Finite Group Databasebeta

← C2×#22573191 #31736733

Subgroups — C2×#22573191 #31736733

Order 17152.

2238 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16864125C211yesmaximal, characteristic, complemented
2C2×#16864068C211yesmaximal, characteristic
3#22573191C2116yesmaximal, direct product factor
4C2×#16863769C213yesmaximal, complemented
5C2×#16863597C213yesmaximal, complemented
6C2×#16863602C213yesmaximal, complemented
7C2×C4×#11761617C213yesmaximal, complemented
8C2×C8×Q8×C67C211yesmaximal, characteristic, complemented
9C22×#11761629—21no—
10C22×#11761617C2213yes—
11C22×#11761617C2216yes—
12#16864125—24no—
13#16864125C2218yescomplemented
14#16864068—24no—
15#16864068C2218yes—
16C2×#14524043C2213yes—
17C2×#14524040C2213yes—
18C2×#14524034—23no—
19C2×#14524018—23no—
20C2×Q8×Dic67—21no—
21C2×C4×Dic134C2213yes—
22C2×C4×D268C2213yes—
23#16863769C22124yescomplemented
24#16863602C22124yescomplemented
25#16863597C22124yescomplemented
26C4×#11761617C22124yescomplemented
27C2×#14523975C2211yescharacteristic
28C2×#14523968C2213yes—
29C2×#14523965C2213yes—
30C2×C4×Q8×C67C2211yescharacteristic
31C8×Q8×C67C2218yescomplemented
32C2×C4⋊C8×C67C2213yes—
33C2×C4×C8×C67C2213yes—
34C2×#11761629—28no—
35C2×#11761629—48no—
36C22×Dic134C2313yes—
37C22×D268—23no—
38C22×D268—23no—
39C22×D268C2313yes—
40C22×C4×D67—23no—
41C2×#11761617—212no—
42C2×#11761617—412no—
43C2×#11761617C23112yes—
44C2×#11761617C23124yes—
45#14524043—26no—
46#14524043C23112yes—
47#14524040—26no—
48#14524040C23112yes—
49#14524034—224no—
50#14524018—224no—
51Q8×Dic67—28no—
52C4×Dic134—26no—
53C4×Dic134C23112yes—
54C4×D268—26no—
55C4×D268C23112yes—
56C2×C4⋊Dic67—23no—
57C2×C4⋊Dic67—23no—
58C2×C4⋊Dic67C2311yescharacteristic
59C2×C4⋊Dic67C2311yescharacteristic
60C2×Dic67⋊C4—23no—
61C2×(C2×D67)⋊C4—23no—
62C2×C4×Dic67—23no—
63#14523975C2314yes—
64#14523968C23112yes—
65#14523965C23112yes—
66C22×Q8×C67D411yescharacteristic
67C22×C8×C67C2313yes—
68C22×C8×C67C2311yescharacteristic
69C4×Q8×C67D414yes—
70C4×Q8×C67C2314yes—
71C2×C4⋊C4×C67D413yes—
72C2×C42×C67C2313yes—
73C4⋊C8×C67C23112yes—
74C4×C8×C67C23112yes—
75C23×D67—23no—
76#11761629—432no—
77C2×Dic134—26no—
78C2×Dic134—26no—
79C2×Dic134—23no—
80C2×Dic134C2416yes—
81C2×D268—26no—
82C2×D268—26no—
83C2×D268—23no—
84C2×D268—212no—
85C2×D268—212no—
86C2×D268—412no—
87C2×D268—86no—
88C2×D268C2416yes—
89C22×Dic67—21no—
90C22×Dic67—23no—
91C22×Dic67—21no—
92C2×C4×D67—212no—
93C2×C4×D67—412no—
94#11761617—224no—
95#11761617—424no—
96C4⋊Dic67—212no—
97C4⋊Dic67—212no—
98C4⋊Dic67—21no—
99C4⋊Dic67—21no—
100C4⋊Dic67C2412yes—
101C4⋊Dic67C2412yes—
102Dic67⋊C4—212no—
103(C2×D67)⋊C4—212no—
104C4×Dic67—212no—
105C2×Q8×C67SD1618yescomplemented
106C2×Q8×C67C2×D414yes—
107C22×C4×C67C2×D413yes—
108C22×C4×C67C2×D413yes—
109C22×C4×C67C2411yescharacteristic
110C2×C8×C67—26no—
111C2×C8×C67#4114yescomplemented
112C2×C8×C67C2416yes—
113C2×C8×C67C2412yes—
114C4⋊C4×C67C2×D4112yes—
115C42×C67C2×D416yes—
116C42×C67C2416yes—
117C23×C67C22×D411yescharacteristic
118C22×D67—26no—
119C22×D67—46no—
120C22×D67—46no—
121C22×D67—46no—
122Q8×C67C2×SD16116yescomplemented
123C2×C4×C67—26no—
124C2×C4×C67C2×SD16112yes—
125C2×C4×C67C22×D416yes—
126C2×C4×C67C22×D411yescharacteristic
127C2×C4×C67C22×D416yes—
128C2×C4×C67C2×#4112yes—
129C2×C4×C67H(5,𝔽2)12yes—
130C2×C4×C67C2511yescharacteristic
131C2×Dic67—22no—
132C2×Dic67—26no—
133C2×Dic67—22no—
134C2×Dic67—46no—
135C2×Dic67—42no—
136C2×Dic67—42no—
137D268—26no—
138D268—26no—
139D268—46no—
140D268—424no—
141D268—812no—
142C4×D67—424no—
143Dic134—26no—
144Dic134—26no—
145Dic134—46no—
146C8×C67—26no—
147C8×C67C2×#4114yescomplemented
148C22×C67#55012yes—
149C22×C67C22×SD1612yes—
150C22×C67#57712yes—
151C22×C67C23×D411yescharacteristic
152C2×D67—46no—
153C2×D67—46no—
154C2×D67—46no—
155C2×D67—812no—
156C4×C67—26no—
157C4×C67#55012yes—
158C4×C67C22×SD16112yes—
159C4×C67C22×#4111yescharacteristic
160C4×C67C2×H(5,𝔽2)11yescharacteristic
161Dic67—46no—
162Dic67—42no—
163Dic67—42no—
164C2×#3392—671nomaximal, Sylow
165C2×C67#339214yescomplemented
166C2×C67C2×#55011yescharacteristic
167C2×C67C23×SD1611yescharacteristic
168C2×C67C2×#57711yescharacteristic
169D67—86no—
170C2×#557—671no—
171C2×#550—671no—
172#3392—6716no—
173C2×#492—673no—
174C2×#460—673no—
175C2×#461—673no—
176C2×C8×Q8D6711yescharacteristic, complemented
177C2×C4×SD16—673no—
178C67C2×#339211yescharacteristic, complemented, Sylow
179C22×#41—1341no—
180C22×SD16—676no—
181C22×SD16—673no—
182#557—678no—
183#557—1344no—
184#550—678no—
185#550—1344no—
186C2×C4⋊Q8—673no—
187C2×#127—673no—
188C2×#125—1343no—
189C2×#121—1343no—
190C2×C4×Q8—673no—
191C2×C4×Q8—1341no—
192C2×C4×Q8C2×D6711yescharacteristic
193C2×C4×D4—673no—
194#492—6724no—
195#461—6724no—
196#460—6724no—
197C8×Q8C2×D6718yescomplemented
198C4×SD16—6724no—
199C2×#106—671no—
200C2×C4⋊C8C2×D6713yes—
201C2×#103—673no—
202C2×#102—673no—
203C2×C4×C8C2×D6713yes—
204C2×#41—2688no—
205C2×#41—1348no—
206C22×Q8—673no—
207C22×Q8D26811yescharacteristic
208C22×D4—673no—
209C22×D4—1343no—
210C22×D4—1343no—
211C23×C4—1343no—
212C2×SD16—26812no—
213C2×SD16—6724no—
214C2×SD16—6712no—
215C2×SD16—13412no—
216C22×C8C22×D6713yes—
217C22×C8C22×D6711yescharacteristic
218C4⋊Q8—6712no—
219C4⋊Q8—1346no—
220#127—6712no—
221#127—1346no—
222#125—13424no—
223#121—13424no—
224C4×Q8—6712no—
225C4×Q8—1348no—
226C4×Q8—1346no—
227C4×Q8D26814yes—
228C4×Q8C22×D6714yes—
229C4×D4—6712no—
230C4×D4—1346no—
231C2×C4⋊C4—671no—
232C2×C4⋊C4—671no—
233C2×C4⋊C4—1343no—
234C2×C4⋊C4—1343no—
235C2×C4⋊C4—1343no—
236C2×C4⋊C4D26813yes—
237C2×C22⋊C4—1343no—
238C2×C42—1343no—
239C2×C42C22×D6713yes—
240#106—674no—
241C4⋊C8C22×D67112yes—
242#103—6712no—
243#102—6712no—
244C4×C8C22×D67112yes—
245C24—1343no—
246#41—26832no—
247C2×Q8—676no—
248C2×Q8—1346no—
249C2×Q8—1346no—
250C2×Q8—1343no—
251C2×Q8#1176161718yescomplemented
252C2×Q8C2×D26814yes—
253C2×D4—26812no—
254C2×D4—5366no—
255C2×D4—676no—
256C2×D4—1343no—
257C2×D4—13412no—
258C2×D4—13412no—
259C2×D4—1346no—
260C2×D4—1346no—
261C22×C4—26812no—
262C22×C4—1341no—
263C22×C4—1343no—
264C22×C4—1341no—
265C22×C4—13412no—
266C22×C4C2×D26813yes—
267C22×C4C2×D26813yes—
268C22×C4C23×D6711yescharacteristic
269SD16—26824no—
270SD16—13424no—
271C2×C8—26no—
272C2×C8#1176162914yescomplemented
273C2×C8C23×D6716yes—
274C2×C8C23×D6712yes—
275C4⋊C4—672no—
276C4⋊C4—672no—
277C4⋊C4—1341no—
278C4⋊C4—1341no—
279C4⋊C4—13412no—
280C4⋊C4—13412no—
281C4⋊C4—13412no—
282C4⋊C4C2×D268112yes—
283C22⋊C4—13412no—
284C42—13412no—
285C42C2×D26816yes—
286C42C23×D6716yes—
287C23—2686no—
288C23—2686no—
289C23—2686no—
290C23—1346no—
291C23C22×D26811yescharacteristic
292Q8—2686no—
293Q8—1346no—
294Q8—1346no—
295Q8C2×#11761617116yescomplemented
296D4—2686no—
297D4—26824no—
298D4—53612no—
299D4—1346no—
300D4—1346no—
301C2×C4—26no—
302C2×C4—2686no—
303C2×C4—2682no—
304C2×C4—2682no—
305C2×C4—26824no—
306C2×C4—1342no—
307C2×C4—1346no—
308C2×C4—1342no—
309C2×C4C2×#11761617112yes—
310C2×C4C22×D26816yes—
311C2×C4C22×D26811yescharacteristic
312C2×C4C22×D26816yes—
313C2×C4C2×#1176162912yes—
314C2×C4#1452408912yes—
315C2×C4C24×D6711yescharacteristic
316C8—26no—
317C8C2×#1176162914yescomplemented
318C22—2686no—
319C22—2686no—
320C22—2686no—
321C22—53612no—
322C22#1686406812yes—
323C22C22×#1176161712yes—
324C22#1686420812yes—
325C22C23×D26811yescharacteristic
326C4—26no—
327C4—2682no—
328C4—2686no—
329C4—2682no—
330C4#1686406812yes—
331C4C22×#11761617112yes—
332C4C22×#1176162911yescharacteristic
333C4C2×#1452408911yescharacteristic
334C2—5366no—
335C2#2257319114yesdirect product factor
336C2C2×#1686406811yescharacteristic
337C2C23×#1176161711yescharacteristic
338C2C2×#1686420811yescharacteristic
subgroupG/H
centerC23C22×D268
commutator, cosocleC2×C4×C67C25
FrattiniC2×C4C24×D67
FittingC2×C8×Q8×C67C2
socleC23×C67C22×D4
prime radicalC2×#16864068C2
hypercenterC2×C8×Q8D67
hypocenterC67C2×#3392
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×C8×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#16864125C2semidirect
3. #22573191C2direct (N × H)
4. C2×#16863769C2semidirect
5. C2×#16863597C2semidirect
6. C2×#16863602C2semidirect
7. C2×C4×#11761617C2semidirect
8. C2×C8×Q8×C67C2semidirect
13. #16864125C22semidirect
23. #16863769C22semidirect
24. #16863602C22semidirect
25. #16863597C22semidirect
26. C4×#11761617C22semidirect
31. C8×Q8×C67C22semidirect
105. C2×Q8×C67SD16semidirect
111. C2×C8×C67#41semidirect
122. Q8×C67C2×SD16semidirect
147. C8×C67C2×#41semidirect
165. C2×C67#3392semidirect
176. C2×C8×Q8D67semidirect
178. C67C2×#3392semidirect
197. C8×Q8C2×D67semidirect
251. C2×Q8#11761617semidirect
272. C2×C8#11761629semidirect
295. Q8C2×#11761617semidirect
317. C8C2×#11761629semidirect
335. C2#22573191direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168641252
C2×#168640682
#225731912
C2×#168637692
C2×#168635972
C2×#168636022
C2×C4×#117616172
C2×C8×Q8×C672
C2×#339267