Finite Group Databasebeta

← #30826502

Subgroups — #30826502

Order 17152.

600 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22570768C211yesmaximal, characteristic
2#22569303C211yesmaximal, characteristic, complemented
3#22569284C211yesmaximal, characteristic
4#22559113C212yesmaximal, complemented
5#22559101C212yesmaximal, complemented
6#22559023C212yesmaximal, complemented
7C8×#11761617C212yesmaximal, complemented
8#22558972C211yesmaximal, characteristic, complemented
9#22558683C211yesmaximal, characteristic, complemented
10#22558679C211yesmaximal, characteristic
11C67×#1450C211yesmaximal, characteristic, complemented
12#16863953C2211yescharacteristic
13#16863946C2211yescharacteristic
14#16863794C411yescharacteristic
15#16863769C411yescharacteristic
16#16863747C411yescharacteristic
17#16863743C411yescharacteristic
18#16863625C412yes—
19#16863625C412yes—
20#16863526C2212yes—
21#16863525—22no—
22C8×Dic134C2212yes—
23C4×#11761617C2212yes—
24C4×#11761617C2212yes—
25#16863475C2212yes—
26#16863466—22no—
27C8×D268C2212yes—
28#16863454—22no—
29#16863453C2211yescharacteristic
30#16863453C2211yescharacteristic
31#16863361—22no—
32#16863149C2212yes—
33#16863147C2212yes—
34#16863128C2212yes—
35#16863128C2212yes—
36#16863125C2212yes—
37#16863125C2212yes—
38#431×C67C2211yescharacteristic
39#431×C67C2211yescharacteristic
40C2×C4×C8×C67C2211yescharacteristic
41#322×C67C2211yescharacteristic
42#322×C67C2211yescharacteristic
43C82×C67C2211yescharacteristic
44C82×C67C2211yescharacteristic
45C2×#11761628—21no—
46C2×#11761617C2×C412yes—
47C2×#11761617C2×C412yes—
48C2×#11761611—22no—
49C2×C8×D67—22no—
50#14524043C2×C411yescharacteristic
51#14524043C2×C411yescharacteristic
52#14524043C2×C411yescharacteristic
53#14524040C2×C411yescharacteristic
54#14524035—21no—
55#14524031—21no—
56#14524022—21no—
57#14524022C2×C412yes—
58#14524017—21no—
59#14524017C2×C412yes—
60C4×Dic134—21no—
61C4×Dic134C2312yes—
62C4×D268—21no—
63C4×D268C2312yes—
64#14524002—21no—
65#14523975C2×C412yes—
66#14523975C2×C412yes—
67C4⋊(C67⋊C8)C2312yes—
68C4⋊(C67⋊C8)C2312yes—
69Dic67⋊C8—22no—
70Dic67⋊C8—22no—
71#14523968C2×C412yes—
72#14523968C2×C412yes—
73#14523965C2×C412yes—
74#14523965C2×C412yes—
75C22⋊(C67⋊C8)—21no—
76C22⋊(C67⋊C8)—21no—
77(C2×D67)⋊C8—22no—
78(C2×D67)⋊C8—22no—
79#14523954—21no—
80#14523954—21no—
81#14523953—22no—
82C8×Dic67—22no—
83C22×C8×C67C2×C411yescharacteristic
84C22×C8×C67C2×C411yescharacteristic
85C2×C42×C67C2311yescharacteristic
86C4⋊C8×C67—21no—
87C4⋊C8×C67D412yes—
88C22⋊C8×C67—21no—
89C22⋊C8×C67D412yes—
90C4×C8×C67C2×C411yescharacteristic
91C4×C8×C67C2×C411yescharacteristic
92C4×C8×C67C2311yescharacteristic
93C4×C8×C67C2311yescharacteristic
94C4×C8×C67C2311yescharacteristic
95C4×C8×C67C2311yescharacteristic
96C4×C8×C67C2311yescharacteristic
97C4×C8×C67C2311yescharacteristic
98#11761628—82no—
99C2×Dic134—21no—
100C2×Dic134—21no—
101C2×Dic134C22×C412yes—
102C2×D268—21no—
103C2×D268—21no—
104C2×D268C22×C412yes—
105C2×C22⋊D67—22no—
106C2×C4×D67—22no—
107#11761617—24no—
108#11761617—24no—
109#11761611—82no—
110C2×C67⋊C8—22no—
111C2×C67⋊C8—22no—
112C8×D67—82no—
113C4⋊Dic67—21no—
114C4⋊Dic67—21no—
115C4⋊Dic67—21no—
116C4⋊Dic67—21no—
117C4⋊Dic67C22×C412yes—
118C4⋊Dic67C22×C412yes—
119Dic67⋊C4—22no—
120Dic67⋊C4—22no—
121C22⋊Dic67—22no—
122(C2×D67)⋊C4—22no—
123(C2×D67)⋊C4—22no—
124C4×Dic67—22no—
125C22×C4×C67C22×C411yescharacteristic
126C22×C4×C67C2×D411yescharacteristic
127C22×C4×C67#4111yescharacteristic
128C2×C8×C67—22no—
129C2×C8×C67—22no—
130C2×C8×C67—21no—
131C2×C8×C67—21no—
132C2×C8×C67C22×C411yescharacteristic
133C2×C8×C67C22×C411yescharacteristic
134C2×C8×C67C22×C411yescharacteristic
135C2×C8×C67C22×C411yescharacteristic
136C2×C8×C67C2×D412yes—
137C2×C8×C67C2×D412yes—
138C42×C67C22×C411yescharacteristic
139C42×C67C2×D411yescharacteristic
140C42×C67#4111yescharacteristic
141C42×C67C2411yescharacteristic
142C23×C67C4×D411yescharacteristic
143C22×D67—22no—
144C2×C4×C67—21no—
145C2×C4×C67—21no—
146C2×C4×C67—22no—
147C2×C4×C67—41no—
148C2×C4×C67C4×D411yescharacteristic
149C2×C4×C67C4×D411yescharacteristic
150C2×C4×C67C4×D411yescharacteristic
151C2×C4×C67C23×C411yescharacteristic
152C2×C4×C67C22×D411yescharacteristic
153C2×C4×C67C2×#4111yescharacteristic
154C22⋊D67—82no—
155C2×Dic67—22no—
156C2×Dic67—22no—
157C2×Dic67—22no—
158D268—22no—
159D268—22no—
160D268—81no—
161D268—81no—
162C4×D67—42no—
163C4×D67—42no—
164Dic134—22no—
165Dic134—22no—
166Dic134—81no—
167Dic134—81no—
168C8×C67—21no—
169C8×C67—21no—
170C8×C67—22no—
171C8×C67—22no—
172C8×C67—41no—
173C8×C67—41no—
174C8×C67#13112yes—
175C8×C67#13112yes—
176C67⋊C8—42no—
177C67⋊C8—42no—
178C22×C67—21no—
179C22×C67—21no—
180C22×C67—21no—
181C22×C67C2×C4×D411yescharacteristic
182C2×D67—42no—
183C2×D67—42no—
184C2×D67—42no—
185C4×C67—21no—
186C4×C67—21no—
187C4×C67—21no—
188C4×C67—21no—
189C4×C67C2×#13111yescharacteristic
190C4×C67C2×#13111yescharacteristic
191C4×C67#57712yes—
192Dic67—42no—
193Dic67—42no—
194Dic67—42no—
195#17971—671nomaximal, Sylow
196C2×C67—41no—
197C2×C67#293111yescharacteristic
198C2×C67#294911yescharacteristic
199C2×C67#295711yescharacteristic
200D67—82no—
201#3167—671no—
202#2985—671no—
203#2981—671no—
204#1592—672no—
205#1590—672no—
206SD16⋊C8—672no—
207C8×SD16—672no—
208#1568—671no—
209#1497—671no—
210#1496—671no—
211#1450D6711yescharacteristic, complemented
212C67#1797111yescharacteristic, complemented, Sylow
213#533—671no—
214#532—671no—
215#498—671no—
216#492—671no—
217#487—671no—
218#486—671no—
219#465—672no—
220#465—672no—
221#446—672no—
222#446—1342no—
223C8×Q8—672no—
224C4×SD16—672no—
225C4×SD16—672no—
226#436—672no—
227#435—1342no—
228C8×D4—672no—
229#433—1342no—
230#432—671no—
231#432—671no—
232#431C2×D6711yescharacteristic
233#431C2×D6711yescharacteristic
234#408—1342no—
235C2×C4×C8C2×D6711yescharacteristic
236#335—672no—
237#334—672no—
238Q8⋊C8—672no—
239Q8⋊C8—672no—
240D4⋊C8—672no—
241D4⋊C8—672no—
242#322C2×D6711yescharacteristic
243#322C2×D6711yescharacteristic
244C82C2×D6711yescharacteristic
245C82C2×D6711yescharacteristic
246C2×#41—1341no—
247C2×SD16—672no—
248C2×SD16—672no—
249C2×M16—1342no—
250C22×C8—1342no—
251C22×C8C4×D6711yescharacteristic
252C22×C8C4×D6711yescharacteristic
253C4⋊Q8—671no—
254C4⋊Q8—671no—
255C4⋊Q8—671no—
256#127—671no—
257#125—1341no—
258#124—1341no—
259C22⋊Q8—672no—
260C22⋊Q8—1341no—
261#121—672no—
262#121—1341no—
263C4×Q8—672no—
264C4×Q8—1341no—
265C4×D4—672no—
266C4×D4—1341no—
267#117—1341no—
268C2×C42C22×D6711yescharacteristic
269#106—672no—
270#106—672no—
271C4⋊C8—21no—
272C4⋊C8—672no—
273C4⋊C8—672no—
274C4⋊C8—1342no—
275C4⋊C8—1342no—
276C4⋊C8D26812yes—
277#103—672no—
278#103—672no—
279#102—672no—
280#102—672no—
281C22⋊C8—21no—
282C22⋊C8—1341no—
283C22⋊C8—1341no—
284C22⋊C8—1342no—
285C22⋊C8—1342no—
286C22⋊C8D26812yes—
287#97—1342no—
288#97—1341no—
289#97—1341no—
290C4×C8—1342no—
291C4×C8C4×D6711yescharacteristic
292C4×C8C4×D6711yescharacteristic
293C4×C8C22×D6711yescharacteristic
294C4×C8C22×D6711yescharacteristic
295C4×C8C22×D6711yescharacteristic
296C4×C8C22×D6711yescharacteristic
297C4×C8C22×D6711yescharacteristic
298C4×C8C22×D6711yescharacteristic
299#41—5362no—
300C2×Q8—672no—
301C2×Q8—1341no—
302C2×Q8—1341no—
303C2×D4—672no—
304C2×D4—1342no—
305C2×D4—1341no—
306C2×D4—1341no—
307C22×C4—1342no—
308C22×C4C2×C4×D6711yescharacteristic
309C22×C4C2×D26811yescharacteristic
310C22×C4#1176162811yescharacteristic
311SD16—1344no—
312SD16—1344no—
313M16—5362no—
314C2×C8—22no—
315C2×C8—22no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—5362no—
319C2×C8—1342no—
320C2×C8—1342no—
321C2×C8C2×C4×D6711yescharacteristic
322C2×C8C2×C4×D6711yescharacteristic
323C2×C8C2×C4×D6711yescharacteristic
324C2×C8C2×C4×D6711yescharacteristic
325C2×C8C2×D26812yes—
326C2×C8C2×D26812yes—
327C4⋊C4—672no—
328C4⋊C4—672no—
329C4⋊C4—1341no—
330C4⋊C4—1341no—
331C4⋊C4—1341no—
332C4⋊C4—1342no—
333C4⋊C4—1341no—
334C4⋊C4—1342no—
335C22⋊C4—1342no—
336C22⋊C4—1342no—
337C22⋊C4—1342no—
338C42—1342no—
339C42C2×C4×D6711yescharacteristic
340C42C2×D26811yescharacteristic
341C42#1176162811yescharacteristic
342C42C23×D6711yescharacteristic
343C23—1342no—
344C23C4×D26811yescharacteristic
345Q8—5361no—
346Q8—5361no—
347Q8—1342no—
348Q8—1342no—
349D4—5361no—
350D4—5361no—
351D4—5362no—
352D4—1342no—
353D4—1342no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—22no—
357C2×C4—41no—
358C2×C4—2682no—
359C2×C4—2682no—
360C2×C4—1342no—
361C2×C4—1342no—
362C2×C4—1342no—
363C2×C4C4×D26811yescharacteristic
364C2×C4C4×D26811yescharacteristic
365C2×C4C4×D26811yescharacteristic
366C2×C4C22×C4×D6711yescharacteristic
367C2×C4C22×D26811yescharacteristic
368C2×C4C2×#1176162811yescharacteristic
369C8—21no—
370C8—21no—
371C8—22no—
372C8—22no—
373C8—41no—
374C8—41no—
375C8—2682no—
376C8—2682no—
377C8#1452404912yes—
378C8#1452404912yes—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—2682no—
383C22—2682no—
384C22—2682no—
385C22C2×C4×D26811yescharacteristic
386C4—21no—
387C4—21no—
388C4—21no—
389C4—21no—
390C4—2682no—
391C4—2682no—
392C4—2682no—
393C4C2×#1452404911yescharacteristic
394C4C2×#1452404911yescharacteristic
395C4#1686420812yes—
396C2—41no—
397C2—5362no—
398C2#2256892211yescharacteristic
399C2#2256907011yescharacteristic
400C2#2256912711yescharacteristic
subgroupG/H
centerC2×C4C22×D268
commutatorC2×C4×C67C23×C4
FrattiniC42C23×D67
FittingC67×#1450C2
socleC22×C67C2×C4×D4
prime radical#16863946C22
cosocleC42×C67C24
hypercenter#1450D67
hypocenterC67#17971
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2×C4 ◁ #431 ◁ #1450

Complements

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

NH ≅ G/Ntype
2. #22569303C2semidirect
4. #22559113C2semidirect
5. #22559101C2semidirect
6. #22559023C2semidirect
7. C8×#11761617C2semidirect
8. #22558972C2semidirect
9. #22558683C2semidirect
11. C67×#1450C2semidirect
211. #1450D67semidirect
212. C67#17971semidirect

Maximal subgroups

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

subgroupindex
#225707682
#225693032
#225692842
#225591132
#225591012
#225590232
C8×#117616172
#225589722
#225586832
#225586792
C67×#14502
#1797167