Finite Group Databasebeta

← #30754136

Subgroups — #30754136

Order 17152.

446 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22561003C211yesmaximal, characteristic
2C4×#14523963C211yesmaximal, characteristic
3C2×#16863123C211yesmaximal, characteristic
4C2×#16863118C211yesmaximal, characteristic, complemented
5#22558050C211yesmaximal, characteristic, complemented
6#22558043C211yesmaximal, characteristic, complemented
7C67×#1939C211yesmaximal, characteristic, complemented
8C2×C4×D268C2211yescharacteristic
9C2×C4⋊(C67⋊C8)C2211yescharacteristic
10C2×#14523963C411yescharacteristic
11C2×#14523963C411yescharacteristic
12C2×(C2×D67)⋊C8—21no—
13C2×(C2×D67)⋊C8—21no—
14C2×C4×C67⋊C8C2211yescharacteristic
15C2×C4×C67⋊C8C2211yescharacteristic
16#16863158C411yescharacteristic
17#16863158C411yescharacteristic
18#16863123—22no—
19#16863118—22no—
20C2×C4⋊C8×C67—21no—
21C2×C4⋊C8×C67C2211yescharacteristic
22C2×C4×C8×C67C2211yescharacteristic
23C4×C4⋊C4×C67C2211yescharacteristic
24#336×C67—21no—
25C22×D268C2×C411yescharacteristic
26C22×C4×D67—21no—
27C22×C67⋊C8—21no—
28C22×C67⋊C8—21no—
29C22×C67⋊C8C2×C411yescharacteristic
30C22×C67⋊C8C2×C411yescharacteristic
31C4×D268—21no—
32C4×D268—21no—
33C4×D268—21no—
34C4×D268—21no—
35C2×C4⋊Dic67C2×C411yescharacteristic
36C2×(C2×D67)⋊C4—21no—
37C4⋊(C67⋊C8)—21no—
38C4⋊(C67⋊C8)—21no—
39#14523963—22no—
40#14523963—22no—
41(C2×D67)⋊C8—42no—
42(C2×D67)⋊C8—42no—
43C4×C67⋊C8—21no—
44C4×C67⋊C8—21no—
45C4×C67⋊C8—21no—
46C4×C67⋊C8D412yes—
47C22×C8×C67—21no—
48C22×C8×C67D411yescharacteristic
49C22×C8×C67D411yescharacteristic
50C2×C4⋊C4×C67C2×C411yescharacteristic
51C2×C4⋊C4×C67C2×C411yescharacteristic
52C2×C42×C67—21no—
53C2×C42×C67C2311yescharacteristic
54C4⋊C8×C67—21no—
55C4⋊C8×C67—21no—
56C4⋊C8×C67—42no—
57C4×C8×C67—21no—
58C4×C8×C67—21no—
59#95×C67—21no—
60C23×D67—21no—
61C2×D268—21no—
62C2×D268—21no—
63C2×D268—21no—
64C2×D268—21no—
65C2×D268—21no—
66C2×D268M1612yes—
67C22×Dic67—21no—
68C2×C4×D67—41no—
69C2×C4×D67—41no—
70C2×C4×D67—41no—
71C2×C4×D67—41no—
72C2×C4×D67—41no—
73C2×C4×D67—41no—
74C2×C67⋊C8—22no—
75C2×C67⋊C8—22no—
76C2×C67⋊C8—22no—
77C2×C67⋊C8—21no—
78C2×C67⋊C8—21no—
79C2×C67⋊C8—21no—
80C2×C67⋊C8—21no—
81C2×C67⋊C8—21no—
82C2×C67⋊C8—21no—
83C2×C67⋊C8—41no—
84C2×C67⋊C8—41no—
85C2×C67⋊C8—41no—
86C4⋊Dic67—21no—
87C4⋊Dic67—21no—
88(C2×D67)⋊C4—41no—
89(C2×D67)⋊C4—41no—
90C22×C4×C67—21no—
91C22×C4×C67—21no—
92C22×C4×C67C22×C411yescharacteristic
93C22×C4×C67C2×D411yescharacteristic
94C22×C4×C67#4111yescharacteristic
95C2×C8×C67—21no—
96C2×C8×C67—21no—
97C2×C8×C67—21no—
98C2×C8×C67—21no—
99C2×C8×C67—21no—
100C2×C8×C67—21no—
101C2×C8×C67—41no—
102C2×C8×C67—41no—
103C2×C8×C67—41no—
104C4⋊C4×C67—21no—
105C4⋊C4×C67—21no—
106C4⋊C4×C67—21no—
107C4⋊C4×C67M1612yes—
108C42×C67—41no—
109C42×C67—41no—
110C42×C67C2×D411yescharacteristic
111C42×C67#4111yescharacteristic
112C42×C67#4111yescharacteristic
113C42×C67#4111yescharacteristic
114C23×C67C4×D411yescharacteristic
115C22×D67—22no—
116C22×D67—41no—
117C22×D67—41no—
118C22×D67—41no—
119C22×D67—41no—
120C22×D67—41no—
121C22×D67—41no—
122C2×C4×C67—21no—
123C2×C4×C67—21no—
124C2×C4×C67—22no—
125C2×C4×C67—41no—
126C2×C4×C67—41no—
127C2×C4×C67—41no—
128C2×C4×C67—41no—
129C2×C4×C67—41no—
130C2×C4×C67#11711yescharacteristic
131C2×C4×C67C4×D411yescharacteristic
132C2×C4×C67#12111yescharacteristic
133C2×C4×C67#12311yescharacteristic
134C2×C4×C67#12411yescharacteristic
135C2×C4×C67#12611yescharacteristic
136C2×C4×C67C2×M1611yescharacteristic
137C2×C4×C67C2×M1611yescharacteristic
138C2×C4×C67#13111yescharacteristic
139C2×C4×C67#13111yescharacteristic
140C2×C4×C67AGL(1,ℤ/8)11yescharacteristic
141C2×C4×C67AGL(1,ℤ/8)11yescharacteristic
142C2×C4×C67AGL(1,ℤ/8)11yescharacteristic
143C2×C4×C67#13711yescharacteristic
144C2×Dic67—41no—
145C2×Dic67—41no—
146C2×Dic67—41no—
147D268—22no—
148D268—22no—
149D268—41no—
150D268—41no—
151C4×D67—81no—
152C4×D67—81no—
153C4×D67—81no—
154C4×D67—81no—
155C8×C67—41no—
156C8×C67—41no—
157C8×C67—81no—
158C67⋊C8—24no—
159C67⋊C8—41no—
160C67⋊C8—41no—
161C67⋊C8—81no—
162C22×C67#38811yescharacteristic
163C22×C67#43511yescharacteristic
164C22×C67#43511yescharacteristic
165C22×C67#44011yescharacteristic
166C22×C67#44211yescharacteristic
167C22×C67#44311yescharacteristic
168C22×C67#44311yescharacteristic
169C2×D67—42no—
170C2×D67—42no—
171C2×D67—42no—
172C2×D67—81no—
173C2×D67—81no—
174C2×D67—81no—
175C2×D67—81no—
176C4×C67—21no—
177C4×C67—21no—
178C4×C67—42no—
179C4×C67—81no—
180C4×C67#43211yescharacteristic
181C4×C67#43311yescharacteristic
182C4×C67#43511yescharacteristic
183C4×C67#43611yescharacteristic
184C4×C67#46811yescharacteristic
185C4×C67#46911yescharacteristic
186C4×C67#48911yescharacteristic
187C4×C67#49011yescharacteristic
188Dic67—81no—
189#9560—671nomaximal, Sylow
190C2×C67#158511yescharacteristic
191C2×C67#158611yescharacteristic
192C2×C67#158611yescharacteristic
193C2×C67#158811yescharacteristic
194C2×C67#192411yescharacteristic
195C2×C67#193611yescharacteristic
196C2×C67#193711yescharacteristic
197D67—82no—
198#1954—671no—
199#1939D6711yescharacteristic, complemented
200C4×#102—671no—
201C2×D4⋊C8—671no—
202C2×#322—671no—
203#1279—671no—
204#1277—671no—
205C67#956011yescharacteristic, complemented, Sylow
206C2×C4×D4—671no—
207C2×C4⋊C8—21no—
208C2×C4⋊C8—671no—
209C2×C4⋊C8C2×D6711yescharacteristic
210C2×#102—671no—
211C2×#102—671no—
212C2×C22⋊C8—1341no—
213C2×C22⋊C8—1341no—
214C2×C4×C8—671no—
215C2×C4×C8—671no—
216C2×C4×C8C2×D6711yescharacteristic
217C4×C4⋊C4C2×D6711yescharacteristic
218#340—671no—
219#340—671no—
220#336—21no—
221D4⋊C8—1342no—
222#322—1342no—
223C22×D4—671no—
224C23×C4—1341no—
225C22×C8—21no—
226C22×C8—671no—
227C22×C8—671no—
228C22×C8—1341no—
229C22×C8—1341no—
230C22×C8C22⋊D6711yescharacteristic
231C22×C8C22⋊D6711yescharacteristic
232C4×D4—1341no—
233C4×D4—1341no—
234C4×D4—1341no—
235C4×D4—1341no—
236C2×C4⋊C4—671no—
237C2×C4⋊C4C4×D6711yescharacteristic
238C2×C4⋊C4C4×D6711yescharacteristic
239C2×C22⋊C4—1341no—
240C2×C42—21no—
241C2×C42C22×D6711yescharacteristic
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—42no—
245C4⋊C8—1341no—
246C4⋊C8—1341no—
247#102—1342no—
248#102—1342no—
249C22⋊C8—2682no—
250C22⋊C8—2682no—
251C4×C8—21no—
252C4×C8—21no—
253C4×C8—672no—
254C4×C8—1341no—
255C4×C8—1341no—
256C4×C8—1341no—
257#95—21no—
258C24—1341no—
259C2×D4—672no—
260C2×D4—1341no—
261C2×D4—1341no—
262C2×D4—1341no—
263C2×D4—1341no—
264C2×D4—1341no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—2681no—
268C22×C4—2681no—
269C22×C4—2681no—
270C22×C4—2681no—
271C22×C4—2681no—
272C22×C4—2681no—
273C22×C4—1341no—
274C22×C4C2×C4×D6711yescharacteristic
275C22×C4C2×C22⋊D6711yescharacteristic
276C22×C4#1176162811yescharacteristic
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—2681no—
287C2×C8—2681no—
288C2×C8—2681no—
289C2×C8—1341no—
290C2×C8—1341no—
291C2×C8—1341no—
292C2×C8—1341no—
293C2×C8—1341no—
294C2×C8—1341no—
295C2×C8—1342no—
296C2×C8—1342no—
297C2×C8—1342no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—1341no—
302C4⋊C4—1341no—
303C4⋊C4#1176161112yes—
304C22⋊C4—2681no—
305C22⋊C4—2681no—
306C42—41no—
307C42—41no—
308C42D4×D6711yescharacteristic
309C42#1176162711yescharacteristic
310C42#1176162911yescharacteristic
311C42#1176162911yescharacteristic
312C23—2681no—
313C23—2681no—
314C23—2681no—
315C23—2681no—
316C23—2681no—
317C23—2681no—
318C23—1342no—
319C23C4×C22⋊D6711yescharacteristic
320D4—2681no—
321D4—2681no—
322D4—1342no—
323D4—1342no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—22no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—2681no—
333C2×C4—2681no—
334C2×C4—2681no—
335C2×C4—5361no—
336C2×C4—5361no—
337C2×C4—5361no—
338C2×C4—5361no—
339C2×C4#1452400111yescharacteristic
340C2×C4#1452400511yescharacteristic
341C2×C4#1452401511yescharacteristic
342C2×C4#1452402811yescharacteristic
343C2×C4#1452403211yescharacteristic
344C2×C4#1452403811yescharacteristic
345C2×C4M16×D6711yescharacteristic
346C2×C4C2×#1176161111yescharacteristic
347C2×C4#1452404911yescharacteristic
348C2×C4#1452405111yescharacteristic
349C2×C4#1452406811yescharacteristic
350C2×C4#1452406811yescharacteristic
351C2×C4#1452406911yescharacteristic
352C2×C4#1452407411yescharacteristic
353C8—41no—
354C8—41no—
355C8—81no—
356C8—2681no—
357C8—2681no—
358C8—5361no—
359C8—1344no—
360C22—2682no—
361C22—2682no—
362C22—2682no—
363C22—5361no—
364C22—5361no—
365C22—5361no—
366C22—5361no—
367C22#1686328611yescharacteristic
368C22#1686346511yescharacteristic
369C22#1686346711yescharacteristic
370C22#1686349411yescharacteristic
371C22#1686350611yescharacteristic
372C22#1686351111yescharacteristic
373C22#1686351111yescharacteristic
374C4—21no—
375C4—21no—
376C4—42no—
377C4—81no—
378C4—5361no—
379C4#1686345211yescharacteristic
380C4#1686345611yescharacteristic
381C4#1686346611yescharacteristic
382C4#1686347311yescharacteristic
383C4#1686364411yescharacteristic
384C4#1686364711yescharacteristic
385C4#1686375811yescharacteristic
386C4#1686376211yescharacteristic
387C2—5362no—
388C2#2255906411yescharacteristic
389C2#2255907111yescharacteristic
390C2#2255907311yescharacteristic
391C2#2255908711yescharacteristic
392C2#2256082111yescharacteristic
393C2#2256090511yescharacteristic
394C2#2256091211yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊D67
commutatorC22×C4×C67C22×C4
FrattiniC2×C42C22×D67
FittingC67×#1939C2
socleC23×C67C4×D4
prime radicalC22×D268C2×C4
cosocleC2×C42×C67C23
hypercenter#1939D67
hypocenterC67#9560
LCSG ▷ C22×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22×C4 ◁ C2×C4×C8 ◁ #1939

Complements

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

NH ≅ G/Ntype
4. C2×#16863118C2semidirect
5. #22558050C2semidirect
6. #22558043C2semidirect
7. C67×#1939C2semidirect
199. #1939D67semidirect
205. C67#9560semidirect

Maximal subgroups

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

subgroupindex
#225610032
C4×#145239632
C2×#168631232
C2×#168631182
#225580502
#225580432
C67×#19392
#956067