Finite Group Databasebeta

← #13127870

Subgroups — #13127870

Order 1664.

350 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11719376C211yesmaximal, characteristic
2#11719272C211yesmaximal, characteristic, complemented
3C13×#464C211yesmaximal, characteristic
4#11718843C211yesmaximal, characteristic, complemented
5#11718842C211yesmaximal, characteristic, complemented
6#11718828C211yesmaximal, characteristic
7#11718823C211yesmaximal, characteristic
8#11718721C211yesmaximal, characteristic
9#11718691C211yesmaximal, characteristic, complemented
10#11718510C211yesmaximal, characteristic, complemented
11#11718489C211yesmaximal, characteristic, complemented
12#11718454C211yesmaximal, characteristic, complemented
13#11718450C211yesmaximal, characteristic, complemented
14#11718428C211yesmaximal, characteristic
15#11718419C211yesmaximal, characteristic
16C2×C13×Q16C2211yescharacteristic
17C2×C13×SD16C2211yescharacteristic
18C13×#124C2211yescharacteristic
19C4×Q8×C13C2211yescharacteristic
20Q8×Dic13C2211yescharacteristic
21#88198C2211yescharacteristic
22#88198C2211yescharacteristic
23#88198C2211yescharacteristic
24C2×#6107C2211yescharacteristic
25C2×#6107C2211yescharacteristic
26C2×#6107C2211yescharacteristic
27#88190C2211yescharacteristic
28D4×Dic13C2211yescharacteristic
29C2×#6105C2211yescharacteristic
30C2×#6105C2211yescharacteristic
31C2×Dic52C2211yescharacteristic
32C2×Dic52C2211yescharacteristic
33#88143—21no—
34#88143—21no—
35#88141C2211yescharacteristic
36#88141C2211yescharacteristic
37#88133—21no—
38#88133—21no—
39#88132—21no—
40#88131—21no—
41#88124—21no—
42#88123C2211yescharacteristic
43C4×Dic26—21no—
44C4×Dic26C2211yescharacteristic
45C4×Dic26C2211yescharacteristic
46C13×C4⋊C8C2211yescharacteristic
47C13×#103C2211yescharacteristic
48C13×#102C2211yescharacteristic
49#88075C2211yescharacteristic
50#88056C2211yescharacteristic
51#88056C2211yescharacteristic
52#88055C2211yescharacteristic
53#88054C2211yescharacteristic
54#88054C2211yescharacteristic
55C8×Dic13C2211yescharacteristic
56#88050C2211yescharacteristic
57#88048C2211yescharacteristic
58#88043C2211yescharacteristic
59C2×Q8×C13C2311yescharacteristic
60C2×Q8×C13C2311yescharacteristic
61C2×D4×C13C2311yescharacteristic
62C22×Dic13—21no—
63C2×Dic26D411yescharacteristic
64C2×Dic26C2311yescharacteristic
65C2×Dic26C2311yescharacteristic
66C2×Dic26C2311yescharacteristic
67C13×Q16—41no—
68C13×SD16—41no—
69C2×C8×C13C2311yescharacteristic
70C2×C8×C13C2311yescharacteristic
71C13×C4⋊C4—21no—
72C13×C4⋊C4C2311yescharacteristic
73C13×C22⋊C4—21no—
74C13×C22⋊C4—21no—
75C42×C13—21no—
76C42×C13C2311yescharacteristic
77C22⋊Dic13—21no—
78C22⋊Dic13—21no—
79C22⋊Dic13—21no—
80#6107—22no—
81#6107—22no—
82#6107—41no—
83#6105—22no—
84#6105—41no—
85C4⋊Dic13—21no—
86C4⋊Dic13—21no—
87C4⋊Dic13—21no—
88C4⋊Dic13D411yescharacteristic
89C4⋊Dic13C2311yescharacteristic
90#6101—21no—
91#6101—21no—
92#6101—21no—
93#6101—21no—
94#6101—21no—
95#6101—21no—
96#6101—21no—
97#6101—21no—
98#6101—21no—
99#6101D411yescharacteristic
100#6101D411yescharacteristic
101C4×Dic13—21no—
102C4×Dic13—21no—
103C4×Dic13—21no—
104C4×Dic13—21no—
105C4×Dic13C2311yescharacteristic
106C4×Dic13C2311yescharacteristic
107C2×#1055C2311yescharacteristic
108C2×#1055C2311yescharacteristic
109Dic52—41no—
110Dic52—41no—
111#3398—131nomaximal, Sylow
112C23×C13—21no—
113Q8×C13—21no—
114Q8×C13—21no—
115Q8×C13—21no—
116Q8×C13—21no—
117D4×C13—21no—
118D4×C13—21no—
119C2×C4×C13—21no—
120C2×C4×C13—21no—
121C2×C4×C13—21no—
122C2×C4×C13C2×D411yescharacteristic
123C2×C4×C13C2×D411yescharacteristic
124C2×C4×C13C2411yescharacteristic
125C2×Dic13—21no—
126C2×Dic13—21no—
127C2×Dic13—21no—
128C2×Dic13—21no—
129C2×Dic13—42no—
130C2×Dic13C2×D411yescharacteristic
131C2×Dic13C2×D411yescharacteristic
132C2×Dic13C2×D411yescharacteristic
133C2×Dic13C2×D411yescharacteristic
134Dic26—22no—
135Dic26—21no—
136Dic26—21no—
137Dic26—21no—
138Dic26—21no—
139Dic26—21no—
140Dic26—21no—
141C8×C13—21no—
142C8×C13—21no—
143#1055—21no—
144#1055#4112yes—
145#557—131no—
146#556—131no—
147#497—131no—
148#497—131no—
149#494—131no—
150#464—131no—
151#464D1311yescharacteristic
152#462—131no—
153#462—131no—
154#462—131no—
155#461—131no—
156#446—131no—
157#441—131no—
158#441—131no—
159C4×SD16—131no—
160C22×C13—41no—
161C22×C13—41no—
162C22×C13—41no—
163C22×C13C22×D411yescharacteristic
164C4×C13—21no—
165C4×C13—21no—
166C4×C13—41no—
167C4×C13—41no—
168C4×C13—41no—
169C4×C13C2×#4111yescharacteristic
170C4×C13H(5,𝔽2)11yescharacteristic
171Dic13—21no—
172Dic13—21no—
173Dic13—21no—
174Dic13—41no—
175Dic13—41no—
176Dic13—41no—
177Dic13—41no—
178Dic13#13712yes—
179C2×Q16—131no—
180C2×Q16—131no—
181C2×Q16—131no—
182C2×Q16—131no—
183C2×Q16—131no—
184C2×Q16C2×D1311yescharacteristic
185C2×SD16—131no—
186C2×SD16—131no—
187C2×SD16C2×D1311yescharacteristic
188C4⋊Q8—131no—
189C4⋊Q8—131no—
190C4⋊Q8—131no—
191C4⋊Q8—131no—
192#126—261no—
193#126—261no—
194#125—261no—
195#125—261no—
196#125—261no—
197#124—131no—
198#124C2×D1311yescharacteristic
199C22⋊Q8—261no—
200C4×Q8—131no—
201C4×Q8—131no—
202C4×Q8—131no—
203C4×Q8—131no—
204C4×Q8—131no—
205C4×Q8—261no—
206C4×Q8C2×D1311yescharacteristic
207C4×D4—131no—
208#117—261no—
209#106—131no—
210C4⋊C8—131no—
211C4⋊C8—131no—
212C4⋊C8C2×D1311yescharacteristic
213#103—131no—
214#103—131no—
215#103—131no—
216#103—131no—
217#103C2×D1311yescharacteristic
218#102C2×D1311yescharacteristic
219#97—131no—
220#97—131no—
221C4×C8—131no—
222C2×C13—81no—
223C2×C13#55011yescharacteristic
224C2×C13C2×#13711yescharacteristic
225C2×C13#57811yescharacteristic
226C2×Q8—131no—
227C2×Q8—131no—
228C2×Q8—131no—
229C2×Q8—131no—
230C2×Q8C22×D1311yescharacteristic
231C2×Q8C22×D1311yescharacteristic
232C2×D4C22×D1311yescharacteristic
233C22×C4—261no—
234Q16—41no—
235Q16—262no—
236Q16—262no—
237Q16—521no—
238Q16—521no—
239Q16—521no—
240SD16—41no—
241SD16—262no—
242SD16—521no—
243C2×C8—131no—
244C2×C8—131no—
245C2×C8C22×D1311yescharacteristic
246C2×C8C22×D1311yescharacteristic
247C4⋊C4—21no—
248C4⋊C4—131no—
249C4⋊C4—131no—
250C4⋊C4—131no—
251C4⋊C4—131no—
252C4⋊C4—261no—
253C4⋊C4—261no—
254C4⋊C4—261no—
255C4⋊C4—261no—
256C4⋊C4—261no—
257C4⋊C4—261no—
258C4⋊C4—261no—
259C4⋊C4—261no—
260C4⋊C4—261no—
261C4⋊C4—261no—
262C4⋊C4—261no—
263C4⋊C4—261no—
264C4⋊C4C22×D1311yescharacteristic
265C22⋊C4—21no—
266C22⋊C4—21no—
267C22⋊C4—261no—
268C22⋊C4—261no—
269C22⋊C4—261no—
270C42—21no—
271C42—131no—
272C42—131no—
273C42—261no—
274C42—261no—
275C42—261no—
276C42—261no—
277C42C22×D1311yescharacteristic
278C13#339811yescharacteristic, complemented, Sylow
279C23—21no—
280Q8—21no—
281Q8—21no—
282Q8—21no—
283Q8—21no—
284Q8—262no—
285Q8—261no—
286Q8—261no—
287Q8—261no—
288Q8—261no—
289Q8—261no—
290Q8—261no—
291D4—21no—
292D4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—131no—
297C2×C4—131no—
298C2×C4—131no—
299C2×C4—131no—
300C2×C4—261no—
301C2×C4—261no—
302C2×C4—261no—
303C2×C4—261no—
304C2×C4—522no—
305C2×C4D4×D1311yescharacteristic
306C2×C4D4×D1311yescharacteristic
307C2×C4C23×D1311yescharacteristic
308C8—21no—
309C8—21no—
310C8—132no—
311C8—261no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22C2×D4×D1311yescharacteristic
316C4—21no—
317C4—21no—
318C4—41no—
319C4—41no—
320C4—41no—
321C4—132no—
322C4—261no—
323C4—261no—
324C4—261no—
325C4—521no—
326C4—521no—
327C4—521no—
328C4—521no—
329C4#8825111yescharacteristic
330C4#41×D1311yescharacteristic
331C2—81no—
332C2#1171930411yescharacteristic
333C2#1171947611yescharacteristic
334C2D13×#13711yescharacteristic
subgroupG/H
centerC22C2×D4×D13
commutator, cosocleC2×C4×C13C24
FrattiniC2×C4C23×D13
FittingC13×#464C2
socleC22×C13C22×D4
prime radicalC2×D4×C13C23
hypercenter#464D13
hypocenterC13#3398
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
2. #11719272C2semidirect
4. #11718843C2semidirect
5. #11718842C2semidirect
9. #11718691C2semidirect
10. #11718510C2semidirect
11. #11718489C2semidirect
12. #11718454C2semidirect
13. #11718450C2semidirect
278. C13#3398semidirect

Maximal subgroups

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

subgroupindex
#117193762
#117192722
C13×#4642
#117188432
#117188422
#117188282
#117188232
#117187212
#117186912
#117185102
#117184892
#117184542
#117184502
#117184282
#117184192
#339813