Finite Group Databasebeta

← #15789687

Subgroups — #15789687

Order 3328.

1830 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718892C211yesmaximal, characteristic
2C2×#11718601C213yesmaximal
3#13113107C211yesmaximal, characteristic, complemented
4#13113099C213yesmaximal, complemented
5#13113078C213yesmaximal
6#13113063C213yesmaximal, complemented
7C22×C4×Q8×C13C211yesmaximal, characteristic, complemented
8C23×Q8×C13C2211yescharacteristic
9C2×C4×Q8×C13—212no—
10C22×C13×C4⋊C4C2213yes—
11C22×C42×C13C2213yes—
12C22×#6103C2211yescharacteristic
13C22×#6103C2213yes—
14C22×#6103C2213yes—
15C22×#6101C2213yes—
16#11718892C418yescomplemented
17C2×#88071C2213yes—
18C2×#88071C2213yes—
19C2×#88071C2213yes—
20C2×#88071C2216yes—
21C2×#88071C2213yes—
22#11718601—212no—
23C24×D13C2311yescharacteristic
24C22×Q8×C13—26no—
25C22×Q8×C13—26no—
26C22×Q8×C13C2×C414yes—
27C23×C4×C13C2313yes—
28C23×C4×C13C2313yes—
29C23×C4×C13C2311yescharacteristic
30C23×Dic13C2311yescharacteristic
31C23×Dic13C2313yes—
32C23×Dic13C2313yes—
33C4×Q8×C13—232no—
34C2×C13×C4⋊C4—212no—
35C2×C13×C4⋊C4—26no—
36C2×C42×C13—212no—
37C2×C42×C13D4112yes—
38C2×#6103—26no—
39C2×#6103—26no—
40C2×#6103—26no—
41C2×#6103—26no—
42C2×#6103—46no—
43C2×#6103C2×C4112yes—
44C2×#6101—26no—
45C2×#6101—46no—
46#88071—26no—
47#88071—26no—
48#88071—26no—
49#88071—46no—
50#88071C2×C4112yes—
51#14301—131nomaximal, Sylow
52C24×C13C2411yescharacteristic
53C23×D13—23no—
54C23×D13—23no—
55C23×D13—23no—
56C23×D13—21no—
57C23×D13—23no—
58C23×D13—21no—
59C23×D13C22×C412yes—
60C2×Q8×C13—28no—
61C2×Q8×C13—224no—
62C2×Q8×C13—224no—
63C22×C4×C13—26no—
64C22×C4×C13—26no—
65C22×C4×C13—21no—
66C22×C4×C13—23no—
67C22×C4×C13—26no—
68C22×C4×C13—23no—
69C22×C4×C13—26no—
70C22×C4×C13—23no—
71C22×C4×C13C22×C416yes—
72C22×C4×C13C2×D416yes—
73C22×C4×C13C2×D416yes—
74C22×C4×C13C2×D416yes—
75C22×C4×C13#4116yes—
76C22×Dic13—23no—
77C22×Dic13—23no—
78C22×Dic13—26no—
79C22×Dic13—26no—
80C22×Dic13—23no—
81C22×Dic13—23no—
82C22×Dic13—26no—
83C22×Dic13—26no—
84C22×Dic13—23no—
85C22×Dic13—23no—
86C22×Dic13—23no—
87C22×Dic13—21no—
88C22×Dic13C22×C416yes—
89C13×C4⋊C4—224no—
90C42×C13—224no—
91#6103—224no—
92#6103—412no—
93#6103—44no—
94#6101—412no—
95C22×C4×Q8D1311yescharacteristic, complemented
96C2×#393—131no—
97C2×#388—133no—
98#1446—131no—
99#1444—133no—
100#1439—133no—
101#1434—133no—
102C23×C13C23×C411yescharacteristic
103C23×C13C22×D413yes—
104C23×C13C2×#4113yes—
105C23×C13H(5,𝔽2)13yes—
106C23×C132−1+411yescharacteristic
107C23×C132−1+411yescharacteristic
108C23×C132−1+413yes—
109C22×D13—26no—
110C22×D13—22no—
111C22×D13—26no—
112C22×D13—41no—
113C22×D13—43no—
114C22×D13—43no—
115C22×D13—43no—
116C22×D13—43no—
117C22×D13—43no—
118C22×D13—43no—
119C22×D13—43no—
120C22×D13—46no—
121Q8×C13—232no—
122C2×C4×C13—212no—
123C2×C4×C13—26no—
124C2×C4×C13—26no—
125C2×C4×C13—212no—
126C2×C4×C13—26no—
127C2×C4×C13—26no—
128C2×C4×C13—26no—
129C2×C4×C13—26no—
130C2×C4×C13—26no—
131C2×C4×C13—46no—
132C2×C4×C13—46no—
133C2×C4×C13C4×D4112yes—
134C2×C4×C13#12014yes—
135C2×Dic13—26no—
136C2×Dic13—212no—
137C2×Dic13—43no—
138C2×Dic13—43no—
139C2×Dic13—46no—
140C2×Dic13—41no—
141C2×Dic13—43no—
142C2×Dic13—43no—
143C2×Dic13—43no—
144C2×Dic13—43no—
145C2×Dic13—43no—
146C2×Dic13—46no—
147C2×Dic13—46no—
148C23×Q8C2×D1311yescharacteristic
149C2×C4×Q8—212no—
150C22×C4⋊C4—133no—
151C22×C4⋊C4C2×D1313yes—
152C22×C22⋊C4—133no—
153C22×C22⋊C4—131no—
154C22×C22⋊C4—133no—
155C22×C42C2×D1313yes—
156#393—138no—
157#388—2612no—
158C2×#95—133no—
159C2×#95—133no—
160C2×#95—133no—
161C2×#95—133no—
162C2×#95—136no—
163C22×C13C2×C4×D413yes—
164C22×C13#51911yescharacteristic
165C22×C13#52013yes—
166C22×C13C2×#12011yescharacteristic
167C22×C13#52513yes—
168C22×C13#53513yes—
169C22×C13#53613yes—
170C22×C13#53613yes—
171C22×C13#53713yes—
172C22×C13#54113yes—
173C22×C13#54213yes—
174C22×C13#54216yes—
175C2×D13—46no—
176C2×D13—46no—
177C2×D13—42no—
178C2×D13—81no—
179C2×D13—81no—
180C2×D13—83no—
181C2×D13—83no—
182C4×C13—24no—
183C4×C13—212no—
184C4×C13—46no—
185Dic13—46no—
186Dic13—81no—
187Dic13—83no—
188C25—131no—
189C22×Q8—26no—
190C22×Q8—26no—
191C22×Q8C4×D1314yes—
192C23×C4—133no—
193C23×C4—131no—
194C23×C4—133no—
195C23×C4C22×D1313yes—
196C23×C4C22×D1313yes—
197C23×C4C22×D1311yescharacteristic
198C4×Q8—232no—
199C2×C4⋊C4—212no—
200C2×C4⋊C4—26no—
201C2×C4⋊C4—266no—
202C2×C4⋊C4—526no—
203C2×C22⋊C4—1312no—
204C2×C22⋊C4—266no—
205C2×C22⋊C4—266no—
206C2×C22⋊C4—266no—
207C2×C22⋊C4—266no—
208C2×C22⋊C4—526no—
209C2×C42—212no—
210C2×C42C22⋊D13112yes—
211#95—1312no—
212#95—266no—
213#95—266no—
214#95—266no—
215#95—526no—
216C2×C13C4×#12011yescharacteristic
217C2×C13#232313yes—
218C2×C13#232613yes—
219C2×C13#261411yescharacteristic
220C2×C13#261511yescharacteristic
221C2×C13#263213yes—
222C2×C13#263713yes—
223D13—82no—
224C24—132no—
225C24—263no—
226C24—261no—
227C24—263no—
228C24—263no—
229C24—261no—
230C24—263no—
231C24C23×D1311yescharacteristic
232C2×Q8—28no—
233C2×Q8—224no—
234C2×Q8—224no—
235C22×C4—26no—
236C22×C4—21no—
237C22×C4—23no—
238C22×C4—26no—
239C22×C4—23no—
240C22×C4—26no—
241C22×C4—23no—
242C22×C4—26no—
243C22×C4—136no—
244C22×C4—263no—
245C22×C4—266no—
246C22×C4—266no—
247C22×C4—263no—
248C22×C4—266no—
249C22×C4—263no—
250C22×C4—263no—
251C22×C4—266no—
252C22×C4—263no—
253C22×C4—261no—
254C22×C4—263no—
255C22×C4—263no—
256C22×C4C2×C4×D1316yes—
257C22×C4#612716yes—
258C22×C4C2×C22⋊D1316yes—
259C22×C4C2×C22⋊D1316yes—
260C22×C4C2×C22⋊D1316yes—
261C4⋊C4—224no—
262C4⋊C4—5212no—
263C22⋊C4—2624no—
264C22⋊C4—5212no—
265C22⋊C4—524no—
266C42—224no—
267C13#1430111yescharacteristic, complemented, Sylow
268C23—266no—
269C23—262no—
270C23—266no—
271C23—523no—
272C23—523no—
273C23—523no—
274C23—523no—
275C23—523no—
276C23—523no—
277C23—526no—
278C23—523no—
279C23—521no—
280C23C22×C4×D1311yescharacteristic
281C23C2×#612713yes—
282C23#8825411yescharacteristic
283C23#8825413yes—
284C23#8825613yes—
285C23#8825711yescharacteristic
286C23C22×C22⋊D1313yes—
287Q8—232no—
288C2×C4—212no—
289C2×C4—212no—
290C2×C4—26no—
291C2×C4—26no—
292C2×C4—26no—
293C2×C4—26no—
294C2×C4—26no—
295C2×C4—26no—
296C2×C4—26no—
297C2×C4—46no—
298C2×C4—46no—
299C2×C4—266no—
300C2×C4—2612no—
301C2×C4—523no—
302C2×C4—523no—
303C2×C4—526no—
304C2×C4—523no—
305C2×C4—523no—
306C2×C4—523no—
307C2×C4—523no—
308C2×C4—523no—
309C2×C4—526no—
310C2×C4—526no—
311C2×C4—521no—
312C2×C4C4×C22⋊D13112yes—
313C2×C4#8820714yes—
314C22—526no—
315C22—526no—
316C22—522no—
317C22—1041no—
318C22—1041no—
319C22—1043no—
320C22—1043no—
321C22#1171927711yescharacteristic
322C22#1171927913yes—
323C22#1171928713yes—
324C22#1171928716yes—
325C22#1171928813yes—
326C22C2×C4×C22⋊D1313yes—
327C22#1171949613yes—
328C22#1171952113yes—
329C22#1171952113yes—
330C22#1171953513yes—
331C22#1171953813yes—
332C22C2×#8820711yescharacteristic
333C4—24no—
334C4—212no—
335C4—46no—
336C4—526no—
337C4—1041no—
338C4—1043no—
339C2—1042no—
340C2C4×#8820711yescharacteristic
341C2#1311751213yes—
342C2#1311754313yes—
343C2#1312055511yescharacteristic
344C2#1312056011yescharacteristic
345C2#1312069613yes—
346C2#1312074213yes—
subgroupG/H
center, FrattiniC24C23×D13
commutatorC23×C13C23×C4
FittingC22×C4×Q8×C13C2
socle, cosocleC24×C13C24
prime radicalC24×D13C23
hypercenterC22×C4×Q8D13
hypocenterC13#14301
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C24 ◁ C22×C4×Q8

Complements

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

NH ≅ G/Ntype
3. #13113107C2semidirect
4. #13113099C2semidirect
6. #13113063C2semidirect
7. C22×C4×Q8×C13C2semidirect
16. #11718892C4semidirect
95. C22×C4×Q8D13semidirect
267. C13#14301semidirect

Maximal subgroups

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

subgroupindex
C2×#117188922
C2×#117186012
#131131072
#131130992
#131130782
#131130632
C22×C4×Q8×C132
#1430113