Finite Group Databasebeta

← #16822734

Subgroups — #16822734

Order 3712.

696 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C22⋊D29C211yesmaximal, characteristic, complemented
2C2×C4×C22⋊D29C211yesmaximal, characteristic, complemented
3C2×#11745288C211yesmaximal, characteristic, complemented
4#14271096C211yesmaximal, characteristic, complemented
5#14271091C211yesmaximal, characteristic, complemented
6#14271086C211yesmaximal, characteristic, complemented
7#14271085C211yesmaximal, characteristic, complemented
8#14271079C211yesmaximal, characteristic, complemented
9#14271078C211yesmaximal, characteristic, complemented
10#14271077C211yesmaximal, characteristic, complemented
11#14271071C211yesmaximal, characteristic
12C4×#90779C211yesmaximal, characteristic, complemented
13C4×#90781C211yesmaximal, characteristic, complemented
14C4×C22⋊Dic29C211yesmaximal, characteristic, complemented
15C2×C4×D4×C29C211yesmaximal, characteristic, complemented
16C22×D4×C29C2211yescharacteristic
17C23×C4×C29C2211yescharacteristic
18C23×C4×C29C2211yescharacteristic
19C22×C22⋊D29C2211yescharacteristic
20C22×C22⋊D29C2211yescharacteristic
21C22×C4×D29C2211yescharacteristic
22C4×D4×C29—24no—
23C2×C4⋊C4×C29C2211yescharacteristic
24C2×C22⋊C4×C29C2211yescharacteristic
25C2×C22⋊C4×C29C2211yescharacteristic
26C2×C42×C29C2211yescharacteristic
27C2×C22⋊Dic29C2211yescharacteristic
28C2×C22⋊Dic29C2211yescharacteristic
29C2×C22⋊Dic29C2211yescharacteristic
30C2×C22⋊Dic29C2211yescharacteristic
31C2×C22⋊Dic29C2211yescharacteristic
32C2×C22⋊Dic29C2211yescharacteristic
33C4×C22⋊D29—24no—
34C4×C22⋊D29—24no—
35C2×#90781C2211yescharacteristic
36C2×#90781C2211yescharacteristic
37C2×#90781C2211yescharacteristic
38C2×#90781C2211yescharacteristic
39#11745288—24no—
40C2×C4⋊Dic29C2211yescharacteristic
41C2×C4⋊Dic29C2211yescharacteristic
42C2×#90779C2211yescharacteristic
43C2×#90779C2211yescharacteristic
44C2×#90779C2211yescharacteristic
45C2×#90779C2211yescharacteristic
46C2×C4×Dic29C2211yescharacteristic
47C2×C4×Dic29C2211yescharacteristic
48C2×C4×Dic29C2211yescharacteristic
49#11745179C2211yescharacteristic
50#11745179C2211yescharacteristic
51#11745179C2211yescharacteristic
52#11745179C2211yescharacteristic
53#11745179C2211yescharacteristic
54#11745179C2211yescharacteristic
55C24×C29C2311yescharacteristic
56C24×C29C2311yescharacteristic
57C23×D29C2311yescharacteristic
58C2×D4×C29—22no—
59C2×D4×C29—22no—
60C2×D4×C29—22no—
61C22×C4×C29—22no—
62C22×C4×C29—22no—
63C22×C4×C29—22no—
64C22×C4×C29—22no—
65C22×C4×C29—22no—
66C22×C4×C29D414yescomplemented
67C22×C4×C29C2311yescharacteristic
68C22×C4×C29C2311yescharacteristic
69C22×C4×C29C2311yescharacteristic
70C22×C4×C29C2311yescharacteristic
71C22×C4×C29C2311yescharacteristic
72C2×C22⋊D29—22no—
73C2×C22⋊D29—22no—
74C2×C22⋊D29—22no—
75C2×C22⋊D29—22no—
76C2×C22⋊D29—22no—
77C2×C22⋊D29—22no—
78C22×Dic29C2311yescharacteristic
79C22×Dic29C2311yescharacteristic
80C22×Dic29C2311yescharacteristic
81C22×Dic29C2311yescharacteristic
82C22×Dic29C2311yescharacteristic
83C22×Dic29C2311yescharacteristic
84C22×Dic29C2311yescharacteristic
85C2×C4×D29—22no—
86C2×C4×D29—22no—
87C2×C4×D29—22no—
88C4⋊C4×C29—22no—
89C22⋊C4×C29—22no—
90C22⋊C4×C29—22no—
91C42×C29—22no—
92C22⋊Dic29—22no—
93C22⋊Dic29—22no—
94C22⋊Dic29—22no—
95C22⋊Dic29—22no—
96C22⋊Dic29—22no—
97C22⋊Dic29—22no—
98#90781—22no—
99#90781—22no—
100#90781—22no—
101#90781—22no—
102C4⋊Dic29—22no—
103C4⋊Dic29—22no—
104#90779—22no—
105#90779—22no—
106#90779—22no—
107#90779—22no—
108C4×Dic29—22no—
109C4×Dic29—22no—
110C4×Dic29D414yescomplemented
111C23×C29—21no—
112C23×C29—21no—
113C23×C29—21no—
114C23×C29—21no—
115C23×C29—21no—
116C23×C29—21no—
117C23×C29—21no—
118C23×C29—21no—
119C23×C29—21no—
120C23×C29—21no—
121C23×C29—21no—
122C23×C29—21no—
123C23×C29C2×D412yes—
124C23×C29#4112yes—
125C23×C29C2411yescharacteristic
126C22×D29—21no—
127C22×D29—21no—
128C22×D29—21no—
129C22×D29—21no—
130C22×D29—21no—
131C22×D29—21no—
132C22×D29#4112yes—
133D4×C29—28no—
134C2×C4×C29—21no—
135C2×C4×C29—21no—
136C2×C4×C29—21no—
137C2×C4×C29—21no—
138C2×C4×C29—28no—
139C2×C4×C29—21no—
140C2×C4×C29—21no—
141C2×C4×C29—21no—
142C2×C4×C29—21no—
143C2×C4×C29—21no—
144C2×C4×C29—21no—
145C2×C4×C29—44no—
146C2×C4×C29C2×D412yes—
147C2×C4×C29C2×D412yes—
148C2×C4×C29C2×D412yes—
149C2×C4×C29#4112yes—
150C2×C4×C29#4112yes—
151C22⋊D29—28no—
152C22⋊D29—28no—
153C2×Dic29—21no—
154C2×Dic29—21no—
155C2×Dic29—21no—
156C2×Dic29—21no—
157C2×Dic29—21no—
158C2×Dic29—21no—
159C2×Dic29—21no—
160C2×Dic29—21no—
161C2×Dic29—21no—
162C2×Dic29—21no—
163C2×Dic29—21no—
164C2×Dic29—21no—
165C2×Dic29—21no—
166C2×Dic29—21no—
167C2×Dic29—21no—
168C2×Dic29—21no—
169C2×Dic29—21no—
170C2×Dic29—21no—
171C2×Dic29C2×D412yes—
172C2×Dic29C2×D412yes—
173C2×Dic29#4112yes—
174C4×D29—44no—
175#2392—291nomaximal, Sylow
176C22×C29—22no—
177C22×C29—22no—
178C22×C29—22no—
179C22×C29—22no—
180C22×C29—22no—
181C22×C29—22no—
182C22×C29—41no—
183C22×C29—41no—
184C22×C29—41no—
185C22×C29—41no—
186C22×C29—41no—
187C22×C29—41no—
188C22×C29—41no—
189C22×C29—41no—
190C22×C29C22×D411yescharacteristic
191C22×C29C22×D411yescharacteristic
192C22×C29C2×#4111yescharacteristic
193C22×C29C2×#4111yescharacteristic
194C22×C29C2×#4111yescharacteristic
195C22×C29C2×#4111yescharacteristic
196C22×C29C2×#4111yescharacteristic
197C2×D29—22no—
198C2×D29—22no—
199C2×D29—22no—
200C2×D29—41no—
201C2×D29—41no—
202C2×D29—41no—
203C2×D29—41no—
204C4×C29—22no—
205C4×C29—22no—
206C4×C29—41no—
207C4×C29—41no—
208C4×C29#12114yes—
209Dic29—22no—
210Dic29—22no—
211Dic29—22no—
212Dic29—41no—
213Dic29—41no—
214Dic29—41no—
215Dic29—41no—
216C2×C4×D4—291no—
217C2×C4×D4—291no—
218C2×C4×D4D2911yescharacteristic, complemented
219C2×#117—291no—
220#398—291no—
221#397—291no—
222#396—291no—
223#396—291no—
224#394—291no—
225#394—291no—
226#394—291no—
227#392—291no—
228C4×C4⋊C4—291no—
229C4×C22⋊C4—291no—
230C4×C22⋊C4—291no—
231C2×C29—42no—
232C2×C29—42no—
233C2×C29C2×#12111yescharacteristic
234C2×C29#52511yescharacteristic
235C2×C29#52511yescharacteristic
236C2×C29#52511yescharacteristic
237C2×C29#52911yescharacteristic
238C2×C29#52911yescharacteristic
239C2×C29#53211yescharacteristic
240D29—42no—
241C22×D4—291no—
242C22×D4—291no—
243C22×D4C2×D2911yescharacteristic
244C23×C4—291no—
245C23×C4C2×D2911yescharacteristic
246C23×C4C2×D2911yescharacteristic
247C4×D4—24no—
248C4×D4—584no—
249C4×D4—584no—
250#117—584no—
251C2×C4⋊C4—291no—
252C2×C4⋊C4—291no—
253C2×C4⋊C4—291no—
254C2×C4⋊C4—291no—
255C2×C4⋊C4—291no—
256C2×C4⋊C4—291no—
257C2×C4⋊C4C2×D2911yescharacteristic
258C2×C22⋊C4—291no—
259C2×C22⋊C4—291no—
260C2×C22⋊C4—291no—
261C2×C22⋊C4—291no—
262C2×C22⋊C4—291no—
263C2×C22⋊C4—291no—
264C2×C22⋊C4—291no—
265C2×C22⋊C4—291no—
266C2×C22⋊C4—291no—
267C2×C22⋊C4—291no—
268C2×C22⋊C4C2×D2911yescharacteristic
269C2×C22⋊C4C2×D2911yescharacteristic
270C2×C42—291no—
271C2×C42—291no—
272C2×C42—291no—
273C2×C42C2×D2911yescharacteristic
274#95—291no—
275#95—291no—
276#95—291no—
277#95—291no—
278#95—291no—
279#95—291no—
280C29#239211yescharacteristic, complemented, Sylow
281C24—291no—
282C24C22×D2911yescharacteristic
283C24C22×D2911yescharacteristic
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—582no—
288C2×D4—582no—
289C2×D4—582no—
290C2×D4—582no—
291C2×D4—582no—
292C2×D4—582no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—291no—
299C22×C4—291no—
300C22×C4—291no—
301C22×C4—291no—
302C22×C4—291no—
303C22×C4—291no—
304C22×C4—291no—
305C22×C4—582no—
306C22×C4—582no—
307C22×C4—582no—
308C22×C4C22⋊D2914yescomplemented
309C22×C4C22×D2911yescharacteristic
310C22×C4C22×D2911yescharacteristic
311C22×C4C22×D2911yescharacteristic
312C22×C4C22×D2911yescharacteristic
313C22×C4C22×D2911yescharacteristic
314C4⋊C4—22no—
315C4⋊C4—582no—
316C4⋊C4—582no—
317C4⋊C4—582no—
318C4⋊C4—582no—
319C4⋊C4—582no—
320C4⋊C4—582no—
321C22⋊C4—22no—
322C22⋊C4—22no—
323C22⋊C4—582no—
324C22⋊C4—582no—
325C22⋊C4—582no—
326C22⋊C4—582no—
327C22⋊C4—582no—
328C22⋊C4—582no—
329C22⋊C4—582no—
330C22⋊C4—582no—
331C22⋊C4—582no—
332C22⋊C4—582no—
333C42—22no—
334C42—294no—
335C42—582no—
336C42—582no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—292no—
350C23—581no—
351C23—581no—
352C23—581no—
353C23—581no—
354C23—581no—
355C23—581no—
356C23#9080512yes—
357C23C2×C22⋊D2912yes—
358C23C23×D2911yescharacteristic
359D4—28no—
360D4—588no—
361D4—588no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—28no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—44no—
374C2×C4—292no—
375C2×C4—292no—
376C2×C4—292no—
377C2×C4—581no—
378C2×C4—581no—
379C2×C4—581no—
380C2×C4—581no—
381C2×C4—581no—
382C2×C4—581no—
383C2×C4—581no—
384C2×C4—581no—
385C2×C4—581no—
386C2×C4—581no—
387C2×C4—581no—
388C2×C4—581no—
389C2×C4—581no—
390C2×C4—581no—
391C2×C4—581no—
392C2×C4—581no—
393C2×C4—581no—
394C2×C4—581no—
395C2×C4—1164no—
396C2×C4#9080512yes—
397C2×C4D4×D2912yes—
398C2×C4#9080712yes—
399C2×C4C2×C22⋊D2912yes—
400C2×C4C2×C22⋊D2912yes—
401C22—22no—
402C22—22no—
403C22—22no—
404C22—22no—
405C22—22no—
406C22—22no—
407C22—41no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—582no—
416C22—582no—
417C22—582no—
418C22—1161no—
419C22—1161no—
420C22—1161no—
421C22—1161no—
422C22C2×#9080511yescharacteristic
423C22C2×#9080511yescharacteristic
424C22C2×D4×D2911yescharacteristic
425C22C2×#9080711yescharacteristic
426C22#41×D2911yescharacteristic
427C22#41×D2911yescharacteristic
428C22C22×C22⋊D2911yescharacteristic
429C4—22no—
430C4—22no—
431C4—41no—
432C4—41no—
433C4—582no—
434C4—582no—
435C4—582no—
436C4—1161no—
437C4—1161no—
438C4—1161no—
439C4—1161no—
440C4#1174530114yes—
441C2—42no—
442C2—42no—
443C2—1162no—
444C2C2×#1174530111yescharacteristic
445C2#1427167011yescharacteristic
446C2#1427167211yescharacteristic
447C2#1427167411yescharacteristic
448C2#1427169311yescharacteristic
449C2#1427169511yescharacteristic
450C2#1427171011yescharacteristic
subgroupG/H
centerC22×C4C22×D29
commutator, socle, cosocleC23×C29C24
FrattiniC23C23×D29
FittingC2×C4×D4×C29C2
prime radical#14271071C2
hypercenterC2×C4×D4D29
hypocenterC29#2392
LCSG ▷ C23×C29 ▷ C29
UCSC1 ◁ C22×C4 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×C4×C22⋊D29C2semidirect
2. C2×C4×C22⋊D29C2semidirect
3. C2×#11745288C2semidirect
4. #14271096C2semidirect
5. #14271091C2semidirect
6. #14271086C2semidirect
7. #14271085C2semidirect
8. #14271079C2semidirect
9. #14271078C2semidirect
10. #14271077C2semidirect
12. C4×#90779C2semidirect
13. C4×#90781C2semidirect
14. C4×C22⋊Dic29C2semidirect
15. C2×C4×D4×C29C2semidirect
66. C22×C4×C29D4semidirect
110. C4×Dic29D4semidirect
218. C2×C4×D4D29semidirect
280. C29#2392semidirect
308. C22×C4C22⋊D29semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊D292
C2×#117452882
#142710962
#142710912
#142710862
#142710852
#142710792
#142710782
#142710772
#142710712
C4×#907792
C4×#907812
C4×C22⋊Dic292
C2×C4×D4×C292
#239229