Finite Group Databasebeta

← #16778823

Subgroups — #16778823

Order 3328.

494 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 100 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#13130727C211yesmaximal, characteristic
2#13130690C211yesmaximal, characteristic
3#13129655C212yesmaximal, complemented
4#13129654C212yesmaximal, complemented
5#13123186C211yesmaximal, characteristic
6C22×C4×Dic13C2211yescharacteristic
7C2×#88142C411yescharacteristic
8C2×#88132—21no—
9C2×C22⋊#1057—21no—
10C2×C22⋊#1057—21no—
11C2×C4⋊#1057—21no—
12C13×#380C411yescharacteristic
13#11718873C411yescharacteristic
14#11718646C2211yescharacteristic
15#11718637C411yescharacteristic
16#11718634C2211yescharacteristic
17#11718603—21no—
18#11718308—21no—
19#11718307—21no—
20#11718290—21no—
21#11718290C2212yes—
22#11718290C2212yes—
23C23×C4×C13C2×C411yescharacteristic
24C23×Dic13D411yescharacteristic
25C23×Dic13D411yescharacteristic
26C22×Dic26—21no—
27C22×#1057—22no—
28C22×#1057—22no—
29C2×C13×C4⋊C4C2×C411yescharacteristic
30C2×C13×C4⋊C4C2×C411yescharacteristic
31C2×C13×C22⋊C4—21no—
32C2×C22⋊Dic13—21no—
33C2×C4⋊Dic13—21no—
34C2×C4⋊Dic13C2×C411yescharacteristic
35C2×C4⋊Dic13C2×C411yescharacteristic
36C2×#6101—21no—
37C2×#6101—21no—
38C2×#6101—21no—
39C2×#6101—21no—
40C2×C4×Dic13—42no—
41C2×C4×Dic13C2×C411yescharacteristic
42C2×C4×Dic13D411yescharacteristic
43C2×C4×Dic13D411yescharacteristic
44C2×C4×Dic13C2311yescharacteristic
45#88142—42no—
46#88132—44no—
47C22⋊#1057—42no—
48C22⋊#1057—42no—
49C4⋊#1057—42no—
50C13×#95—21no—
51#88071—21no—
52#88071—21no—
53#88071—21no—
54#7725—131nomaximal, Sylow
55C24×C13C22⋊C411yescharacteristic
56C22×C4×C13—22no—
57C22×C4×C13—22no—
58C22×C4×C13—21no—
59C22×C4×C13—21no—
60C22×C4×C13—41no—
61C22×C4×C13C22⋊C411yescharacteristic
62C22×C4×C13C22×C411yescharacteristic
63C22×Dic13—22no—
64C22×Dic13—22no—
65C22×Dic13—21no—
66C22×Dic13—21no—
67C22×Dic13—41no—
68C22×Dic13—41no—
69C22×Dic13—41no—
70C22×Dic13C22⋊C411yescharacteristic
71C22×Dic13C22⋊C411yescharacteristic
72C22×Dic13SD1612yes—
73C22×Dic13Q1612yes—
74C22×Dic13C2×D411yescharacteristic
75C22×Dic13C2×D411yescharacteristic
76C2×Dic26—42no—
77C2×Dic26—42no—
78C2×Dic26—42no—
79C2×#1057—42no—
80C2×#1057—42no—
81C2×#1057—42no—
82C2×#1057—42no—
83C2×#1057—42no—
84C2×#1057—42no—
85C13×C4⋊C4—22no—
86C13×C4⋊C4—41no—
87C13×C22⋊C4—42no—
88C22⋊Dic13—42no—
89C4⋊Dic13—22no—
90C4⋊Dic13—42no—
91C4⋊Dic13—41no—
92#6101—42no—
93#6101—42no—
94#6101—42no—
95#6101—42no—
96C4×Dic13—22no—
97C4×Dic13—22no—
98C4×Dic13—41no—
99C4×Dic13—41no—
100#2857—131no—
101#1817—131no—
102#1797—131no—
103#1271—132no—
104#1271—132no—
105C23×C13—21no—
106C23×C13—21no—
107C23×C13—21no—
108C23×C13—21no—
109C23×C13—21no—
110C23×C13—21no—
111C23×C13#10312yes—
112C23×C13C2×C22⋊C411yescharacteristic
113C2×C4×C13—21no—
114C2×C4×C13—21no—
115C2×C4×C13—21no—
116C2×C4×C13—21no—
117C2×C4×C13—44no—
118C2×C4×C13—41no—
119C2×C4×C13—41no—
120C2×C4×C13—41no—
121C2×C4×C13—41no—
122C2×C4×C13—41no—
123C2×C4×C13—41no—
124C2×C4×C13#10112yes—
125C2×C4×C13#13112yes—
126C2×Dic13—21no—
127C2×Dic13—21no—
128C2×Dic13—21no—
129C2×Dic13—21no—
130C2×Dic13—21no—
131C2×Dic13—21no—
132C2×Dic13—21no—
133C2×Dic13—21no—
134C2×Dic13—42no—
135C2×Dic13—42no—
136C2×Dic13—44no—
137C2×Dic13—41no—
138C2×Dic13—41no—
139C2×Dic13—41no—
140C2×Dic13—41no—
141C2×Dic13—41no—
142C2×Dic13—41no—
143C2×Dic13#9912yes—
144C2×Dic13#10312yes—
145C2×Dic13C2×SD1611yescharacteristic
146C2×Dic13C2×Q1611yescharacteristic
147C2×Dic13AGL(1,ℤ/8)12yes—
148Dic26—84no—
149#1057—82no—
150#1057—82no—
151C2×C4⋊Q8—131no—
152C2×C22⋊Q8—261no—
153C22×C42—131no—
154C2×C4⋊C8—261no—
155C2×C22⋊C8—261no—
156C2×C22⋊C8—261no—
157#391—261no—
158#389—131no—
159#389—131no—
160#389—261no—
161#384—131no—
162#380—131no—
163#380—261no—
164#380#26111yescharacteristic
165#336—132no—
166#336—132no—
167#336—261no—
168C22×C13—22no—
169C22×C13—22no—
170C22×C13—22no—
171C22×C13—41no—
172C22×C13—41no—
173C22×C13—41no—
174C22×C13—41no—
175C22×C13#40811yescharacteristic
176C22×C13C2×#9911yescharacteristic
177C22×C13C2×#10111yescharacteristic
178C22×C13C2×#10311yescharacteristic
179C22×C13#41811yescharacteristic
180C22×C13#42111yescharacteristic
181C22×C13#42111yescharacteristic
182C4×C13—22no—
183C4×C13—22no—
184C4×C13—81no—
185C4×C13—81no—
186Dic13—22no—
187Dic13—22no—
188Dic13—21no—
189Dic13—22no—
190Dic13—21no—
191Dic13—81no—
192Dic13—81no—
193C22×Q8—261no—
194C23×C4—131no—
195C23×C4—131no—
196C23×C4C2×#26111yescharacteristic
197C22×C8—262no—
198C22×C8—262no—
199C4⋊Q8—522no—
200C22⋊Q8—524no—
201C2×C4⋊C4—131no—
202C2×C4⋊C4—131no—
203C2×C4⋊C4—261no—
204C2×C4⋊C4—261no—
205C2×C4⋊C4—261no—
206C2×C4⋊C4—261no—
207C2×C4⋊C4—261no—
208C2×C4⋊C4C2×#26111yescharacteristic
209C2×C4⋊C4C2×#26111yescharacteristic
210C2×C22⋊C4—21no—
211C2×C22⋊C4—261no—
212C2×C42—131no—
213C2×C42—131no—
214C2×C42—131no—
215C2×C42—131no—
216C2×C42—522no—
217C4⋊C8—522no—
218C22⋊C8—522no—
219C22⋊C8—522no—
220#95—21no—
221#95—261no—
222#95—261no—
223#95—261no—
224C2×C13—42no—
225C2×C13#147211yescharacteristic
226C2×C13#148111yescharacteristic
227C2×C13#149611yescharacteristic
228C2×C13#151011yescharacteristic
229C2×C13#152011yescharacteristic
230C2×C13#153711yescharacteristic
231C2×C13#154911yescharacteristic
232C24C22⋊#26111yescharacteristic
233C2×Q8—522no—
234C2×Q8—522no—
235C2×Q8—522no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—41no—
241C22×C4—131no—
242C22×C4—131no—
243C22×C4—131no—
244C22×C4—131no—
245C22×C4—132no—
246C22×C4—132no—
247C22×C4—262no—
248C22×C4—262no—
249C22×C4—261no—
250C22×C4—261no—
251C22×C4—521no—
252C22×C4—521no—
253C22×C4—521no—
254C22×C4C22⋊#26111yescharacteristic
255C22×C4C22×#26111yescharacteristic
256C2×C8—522no—
257C2×C8—522no—
258C2×C8—522no—
259C2×C8—522no—
260C2×C8—522no—
261C2×C8—522no—
262C4⋊C4—22no—
263C4⋊C4—41no—
264C4⋊C4—262no—
265C4⋊C4—521no—
266C4⋊C4—522no—
267C4⋊C4—522no—
268C4⋊C4—522no—
269C4⋊C4—522no—
270C4⋊C4—522no—
271C22⋊C4—42no—
272C22⋊C4—522no—
273C42—262no—
274C42—262no—
275C42—521no—
276C42—521no—
277C13#772511yescharacteristic, complemented, Sylow
278C23—21no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—21no—
283C23—21no—
284C23#8811712yes—
285C23C2×C22⋊#26111yescharacteristic
286Q8—1044no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—44no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—132no—
299C2×C4—132no—
300C2×C4—131no—
301C2×C4—132no—
302C2×C4—131no—
303C2×C4—261no—
304C2×C4—261no—
305C2×C4—261no—
306C2×C4—261no—
307C2×C4—261no—
308C2×C4—261no—
309C2×C4—261no—
310C2×C4—261no—
311C2×C4—522no—
312C2×C4—522no—
313C2×C4—524no—
314C2×C4—521no—
315C2×C4—521no—
316C2×C4—521no—
317C2×C4—521no—
318C2×C4—521no—
319C2×C4—521no—
320C2×C4#8811312yes—
321C2×C4#8823812yes—
322C8—1042no—
323C8—1042no—
324C22—22no—
325C22—22no—
326C22—22no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22C2×#8811311yescharacteristic
332C22#1171913811yescharacteristic
333C22D13⋊#9911yescharacteristic
334C22#1171914611yescharacteristic
335C22C2×#8811711yescharacteristic
336C22#1171916611yescharacteristic
337C22#1171916611yescharacteristic
338C4—22no—
339C4—22no—
340C4—81no—
341C4—81no—
342C4—262no—
343C4—262no—
344C4—261no—
345C4—262no—
346C4—261no—
347C4—1041no—
348C4—1041no—
349C2—42no—
350C2#1313011511yescharacteristic
351C2#1313014411yescharacteristic
352C2#1313019511yescharacteristic
353C2#1313024211yescharacteristic
354C2#1313027411yescharacteristic
355C2#1313032711yescharacteristic
356C2#1313035111yescharacteristic
subgroupG/H
centerC23C2×C22⋊#261
commutatorC22×C4×C13C22×C4
FrattiniC22×C4C22×#261
FittingC13×#380C4
socleC23×C13C2×C22⋊C4
prime radicalC24×C13C22⋊C4
cosocleC2×C4×Dic13C23
hypercenter#380#261
hypocenterC13#7725
LCSG ▷ C22×C4×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C23 ◁ C23×C4 ◁ #380

Complements

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

NH ≅ G/Ntype
3. #13129655C2semidirect
4. #13129654C2semidirect
277. C13#7725semidirect

Maximal subgroups

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

subgroupindex
#131307272
#131306902
#131296552
#131296542
#131231862
#772513