Finite Group Databasebeta

← #35218218

Subgroups — #35218218

Order 20864.

528 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19104712C211yesmaximal, characteristic
2C2×#19104691C211yesmaximal, characteristic
3#23781606C211yesmaximal, characteristic, complemented
4#23781599C211yesmaximal, characteristic, complemented
5#23781596C211yesmaximal, characteristic, complemented
6#23781585C211yesmaximal, characteristic, complemented
7#23781569C211yesmaximal, characteristic, complemented
8#23781560C211yesmaximal, characteristic, complemented
9#23781551C211yesmaximal, characteristic, complemented
10#23781547C211yesmaximal, characteristic, complemented
11#23781538C211yesmaximal, characteristic
12#23781537C211yesmaximal, characteristic
13#23781526C211yesmaximal, characteristic
14C2×C163⋊#95C211yesmaximal, characteristic
15#380×C163C211yesmaximal, characteristic
16C22×Dic326C2211yescharacteristic
17C23×Dic163C2211yescharacteristic
18C23×Dic163C2211yescharacteristic
19#19104712—24no—
20#19104691—24no—
21C2×C4⋊Dic163C2211yescharacteristic
22C2×Dic163⋊C4C2211yescharacteristic
23C2×Dic163⋊C4C2211yescharacteristic
24C2×Dic163⋊C4C2211yescharacteristic
25C2×Dic163⋊C4C2211yescharacteristic
26C2×Dic163⋊C4C2211yescharacteristic
27C2×Dic163⋊C4C2211yescharacteristic
28C2×Dic163⋊C4C2211yescharacteristic
29C2×Dic163⋊C4C2211yescharacteristic
30C2×C22⋊Dic163C2211yescharacteristic
31C2×C22⋊Dic163C2211yescharacteristic
32C2×C4×Dic163C2211yescharacteristic
33C2×C4×Dic163C2211yescharacteristic
34C163⋊#95C2211yescharacteristic
35C163⋊#95C2211yescharacteristic
36C163⋊#95C2211yescharacteristic
37C163⋊#95C2211yescharacteristic
38C163⋊#95C2211yescharacteristic
39C163⋊#95C2211yescharacteristic
40C163⋊#95C2211yescharacteristic
41C163⋊#95C2211yescharacteristic
42C163⋊#95C2211yescharacteristic
43C163⋊#95C2211yescharacteristic
44C163⋊#95C2211yescharacteristic
45C163⋊#95C2211yescharacteristic
46C23×C4×C163C2211yescharacteristic
47C2×C4⋊C4×C163C2211yescharacteristic
48C2×C4⋊C4×C163C2211yescharacteristic
49C2×C22⋊C4×C163C2211yescharacteristic
50C2×C22⋊C4×C163C2211yescharacteristic
51#95×C163C2211yescharacteristic
52#95×C163C2211yescharacteristic
53C2×Dic326—22no—
54C2×Dic326—22no—
55C2×Dic326—41no—
56C22×Dic163—22no—
57C22×Dic163—22no—
58C22×Dic163—22no—
59C22×Dic163—22no—
60C22×Dic163—41no—
61C22×Dic163Q814yescomplemented
62C22×Dic163C2311yescharacteristic
63C22×Dic163C2311yescharacteristic
64C22×Dic163C2311yescharacteristic
65C22×Dic163C2311yescharacteristic
66C22×Dic163C2311yescharacteristic
67C22×Dic163C2311yescharacteristic
68C22×Dic163C2311yescharacteristic
69C22×Dic163C2311yescharacteristic
70C4⋊Dic163—22no—
71Dic163⋊C4—22no—
72Dic163⋊C4—22no—
73Dic163⋊C4—22no—
74Dic163⋊C4—22no—
75Dic163⋊C4—22no—
76Dic163⋊C4—22no—
77Dic163⋊C4—41no—
78Dic163⋊C4—41no—
79C22⋊Dic163—22no—
80C22⋊Dic163D414yes—
81C4×Dic163—22no—
82C4×Dic163—22no—
83C24×C163C2311yescharacteristic
84C22×C4×C163—22no—
85C22×C4×C163—22no—
86C22×C4×C163—41no—
87C22×C4×C163C2311yescharacteristic
88C22×C4×C163C2311yescharacteristic
89C22×C4×C163C2311yescharacteristic
90C22×C4×C163C2311yescharacteristic
91C22×C4×C163C2311yescharacteristic
92C22×C4×C163C2311yescharacteristic
93C4⋊C4×C163—22no—
94C4⋊C4×C163—41no—
95C22⋊C4×C163—22no—
96C22⋊C4×C163—22no—
97Dic326—44no—
98C2×Dic163—28no—
99C2×Dic163—21no—
100C2×Dic163—21no—
101C2×Dic163—21no—
102C2×Dic163—21no—
103C2×Dic163—21no—
104C2×Dic163—21no—
105C2×Dic163—21no—
106C2×Dic163—21no—
107C2×Dic163—21no—
108C2×Dic163—21no—
109C2×Dic163—21no—
110C2×Dic163—21no—
111C2×Dic163—21no—
112C2×Dic163—21no—
113C2×Dic163—21no—
114C2×Dic163—21no—
115C2×Dic163—21no—
116C2×Dic163—21no—
117C2×Dic163—21no—
118C2×Dic163—44no—
119C2×Dic163C2×D412yes—
120C2×Dic163C2×D412yes—
121C2×Dic163C2×Q812yes—
122C2×Dic163C2×Q812yes—
123C2×Dic163#4112yes—
124C23×C163—21no—
125C23×C163—21no—
126C23×C163—21no—
127C23×C163—21no—
128C23×C163C2×D412yes—
129C23×C163C2×Q812yes—
130C23×C163#4112yes—
131C23×C163C2411yescharacteristic
132C2×C4×C163—21no—
133C2×C4×C163—21no—
134C2×C4×C163—21no—
135C2×C4×C163—21no—
136C2×C4×C163—21no—
137C2×C4×C163—21no—
138C2×C4×C163—21no—
139C2×C4×C163—21no—
140C2×C4×C163—21no—
141C2×C4×C163—21no—
142C2×C4×C163—21no—
143C2×C4×C163—21no—
144C2×C4×C163—21no—
145C2×C4×C163—21no—
146C2×C4×C163—21no—
147C2×C4×C163—21no—
148C2×C4×C163—21no—
149C2×C4×C163—44no—
150C2×C4×C163#4112yes—
151C22×C163—22no—
152C22×C163—22no—
153C22×C163—22no—
154C22×C163—22no—
155C22×C163—22no—
156C22×C163—22no—
157C22×C163C22⋊Q814yes—
158C22×C163C22×D411yescharacteristic
159C22×C163C22×Q811yescharacteristic
160C22×C163C2×#4111yescharacteristic
161C22×C163C2×#4111yescharacteristic
162C22×C163C2×#4111yescharacteristic
163C22×C163H(5,𝔽2)11yescharacteristic
164C22×C1632−1+411yescharacteristic
165C4×C163—22no—
166C4×C163—41no—
167C4×C163—41no—
168C4×C163—41no—
169C4×C163—41no—
170C4×C163—41no—
171Dic163—22no—
172Dic163—22no—
173Dic163—22no—
174Dic163—22no—
175Dic163—22no—
176Dic163—41no—
177Dic163—41no—
178Dic163—41no—
179C2×C163—24no—
180C2×C163C2×C22⋊Q811yescharacteristic
181C2×C163#53311yescharacteristic
182C2×C163#54411yescharacteristic
183C2×C163#54611yescharacteristic
184C2×C163#54711yescharacteristic
185C2×C163#55111yescharacteristic
186C2×C163#55211yescharacteristic
187C163#243811yescharacteristic, complemented, Sylow
188#2438—1631nomaximal, Sylow
189C2×C22⋊Q8—1631no—
190C2×#117—1631no—
191#400—1631no—
192#398—1631no—
193#398—1631no—
194#395—1631no—
195#391—1631no—
196#389—1631no—
197#387—1631no—
198#387—1631no—
199#385—1631no—
200#385—1631no—
201#381—1631no—
202#380D16311yescharacteristic
203C2×#95—1631no—
204C22×Q8—1631no—
205C23×C4—1631no—
206C23×C4—1631no—
207C23×C4C2×D16311yescharacteristic
208C22⋊Q8—3264no—
209#117—3264no—
210C2×C4⋊C4—1631no—
211C2×C4⋊C4—1631no—
212C2×C4⋊C4—1631no—
213C2×C4⋊C4—1631no—
214C2×C4⋊C4—1631no—
215C2×C4⋊C4—1631no—
216C2×C4⋊C4—1631no—
217C2×C4⋊C4—1631no—
218C2×C4⋊C4—1631no—
219C2×C4⋊C4C2×D16311yescharacteristic
220C2×C4⋊C4C2×D16311yescharacteristic
221C2×C22⋊C4—1631no—
222C2×C22⋊C4—1631no—
223C2×C22⋊C4C2×D16311yescharacteristic
224C2×C22⋊C4C2×D16311yescharacteristic
225C2×C42—1631no—
226C2×C42—1631no—
227#95—1631no—
228#95—1631no—
229#95—1631no—
230#95—1631no—
231#95—1631no—
232#95—1631no—
233#95—1631no—
234#95—1631no—
235#95—1631no—
236#95—1631no—
237#95—1631no—
238#95—1631no—
239#95C2×D16311yescharacteristic
240#95C2×D16311yescharacteristic
241C24C22×D16311yescharacteristic
242C2×Q8—3262no—
243C2×Q8—3262no—
244C2×Q8—6521no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—41no—
248C22×C4—3262no—
249C22×C4—3262no—
250C22×C4—3262no—
251C22×C4—3262no—
252C22×C4—6521no—
253C22×C4—1631no—
254C22×C4—1631no—
255C22×C4—1631no—
256C22×C4—1634no—
257C22×C4—1631no—
258C22×C4—1631no—
259C22×C4—1631no—
260C22×C4—1631no—
261C22×C4—1631no—
262C22×C4C22×D16311yescharacteristic
263C22×C4C22×D16311yescharacteristic
264C22×C4C22×D16311yescharacteristic
265C22×C4C22×D16311yescharacteristic
266C22×C4C22×D16311yescharacteristic
267C22×C4C22×D16311yescharacteristic
268C4⋊C4—22no—
269C4⋊C4—41no—
270C4⋊C4—3262no—
271C4⋊C4—3262no—
272C4⋊C4—3262no—
273C4⋊C4—3262no—
274C4⋊C4—3262no—
275C4⋊C4—3262no—
276C4⋊C4—3262no—
277C4⋊C4—6521no—
278C4⋊C4—6521no—
279C22⋊C4—22no—
280C22⋊C4—22no—
281C22⋊C4—3262no—
282C22⋊C4—1634no—
283C42—3262no—
284C42—3262no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23D4×D16312yes—
290C23Q8×D16312yes—
291C23#1457969912yes—
292C23C23×D16311yescharacteristic
293Q8—6524no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—44no—
312C2×C4—3261no—
313C2×C4—3261no—
314C2×C4—3261no—
315C2×C4—3261no—
316C2×C4—3261no—
317C2×C4—3261no—
318C2×C4—3261no—
319C2×C4—3261no—
320C2×C4—3261no—
321C2×C4—3261no—
322C2×C4—3268no—
323C2×C4—3261no—
324C2×C4—3261no—
325C2×C4—3261no—
326C2×C4—3261no—
327C2×C4—3261no—
328C2×C4—3261no—
329C2×C4—3261no—
330C2×C4—3261no—
331C2×C4—3261no—
332C2×C4—6524no—
333C2×C4—1632no—
334C2×C4—1632no—
335C2×C4—1632no—
336C2×C4—1632no—
337C2×C4—1632no—
338C2×C4#1457969812yes—
339C22—22no—
340C22—22no—
341C22—22no—
342C22—22no—
343C22—22no—
344C22—22no—
345C22#1910471314yes—
346C22C2×D4×D16311yescharacteristic
347C22C2×Q8×D16311yescharacteristic
348C22#41×D16311yescharacteristic
349C22C2×#1457969811yescharacteristic
350C22C2×#1457969911yescharacteristic
351C22#1910478011yescharacteristic
352C22#1910478211yescharacteristic
353C4—22no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—3262no—
360C4—3262no—
361C4—3262no—
362C4—3262no—
363C4—3262no—
364C4—6521no—
365C4—6521no—
366C4—6521no—
367C2—24no—
368C2C2×#1910471311yescharacteristic
369C2#2378221711yescharacteristic
370C2#2378229411yescharacteristic
371C2#2378230411yescharacteristic
372C2#2378231511yescharacteristic
373C2#2378234111yescharacteristic
374C2#2378234811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D163
commutator, socle, cosocleC23×C163C24
Fitting#380×C163C2
prime radicalC24×C163C23
hypercenter#380D163
hypocenterC163#2438
LCSG ▷ C23×C163 ▷ C163
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
3. #23781606C2semidirect
4. #23781599C2semidirect
5. #23781596C2semidirect
6. #23781585C2semidirect
7. #23781569C2semidirect
8. #23781560C2semidirect
9. #23781551C2semidirect
10. #23781547C2semidirect
61. C22×Dic163Q8semidirect
187. C163#2438semidirect

Maximal subgroups

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

subgroupindex
C2×#191047122
C2×#191046912
#237816062
#237815992
#237815962
#237815852
#237815692
#237815602
#237815512
#237815472
#237815382
#237815372
#237815262
C2×C163⋊#952
#380×C1632
#2438163