Finite Group Databasebeta

← C2×#22569319 #31720877

Subgroups — C2×#22569319 #31720877

Order 17152.

2434 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#11761629C211yesmaximal, characteristic
2#22569319C2116yesmaximal, direct product factor
3C2×Q8×C67⋊C8C211yesmaximal, characteristic, complemented
4C2×C8×D268C213yesmaximal, complemented
5C2×#16863461C213yesmaximal, complemented
6C2×#16863447C213yesmaximal, complemented
7C2×C4⋊C8×D67C213yesmaximal, complemented
8C2×C8×Q8×C67C211yesmaximal, characteristic, complemented
9C22×#11761629C411yescharacteristic
10C22×C8×D67C2216yes—
11C4×#11761629C418yes—
12C4×#11761629C2218yes—
13C2×Q8×Dic67C411yescharacteristic
14C2×C4×D268C2213yes—
15C2×#14524005C413yes—
16C2×#14524001C413yes—
17C2×C42×D67C2213yes—
18Q8×C67⋊C8C2218yescomplemented
19C8×D268C22124yescomplemented
20#16863461C22124yescomplemented
21#16863447C22124yescomplemented
22C2×C4⋊(C67⋊C8)C2213yes—
23C2×Dic67⋊C8C2213yes—
24C2×Dic67⋊C8C2211yescharacteristic
25C4⋊C8×D67C22124yescomplemented
26C2×(C2×D67)⋊C8C2213yes—
27C2×(C2×D67)⋊C8C2213yes—
28C2×C4×C67⋊C8C2213yes—
29C2×C4×Q8×C67C2211yescharacteristic
30C8×Q8×C67C2218yescomplemented
31C2×C4⋊C8×C67C2213yes—
32C2×C4×C8×C67C2213yes—
33C2×#11761629C8116yescomplemented
34C2×#11761629C2×C418yes—
35C22×D268C2×C413yes—
36C22×C4×D67C2×C413yes—
37C22×C4×D67C2×C413yes—
38C22×C4×D67C2313yes—
39C22×C67⋊C8C2311yescharacteristic
40C22×C67⋊C8C2313yes—
41C2×C8×D67—412no—
42C2×C8×D67C23124yes—
43Q8×Dic67C2×C418yes—
44C4×D268C2×C4112yes—
45C4×D268C23112yes—
46#14524005C2×C4124yes—
47#14524001C2×C4124yes—
48C2×C4⋊Dic67C2×C413yes—
49C2×(C2×D67)⋊C4C2×C416yes—
50C2×C4×Dic67C2×C413yes—
51C2×C4×Dic67C2×C413yes—
52C2×C4×Dic67C2311yescharacteristic
53C42×D67C2×C4112yes—
54C42×D67C23112yes—
55C4⋊(C67⋊C8)C23112yes—
56Dic67⋊C8C23112yes—
57Dic67⋊C8C2314yes—
58(C2×D67)⋊C8C23112yes—
59(C2×D67)⋊C8C23112yes—
60C4×C67⋊C8C23112yes—
61C22×Q8×C67C2×C411yescharacteristic
62C22×C8×C67C2311yescharacteristic
63C22×C8×C67C2313yes—
64C4×Q8×C67C2×C414yes—
65C4×Q8×C67C2314yes—
66C2×C4⋊C4×C67C2×C413yes—
67C2×C42×C67C2313yes—
68C4⋊C8×C67C23112yes—
69C4×C8×C67C23112yes—
70C23×D67C22×C413yes—
71#11761629C2×C8164yescomplemented
72C2×D268C2×C8124yescomplemented
73C2×D268C22×C4112yes—
74C22×Dic67C22×C411yescharacteristic
75C22×Dic67C22×C413yes—
76C22×Dic67C22×C411yescharacteristic
77C2×C4×D67—26no—
78C2×C4×D67—26no—
79C2×C4×D67—46no—
80C2×C4×D67C2×C8124yescomplemented
81C2×C4×D67C22×C416yes—
82C2×C4×D67C22×C4112yes—
83C2×C4×D67C22×C4112yes—
84C2×C4×D67C2416yes—
85C2×C67⋊C8—26no—
86C2×C67⋊C8—22no—
87C2×C67⋊C8C2412yes—
88C2×C67⋊C8C2416yes—
89C8×D67—424no—
90C4⋊Dic67C22×C4112yes—
91(C2×D67)⋊C4C22×C4124yes—
92C4×Dic67C22×C412yes—
93C4×Dic67C22×C4112yes—
94C4×Dic67C22×C4112yes—
95C4×Dic67C2412yes—
96C2×Q8×C67C2×C818yescomplemented
97C2×Q8×C67C22×C414yes—
98C22×C4×C67C22×C413yes—
99C22×C4×C67C22×C413yes—
100C22×C4×C67C2411yescharacteristic
101C2×C8×C67—26no—
102C2×C8×C67—22no—
103C2×C8×C67C2416yes—
104C2×C8×C67C2412yes—
105C4⋊C4×C67C22×C4112yes—
106C42×C67C22×C416yes—
107C42×C67C2416yes—
108C23×C67C23×C411yescharacteristic
109C22×D67—26no—
110C22×D67—26no—
111C22×D67C22×C8112yes—
112C22×D67C23×C416yes—
113Q8×C67C22×C8116yescomplemented
114C2×C4×C67—26no—
115C2×C4×C67C22×C8112yes—
116C2×C4×C67C23×C416yes—
117C2×C4×C67C23×C411yescharacteristic
118C2×C4×C67C23×C416yes—
119C2×C4×C67H(5,𝔽2)12yes—
120C2×C4×C672−1+412yes—
121C2×C4×C67C2511yescharacteristic
122C2×Dic67—26no—
123C2×Dic67—22no—
124C2×Dic67C22×C814yes—
125C2×Dic67C23×C412yes—
126C2×Dic67C23×C416yes—
127C2×Dic67C23×C412yes—
128D268C22×C8148yescomplemented
129C4×D67—212no—
130C4×D67—26no—
131C4×D67—26no—
132C4×D67—412no—
133C4×D67C22×C8148yescomplemented
134C8×C67—26no—
135C8×C67—22no—
136C67⋊C8—26no—
137C67⋊C8—22no—
138C22×C67#52012yes—
139C22×C67C23×C812yes—
140C22×C67#56812yes—
141C22×C67C24×C411yescharacteristic
142C2×D67—26no—
143C2×D67—224no—
144C2×D67—26no—
145C2×D67C23×C8112yes—
146C4×C67—26no—
147C4×C67#52012yes—
148C4×C67C23×C8112yes—
149C4×C67C2×H(5,𝔽2)11yescharacteristic
150C4×C67C2×2−1+411yescharacteristic
151Dic67—26no—
152Dic67—22no—
153Dic67C23×C814yes—
154C2×#2988—671nomaximal, Sylow
155C2×C67#298814yescomplemented
156C2×C67C2×#52011yescharacteristic
157C2×C67C24×C811yescharacteristic
158C2×C67C2×#56811yescharacteristic
159D67—212no—
160C2×C4×#41—671no—
161#2988—6716no—
162C2×C8×Q8—671no—
163C2×C8×Q8D6711yescharacteristic, complemented
164C2×C8×D4—673no—
165C2×C8×D4—673no—
166C2×#431—673no—
167C22×C4⋊C8—673no—
168C67C2×#298811yescharacteristic, complemented, Sylow
169C22×#41—671no—
170C23×C8—676no—
171C4×#41—678no—
172C4×#41—678no—
173C2×C4×Q8—671no—
174C2×C4×Q8C2×D6711yescharacteristic
175C2×C4×D4—673no—
176C2×C4×D4—673no—
177C2×#117—673no—
178C22×C42—673no—
179C8×Q8—678no—
180C8×Q8C2×D6718yescomplemented
181C8×D4—6724no—
182C8×D4—6724no—
183#431—6724no—
184C2×C4⋊C8—673no—
185C2×C4⋊C8—671no—
186C2×C4⋊C8—673no—
187C2×C4⋊C8—6724no—
188C2×C4⋊C8C2×D6713yes—
189C2×C22⋊C8—673no—
190C2×C22⋊C8—673no—
191C2×C4×C8—673no—
192C2×C4×C8C2×D6713yes—
193C2×#41—6716no—
194C2×#41—678no—
195C22×Q8C4×D6711yescharacteristic
196C22×D4—673no—
197C23×C4—673no—
198C23×C4—673no—
199C23×C4—673no—
200C22×C8—26812no—
201C22×C8—6724no—
202C22×C8—673no—
203C22×C8—671no—
204C22×C8C22×D6713yes—
205C22×C8C22×D6711yescharacteristic
206C4×Q8—678no—
207C4×Q8C4×D6714yes—
208C4×Q8C22×D6714yes—
209C4×D4—6724no—
210C4×D4—6712no—
211C4×D4—6712no—
212#117—6724no—
213C2×C4⋊C4—673no—
214C2×C4⋊C4C4×D6713yes—
215C2×C22⋊C4—676no—
216C2×C42—673no—
217C2×C42—673no—
218C2×C42—671no—
219C2×C42—6712no—
220C2×C42—6712no—
221C2×C42C22×D6713yes—
222C4⋊C8—6712no—
223C4⋊C8—674no—
224C4⋊C8—6712no—
225C4⋊C8C22×D67112yes—
226C22⋊C8—6712no—
227C22⋊C8—6712no—
228C4×C8—6712no—
229C4×C8C22×D67112yes—
230C24—673no—
231#41—6764no—
232C2×Q8C8×D6718yescomplemented
233C2×Q8C2×C4×D6714yes—
234C2×D4—6724no—
235C2×D4—6712no—
236C22×C4—2686no—
237C22×C4—6712no—
238C22×C4—671no—
239C22×C4—6724no—
240C22×C4—6712no—
241C22×C4—671no—
242C22×C4—673no—
243C22×C4—676no—
244C22×C4—676no—
245C22×C4—1346no—
246C22×C4—1346no—
247C22×C4C2×C4×D6713yes—
248C22×C4C2×C4×D6713yes—
249C22×C4C23×D6711yescharacteristic
250C2×C8—26no—
251C2×C8—22no—
252C2×C8—26824no—
253C2×C8—672no—
254C2×C8—676no—
255C2×C8—1346no—
256C2×C8—1342no—
257C2×C8C23×D6712yes—
258C2×C8C23×D6716yes—
259C4⋊C4—6712no—
260C4⋊C4C2×C4×D67112yes—
261C22⋊C4—6724no—
262C42—672no—
263C42—6712no—
264C42—6712no—
265C42—672no—
266C42C2×C4×D6716yes—
267C42C23×D6716yes—
268C23—6712no—
269C23—676no—
270C23—1346no—
271C23—1346no—
272C23C22×C4×D6711yescharacteristic
273Q8C2×C8×D67116yescomplemented
274D4—6748no—
275C2×C4—26no—
276C2×C4—26812no—
277C2×C4—674no—
278C2×C4—672no—
279C2×C4—6748no—
280C2×C4—672no—
281C2×C4—676no—
282C2×C4—1346no—
283C2×C4—1342no—
284C2×C4—13412no—
285C2×C4—1346no—
286C2×C4—1346no—
287C2×C4C2×C8×D67112yes—
288C2×C4C22×C4×D6716yes—
289C2×C4C22×C4×D6711yescharacteristic
290C2×C4C22×C4×D6716yes—
291C2×C4#1452408912yes—
292C2×C4#1452409112yes—
293C2×C4C24×D6711yescharacteristic
294C8—26no—
295C8—22no—
296C8—1346no—
297C8—1342no—
298C22—6712no—
299C22—1346no—
300C22—1346no—
301C22—13424no—
302C22#1686387812yes—
303C22C22×C8×D6712yes—
304C22#1686416112yes—
305C22C23×C4×D6711yescharacteristic
306C4—26no—
307C4—674no—
308C4—1342no—
309C4—1346no—
310C4#1686387812yes—
311C4C22×C8×D67112yes—
312C4C2×#1452408911yescharacteristic
313C4C2×#1452409111yescharacteristic
314C2—13412no—
315C2#2256931914yesdirect product factor
316C2C2×#1686387811yescharacteristic
317C2C23×C8×D6711yescharacteristic
318C2C2×#1686416111yescharacteristic
subgroupG/H
centerC22×C4C23×D67
commutatorC2×C67C24×C8
FrattiniC2×C4C24×D67
FittingC2×C8×Q8×C67C2
socleC23×C67C23×C4
prime radicalC22×#11761629C4
cosocleC2×C4×C67C25
hypercenterC2×C8×Q8D67
hypocenterC67C2×#2988
LCSG ▷ C2×C67 ▷ C67
UCSC1 ◁ C22×C4 ◁ C2×C8×Q8

Complements

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

NH ≅ G/Ntype
2. #22569319C2direct (N × H)
3. C2×Q8×C67⋊C8C2semidirect
4. C2×C8×D268C2semidirect
5. C2×#16863461C2semidirect
6. C2×#16863447C2semidirect
7. C2×C4⋊C8×D67C2semidirect
8. C2×C8×Q8×C67C2semidirect
18. Q8×C67⋊C8C22semidirect
19. C8×D268C22semidirect
20. #16863461C22semidirect
21. #16863447C22semidirect
25. C4⋊C8×D67C22semidirect
30. C8×Q8×C67C22semidirect
33. C2×#11761629C8semidirect
71. #11761629C2×C8semidirect
72. C2×D268C2×C8semidirect
80. C2×C4×D67C2×C8semidirect
96. C2×Q8×C67C2×C8semidirect
113. Q8×C67C22×C8semidirect
128. D268C22×C8semidirect
133. C4×D67C22×C8semidirect
155. C2×C67#2988semidirect
163. C2×C8×Q8D67semidirect
168. C67C2×#2988semidirect
180. C8×Q8C2×D67semidirect
232. C2×Q8C8×D67semidirect
273. Q8C2×C8×D67semidirect
315. C2#22569319direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×#117616292
#225693192
C2×Q8×C67⋊C82
C2×C8×D2682
C2×#168634612
C2×#168634472
C2×C4⋊C8×D672
C2×C8×Q8×C672
C2×#298867