Finite Group Databasebeta

← #2255×D67 #31011324

Subgroups — #2255×D67 #31011324

Order 17152.

518 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#500×D67C211yesmaximal, characteristic
2#499×D67C211yesmaximal, characteristic
3#22562496C211yesmaximal, characteristic, complemented
4#22562495C211yesmaximal, characteristic, complemented
5#22562494C212yesmaximal, complemented
6#22562493C212yesmaximal, complemented
7#22562492C211yesmaximal, characteristic, complemented
8#367×D67C212yesmaximal
9#366×D67C212yesmaximal
10#346×D67C211yesmaximal, characteristic
11C67×#2255C211yesmaximal, characteristic, complemented
12C4⋊Q8×D67—21no—
13#125×D67—21no—
14C2×C16×D67C2212yes—
15#16863801C2211yescharacteristic
16#16863800—21no—
17#16863800C2212yes—
18#16863799—21no—
19#16863798C2211yescharacteristic
20#16863797C2211yescharacteristic
21#16863797C2211yescharacteristic
22#16863796—21no—
23#16863795—21no—
24#107×D67—21no—
25#107×D67C2211yescharacteristic
26#107×D67C2212yes—
27#107×D67C2211yescharacteristic
28#106×D67—21no—
29C4×C8×D67C2211yescharacteristic
30#16863224C2212yes—
31#16863223C2212yes—
32#16863223C2212yes—
33#16863222C2212yes—
34#16863221C2212yes—
35#16863221C2212yes—
36#16863172C2211yescharacteristic
37#16863171C2212yes—
38#500×C67C2211yescharacteristic
39#499×C67C2211yescharacteristic
40#367×C67C2212yes—
41#366×C67C2212yes—
42#346×C67C2211yescharacteristic
43C2×Q8×D67—41no—
44C2×C8×D67D411yescharacteristic
45C2×C8×D67C2311yescharacteristic
46#14524043—21no—
47#14524042—22no—
48#14524042—41no—
49#14524041—41no—
50#14524035—21no—
51#14524034—21no—
52#14524034—21no—
53#14524033—41no—
54#14524033—41no—
55C4⋊C4×D67—21no—
56C4⋊C4×D67—21no—
57C4⋊C4×D67—22no—
58C4⋊C4×D67—41no—
59C4⋊C4×D67—41no—
60C4⋊C4×D67—41no—
61C42×D67D411yescharacteristic
62C2×C67⋊C16C2312yes—
63C16×D67Q818yescomplemented
64#14523977—21no—
65#14523977C2311yescharacteristic
66#14523977C2311yescharacteristic
67#14523977C2312yes—
68#14523976—21no—
69#14523976—21no—
70#14523976—22no—
71#14523976—22no—
72#14523975—21no—
73#14523974—21no—
74#14523974—21no—
75C4×C67⋊C8D411yescharacteristic
76C8×Dic67D411yescharacteristic
77C8×Dic67C2311yescharacteristic
78C4⋊Q8×C67—21no—
79#125×C67—21no—
80C2×C16×C67C2312yes—
81#107×C67—21no—
82#107×C67C2311yescharacteristic
83#107×C67C2311yescharacteristic
84#107×C67C2312yes—
85#106×C67—21no—
86C4×C8×C67C2311yescharacteristic
87C2×Dic134—41no—
88C2×Dic134—42no—
89Q8×D67—84no—
90C2×C4×D67—41no—
91C2×C4×D67—41no—
92C2×C4×D67—42no—
93C2×C4×D67D811yescharacteristic
94C2×C4×D67D811yescharacteristic
95C2×C4×D67C2×D411yescharacteristic
96C2×C67⋊C8C2×D411yescharacteristic
97C2×C67⋊C8C2×D411yescharacteristic
98C8×D67Q1614yes—
99C8×D67C2×Q814yes—
100C4⋊Dic67—21no—
101C4⋊Dic67—21no—
102C4⋊Dic67—22no—
103C4⋊Dic67—41no—
104C4⋊Dic67—41no—
105C4⋊Dic67—41no—
106Dic67⋊C4—41no—
107Dic67⋊C4—41no—
108Dic67⋊C4—41no—
109Dic67⋊C4—41no—
110Dic67⋊C4—42no—
111Dic67⋊C4—41no—
112Dic67⋊C4—41no—
113Dic67⋊C4—42no—
114C4×Dic67D811yescharacteristic
115C4×Dic67D811yescharacteristic
116C4×Dic67C2×D411yescharacteristic
117C67⋊C16C2×Q814yescomplemented
118C2×Q8×C67—41no—
119C2×C8×C67C2×D411yescharacteristic
120C2×C8×C67C2411yescharacteristic
121C4⋊C4×C67—21no—
122C4⋊C4×C67—21no—
123C4⋊C4×C67—22no—
124C4⋊C4×C67—41no—
125C4⋊C4×C67—41no—
126C4⋊C4×C67—41no—
127C42×C67C2×D411yescharacteristic
128C16×C67C2×Q814yescomplemented
129C22×D67C2×D811yescharacteristic
130Q8×C67—82no—
131C2×C4×C67—41no—
132C2×C4×C67—41no—
133C2×C4×C67—42no—
134C2×C4×C67C2×D811yescharacteristic
135C2×C4×C67C2×D811yescharacteristic
136C2×C4×C67C22×D411yescharacteristic
137C2×Dic67—41no—
138C2×Dic67—41no—
139C2×Dic67—42no—
140C2×Dic67C2×D811yescharacteristic
141C2×Dic67C2×D811yescharacteristic
142C2×Dic67C2×D811yescharacteristic
143C4×D67—22no—
144C4×D67—22no—
145C4×D67—82no—
146C4×D67—84no—
147C4×D67—82no—
148C4×D67C4⋊Q812yes—
149C4×D67C2×Q1612yes—
150Dic134—84no—
151Dic134—82no—
152C8×C67C2×Q1612yes—
153C8×C67C22×Q812yes—
154C67⋊C8C4⋊Q812yes—
155C67⋊C8C2×Q1612yes—
156C22×C67C22×D811yescharacteristic
157C2×D67#50012yes—
158C2×D67#50912yes—
159C2×D67#51012yes—
160C4×C67—21no—
161C4×C67—21no—
162C4×C67—82no—
163C4×C67—81no—
164C4×C67—81no—
165C4×C67C2×C4⋊Q811yescharacteristic
166C4×C67C22×Q1611yescharacteristic
167Dic67—21no—
168Dic67—21no—
169Dic67—82no—
170Dic67—81no—
171Dic67—81no—
172Dic67#50012yes—
173C2×#2255—671nomaximal, Sylow
174C2×C67C2×#50011yescharacteristic
175C2×C67C2×#50911yescharacteristic
176C2×C67C2×#51011yescharacteristic
177D67#225514yesdirect product factor
178C2×#500—671no—
179C2×#499—671no—
180#2255—671no—
181#2255—671no—
182#2255—672no—
183#2255—672no—
184#2255—671no—
185#2255D6711yescharacteristic, direct product factor
186C2×#367—672no—
187C2×#366—672no—
188C2×#346—671no—
189C67C2×#225511yescharacteristic, complemented, Sylow
190C2×C4⋊Q8—1341no—
191C2×#125—1341no—
192C22×C16—672no—
193#500—672no—
194#500—671no—
195#500—1341no—
196#500—1341no—
197#500C2×D6711yescharacteristic
198#499—671no—
199#499—671no—
200#499—671no—
201#499—1341no—
202#499—1341no—
203#499C2×D6711yescharacteristic
204C2×#107—671no—
205C2×#107—671no—
206C2×#107—672no—
207C2×#107—1341no—
208C2×#106—1341no—
209C2×C4×C8—671no—
210#367—672no—
211#367—672no—
212#367—672no—
213#367C2×D6712yes—
214#366—672no—
215#366—672no—
216#366—672no—
217#366C2×D6712yes—
218#346—672no—
219#346—671no—
220#346C2×D6711yescharacteristic
221C22×Q8—2681no—
222C22×C8—671no—
223C22×C8—671no—
224C4⋊Q8—21no—
225C4⋊Q8—2681no—
226C4⋊Q8—2681no—
227C4⋊Q8—1342no—
228C4⋊Q8—1341no—
229#125—21no—
230#125—2681no—
231#125—2681no—
232#125—1341no—
233#125—1341no—
234#125—1341no—
235C2×C4⋊C4—2681no—
236C2×C4⋊C4—2681no—
237C2×C4⋊C4—2681no—
238C2×C4⋊C4—1341no—
239C2×C4⋊C4—1341no—
240C2×C4⋊C4—1342no—
241C2×C42—671no—
242C2×C16—672no—
243C2×C16—678no—
244C2×C16C22×D6712yes—
245#107—21no—
246#107—671no—
247#107—672no—
248#107—671no—
249#107—1341no—
250#107—1341no—
251#107—1342no—
252#107—1342no—
253#107—1341no—
254#107C22×D6711yescharacteristic
255#107C22×D6712yes—
256#107C22×D6711yescharacteristic
257#106—21no—
258#106—1341no—
259#106—1341no—
260#106—1341no—
261C4×C8—671no—
262C4×C8—671no—
263C4×C8—671no—
264C4×C8C22×D6711yescharacteristic
265C2×Q8—41no—
266C2×Q8—2682no—
267C2×Q8—2681no—
268C2×Q8—5364no—
269C22×C4—2681no—
270C22×C4—2681no—
271C22×C4—2682no—
272C22×C4—671no—
273C22×C4—671no—
274C22×C4—671no—
275C2×C8—671no—
276C2×C8—671no—
277C2×C8—674no—
278C2×C8—674no—
279C2×C8D4×D6711yescharacteristic
280C2×C8C23×D6711yescharacteristic
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—22no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—2681no—
288C4⋊C4—2681no—
289C4⋊C4—2681no—
290C4⋊C4—2681no—
291C4⋊C4—2682no—
292C4⋊C4—2681no—
293C4⋊C4—2681no—
294C4⋊C4—2682no—
295C4⋊C4—2681no—
296C4⋊C4—2681no—
297C4⋊C4—2681no—
298C4⋊C4—1341no—
299C4⋊C4—1341no—
300C4⋊C4—1342no—
301C42—671no—
302C42—671no—
303C42—671no—
304C42D4×D6711yescharacteristic
305C16—674no—
306C16Q8×D6714yescomplemented
307C23—671no—
308Q8—82no—
309Q8—5362no—
310Q8—5364no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—2681no—
315C2×C4—2681no—
316C2×C4—2682no—
317C2×C4—5364no—
318C2×C4—5362no—
319C2×C4—5362no—
320C2×C4—671no—
321C2×C4—672no—
322C2×C4—672no—
323C2×C4—671no—
324C2×C4—671no—
325C2×C4—1342no—
326C2×C4—1342no—
327C2×C4D8×D6711yescharacteristic
328C2×C4D8×D6711yescharacteristic
329C2×C4C2×D4×D6711yescharacteristic
330C8—672no—
331C8—672no—
332C8Q16×D6712yes—
333C8C2×Q8×D6712yes—
334C22—672no—
335C22—672no—
336C22—672no—
337C22C2×D8×D6711yescharacteristic
338C4—21no—
339C4—21no—
340C4—82no—
341C4—81no—
342C4—81no—
343C4—5362no—
344C4—5361no—
345C4—5361no—
346C4—672no—
347C4—1341no—
348C4—1341no—
349C4C4⋊Q8×D6711yescharacteristic
350C4C2×Q16×D6711yescharacteristic
351C2—674no—
352C2#500×D6711yescharacteristic
353C2#509×D6711yescharacteristic
354C2#510×D6711yescharacteristic
subgroupG/H
centerC22C2×D8×D67
commutator, cosocleC2×C8×C67C24
FrattiniC2×C8C23×D67
FittingC67×#2255C2
socleC22×C67C22×D8
prime radicalC22×D67C2×D8
hypercenter#2255D67
hypocenterC67C2×#2255
LCSG ▷ C2×C8×C67 ▷ C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2255

Complements

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

NH ≅ G/Ntype
3. #22562496C2semidirect
4. #22562495C2semidirect
5. #22562494C2semidirect
6. #22562493C2semidirect
7. #22562492C2semidirect
11. C67×#2255C2semidirect
63. C16×D67Q8semidirect
117. C67⋊C16C2×Q8semidirect
128. C16×C67C2×Q8semidirect
177. D67#2255direct (N × H)
185. #2255D67direct (N × H)
189. C67C2×#2255semidirect
306. C16Q8×D67semidirect

Maximal subgroups

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

subgroupindex
#500×D672
#499×D672
#225624962
#225624952
#225624942
#225624932
#225624922
#367×D672
#366×D672
#346×D672
C67×#22552
C2×#225567