Finite Group Databasebeta

← #30869339

Subgroups — #30869339

Order 17152.

714 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22570534C211yesmaximal, characteristic
2#22569144C211yesmaximal, characteristic, complemented
3#22568920C211yesmaximal, characteristic
4#22561037C211yesmaximal, characteristic
5#22560855C214yesmaximal, complemented
6#22560485C214yesmaximal, complemented
7#22560380C211yesmaximal, characteristic, complemented
8#22559955C211yesmaximal, characteristic, complemented
9C67×#1994C211yesmaximal, characteristic, complemented
10C2×#14524051C2212yes—
11#16863946C2211yescharacteristic
12#16863753C414yescomplemented
13#16863748C411yescharacteristic, complemented
14#16863748C411yescharacteristic, complemented
15#16863744C411yescharacteristic, complemented
16#16863744C411yescharacteristic, complemented
17#16863527—21no—
18#16863527—21no—
19Q8×C67⋊C8—21no—
20Q8×C67⋊C8—21no—
21#16863473—21no—
22#16863473—21no—
23#16863461—21no—
24#16863461—21no—
25C2×#14523978C2214yes—
26#16863423C2212yes—
27#16863420C2211yescharacteristic
28C2×#14523971C2214yes—
29#16863390C2214yes—
30#16863364C2212yes—
31#16863352C2212yes—
32C2×C4×C67⋊C8C2211yescharacteristic
33C2×C4×C67⋊C8C2211yescharacteristic
34#16863151C2212yes—
35#16863151C2212yes—
36#533×C67C2211yescharacteristic
37#423×C67C2211yescharacteristic
38C43×C67C2211yescharacteristic
39#339×C67C2214yes—
40C2×#11761628C2312yes—
41#14524051—44no—
42C2×#11761612C2312yes—
43C2×#11761611—22no—
44C22×C67⋊C8C2312yes—
45C22×C67⋊C8C2311yescharacteristic
46C22×C67⋊C8C2311yescharacteristic
47C2×C8×D67—22no—
48#14524043C2×C411yescharacteristic
49#14524040C2×C411yescharacteristic
50#14524031C2×C412yes—
51#14524017—21no—
52#14524017—21no—
53C4×D268—21no—
54C4×D268—21no—
55#14523978—44no—
56C4⋊(C67⋊C8)—22no—
57C4⋊(C67⋊C8)—22no—
58C4⋊(C67⋊C8)—21no—
59C4⋊(C67⋊C8)—21no—
60#14523971—44no—
61#14523966C2×C414yes—
62#14523966C2×C414yes—
63#14523963C2×C414yes—
64#14523963C2×C414yes—
65(C2×D67)⋊C8—22no—
66(C2×D67)⋊C8—22no—
67#14523954—22no—
68C4×C67⋊C8—22no—
69C4×C67⋊C8C2×C411yescharacteristic
70C4×C67⋊C8C2×C411yescharacteristic
71C4×C67⋊C8C2×C412yes—
72C4×C67⋊C8D411yescharacteristic
73C4×C67⋊C8D411yescharacteristic
74C4×C67⋊C8D411yescharacteristic
75C4×C67⋊C8D411yescharacteristic
76C2×M16×C67C2312yes—
77C4⋊Q8×C67C2×C411yescharacteristic
78C4⋊Q8×C67C2×C411yescharacteristic
79#125×C67C2×C412yes—
80C22⋊Q8×C67—21no—
81C22⋊Q8×C67—21no—
82C4×Q8×C67—21no—
83C4×Q8×C67—21no—
84#117×C67C2312yes—
85C2×C42×C67—21no—
86C2×C42×C67—21no—
87C2×C42×C67C2312yes—
88C2×C42×C67C2311yescharacteristic
89C4⋊C8×C67—21no—
90C4⋊C8×C67—21no—
91#11761628—24no—
92#11761628—24no—
93C2×Dic134C22×C412yes—
94C2×D268—21no—
95C2×D268—21no—
96C2×D268C22×C412yes—
97C2×C22⋊D67—22no—
98C2×C4×D67—22no—
99#11761612—24no—
100#11761611—44no—
101C2×C67⋊C8—22no—
102C2×C67⋊C8—22no—
103C2×C67⋊C8—22no—
104C2×C67⋊C8—22no—
105C2×C67⋊C8—22no—
106C2×C67⋊C8C22×C412yes—
107C2×C67⋊C8C22×C412yes—
108C2×C67⋊C8C2×D411yescharacteristic
109C2×C67⋊C8C2×D411yescharacteristic
110C2×C67⋊C8C2×D411yescharacteristic
111C2×C67⋊C8C2×D411yescharacteristic
112C8×D67—44no—
113C4⋊Dic67—21no—
114C4⋊Dic67—21no—
115(C2×D67)⋊C4—22no—
116(C2×D67)⋊C4—22no—
117C2×Q8×C67—21no—
118C2×Q8×C67—21no—
119C22×C4×C67—22no—
120C22×C4×C67#4111yescharacteristic
121C22×C4×C67#4111yescharacteristic
122C22×C4×C67C2411yescharacteristic
123M16×C67—24no—
124C2×C8×C67—22no—
125C4⋊C4×C67—22no—
126C4⋊C4×C67—22no—
127C4⋊C4×C67—21no—
128C4⋊C4×C67—21no—
129C4⋊C4×C67C22×C412yes—
130C4⋊C4×C67C22×C412yes—
131C22⋊C4×C67—22no—
132C42×C67—24no—
133C42×C67—22no—
134C42×C67—22no—
135C42×C67—22no—
136C42×C67—42no—
137C42×C67C22×C411yescharacteristic
138C42×C67C22×C411yescharacteristic
139C42×C67C2×D411yescharacteristic
140C42×C67C2×D411yescharacteristic
141C23×C67C2×#4111yescharacteristic
142C22×D67—22no—
143Q8×C67—42no—
144Q8×C67—42no—
145C2×C4×C67—22no—
146C2×C4×C67—22no—
147C2×C4×C67—24no—
148C2×C4×C67—22no—
149C2×C4×C67—22no—
150C2×C4×C67—42no—
151C2×C4×C67—42no—
152C2×C4×C67C4×D411yescharacteristic
153C2×C4×C67C4×D411yescharacteristic
154C2×C4×C67C4×D411yescharacteristic
155C2×C4×C67C4×D411yescharacteristic
156C2×C4×C67C23×C411yescharacteristic
157C2×C4×C67C22×D411yescharacteristic
158C2×C4×C67H(5,𝔽2)12yes—
159C2×C4×C672−1+412yes—
160C22⋊D67—44no—
161C2×Dic67—22no—
162D268—24no—
163D268—42no—
164D268—42no—
165C4×D67—42no—
166C4×D67—42no—
167Dic134—24no—
168C8×C67—42no—
169C67⋊C8—22no—
170C67⋊C8—22no—
171C67⋊C8—21no—
172C67⋊C8—21no—
173C67⋊C8—21no—
174C67⋊C8—21no—
175C67⋊C8—42no—
176C22×C67—21no—
177C22×C67—21no—
178C22×C67C2×C4×D411yescharacteristic
179C22×C67#54212yes—
180C2×D67—42no—
181C2×D67—42no—
182C2×D67—42no—
183C4×C67—21no—
184C4×C67—21no—
185C4×C67—44no—
186C4×C67—42no—
187C4×C67—42no—
188C4×C67#44312yes—
189C4×C67#44412yes—
190C4×C67#52012yes—
191C4×C67#53511yescharacteristic
192C4×C67#53611yescharacteristic
193Dic67—42no—
194#22234—671nomaximal, Sylow
195C2×C67—22no—
196C2×C67#232611yescharacteristic
197C2×C67C2×#44311yescharacteristic
198C2×C67C2×#44411yescharacteristic
199D67—82no—
200#3131—671no—
201#2960—671no—
202#2931—671no—
203#1994D6711yescharacteristic, complemented
204#1962—671no—
205#1929—674no—
206#1866—674no—
207#1848—671no—
208#1744—671no—
209C67#2223411yescharacteristic, complemented, Sylow
210C2×#131—672no—
211#533C2×D6711yescharacteristic
212#532—671no—
213#488—674no—
214#487—671no—
215#487—671no—
216#486—671no—
217#486—671no—
218#446—1341no—
219#446—1341no—
220C8×Q8—1341no—
221C8×Q8—1341no—
222#436—1341no—
223#436—1341no—
224C8×D4—1341no—
225C8×D4—1341no—
226C2×#108—674no—
227#424—672no—
228#423—671no—
229#423C2×D6711yescharacteristic
230C2×#104—674no—
231#416—674no—
232#408—672no—
233#405—672no—
234C2×C4×C8—671no—
235C2×C4×C8—671no—
236C43C2×D6711yescharacteristic
237#339C2×D6714yes—
238#336—672no—
239#336—672no—
240C2×#41—672no—
241#131—2684no—
242C2×M16—672no—
243C2×M16—1342no—
244C2×M16C22×D6712yes—
245C22×C8—671no—
246C22×C8—671no—
247C22×C8—672no—
248C22×C8—1342no—
249C4⋊Q8—671no—
250C4⋊Q8C4×D6711yescharacteristic
251C4⋊Q8C4×D6711yescharacteristic
252#127—671no—
253#125C4×D6712yes—
254#124—672no—
255C22⋊Q8—21no—
256C22⋊Q8—21no—
257#121—1341no—
258#121—1341no—
259C4×Q8—21no—
260C4×Q8—21no—
261C4×D4—1341no—
262C4×D4—1341no—
263#117C22×D6712yes—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42C22×D6712yes—
267C2×C42C22×D6711yescharacteristic
268#108—2684no—
269C4⋊C8—21no—
270C4⋊C8—21no—
271C4⋊C8—1342no—
272C4⋊C8—1342no—
273C4⋊C8—1341no—
274C4⋊C8—1341no—
275#104—2684no—
276#103—674no—
277#103—674no—
278#102—674no—
279#102—674no—
280C22⋊C8—1342no—
281C22⋊C8—1342no—
282#97—1342no—
283C4×C8—671no—
284C4×C8—671no—
285C4×C8—672no—
286C4×C8—671no—
287C4×C8—671no—
288C4×C8—671no—
289C4×C8—671no—
290C4×C8—1342no—
291#41—1344no—
292#41—1344no—
293C2×Q8—21no—
294C2×Q8—21no—
295C2×Q8—672no—
296C2×D4—672no—
297C2×D4—1342no—
298C2×D4—1341no—
299C2×D4—1341no—
300C22×C4—22no—
301C22×C4—1342no—
302C22×C4#1176162711yescharacteristic
303C22×C4#1176162711yescharacteristic
304C22×C4C23×D6711yescharacteristic
305M16—24no—
306M16—2684no—
307M16—1344no—
308C2×C8—22no—
309C2×C8—2684no—
310C2×C8—672no—
311C2×C8—672no—
312C2×C8—671no—
313C2×C8—671no—
314C2×C8—671no—
315C2×C8—671no—
316C2×C8—1342no—
317C2×C8—1342no—
318C2×C8—1342no—
319C2×C8—1342no—
320C2×C8—1342no—
321C4⋊C4—22no—
322C4⋊C4—22no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—1341no—
326C4⋊C4—1341no—
327C4⋊C4C2×C4×D6712yes—
328C4⋊C4C2×C4×D6712yes—
329C22⋊C4—22no—
330C22⋊C4—1342no—
331C22⋊C4—1342no—
332C42—24no—
333C42—22no—
334C42—22no—
335C42—22no—
336C42—42no—
337C42C2×C4×D6711yescharacteristic
338C42C2×C4×D6711yescharacteristic
339C42D4×D6711yescharacteristic
340C42D4×D6711yescharacteristic
341C23—1342no—
342C23C2×#1176162711yescharacteristic
343Q8—42no—
344Q8—42no—
345Q8—1344no—
346D4—2684no—
347D4—2682no—
348D4—2682no—
349D4—1344no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—24no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—2682no—
358C2×C4—2682no—
359C2×C4—1342no—
360C2×C4(C22⋊D67)⋊C411yescharacteristic
361C2×C4(C22⋊D67)⋊C411yescharacteristic
362C2×C4(C22⋊D67)⋊C411yescharacteristic
363C2×C4(C22⋊D67)⋊C411yescharacteristic
364C2×C4C22×C4×D6711yescharacteristic
365C2×C4C2×D4×D6711yescharacteristic
366C2×C4#1452408912yes—
367C2×C4#1452409112yes—
368C8—42no—
369C8—2682no—
370C8—1342no—
371C8—1342no—
372C8—1341no—
373C8—1341no—
374C8—1341no—
375C8—1341no—
376C22—21no—
377C22—21no—
378C22—2682no—
379C22—2682no—
380C22—2682no—
381C22C2×(C22⋊D67)⋊C411yescharacteristic
382C22#1686401912yes—
383C4—21no—
384C4—21no—
385C4—44no—
386C4—42no—
387C4—42no—
388C4—2682no—
389C4#1686351012yes—
390C4#1686351512yes—
391C4#1686387812yes—
392C4#1686396411yescharacteristic
393C4#1686397111yescharacteristic
394C2—22no—
395C2—5362no—
396C2#2256292311yescharacteristic
397C2C2×#1686351011yescharacteristic
398C2C2×#1686351511yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D67
commutatorC2×C4×C67C23×C4
FrattiniC22×C4C23×D67
FittingC67×#1994C2
socleC22×C67C2×C4×D4
prime radical#16863946C22
cosocleC22×C4×C67C24
hypercenter#1994D67
hypocenterC67#22234
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1994

Complements

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

NH ≅ G/Ntype
2. #22569144C2semidirect
5. #22560855C2semidirect
6. #22560485C2semidirect
7. #22560380C2semidirect
8. #22559955C2semidirect
9. C67×#1994C2semidirect
12. #16863753C4semidirect
13. #16863748C4semidirect
14. #16863748C4semidirect
15. #16863744C4semidirect
16. #16863744C4semidirect
203. #1994D67semidirect
209. C67#22234semidirect

Maximal subgroups

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

subgroupindex
#225705342
#225691442
#225689202
#225610372
#225608552
#225604852
#225603802
#225599552
C67×#19942
#2223467