Finite Group Databasebeta

← #15767333

Subgroups — #15767333

Order 3328.

560 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 121 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#13120950C211yesmaximal, characteristic
2#13115026C212yesmaximal
3#13113092C211yesmaximal, characteristic
4#13112716C212yesmaximal
5C13×#1280C211yesmaximal, characteristic
6C22×C4⋊Dic13C2211yescharacteristic
7C2×#88178—21no—
8C2×C13×C22⋊C8C2212yes—
9C2×C13×#95C2211yescharacteristic
10#11718873—21no—
11#11718873C2212yes—
12C2×#88071—21no—
13C2×#88071—21no—
14C2×#88071C2211yescharacteristic
15C2×#88058—22no—
16#11718322—22no—
17#11718317—22no—
18#11718143—22no—
19C23×C4×C13D411yescharacteristic
20C23×C4×C13D411yescharacteristic
21C23×C4×C13C2311yescharacteristic
22C23×Dic13—21no—
23C23×Dic13—21no—
24C22×Dic26—21no—
25C22×C8×C13—22no—
26C2×C22⋊Dic13—22no—
27C2×C4⋊Dic13—22no—
28C2×C4⋊Dic13—21no—
29C2×C4⋊Dic13—21no—
30C2×C4⋊Dic13—21no—
31C2×C4⋊Dic13—42no—
32C2×C4⋊Dic13D412yes—
33C2×C4⋊Dic13D412yes—
34#88178—42no—
35#88178—42no—
36C2×#6101—22no—
37C13×C22⋊C8—22no—
38C13×C22⋊C8Q814yescomplemented
39C13×#95—22no—
40#88071—22no—
41#88071—22no—
42#88071—21no—
43#88071—42no—
44#88071—42no—
45#88071Q812yescomplemented
46#88058—44no—
47#10569—131nomaximal, Sylow
48C24×C13C2×D411yescharacteristic
49C22×C4×C13—21no—
50C22×C4×C13—21no—
51C22×C4×C13—21no—
52C22×C4×C13—21no—
53C22×C4×C13—41no—
54C22×C4×C13—41no—
55C22×C4×C13—41no—
56C22×C4×C13—41no—
57C22×C4×C13—41no—
58C22×C4×C13—41no—
59C22×C4×C13D812yes—
60C22×C4×C13Q1612yes—
61C22×C4×C13C2×D412yes—
62C22×C4×C13C2×Q812yes—
63C22×C4×C13C2×Q811yescharacteristic
64C22×C4×C13#4111yescharacteristic
65C22×Dic13—22no—
66C22×Dic13—22no—
67C22×Dic13—22no—
68C22×Dic13—22no—
69C22×Dic13—41no—
70C22×Dic13—41no—
71C22×Dic13—41no—
72C22×Dic13—41no—
73C22×Dic13—41no—
74C22×Dic13—41no—
75C22×Dic13—41no—
76C22×Dic13—41no—
77C22×Dic13—41no—
78C22×Dic13—41no—
79C22×Dic13—41no—
80C22×Dic13—41no—
81C2×Dic26—42no—
82C2×Dic26—81no—
83C2×Dic26—81no—
84C2×C8×C13—24no—
85C2×C8×C13—42no—
86C2×C8×C13—42no—
87C22⋊Dic13—42no—
88C22⋊Dic13—42no—
89C4⋊Dic13—24no—
90C4⋊Dic13—42no—
91C4⋊Dic13—42no—
92C4⋊Dic13—42no—
93C4⋊Dic13—42no—
94C4⋊Dic13—81no—
95C4⋊Dic13—81no—
96#6101—44no—
97#2659—131no—
98#1852—132no—
99#1442—131no—
100#1294—132no—
101#1280D1311yescharacteristic
102C23×C13—21no—
103C23×C13—21no—
104C23×C13—21no—
105C23×C13—21no—
106C23×C13#12011yescharacteristic
107C23×C13C22⋊Q811yescharacteristic
108C23×C13C4⋊Q811yescharacteristic
109C23×C13C2×D811yescharacteristic
110C23×C13C2×Q1611yescharacteristic
111C23×C13AGL(1,ℤ/8)11yescharacteristic
112C23×C13#13711yescharacteristic
113C2×C4×C13—22no—
114C2×C4×C13—22no—
115C2×C4×C13—22no—
116C2×C4×C13—22no—
117C2×C4×C13—42no—
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—42no—
125C2×C4×C13—42no—
126C2×C4×C13—81no—
127C2×C4×C13—81no—
128C2×C4×C13C22⋊Q812yes—
129C2×C4×C13C4⋊Q812yes—
130C2×Dic13—42no—
131C2×Dic13—42no—
132C2×Dic13—42no—
133C2×Dic13—42no—
134C2×Dic13—42no—
135C2×Dic13—42no—
136C2×Dic13—42no—
137C2×Dic13—42no—
138C2×Dic13—42no—
139C2×Dic13—42no—
140C2×Dic13—42no—
141C2×Dic13—42no—
142C2×Dic13—81no—
143C2×Dic13—81no—
144C2×Dic13—81no—
145C2×Dic13—81no—
146C2×Dic13—81no—
147C2×Dic13—81no—
148C2×Dic13—81no—
149C2×Dic13—81no—
150Dic26—84no—
151C8×C13—44no—
152C2×C22⋊Q8—261no—
153C22×C4⋊C4—131no—
154C2×#107—262no—
155C2×C22⋊C8C2×D1312yes—
156#398—262no—
157#395—262no—
158#380—132no—
159#380—261no—
160C2×#95—131no—
161C2×#95—261no—
162C2×#95—261no—
163C2×#95C2×D1311yescharacteristic
164#340—262no—
165C22×C13—21no—
166C22×C13—21no—
167C22×C13—21no—
168C22×C13—21no—
169C22×C13—21no—
170C22×C13—21no—
171C22×C13—21no—
172C22×C13—21no—
173C22×C13—21no—
174C22×C13—21no—
175C22×C13—21no—
176C22×C13—21no—
177C22×C13#39511yescharacteristic
178C22×C13#44711yescharacteristic
179C22×C13#45111yescharacteristic
180C22×C13#45411yescharacteristic
181C22×C13#45511yescharacteristic
182C22×C13B(4,𝔽2)11yescharacteristic
183C22×C13#45811yescharacteristic
184C22×C13#47411yescharacteristic
185C22×C13#47711yescharacteristic
186C22×C13#50011yescharacteristic
187C22×C13#50111yescharacteristic
188C4×C13—22no—
189C4×C13—22no—
190C4×C13—42no—
191C4×C13—42no—
192Dic13—82no—
193Dic13—82no—
194Dic13—82no—
195Dic13—82no—
196C22×Q8—261no—
197C23×C4—261no—
198C23×C4—261no—
199C23×C4D5211yescharacteristic
200C23×C4D5211yescharacteristic
201C23×C4C22×D1311yescharacteristic
202C22×C8—22no—
203C22⋊Q8—522no—
204C22⋊Q8—522no—
205C2×C4⋊C4—132no—
206C2×C4⋊C4—132no—
207C2×C4⋊C4—262no—
208C2×C4⋊C4—262no—
209C2×C4⋊C4—261no—
210C2×C4⋊C4—261no—
211C2×C4⋊C4—261no—
212C2×C4⋊C4—522no—
213C2×C22⋊C4—262no—
214#107—524no—
215C22⋊C8—22no—
216C22⋊C8Dic2614yescomplemented
217#95—22no—
218#95—132no—
219#95—262no—
220#95—262no—
221#95—261no—
222#95—522no—
223#95—522no—
224C2×C13—21no—
225C2×C13—21no—
226C2×C13—21no—
227C2×C13—21no—
228C2×C13#159811yescharacteristic
229C2×C13#160211yescharacteristic
230C2×C13#160711yescharacteristic
231C2×C13#161711yescharacteristic
232C2×C13#205711yescharacteristic
233C2×C13#206111yescharacteristic
234C2×C13#206211yescharacteristic
235C24C2×D5211yescharacteristic
236C2×Q8—522no—
237C2×Q8—1041no—
238C2×Q8—1041no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—262no—
250C22×C4—262no—
251C22×C4—262no—
252C22×C4—262no—
253C22×C4—521no—
254C22×C4—521no—
255C22×C4—521no—
256C22×C4—521no—
257C22×C4—521no—
258C22×C4—521no—
259C22×C4—521no—
260C22×C4—521no—
261C22×C4—521no—
262C22×C4—521no—
263C22×C4—521no—
264C22×C4—521no—
265C22×C4D10412yes—
266C22×C4Dic5212yes—
267C22×C4C2×Dic2612yes—
268C22×C4D4×D1312yes—
269C22×C4#612911yescharacteristic
270C22×C4Q8×D1311yescharacteristic
271C2×C8—24no—
272C2×C8—42no—
273C2×C8—42no—
274C4⋊C4—264no—
275C4⋊C4—522no—
276C4⋊C4—522no—
277C4⋊C4—522no—
278C4⋊C4—524no—
279C4⋊C4—522no—
280C4⋊C4—1041no—
281C4⋊C4—1041no—
282C22⋊C4—522no—
283C22⋊C4—522no—
284C13#1056911yescharacteristic, complemented, Sylow
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23#8812311yescharacteristic
290C23#8813611yescharacteristic
291C23#8815111yescharacteristic
292C23C2×D10411yescharacteristic
293C23C2×Dic5211yescharacteristic
294C23#8816211yescharacteristic
295C23#8816311yescharacteristic
296Q8—1044no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—42no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—522no—
313C2×C4—522no—
314C2×C4—522no—
315C2×C4—522no—
316C2×C4—522no—
317C2×C4—522no—
318C2×C4—522no—
319C2×C4—522no—
320C2×C4—522no—
321C2×C4—522no—
322C2×C4—522no—
323C2×C4—522no—
324C2×C4—1041no—
325C2×C4—1041no—
326C2×C4—1041no—
327C2×C4—1041no—
328C2×C4—1041no—
329C2×C4—1041no—
330C2×C4—1041no—
331C2×C4—1041no—
332C2×C4#8813212yes—
333C2×C4#8814212yes—
334C8—44no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22#1171831711yescharacteristic
348C22#1171834311yescharacteristic
349C22#1171836311yescharacteristic
350C22#1171839311yescharacteristic
351C22#1171840011yescharacteristic
352C22#1171840211yescharacteristic
353C22#1171840311yescharacteristic
354C22#1171840711yescharacteristic
355C22#1171841411yescharacteristic
356C22#1171850711yescharacteristic
357C22#1171851111yescharacteristic
358C4—22no—
359C4—22no—
360C4—42no—
361C4—42no—
362C4—1042no—
363C4—1042no—
364C4—1042no—
365C4—1042no—
366C2—21no—
367C2—21no—
368C2—21no—
369C2—21no—
370C2#1311377311yescharacteristic
371C2#1311379511yescharacteristic
372C2#1311382411yescharacteristic
373C2#1311388011yescharacteristic
374C2#1311616511yescharacteristic
375C2#1311618711yescharacteristic
376C2#1311619011yescharacteristic
subgroupG/H
centerC23#88136
commutator, cosocleC23×C4×C13C23
FrattiniC23×C4C22×D13
FittingC13×#1280C2
socleC23×C13#120
prime radicalC24×C13C2×D4
hypercenter#1280D13
hypocenterC13#10569
LCSG ▷ C23×C4×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1280

Complements

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

NH ≅ G/Ntype
38. C13×C22⋊C8Q8semidirect
45. #88071Q8semidirect
216. C22⋊C8Dic26semidirect
284. C13#10569semidirect

Maximal subgroups

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

subgroupindex
#131209502
#131150262
#131130922
#131127162
C13×#12802
#1056913