Finite Group Databasebeta

← #30766899

Subgroups — #30766899

Order 17152.

422 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C22⋊(C67⋊C16)C211yesmaximal, characteristic, complemented
2#22560234C211yesmaximal, characteristic, complemented
3#22558221C212yesmaximal, complemented
4#22558218C212yesmaximal, complemented
5C67×#3050C211yesmaximal, characteristic
6C22×C67⋊C16—21no—
7C2×#14523975—21no—
8#16863256—21no—
9C22⋊(C67⋊C16)—21no—
10C22⋊(C67⋊C16)C2212yes—
11#16863159C2212yes—
12C2×#135×C67C411yescharacteristic, complemented
13C23×C8×C67C2211yescharacteristic
14#525×C67—21no—
15#467×C67C411yescharacteristic, complemented
16#467×C67C2211yescharacteristic
17#466×C67C411yescharacteristic, complemented
18#466×C67C2211yescharacteristic
19#465×C67—21no—
20#465×C67—21no—
21#427×C67C411yescharacteristic, complemented
22#416×C67—21no—
23C2×C4⋊Dic67—22no—
24C2×C22⋊Dic67—41no—
25C2×C67⋊C16—22no—
26C2×C67⋊C16—42no—
27#14523975—42no—
28C67⋊#95—41no—
29C2×#41×C67—21no—
30C23×C4×C67D411yescharacteristic
31#135×C67—42no—
32C2×Q16×C67C2×C411yescharacteristic
33C2×SD16×C67—21no—
34C2×D8×C67C2×C411yescharacteristic
35C22×C8×C67—42no—
36C22×C8×C67C2×C411yescharacteristic
37C22×C8×C67C2×C411yescharacteristic
38C22×C8×C67D411yescharacteristic
39C22×C8×C67D411yescharacteristic
40C22×C8×C67D411yescharacteristic
41C22×C8×C67C2311yescharacteristic
42#123×C67—41no—
43C22⋊Q8×C67—21no—
44C22⋊Q8×C67—21no—
45#121×C67—21no—
46#121×C67—21no—
47#120×C67—41no—
48C4×D4×C67—41no—
49C4×D4×C67—41no—
50#117×C67—21no—
51#107×C67C2×C411yescharacteristic
52#107×C67C2×C411yescharacteristic
53#106×C67—21no—
54#103×C67—21no—
55#103×C67—21no—
56#102×C67—21no—
57#102×C67—21no—
58C22×Dic67—42no—
59C4⋊Dic67—42no—
60C4⋊Dic67—42no—
61C22⋊Dic67—82no—
62C67⋊C16—42no—
63C24×C67D811yescharacteristic
64#41×C67—42no—
65#41×C67—42no—
66C2×Q8×C67—21no—
67C2×D4×C67—21no—
68C2×D4×C67—41no—
69C2×D4×C67—41no—
70C2×D4×C67—41no—
71C22×C4×C67—41no—
72C22×C4×C67—41no—
73C22×C4×C67—41no—
74C22×C4×C67C22⋊C411yescharacteristic
75C22×C4×C67C22⋊C411yescharacteristic
76C22×C4×C67D811yescharacteristic
77C22×C4×C67SD1611yescharacteristic
78C22×C4×C67SD1611yescharacteristic
79C22×C4×C67C2×D411yescharacteristic
80Q16×C67—41no—
81SD16×C67—42no—
82D8×C67—41no—
83C2×C8×C67—22no—
84C2×C8×C67—22no—
85C2×C8×C67—22no—
86C2×C8×C67—41no—
87C2×C8×C67—41no—
88C2×C8×C67—41no—
89C2×C8×C67C22⋊C411yescharacteristic
90C2×C8×C67C22⋊C411yescharacteristic
91C2×C8×C67C22×C411yescharacteristic
92C2×C8×C67C2×D411yescharacteristic
93C4⋊C4×C67—21no—
94C4⋊C4×C67—21no—
95C4⋊C4×C67—41no—
96C4⋊C4×C67—41no—
97C22⋊C4×C67—41no—
98C22⋊C4×C67—41no—
99C22⋊C4×C67—41no—
100C22⋊C4×C67—41no—
101C22⋊C4×C67—41no—
102C42×C67—41no—
103C23×C67—22no—
104C23×C67—22no—
105C23×C67—41no—
106C23×C67—41no—
107C23×C67#10211yescharacteristic
108C23×C67#10211yescharacteristic
109C23×C67C2×D811yescharacteristic
110Q8×C67—41no—
111Q8×C67—41no—
112D4×C67—41no—
113D4×C67—41no—
114D4×C67—82no—
115D4×C67—82no—
116D4×C67—82no—
117C2×C4×C67—21no—
118C2×C4×C67—21no—
119C2×C4×C67—21no—
120C2×C4×C67—21no—
121C2×C4×C67—21no—
122C2×C4×C67—21no—
123C2×C4×C67—41no—
124C2×C4×C67—41no—
125C2×C4×C67—41no—
126C2×C4×C67—41no—
127C2×C4×C67—41no—
128C2×C4×C67—41no—
129C2×C4×C67—82no—
130C2×C4×C67#10211yescharacteristic
131C2×C4×C67#10211yescharacteristic
132C2×C4×C67C2×C22⋊C411yescharacteristic
133C2×C4×C67C2×SD1611yescharacteristic
134C2×Dic67—82no—
135C2×Dic67—82no—
136C2×Dic67—82no—
137C8×C67—21no—
138C8×C67—21no—
139C8×C67—21no—
140C8×C67#9912yes—
141C22×C67—21no—
142C22×C67—21no—
143C22×C67—21no—
144C22×C67—21no—
145C22×C67—21no—
146C22×C67—21no—
147C22×C67—21no—
148C22×C67—44no—
149C22×C67—81no—
150C22×C67—81no—
151C22×C67—81no—
152C22×C67#36012yes—
153C22×C67C2×#10211yescharacteristic
154C22×C67#50812yes—
155C4×C67—21no—
156C4×C67—21no—
157C4×C67—21no—
158C4×C67—81no—
159C4×C67—81no—
160C4×C67—81no—
161C4×C67C2×#9911yescharacteristic
162C4×C67#42111yescharacteristic
163Dic67—162no—
164#11885—671nomaximal, Sylow
165C2×C67—22no—
166C2×C67—22no—
167C2×C67—41no—
168C2×C67—161no—
169C2×C67#151911yescharacteristic
170C2×C67#213811yescharacteristic
171C2×C67C2×#36011yescharacteristic
172#3050D6711yescharacteristic
173C2×C22⋊C16—671no—
174#1807—671no—
175#1348—672no—
176#1347—672no—
177C67#1188511yescharacteristic, complemented, Sylow
178C2×#135Dic6711yescharacteristic, complemented
179C23×C8C2×D6711yescharacteristic
180#525—21no—
181C22×C16—1341no—
182#467Dic6711yescharacteristic, complemented
183#467C2×D6711yescharacteristic
184#466Dic6711yescharacteristic, complemented
185#466C2×D6711yescharacteristic
186#465—21no—
187#465—21no—
188#427Dic6711yescharacteristic, complemented
189C2×#106—1341no—
190#416—21no—
191#380—1341no—
192C22⋊C16—672no—
193C22⋊C16—1341no—
194#340—672no—
195C2×#41—21no—
196C23×C4C22⋊D6711yescharacteristic
197#135—42no—
198C2×Q16C2×Dic6711yescharacteristic
199C2×SD16—21no—
200C2×D8C2×Dic6711yescharacteristic
201C22×C8—42no—
202C22×C8C2×Dic6711yescharacteristic
203C22×C8C2×Dic6711yescharacteristic
204C22×C8C22⋊D6711yescharacteristic
205C22×C8C22⋊D6711yescharacteristic
206C22×C8C22⋊D6711yescharacteristic
207C22×C8C22×D6711yescharacteristic
208#123—41no—
209C22⋊Q8—21no—
210C22⋊Q8—21no—
211#121—21no—
212#121—21no—
213#120—41no—
214C4×D4—41no—
215C4×D4—41no—
216#117—21no—
217C2×C4⋊C4—1342no—
218C2×C22⋊C4—2681no—
219C2×C16—2682no—
220C2×C16—1342no—
221#107C2×Dic6711yescharacteristic
222#107C2×Dic6711yescharacteristic
223#106—21no—
224#106—2682no—
225#103—21no—
226#103—21no—
227#102—21no—
228#102—21no—
229#95—2681no—
230C24#1176161311yescharacteristic
231#41—42no—
232#41—42no—
233C2×Q8—21no—
234C2×D4—21no—
235C2×D4—41no—
236C2×D4—41no—
237C2×D4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—2682no—
242C22×C4C22⋊Dic6711yescharacteristic
243C22×C4C22⋊Dic6711yescharacteristic
244C22×C4#1176161311yescharacteristic
245C22×C4#1176161611yescharacteristic
246C22×C4#1176161611yescharacteristic
247C22×C4C2×C22⋊D6711yescharacteristic
248Q16—41no—
249SD16—42no—
250D8—41no—
251C2×C8—22no—
252C2×C8—22no—
253C2×C8—22no—
254C2×C8—41no—
255C2×C8—41no—
256C2×C8—41no—
257C2×C8C22⋊Dic6711yescharacteristic
258C2×C8C22⋊Dic6711yescharacteristic
259C2×C8C22×Dic6711yescharacteristic
260C2×C8C2×C22⋊D6711yescharacteristic
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—2682no—
266C4⋊C4—2682no—
267C22⋊C4—41no—
268C22⋊C4—41no—
269C22⋊C4—41no—
270C22⋊C4—41no—
271C22⋊C4—41no—
272C22⋊C4—5362no—
273C42—41no—
274C16—2682no—
275C23—22no—
276C23—22no—
277C23—41no—
278C23—41no—
279C23#1452396411yescharacteristic
280C23#1452396411yescharacteristic
281C23C2×#1176161311yescharacteristic
282Q8—41no—
283Q8—41no—
284D4—41no—
285D4—41no—
286D4—82no—
287D4—82no—
288D4—82no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—82no—
302C2×C4—5362no—
303C2×C4—5362no—
304C2×C4—5362no—
305C2×C4#1452396411yescharacteristic
306C2×C4#1452396411yescharacteristic
307C2×C4C2×C22⋊Dic6711yescharacteristic
308C2×C4C2×#1176161611yescharacteristic
309C8—21no—
310C8—21no—
311C8—21no—
312C8#1452395812yes—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—44no—
321C22—81no—
322C22—81no—
323C22—81no—
324C22#1686320612yes—
325C22C2×#1452396411yescharacteristic
326C22#1686382612yes—
327C4—21no—
328C4—21no—
329C4—21no—
330C4—81no—
331C4—81no—
332C4—81no—
333C4—10722no—
334C4C2×#1452395811yescharacteristic
335C4#1686341311yescharacteristic
336C2—22no—
337C2—22no—
338C2—41no—
339C2—161no—
340C2#2255877711yescharacteristic
341C2#2256191611yescharacteristic
342C2C2×#1686320611yescharacteristic
subgroupG/H
centerC22C2×#14523964
commutatorC2×C8×C67C22×C4
FrattiniC22×C8C22×D67
Fitting, prime radicalC67×#3050C2
socleC22×C67C2×#102
cosocleC22×C8×C67C23
hypercenter#3050D67
hypocenterC67#11885
LCSG ▷ C2×C8×C67 ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ #3050

Complements

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

NH ≅ G/Ntype
1. C2×C22⋊(C67⋊C16)C2semidirect
2. #22560234C2semidirect
3. #22558221C2semidirect
4. #22558218C2semidirect
12. C2×#135×C67C4semidirect
15. #467×C67C4semidirect
17. #466×C67C4semidirect
21. #427×C67C4semidirect
177. C67#11885semidirect
178. C2×#135Dic67semidirect
182. #467Dic67semidirect
184. #466Dic67semidirect
188. #427Dic67semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊(C67⋊C16)2
#225602342
#225582212
#225582182
C67×#30502
#1188567