Finite Group Databasebeta

← #30772331

Subgroups — #30772331

Order 17152.

486 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#107×D67C211yesmaximal, characteristic
2C2×#16863222C211yesmaximal, characteristic, complemented
3C2×#16863221C211yesmaximal, characteristic, complemented
4C2×(C2×D67)⋊C16C211yesmaximal, characteristic
5#22560235C211yesmaximal, characteristic
6#22558300C211yesmaximal, characteristic, complemented
7C67×#1380C211yesmaximal, characteristic, complemented
8C22×C8×D67C2211yescharacteristic
9C2×C4⋊C4×D67—21no—
10C22×C67⋊C16C2211yescharacteristic
11C2×#14523977C2211yescharacteristic
12C2×#14523977C2211yescharacteristic
13C2×#14523976—21no—
14#107×D67—22no—
15#107×D67—22no—
16#16863253—21no—
17#16863222—22no—
18#16863221—22no—
19(C2×D67)⋊C16C414yescomplemented
20#16863158—21no—
21C22×C16×C67C2211yescharacteristic
22C2×#107×C67C2211yescharacteristic
23C2×#107×C67C2211yescharacteristic
24C22×C4×D67—41no—
25C22×C4×D67D411yescharacteristic
26C22×C67⋊C8D411yescharacteristic
27C2×C8×D67—21no—
28C2×C8×D67—21no—
29C2×C8×D67—21no—
30C2×C8×D67—21no—
31C2×C8×D67C2×C412yes—
32C2×C8×D67Q812yes—
33C2×C4⋊Dic67—21no—
34C2×C4⋊Dic67—21no—
35C2×Dic67⋊C4—41no—
36C4⋊C4×D67—24no—
37C4⋊C4×D67—42no—
38C2×(C2×D67)⋊C4—41no—
39C2×C67⋊C16—21no—
40C2×C67⋊C16—21no—
41C2×C67⋊C16C2×C412yescomplemented
42#14523977—21no—
43#14523977—21no—
44#14523977—21no—
45#14523977D412yes—
46#14523976—24no—
47C67⋊#95—41no—
48C22×C8×C67C2311yescharacteristic
49C2×C4⋊C4×C67—21no—
50C2×C4⋊C4×C67—21no—
51C2×C16×C67—21no—
52C2×C16×C67—21no—
53C2×C16×C67C2×C412yescomplemented
54#107×C67—21no—
55#107×C67—21no—
56#107×C67—21no—
57#107×C67D412yes—
58C23×D67D811yescharacteristic
59C22×Dic67—41no—
60C22×Dic67—41no—
61C22×Dic67D811yescharacteristic
62C2×C4×D67—21no—
63C2×C4×D67—21no—
64C2×C4×D67—21no—
65C2×C4×D67—21no—
66C2×C4×D67—44no—
67C2×C4×D67—82no—
68C2×C4×D67—82no—
69C2×C4×D67C4⋊C412yes—
70C2×C4×D67Q1612yes—
71C2×C67⋊C8—21no—
72C2×C67⋊C8—21no—
73C2×C67⋊C8C4⋊C412yes—
74C8×D67—22no—
75C8×D67—22no—
76C8×D67—22no—
77C8×D67—22no—
78C4⋊Dic67—22no—
79C4⋊Dic67—41no—
80C4⋊Dic67—41no—
81C4⋊Dic67—41no—
82Dic67⋊C4—44no—
83(C2×D67)⋊C4—82no—
84C67⋊C16—22no—
85C22×C4×C67—41no—
86C22×C4×C67—41no—
87C22×C4×C67C2×D411yescharacteristic
88C2×C8×C67C22×C411yescharacteristic
89C2×C8×C67C2×D411yescharacteristic
90C2×C8×C67C2×D411yescharacteristic
91C2×C8×C67C2×Q811yescharacteristic
92C2×C8×C67#4111yescharacteristic
93C2×C8×C67#4111yescharacteristic
94C4⋊C4×C67—22no—
95C4⋊C4×C67—41no—
96C4⋊C4×C67—41no—
97C4⋊C4×C67—41no—
98C16×C67—22no—
99C23×C67C2×D811yescharacteristic
100C22×D67—21no—
101C22×D67—21no—
102C22×D67—21no—
103C22×D67—21no—
104C22×D67#10712yes—
105C22×D67D1612yes—
106C22×D67Q3212yes—
107C2×C4×C67—42no—
108C2×C4×C67—81no—
109C2×C4×C67—81no—
110C2×C4×C67—81no—
111C2×C4×C67—81no—
112C2×C4×C67—81no—
113C2×C4×C67C2×C4⋊C411yescharacteristic
114C2×C4×C67#12011yescharacteristic
115C2×C4×C67#12311yescharacteristic
116C2×C4×C67C2×Q1611yescharacteristic
117C2×C4×C67AGL(1,ℤ/8)11yescharacteristic
118C2×C4×C67#13711yescharacteristic
119C2×Dic67—21no—
120C2×Dic67—21no—
121C2×Dic67—42no—
122C2×Dic67—81no—
123C2×Dic67—81no—
124C2×Dic67—81no—
125C2×Dic67—81no—
126C2×Dic67—81no—
127C2×Dic67#10712yes—
128C4×D67—22no—
129C4×D67—22no—
130C4×D67—22no—
131C4×D67—22no—
132C4×D67—88no—
133C8×C67C4×D411yescharacteristic
134C8×C67C4×D411yescharacteristic
135C8×C67C22⋊Q811yescharacteristic
136C8×C67C22⋊Q811yescharacteristic
137C67⋊C8—22no—
138C22×C67C2×#10711yescharacteristic
139C22×C67#44711yescharacteristic
140C22×C67#48011yescharacteristic
141C22×C67C2×D1611yescharacteristic
142C22×C67C2×Q3211yescharacteristic
143C22×C67#50911yescharacteristic
144C22×C67#51011yescharacteristic
145C2×D67—22no—
146C2×D67—22no—
147C2×D67—22no—
148C2×D67—22no—
149C2×D67—22no—
150C2×D67—22no—
151C2×D67#36614yescomplemented
152C4×C67—82no—
153C4×C67—161no—
154C4×C67#38511yescharacteristic
155C4×C67#42811yescharacteristic
156C4×C67#45111yescharacteristic
157C4×C67#48411yescharacteristic
158Dic67—22no—
159Dic67—82no—
160Dic67—161no—
161#12919—671nomaximal, Sylow
162C2×C67#185211yescharacteristic
163C2×C67C2×#36611yescharacteristic
164C2×C67#215911yescharacteristic
165C2×C67#218411yescharacteristic
166C2×C67#218811yescharacteristic
167C2×C67#223211yescharacteristic
168C2×C67#223611yescharacteristic
169D67—24no—
170C22×#107—671no—
171C2×#366—671no—
172C2×#366—671no—
173C2×C22⋊C16—671no—
174#1808—671no—
175#1380—671no—
176#1380D6711yescharacteristic, complemented
177C67#1291911yescharacteristic, complemented, Sylow
178C23×C8—671no—
179C22×C4⋊C4—1341no—
180C22×C16—671no—
181C22×C16C2×D6711yescharacteristic
182C2×#107—671no—
183C2×#107—671no—
184C2×#107—1342no—
185C2×#107—1341no—
186C2×#107—1342no—
187C2×#107C2×D6711yescharacteristic
188C2×#107C2×D6711yescharacteristic
189#380—1341no—
190#366—1342no—
191#366—1342no—
192C22⋊C16—674no—
193#340—1341no—
194C23×C4—2681no—
195C23×C4—671no—
196C22×C8—671no—
197C22×C8—672no—
198C22×C8—672no—
199C22×C8—1341no—
200C22×C8—1341no—
201C22×C8—1341no—
202C22×C8—1341no—
203C22×C8C22×D6711yescharacteristic
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—2682no—
207C2×C4⋊C4—2681no—
208C2×C4⋊C4—1344no—
209C2×C4⋊C4—1341no—
210C2×C4⋊C4—1341no—
211C2×C22⋊C4—2681no—
212C2×C16—21no—
213C2×C16—21no—
214C2×C16—672no—
215C2×C16—1341no—
216C2×C16—1341no—
217C2×C16C4×D6712yescomplemented
218#107—21no—
219#107—21no—
220#107—21no—
221#107—672no—
222#107—1344no—
223#107—1341no—
224#107—1341no—
225#107—1341no—
226#107D26812yes—
227#95—2681no—
228C24—671no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—2684no—
232C22×C4—2681no—
233C22×C4—2681no—
234C22×C4—5362no—
235C22×C4—5362no—
236C22×C4—671no—
237C22×C4—672no—
238C22×C4—672no—
239C22×C4—1341no—
240C22×C4—1341no—
241C22×C4—1341no—
242C22×C4—1341no—
243C22×C4D4×D6711yescharacteristic
244C2×C8—672no—
245C2×C8—1341no—
246C2×C8—1341no—
247C2×C8—1342no—
248C2×C8—1342no—
249C2×C8—1342no—
250C2×C8—1342no—
251C2×C8C2×C4×D6711yescharacteristic
252C2×C8D4×D6711yescharacteristic
253C2×C8C2×D26811yescharacteristic
254C2×C8Q8×D6711yescharacteristic
255C2×C8#1176162711yescharacteristic
256C2×C8#1176162811yescharacteristic
257C4⋊C4—22no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—2684no—
262C4⋊C4—2681no—
263C4⋊C4—2681no—
264C4⋊C4—2681no—
265C4⋊C4—1342no—
266C22⋊C4—5362no—
267C16—22no—
268C16—1342no—
269C23—672no—
270C23—672no—
271C23—672no—
272C23—1341no—
273C23—1341no—
274C23—1341no—
275C23—1341no—
276C23D8×D6711yescharacteristic
277C2×C4—42no—
278C2×C4—81no—
279C2×C4—81no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—2682no—
284C2×C4—5368no—
285C2×C4—5361no—
286C2×C4—5361no—
287C2×C4—5361no—
288C2×C4—5361no—
289C2×C4—5361no—
290C2×C4—672no—
291C2×C4—1341no—
292C2×C4—1341no—
293C2×C4—1342no—
294C2×C4—1342no—
295C2×C4—1342no—
296C2×C4—1342no—
297C2×C4C4⋊C4×D6711yescharacteristic
298C2×C4#1452401311yescharacteristic
299C2×C4#1452402511yescharacteristic
300C2×C4Q16×D6711yescharacteristic
301C2×C4#1452407011yescharacteristic
302C2×C4#1452407211yescharacteristic
303C8—1342no—
304C8(C22⋊D67)⋊C411yescharacteristic
305C8C4×D26811yescharacteristic
306C8#1452402111yescharacteristic
307C8#1452402311yescharacteristic
308C22—674no—
309C22—1342no—
310C22—1342no—
311C22—1342no—
312C22—1342no—
313C22—1342no—
314C22—1342no—
315C22#107×D6711yescharacteristic
316C22#1686353111yescharacteristic
317C22#1686370811yescharacteristic
318C22D16×D6711yescharacteristic
319C22Q32×D6711yescharacteristic
320C22#1686383311yescharacteristic
321C22#1686383511yescharacteristic
322C4—82no—
323C4—161no—
324C4—5362no—
325C4—10721no—
326C4—1342no—
327C4#1686327011yescharacteristic
328C4#1686343511yescharacteristic
329C4#1686355311yescharacteristic
330C4#1686373011yescharacteristic
331C2—1344no—
332C2#2256039811yescharacteristic
333C2#366×D6711yescharacteristic
334C2#2256199611yescharacteristic
335C2#2256212211yescharacteristic
336C2#2256214411yescharacteristic
337C2#2256238811yescharacteristic
338C2#2256241011yescharacteristic
subgroupG/H
centerC23D8×D67
commutatorC2×C8×C67C22×C4
FrattiniC22×C8C22×D67
FittingC67×#1380C2
socleC23×C67C2×D8
prime radicalC23×D67D8
cosocleC22×C8×C67C23
hypercenter#1380D67
hypocenterC67#12919
LCSG ▷ C2×C8×C67 ▷ C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ #1380

Complements

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

NH ≅ G/Ntype
2. C2×#16863222C2semidirect
3. C2×#16863221C2semidirect
6. #22558300C2semidirect
7. C67×#1380C2semidirect
19. (C2×D67)⋊C16C4semidirect
41. C2×C67⋊C16C2×C4semidirect
53. C2×C16×C67C2×C4semidirect
151. C2×D67#366semidirect
176. #1380D67semidirect
177. C67#12919semidirect
217. C2×C16C4×D67semidirect

Maximal subgroups

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

subgroupindex
C2×#107×D672
C2×#168632222
C2×#168632212
C2×(C2×D67)⋊C162
#225602352
#225583002
C67×#13802
#1291967