Finite Group Databasebeta

← #27181331

Subgroups — #27181331

Order 12928.

396 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20265782C211yesmaximal, characteristic, complemented
2#20265495C211yesmaximal, characteristic, complemented
3#20265490C211yesmaximal, characteristic, complemented
4#20265484C211yesmaximal, characteristic, complemented
5#20265477C211yesmaximal, characteristic, complemented
6#20265377C211yesmaximal, characteristic, complemented
7#20265372C211yesmaximal, characteristic, complemented
8#20265322C211yesmaximal, characteristic, complemented
9#20265322C211yesmaximal, characteristic, complemented
10#20265287C211yesmaximal, characteristic, complemented
11#20265275C211yesmaximal, characteristic, complemented
12C4×#13108061C211yesmaximal, characteristic, complemented
13C4×#13108059C211yesmaximal, characteristic, complemented
14#20265238C211yesmaximal, characteristic, complemented
15#520×C101C211yesmaximal, characteristic, complemented
16C2×#13108061C2211yescharacteristic
17C2×#13108061C2211yescharacteristic
18C2×#13108059C2211yescharacteristic
19C2×#13108059C2211yescharacteristic
20#15692252C2211yescharacteristic
21#15692247C2211yescharacteristic
22#15692240—21no—
23#15692240—21no—
24#15692229C2211yescharacteristic
25#15692229C2211yescharacteristic
26#15692228—21no—
27#15692228—21no—
28C4×D404C2211yescharacteristic
29C4×D404C2211yescharacteristic
30#15692188C2211yescharacteristic
31#15692188C2211yescharacteristic
32#15692186C2211yescharacteristic
33#15692186C2211yescharacteristic
34C4⋊#11717579C2211yescharacteristic
35C4⋊#11717579C2211yescharacteristic
36#15692179C2211yescharacteristic
37#15692179C2211yescharacteristic
38#15692176C2211yescharacteristic
39#15692176C2211yescharacteristic
40#15692175C2211yescharacteristic
41#15692175C2211yescharacteristic
42#15692175C2211yescharacteristic
43#15692175C2211yescharacteristic
44C22⋊#11717579C2211yescharacteristic
45C22⋊#11717579C2211yescharacteristic
46#15692166C2211yescharacteristic
47C4×#11717579C2211yescharacteristic
48C2×#41×C101C2211yescharacteristic
49C4×Q8×C101C2211yescharacteristic
50C4×Q8×C101C2211yescharacteristic
51C4×D4×C101—21no—
52C4×D4×C101—21no—
53C4×D4×C101C2211yescharacteristic
54C4×D4×C101C2211yescharacteristic
55#117×C101—21no—
56#117×C101C2211yescharacteristic
57C2×C4⋊C4×C101—21no—
58C2×C4⋊C4×C101C2211yescharacteristic
59C2×D404—21no—
60C2×D404—21no—
61C2×D404C2311yescharacteristic
62C2×D404C2311yescharacteristic
63C2×C22⋊D101—21no—
64C2×C22⋊D101—21no—
65C2×C4×D101—21no—
66C2×C4×D101—21no—
67#13108061—22no—
68#13108061—22no—
69#13108059—22no—
70#13108059—22no—
71C2×#11717579C2311yescharacteristic
72C2×#11717579C2311yescharacteristic
73C2×#11717579C2311yescharacteristic
74C2×#11717579C2311yescharacteristic
75C4⋊Dic101C2311yescharacteristic
76C4⋊Dic101C2311yescharacteristic
77#13108053—21no—
78#13108053—21no—
79#13108051—21no—
80#13108051—21no—
81#13108051—21no—
82#13108051—21no—
83#41×C101—22no—
84#41×C101—22no—
85C2×Q8×C101C2311yescharacteristic
86C2×D4×C101—21no—
87C2×D4×C101C2311yescharacteristic
88C22×C4×C101—21no—
89C22×C4×C101—21no—
90C22×C4×C101—21no—
91C22×C4×C101—21no—
92C22×C4×C101C2311yescharacteristic
93C4⋊C4×C101—21no—
94C4⋊C4×C101—21no—
95C4⋊C4×C101—21no—
96C4⋊C4×C101D411yescharacteristic
97C4⋊C4×C101D411yescharacteristic
98C4⋊C4×C101C2311yescharacteristic
99C4⋊C4×C101C2311yescharacteristic
100C22⋊C4×C101—21no—
101C22⋊C4×C101—21no—
102C22⋊C4×C101D411yescharacteristic
103C22⋊C4×C101D411yescharacteristic
104C42×C101—21no—
105C42×C101—21no—
106C42×C101C2311yescharacteristic
107C42×C101C2311yescharacteristic
108C23×C101—21no—
109C23×C101C2×D411yescharacteristic
110C22×D101—21no—
111C22×D101—21no—
112Q8×C101—21no—
113Q8×C101#4112yes—
114D4×C101—24no—
115D4×C101—21no—
116D4×C101#4112yes—
117C2×C4×C101—24no—
118C2×C4×C101—21no—
119C2×C4×C101—21no—
120C2×C4×C101—21no—
121C2×C4×C101—21no—
122C2×C4×C101—21no—
123C2×C4×C101—81no—
124C2×C4×C101—81no—
125C2×C4×C101—81no—
126C2×C4×C101C2×D411yescharacteristic
127C2×C4×C101C2×D411yescharacteristic
128C2×C4×C101C2×D411yescharacteristic
129C2×C4×C101C2×D411yescharacteristic
130C2×C4×C101C2×D411yescharacteristic
131C2×C4×C101C2411yescharacteristic
132C22⋊D101—81no—
133C22⋊D101—81no—
134C2×Dic101—21no—
135C2×Dic101—21no—
136D404—21no—
137D404—21no—
138D404—21no—
139D404—21no—
140D404—81no—
141D404—81no—
142C4×D101—81no—
143C4×D101—81no—
144#11717579—21no—
145#11717579—21no—
146#11717579—21no—
147#11717579—21no—
148C22×C101—21no—
149C22×C101—22no—
150C22×C101—21no—
151C22×C101—21no—
152C22×C101—41no—
153C22×C101—41no—
154C22×C101C22×D411yescharacteristic
155C2×D101—41no—
156C2×D101—41no—
157C2×D101—41no—
158C2×D101—41no—
159C2×D101—41no—
160C2×D101—41no—
161C4×C101—21no—
162C4×C101—22no—
163C4×C101—21no—
164C4×C101—21no—
165C4×C101—21no—
166C4×C101—21no—
167C4×C101—41no—
168C4×C101—41no—
169C4×C101C2×#4111yescharacteristic
170C4×C101C2×#4111yescharacteristic
171Dic101—41no—
172Dic101—41no—
173C2×C101—41no—
174C2×C101—42no—
175C2×C101#52511yescharacteristic
176C2×C101#57611yescharacteristic
177C2×C101#57711yescharacteristic
178D101—81no—
179D101—81no—
180#3119—1011nomaximal, Sylow
181C101#311911yescharacteristic, complemented, Sylow
182#540—1011no—
183#520D10111yescharacteristic, complemented
184#481—1011no—
185#480—1011no—
186#479—1011no—
187#478—1011no—
188#460—1011no—
189#459—1011no—
190#449—1011no—
191#449—1011no—
192#442—1011no—
193#440—1011no—
194C4×SD16—1011no—
195C4×D8—1011no—
196#433—1011no—
197C2×#41C2×D10111yescharacteristic
198C2×SD16—1011no—
199C2×SD16—1011no—
200C2×D8—1011no—
201C2×D8—1011no—
202#127—1011no—
203#125—1011no—
204#123—2021no—
205#123—2021no—
206#121—1011no—
207#121—1011no—
208#121—2021no—
209#121—2021no—
210C4×Q8C2×D10111yescharacteristic
211C4×Q8C2×D10111yescharacteristic
212C4×D4—21no—
213C4×D4—21no—
214C4×D4—1011no—
215C4×D4—1011no—
216C4×D4C2×D10111yescharacteristic
217C4×D4C2×D10111yescharacteristic
218#117—21no—
219#117C2×D10111yescharacteristic
220C2×C4⋊C4—21no—
221C2×C4⋊C4C2×D10111yescharacteristic
222#107—1011no—
223#107—1011no—
224#106—1011no—
225#106—1011no—
226C4⋊C8—1011no—
227C4⋊C8—1011no—
228#103—1011no—
229#103—1011no—
230#102—1011no—
231#102—1011no—
232#102—1011no—
233#102—1011no—
234#102—1011no—
235#102—1011no—
236C22⋊C8—1011no—
237C22⋊C8—1011no—
238#97—1011no—
239C4×C8—1011no—
240#41—22no—
241#41—22no—
242C2×Q8C22×D10111yescharacteristic
243C2×D4—21no—
244C2×D4—1011no—
245C2×D4—1011no—
246C2×D4—2021no—
247C2×D4—2021no—
248C2×D4—2021no—
249C2×D4—2021no—
250C2×D4C22×D10111yescharacteristic
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—2021no—
256C22×C4—2021no—
257C22×C4C22×D10111yescharacteristic
258SD16—2022no—
259SD16—2022no—
260D8—2022no—
261D8—2022no—
262C2×C8—1011no—
263C2×C8—1011no—
264C2×C8—1011no—
265C2×C8—1011no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—1011no—
270C4⋊C4—1011no—
271C4⋊C4—2021no—
272C4⋊C4—2021no—
273C4⋊C4C22⋊D10111yescharacteristic
274C4⋊C4C22⋊D10111yescharacteristic
275C4⋊C4C22×D10111yescharacteristic
276C4⋊C4C22×D10111yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—2021no—
280C22⋊C4—2021no—
281C22⋊C4—2021no—
282C22⋊C4—2021no—
283C22⋊C4C22⋊D10111yescharacteristic
284C22⋊C4C22⋊D10111yescharacteristic
285C42—21no—
286C42—21no—
287C42C22×D10111yescharacteristic
288C42C22×D10111yescharacteristic
289C23—21no—
290C23—2021no—
291C23—2021no—
292C23C2×C22⋊D10111yescharacteristic
293Q8—21no—
294Q8#1310807412yes—
295D4—24no—
296D4—21no—
297D4—8081no—
298D4—8081no—
299D4—8081no—
300D4—8081no—
301D4—2021no—
302D4—2021no—
303D4—2021no—
304D4—2021no—
305D4#1310807412yes—
306C2×C4—24no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—8081no—
316C2×C4—8081no—
317C2×C4—2021no—
318C2×C4—2021no—
319C2×C4C2×C22⋊D10111yescharacteristic
320C2×C4C2×C22⋊D10111yescharacteristic
321C2×C4C2×C22⋊D10111yescharacteristic
322C2×C4C2×C22⋊D10111yescharacteristic
323C2×C4C2×C22⋊D10111yescharacteristic
324C2×C4C23×D10111yescharacteristic
325C8—2021no—
326C8—2021no—
327C8—2021no—
328C8—2021no—
329C22—21no—
330C22—21no—
331C22—22no—
332C22—21no—
333C22—41no—
334C22—41no—
335C22—4041no—
336C22—4041no—
337C22—4041no—
338C22—4041no—
339C22—4041no—
340C22—4041no—
341C22C22×C22⋊D10111yescharacteristic
342C4—21no—
343C4—21no—
344C4—21no—
345C4—22no—
346C4—21no—
347C4—21no—
348C4—41no—
349C4—41no—
350C4—4041no—
351C4—4041no—
352C4C2×#1310807411yescharacteristic
353C4C2×#1310807411yescharacteristic
354C2—41no—
355C2—42no—
356C2—8081no—
357C2—8081no—
358C2#2026568811yescharacteristic
359C2#2026598011yescharacteristic
360C2#2026598511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#520×C101C2
socleC22×C101C22×D4
hypercenter#520D101
hypocenterC101#3119
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ #117 ◁ #520

Complements

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

NH ≅ G/Ntype
1. #20265782C2semidirect
2. #20265495C2semidirect
3. #20265490C2semidirect
4. #20265484C2semidirect
5. #20265477C2semidirect
6. #20265377C2semidirect
7. #20265372C2semidirect
8. #20265322C2semidirect
9. #20265322C2semidirect
10. #20265287C2semidirect
11. #20265275C2semidirect
12. C4×#13108061C2semidirect
13. C4×#13108059C2semidirect
14. #20265238C2semidirect
15. #520×C101C2semidirect
181. C101#3119semidirect
183. #520D101semidirect

Maximal subgroups

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

subgroupindex
#202657822
#202654952
#202654902
#202654842
#202654772
#202653772
#202653722
#202653222
#202652872
#202652752
C4×#131080612
C4×#131080592
#202652382
#520×C1012
#3119101