Finite Group Databasebeta

← #30770947

Subgroups — #30770947

Order 17152.

420 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
1C2×M32×D67C211yesmaximal, characteristic
2#22561819C211yesmaximal, characteristic
3C2×#16863173C211yesmaximal, characteristic, complemented
4#22559986C211yesmaximal, characteristic
5#22558289C211yesmaximal, characteristic, complemented
6#22558288C211yesmaximal, characteristic, complemented
7C2×#347×C67C211yesmaximal, characteristic, complemented
8C22×C8×D67C2211yescharacteristic
9C2×#14523983C2211yescharacteristic
10C2×#14523983C2211yescharacteristic
11C2×#14523982C412yescomplemented
12M32×D67—22no—
13M32×D67—22no—
14C2×C16×D67C412yescomplemented
15C2×(C2×D67)⋊C8C411yescharacteristic
16C2×#14523953C2211yescharacteristic
17C4×(C2×D67)⋊C4C411yescharacteristic
18(C2×D67)⋊C16C412yescomplemented
19(C2×D67)⋊C16C412yescomplemented
20#16863173—22no—
21#16863151C411yescharacteristic
22#16863150C411yescharacteristic
23C2×M32×C67C2211yescharacteristic
24C2×M32×C67C2211yescharacteristic
25C2×#97×C67C2211yescharacteristic
26#347×C67—22no—
27C22×C4×D67C2×C411yescharacteristic
28C22×C67⋊C8C2×C411yescharacteristic
29C22×C67⋊C8C2×C411yescharacteristic
30C2×C8×D67—21no—
31C2×C8×D67—21no—
32C2×C8×D67—21no—
33C2×C8×D67—21no—
34C2×C8×D67C2×C411yescharacteristic
35C2×C8×D67C2×C411yescharacteristic
36C2×C8×D67C2×C411yescharacteristic
37C2×C8×D67C2×C411yescharacteristic
38C2×(C2×D67)⋊C4—21no—
39C2×C4×Dic67C2×C411yescharacteristic
40#14523983—21no—
41#14523983—21no—
42#14523983—21no—
43#14523983—21no—
44#14523982—24no—
45C2×C67⋊C16C2×C412yes—
46C2×C67⋊C16C2×C412yes—
47C16×D67—24no—
48(C2×D67)⋊C8—21no—
49(C2×D67)⋊C8—21no—
50#14523953—21no—
51#14523953—21no—
52C67⋊#95—21no—
53C22×C8×C67C2×C411yescharacteristic
54C22×C8×C67C2311yescharacteristic
55C2×C42×C67C2×C411yescharacteristic
56M32×C67—21no—
57M32×C67—21no—
58M32×C67D412yes—
59M32×C67D412yes—
60C2×C16×C67C2×C412yes—
61C2×C16×C67C2×C412yes—
62#97×C67—21no—
63#97×C67—21no—
64C23×D67M1611yescharacteristic
65C22×Dic67—21no—
66C22×Dic67M1611yescharacteristic
67C2×C4×D67—21no—
68C2×C4×D67—21no—
69C2×C4×D67—21no—
70C2×C4×D67—21no—
71C2×C4×D67C4211yescharacteristic
72C2×C4×D67C4211yescharacteristic
73C2×C4×D67M1611yescharacteristic
74C2×C4×D67M1611yescharacteristic
75C2×C67⋊C8—21no—
76C2×C67⋊C8—21no—
77C2×C67⋊C8—21no—
78C2×C67⋊C8—21no—
79C2×C67⋊C8—21no—
80C2×C67⋊C8C4211yescharacteristic
81C2×C67⋊C8C4211yescharacteristic
82C8×D67—21no—
83C8×D67—21no—
84C8×D67—21no—
85C8×D67—21no—
86C8×D67—21no—
87C8×D67—21no—
88C8×D67—21no—
89C8×D67—21no—
90(C2×D67)⋊C4—42no—
91C4×Dic67—21no—
92C4×Dic67—21no—
93C67⋊C16—22no—
94C67⋊C16—22no—
95C22×C4×C67—21no—
96C22×C4×C67C22×C411yescharacteristic
97C2×C8×C67—21no—
98C2×C8×C67—21no—
99C2×C8×C67—21no—
100C2×C8×C67C22×C411yescharacteristic
101C2×C8×C67C22×C411yescharacteristic
102C2×C8×C67C2×D411yescharacteristic
103C2×C8×C67C2×D411yescharacteristic
104C2×C8×C67#4111yescharacteristic
105C2×C8×C67#4111yescharacteristic
106C42×C67—21no—
107C42×C67—21no—
108C16×C67—22no—
109C16×C67C22⋊C414yescomplemented
110C23×C67C2×M1611yescharacteristic
111C22×D67—21no—
112C22×D67—21no—
113C22×D67—22no—
114C22×D67—41no—
115C22×D67#9711yescharacteristic
116C22×D67#9711yescharacteristic
117C2×C4×C67—41no—
118C2×C4×C67—41no—
119C2×C4×C67—41no—
120C2×C4×C67C2×C4211yescharacteristic
121C2×C4×C67C4×D411yescharacteristic
122C2×C4×C67C4×D411yescharacteristic
123C2×C4×C67C2×M1611yescharacteristic
124C2×C4×C67#13111yescharacteristic
125C2×C4×C67#13111yescharacteristic
126C2×Dic67—21no—
127C2×Dic67—21no—
128C2×Dic67—41no—
129C2×Dic67—41no—
130C2×Dic67—41no—
131C2×Dic67#9711yescharacteristic
132C2×Dic67#9711yescharacteristic
133C4×D67—21no—
134C4×D67—21no—
135C4×D67—21no—
136C4×D67—21no—
137C4×D67—21no—
138C4×D67—21no—
139C4×D67—21no—
140C4×D67—21no—
141C8×C67—41no—
142C8×C67C2×C22⋊C411yescharacteristic
143C8×C67#11711yescharacteristic
144C8×C67C4×D411yescharacteristic
145C8×C67C4×D411yescharacteristic
146C67⋊C8—21no—
147C67⋊C8—21no—
148C67⋊C8—41no—
149C22×C67—21no—
150C22×C67—21no—
151C22×C67C2×#9711yescharacteristic
152C22×C67#43511yescharacteristic
153C22×C67#43511yescharacteristic
154C2×D67—21no—
155C2×D67—21no—
156C2×D67—21no—
157C2×D67—21no—
158C2×D67—21no—
159C2×D67—42no—
160C2×D67—42no—
161C2×D67#34712yescomplemented
162C4×C67—81no—
163C4×C67C4×C22⋊C411yescharacteristic
164C4×C67#40511yescharacteristic
165C4×C67#43511yescharacteristic
166C4×C67#43511yescharacteristic
167Dic67—21no—
168Dic67—21no—
169Dic67—81no—
170#12616—671nomaximal, Sylow
171C2×C67—21no—
172C2×C67—21no—
173C2×C67#175211yescharacteristic
174C2×C67C2×#34711yescharacteristic
175C2×C67#211011yescharacteristic
176D67—22no—
177D67—41no—
178C22×M32—671no—
179#2112—671no—
180C2×#347—671no—
181C2×#347D6711yescharacteristic, complemented
182#1752—671no—
183#1375—671no—
184#1375—671no—
185C67#1261611yescharacteristic, complemented, Sylow
186C23×C8—671no—
187C2×M32—671no—
188C2×M32—671no—
189C2×M32—672no—
190C2×M32—1342no—
191C2×M32—1342no—
192C2×M32C2×D6711yescharacteristic
193C2×M32C2×D6711yescharacteristic
194C22×C16—672no—
195C2×C22⋊C8—671no—
196C2×#97—671no—
197C2×#97C2×D6711yescharacteristic
198C4×C22⋊C4—671no—
199C22⋊C16—672no—
200C22⋊C16—672no—
201#347—22no—
202#347—1342no—
203#336—671no—
204#336—671no—
205C23×C4—671no—
206C22×C8—671no—
207C22×C8—671no—
208C22×C8—671no—
209C22×C8—671no—
210C22×C8—671no—
211C22×C8—671no—
212C22×C8—1341no—
213C22×C8—1341no—
214C22×C8—1341no—
215C22×C8—1341no—
216C22×C8C4×D6711yescharacteristic
217C22×C8C22×D6711yescharacteristic
218C2×C22⋊C4—1341no—
219C2×C42—671no—
220C2×C42C4×D6711yescharacteristic
221M32—21no—
222M32—21no—
223M32—1341no—
224M32—1341no—
225M32—1341no—
226M32—1341no—
227M32—1344no—
228M32D26812yes—
229M32C22⋊D6712yes—
230C2×C16—672no—
231C2×C16—672no—
232C2×C16—1344no—
233C2×C16C4×D6712yes—
234C2×C16C4×D6712yes—
235C22⋊C8—1341no—
236C22⋊C8—1341no—
237#97—21no—
238#97—21no—
239#97—1341no—
240#97—1341no—
241#95—1341no—
242C24—671no—
243C22×C4—21no—
244C22×C4—671no—
245C22×C4—671no—
246C22×C4—671no—
247C22×C4—671no—
248C22×C4—671no—
249C22×C4—1341no—
250C22×C4—1341no—
251C22×C4—1341no—
252C22×C4—1341no—
253C22×C4—1341no—
254C22×C4C2×C4×D6711yescharacteristic
255C2×C8—21no—
256C2×C8—21no—
257C2×C8—21no—
258C2×C8—671no—
259C2×C8—671no—
260C2×C8—1341no—
261C2×C8—1341no—
262C2×C8—1341no—
263C2×C8—1341no—
264C2×C8—1341no—
265C2×C8—1341no—
266C2×C8—1341no—
267C2×C8—1341no—
268C2×C8—1341no—
269C2×C8—1341no—
270C2×C8—1341no—
271C2×C8—1341no—
272C2×C8—1341no—
273C2×C8C2×C4×D6711yescharacteristic
274C2×C8C2×C4×D6711yescharacteristic
275C2×C8C2×C22⋊D6711yescharacteristic
276C2×C8C2×D26811yescharacteristic
277C2×C8#1176162811yescharacteristic
278C2×C8#1176162811yescharacteristic
279C22⋊C4—2682no—
280C42—21no—
281C42—21no—
282C42—1341no—
283C42—1341no—
284C16—22no—
285C16—1342no—
286C16—1342no—
287C16(C2×D67)⋊C414yescomplemented
288C23—2681no—
289C23—671no—
290C23—671no—
291C23—1342no—
292C23—1341no—
293C23—1341no—
294C23M16×D6711yescharacteristic
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—2681no—
299C2×C4—2681no—
300C2×C4—2681no—
301C2×C4—671no—
302C2×C4—671no—
303C2×C4—1341no—
304C2×C4—1341no—
305C2×C4—1341no—
306C2×C4—1341no—
307C2×C4—1341no—
308C2×C4—1341no—
309C2×C4—1341no—
310C2×C4—1341no—
311C2×C4—1341no—
312C2×C4—1341no—
313C2×C4C42×D6711yescharacteristic
314C2×C4C4×C22⋊D6711yescharacteristic
315C2×C4C4×D26811yescharacteristic
316C2×C4M16×D6711yescharacteristic
317C2×C4#1452405111yescharacteristic
318C2×C4#1452405111yescharacteristic
319C8—41no—
320C8—2681no—
321C8—1341no—
322C8—1341no—
323C8C2×(C2×D67)⋊C411yescharacteristic
324C8#1452400011yescharacteristic
325C8C4×C22⋊D6711yescharacteristic
326C8C4×D26811yescharacteristic
327C22—21no—
328C22—21no—
329C22—2682no—
330C22—2682no—
331C22—672no—
332C22—1341no—
333C22—1341no—
334C22—1341no—
335C22—1341no—
336C22—1341no—
337C22#97×D6711yescharacteristic
338C22#1686346511yescharacteristic
339C22#1686346911yescharacteristic
340C4—81no—
341C4—5361no—
342C4—1341no—
343C4—1341no—
344C4C4×(C2×D67)⋊C411yescharacteristic
345C4#1686335311yescharacteristic
346C4#1686346511yescharacteristic
347C4#1686346911yescharacteristic
348C2—21no—
349C2—21no—
350C2—2681no—
351C2—1342no—
352C2#2255998611yescharacteristic
353C2#347×D6711yescharacteristic
354C2#2256181211yescharacteristic
subgroupG/H
centerC2×C4C42×D67
commutatorC2×C4×C67C2×C42
FrattiniC22×C8C22×D67
FittingC2×#347×C67C2
socleC22×C67C2×#97
prime radicalC23×D67M16
cosocleC22×C8×C67C23
hypercenterC2×#347D67
hypocenterC67#12616
LCSG ▷ C2×C4×C67 ▷ C67
UCSC1 ◁ C2×C4 ◁ C2×#347

Complements

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

NH ≅ G/Ntype
3. C2×#16863173C2semidirect
5. #22558289C2semidirect
6. #22558288C2semidirect
7. C2×#347×C67C2semidirect
11. C2×#14523982C4semidirect
14. C2×C16×D67C4semidirect
18. (C2×D67)⋊C16C4semidirect
19. (C2×D67)⋊C16C4semidirect
109. C16×C67C22⋊C4semidirect
161. C2×D67#347semidirect
181. C2×#347D67semidirect
185. C67#12616semidirect
287. C16(C2×D67)⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×M32×D672
#225618192
C2×#168631732
#225599862
#225582892
#225582882
C2×#347×C672
#1261667