Finite Group Databasebeta

← #31010657

Subgroups — #31010657

Order 17152.

632 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16863632C211yesmaximal, characteristic
2#22562400C214yesmaximal, complemented
3#22562394C212yesmaximal, complemented
4#22562389C212yesmaximal, complemented
5#22561998C212yesmaximal, complemented
6#22561924C212yesmaximal, complemented
7#22561820C211yesmaximal, characteristic, complemented
8C2×#427×C67C211yesmaximal, characteristic, complemented
9C22×D536C2211yescharacteristic
10C2×#14524017—21no—
11C2×#14523983C2212yes—
12#16863632—21no—
13#16863632—21no—
14#16863632C2212yes—
15#16863632C2212yes—
16C2×#14523977C2211yescharacteristic
17C2×#14523965—21no—
18#16863223C2214yescomplemented
19#16863221C2214yescomplemented
20#16863196C2214yescomplemented
21#16863196C2214yescomplemented
22C22⋊(C67⋊C16)C2212yescomplemented
23C22⋊(C67⋊C16)C2212yescomplemented
24C23×C8×C67C2211yescharacteristic
25C2×#117×C67—21no—
26#427×C67—21no—
27#427×C67—21no—
28#427×C67C2212yes—
29#427×C67C2212yescomplemented
30C2×#107×C67C2211yescharacteristic, complemented
31C2×#107×C67C2211yescharacteristic, complemented
32C2×#106×C67—21no—
33C22×D268—21no—
34C22×C22⋊D67—41no—
35C2×D536—21no—
36C2×D536—24no—
37C2×D536C2312yes—
38#14524017—22no—
39#14524017—22no—
40#14524017—41no—
41#14524017—41no—
42C2×C4⋊Dic67—21no—
43C2×(C2×D67)⋊C4—41no—
44#14523983—24no—
45C2×C67⋊C16C2314yes—
46#14523977—21no—
47#14523977C2312yes—
48#14523965—22no—
49#14523965—22no—
50C23×C4×C67D411yescharacteristic
51C22×C8×C67—24no—
52C22×C8×C67D411yescharacteristic
53C22×C8×C67D411yescharacteristic
54C22×C8×C67D411yescharacteristic
55C22×C8×C67C2311yescharacteristic
56C22×C8×C67C2311yescharacteristic
57C22×C8×C67C2311yescharacteristic
58#117×C67—22no—
59#117×C67—22no—
60#117×C67—41no—
61#117×C67—41no—
62C2×C4⋊C4×C67—21no—
63C2×C4⋊C4×C67—21no—
64C2×C22⋊C4×C67—41no—
65C2×C42×C67—41no—
66#107×C67—21no—
67#107×C67—21no—
68#107×C67C2312yes—
69#107×C67C2312yes—
70#106×C67—22no—
71#106×C67—22no—
72C23×D67—41no—
73C2×D268—22no—
74C2×D268—42no—
75C2×D268—42no—
76C2×D268—41no—
77C2×C22⋊D67—42no—
78C2×C22⋊D67—42no—
79C2×C22⋊D67—84no—
80C22×Dic67—41no—
81D536—24no—
82D536—42no—
83C4⋊Dic67—22no—
84C4⋊Dic67—41no—
85(C2×D67)⋊C4—42no—
86(C2×D67)⋊C4—42no—
87C67⋊C16—24no—
88C24×C67D811yescharacteristic
89C22×C4×C67—22no—
90C22×C4×C67—22no—
91C22×C4×C67—41no—
92C22×C4×C67—41no—
93C22×C4×C67D811yescharacteristic
94C22×C4×C67D811yescharacteristic
95C22×C4×C67D811yescharacteristic
96C22×C4×C67C2×D411yescharacteristic
97C22×C4×C67C2×D411yescharacteristic
98C22×C4×C67C2×D411yescharacteristic
99C2×C8×C67—22no—
100C2×C8×C67—22no—
101C2×C8×C67—22no—
102C2×C8×C67—22no—
103C2×C8×C67—22no—
104C2×C8×C67C2×D411yescharacteristic
105C2×C8×C67C2×D411yescharacteristic
106C2×C8×C67C2×D411yescharacteristic
107C2×C8×C67#4114yes—
108C2×C8×C67C2411yescharacteristic
109C4⋊C4×C67—22no—
110C4⋊C4×C67—22no—
111C4⋊C4×C67—41no—
112C4⋊C4×C67—41no—
113C22⋊C4×C67—42no—
114C22⋊C4×C67—42no—
115C42×C67—42no—
116C42×C67—42no—
117C23×C67—22no—
118C23×C67—22no—
119C23×C67—41no—
120C23×C67C2×D811yescharacteristic
121C23×C67C2×D811yescharacteristic
122C23×C67C2×D811yescharacteristic
123C22×D67—42no—
124C22×D67—82no—
125C22×D67—82no—
126C22×D67—82no—
127C2×C4×C67—22no—
128C2×C4×C67—21no—
129C2×C4×C67—21no—
130C2×C4×C67—21no—
131C2×C4×C67—21no—
132C2×C4×C67—22no—
133C2×C4×C67—22no—
134C2×C4×C67—42no—
135C2×C4×C67—42no—
136C2×C4×C67—82no—
137C2×C4×C67—82no—
138C2×C4×C67#12312yes—
139C2×C4×C67C2×D811yescharacteristic
140C2×C4×C67C2×D811yescharacteristic
141C2×C4×C67C2×D811yescharacteristic
142C2×C4×C67#13712yes—
143C2×C4×C67C22×D411yescharacteristic
144C22⋊D67—84no—
145C22⋊D67—84no—
146C2×Dic67—42no—
147C2×Dic67—82no—
148D268—42no—
149D268—42no—
150D268—82no—
151C8×C67—21no—
152C8×C67—21no—
153C8×C67#12312yes—
154C8×C67C2×#4112yes—
155C22×C67—21no—
156C22×C67—21no—
157C22×C67—21no—
158C22×C67—21no—
159C22×C67—21no—
160C22×C67—21no—
161C22×C67—21no—
162C22×C67—44no—
163C22×C67#48012yes—
164C22×C67#50912yes—
165C22×C67C22×D811yescharacteristic
166C2×D67—82no—
167C2×D67—82no—
168C2×D67—82no—
169C2×D67—164no—
170C4×C67—21no—
171C4×C67—21no—
172C4×C67—82no—
173C4×C67—82no—
174C4×C67#48012yes—
175C4×C67C2×#12311yescharacteristic
176C4×C67C2×#13711yescharacteristic
177Dic67—82no—
178#33648—671nomaximal, Sylow
179C2×C67—22no—
180C2×C67—22no—
181C2×C67—41no—
182C2×C67C2×#48011yescharacteristic
183C2×C67C2×#50911yescharacteristic
184C2×C67#341411yescharacteristic
185D67—162no—
186C2×#466—671no—
187C2×#427D6711yescharacteristic, complemented
188#2234—674no—
189#2233—672no—
190#2232—672no—
191#2159—672no—
192#2140—672no—
193#2112—671no—
194C67#3364811yescharacteristic, complemented, Sylow
195C22×D8—671no—
196C23×C8C2×D6711yescharacteristic
197C2×#121—1341no—
198C2×#117—21no—
199C2×M32—672no—
200#466—672no—
201#466—672no—
202#466—1341no—
203#466—1341no—
204#427—21no—
205#427—21no—
206#427C2×D6712yes—
207#427C2×D6712yescomplemented
208C2×#107—671no—
209C2×#107C2×D6711yescharacteristic, complemented
210C2×#107C2×D6711yescharacteristic, complemented
211C2×#106—21no—
212C2×#102—1341no—
213#367—674no—
214#366—674no—
215#357—674no—
216#357—674no—
217C22⋊C16—672no—
218C22⋊C16—672no—
219C22×D4—2681no—
220C22×D4—1341no—
221C23×C4C22⋊D6711yescharacteristic
222C2×D8—672no—
223C2×D8—1341no—
224C2×D8—1344no—
225C22×C8—24no—
226C22×C8C22⋊D6711yescharacteristic
227C22×C8C22⋊D6711yescharacteristic
228C22×C8C22⋊D6711yescharacteristic
229C22×C8C22×D6711yescharacteristic
230C22×C8C22×D6711yescharacteristic
231C22×C8C22×D6711yescharacteristic
232#121—2681no—
233#121—2681no—
234#121—1342no—
235#121—1342no—
236#117—22no—
237#117—22no—
238#117—41no—
239#117—41no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—1341no—
243C2×C22⋊C4—41no—
244C2×C22⋊C4—2681no—
245C2×C42—41no—
246M32—1344no—
247C2×C16—674no—
248#107—21no—
249#107—21no—
250#107—672no—
251#107—1341no—
252#107C22×D6712yes—
253#107C22×D6712yes—
254#106—22no—
255#106—22no—
256#102—1342no—
257#102—1342no—
258C24—2681no—
259C24#1176161311yescharacteristic
260C2×D4—2682no—
261C2×D4—2682no—
262C2×D4—2682no—
263C2×D4—2681no—
264C2×D4—2682no—
265C2×D4—5364no—
266C2×D4—1342no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—2681no—
272C22×C4#1176161311yescharacteristic
273C22×C4#1176161311yescharacteristic
274C22×C4#1176161311yescharacteristic
275C22×C4C2×C22⋊D6711yescharacteristic
276C22×C4C2×C22⋊D6711yescharacteristic
277C22×C4C2×C22⋊D6711yescharacteristic
278D8—2682no—
279D8—1344no—
280C2×C8—22no—
281C2×C8—22no—
282C2×C8—22no—
283C2×C8—22no—
284C2×C8—22no—
285C2×C8C2×C22⋊D6711yescharacteristic
286C2×C8C2×C22⋊D6711yescharacteristic
287C2×C8C2×C22⋊D6711yescharacteristic
288C2×C8#1176162814yes—
289C2×C8C23×D6711yescharacteristic
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—2681no—
295C4⋊C4—1342no—
296C22⋊C4—42no—
297C22⋊C4—42no—
298C22⋊C4—2682no—
299C22⋊C4—2682no—
300C42—42no—
301C42—42no—
302C16—1344no—
303C23—22no—
304C23—22no—
305C23—41no—
306C23—2682no—
307C23—5362no—
308C23—5362no—
309C23—5362no—
310C23C2×#1176161311yescharacteristic
311C23C2×#1176161311yescharacteristic
312C23C2×#1176161311yescharacteristic
313D4—2682no—
314D4—2682no—
315D4—5364no—
316D4—5364no—
317D4—5362no—
318C2×C4—22no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—2682no—
330C2×C4—5362no—
331C2×C4#1452402612yes—
332C2×C4C2×#1176161311yescharacteristic
333C2×C4C2×#1176161311yescharacteristic
334C2×C4C2×#1176161311yescharacteristic
335C2×C4#1452407412yes—
336C2×C4C22×C22⋊D6711yescharacteristic
337C8—21no—
338C8—21no—
339C8#1452402612yes—
340C8C2×#1176162812yes—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—44no—
349C22—5362no—
350C22—5362no—
351C22—5362no—
352C22—10724no—
353C22#1686370912yes—
354C22#1686383112yes—
355C22C22×#1176161311yescharacteristic
356C4—21no—
357C4—21no—
358C4—82no—
359C4—82no—
360C4—5362no—
361C4#1686370912yes—
362C4C2×#1452402611yescharacteristic
363C4C2×#1452407411yescharacteristic
364C2—22no—
365C2—22no—
366C2—41no—
367C2—10722no—
368C2C2×#1686370911yescharacteristic
369C2C2×#1686383111yescharacteristic
370C2#2257334311yescharacteristic
subgroupG/H
centerC22C22×#11761613
commutator, cosocleC2×C8×C67C24
FrattiniC2×C8C23×D67
FittingC2×#427×C67C2
socleC22×C67C22×D8
prime radicalC2×#16863632C2
hypercenterC2×#427D67
hypocenterC67#33648
LCSG ▷ C2×C8×C67 ▷ C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ C2×#427

Complements

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

NH ≅ G/Ntype
2. #22562400C2semidirect
3. #22562394C2semidirect
4. #22562389C2semidirect
5. #22561998C2semidirect
6. #22561924C2semidirect
7. #22561820C2semidirect
8. C2×#427×C67C2semidirect
18. #16863223C22semidirect
19. #16863221C22semidirect
20. #16863196C22semidirect
21. #16863196C22semidirect
22. C22⋊(C67⋊C16)C22semidirect
23. C22⋊(C67⋊C16)C22semidirect
29. #427×C67C22semidirect
30. C2×#107×C67C22semidirect
31. C2×#107×C67C22semidirect
187. C2×#427D67semidirect
194. C67#33648semidirect
207. #427C2×D67semidirect
209. C2×#107C2×D67semidirect
210. C2×#107C2×D67semidirect

Maximal subgroups

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

subgroupindex
C2×#168636322
#225624002
#225623942
#225623892
#225619982
#225619242
#225618202
C2×#427×C672
#3364867