Finite Group Databasebeta

← C2×#23769231 #34455337

Subgroups — C2×#23769231 #34455337

Order 20224.

2614 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14579118C211yesmaximal, characteristic
2#23769231C2116yesmaximal, direct product factor
3C2×#19098496C214yesmaximal, complemented
4C2×#19098485C214yesmaximal
5C2×#19098459C212yesmaximal, complemented
6C2×#19098433C211yesmaximal, characteristic, complemented
7C2×#19098420C212yesmaximal
8C22×#125×C79C211yesmaximal, characteristic, complemented
9C23×C4×D79C2211yescharacteristic
10C2×#14579118—28no—
11C2×#14579118C2218yes—
12C22×C4⋊Dic79C2212yes—
13C22×C4⋊Dic79C2214yes—
14C22×Dic79⋊C4C2212yes—
15C22×Dic79⋊C4C2214yes—
16C22×(C2×D79)⋊C4C2212yes—
17C22×(C2×D79)⋊C4C2214yes—
18C22×C4×Dic79C2211yescharacteristic
19#19098496C22132yescomplemented
20#19098485C22132yes—
21#19098459C22116yescomplemented
22#19098433C2218yescomplemented
23#19098420C22116yes—
24C2×C79⋊#95C2214yes—
25C2×C79⋊#95C2214yes—
26C2×#125×C79—28no—
27C2×#125×C79C2218yescomplemented
28C22×C4⋊C4×C79C2212yes—
29C22×C4⋊C4×C79C2214yes—
30C22×C42×C79C2211yescharacteristic
31C24×D79C2311yescharacteristic
32C23×Dic79C2312yes—
33C23×Dic79C2314yes—
34C23×Dic79C2311yescharacteristic
35C22×C4×D79—24no—
36C22×C4×D79—24no—
37C22×C4×D79Q818yes—
38C22×C4×D79C2314yes—
39#14579118—232no—
40C2×C4⋊Dic79—28no—
41C2×C4⋊Dic79—48no—
42C2×C4⋊Dic79C2318yes—
43C2×C4⋊Dic79C23116yes—
44C2×Dic79⋊C4—28no—
45C2×Dic79⋊C4—48no—
46C2×Dic79⋊C4C2318yes—
47C2×Dic79⋊C4C23116yes—
48C2×(C2×D79)⋊C4—28no—
49C2×(C2×D79)⋊C4—48no—
50C2×(C2×D79)⋊C4C2318yes—
51C2×(C2×D79)⋊C4C23116yes—
52C2×C4×Dic79—24no—
53C2×C4×Dic79C2314yes—
54C79⋊#95C23116yes—
55C79⋊#95C23116yes—
56C23×C4×C79C2314yes—
57C23×C4×C79C2312yes—
58C23×C4×C79C2311yescharacteristic
59#125×C79—232no—
60C2×C4⋊C4×C79—216no—
61C2×C4⋊C4×C79—28no—
62C2×C4⋊C4×C79C2318yes—
63C2×C4⋊C4×C79C23116yes—
64C2×C42×C79D418yes—
65C2×C42×C79C2314yes—
66C23×D79—22no—
67C23×D79—22no—
68C23×D79—24no—
69C23×D79—24no—
70C23×D79C2×Q814yes—
71C23×D79C2412yes—
72C22×Dic79—22no—
73C22×Dic79—24no—
74C22×Dic79—22no—
75C22×Dic79—28no—
76C22×Dic79—28no—
77C22×Dic79—24no—
78C22×Dic79—28no—
79C22×Dic79—24no—
80C22×Dic79C2×Q814yes—
81C22×Dic79C2414yes—
82C22×Dic79C2418yes—
83C22×Dic79C2412yes—
84C2×C4×D79—232no—
85C2×C4×D79—28no—
86C2×C4×D79—28no—
87C2×C4×D79C2×Q8116yes—
88C4⋊Dic79—216no—
89C4⋊Dic79—416no—
90Dic79⋊C4—216no—
91Dic79⋊C4—416no—
92(C2×D79)⋊C4—216no—
93(C2×D79)⋊C4—416no—
94C4×Dic79—28no—
95C24×C79C2411yescharacteristic
96C22×C4×C79—24no—
97C22×C4×C79—24no—
98C22×C4×C79—28no—
99C22×C4×C79—28no—
100C22×C4×C79—28no—
101C22×C4×C79C2×D418yes—
102C22×C4×C79C2×D414yes—
103C22×C4×C79C2×Q814yes—
104C22×C4×C79#4114yes—
105C22×C4×C79C2418yes—
106C22×C4×C79C2414yes—
107C22×C4×C79C2412yes—
108C4⋊C4×C79—232no—
109C4⋊C4×C79—216no—
110C42×C79C2×D4116yes—
111C23×C79C22×D412yes—
112C23×C79C22×Q812yes—
113C23×C79C2×#4112yes—
114C23×C79H(5,𝔽2)14yes—
115C23×C792−1+414yes—
116C23×C79C2511yescharacteristic
117C22×D79—28no—
118C22×D79—216no—
119C22×D79—22no—
120C22×D79—22no—
121C22×D79—24no—
122C22×D79—24no—
123C22×D79—48no—
124C22×D79—44no—
125C22×D79—44no—
126C22×D79C22×Q814yes—
127C2×C4×C79—216no—
128C2×C4×C79—24no—
129C2×C4×C79—24no—
130C2×C4×C79—28no—
131C2×C4×C79—28no—
132C2×C4×C79—28no—
133C2×C4×C79—416no—
134C2×C4×C79C22⋊Q8116yes—
135C2×C4×C79C22×D418yes—
136C2×C4×C79C22×D414yes—
137C2×C4×C79C22×Q814yes—
138C2×C4×C79C2×#4114yes—
139C2×Dic79—28no—
140C2×Dic79—22no—
141C2×Dic79—22no—
142C2×Dic79—24no—
143C2×Dic79—28no—
144C2×Dic79—28no—
145C2×Dic79—24no—
146C2×Dic79—28no—
147C2×Dic79—24no—
148C2×Dic79—48no—
149C2×Dic79—416no—
150C2×Dic79C22×Q814yes—
151C4×D79—232no—
152C22×C79C2×C22⋊Q814yes—
153C22×C79#53718yes—
154C22×C79#54014yes—
155C22×C79#54114yes—
156C22×C79#54418yes—
157C22×C79C23×D411yescharacteristic
158C22×C79C23×Q811yescharacteristic
159C22×C79C22×#4111yescharacteristic
160C22×C79C2×H(5,𝔽2)12yes—
161C22×C79C2×2−1+412yes—
162C2×D79—24no—
163C2×D79—28no—
164C2×D79—416no—
165C2×D79—44no—
166C2×D79—42no—
167C2×D79—42no—
168C4×C79—28no—
169C4×C79—48no—
170C4×C79C2×C22⋊Q818yes—
171Dic79—24no—
172Dic79—44no—
173Dic79—48no—
174C2×#2629—791nomaximal, Sylow
175C2×C79#262918yescomplemented
176C2×C79C22×C22⋊Q811yescharacteristic
177C2×C79C2×#53712yes—
178C2×C79C2×#54011yescharacteristic
179C2×C79C2×#54111yescharacteristic
180C2×C79C2×#54412yes—
181D79—44no—
182C22×#125D7911yescharacteristic, complemented
183C22×#117—791no—
184#2629—7916no—
185C2×#398—794no—
186C2×#396—794no—
187C2×#389—792no—
188C2×#384—791no—
189C2×#380—792no—
190C79C2×#262911yescharacteristic, complemented, Sylow
191C24×C4—791no—
192C2×#125—28no—
193C2×#125C2×D7918yescomplemented
194C2×#117—798no—
195C2×#117—1588no—
196C22×C4⋊C4—794no—
197C22×C4⋊C4—794no—
198C22×C4⋊C4—792no—
199C22×C4⋊C4—792no—
200C22×C4⋊C4C2×D7914yes—
201C22×C4⋊C4C2×D7912yes—
202C22×C22⋊C4—794no—
203C22×C22⋊C4—792no—
204C22×C42—791no—
205C22×C42C2×D7911yescharacteristic
206#398—7932no—
207#396—7932no—
208#389—7916no—
209#384—798no—
210#380—7916no—
211C2×#95—794no—
212C2×#95—794no—
213C25—791no—
214C23×C4—791no—
215C23×C4—798no—
216C23×C4—794no—
217C23×C4—794no—
218C23×C4—792no—
219C23×C4—1584no—
220C23×C4—1584no—
221C23×C4C22×D7914yes—
222C23×C4C22×D7912yes—
223C23×C4C22×D7911yescharacteristic
224#125—232no—
225#117—15832no—
226C2×C4⋊C4—216no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—3168no—
229C2×C4⋊C4—3168no—
230C2×C4⋊C4—7916no—
231C2×C4⋊C4—7916no—
232C2×C4⋊C4—798no—
233C2×C4⋊C4—798no—
234C2×C4⋊C4—1588no—
235C2×C4⋊C4—1588no—
236C2×C4⋊C4C22×D7918yes—
237C2×C4⋊C4C22×D79116yes—
238C2×C22⋊C4—3168no—
239C2×C22⋊C4—7916no—
240C2×C22⋊C4—798no—
241C2×C22⋊C4—1588no—
242C2×C42—794no—
243C2×C42—1584no—
244C2×C42C22⋊D7918yes—
245C2×C42C22×D7914yes—
246#95—7916no—
247#95—7916no—
248C24—794no—
249C24—792no—
250C24—1582no—
251C24—1584no—
252C24—1584no—
253C24—1582no—
254C24C23×D7911yescharacteristic
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—28no—
258C22×C4—28no—
259C22×C4—28no—
260C22×C4—794no—
261C22×C4—792no—
262C22×C4—7916no—
263C22×C4—794no—
264C22×C4—798no—
265C22×C4—1588no—
266C22×C4—1588no—
267C22×C4—1584no—
268C22×C4—1588no—
269C22×C4—1584no—
270C22×C4—1584no—
271C22×C4—1582no—
272C22×C4—1582no—
273C22×C4—1588no—
274C22×C4—15832no—
275C22×C4—1588no—
276C22×C4C2×C22⋊D7918yes—
277C22×C4C2×C22⋊D7914yes—
278C22×C4Q8×D7914yes—
279C22×C4#1192962514yes—
280C22×C4C23×D7918yes—
281C22×C4C23×D7914yes—
282C22×C4C23×D7912yes—
283C4⋊C4—232no—
284C4⋊C4—216no—
285C4⋊C4—31616no—
286C4⋊C4—31616no—
287C4⋊C4—15816no—
288C4⋊C4—15816no—
289C22⋊C4—31616no—
290C22⋊C4—15816no—
291C42—1588no—
292C42C2×C22⋊D79116yes—
293C23—3168no—
294C23—3164no—
295C23—3164no—
296C23—794no—
297C23—1584no—
298C23—1584no—
299C23—15816no—
300C23—1588no—
301C23—1582no—
302C23—1582no—
303C23C22×C22⋊D7912yes—
304C23C2×Q8×D7912yes—
305C23C2×#1192962512yes—
306C23#1457920714yes—
307C23#1457920814yes—
308C23C24×D7911yescharacteristic
309C2×C4—24no—
310C2×C4—24no—
311C2×C4—216no—
312C2×C4—28no—
313C2×C4—28no—
314C2×C4—28no—
315C2×C4—416no—
316C2×C4—3168no—
317C2×C4—31616no—
318C2×C4—794no—
319C2×C4—1588no—
320C2×C4—1584no—
321C2×C4—1588no—
322C2×C4—1584no—
323C2×C4—1584no—
324C2×C4—1588no—
325C2×C4—1588no—
326C2×C4—1582no—
327C2×C4—1582no—
328C2×C4—15832no—
329C2×C4#14579137116yes—
330C2×C4C22×C22⋊D7918yes—
331C2×C4C22×C22⋊D7914yes—
332C2×C4C2×Q8×D7914yes—
333C2×C4C2×#1192962514yes—
334C22—3162no—
335C22—31616no—
336C22—3164no—
337C22—3162no—
338C22—1588no—
339C22—1584no—
340C22C2×#1457913714yes—
341C22#1909914118yes—
342C22#1909916914yes—
343C22#1909917614yes—
344C22#1909919318yes—
345C22C23×C22⋊D7911yescharacteristic
346C22C22×Q8×D7911yescharacteristic
347C22C22×#1192962511yescharacteristic
348C22C2×#1457920712yes—
349C22C2×#1457920812yes—
350C4—28no—
351C4—48no—
352C4—3164no—
353C4—3168no—
354C4—1584no—
355C4C2×#1457913718yes—
356C2—3164no—
357C2#2376923118yesdirect product factor
358C2C22×#1457913711yescharacteristic
359C2C2×#1909914112yes—
360C2C2×#1909916911yescharacteristic
361C2C2×#1909917611yescharacteristic
362C2C2×#1909919312yes—
subgroupG/H
centerC24C23×D79
commutator, cosocleC23×C79C25
FrattiniC23C24×D79
FittingC22×#125×C79C2
socleC24×C79C24
prime radicalC24×D79C23
hypercenterC22×#125D79
hypocenterC79C2×#2629
LCSG ▷ C23×C79 ▷ C79
UCSC1 ◁ C24 ◁ C22×#125

Complements

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

NH ≅ G/Ntype
2. #23769231C2direct (N × H)
3. C2×#19098496C2semidirect
5. C2×#19098459C2semidirect
6. C2×#19098433C2semidirect
8. C22×#125×C79C2semidirect
19. #19098496C22semidirect
21. #19098459C22semidirect
22. #19098433C22semidirect
27. C2×#125×C79C22semidirect
175. C2×C79#2629semidirect
182. C22×#125D79semidirect
190. C79C2×#2629semidirect
193. C2×#125C2×D79semidirect
357. C2#23769231direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145791182
#237692312
C2×#190984962
C2×#190984852
C2×#190984592
C2×#190984332
C2×#190984202
C22×#125×C792
C2×#262979