Finite Group Databasebeta

← #16833571

Subgroups — #16833571

Order 3712.

1104 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#14271896C213yesmaximal, complemented
2Q8×Dic58C211yesmaximal, characteristic, complemented
3#14271868C213yesmaximal, complemented
4#14271832C211yesmaximal, characteristic, complemented
5#14271825C211yesmaximal, characteristic, complemented
6#14271820C211yesmaximal, characteristic, complemented
7#14271818C213yesmaximal, complemented
8#14271814C213yesmaximal, complemented
9#14271784C213yesmaximal, complemented
10#14271778C213yesmaximal, complemented
11#14271710C213yesmaximal, complemented
12#14271691C213yesmaximal, complemented
13#14271645C211yesmaximal, characteristic, complemented
14C4×Q8×D29C211yesmaximal, characteristic, complemented
15C29×#520C211yesmaximal, characteristic, complemented
16C2×#41×C29C2211yescharacteristic
17C2×#90809C2211yescharacteristic
18C2×Q8×D29C2211yescharacteristic
19C2×#90805C2216yes—
20C4×Q8×C29C2211yescharacteristic
21C4×Q8×C29C2211yescharacteristic
22C4×D4×C29C2213yes—
23C4×D4×C29C2213yes—
24C29×#117C2213yes—
25C2×C4⋊C4×C29C2213yes—
26Q8×Dic29C2211yescharacteristic
27Q8×Dic29C2211yescharacteristic
28#11745302C2213yes—
29#11745300C2213yes—
30#11745300C2213yes—
31#11745298C2213yes—
32#11745292C2213yes—
33#11745291C2216yes—
34C4×C22⋊D29C2213yes—
35C4×C22⋊D29C2213yes—
36C4×C22⋊D29C2216yes—
37#11745286C2216yes—
38#11745286C2213yes—
39#11745260C2213yes—
40#11745259C2213yes—
41#11745258C2216yes—
42#11745258C2211yescharacteristic
43#11745257C2213yes—
44#11745256C2213yes—
45#11745255C2213yes—
46C4⋊C4×D29C2213yes—
47C4⋊C4×D29C2213yes—
48#11745251C2213yes—
49#11745250C2213yes—
50#11745250C2213yes—
51#11745249C2213yes—
52#11745247C2216yes—
53#11745247C2213yes—
54#11745246C2213yes—
55#11745245C2216yes—
56#11745245C2213yes—
57#11745241C2213yes—
58#11745237C2213yes—
59C4×D116C2213yes—
60C4×D116C2213yes—
61#11745234C2213yes—
62C42×D29C2213yes—
63#11745231C2213yes—
64C4×Dic58C2213yes—
65C4×Dic58C2213yes—
66#41×C29—24no—
67C2×Q8×C29C2311yescharacteristic
68C2×D4×C29C2313yes—
69C22×C4×C29C2313yes—
70C22×C4×C29C2316yes—
71C2×C22⋊D29C2313yes—
72C2×C22⋊D29C2313yes—
73C2×C22⋊D29C2316yes—
74#90809—24no—
75Q8×D29—24no—
76#90805—412no—
77C2×D116C2313yes—
78C2×D116C2313yes—
79C2×C4×D29C2313yes—
80C2×C4×D29C2313yes—
81C2×C4×D29C2313yes—
82C2×C4×D29C2311yescharacteristic
83C2×C4×D29C2316yes—
84C2×Dic58C2313yes—
85C2×Dic58C2311yescharacteristic
86C2×Dic58C2316yes—
87C4⋊C4×C29C2313yes—
88C4⋊C4×C29C2313yes—
89C4⋊C4×C29C2313yes—
90C4⋊C4×C29C2311yescharacteristic
91C22⋊C4×C29C2313yes—
92C22⋊C4×C29C2313yes—
93C42×C29C2313yes—
94C42×C29C2313yes—
95C22⋊Dic29C2316yes—
96C22⋊Dic29C2313yes—
97C22⋊Dic29C2313yes—
98#90781C2311yescharacteristic
99#90781C2316yes—
100#90781C2313yes—
101#90781C2313yes—
102#90781C2313yes—
103#90781C2311yescharacteristic
104#90781C2313yes—
105C4⋊Dic29C2313yes—
106C4⋊Dic29C2313yes—
107C4⋊Dic29C2313yes—
108C4⋊Dic29C2313yes—
109#90779C2313yes—
110#90779C2313yes—
111#90779C2316yes—
112#90779C2311yescharacteristic
113#90779C2313yes—
114#90779C2313yes—
115#90779C2311yescharacteristic
116C4×Dic29C2313yes—
117C4×Dic29C2313yes—
118C4×Dic29C2313yes—
119C4×Dic29C2313yes—
120C23×C29C2413yes—
121C22×D29C2413yes—
122C22×D29C2411yescharacteristic
123Q8×C29#4114yescomplemented
124D4×C29—26no—
125C2×C4×C29—26no—
126C2×C4×C29—46no—
127C2×C4×C29C2411yescharacteristic
128C2×C4×C29C2413yes—
129C2×C4×C29C2411yescharacteristic
130C2×C4×C29C2413yes—
131C2×C4×C29C2411yescharacteristic
132C2×C4×C29C2413yes—
133C22⋊D29—26no—
134C22⋊D29—26no—
135C22⋊D29—46no—
136C2×Dic29C2411yescharacteristic
137C2×Dic29C2413yes—
138C2×Dic29C2413yes—
139C2×Dic29C2413yes—
140C2×Dic29C2411yescharacteristic
141C2×Dic29C2411yescharacteristic
142D116—26no—
143D116—26no—
144C4×D29—26no—
145C4×D29—26no—
146C4×D29—26no—
147C4×D29—22no—
148C4×D29—46no—
149Dic58—26no—
150Dic58—22no—
151Dic58—46no—
152#3522—291nomaximal, Sylow
153C22×C29—23no—
154C22×C29—23no—
155C22×C29—23no—
156C22×C29C2511yescharacteristic
157C2×D29—21no—
158C2×D29—21no—
159C2×D29—23no—
160C2×D29—23no—
161C2×D29—23no—
162C2×D29H(5,𝔽2)12yes—
163C4×C29—21no—
164C4×C29—23no—
165C4×C29—23no—
166C4×C29—21no—
167C4×C29—21no—
168C4×C29C2×#4116yes—
169Dic29—23no—
170Dic29—21no—
171Dic29—23no—
172Dic29—23no—
173Dic29—21no—
174Dic292−1+412yes—
175#559—293no—
176Q82—291no—
177#554—293no—
178#550—291no—
179#548—291no—
180#548—291no—
181#547—293no—
182#547—293no—
183#542—293no—
184#542—293no—
185#532—293no—
186#529—293no—
187#520—291no—
188#520D2911yescharacteristic, complemented
189C2×C4×Q8—291no—
190C2×C29—43no—
191C2×C29C22×#4111yescharacteristic
192C2×C29C2×H(5,𝔽2)11yescharacteristic
193C2×C29C2×2−1+411yescharacteristic
194D29—22no—
195D29—43no—
196C2×#41—291no—
197C2×#41—296no—
198C2×#41C2×D2911yescharacteristic
199C22×Q8—291no—
200C4⋊Q8—293no—
201C4⋊Q8—293no—
202C4⋊Q8—293no—
203#127—293no—
204#126—293no—
205#126—293no—
206#125—293no—
207#124—293no—
208#124—293no—
209#124—296no—
210#124—293no—
211#123—296no—
212#123—293no—
213#123—293no—
214#123—296no—
215C22⋊Q8—296no—
216C22⋊Q8—293no—
217C22⋊Q8—291no—
218C22⋊Q8—296no—
219C22⋊Q8—293no—
220#121—293no—
221#121—293no—
222#121—293no—
223#121—293no—
224#121—293no—
225C4×Q8—291no—
226C4×Q8—291no—
227C4×Q8—293no—
228C4×Q8—293no—
229C4×Q8—293no—
230C4×Q8C2×D2911yescharacteristic
231C4×Q8C2×D2911yescharacteristic
232C4×D4—293no—
233C4×D4—293no—
234C4×D4—296no—
235C4×D4—293no—
236C4×D4—293no—
237C4×D4—293no—
238C4×D4C2×D2913yes—
239C4×D4C2×D2913yes—
240#117—293no—
241#117C2×D2913yes—
242C2×C4⋊C4—293no—
243C2×C4⋊C4—293no—
244C2×C4⋊C4C2×D2913yes—
245C2×C42—293no—
246C29#352211yescharacteristic, complemented, Sylow
247#41—24no—
248#41—584no—
249#41—11612no—
250C2×Q8—291no—
251C2×Q8—293no—
252C2×Q8—296no—
253C2×Q8—584no—
254C2×Q8C22×D2911yescharacteristic
255C2×D4—293no—
256C2×D4—293no—
257C2×D4—296no—
258C2×D4—293no—
259C2×D4—293no—
260C2×D4C22×D2913yes—
261C22×C4—291no—
262C22×C4—293no—
263C22×C4—293no—
264C22×C4—293no—
265C22×C4—296no—
266C22×C4C22×D2913yes—
267C22×C4C22×D2916yes—
268C4⋊C4—293no—
269C4⋊C4—293no—
270C4⋊C4—293no—
271C4⋊C4—296no—
272C4⋊C4—291no—
273C4⋊C4—291no—
274C4⋊C4—293no—
275C4⋊C4—293no—
276C4⋊C4—293no—
277C4⋊C4—293no—
278C4⋊C4—293no—
279C4⋊C4C22×D2913yes—
280C4⋊C4C22×D2913yes—
281C4⋊C4C22×D2913yes—
282C4⋊C4C22×D2911yescharacteristic
283C22⋊C4—293no—
284C22⋊C4—296no—
285C22⋊C4—293no—
286C22⋊C4—293no—
287C22⋊C4—293no—
288C22⋊C4—291no—
289C22⋊C4—291no—
290C22⋊C4—293no—
291C22⋊C4—296no—
292C22⋊C4—293no—
293C22⋊C4C22×D2913yes—
294C22⋊C4C22×D2913yes—
295C42—293no—
296C42—293no—
297C42—293no—
298C42—293no—
299C42C22×D2913yes—
300C42C22×D2913yes—
301C23—291no—
302C23—293no—
303C23C23×D2913yes—
304Q8—582no—
305Q8—586no—
306Q8—1166no—
307Q8#9080514yescomplemented
308D4—26no—
309D4—586no—
310D4—586no—
311D4—586no—
312D4—586no—
313D4—1166no—
314C2×C4—26no—
315C2×C4—46no—
316C2×C4—291no—
317C2×C4—291no—
318C2×C4—293no—
319C2×C4—293no—
320C2×C4—293no—
321C2×C4—291no—
322C2×C4—582no—
323C2×C4—586no—
324C2×C4—586no—
325C2×C4—586no—
326C2×C4—1166no—
327C2×C4C23×D2911yescharacteristic
328C2×C4C23×D2913yes—
329C2×C4C23×D2911yescharacteristic
330C2×C4C23×D2913yes—
331C2×C4C23×D2911yescharacteristic
332C2×C4C23×D2913yes—
333C22—23no—
334C22—23no—
335C22—23no—
336C22—292no—
337C22—581no—
338C22—581no—
339C22—583no—
340C22—583no—
341C22—583no—
342C22C24×D2911yescharacteristic
343C4—21no—
344C4—23no—
345C4—23no—
346C4—21no—
347C4—21no—
348C4—292no—
349C4—581no—
350C4—583no—
351C4—583no—
352C4—583no—
353C4—581no—
354C4C2×#9080516yes—
355C2—43no—
356C2—582no—
357C2—1163no—
358C2C22×#9080511yescharacteristic
359C2H(5,𝔽2)×D2911yescharacteristic
360C22−1+4×D2911yescharacteristic
subgroupG/H
center, FrattiniC22C24×D29
commutator, socle, cosocleC22×C29C25
FittingC29×#520C2
hypercenter#520D29
hypocenterC29#3522
LCSG ▷ C22×C29 ▷ C29
UCSC1 ◁ C22 ◁ #520

Complements

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

NH ≅ G/Ntype
1. #14271896C2semidirect
2. Q8×Dic58C2semidirect
3. #14271868C2semidirect
4. #14271832C2semidirect
5. #14271825C2semidirect
6. #14271820C2semidirect
7. #14271818C2semidirect
8. #14271814C2semidirect
9. #14271784C2semidirect
10. #14271778C2semidirect
11. #14271710C2semidirect
12. #14271691C2semidirect
13. #14271645C2semidirect
14. C4×Q8×D29C2semidirect
15. C29×#520C2semidirect
123. Q8×C29#41semidirect
188. #520D29semidirect
246. C29#3522semidirect
307. Q8#90805semidirect

Maximal subgroups

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

subgroupindex
#142718962
Q8×Dic582
#142718682
#142718322
#142718252
#142718202
#142718182
#142718142
#142717842
#142717782
#142717102
#142716912
#142716452
C4×Q8×D292
C29×#5202
#352229