Finite Group Databasebeta

← #46850465

Subgroups — #46850465

Order 34688.

1030 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774630C213yesmaximal, complemented
2#31774626C211yesmaximal, characteristic, complemented
3#31774612C213yesmaximal, complemented
4#31774595C213yesmaximal, complemented
5#31774582C213yesmaximal, complemented
6#31774566C211yesmaximal, characteristic, complemented
7#31774559C211yesmaximal, characteristic, complemented
8#31774554C211yesmaximal, characteristic, complemented
9#31774518C213yesmaximal, complemented
10#31774498C213yesmaximal, complemented
11#31774429C213yesmaximal, complemented
12#31774425C213yesmaximal, complemented
13#31774373C211yesmaximal, characteristic, complemented
14C4×#16864655C211yesmaximal, characteristic, complemented
15#520×C271C211yesmaximal, characteristic, complemented
16C2×#16864655C2211yescharacteristic
17C2×Q8×D271C2211yescharacteristic
18#22576160C2213yes—
19#22576156C2213yes—
20#22576155C2216yes—
21#22576155C2211yescharacteristic
22#22576153C2213yes—
23#22576151C2213yes—
24#22576151C2213yes—
25#22576150C2213yes—
26#22576148C2213yes—
27#22576147C2213yes—
28#22576146C2216yes—
29#22576146C2213yes—
30#22576145C2213yes—
31#22576143C2216yes—
32#22576142C2216yes—
33#22576142C2213yes—
34#22576140C2213yes—
35#22576139C2213yes—
36#22576138C2213yes—
37#22576135C2213yes—
38#22576134C2216yes—
39#22576134C2213yes—
40#22576133C2213yes—
41#22576131C2213yes—
42#22576131C2213yes—
43Q8×Dic271C2211yescharacteristic
44Q8×Dic271C2211yescharacteristic
45C4×Dic542C2213yes—
46C4×Dic542C2213yes—
47#22576125C2213yes—
48C4×D1084C2213yes—
49C4×D1084C2213yes—
50#22576122C2213yes—
51C4×C22⋊D271C2213yes—
52C4×C22⋊D271C2213yes—
53C4×C22⋊D271C2216yes—
54#22576119C2216yes—
55#22576118C2213yes—
56#22576118C2213yes—
57#22576117C2213yes—
58C42×D271C2213yes—
59C2×#41×C271C2211yescharacteristic
60C4×Q8×C271C2211yescharacteristic
61C4×Q8×C271C2211yescharacteristic
62C4×D4×C271C2213yes—
63C4×D4×C271C2213yes—
64#117×C271C2213yes—
65C2×C4⋊C4×C271C2213yes—
66#16864655—24no—
67C2×Dic542C2313yes—
68C2×Dic542C2313yes—
69Q8×D271—24no—
70C2×D1084C2313yes—
71C2×D1084C2313yes—
72C2×C22⋊D271C2313yes—
73C2×C22⋊D271C2316yes—
74C2×C22⋊D271C2313yes—
75C2×C4×D271C2313yes—
76C2×C4×D271C2313yes—
77C2×C4×D271C2313yes—
78C2×C4×D271C2313yes—
79C4⋊Dic271C2313yes—
80C4⋊Dic271C2313yes—
81C4⋊Dic271C2316yes—
82C4⋊Dic271C2313yes—
83C4⋊Dic271C2311yescharacteristic
84Dic271⋊C4C2313yes—
85Dic271⋊C4C2313yes—
86Dic271⋊C4C2313yes—
87Dic271⋊C4C2313yes—
88Dic271⋊C4C2313yes—
89Dic271⋊C4C2311yescharacteristic
90C22⋊Dic271C2313yes—
91C22⋊Dic271C2316yes—
92C22⋊Dic271C2313yes—
93(C2×D271)⋊C4C2311yescharacteristic
94(C2×D271)⋊C4C2313yes—
95(C2×D271)⋊C4C2313yes—
96(C2×D271)⋊C4C2316yes—
97(C2×D271)⋊C4C2316yes—
98(C2×D271)⋊C4C2313yes—
99(C2×D271)⋊C4C2311yescharacteristic
100(C2×D271)⋊C4C2311yescharacteristic
101C4×Dic271C2313yes—
102C4×Dic271C2316yes—
103C4×Dic271C2313yes—
104C4×Dic271C2313yes—
105C4×Dic271C2311yescharacteristic
106#41×C271—24no—
107C2×Q8×C271C2311yescharacteristic
108C2×D4×C271C2313yes—
109C22×C4×C271C2313yes—
110C22×C4×C271C2316yes—
111C4⋊C4×C271C2313yes—
112C4⋊C4×C271C2311yescharacteristic
113C4⋊C4×C271C2313yes—
114C4⋊C4×C271C2313yes—
115C22⋊C4×C271C2313yes—
116C22⋊C4×C271C2313yes—
117C42×C271C2313yes—
118C42×C271C2313yes—
119C22×D271C2411yescharacteristic
120C22×D271C2413yes—
121Dic542—26no—
122Dic542—43no—
123D1084—26no—
124D1084—43no—
125C22⋊D271—43no—
126C22⋊D271—43no—
127C22⋊D271—46no—
128C2×Dic271C2411yescharacteristic
129C2×Dic271C2411yescharacteristic
130C2×Dic271C2413yes—
131C2×Dic271C2411yescharacteristic
132C2×Dic271C2413yes—
133C2×Dic271C2413yes—
134C4×D271—26no—
135C4×D271—26no—
136C4×D271—43no—
137C4×D271—43no—
138C23×C271C2413yes—
139Q8×C271#4114yes—
140D4×C271—26no—
141C2×C4×C271—26no—
142C2×C4×C271—46no—
143C2×C4×C271C2411yescharacteristic
144C2×C4×C271C2411yescharacteristic
145C2×C4×C271C2413yes—
146C2×C4×C271C2413yes—
147C2×C4×C271C2411yescharacteristic
148C2×C4×C271C2413yes—
149C22×C271—23no—
150C22×C271—23no—
151C22×C271—23no—
152C22×C271C2511yescharacteristic
153C2×D271—21no—
154C2×D271—23no—
155C2×D271—23no—
156C2×D271—23no—
157C2×D271—21no—
158C2×D271—21no—
159C4×C271—21no—
160C4×C271—23no—
161C4×C271—23no—
162C4×C271—21no—
163C4×C271—21no—
164C4×C271C2×#4116yes—
165Dic271—21no—
166Dic271—23no—
167Dic271—21no—
168Dic271—23no—
169Dic271—21no—
170Dic271—23no—
171C2×C271—43no—
172C2×C271C22×#4111yescharacteristic
173C2×C271#58511yescharacteristic
174C2×C271#58511yescharacteristic
175D271—41no—
176D271—43no—
177C271#352411yescharacteristic, complemented, Sylow
178#3524—2711nomaximal, Sylow
179#559—2713no—
180#557—2711no—
181#555—2713no—
182#553—2713no—
183#552—2713no—
184#550—2711no—
185#548—2711no—
186#548—2711no—
187#542—2713no—
188#540—2713no—
189#529—2713no—
190#529—2713no—
191#520D27111yescharacteristic, complemented
192#519—2711no—
193C4×#41—2711no—
194C2×#41—2711no—
195C2×#41C2×D27111yescharacteristic
196C22×Q8—2711no—
197C4⋊Q8—2713no—
198#127—2713no—
199#126—2711no—
200#126—2716no—
201#126—2713no—
202#125—2713no—
203#125—2713no—
204#125—2713no—
205#124—2713no—
206#124—2713no—
207#124—2716no—
208#124—2713no—
209#123—2716no—
210#123—2713no—
211#123—2716no—
212#123—2713no—
213C22⋊Q8—2713no—
214C22⋊Q8—2713no—
215C22⋊Q8—2713no—
216#121—2713no—
217#121—2713no—
218#121—2716no—
219#121—2713no—
220#121—2713no—
221#121—2713no—
222C4×Q8—2711no—
223C4×Q8—2711no—
224C4×Q8—2713no—
225C4×Q8—2713no—
226C4×Q8—2713no—
227C4×Q8C2×D27111yescharacteristic
228C4×Q8C2×D27111yescharacteristic
229C4×D4—2713no—
230C4×D4—2713no—
231C4×D4—2716no—
232C4×D4—2713no—
233C4×D4—2713no—
234C4×D4—2713no—
235C4×D4C2×D27113yes—
236C4×D4C2×D27113yes—
237#117—2716no—
238#117—2713no—
239#117—2713no—
240#117—2713no—
241#117C2×D27113yes—
242C2×C4⋊C4C2×D27113yes—
243C2×C42—2713no—
244#41—24no—
245#41—5424no—
246C2×Q8—2713no—
247C2×Q8—2713no—
248C2×Q8—5424no—
249C2×Q8C22×D27111yescharacteristic
250C2×D4—2713no—
251C2×D4—2716no—
252C2×D4—2713no—
253C2×D4—2713no—
254C2×D4—2713no—
255C2×D4C22×D27113yes—
256C22×C4—2713no—
257C22×C4—2713no—
258C22×C4—2713no—
259C22×C4—2713no—
260C22×C4C22×D27113yes—
261C22×C4C22×D27116yes—
262C4⋊C4—2713no—
263C4⋊C4—2713no—
264C4⋊C4—2713no—
265C4⋊C4—2713no—
266C4⋊C4—2713no—
267C4⋊C4—2716no—
268C4⋊C4—2711no—
269C4⋊C4—2713no—
270C4⋊C4—2713no—
271C4⋊C4—2711no—
272C4⋊C4—2713no—
273C4⋊C4C22×D27113yes—
274C4⋊C4C22×D27113yes—
275C4⋊C4C22×D27113yes—
276C4⋊C4C22×D27111yescharacteristic
277C22⋊C4—2713no—
278C22⋊C4—2716no—
279C22⋊C4—2713no—
280C22⋊C4—2711no—
281C22⋊C4—2713no—
282C22⋊C4—2716no—
283C22⋊C4—2711no—
284C22⋊C4—2711no—
285C22⋊C4—2713no—
286C22⋊C4—2716no—
287C22⋊C4—2713no—
288C22⋊C4C22×D27113yes—
289C22⋊C4C22×D27113yes—
290C42—2711no—
291C42—2713no—
292C42—2713no—
293C42—2716no—
294C42—2713no—
295C42C22×D27113yes—
296C42C22×D27113yes—
297C23—2711no—
298C23—2713no—
299C23C23×D27113yes—
300Q8—5426no—
301Q8—10843no—
302Q8#1686465414yes—
303D4—26no—
304D4—5426no—
305D4—10843no—
306D4—10843no—
307D4—10843no—
308D4—10846no—
309C2×C4—26no—
310C2×C4—46no—
311C2×C4—2711no—
312C2×C4—2711no—
313C2×C4—2713no—
314C2×C4—2711no—
315C2×C4—2713no—
316C2×C4—2713no—
317C2×C4—5426no—
318C2×C4—5426no—
319C2×C4—10843no—
320C2×C4—10843no—
321C2×C4C23×D27113yes—
322C2×C4C23×D27111yescharacteristic
323C2×C4C23×D27111yescharacteristic
324C2×C4C23×D27113yes—
325C2×C4C23×D27111yescharacteristic
326C2×C4C23×D27113yes—
327C22—23no—
328C22—23no—
329C22—23no—
330C22—5421no—
331C22—5423no—
332C22—5423no—
333C22—5421no—
334C22—5423no—
335C22—5421no—
336C22C24×D27111yescharacteristic
337C4—21no—
338C4—23no—
339C4—23no—
340C4—21no—
341C4—21no—
342C4—5421no—
343C4—5423no—
344C4—5421no—
345C4—5421no—
346C4—5423no—
347C4—5423no—
348C4C2×#1686465416yes—
349C2—43no—
350C2—10841no—
351C2—10843no—
352C2C22×#1686465411yescharacteristic
353C2#3177473211yescharacteristic
354C2#3177473411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D271
commutator, socle, cosocleC22×C271C25
Fitting#520×C271C2
hypercenter#520D271
hypocenterC271#3524
LCSG ▷ C22×C271 ▷ C271
UCSC1 ◁ C22 ◁ #520

Complements

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

NH ≅ G/Ntype
1. #31774630C2semidirect
2. #31774626C2semidirect
3. #31774612C2semidirect
4. #31774595C2semidirect
5. #31774582C2semidirect
6. #31774566C2semidirect
7. #31774559C2semidirect
8. #31774554C2semidirect
9. #31774518C2semidirect
10. #31774498C2semidirect
11. #31774429C2semidirect
12. #31774425C2semidirect
13. #31774373C2semidirect
14. C4×#16864655C2semidirect
15. #520×C271C2semidirect
177. C271#3524semidirect
191. #520D271semidirect

Maximal subgroups

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

subgroupindex
#317746302
#317746262
#317746122
#317745952
#317745822
#317745662
#317745592
#317745542
#317745182
#317744982
#317744292
#317744252
#317743732
C4×#168646552
#520×C2712
#3524271