Finite Group Databasebeta

← C2×#23776038 #35167829

Subgroups — C2×#23776038 #35167829

Order 20224.

2058 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×D316C211yesmaximal, characteristic
2#23776038C2116yesmaximal, direct product factor
3C2×#19098913C213yesmaximal, complemented
4C2×#19098848C213yesmaximal, complemented
5C2×#19098765C213yesmaximal
6C2×Q8×C79⋊C8C211yesmaximal, characteristic, complemented
7C2×C4×#11929611C213yesmaximal
8C2×#557×C79C211yesmaximal, characteristic, complemented
9C22×Q8×D79—21no—
10C22×#11929611C2213yes—
11Q8×D316—24no—
12Q8×D316C2218yes—
13C2×#14579161C2213yes—
14C2×#14579141—23no—
15C2×C4×D316C2213yes—
16#19098913C22124yescomplemented
17#19098848C22124yescomplemented
18#19098765C22124yes—
19Q8×C79⋊C8C2218yescomplemented
20C4×#11929611C22124yes—
21C2×#14579092C2213yes—
22C2×C4⋊(C79⋊C8)C2213yes—
23C2×#14579085C2213yes—
24C2×#14579085C2216yes—
25C2×#14579081C2211yescharacteristic
26C2×C4×C79⋊C8C2213yes—
27#557×C79—24no—
28#557×C79C2218yescomplemented
29C2×C4⋊Q8×C79C2213yes—
30C2×#125×C79—23no—
31C2×C4×Q8×C79—21no—
32C2×C4×Q8×C79C2213yes—
33C2×C4×Q8×C79C2211yescharacteristic
34C22×Dic158—23no—
35C2×Q8×D79—28no—
36C2×Q8×D79—48no—
37C22×D316C2311yescharacteristic
38C22×C4×D79—23no—
39C2×#11929611—212no—
40C2×#11929611C23112yes—
41C22×C79⋊C8C2311yescharacteristic
42C22×C79⋊C8C2313yes—
43#14579161—26no—
44#14579161C23112yes—
45#14579141—224no—
46C4×D316—26no—
47C4×D316C23112yes—
48C2×C4⋊Dic79—23no—
49C2×C4⋊Dic79C2313yes—
50C2×(C2×D79)⋊C4—23no—
51#14579092C23112yes—
52C4⋊(C79⋊C8)C23112yes—
53#14579085C23112yes—
54#14579085C23124yes—
55#14579081C2314yes—
56C4×C79⋊C8C23112yes—
57C22×Q8×C79D411yescharacteristic
58C22×Q8×C79C2313yes—
59C4⋊Q8×C79—26no—
60C4⋊Q8×C79C23112yes—
61#125×C79—224no—
62C4×Q8×C79—28no—
63C4×Q8×C79—26no—
64C4×Q8×C79D414yes—
65C4×Q8×C79C23112yes—
66C4×Q8×C79C2314yes—
67C2×C4⋊C4×C79—23no—
68C2×C4⋊C4×C79—23no—
69C2×C4⋊C4×C79—23no—
70C2×C4⋊C4×C79D413yes—
71C2×C4⋊C4×C79C2311yescharacteristic
72C2×C42×C79—23no—
73C2×C42×C79C2313yes—
74C23×D79—21no—
75C2×Dic158—212no—
76C2×Dic158—412no—
77Q8×D79—432no—
78C2×D316—21no—
79C2×D316—22no—
80C2×D316#4114yes—
81C2×D316C2412yes—
82C22×Dic79—23no—
83C2×C4×D79—212no—
84C2×C4×D79—412no—
85#11929611—224no—
86C2×C79⋊C8—26no—
87C2×C79⋊C8—22no—
88C2×C79⋊C8C2412yes—
89C2×C79⋊C8C2416yes—
90C4⋊Dic79—212no—
91C4⋊Dic79—23no—
92C4⋊Dic79C2416yes—
93(C2×D79)⋊C4—212no—
94C2×Q8×C79—23no—
95C2×Q8×C79—26no—
96C2×Q8×C79—26no—
97C2×Q8×C79SD1618yescomplemented
98C2×Q8×C79C2×D414yes—
99C2×Q8×C79C2416yes—
100C22×C4×C79—21no—
101C22×C4×C79—23no—
102C22×C4×C79C2×D413yes—
103C22×C4×C79C2×D413yes—
104C22×C4×C79C2411yescharacteristic
105C4⋊C4×C79—212no—
106C4⋊C4×C79—212no—
107C4⋊C4×C79—212no—
108C4⋊C4×C79—21no—
109C4⋊C4×C79C2×D4112yes—
110C4⋊C4×C79C2412yes—
111C42×C79—212no—
112C42×C79C2×D416yes—
113C42×C79C2416yes—
114C23×C79C22×D411yescharacteristic
115C22×D79—24no—
116C22×D79—22no—
117C22×D79—42no—
118C22×D79—42no—
119Q8×C79—26no—
120Q8×C79—26no—
121Q8×C79—46no—
122Q8×C79C2×SD16116yescomplemented
123C2×C4×C79—26no—
124C2×C4×C79—22no—
125C2×C4×C79—26no—
126C2×C4×C79—46no—
127C2×C4×C79—42no—
128C2×C4×C79C2×SD16112yes—
129C2×C4×C79C22×D416yes—
130C2×C4×C79C22×D411yescharacteristic
131C2×C4×C79C22×D416yes—
132C2×C4×C79C2×#4112yes—
133C2×C4×C792−1+412yes—
134C2×C4×C79C2511yescharacteristic
135C2×Dic79—26no—
136C2×Dic79—46no—
137D316—24no—
138D316—22no—
139D316C2×#4114yes—
140C4×D79—424no—
141Dic158—424no—
142C79⋊C8—26no—
143C79⋊C8—22no—
144C22×C79#54812yes—
145C22×C79C22×SD1612yes—
146C22×C79#57812yes—
147C22×C79C23×D411yescharacteristic
148C2×D79—24no—
149C2×D79—48no—
150C2×D79—42no—
151C2×D79—42no—
152C4×C79—26no—
153C4×C79—46no—
154C4×C79—42no—
155C4×C79#54812yes—
156C4×C79C22×SD16112yes—
157C4×C79C22×#4111yescharacteristic
158C4×C79C2×2−1+411yescharacteristic
159Dic79—46no—
160C2×#3359—791nomaximal, Sylow
161C2×C79#335914yescomplemented
162C2×C79C2×#54811yescharacteristic
163C2×C79C23×SD1611yescharacteristic
164C2×C79C2×#57811yescharacteristic
165D79—44no—
166C2×#557D7911yescharacteristic, complemented
167C2×D4×Q8—791no—
168#3359—7916no—
169C2×#487—793no—
170C2×#475—793no—
171C2×#461—793no—
172C2×C8×Q8—791no—
173C2×C4×SD16—793no—
174C79C2×#335911yescharacteristic, complemented, Sylow
175C23×Q8—1581no—
176C22×SD16—793no—
177#557—24no—
178#557C2×D7918yescomplemented
179D4×Q8—798no—
180D4×Q8—1584no—
181C2×C4⋊Q8—793no—
182C2×C4⋊Q8C2×D7913yes—
183C2×#125—23no—
184C2×C22⋊Q8—1583no—
185C2×C4×Q8—21no—
186C2×C4×Q8C2×D7913yes—
187C2×C4×Q8C2×D7911yescharacteristic
188C2×C4×D4—793no—
189#487—7924no—
190#475—7924no—
191#461—7924no—
192C8×Q8—798no—
193C4×SD16—7924no—
194C2×#106—793no—
195C2×C4⋊C8—793no—
196C2×#103—796no—
197C2×#103—793no—
198C2×#102—791no—
199C2×C4×C8—793no—
200C22×Q8—3168no—
201C22×Q8—1588no—
202C22×Q8—1583no—
203C22×Q8C22⋊D7911yescharacteristic
204C22×Q8C22×D7913yes—
205C22×D4—791no—
206C23×C4—1583no—
207C2×SD16—7912no—
208C2×SD16—15812no—
209C22×C8—793no—
210C22×C8—791no—
211C4⋊Q8—26no—
212C4⋊Q8—7912no—
213C4⋊Q8—1586no—
214C4⋊Q8C22×D79112yes—
215#125—224no—
216C22⋊Q8—15824no—
217C4×Q8—28no—
218C4×Q8—26no—
219C4×Q8C22⋊D7914yes—
220C4×Q8C22×D79112yes—
221C4×Q8C22×D7914yes—
222C4×D4—7912no—
223C4×D4—1586no—
224C2×C4⋊C4—23no—
225C2×C4⋊C4—23no—
226C2×C4⋊C4—23no—
227C2×C4⋊C4—793no—
228C2×C4⋊C4—1583no—
229C2×C4⋊C4C22⋊D7913yes—
230C2×C4⋊C4C22×D7911yescharacteristic
231C2×C22⋊C4—1583no—
232C2×C42—23no—
233C2×C42C22×D7913yes—
234#106—7912no—
235C4⋊C8—7912no—
236#103—7912no—
237#103—7924no—
238#102—794no—
239C4×C8—7912no—
240C24—1581no—
241C2×Q8—23no—
242C2×Q8—26no—
243C2×Q8—26no—
244C2×Q8—31632no—
245C2×Q8—31612no—
246C2×Q8—15812no—
247C2×Q8#1192961118yescomplemented
248C2×Q8C2×C22⋊D7914yes—
249C2×Q8C23×D7916yes—
250C2×D4—794no—
251C2×D4—792no—
252C2×D4—1582no—
253C2×D4—1581no—
254C22×C4—21no—
255C22×C4—23no—
256C22×C4—31612no—
257C22×C4—1583no—
258C22×C4—15812no—
259C22×C4C2×C22⋊D7913yes—
260C22×C4C2×C22⋊D7913yes—
261C22×C4C23×D7911yescharacteristic
262SD16—15824no—
263C2×C8—792no—
264C2×C8—796no—
265C2×C8—1586no—
266C2×C8—1582no—
267C4⋊C4—212no—
268C4⋊C4—212no—
269C4⋊C4—212no—
270C4⋊C4—21no—
271C4⋊C4—796no—
272C4⋊C4—1583no—
273C4⋊C4—15812no—
274C4⋊C4C2×C22⋊D79112yes—
275C4⋊C4C23×D7912yes—
276C22⋊C4—15812no—
277C42—212no—
278C42C2×C22⋊D7916yes—
279C42C23×D7916yes—
280C23—3162no—
281C23—3162no—
282C23—1584no—
283C23—1582no—
284C23C22×C22⋊D7911yescharacteristic
285Q8—26no—
286Q8—26no—
287Q8—46no—
288Q8—31624no—
289Q8C2×#11929611116yescomplemented
290D4—794no—
291D4—1582no—
292D4—1584no—
293C2×C4—26no—
294C2×C4—22no—
295C2×C4—26no—
296C2×C4—46no—
297C2×C4—42no—
298C2×C4—3166no—
299C2×C4—31624no—
300C2×C4—1586no—
301C2×C4C2×#11929611112yes—
302C2×C4C22×C22⋊D7916yes—
303C2×C4C22×C22⋊D7911yescharacteristic
304C2×C4C22×C22⋊D7916yes—
305C2×C4#41×D7912yes—
306C2×C4#1457920812yes—
307C2×C4C24×D7911yescharacteristic
308C8—1582no—
309C8—1586no—
310C22—3168no—
311C22—3162no—
312C22—3162no—
313C22—1584no—
314C22#1909922012yes—
315C22C22×#1192961112yes—
316C22#1909937812yes—
317C22C23×C22⋊D7911yescharacteristic
318C4—26no—
319C4—46no—
320C4—42no—
321C4—3166no—
322C4#1909922012yes—
323C4C22×#11929611112yes—
324C4C2×#41×D7911yescharacteristic
325C4C2×#1457920811yescharacteristic
326C2—3164no—
327C2#2377603814yesdirect product factor
328C2C2×#1909922011yescharacteristic
329C2C23×#1192961111yescharacteristic
330C2C2×#1909937811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D79
commutator, cosocleC2×C4×C79C25
FrattiniC2×C4C24×D79
FittingC2×#557×C79C2
socleC23×C79C22×D4
prime radicalC22×D316C23
hypercenterC2×#557D79
hypocenterC79C2×#3359
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#557

Complements

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

NH ≅ G/Ntype
2. #23776038C2direct (N × H)
3. C2×#19098913C2semidirect
4. C2×#19098848C2semidirect
6. C2×Q8×C79⋊C8C2semidirect
8. C2×#557×C79C2semidirect
16. #19098913C22semidirect
17. #19098848C22semidirect
19. Q8×C79⋊C8C22semidirect
28. #557×C79C22semidirect
97. C2×Q8×C79SD16semidirect
122. Q8×C79C2×SD16semidirect
161. C2×C79#3359semidirect
166. C2×#557D79semidirect
174. C79C2×#3359semidirect
178. #557C2×D79semidirect
247. C2×Q8#11929611semidirect
289. Q8C2×#11929611semidirect
327. C2#23776038direct (N × H)

Maximal subgroups

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

subgroupindex
C2×Q8×D3162
#237760382
C2×#190989132
C2×#190988482
C2×#190987652
C2×Q8×C79⋊C82
C2×C4×#119296112
C2×#557×C792
C2×#335979