Finite Group Databasebeta

← #30760347

Subgroups — #30760347

Order 17152.

454 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22561136C211yesmaximal, characteristic
2#22561015C211yesmaximal, characteristic
3#22560967C211yesmaximal, characteristic, complemented
4C2×#16863147C211yesmaximal, characteristic, complemented
5C2×#16863139C211yesmaximal, characteristic
6#22558093C211yesmaximal, characteristic, complemented
7C67×#1939C211yesmaximal, characteristic, complemented
8C2×#14524040C2211yescharacteristic
9C2×#14523976—21no—
10C2×C4⋊(C67⋊C8)C2211yescharacteristic
11C2×C4⋊(C67⋊C8)C2211yescharacteristic
12C2×#14523965—21no—
13C2×#14523965—21no—
14C2×#14523963—21no—
15C2×#14523963—21no—
16#16863266C2211yescharacteristic
17#16863158—21no—
18#16863147—22no—
19#16863139—22no—
20C2×C4⋊C8×C67—21no—
21C2×C4⋊C8×C67C2211yescharacteristic
22C2×C4×C8×C67C2211yescharacteristic
23C4×C4⋊C4×C67C2211yescharacteristic
24#336×C67—21no—
25C22×D268—21no—
26C22×D268—21no—
27C22×D268—21no—
28C22×C67⋊C8—21no—
29C22×C67⋊C8—21no—
30#14524040—21no—
31#14524040—21no—
32#14524040—21no—
33#14524040Q812yes—
34C2×C4⋊Dic67—21no—
35C2×C4⋊Dic67—21no—
36C2×C4⋊Dic67—21no—
37#14523976—42no—
38C4⋊(C67⋊C8)—21no—
39C4⋊(C67⋊C8)—21no—
40C4⋊(C67⋊C8)—21no—
41C4⋊(C67⋊C8)D412yes—
42#14523965—42no—
43#14523965—42no—
44#14523963—42no—
45#14523963—42no—
46C22×C8×C67—21no—
47C22×C8×C67D411yescharacteristic
48C22×C8×C67D411yescharacteristic
49C2×C4⋊C4×C67D411yescharacteristic
50C2×C4⋊C4×C67D411yescharacteristic
51C2×C42×C67—21no—
52C2×C42×C67C2311yescharacteristic
53C4⋊C8×C67—21no—
54C4⋊C8×C67—21no—
55C4⋊C8×C67—42no—
56C4×C8×C67—21no—
57C4×C8×C67—21no—
58#95×C67—21no—
59C23×D67—41no—
60C2×D268—22no—
61C2×D268—22no—
62C2×D268—22no—
63C2×D268—41no—
64C2×D268—41no—
65C2×D268—41no—
66C2×D268—41no—
67C2×D268—41no—
68C2×D268—41no—
69C2×D268—41no—
70C2×D268—41no—
71C2×D268—41no—
72C2×D268—41no—
73C2×D268—41no—
74C2×D268—81no—
75C2×D268—81no—
76C22×Dic67—41no—
77C2×C67⋊C8—22no—
78C2×C67⋊C8—41no—
79C2×C67⋊C8—41no—
80C2×C67⋊C8—41no—
81C2×C67⋊C8—41no—
82C2×C67⋊C8—41no—
83C4⋊Dic67—41no—
84C4⋊Dic67—41no—
85C4⋊Dic67—41no—
86C4⋊Dic67—41no—
87C4⋊Dic67—81no—
88C22×C4×C67—21no—
89C22×C4×C67—21no—
90C22×C4×C67C2×D411yescharacteristic
91C22×C4×C67C2×D411yescharacteristic
92C22×C4×C67#4111yescharacteristic
93C2×C8×C67—21no—
94C2×C8×C67—21no—
95C2×C8×C67—21no—
96C2×C8×C67—21no—
97C2×C8×C67—21no—
98C2×C8×C67—41no—
99C2×C8×C67—41no—
100C2×C8×C67—41no—
101C2×C8×C67D812yes—
102C4⋊C4×C67—21no—
103C4⋊C4×C67—21no—
104C4⋊C4×C67—21no—
105C4⋊C4×C67SD1612yes—
106C42×C67—41no—
107C42×C67—41no—
108C42×C67C2×D411yescharacteristic
109C42×C67C2×Q811yescharacteristic
110C42×C67#4111yescharacteristic
111C42×C67#4111yescharacteristic
112C23×C67#12111yescharacteristic
113C22×D67—42no—
114C22×D67—81no—
115C22×D67—81no—
116C22×D67—81no—
117C22×D67—81no—
118C22×D67—81no—
119C22×D67—81no—
120C2×C4×C67—21no—
121C2×C4×C67—21no—
122C2×C4×C67—22no—
123C2×C4×C67—41no—
124C2×C4×C67—41no—
125C2×C4×C67—41no—
126C2×C4×C67—41no—
127C2×C4×C67—41no—
128C2×C4×C67#12111yescharacteristic
129C2×C4×C67#12111yescharacteristic
130C2×C4×C67C22⋊Q811yescharacteristic
131C2×C4×C67C22⋊Q811yescharacteristic
132C2×C4×C67C22⋊Q811yescharacteristic
133C2×C4×C67#12611yescharacteristic
134C2×C4×C67C2×D811yescharacteristic
135C2×C4×C67C2×SD1611yescharacteristic
136C2×C4×C67#13511yescharacteristic
137C2×C4×C67#13511yescharacteristic
138C2×C4×C67AGL(1,ℤ/8)11yescharacteristic
139C2×C4×C67AGL(1,ℤ/8)11yescharacteristic
140C2×C4×C67AGL(1,ℤ/8)11yescharacteristic
141C2×C4×C67#13711yescharacteristic
142C2×Dic67—81no—
143C2×Dic67—81no—
144C2×Dic67—81no—
145D268—42no—
146D268—42no—
147D268—42no—
148D268—42no—
149D268—82no—
150D268—82no—
151D268—81no—
152D268—81no—
153D268—81no—
154D268—81no—
155C8×C67—22no—
156C8×C67—41no—
157C8×C67—81no—
158C67⋊C8—42no—
159C67⋊C8—81no—
160C22×C67#39611yescharacteristic
161C22×C67#45911yescharacteristic
162C22×C67#46411yescharacteristic
163C22×C67#46511yescharacteristic
164C22×C67#46911yescharacteristic
165C22×C67#47111yescharacteristic
166C22×C67#47211yescharacteristic
167C2×D67—82no—
168C2×D67—82no—
169C2×D67—82no—
170C2×D67—161no—
171C2×D67—161no—
172C2×D67—161no—
173C2×D67—161no—
174C4×C67—21no—
175C4×C67—21no—
176C4×C67—42no—
177C4×C67—81no—
178C4×C67#46011yescharacteristic
179C4×C67#46311yescharacteristic
180C4×C67#46611yescharacteristic
181C4×C67#46811yescharacteristic
182C4×C67#47411yescharacteristic
183C4×C67#47611yescharacteristic
184C4×C67#47811yescharacteristic
185C4×C67#47811yescharacteristic
186Dic67—161no—
187#10587—671nomaximal, Sylow
188C2×C67#166511yescharacteristic
189C2×C67#167811yescharacteristic
190C2×C67#168711yescharacteristic
191C2×C67#170211yescharacteristic
192C2×C67#206311yescharacteristic
193C2×C67#206511yescharacteristic
194C2×C67#206711yescharacteristic
195D67—162no—
196#1982—671no—
197#1957—671no—
198#1947—671no—
199#1939D6711yescharacteristic, complemented
200C2×#334—671no—
201C2×#331—671no—
202#1296—671no—
203C67#1058711yescharacteristic, complemented, Sylow
204C2×#127—671no—
205C2×#107—1341no—
206C2×C4⋊C8—21no—
207C2×C4⋊C8—671no—
208C2×C4⋊C8—671no—
209C2×C4⋊C8C2×D6711yescharacteristic
210C2×#102—1341no—
211C2×#102—1341no—
212C2×#102—1341no—
213C2×#102—1341no—
214C2×C4×C8C2×D6711yescharacteristic
215#384—671no—
216C4×C4⋊C4C2×D6711yescharacteristic
217#340—1341no—
218#336—21no—
219#334—1342no—
220#331—1342no—
221C22×D4—1341no—
222C22×D4—1341no—
223C22×D4—1341no—
224C22×C8—21no—
225C22×C8—1341no—
226C22×C8—1341no—
227C22×C8C22⋊D6711yescharacteristic
228C22×C8C22⋊D6711yescharacteristic
229#127—672no—
230#127—1341no—
231#127—1341no—
232#127—1341no—
233C2×C4⋊C4—1341no—
234C2×C4⋊C4—1341no—
235C2×C4⋊C4—1341no—
236C2×C4⋊C4D26811yescharacteristic
237C2×C4⋊C4D26811yescharacteristic
238C2×C42—21no—
239C2×C42C22×D6711yescharacteristic
240#107—2682no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—42no—
244C4⋊C8—672no—
245C4⋊C8—1341no—
246C4⋊C8—1341no—
247C4⋊C8—1341no—
248#102—2682no—
249#102—2682no—
250#102—2682no—
251#102—2682no—
252C4×C8—21no—
253C4×C8—21no—
254#95—21no—
255C24—2681no—
256C2×D4—2681no—
257C2×D4—2681no—
258C2×D4—2681no—
259C2×D4—2681no—
260C2×D4—2681no—
261C2×D4—2681no—
262C2×D4—2681no—
263C2×D4—2681no—
264C2×D4—2681no—
265C2×D4—2681no—
266C2×D4—2681no—
267C2×D4—5361no—
268C2×D4—5361no—
269C2×D4—1342no—
270C2×D4—1342no—
271C2×D4—1342no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—2681no—
275C22×C4C2×C22⋊D6711yescharacteristic
276C22×C4C2×D26811yescharacteristic
277C22×C4#1176162811yescharacteristic
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—2681no—
290C2×C8—2681no—
291C2×C8—1342no—
292C2×C8#1176161312yes—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—2681no—
297C4⋊C4—2681no—
298C4⋊C4—2681no—
299C4⋊C4—2681no—
300C4⋊C4—5361no—
301C4⋊C4#1176161712yes—
302C42—41no—
303C42—41no—
304C42D4×D6711yescharacteristic
305C42Q8×D6711yescharacteristic
306C42#1176162711yescharacteristic
307C42#1176162911yescharacteristic
308C23—2682no—
309C23—5361no—
310C23—5361no—
311C23—5361no—
312C23—5361no—
313C23—5361no—
314C23—5361no—
315C23#1452401711yescharacteristic
316D4—2682no—
317D4—2682no—
318D4—2682no—
319D4—2682no—
320D4—5362no—
321D4—5362no—
322D4—5361no—
323D4—5361no—
324D4—5361no—
325D4—5361no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—22no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—5361no—
335C2×C4—5361no—
336C2×C4—5361no—
337C2×C4#1452401511yescharacteristic
338C2×C4#1452401811yescharacteristic
339C2×C4#1452401911yescharacteristic
340C2×C4#1452402111yescharacteristic
341C2×C4#1452402311yescharacteristic
342C2×C4#1452403811yescharacteristic
343C2×C4C2×#1176161311yescharacteristic
344C2×C4C2×#1176161711yescharacteristic
345C2×C4#1452406311yescharacteristic
346C2×C4#1452406511yescharacteristic
347C2×C4#1452406811yescharacteristic
348C2×C4#1452407111yescharacteristic
349C2×C4#1452407111yescharacteristic
350C2×C4#1452407411yescharacteristic
351C8—22no—
352C8—41no—
353C8—81no—
354C8—2682no—
355C8—5361no—
356C22—5362no—
357C22—5362no—
358C22—5362no—
359C22—10721no—
360C22—10721no—
361C22—10721no—
362C22—10721no—
363C22#1686331811yescharacteristic
364C22#1686359111yescharacteristic
365C22#1686361811yescharacteristic
366C22#1686362511yescharacteristic
367C22#1686364911yescharacteristic
368C22#1686365911yescharacteristic
369C22#1686366611yescharacteristic
370C4—21no—
371C4—21no—
372C4—42no—
373C4—81no—
374C4—10721no—
375C4#1686359711yescharacteristic
376C4#1686361211yescharacteristic
377C4#1686363011yescharacteristic
378C4#1686364411yescharacteristic
379C4#1686367311yescharacteristic
380C4#1686368711yescharacteristic
381C4#1686369311yescharacteristic
382C4#1686369711yescharacteristic
383C2—10722no—
384C2#2255955011yescharacteristic
385C2#2255962511yescharacteristic
386C2#2255968811yescharacteristic
387C2#2255977711yescharacteristic
388C2#2256156811yescharacteristic
389C2#2256158211yescharacteristic
390C2#2256159411yescharacteristic
subgroupG/H
centerC23#14524017
commutator, cosocleC2×C42×C67C23
FrattiniC2×C42C22×D67
FittingC67×#1939C2
socleC23×C67#121
prime radicalC2×#14524040C22
hypercenter#1939D67
hypocenterC67#10587
LCSG ▷ C2×C42×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1939

Complements

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

NH ≅ G/Ntype
3. #22560967C2semidirect
4. C2×#16863147C2semidirect
6. #22558093C2semidirect
7. C67×#1939C2semidirect
199. #1939D67semidirect
203. C67#10587semidirect

Maximal subgroups

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

subgroupindex
#225611362
#225610152
#225609672
C2×#168631472
C2×#168631392
#225580932
C67×#19392
#1058767