Finite Group Databasebeta

← C2×#23766318 #34447575

Subgroups — C2×#23766318 #34447575

Order 20224.

3158 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×Q8×Dic79C211yesmaximal, characteristic
2C22×Q8×Dic79C211yesmaximal, characteristic
3#23766318C2116yesmaximal, direct product factor
4C2×#19098461C213yesmaximal
5C2×#19098459C213yesmaximal
6C2×#19098428C213yesmaximal
7C2×C4×Dic79⋊C4C213yesmaximal
8C22×C4×Q8×C79C211yesmaximal, characteristic
9C2×Q8×Dic79—28no—
10C2×Q8×Dic79C4116yescomplemented
11C2×Q8×Dic79C2218yes—
12C2×Q8×Dic79C2218yes—
13C22×C4⋊Dic79C2213yes—
14C22×C4⋊Dic79C2213yes—
15C22×Dic79⋊C4C2213yes—
16C22×Dic79⋊C4C2213yes—
17C22×C4×Dic79C2213yes—
18C22×C4×Dic79C2213yes—
19#19098461C22124yes—
20#19098459C22124yes—
21#19098428C22124yes—
22C4×Dic79⋊C4C22124yes—
23C2×C79⋊#95C2213yes—
24C2×C79⋊#95C2216yes—
25C2×C79⋊#95C2211yescharacteristic
26C23×Q8×C79C2211yescharacteristic
27C2×C4×Q8×C79—28no—
28C2×C4×Q8×C79C2218yes—
29C22×C4⋊C4×C79C2213yes—
30C22×C42×C79C2213yes—
31C23×Dic79C2313yes—
32C23×Dic79C2311yescharacteristic
33C23×Dic79C2313yes—
34C23×Dic79C2311yescharacteristic
35Q8×Dic79—232no—
36Q8×Dic79C2×C4164yescomplemented
37C2×C4⋊Dic79—212no—
38C2×C4⋊Dic79C2×C4124yes—
39C2×C4⋊Dic79C23112yes—
40C2×C4⋊Dic79C23112yes—
41C2×Dic79⋊C4—212no—
42C2×Dic79⋊C4—212no—
43C2×Dic79⋊C4C23112yes—
44C2×Dic79⋊C4C23112yes—
45C2×C4×Dic79—212no—
46C2×C4×Dic79C2×C4124yes—
47C2×C4×Dic79C23112yes—
48C2×C4×Dic79C23112yes—
49C79⋊#95C2314yes—
50C79⋊#95C23112yes—
51C79⋊#95C23124yes—
52C22×Q8×C79C2×C418yes—
53C22×Q8×C79D418yes—
54C22×Q8×C79Q818yes—
55C22×Q8×C79C2314yes—
56C23×C4×C79C2313yes—
57C23×C4×C79C2311yescharacteristic
58C23×C4×C79C2313yes—
59C4×Q8×C79—232no—
60C2×C4⋊C4×C79—212no—
61C2×C4⋊C4×C79C23112yes—
62C2×C42×C79—212no—
63C2×C42×C79C23112yes—
64C22×Dic79—22no—
65C22×Dic79—26no—
66C22×Dic79—26no—
67C22×Dic79—26no—
68C22×Dic79—26no—
69C22×Dic79—22no—
70C22×Dic79—22no—
71C22×Dic79—22no—
72C22×Dic79—26no—
73C22×Dic79C22×C414yes—
74C22×Dic79C22×C4112yes—
75C22×Dic79#4114yes—
76C22×Dic79C2416yes—
77C22×Dic79C2412yes—
78C22×Dic79C2416yes—
79C22×Dic79C2412yes—
80C4⋊Dic79—224no—
81C4⋊Dic79C22×C4148yes—
82Dic79⋊C4—224no—
83Dic79⋊C4—224no—
84C4×Dic79—224no—
85C4×Dic79C22×C4148yes—
86C24×C79C2411yescharacteristic
87C2×Q8×C79C4⋊C4164yescomplemented
88C2×Q8×C79C22×C4116yes—
89C2×Q8×C79C2×D4116yes—
90C2×Q8×C79C2×Q8116yes—
91C22×C4×C79—22no—
92C22×C4×C79—22no—
93C22×C4×C79—26no—
94C22×C4×C79—26no—
95C22×C4×C79—26no—
96C22×C4×C79C22×C4112yes—
97C22×C4×C79C2×D4112yes—
98C22×C4×C79C2×Q8112yes—
99C22×C4×C79#4114yes—
100C22×C4×C79C2416yes—
101C22×C4×C79C2412yes—
102C22×C4×C79C2416yes—
103C4⋊C4×C79—224no—
104C42×C79—224no—
105C23×C79C23×C412yes—
106C23×C79C22×D412yes—
107C23×C79C22×Q812yes—
108C23×C79C2×#4112yes—
109C23×C79C2×#4112yes—
110C23×C792−1+412yes—
111C23×C792−1+412yes—
112C23×C79C2511yescharacteristic
113Q8×C79C2×C4⋊C4164yescomplemented
114C2×C4×C79—28no—
115C2×C4×C79—22no—
116C2×C4×C79—22no—
117C2×C4×C79—26no—
118C2×C4×C79—26no—
119C2×C4×C79—26no—
120C2×C4×C79—412no—
121C2×C4×C79C2×C4⋊C4148yes—
122C2×C4×C79C23×C4112yes—
123C2×C4×C79C22×D4112yes—
124C2×C4×C79C22×Q8112yes—
125C2×C4×C79C2×#4114yes—
126C2×Dic79—224no—
127C2×Dic79—28no—
128C2×Dic79—28no—
129C2×Dic79—22no—
130C2×Dic79—26no—
131C2×Dic79—26no—
132C2×Dic79—22no—
133C2×Dic79—22no—
134C2×Dic79—22no—
135C2×Dic79—26no—
136C2×Dic79—26no—
137C2×Dic79—26no—
138C2×Dic79—412no—
139C2×Dic79C23×C414yes—
140C2×Dic79C23×C4112yes—
141C2×Dic79C2×#4114yes—
142C22×C79C22×C4⋊C414yes—
143C22×C79C4×#4114yes—
144C22×C79#51914yes—
145C22×C79#54814yes—
146C22×C79#54814yes—
147C22×C79#55714yes—
148C22×C79#55714yes—
149C22×C79C24×C411yescharacteristic
150C22×C79C23×D411yescharacteristic
151C22×C79C23×Q811yescharacteristic
152C22×C79C22×#4111yescharacteristic
153C22×C79C22×#4111yescharacteristic
154C22×C79C2×2−1+411yescharacteristic
155C22×C79C2×2−1+411yescharacteristic
156C4×C79—24no—
157C4×C79—46no—
158C4×C79C22×C4⋊C4124yes—
159Dic79—212no—
160Dic79—24no—
161Dic79—24no—
162Dic79—46no—
163C2×#2351—791nomaximal, Sylow
164C2×C79#235118yescomplemented
165C2×C79C23×C4⋊C411yescharacteristic
166C2×C79C2×C4×#4111yescharacteristic
167C2×C79C2×#51911yescharacteristic
168C2×C79C2×#54811yescharacteristic
169C2×C79C2×#54811yescharacteristic
170C2×C79C2×#55711yescharacteristic
171C2×C79C2×#55711yescharacteristic
172C22×C4×Q8—791no—
173C22×C4×Q8—791no—
174C22×C4×Q8D7911yescharacteristic
175#2351—7916no—
176C2×#389—793no—
177C2×#389—793no—
178C2×#382—793no—
179C2×C4×C4⋊C4—793no—
180C79C2×#235111yescharacteristic, complemented, Sylow
181C23×Q8C2×D7911yescharacteristic
182C2×C4×Q8—28no—
183C2×C4×Q8—798no—
184C2×C4×Q8—7916no—
185C2×C4×Q8—798no—
186C2×C4×Q8—1588no—
187C2×C4×Q8C2×D7918yes—
188C22×C4⋊C4—793no—
189C22×C4⋊C4—793no—
190C22×C4⋊C4—793no—
191C22×C4⋊C4—793no—
192C22×C4⋊C4C2×D7913yes—
193C22×C42—793no—
194C22×C42—793no—
195C22×C42C2×D7913yes—
196#389—7924no—
197#389—7924no—
198#382—7924no—
199C4×C4⋊C4—7924no—
200C2×#95—791no—
201C2×#95—793no—
202C2×#95—796no—
203C22×Q8Dic15818yes—
204C22×Q8C4×D7918yes—
205C22×Q8C22⋊D7918yes—
206C22×Q8C22×D7914yes—
207C23×C4—793no—
208C23×C4—791no—
209C23×C4—791no—
210C23×C4—793no—
211C23×C4C22×D7913yes—
212C23×C4C22×D7911yescharacteristic
213C23×C4C22×D7913yes—
214C4×Q8—232no—
215C4×Q8—7964no—
216C4×Q8—15832no—
217C2×C4⋊C4—212no—
218C2×C4⋊C4—7924no—
219C2×C4⋊C4—7912no—
220C2×C4⋊C4—7912no—
221C2×C4⋊C4—7912no—
222C2×C4⋊C4—7912no—
223C2×C4⋊C4—15812no—
224C2×C4⋊C4—15812no—
225C2×C4⋊C4—15812no—
226C2×C4⋊C4C22×D79112yes—
227C2×C42—212no—
228C2×C42—7924no—
229C2×C42—7912no—
230C2×C42—7912no—
231C2×C42—15812no—
232C2×C42C22×D79112yes—
233#95—794no—
234#95—7912no—
235#95—7924no—
236C24C23×D7911yescharacteristic
237C2×Q8Dic79⋊C4164yescomplemented
238C2×Q8C2×C4×D79116yes—
239C2×Q8C2×C22⋊D79116yes—
240C2×Q8C2×Dic158116yes—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—26no—
244C22×C4—26no—
245C22×C4—26no—
246C22×C4—794no—
247C22×C4—792no—
248C22×C4—796no—
249C22×C4—792no—
250C22×C4—794no—
251C22×C4—7912no—
252C22×C4—796no—
253C22×C4—1586no—
254C22×C4—1586no—
255C22×C4—1586no—
256C22×C4—1586no—
257C22×C4—1586no—
258C22×C4—1582no—
259C22×C4—1582no—
260C22×C4—1582no—
261C22×C4—1582no—
262C22×C4C2×C4×D79112yes—
263C22×C4C2×C22⋊D79112yes—
264C22×C4C2×Dic158112yes—
265C22×C4#1192962514yes—
266C22×C4C23×D7916yes—
267C22×C4C23×D7912yes—
268C22×C4C23×D7916yes—
269C4⋊C4—224no—
270C4⋊C4—7948no—
271C4⋊C4—15824no—
272C4⋊C4—15824no—
273C4⋊C4—15824no—
274C42—224no—
275C42—7948no—
276C42—15824no—
277C23C22×C4×D7912yes—
278C23C22×C22⋊D7912yes—
279C23C22×Dic15812yes—
280C23#41×D7912yes—
281C23C2×#1192962512yes—
282C23#1457920812yes—
283C23#1457920912yes—
284C23C24×D7911yescharacteristic
285Q8C2×Dic79⋊C4164yescomplemented
286C2×C4—22no—
287C2×C4—28no—
288C2×C4—22no—
289C2×C4—26no—
290C2×C4—26no—
291C2×C4—26no—
292C2×C4—412no—
293C2×C4—31612no—
294C2×C4—794no—
295C2×C4—7912no—
296C2×C4—794no—
297C2×C4—1586no—
298C2×C4—1586no—
299C2×C4—1586no—
300C2×C4—1582no—
301C2×C4—1582no—
302C2×C4—1582no—
303C2×C4—1588no—
304C2×C4—15824no—
305C2×C4—1588no—
306C2×C4—1582no—
307C2×C4—1586no—
308C2×C4—1586no—
309C2×C4C2×Dic79⋊C4148yes—
310C2×C4C22×C4×D79112yes—
311C2×C4C22×C22⋊D79112yes—
312C2×C4C22×Dic158112yes—
313C2×C4C2×#1192962514yes—
314C22C22×Dic79⋊C414yes—
315C22C4×#1192962514yes—
316C22#1909903914yes—
317C22#1909922014yes—
318C22#1909922414yes—
319C22#1909928914yes—
320C22#1909929214yes—
321C22C23×C4×D7911yescharacteristic
322C22C23×C22⋊D7911yescharacteristic
323C22C23×Dic15811yescharacteristic
324C22C2×#41×D7911yescharacteristic
325C22C22×#1192962511yescharacteristic
326C22C2×#1457920811yescharacteristic
327C22C2×#1457920911yescharacteristic
328C4—24no—
329C4—46no—
330C4—3166no—
331C4—15812no—
332C4—1584no—
333C4—1584no—
334C4C22×Dic79⋊C4124yes—
335C2#2376631818yesdirect product factor
336C2C23×Dic79⋊C411yescharacteristic
337C2C2×C4×#1192962511yescharacteristic
338C2C2×#1909903911yescharacteristic
339C2C2×#1909922011yescharacteristic
340C2C2×#1909922411yescharacteristic
341C2C2×#1909928911yescharacteristic
342C2C2×#1909929211yescharacteristic
subgroupG/H
centerC24C23×D79
commutatorC22×C79C24×C4
FrattiniC23C24×D79
FittingC22×C4×Q8×C79C2
socle, prime radicalC24×C79C24
cosocleC23×C79C25
hypercenterC22×C4×Q8D79
hypocenterC79C2×#2351
LCSG ▷ C22×C79 ▷ C79
UCSC1 ◁ C24 ◁ C22×C4×Q8

Complements

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

NH ≅ G/Ntype
3. #23766318C2direct (N × H)
10. C2×Q8×Dic79C4semidirect
36. Q8×Dic79C2×C4semidirect
87. C2×Q8×C79C4⋊C4semidirect
113. Q8×C79C2×C4⋊C4semidirect
164. C2×C79#2351semidirect
180. C79C2×#2351semidirect
237. C2×Q8Dic79⋊C4semidirect
285. Q8C2×Dic79⋊C4semidirect
335. C2#23766318direct (N × H)

Maximal subgroups

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

subgroupindex
C22×Q8×Dic792
#237663182
C2×#190984612
C2×#190984592
C2×#190984282
C2×C4×Dic79⋊C42
C22×C4×Q8×C792
C2×#235179