Finite Group Databasebeta

← #15766151

Subgroups — #15766151

Order 3328.

602 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 602 conjugacy classes of subgroups — too many to draw. Only its 75 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#13120501C211yesmaximal, characteristic, complemented
2#13114780C212yesmaximal
3#13112744C211yesmaximal, characteristic, complemented
4#13112688C212yesmaximal, complemented
5C13×#1782C211yesmaximal, characteristic
6C22×C13×#41C2211yescharacteristic
7C2×C13×#99—21no—
8C2×C13×#99—21no—
9C13×#381C2211yescharacteristic
10C13×#381C2211yescharacteristic
11C2×#88077—22no—
12C2×#88077—22no—
13C4×C22⋊Dic13C2212yes—
14#11718868C2212yes—
15#11718610—21no—
16#11718608—21no—
17#11718316—22no—
18#11718307—22no—
19C2×C13×#41—22no—
20C2×C13×#41—22no—
21C2×C13×#41—42no—
22C2×C13×#41—42no—
23C22×Q8×C13Q811yescharacteristic, complemented
24C22×D4×C13—21no—
25C22×D4×C13Q811yescharacteristic, complemented
26C23×C4×C13—21no—
27C23×C4×C13C2311yescharacteristic
28C22×Dic26—21no—
29C2×C13×C22⋊C4—21no—
30C2×C13×C22⋊C4D411yescharacteristic
31C2×C13×C22⋊C4D411yescharacteristic
32C2×C22⋊Dic13—22no—
33C2×C4⋊Dic13—21no—
34C2×#6101—22no—
35C2×C4×Dic13D412yes—
36C2×C4×Dic13D412yes—
37C2×#6099—22no—
38C13×#99—42no—
39C13×#99—42no—
40C13×#95—22no—
41C13×#95—22no—
42#88077—44no—
43#88077—44no—
44#88071—21no—
45#88071—22no—
46#88071—21no—
47#88071—22no—
48C22⋊#1055—22no—
49C22⋊#1055—22no—
50#10386—131nomaximal, Sylow
51C24×C13—21no—
52C24×C13C2×D411yescharacteristic
53C13×#41—24no—
54C13×#41—24no—
55C13×#41—42no—
56C13×#41—42no—
57C13×#41—44no—
58C13×#41—44no—
59C2×Q8×C13—22no—
60C2×Q8×C13—42no—
61C2×D4×C13—22no—
62C2×D4×C13—22no—
63C2×D4×C13—22no—
64C2×D4×C13—42no—
65C2×D4×C13—41no—
66C2×D4×C13—41no—
67C2×D4×C13—41no—
68C2×D4×C13—41no—
69C22×C4×C13—21no—
70C22×C4×C13—21no—
71C22×C4×C13—21no—
72C22×C4×C13—41no—
73C22×C4×C13—41no—
74C22×C4×C13—42no—
75C22×C4×C13—42no—
76C22×C4×C13—42no—
77C22×C4×C13—41no—
78C22×C4×C13C2×D412yes—
79C22×C4×C13C2×Q811yescharacteristic
80C22×C4×C13#4111yescharacteristic
81C22×C4×C13#4112yes—
82C22×Dic13—22no—
83C22×Dic13—22no—
84C2×Dic26—44no—
85C2×Dic26—81no—
86C13×C22⋊C4—41no—
87C13×C22⋊C4—41no—
88C13×C22⋊C4—41no—
89C13×C22⋊C4—41no—
90C22⋊Dic13—44no—
91C4⋊Dic13—81no—
92#6101—44no—
93C4×Dic13—42no—
94C4×Dic13—42no—
95#6099—42no—
96#6099—42no—
97C2×#1055—42no—
98#2607—131no—
99#1789—132no—
100#1782D1311yescharacteristic
101#1306—131no—
102#1284—132no—
103C23×C13—21no—
104C23×C13—21no—
105C23×C13—22no—
106C23×C13—42no—
107C23×C13—41no—
108C23×C13—41no—
109C23×C13—82no—
110C23×C13#12011yescharacteristic
111C23×C13C22⋊Q811yescharacteristic
112C23×C13#12411yescharacteristic
113Q8×C13—24no—
114Q8×C13—42no—
115D4×C13—24no—
116D4×C13—42no—
117D4×C13—42no—
118D4×C13—42no—
119D4×C13—42no—
120D4×C13—42no—
121C2×C4×C13—22no—
122C2×C4×C13—22no—
123C2×C4×C13—22no—
124C2×C4×C13—41no—
125C2×C4×C13—41no—
126C2×C4×C13—42no—
127C2×C4×C13—44no—
128C2×C4×C13—44no—
129C2×C4×C13—41no—
130C2×C4×C13—41no—
131C2×C4×C13—42no—
132C2×C4×C13—42no—
133C2×C4×C13—41no—
134C2×C4×C13—41no—
135C2×C4×C13—81no—
136C2×C4×C13—82no—
137C2×C4×C13C22⋊Q812yes—
138C2×C4×C13#12412yes—
139C2×Dic13—24no—
140C2×Dic13—42no—
141C2×Dic13—42no—
142C2×Dic13—42no—
143C2×Dic13—42no—
144C2×Dic13—42no—
145Dic26—84no—
146#1055—82no—
147C22×#41C2×D1311yescharacteristic
148C2×#104—262no—
149C2×#104—262no—
150C2×#99—21no—
151C2×#99—21no—
152#407—132no—
153#399—261no—
154#393—261no—
155#391—262no—
156#387—262no—
157#381C2×D1311yescharacteristic
158#381C2×D1311yescharacteristic
159C4×C22⋊C4—132no—
160C22×C13—22no—
161C22×C13—21no—
162C22×C13—21no—
163C22×C13—22no—
164C22×C13—22no—
165C22×C13—42no—
166C22×C13—81no—
167C22×C13—81no—
168C22×C13—82no—
169C22×C13—82no—
170C22×C13—82no—
171C22×C13#39311yescharacteristic
172C22×C13B(4,𝔽2)11yescharacteristic
173C22×C13B(4,𝔽2)11yescharacteristic
174C4×C13—22no—
175C4×C13—22no—
176C4×C13—42no—
177C4×C13—81no—
178C4×C13—81no—
179C4×C13—161no—
180Dic13—44no—
181Dic13—82no—
182C2×#41—22no—
183C2×#41—22no—
184C2×#41—42no—
185C2×#41—42no—
186C22×Q8—261no—
187C22×Q8Dic2611yescharacteristic, complemented
188C22×D4—21no—
189C22×D4Dic2611yescharacteristic, complemented
190C23×C4—21no—
191C23×C4C22×D1311yescharacteristic
192C2×M16—262no—
193C2×C4⋊C4—262no—
194C2×C4⋊C4—261no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—262no—
197C2×C22⋊C4C22⋊D1311yescharacteristic
198C2×C22⋊C4C22⋊D1311yescharacteristic
199C2×C42—132no—
200C2×C42—132no—
201#104—524no—
202#104—524no—
203#99—42no—
204#99—42no—
205C22⋊C8—262no—
206C22⋊C8—262no—
207#95—22no—
208#95—22no—
209#95—261no—
210#95—262no—
211#95—261no—
212#95—262no—
213C2×C13—21no—
214C2×C13—21no—
215C2×C13—41no—
216C2×C13—41no—
217C2×C13—82no—
218C2×C13#202911yescharacteristic
219C2×C13#203112yes—
220C24—21no—
221C24C2×C22⋊D1311yescharacteristic
222#41—24no—
223#41—24no—
224#41—42no—
225#41—42no—
226#41—44no—
227#41—44no—
228C2×Q8—22no—
229C2×Q8—42no—
230C2×Q8—524no—
231C2×Q8—1041no—
232C2×D4—22no—
233C2×D4—22no—
234C2×D4—22no—
235C2×D4—42no—
236C2×D4—41no—
237C2×D4—41no—
238C2×D4—41no—
239C2×D4—41no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—41no—
249C22×C4—262no—
250C22×C4—262no—
251C22×C4C2×Dic2611yescharacteristic
252C22×C4#612711yescharacteristic
253C22×C4D4×D1312yes—
254C22×C4#612912yes—
255M16—522no—
256M16—522no—
257C2×C8—522no—
258C4⋊C4—524no—
259C4⋊C4—1041no—
260C22⋊C4—41no—
261C22⋊C4—41no—
262C22⋊C4—41no—
263C22⋊C4—41no—
264C22⋊C4—524no—
265C42—522no—
266C42—522no—
267C13#1038611yescharacteristic, complemented, Sylow
268C23—21no—
269C23—21no—
270C23—22no—
271C23—42no—
272C23—41no—
273C23—41no—
274C23—82no—
275C23#8817811yescharacteristic
276C23#8819011yescharacteristic
277C23#8819111yescharacteristic
278Q8—24no—
279Q8—42no—
280Q8—1044no—
281D4—24no—
282D4—42no—
283D4—42no—
284D4—42no—
285D4—42no—
286D4—42no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—44no—
294C2×C4—44no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—81no—
302C2×C4—82no—
303C2×C4—264no—
304C2×C4—522no—
305C2×C4—522no—
306C2×C4—522no—
307C2×C4—522no—
308C2×C4—522no—
309C2×C4#8813212yes—
310C2×C4#8813912yes—
311C8—1042no—
312C22—22no—
313C22—21no—
314C22—21no—
315C22—22no—
316C22—22no—
317C22—42no—
318C22—81no—
319C22—81no—
320C22—82no—
321C22—82no—
322C22—82no—
323C22#1171860811yescharacteristic
324C22#1171869311yescharacteristic
325C22#1171869311yescharacteristic
326C4—22no—
327C4—22no—
328C4—42no—
329C4—81no—
330C4—81no—
331C4—161no—
332C4—524no—
333C4—1042no—
334C2—21no—
335C2—21no—
336C2—41no—
337C2—41no—
338C2—82no—
339C2#1311600011yescharacteristic
340C2#1311601012yes—
subgroupG/H
centerC22#11718608
commutator, cosocleC23×C4×C13C23
FrattiniC23×C4C22×D13
FittingC13×#1782C2
socleC22×C13#393
prime radicalC22×C13×#41C22
hypercenter#1782D13
hypocenterC13#10386
LCSG ▷ C23×C4×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1782

Complements

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

NH ≅ G/Ntype
1. #13120501C2semidirect
3. #13112744C2semidirect
4. #13112688C2semidirect
23. C22×Q8×C13Q8semidirect
25. C22×D4×C13Q8semidirect
187. C22×Q8Dic26semidirect
189. C22×D4Dic26semidirect
267. C13#10386semidirect

Maximal subgroups

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

subgroupindex
#131205012
#131147802
#131127442
#131126882
C13×#17822
#1038613