Finite Group Databasebeta

← #30745875

Subgroups — #30745875

Order 17152.

742 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16863449C211yesmaximal, characteristic
2#22562616C211yesmaximal, characteristic
3#22558088C214yesmaximal
4C2×#382×C67C211yesmaximal, characteristic
5C2×#14524002—21no—
6#16863449—22no—
7#16863449C414yescomplemented
8C2×C4⋊(C67⋊C8)—21no—
9C2×C22⋊(C67⋊C8)C2212yes—
10C2×C4×C67⋊C8—21no—
11#16863266C412yescomplemented
12#16863262—21no—
13#16863256—21no—
14#16863256C2212yes—
15C4×C4⋊Dic67—21no—
16C4×C4⋊Dic67C412yescomplemented
17C22×C4⋊C4×C67—21no—
18C22×C42×C67C2211yescharacteristic
19#382×C67—22no—
20#382×C67C414yescomplemented
21C2×#95×C67—21no—
22C2×#95×C67C2212yes—
23C22×C67⋊C8—22no—
24#14524002—42no—
25#14524002—42no—
26C2×C4⋊Dic67—22no—
27C2×C4⋊Dic67—22no—
28C2×C4⋊Dic67C2×C414yes—
29C2×C22⋊Dic67—22no—
30C2×C4×Dic67—22no—
31C4⋊(C67⋊C8)—24no—
32C22⋊(C67⋊C8)—22no—
33C22⋊(C67⋊C8)C2×C414yes—
34C4×C67⋊C8—24no—
35C67⋊#95—22no—
36C67⋊#95—22no—
37C23×C4×C67—22no—
38C23×C4×C67D411yescharacteristic
39C23×C4×C67D411yescharacteristic
40C23×C4×C67C2311yescharacteristic
41C2×C4⋊C4×C67—24no—
42C2×C4⋊C4×C67—42no—
43C2×C4⋊C4×C67—42no—
44C2×C42×C67C2×C412yes—
45C2×C42×C67C2×C411yescharacteristic
46C2×C42×C67C2×C411yescharacteristic
47C2×C42×C67C2×C412yes—
48C2×C42×C67D411yescharacteristic
49C2×C42×C67D411yescharacteristic
50C2×C42×C67D412yes—
51C2×C42×C67Q812yes—
52#95×C67—24no—
53#95×C67—22no—
54#95×C67C2×C414yes—
55C22×Dic67—24no—
56C2×C67⋊C8—24no—
57C2×C67⋊C8—44no—
58C4⋊Dic67—44no—
59C4⋊Dic67—44no—
60C4⋊Dic67—44no—
61C22⋊Dic67—42no—
62C22⋊Dic67—42no—
63C4×Dic67—44no—
64C24×C67C2×D411yescharacteristic
65C22×C4×C67—24no—
66C22×C4×C67—42no—
67C22×C4×C67—42no—
68C22×C4×C67—44no—
69C22×C4×C67—44no—
70C22×C4×C67—42no—
71C22×C4×C67—42no—
72C22×C4×C67C22⋊C411yescharacteristic
73C22×C4×C67C22⋊C411yescharacteristic
74C22×C4×C67C22⋊C411yescharacteristic
75C22×C4×C67C22⋊C411yescharacteristic
76C22×C4×C67C22⋊C412yes—
77C22×C4×C67C22⋊C412yes—
78C22×C4×C67C4⋊C412yes—
79C22×C4×C67C4⋊C412yes—
80C22×C4×C67D812yes—
81C22×C4×C67SD1612yes—
82C22×C4×C67SD1612yes—
83C22×C4×C67Q1612yes—
84C22×C4×C67C22×C411yescharacteristic
85C22×C4×C67C22×C411yescharacteristic
86C22×C4×C67C22×C411yescharacteristic
87C22×C4×C67C2×D411yescharacteristic
88C22×C4×C67C2×D411yescharacteristic
89C22×C4×C67C2×Q811yescharacteristic
90C4⋊C4×C67—48no—
91C42×C67C4212yes—
92C42×C67C4212yes—
93C42×C67C22⋊C412yes—
94C42×C67C22⋊C412yes—
95C42×C67C4⋊C412yes—
96C42×C67C4⋊C412yes—
97C42×C67C4⋊C412yes—
98C42×C67C4⋊C412yes—
99C23×C67—22no—
100C23×C67—22no—
101C23×C67C2×C22⋊C411yescharacteristic
102C23×C67C2×C22⋊C411yescharacteristic
103C23×C67C2×C4⋊C411yescharacteristic
104C23×C67C2×D811yescharacteristic
105C23×C67C2×SD1611yescharacteristic
106C23×C67C2×SD1611yescharacteristic
107C23×C67C2×Q1611yescharacteristic
108C2×C4×C67—22no—
109C2×C4×C67—22no—
110C2×C4×C67—22no—
111C2×C4×C67—22no—
112C2×C4×C67—42no—
113C2×C4×C67—42no—
114C2×C4×C67—41no—
115C2×C4×C67—41no—
116C2×C4×C67—41no—
117C2×C4×C67—41no—
118C2×C4×C67—44no—
119C2×C4×C67—44no—
120C2×C4×C67—44no—
121C2×C4×C67—84no—
122C2×C4×C67—84no—
123C2×C4×C67#9512yes—
124C2×C4×C67#9512yes—
125C2×C4×C67#9512yes—
126C2×C4×C67#9512yes—
127C2×C4×C67#10212yes—
128C2×C4×C67#10212yes—
129C2×C4×C67#10214yes—
130C2×C4×C67#10312yes—
131C2×C4×C67#10312yes—
132C2×C4×C67#10314yes—
133C2×C4×C67#10614yes—
134C2×C4×C67#10714yes—
135C2×C4×C67C2×C4211yescharacteristic
136C2×C4×C67C2×C22⋊C411yescharacteristic
137C2×C4×C67C2×C4⋊C411yescharacteristic
138C2×C4×C67C2×C4⋊C411yescharacteristic
139C2×Dic67—44no—
140C2×Dic67—44no—
141C2×Dic67—44no—
142C67⋊C8—44no—
143C22×C67—21no—
144C22×C67—21no—
145C22×C67—21no—
146C22×C67—21no—
147C22×C67—22no—
148C22×C67—22no—
149C22×C67—22no—
150C22×C67—22no—
151C22×C67C2×#9511yescharacteristic
152C22×C67#41012yes—
153C22×C67C2×#10211yescharacteristic
154C22×C67C2×#10211yescharacteristic
155C22×C67C2×#10311yescharacteristic
156C22×C67C2×#10311yescharacteristic
157C22×C67#42112yes—
158C22×C67C2×#10611yescharacteristic
159C22×C67C2×#10711yescharacteristic
160C4×C67—22no—
161C4×C67—22no—
162C4×C67—21no—
163C4×C67—21no—
164C4×C67—21no—
165C4×C67—21no—
166C4×C67—84no—
167C4×C67#34014yes—
168C4×C67#34014yes—
169Dic67—84no—
170#8026—671nomaximal, Sylow
171C2×C67—21no—
172C2×C67—21no—
173C2×C67—21no—
174C2×C67—21no—
175C2×C67#151712yes—
176C2×C67#151812yes—
177C2×C67#173311yescharacteristic
178C2×C67C2×#34011yescharacteristic
179C2×C67#174011yescharacteristic
180C2×#431—671no—
181#2285—671no—
182C2×#382D6711yescharacteristic
183#1294—674no—
184C67#802611yescharacteristic, complemented, Sylow
185C2×#117—1341no—
186C22×C4⋊C4—21no—
187C22×C42C2×D6711yescharacteristic
188#431—674no—
189#431—1342no—
190C2×C4⋊C8—1341no—
191C2×C22⋊C8—672no—
192C2×C4×C8—1341no—
193#384—672no—
194#382—22no—
195#382—1341no—
196#382Dic6714yescomplemented
197#380—672no—
198#380—1341no—
199C4×C4⋊C4—672no—
200C4×C4⋊C4—1341no—
201C2×#95—21no—
202C2×#95C2×D6712yes—
203C23×C4—22no—
204C23×C4C22⋊D6711yescharacteristic
205C23×C4C22⋊D6711yescharacteristic
206C23×C4C22×D6711yescharacteristic
207C22×C8—1342no—
208#117—2682no—
209#117—2682no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—674no—
214C2×C4⋊C4—1342no—
215C2×C4⋊C4—1342no—
216C2×C22⋊C4—1342no—
217C2×C42—1342no—
218C2×C42Dic13412yes—
219C2×C42C4×D6712yes—
220C2×C42C4×D6711yescharacteristic
221C2×C42C4×D6711yescharacteristic
222C2×C42D26811yescharacteristic
223C2×C42D26811yescharacteristic
224C2×C42C2×Dic6712yes—
225C2×C42C22⋊D6712yes—
226C4⋊C8—1344no—
227C22⋊C8—674no—
228C22⋊C8—1342no—
229C4×C8—1344no—
230#95—24no—
231#95—22no—
232#95—1342no—
233#95—1342no—
234#95C2×Dic6714yes—
235C24C2×C22⋊D6711yescharacteristic
236C22×C4—24no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—44no—
240C22×C4—44no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—1344no—
244C22×C4(C2×D67)⋊C411yescharacteristic
245C22×C4(C2×D67)⋊C411yescharacteristic
246C22×C4(C2×D67)⋊C411yescharacteristic
247C22×C4(C2×D67)⋊C411yescharacteristic
248C22×C4C22⋊Dic6712yes—
249C22×C4C22⋊Dic6712yes—
250C22×C4Dic67⋊C412yes—
251C22×C4Dic67⋊C412yes—
252C22×C4#1176161312yes—
253C22×C4#1176161512yes—
254C22×C4#1176161612yes—
255C22×C4#1176161812yes—
256C22×C4C2×C4×D6711yescharacteristic
257C22×C4C2×C4×D6711yescharacteristic
258C22×C4C22×Dic6711yescharacteristic
259C22×C4C2×C22⋊D6711yescharacteristic
260C22×C4C2×D26811yescharacteristic
261C22×C4C2×Dic13411yescharacteristic
262C2×C8—2684no—
263C2×C8—1344no—
264C4⋊C4—48no—
265C4⋊C4—2684no—
266C4⋊C4—2684no—
267C4⋊C4—2684no—
268C22⋊C4—2682no—
269C22⋊C4—2682no—
270C42—2684no—
271C42C4×Dic6712yes—
272C42C4×Dic6712yes—
273C42(C2×D67)⋊C412yes—
274C42(C2×D67)⋊C412yes—
275C42Dic67⋊C412yes—
276C42Dic67⋊C412yes—
277C42C4⋊Dic6712yes—
278C42C4⋊Dic6712yes—
279C23—22no—
280C23—22no—
281C23C2×(C2×D67)⋊C411yescharacteristic
282C23C2×C22⋊Dic6711yescharacteristic
283C23C2×Dic67⋊C411yescharacteristic
284C23C2×#1176161311yescharacteristic
285C23C2×#1176161511yescharacteristic
286C23C2×#1176161611yescharacteristic
287C23C2×#1176161811yescharacteristic
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—44no—
299C2×C4—44no—
300C2×C4—44no—
301C2×C4—84no—
302C2×C4—84no—
303C2×C4—2684no—
304C2×C4—2684no—
305C2×C4—2684no—
306C2×C4C67⋊#9512yes—
307C2×C4C67⋊#9512yes—
308C2×C4C67⋊#9512yes—
309C2×C4C67⋊#9512yes—
310C2×C4#1452396312yes—
311C2×C4#1452396312yes—
312C2×C4#1452396414yes—
313C2×C4#1452396612yes—
314C2×C4#1452396612yes—
315C2×C4#1452396714yes—
316C2×C4#1452397414yes—
317C2×C4#1452397614yes—
318C2×C4C2×C4×Dic6711yescharacteristic
319C2×C4C2×(C2×D67)⋊C411yescharacteristic
320C2×C4C2×Dic67⋊C411yescharacteristic
321C2×C4C2×C4⋊Dic6711yescharacteristic
322C8—2684no—
323C22—22no—
324C22—22no—
325C22—22no—
326C22—22no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22C2×C67⋊#9511yescharacteristic
332C22#1686336912yes—
333C22C2×#1452396311yescharacteristic
334C22C2×#1452396411yescharacteristic
335C22C2×#1452396611yescharacteristic
336C22C2×#1452396711yescharacteristic
337C22#1686341412yes—
338C22C2×#1452397411yescharacteristic
339C22C2×#1452397611yescharacteristic
340C4—22no—
341C4—22no—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—84no—
347C4—5364no—
348C4#1686315814yes—
349C4#1686315814yes—
350C2—21no—
351C2—21no—
352C2—21no—
353C2—21no—
354C2#2255876812yes—
355C2#2255877212yes—
356C2#2255991911yescharacteristic
357C2C2×#1686315811yescharacteristic
358C2#2255994111yescharacteristic
subgroupG/H
centerC23C2×(C2×D67)⋊C4
commutatorC2×C4×C67C2×C42
FrattiniC23×C4C22×D67
FittingC2×#382×C67C2
socleC23×C67C2×C22⋊C4
prime radicalC24×C67C2×D4
cosocleC23×C4×C67C23
hypercenterC2×#382D67
hypocenterC67#8026
LCSG ▷ C2×C4×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#382

Complements

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

NH ≅ G/Ntype
7. #16863449C4semidirect
11. #16863266C4semidirect
16. C4×C4⋊Dic67C4semidirect
20. #382×C67C4semidirect
184. C67#8026semidirect
196. #382Dic67semidirect

Maximal subgroups

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

subgroupindex
C2×#168634492
#225626162
#225580882
C2×#382×C672
#802667