Finite Group Databasebeta

← C2×#23776072 #35167866

Subgroups — C2×#23776072 #35167866

Order 20224.

2174 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×Q8×Dic158C211yesmaximal, characteristic, complemented
2C2×#19099235C211yesmaximal, characteristic
3#23776072C2116yesmaximal, direct product factor
4C2×#19098910C213yesmaximal, complemented
5C2×#19098854C213yesmaximal, complemented
6C2×#19098764C213yesmaximal, complemented
7C2×C4×#11929613C213yesmaximal, complemented
8C2×C8×Q8×C79C211yesmaximal, characteristic, complemented
9C22×#11929625—21no—
10C22×#11929613C2213yes—
11Q8×Dic158—24no—
12Q8×Dic158C2218yescomplemented
13#19099235—24no—
14#19099235C2218yes—
15C2×#14579161C2213yes—
16C2×#14579160—23no—
17C2×#14579158C2213yes—
18C2×#14579136—23no—
19C2×Q8×Dic79—21no—
20C2×C4×Dic158C2213yes—
21C2×C4×D316C2213yes—
22#19098910C22124yescomplemented
23#19098854C22124yescomplemented
24#19098764C22124yescomplemented
25C4×#11929613C22124yescomplemented
26C2×#14579093C2213yes—
27C2×#14579086C2211yescharacteristic
28C2×#14579083C2213yes—
29C2×#14579083C2216yes—
30C2×C4×Q8×C79C2211yescharacteristic
31C8×Q8×C79C2218yescomplemented
32C2×C4⋊C8×C79C2213yes—
33C2×C4×C8×C79C2213yes—
34C2×#11929625—28no—
35C2×#11929625—48no—
36C22×Dic158—23no—
37C22×Dic158C2311yescharacteristic
38C22×D316—23no—
39C22×D316—23no—
40C22×D316C2313yes—
41C22×C4×D79—23no—
42C2×#11929613—212no—
43C2×#11929613C23112yes—
44#14579161—26no—
45#14579161C23112yes—
46#14579160—224no—
47#14579158—26no—
48#14579158C23112yes—
49#14579136—224no—
50Q8×Dic79—28no—
51C4×Dic158—26no—
52C4×Dic158C23112yes—
53C4×D316—26no—
54C4×D316C23112yes—
55C2×C4⋊Dic79—23no—
56C2×C4⋊Dic79C2311yescharacteristic
57C2×C4⋊Dic79C2313yes—
58C2×Dic79⋊C4—23no—
59C2×(C2×D79)⋊C4—23no—
60C2×C4×Dic79—23no—
61#14579093C23112yes—
62#14579086C2314yes—
63#14579083C23112yes—
64#14579083C23124yes—
65C22×Q8×C79D411yescharacteristic
66C22×C8×C79C2313yes—
67C22×C8×C79C2311yescharacteristic
68C4×Q8×C79D414yes—
69C4×Q8×C79C2314yes—
70C2×C4⋊C4×C79D413yes—
71C2×C42×C79C2313yes—
72C4⋊C8×C79C23112yes—
73C4×C8×C79C23112yes—
74C23×D79—23no—
75#11929625—432no—
76C2×Dic158—21no—
77C2×Dic158—212no—
78C2×Dic158—22no—
79C2×Dic158—86no—
80C2×Dic158#4114yescomplemented
81C2×Dic158C2412yes—
82C2×D316—212no—
83C2×D316—23no—
84C2×D316—212no—
85C2×D316—26no—
86C2×D316—26no—
87C2×D316—412no—
88C2×D316—86no—
89C2×D316C2416yes—
90C22×Dic79—21no—
91C22×Dic79—23no—
92C22×Dic79—21no—
93C2×C4×D79—212no—
94C2×C4×D79—412no—
95#11929613—224no—
96C4⋊Dic79—212no—
97C4⋊Dic79—21no—
98C4⋊Dic79—23no—
99C4⋊Dic79C2416yes—
100C4⋊Dic79C2412yes—
101Dic79⋊C4—212no—
102(C2×D79)⋊C4—212no—
103C4×Dic79—212no—
104C2×Q8×C79SD1618yescomplemented
105C2×Q8×C79C2×D414yes—
106C22×C4×C79C2×D413yes—
107C22×C4×C79C2×D413yes—
108C22×C4×C79C2411yescharacteristic
109C2×C8×C79—26no—
110C2×C8×C79—22no—
111C2×C8×C79C2412yes—
112C2×C8×C79C2416yes—
113C4⋊C4×C79C2×D4112yes—
114C42×C79C2×D416yes—
115C42×C79C2416yes—
116C23×C79C22×D411yescharacteristic
117C22×D79—26no—
118C22×D79—46no—
119C22×D79—46no—
120C22×D79—46no—
121Q8×C79C2×SD16116yescomplemented
122C2×C4×C79—26no—
123C2×C4×C79C2×SD16112yes—
124C2×C4×C79C22×D416yes—
125C2×C4×C79C22×D411yescharacteristic
126C2×C4×C79C22×D416yes—
127C2×C4×C79C2×#4112yes—
128C2×C4×C792−1+412yes—
129C2×C4×C79C2511yescharacteristic
130C2×Dic79—24no—
131C2×Dic79—22no—
132C2×Dic79—26no—
133C2×Dic79—22no—
134C2×Dic79—42no—
135C2×Dic79—46no—
136D316—26no—
137D316—26no—
138D316—46no—
139D316—424no—
140D316—812no—
141C4×D79—424no—
142Dic158—24no—
143Dic158—22no—
144Dic158—812no—
145Dic158C2×#4114yescomplemented
146C8×C79—26no—
147C8×C79—22no—
148C22×C79#54812yes—
149C22×C79C22×SD1612yes—
150C22×C79#57612yes—
151C22×C79C23×D411yescharacteristic
152C2×D79—46no—
153C2×D79—46no—
154C2×D79—46no—
155C2×D79—812no—
156C4×C79—26no—
157C4×C79#54812yes—
158C4×C79C22×SD16112yes—
159C4×C79C22×#4111yescharacteristic
160C4×C79C2×2−1+411yescharacteristic
161Dic79—24no—
162Dic79—42no—
163Dic79—46no—
164C2×#3362—791nomaximal, Sylow
165C2×C79#336214yescomplemented
166C2×C79C2×#54811yescharacteristic
167C2×C79C23×SD1611yescharacteristic
168C2×C79C2×#57611yescharacteristic
169D79—86no—
170C2×Q82—791no—
171C2×#550—791no—
172#3362—7916no—
173C2×#486—793no—
174C2×#476—793no—
175C2×#460—793no—
176C2×C8×Q8D7911yescharacteristic, complemented
177C2×C4×SD16—793no—
178C79C2×#336211yescharacteristic, complemented, Sylow
179C22×#41—1581no—
180C22×SD16—793no—
181Q82—798no—
182Q82—1584no—
183#550—798no—
184#550—1584no—
185C2×C4⋊Q8—793no—
186C2×C4⋊Q8—1583no—
187C2×#127—793no—
188C2×#121—1583no—
189C2×C4×Q8—793no—
190C2×C4×Q8—1581no—
191C2×C4×Q8C2×D7911yescharacteristic
192C2×C4×D4—793no—
193#486—7924no—
194#476—7924no—
195#460—7924no—
196C8×Q8C2×D7918yescomplemented
197C4×SD16—7924no—
198C2×#106—793no—
199C2×C4⋊C8C2×D7913yes—
200C2×#103—791no—
201C2×#102—793no—
202C2×#102—796no—
203C2×C4×C8C2×D7913yes—
204C2×#41—3168no—
205C2×#41—1588no—
206C22×Q8—791no—
207C22×Q8—1583no—
208C22×Q8D31611yescharacteristic
209C22×D4—793no—
210C22×D4—1583no—
211C22×D4—1583no—
212C23×C4—1583no—
213C2×SD16—7912no—
214C2×SD16—15812no—
215C22×C8C22×D7913yes—
216C22×C8C22×D7911yescharacteristic
217C4⋊Q8—7912no—
218C4⋊Q8—15824no—
219C4⋊Q8—1586no—
220#127—7912no—
221#127—1586no—
222#121—15824no—
223C4×Q8—7912no—
224C4×Q8—1588no—
225C4×Q8—1586no—
226C4×Q8D31614yes—
227C4×Q8C22×D7914yes—
228C4×D4—7912no—
229C4×D4—1586no—
230C2×C4⋊C4—791no—
231C2×C4⋊C4—793no—
232C2×C4⋊C4—1583no—
233C2×C4⋊C4—1583no—
234C2×C4⋊C4D31613yes—
235C2×C22⋊C4—1583no—
236C2×C42—1583no—
237C2×C42C22×D7913yes—
238#106—7912no—
239C4⋊C8C22×D79112yes—
240#103—794no—
241#102—7912no—
242#102—7924no—
243C4×C8C22×D79112yes—
244C24—1583no—
245#41—31632no—
246C2×Q8—794no—
247C2×Q8—792no—
248C2×Q8—6326no—
249C2×Q8—1582no—
250C2×Q8—15812no—
251C2×Q8—1581no—
252C2×Q8#1192961318yescomplemented
253C2×Q8C2×D31614yes—
254C2×D4—31612no—
255C2×D4—796no—
256C2×D4—6326no—
257C2×D4—1586no—
258C2×D4—1586no—
259C2×D4—1583no—
260C2×D4—15812no—
261C2×D4—15812no—
262C22×C4—31612no—
263C22×C4—1583no—
264C22×C4—1581no—
265C22×C4—15812no—
266C22×C4—1581no—
267C22×C4C2×D31613yes—
268C22×C4C2×D31613yes—
269C22×C4C23×D7911yescharacteristic
270SD16—15824no—
271C2×C8—26no—
272C2×C8—22no—
273C2×C8C23×D7912yes—
274C2×C8C23×D7916yes—
275C4⋊C4—796no—
276C4⋊C4—792no—
277C4⋊C4—15812no—
278C4⋊C4—1581no—
279C4⋊C4—1583no—
280C4⋊C4—15812no—
281C4⋊C4C2×D316112yes—
282C22⋊C4—15812no—
283C42—15812no—
284C42C2×D31616yes—
285C42C23×D7916yes—
286C23—3166no—
287C23—3166no—
288C23—3166no—
289C23—1586no—
290C23C22×D31611yescharacteristic
291Q8—794no—
292Q8—63212no—
293Q8—1582no—
294Q8—1584no—
295Q8C2×#11929613116yescomplemented
296D4—3166no—
297D4—31624no—
298D4—63212no—
299D4—1586no—
300D4—1586no—
301C2×C4—26no—
302C2×C4—3162no—
303C2×C4—3166no—
304C2×C4—31624no—
305C2×C4—1586no—
306C2×C4—1582no—
307C2×C4—1584no—
308C2×C4—1582no—
309C2×C4C2×#11929613112yes—
310C2×C4C22×D31616yes—
311C2×C4C22×D31611yescharacteristic
312C2×C4C22×D31616yes—
313C2×C4#41×D7912yes—
314C2×C4#1457920912yes—
315C2×C4C24×D7911yescharacteristic
316C8—22no—
317C8—26no—
318C22—3166no—
319C22—3166no—
320C22—3166no—
321C22—63212no—
322C22#1909922512yes—
323C22C22×#1192961312yes—
324C22#1909936812yes—
325C22C23×D31611yescharacteristic
326C4—26no—
327C4—3162no—
328C4—3166no—
329C4—1584no—
330C4#1909922512yes—
331C4C22×#11929613112yes—
332C4C2×#41×D7911yescharacteristic
333C4C2×#1457920911yescharacteristic
334C2—6326no—
335C2#2377607214yesdirect product factor
336C2C2×#1909922511yescharacteristic
337C2C23×#1192961311yescharacteristic
338C2C2×#1909936811yescharacteristic
subgroupG/H
centerC23C22×D316
commutator, cosocleC2×C4×C79C25
FrattiniC2×C4C24×D79
FittingC2×C8×Q8×C79C2
socleC23×C79C22×D4
prime radicalC2×#19099235C2
hypercenterC2×C8×Q8D79
hypocenterC79C2×#3362
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×C8×Q8

Complements

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

NH ≅ G/Ntype
1. C2×Q8×Dic158C2semidirect
3. #23776072C2direct (N × H)
4. C2×#19098910C2semidirect
5. C2×#19098854C2semidirect
6. C2×#19098764C2semidirect
7. C2×C4×#11929613C2semidirect
8. C2×C8×Q8×C79C2semidirect
12. Q8×Dic158C22semidirect
22. #19098910C22semidirect
23. #19098854C22semidirect
24. #19098764C22semidirect
25. C4×#11929613C22semidirect
31. C8×Q8×C79C22semidirect
80. C2×Dic158#41semidirect
104. C2×Q8×C79SD16semidirect
121. Q8×C79C2×SD16semidirect
145. Dic158C2×#41semidirect
165. C2×C79#3362semidirect
176. C2×C8×Q8D79semidirect
178. C79C2×#3362semidirect
196. C8×Q8C2×D79semidirect
252. C2×Q8#11929613semidirect
295. Q8C2×#11929613semidirect
335. C2#23776072direct (N × H)

Maximal subgroups

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

subgroupindex
C2×Q8×Dic1582
C2×#190992352
#237760722
C2×#190989102
C2×#190988542
C2×#190987642
C2×C4×#119296132
C2×C8×Q8×C792
C2×#336279