—42no—
111C23×C13—48no—
112C23×C13—48no—
113C23×C13—48no—
114C23×C13—44no—
115C23×C13—84no—
116C23×C13—82no—
117C23×C13—82no—
118C23×C13—84no—
119C23×C13—84no—
120C23×C13—84no—
121C23×C13—84no—
122C23×C13#9914yes—
123C23×C13C2×C22⋊C411yescharacteristic
124C23×C13#12014yes—
125Q8×C13—42no—
126D4×C13—28no—
127D4×C13—44no—
128D4×C13—42no—
129D4×C13—48no—
130D4×C13—48no—
131D4×C13—48no—
132D4×C13—44no—
133D4×C13—88no—
134C2×C4×C13—22no—
135C2×C4×C13—24no—
136C2×C4×C13—22no—
137C2×C4×C13—21no—
138C2×C4×C13—22no—
139C2×C4×C13—44no—
140C2×C4×C13—44no—
141C2×C4×C13—44no—
142C2×C4×C13—84no—
143C2×C4×C13C2×C22⋊C412yes—
144C2×Dic13—84no—
145C2×Dic13—88no—
146C2×H(5,𝔽2)Dic1311yescharacteristic, complemented
147C23×D4C2×D1311yescharacteristic
148#546—24no—
149D42—24no—
150#535C2×D1312yes—
151#534—22no—
152#525Dic1312yescomplemented
153C2×#120—22no—
154#518Dic1311yescharacteristic, complemented
155C2×#99—134no—
156C2×#99—262no—
157#379—262no—
158#354—264no—
159#353—264no—
160C22×C13—22no—
161C22×C13—24no—
162C22×C13—24no—
163C22×C13—24no—
164C22×C13—41no—
165C22×C13—44no—
166C22×C13—44no—
167C22×C13—44no—
168C22×C13—44no—
169C22×C13—41no—
170C22×C13—42no—
171C22×C13—44no—
172C22×C13—41no—
173C22×C13—42no—
174C22×C13—82no—
175C22×C13—82no—
176C22×C13—816no—
177C22×C13—164no—
178C22×C13#37911yescharacteristic
179C22×C13C2×#9912yes—
180C4×C13—22no—
181C4×C13—44no—
182C4×C13—41no—
183C4×C13—42no—
184Dic13—164no—
185C25C22⋊D1312yes—
186H(5,𝔽2)—44no—
187C2×#41—22no—
188C2×#41C2×Dic1312yes—
189C22×D4—22no—
190C22×D4—24no—
191C22×D4—22no—
192C22×D4—24no—
193C22×D4—44no—
194C22×D4C2×Dic1311yescharacteristic
195C22×D4C22⋊D1312yes—
196C22×D4C22×D1311yescharacteristic
197C23×C4C2×Dic1311yescharacteristic
198#127C22⋊D1314yes—
199#124C22⋊D1314yes—
200#123—24no—
201#123—22no—
202C22⋊Q8—22no—
203#121—28no—
204#121—22no—
205#121—24no—
206#120—28no—
207#120—24no—
208#120—22no—
209#120—24no—
210C4×D4—24no—
211C4×D4—24no—
212#117C2×Dic1312yes—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—264no—
216C2×C22⋊C4—524no—
217#99—268no—
218#99—524no—
219C2×C13—21no—
220C2×C13—22no—
221C2×C13—44no—
222C2×C13—42no—
223C2×C13—44no—
224C2×C13—82no—
225C2×C13#178111yescharacteristic
226C24—22no—
227C24—24no—
228C24—24no—
229C24—42no—
230C24—84no—
231C24C22⋊Dic1311yescharacteristic
232C24C22⋊Dic1312yes—
233C24C2×C22⋊D1312yes—
234#41—44no—
235#41—48no—
236C2×Q8C22⋊Dic1312yescomplemented
237C2×D4—24no—
238C2×D4—24no—
239C2×D4—24no—
240C2×D4—24no—
241C2×D4—22no—
242C2×D4—24no—
243C2×D4—28no—
244C2×D4—22no—
245C2×D4—42no—
246C2×D4—42no—
247C2×D4—416no—
248C2×D4—48no—
249C2×D4—44no—
250C2×D4—44no—
251C2×D4—88no—
252C2×D4C22⋊Dic1312yescomplemented
253C2×D4C2×C22⋊D1314yes—
254C22×C4—22no—
255C22×C4—24no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—42no—
259C22×C4—524no—
260C22×C4C22⋊Dic1311yescharacteristic
261C22×C4C22×Dic1311yescharacteristic
262C4⋊C4—24no—
263C4⋊C4—22no—
264C22⋊C4—24no—
265C22⋊C4—24no—
266C22⋊C4—24no—
267C22⋊C4—22no—
268C22⋊C4—28no—
269C22⋊C4—524no—
270C22⋊C4—524no—
271C22⋊C4—1048no—
272C42C22⋊Dic1314yescomplemented
273C13#1270711yescharacteristic, complemented, Sylow
274C23—22no—
275C23—22no—
276C23—21no—
277C23—22no—
278C23—22no—
279C23—22no—
280C23—22no—
281C23—42no—
282C23—48no—
283C23—48no—
284C23—48no—
285C23—44no—
286C23—84no—
287C23—82no—
288C23—82no—
289C23—84no—
290C23—84no—
291C23—84no—
292C23—84no—
293C23#8807414yes—
294C23C2×C22⋊Dic1311yescharacteristic
295C23#8820714yes—
296Q8—42no—
297D4—28no—
298D4—44no—
299D4—42no—
300D4—48no—
301D4—48no—
302D4—48no—
303D4—44no—
304D4—88no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—24no—
308C2×C4—22no—
309C2×C4—21no—
310C2×C4—44no—
311C2×C4—44no—
312C2×C4—44no—
313C2×C4—84no—
314C2×C4—1044no—
315C2×C4—1048no—
316C2×C4C2×C22⋊Dic1312yes—
317C22—22no—
318C22—24no—
319C22—24no—
320C22—24no—
321C22—41no—
322C22—44no—
323C22—44no—
324C22—44no—
325C22—44no—
326C22—41no—
327C22—42no—
328C22—44no—
329C22—41no—
330C22—42no—
331C22—82no—
332C22—82no—
333C22—816no—
334C22—164no—
335C22C2×#8807412yes—
336C22#1171890811yescharacteristic
337C4—22no—
338C4—44no—
339C4—41no—
340C4—42no—
341C4—2084no—
342C2—21no—
343C2—22no—
344C2—44no—
345C2—42no—
346C2—44no—
347C2—82no—
348C2#1311475111yescharacteristic
subgroupG/H
centerC2#13114751
commutatorC22×C4×C13C22×C4
FrattiniC22×D4C22×D13
Fitting, prime radicalC13×#3484C2
socleC2×C13#1781
cosocleC22×D4×C13C23
hypercenter#3484D13
hypocenterC13#12707
LCSG ▷ C22×C4×C13 ▷ C22×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C2 ◁ C23 ◁ C23×D4 ◁ #3484

Complements

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

NH ≅ G/Ntype
1. #13116505C2semidirect
2. #13114751C2semidirect
4. C2×C13×H(5,𝔽2)C4semidirect
10. C13×#525C4semidirect
12. C13×#518C4semidirect
15. C2×#88074C22semidirect
63. C2×Q8×C13C22⋊C4semidirect
79. C2×D4×C13C22⋊C4semidirect
96. C42×C13C22⋊C4semidirect
146. C2×H(5,𝔽2)Dic13semidirect
152. #525Dic13semidirect
154. #518Dic13semidirect
236. C2×Q8C22⋊Dic13semidirect
252. C2×D4C22⋊Dic13semidirect
272. C42C22⋊Dic13semidirect
273. C13#12707semidirect

Maximal subgroups

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

subgroupindex
#131165052
#131147512
C13×#34842
#1270713