Finite Group Databasebeta

← #30857921

Subgroups — #30857921

Order 17152.

914 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×Dic67⋊C8C211yesmaximal, characteristic
2C2×#16863352C211yesmaximal, characteristic
3#22562691C211yesmaximal, characteristic
4#22560771C214yesmaximal, complemented
5#22560764C214yesmaximal, complemented
6#22559907C211yesmaximal, characteristic
7#22559905C211yesmaximal, characteristic
8#22559902C211yesmaximal, characteristic
9C67×#1729C211yesmaximal, characteristic
10C22×#11761612C2211yescharacteristic
11C23×C67⋊C8C2211yescharacteristic
12C2×#14524002C2211yescharacteristic
13C22×C4×Dic67C2211yescharacteristic
14C2×Dic67⋊C8—24no—
15C2×Dic67⋊C8C2212yes—
16C2×Dic67⋊C8C2212yes—
17#16863352—24no—
18C2×#14523954C2212yes—
19C2×C4×C67⋊C8C2212yes—
20#16863292C412yes—
21#16863292C412yes—
22#16863268C412yes—
23#16863268C412yes—
24C4×Dic67⋊C4C2214yes—
25#16863150C2212yes—
26#16863150C2214yes—
27#16863150C2214yes—
28#16863150C2212yes—
29C22×M16×C67C2211yescharacteristic
30C23×C8×C67C2211yescharacteristic
31C2×#117×C67C2211yescharacteristic
32#336×C67C2214yes—
33C23×Dic67C2×C411yescharacteristic
34C23×Dic67C2×C411yescharacteristic
35C2×#11761612—22no—
36C2×#11761612—22no—
37C2×#11761612Q814yes—
38C22×C67⋊C8—22no—
39C22×C67⋊C8—22no—
40C22×C67⋊C8D414yes—
41C22×C67⋊C8C2311yescharacteristic
42C22×C67⋊C8C2311yescharacteristic
43C22×C67⋊C8C2312yes—
44#14524002—22no—
45#14524002—22no—
46C2×C4⋊Dic67C2×C412yes—
47C2×Dic67⋊C4C2×C414yes—
48C2×Dic67⋊C4C2×C414yes—
49C2×C22⋊Dic67C2×C412yes—
50C2×C4×Dic67—22no—
51C2×C4×Dic67—22no—
52C2×C4×Dic67C2×C412yes—
53C2×C4×Dic67C2312yes—
54C2×C4×Dic67C2312yes—
55Dic67⋊C8—24no—
56Dic67⋊C8—24no—
57#14523954—24no—
58C4×C67⋊C8—24no—
59C67⋊#95C2×C414yes—
60C67⋊#95C2×C414yes—
61C23×C4×C67C2311yescharacteristic
62C2×M16×C67—22no—
63C2×M16×C67—22no—
64C2×M16×C67—22no—
65C22×C8×C67—22no—
66C22×C8×C67—22no—
67C22×C8×C67—22no—
68C22×C8×C67C2312yes—
69C22×C8×C67C2312yes—
70#117×C67—22no—
71#117×C67—22no—
72C2×C4⋊C4×C67C2×C411yescharacteristic
73C2×C4⋊C4×C67C2×C411yescharacteristic
74C2×C22⋊C4×C67C2×C412yes—
75C2×C42×C67C2312yes—
76C22×Dic67—22no—
77C22×Dic67—24no—
78C22×Dic67—22no—
79C22×Dic67—42no—
80C22×Dic67C22×C412yes—
81C22×Dic67C22×C414yes—
82C22×Dic67C22×C412yes—
83#11761612—28no—
84C2×C67⋊C8—28no—
85C2×C67⋊C8—22no—
86C2×C67⋊C8—21no—
87C2×C67⋊C8—21no—
88C2×C67⋊C8—21no—
89C2×C67⋊C8—21no—
90C2×C67⋊C8—22no—
91C2×C67⋊C8C2×D412yes—
92C2×C67⋊C8C2×D412yes—
93C2×C67⋊C8C2×Q814yes—
94C4⋊Dic67—24no—
95Dic67⋊C4—28no—
96Dic67⋊C4—44no—
97C22⋊Dic67—24no—
98C4×Dic67—24no—
99C4×Dic67—22no—
100C4×Dic67—22no—
101C4×Dic67—22no—
102C4×Dic67—22no—
103C24×C67C22×C411yescharacteristic
104C22×C4×C67C22×C412yes—
105C22×C4×C67C22×C412yes—
106C22×C4×C67C22×C411yescharacteristic
107C22×C4×C67C2×D412yes—
108C22×C4×C67C2×Q812yes—
109C22×C4×C67#4112yes—
110C22×C4×C67#4112yes—
111C22×C4×C67#4112yes—
112C22×C4×C67#4112yes—
113C22×C4×C67C2411yescharacteristic
114M16×C67—44no—
115C2×C8×C67—22no—
116C2×C8×C67—22no—
117C2×C8×C67—22no—
118C2×C8×C67—22no—
119C2×C8×C67—22no—
120C2×C8×C67—22no—
121C2×C8×C67—44no—
122C4⋊C4×C67—22no—
123C4⋊C4×C67—22no—
124C22⋊C4×C67—24no—
125C42×C67—22no—
126C42×C67—22no—
127C23×C67—22no—
128C23×C67—22no—
129C23×C67#11712yes—
130C23×C67C4×D412yes—
131C23×C67C4×Q812yes—
132C23×C67C23×C411yescharacteristic
133C2×C4×C67—22no—
134C2×C4×C67—22no—
135C2×C4×C67—24no—
136C2×C4×C67—24no—
137C2×C4×C67#11712yes—
138C2×C4×C67C4×D412yes—
139C2×C4×C67C4×Q812yes—
140C2×C4×C67C22⋊Q814yes—
141C2×C4×C67#12314yes—
142C2×C4×C67#12514yes—
143C2×C4×C67#12614yes—
144C2×C4×C67C22×D411yescharacteristic
145C2×C4×C67C22×Q811yescharacteristic
146C2×C4×C67C2×#4111yescharacteristic
147C2×C4×C67C2×#4111yescharacteristic
148C2×C4×C67C2×#4111yescharacteristic
149C2×C4×C67C2×#4111yescharacteristic
150C2×Dic67—28no—
151C2×Dic67—22no—
152C2×Dic67—22no—
153C2×Dic67—24no—
154C2×Dic67—22no—
155C2×Dic67—24no—
156C2×Dic67—22no—
157C2×Dic67—24no—
158C2×Dic67—22no—
159C2×Dic67—44no—
160C2×Dic67#13114yes—
161C8×C67—42no—
162C8×C67—42no—
163C67⋊C8—24no—
164C67⋊C8—22no—
165C67⋊C8—22no—
166C22×C67—24no—
167C22×C67—22no—
168C22×C67—22no—
169C22×C67—24no—
170C22×C67#38714yes—
171C22×C67C2×#11711yescharacteristic
172C22×C67C2×C4×D411yescharacteristic
173C22×C67C2×C4×Q811yescharacteristic
174C22×C67C2×#13112yes—
175C22×C67#56812yes—
176C4×C67—42no—
177C4×C67—42no—
178C4×C67#38714yes—
179C4×C67C2×C22⋊Q811yescharacteristic
180C4×C67C2×#12311yescharacteristic
181C4×C67C2×#12511yescharacteristic
182C4×C67C2×#12611yescharacteristic
183Dic67—24no—
184Dic67—44no—
185Dic67—42no—
186#21024—671nomaximal, Sylow
187C2×C67—24no—
188C2×C67C2×#38711yescharacteristic
189C2×C67#292512yes—
190C2×C67#292911yescharacteristic
191C2×C67#293011yescharacteristic
192C2×C67#296212yes—
193C22×C4⋊C8—671no—
194C2×#405—671no—
195#2296—671no—
196#1917—674no—
197#1916—674no—
198#1729—671no—
199#1729—671no—
200#1729D6711yescharacteristic
201#1728—671no—
202C67#2102411yescharacteristic, complemented, Sylow
203C22×M16—671no—
204C22×M16C2×D6711yescharacteristic
205C23×C8—671no—
206C23×C8C2×D6711yescharacteristic
207C2×#117—671no—
208C2×#117C2×D6711yescharacteristic
209C22×C42—671no—
210C2×C4⋊C8—672no—
211C2×C4⋊C8—672no—
212C2×C4⋊C8—1344no—
213#405—1344no—
214C2×#97—672no—
215C2×C4×C8—672no—
216#389—672no—
217#389—672no—
218#385—672no—
219#385—672no—
220C4×C4⋊C4—674no—
221#336—672no—
222#336—674no—
223#336—674no—
224#336—672no—
225#336C2×D6714yes—
226C23×C4—671no—
227C23×C4—671no—
228C23×C4C22×D6711yescharacteristic
229C2×M16—22no—
230C2×M16—22no—
231C2×M16—22no—
232C2×M16—674no—
233C2×M16—1342no—
234C2×M16—1342no—
235C22×C8—22no—
236C22×C8—22no—
237C22×C8—22no—
238C22×C8—674no—
239C22×C8—671no—
240C22×C8—671no—
241C22×C8—672no—
242C22×C8—1342no—
243C22×C8—1342no—
244C22×C8C22×D6712yes—
245C22×C8C22×D6712yes—
246#117—22no—
247#117—22no—
248#117—1342no—
249#117—1342no—
250C2×C4⋊C4—674no—
251C2×C4⋊C4—672no—
252C2×C4⋊C4—674no—
253C2×C4⋊C4C4×D6711yescharacteristic
254C2×C4⋊C4C4×D6711yescharacteristic
255C2×C22⋊C4—672no—
256C2×C22⋊C4C4×D6712yes—
257C2×C42—672no—
258C2×C42—672no—
259C2×C42—672no—
260C2×C42—1342no—
261C2×C42—1342no—
262C2×C42C22×D6712yes—
263C4⋊C8—1344no—
264C4⋊C8—1344no—
265#97—1344no—
266C4×C8—1344no—
267#95—674no—
268#95—674no—
269C24C2×C4×D6711yescharacteristic
270C22×C4—2682no—
271C22×C4—672no—
272C22×C4—672no—
273C22×C4—674no—
274C22×C4—1342no—
275C22×C4—1344no—
276C22×C4—1342no—
277C22×C4C2×C4×D6712yes—
278C22×C4C2×C4×D6712yes—
279C22×C4C2×C4×D6711yescharacteristic
280C22×C4D4×D6712yes—
281C22×C4Q8×D6712yes—
282C22×C4#1176162712yes—
283C22×C4#1176162712yes—
284C22×C4#1176162812yes—
285C22×C4#1176162812yes—
286C22×C4C23×D6711yescharacteristic
287M16—44no—
288M16—1348no—
289C2×C8—22no—
290C2×C8—22no—
291C2×C8—22no—
292C2×C8—22no—
293C2×C8—22no—
294C2×C8—22no—
295C2×C8—44no—
296C2×C8—672no—
297C2×C8—672no—
298C2×C8—674no—
299C2×C8—1348no—
300C2×C8—1342no—
301C2×C8—1341no—
302C2×C8—1341no—
303C2×C8—1341no—
304C2×C8—1341no—
305C2×C8—1342no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—2684no—
309C4⋊C4—1348no—
310C4⋊C4—1344no—
311C22⋊C4—24no—
312C22⋊C4—1344no—
313C42—22no—
314C42—22no—
315C42—1344no—
316C42—1342no—
317C42—1342no—
318C42—1342no—
319C42—1342no—
320C23—22no—
321C23—22no—
322C23#1452400012yes—
323C23(C22⋊D67)⋊C412yes—
324C23#1452400812yes—
325C23C22×C4×D6711yescharacteristic
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—24no—
329C2×C4—24no—
330C2×C4—2684no—
331C2×C4—674no—
332C2×C4—1344no—
333C2×C4—1342no—
334C2×C4—1344no—
335C2×C4—1342no—
336C2×C4—1344no—
337C2×C4—1342no—
338C2×C4—1348no—
339C2×C4—1342no—
340C2×C4—1342no—
341C2×C4#1452400012yes—
342C2×C4(C22⋊D67)⋊C412yes—
343C2×C4#1452400812yes—
344C2×C4#1452402114yes—
345C2×C4#1452402514yes—
346C2×C4#1452403314yes—
347C2×C4#1452403614yes—
348C2×C4C2×D4×D6711yescharacteristic
349C2×C4C2×Q8×D6711yescharacteristic
350C2×C4C2×#1176162711yescharacteristic
351C2×C4C2×#1176162711yescharacteristic
352C2×C4C2×#1176162811yescharacteristic
353C2×C4C2×#1176162811yescharacteristic
354C8—42no—
355C8—42no—
356C8—1344no—
357C8—1342no—
358C8—1342no—
359C22—24no—
360C22—24no—
361C22—22no—
362C22—22no—
363C22#1686327814yes—
364C22C2×#1452400011yescharacteristic
365C22C2×(C22⋊D67)⋊C411yescharacteristic
366C22C2×#1452400811yescharacteristic
367C22#131×D6712yes—
368C22#1686416112yes—
369C4—42no—
370C4—42no—
371C4—2684no—
372C4—2682no—
373C4—1344no—
374C4#1686327814yes—
375C4C2×#1452402111yescharacteristic
376C4C2×#1452402511yescharacteristic
377C4C2×#1452403311yescharacteristic
378C4C2×#1452403611yescharacteristic
379C2—24no—
380C2C2×#1686327811yescharacteristic
381C2#2256887312yes—
382C2#2256890611yescharacteristic
383C2#2256891511yescharacteristic
384C2#2256915512yes—
subgroupG/H
center, FrattiniC22×C4C23×D67
commutator, socleC23×C67C23×C4
FittingC67×#1729C2
prime radicalC24×C67C22×C4
cosocleC22×C4×C67C24
hypercenter#1729D67
hypocenterC67#21024
LCSG ▷ C23×C67 ▷ C67
UCSC1 ◁ C22×C4 ◁ #1729

Complements

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

NH ≅ G/Ntype
4. #22560771C2semidirect
5. #22560764C2semidirect
202. C67#21024semidirect

Maximal subgroups

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

subgroupindex
C22×Dic67⋊C82
C2×#168633522
#225626912
#225607712
#225607642
#225599072
#225599052
#225599022
C67×#17292
#2102467