Finite Group Databasebeta

← #30869335

Subgroups — #30869335

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#22570531C211yesmaximal, characteristic
2#22569139C211yesmaximal, characteristic, complemented
3#22568921C211yesmaximal, characteristic
4#22561186C211yesmaximal, characteristic, complemented
5#22560860C214yesmaximal, complemented
6#22560482C214yesmaximal, complemented
7#22560380C211yesmaximal, characteristic, complemented
8#22559955C211yesmaximal, characteristic, complemented
9C67×#1962C211yesmaximal, characteristic
10C2×#14524050C2212yes—
11#16863953C2211yescharacteristic
12#16863750C414yescomplemented
13#16863746C411yescharacteristic, complemented
14#16863746C411yescharacteristic, complemented
15#16863742C411yescharacteristic, complemented
16#16863742C411yescharacteristic, complemented
17#16863523—21no—
18#16863523—21no—
19#16863520—21no—
20#16863520—21no—
21#16863474—21no—
22#16863474—21no—
23D4×C67⋊C8—21no—
24D4×C67⋊C8—21no—
25C2×#14523978C2214yes—
26#16863424C2212yes—
27#16863420C2211yescharacteristic
28#16863392C2214yes—
29#16863363C2212yes—
30#16863354C2212yes—
31C2×C4×C67⋊C8C2211yescharacteristic
32C2×C4×C67⋊C8C2211yescharacteristic
33#16863156C2214yes—
34#16863151C2212yes—
35#16863151C2212yes—
36#532×C67C2211yescharacteristic
37#423×C67C2211yescharacteristic
38C2×#104×C67C2214yes—
39C43×C67C2211yescharacteristic
40#14524050—44no—
41C2×#11761612—22no—
42C2×#11761612C2312yes—
43C22×C67⋊C8—22no—
44C22×C67⋊C8C2312yes—
45C22×C67⋊C8C2311yescharacteristic
46C22×C67⋊C8C2311yescharacteristic
47#14524043C2×C411yescharacteristic
48#14524043C2×C411yescharacteristic
49#14524035C2×C412yes—
50#14524022—21no—
51#14524022—21no—
52C4×Dic134—21no—
53C4×Dic134—21no—
54#14524002C2312yes—
55#14523978—44no—
56C4⋊(C67⋊C8)—21no—
57C4⋊(C67⋊C8)—21no—
58Dic67⋊C8—22no—
59Dic67⋊C8—22no—
60#14523967C2×C414yes—
61#14523967C2×C414yes—
62#14523964C2×C414yes—
63#14523964C2×C414yes—
64C22⋊(C67⋊C8)—22no—
65C22⋊(C67⋊C8)—22no—
66#14523953—22no—
67C4×C67⋊C8C2×C411yescharacteristic
68C4×C67⋊C8C2×C411yescharacteristic
69C4×C67⋊C8C2×C412yes—
70C4×C67⋊C8D411yescharacteristic
71C4×C67⋊C8D411yescharacteristic
72C4×C67⋊C8D411yescharacteristic
73C4×C67⋊C8D411yescharacteristic
74C8×Dic67—22no—
75C2×#41×C67C2312yes—
76C2×M16×C67C2312yes—
77C4⋊Q8×C67C2×C411yescharacteristic
78#127×C67C2×C411yescharacteristic
79#124×C67C2×C412yes—
80#121×C67—21no—
81#121×C67—21no—
82C4×D4×C67—21no—
83C4×D4×C67—21no—
84C2×C42×C67—21no—
85C2×C42×C67—21no—
86C2×C42×C67C2312yes—
87C2×C42×C67C2311yescharacteristic
88C4⋊C8×C67—21no—
89C4⋊C8×C67—21no—
90#104×C67—44no—
91C2×Dic134—21no—
92C2×Dic134—21no—
93#11761612—24no—
94#11761612—44no—
95C2×C67⋊C8—22no—
96C2×C67⋊C8—22no—
97C2×C67⋊C8—22no—
98C2×C67⋊C8—22no—
99C2×C67⋊C8—22no—
100C2×C67⋊C8—44no—
101C2×C67⋊C8C22×C412yes—
102C2×C67⋊C8C22×C412yes—
103C2×C67⋊C8C2×D411yescharacteristic
104C2×C67⋊C8C2×D411yescharacteristic
105C2×C67⋊C8C2×D411yescharacteristic
106C2×C67⋊C8C2×D411yescharacteristic
107C4⋊Dic67—21no—
108C4⋊Dic67—21no—
109C4⋊Dic67C22×C412yes—
110C4⋊Dic67C22×C412yes—
111Dic67⋊C4—22no—
112Dic67⋊C4—22no—
113C22⋊Dic67—22no—
114C4×Dic67—22no—
115#41×C67—24no—
116#41×C67—24no—
117C2×Q8×C67C22×C412yes—
118C2×D4×C67—22no—
119C2×D4×C67—21no—
120C2×D4×C67—21no—
121C2×D4×C67C22×C412yes—
122C22×C4×C67—22no—
123C22×C4×C67—22no—
124C22×C4×C67#4111yescharacteristic
125C22×C4×C67#4111yescharacteristic
126C22×C4×C67C2411yescharacteristic
127M16×C67—24no—
128C2×C8×C67—22no—
129C4⋊C4×C67—21no—
130C4⋊C4×C67—21no—
131C22⋊C4×C67—22no—
132C22⋊C4×C67—22no—
133C42×C67—24no—
134C42×C67—22no—
135C42×C67—22no—
136C42×C67—42no—
137C42×C67C22×C411yescharacteristic
138C42×C67C22×C411yescharacteristic
139C42×C67C2×D411yescharacteristic
140C42×C67C2×D411yescharacteristic
141C23×C67—22no—
142C23×C67C2×#4111yescharacteristic
143Q8×C67—24no—
144D4×C67—24no—
145D4×C67—44no—
146D4×C67—42no—
147D4×C67—42no—
148C2×C4×C67—22no—
149C2×C4×C67—22no—
150C2×C4×C67—24no—
151C2×C4×C67—22no—
152C2×C4×C67—42no—
153C2×C4×C67—42no—
154C2×C4×C67—42no—
155C2×C4×C67—42no—
156C2×C4×C67C4×D411yescharacteristic
157C2×C4×C67C4×D411yescharacteristic
158C2×C4×C67C4×D411yescharacteristic
159C2×C4×C67C4×D411yescharacteristic
160C2×C4×C67C23×C411yescharacteristic
161C2×C4×C67C22×D411yescharacteristic
162C2×C4×C67H(5,𝔽2)12yes—
163C2×C4×C672−1+412yes—
164C2×Dic67—22no—
165C2×Dic67—22no—
166Dic134—42no—
167Dic134—42no—
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×C67—42no—
179C22×C67—42no—
180C22×C67—42no—
181C22×C67C2×C4×D411yescharacteristic
182C22×C67#54212yes—
183C4×C67—21no—
184C4×C67—21no—
185C4×C67—44no—
186C4×C67—42no—
187C4×C67#44312yes—
188C4×C67#44412yes—
189C4×C67#52012yes—
190C4×C67#53511yescharacteristic
191C4×C67#53611yescharacteristic
192Dic67—42no—
193Dic67—42no—
194#22234—671nomaximal, Sylow
195C2×C67—22no—
196C2×C67—82no—
197C2×C67#232611yescharacteristic
198C2×C67C2×#44311yescharacteristic
199C2×C67C2×#44411yescharacteristic
200#3131—671no—
201#2960—671no—
202#2931—671no—
203#1994—671no—
204#1962D6711yescharacteristic
205#1929—674no—
206#1866—674no—
207#1848—671no—
208#1744—671no—
209C67#2223411yescharacteristic, complemented, Sylow
210C2×#131—672no—
211#533—671no—
212#532C2×D6711yescharacteristic
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×#104C2×D6714yes—
231#416—674no—
232#408—672no—
233#405—672no—
234C2×C4×C8—671no—
235C2×C4×C8—671no—
236C43C2×D6711yescharacteristic
237#339—674no—
238#336—672no—
239#336—672no—
240C2×#41C22×D6712yes—
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⋊Q8—671no—
251C4⋊Q8C4×D6711yescharacteristic
252#127C4×D6711yescharacteristic
253#125—672no—
254#124C4×D6712yes—
255C22⋊Q8—1341no—
256C22⋊Q8—1341no—
257#121—21no—
258#121—21no—
259C4×Q8—1341no—
260C4×Q8—1341no—
261C4×D4—21no—
262C4×D4—21no—
263#117—672no—
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—44no—
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—24no—
292#41—24no—
293C2×Q8—1341no—
294C2×Q8—1341no—
295C2×Q8C2×C4×D6712yes—
296C2×D4—22no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4C2×C4×D6712yes—
300C22×C4—22no—
301C22×C4—22no—
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—21no—
322C4⋊C4—21no—
323C4⋊C4—672no—
324C4⋊C4—672no—
325C4⋊C4—1342no—
326C4⋊C4—1342no—
327C4⋊C4—1341no—
328C4⋊C4—1341no—
329C22⋊C4—22no—
330C22⋊C4—22no—
331C22⋊C4—1342no—
332C42—24no—
333C42—22no—
334C42—22no—
335C42—42no—
336C42—1342no—
337C42C2×C4×D6711yescharacteristic
338C42C2×C4×D6711yescharacteristic
339C42D4×D6711yescharacteristic
340C42D4×D6711yescharacteristic
341C23—22no—
342C23C2×#1176162711yescharacteristic
343Q8—24no—
344Q8—2682no—
345Q8—2682no—
346D4—24no—
347D4—44no—
348D4—42no—
349D4—42no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—24no—
353C2×C4—22no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—1342no—
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#1452408812yes—
367C2×C4#1452409012yes—
368C8—42no—
369C8—2682no—
370C8—1342no—
371C8—1342no—
372C8—1341no—
373C8—1341no—
374C8—1341no—
375C8—1341no—
376C22—21no—
377C22—21no—
378C22—42no—
379C22—42no—
380C22—42no—
381C22C2×(C22⋊D67)⋊C411yescharacteristic
382C22#1686401312yes—
383C4—21no—
384C4—21no—
385C4—44no—
386C4—42no—
387C4—2682no—
388C4—2682no—
389C4#1686351012yes—
390C4#1686351512yes—
391C4#1686387512yes—
392C4#1686395911yescharacteristic
393C4#1686396611yescharacteristic
394C2—22no—
395C2—82no—
396C2#2256291911yescharacteristic
397C2C2×#1686351011yescharacteristic
398C2C2×#1686351511yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D67
commutatorC2×C4×C67C23×C4
FrattiniC22×C4C23×D67
FittingC67×#1962C2
socleC22×C67C2×C4×D4
prime radical#532×C67C22
cosocleC22×C4×C67C24
hypercenter#1962D67
hypocenterC67#22234
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1962

Complements

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

NH ≅ G/Ntype
2. #22569139C2semidirect
4. #22561186C2semidirect
5. #22560860C2semidirect
6. #22560482C2semidirect
7. #22560380C2semidirect
8. #22559955C2semidirect
12. #16863750C4semidirect
13. #16863746C4semidirect
14. #16863746C4semidirect
15. #16863742C4semidirect
16. #16863742C4semidirect
209. C67#22234semidirect

Maximal subgroups

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

subgroupindex
#225705312
#225691392
#225689212
#225611862
#225608602
#225604822
#225603802
#225599552
C67×#19622
#2223467