Finite Group Databasebeta

← #16830874

Subgroups — #16830874

Order 3712.

386 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#14271715C211yesmaximal, characteristic, complemented
2#14271708C211yesmaximal, characteristic
3#14271457C211yesmaximal, characteristic, complemented
4#14271456C211yesmaximal, characteristic, complemented
5#14271442C211yesmaximal, characteristic, complemented
6#14271441C211yesmaximal, characteristic, complemented
7#14271411C211yesmaximal, characteristic, complemented
8#14271409C211yesmaximal, characteristic, complemented
9#14271407C211yesmaximal, characteristic, complemented
10#14271367C211yesmaximal, characteristic, complemented
11#14271366C211yesmaximal, characteristic, complemented
12#14271362C211yesmaximal, characteristic, complemented
13#14271361C211yesmaximal, characteristic, complemented
14#14271186C211yesmaximal, characteristic, complemented
15C29×#468C211yesmaximal, characteristic, complemented
16C2×#90807—21no—
17C2×SD16×C29C2211yescharacteristic
18C2×M16×C29C2211yescharacteristic
19C29×C22⋊Q8C2211yescharacteristic
20C29×#121C2211yescharacteristic
21Q8×Dic29C2211yescharacteristic
22#11745306C2211yescharacteristic
23C2×#90784C2211yescharacteristic
24#11745302C2211yescharacteristic
25#11745301—21no—
26D4×Dic29C2211yescharacteristic
27C2×#90783C2211yescharacteristic
28#11745292C2211yescharacteristic
29#11745286C2211yescharacteristic
30C2×C4×Dic29C2211yescharacteristic
31C2×#90777C2211yescharacteristic
32C2×#90773C2211yescharacteristic
33#11745255C2211yescharacteristic
34#11745251—21no—
35#11745250C2211yescharacteristic
36#11745250C2211yescharacteristic
37#11745249—21no—
38#11745249C2211yescharacteristic
39#11745247—21no—
40#11745243—21no—
41#11745240—21no—
42#11745239—21no—
43C29×#107C2211yescharacteristic
44C29×#103C2211yescharacteristic
45C29×#102C2211yescharacteristic
46#11745183C2211yescharacteristic
47#11745180C2211yescharacteristic
48#11745169C2211yescharacteristic
49#11745166C2211yescharacteristic
50#11745164C2211yescharacteristic
51#11745162C2211yescharacteristic
52#11745162C2211yescharacteristic
53#11745162C2211yescharacteristic
54#11745162C2211yescharacteristic
55#11745158C2211yescharacteristic
56#11745157C2211yescharacteristic
57#11745155C2211yescharacteristic
58#11745155C2211yescharacteristic
59C2×Q8×C29C2311yescharacteristic
60C2×D4×C29—21no—
61C2×D4×C29C2311yescharacteristic
62C22×C4×C29C2311yescharacteristic
63C2×C22⋊D29—21no—
64C2×C22⋊D29—21no—
65C2×C22⋊D29—21no—
66C22×Dic29—21no—
67C22×Dic29D411yescharacteristic
68C22×Dic29D411yescharacteristic
69#90807—44no—
70C2×D116C2311yescharacteristic
71C2×C4×D29—21no—
72C2×Dic58—21no—
73C2×Dic58—21no—
74C2×Dic58C2311yescharacteristic
75SD16×C29—41no—
76M16×C29—41no—
77C2×C8×C29C2311yescharacteristic
78C2×C8×C29C2311yescharacteristic
79C4⋊C4×C29—21no—
80C4⋊C4×C29C2311yescharacteristic
81C4⋊C4×C29C2311yescharacteristic
82C22⋊C4×C29—21no—
83C22⋊C4×C29—21no—
84C22⋊Dic29—21no—
85C22⋊Dic29—21no—
86#90784—41no—
87#90783—41no—
88#90781—21no—
89#90781—21no—
90C4⋊Dic29—21no—
91C4⋊Dic29C2311yescharacteristic
92C4⋊Dic29C2311yescharacteristic
93#90779—21no—
94#90779—21no—
95#90779—21no—
96#90779—21no—
97#90779—21no—
98#90779—21no—
99C4×Dic29—21no—
100C4×Dic29—21no—
101C4×Dic29D411yescharacteristic
102C4×Dic29D411yescharacteristic
103C4×Dic29C2311yescharacteristic
104C4×Dic29C2311yescharacteristic
105#90777—41no—
106C2×#6597C2311yescharacteristic
107C2×#6597C2311yescharacteristic
108#90773—41no—
109C23×C29—21no—
110C23×C29C2×D411yescharacteristic
111C22×D29—21no—
112Q8×C29—21no—
113Q8×C29—21no—
114D4×C29—21no—
115D4×C29—21no—
116D4×C29—42no—
117C2×C4×C29—21no—
118C2×C4×C29—21no—
119C2×C4×C29—21no—
120C2×C4×C29—21no—
121C2×C4×C29—21no—
122C2×C4×C29C2×D411yescharacteristic
123C2×C4×C29C2411yescharacteristic
124C22⋊D29—42no—
125C22⋊D29—42no—
126C22⋊D29—42no—
127C2×Dic29—22no—
128C2×Dic29—22no—
129C2×Dic29—21no—
130C2×Dic29—21no—
131C2×Dic29—21no—
132C2×Dic29—42no—
133C2×Dic29C2×D411yescharacteristic
134C2×Dic29C2×D411yescharacteristic
135C2×Dic29C2×D411yescharacteristic
136C2×Dic29C2×D411yescharacteristic
137D116—21no—
138D116—21no—
139C4×D29—42no—
140Dic58—21no—
141Dic58—21no—
142Dic58—42no—
143Dic58—42no—
144C8×C29—21no—
145C8×C29—21no—
146#6597—21no—
147#6597—21no—
148#3254—291nomaximal, Sylow
149C22×C29—21no—
150C22×C29—21no—
151C22×C29—21no—
152C22×C29—41no—
153C22×C29—41no—
154C22×C29—41no—
155C22×C29C22×D411yescharacteristic
156C2×D29—41no—
157C2×D29—41no—
158C2×D29—41no—
159C4×C29—21no—
160C4×C29—41no—
161C4×C29—41no—
162C4×C29—41no—
163C4×C29H(5,𝔽2)11yescharacteristic
164C4×C292−1+411yescharacteristic
165Dic29—21no—
166Dic29—21no—
167Dic29—41no—
168Dic29—41no—
169Dic29—41no—
170Dic29AGL(1,ℤ/8)12yes—
171Dic29#13712yes—
172#533—291no—
173#532—291no—
174#477—291no—
175#477—291no—
176#474—291no—
177#474—291no—
178#468—291no—
179#468—291no—
180#468—291no—
181#468D2911yescharacteristic, complemented
182#461—291no—
183#461—291no—
184#460—291no—
185#460—291no—
186#423—291no—
187C2×C29—41no—
188C2×C29—81no—
189C2×C29#53711yescharacteristic
190C2×C29C2×AGL(1,ℤ/8)11yescharacteristic
191C2×C29C2×#13711yescharacteristic
192D29—81no—
193C2×#41—581no—
194C2×SD16—291no—
195C2×SD16—291no—
196C2×SD16—291no—
197C2×SD16C2×D2911yescharacteristic
198C2×M16—291no—
199C2×M16C2×D2911yescharacteristic
200C4⋊Q8—291no—
201C4⋊Q8—291no—
202C4⋊Q8—291no—
203#127—291no—
204#125—581no—
205#124—581no—
206C22⋊Q8—291no—
207C22⋊Q8—581no—
208C22⋊Q8C2×D2911yescharacteristic
209#121—291no—
210#121—581no—
211#121C2×D2911yescharacteristic
212C4×Q8—291no—
213C4×Q8—291no—
214C4×Q8—581no—
215C4×D4—291no—
216C4×D4—291no—
217C4×D4—581no—
218#117—581no—
219C2×C42—291no—
220#107—291no—
221#107—291no—
222#107—291no—
223#107C2×D2911yescharacteristic
224C4⋊C8—291no—
225C4⋊C8—291no—
226C4⋊C8—291no—
227C4⋊C8—291no—
228#103—291no—
229#103—291no—
230#103—291no—
231#103C2×D2911yescharacteristic
232#102—291no—
233#102—291no—
234#102—291no—
235#102C2×D2911yescharacteristic
236C29#325411yescharacteristic, complemented, Sylow
237#41—1164no—
238C2×Q8—291no—
239C2×Q8—581no—
240C2×Q8—581no—
241C2×Q8C22×D2911yescharacteristic
242C2×D4—21no—
243C2×D4—291no—
244C2×D4—581no—
245C2×D4—581no—
246C2×D4—581no—
247C2×D4C22×D2911yescharacteristic
248C22×C4—291no—
249C22×C4—291no—
250C22×C4—581no—
251C22×C4—581no—
252C22×C4C22×D2911yescharacteristic
253SD16—41no—
254SD16—1161no—
255SD16—1161no—
256SD16—1161no—
257M16—41no—
258M16—1161no—
259C2×C8—291no—
260C2×C8—291no—
261C2×C8C22×D2911yescharacteristic
262C2×C8C22×D2911yescharacteristic
263C4⋊C4—21no—
264C4⋊C4—291no—
265C4⋊C4—291no—
266C4⋊C4—581no—
267C4⋊C4—581no—
268C4⋊C4—581no—
269C4⋊C4—581no—
270C4⋊C4—581no—
271C4⋊C4—581no—
272C4⋊C4—581no—
273C4⋊C4C22×D2911yescharacteristic
274C4⋊C4C22×D2911yescharacteristic
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—581no—
278C22⋊C4—581no—
279C22⋊C4—581no—
280C22⋊C4—581no—
281C42—291no—
282C42—291no—
283C42—291no—
284C42—291no—
285C42—581no—
286C42—581no—
287C23—21no—
288C23—581no—
289C23D4×D2911yescharacteristic
290Q8—21no—
291Q8—21no—
292Q8—581no—
293Q8—581no—
294Q8—1162no—
295Q8—1162no—
296D4—21no—
297D4—21no—
298D4—42no—
299D4—581no—
300D4—581no—
301D4—1162no—
302D4—1162no—
303D4—1162no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—291no—
310C2×C4—291no—
311C2×C4—291no—
312C2×C4—291no—
313C2×C4—582no—
314C2×C4—582no—
315C2×C4—581no—
316C2×C4—581no—
317C2×C4—581no—
318C2×C4—1162no—
319C2×C4—1162no—
320C2×C4D4×D2911yescharacteristic
321C2×C4C23×D2911yescharacteristic
322C8—21no—
323C8—21no—
324C8—581no—
325C8—581no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—1161no—
333C22—1161no—
334C22—1161no—
335C22C2×D4×D2911yescharacteristic
336C4—21no—
337C4—41no—
338C4—41no—
339C4—41no—
340C4—292no—
341C4—292no—
342C4—581no—
343C4—581no—
344C4—1161no—
345C4—1161no—
346C4—1161no—
347C4#1174535911yescharacteristic
348C4#1174536511yescharacteristic
349C2—41no—
350C2—81no—
351C2—2321no—
352C2#1427174011yescharacteristic
353C2AGL(1,ℤ/8)×D2911yescharacteristic
354C2#137×D2911yescharacteristic
subgroupG/H
centerC22C2×D4×D29
commutator, cosocleC2×C4×C29C24
FrattiniC2×C4C23×D29
FittingC29×#468C2
socleC22×C29C22×D4
prime radical#14271708C2
hypercenter#468D29
hypocenterC29#3254
LCSG ▷ C2×C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. #14271715C2semidirect
3. #14271457C2semidirect
4. #14271456C2semidirect
5. #14271442C2semidirect
6. #14271441C2semidirect
7. #14271411C2semidirect
8. #14271409C2semidirect
9. #14271407C2semidirect
10. #14271367C2semidirect
11. #14271366C2semidirect
12. #14271362C2semidirect
13. #14271361C2semidirect
14. #14271186C2semidirect
15. C29×#468C2semidirect
181. #468D29semidirect
236. C29#3254semidirect

Maximal subgroups

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

subgroupindex
#142717152
#142717082
#142714572
#142714562
#142714422
#142714412
#142714112
#142714092
#142714072
#142713672
#142713662
#142713622
#142713612
#142711862
C29×#4682
#325429