Finite Group Databasebeta

← C2×#10617384 #13045190

Subgroups — C2×#10617384 #13045190

Order 1280.

1706 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#66062C211yesmaximal, characteristic
2C2×#65788C214yesmaximal, complemented
3#10617384C2116yesmaximal, direct product factor
4C2×#65707C214yesmaximal, complemented
5C2×#65706C214yesmaximal
6C2×#65684C211yesmaximal, characteristic
7C2×#65684C211yesmaximal, characteristic
8C2×C5×C4⋊Q8C411yescharacteristic
9C2×#4092C412yes—
10C2×Q8×Dic5C2212yes—
11#66062C418yes—
12#66062C2218yes—
13C2×#4043C412yes—
14C2×#4038C2212yes—
15C2×#4036C412yes—
16C2×#4033C2212yes—
17C2×C42×D5C2211yescharacteristic
18C2×#4015C411yescharacteristic
19#65788C22132yescomplemented
20C2×#4004C2212yes—
21C2×#4004C2214yes—
22C2×#4004C2212yes—
23C2×#4003C2212yes—
24C2×#4003C2212yes—
25C2×#4002C2212yes—
26C2×#4002C2214yes—
27C2×#4002C2212yes—
28C2×C4⋊#170C2214yes—
29C2×C4×#170C2214yes—
30#65707C22132yescomplemented
31#65706C22132yes—
32#65684C2218yes—
33#65684C2218yes—
34C2×#2987—51nomaximal, Sylow
35C22×C5×Q8C2×C412yes—
36C22×C4×D5C2×C411yescharacteristic
37C22×C4×D5C2×C411yescharacteristic
38C22×C4×D5C2311yescharacteristic
39C22×Dic10C2×C412yes—
40C22×#170C2314yes—
41C22×#170C2314yes—
42C5×C4⋊Q8C2×C418yes—
43C2×C5×C4⋊C4C2×C412yes—
44C2×C5×C4⋊C4C2×C412yes—
45C2×C42×C5C2×C411yescharacteristic
46#4092C2×C4116yes—
47Q8×Dic5C2×C418yes—
48Q8×Dic5C2318yes—
49C2×#685C2×C414yes—
50C2×C4⋊Dic5C2×C412yes—
51C2×C4⋊Dic5C2×C412yes—
52C2×C4⋊Dic5C2×C414yes—
53C2×#683C2×C414yes—
54C2×C4×Dic5C2×C411yescharacteristic
55C2×C4×Dic5C2311yescharacteristic
56C2×C4×Dic5C2311yescharacteristic
57C2×C4×Dic5C2314yes—
58#4043C2×C4116yes—
59#4038C2×C418yes—
60#4038C2318yes—
61#4036C2×C4116yes—
62#4033C2×C418yes—
63#4033C2318yes—
64C42×D5C2×C414yes—
65C42×D5C2314yes—
66#4015C2×C418yes—
67#4004C2318yes—
68#4004C2318yes—
69#4004C23116yes—
70#4003C2318yes—
71#4003C2318yes—
72#4002C2318yes—
73#4002C2318yes—
74#4002C23116yes—
75C4⋊#170C23116yes—
76C4×#170C23116yes—
77C2×#533—51no—
78#2987—516no—
79C2×#446—54no—
80C2×#446—54no—
81C2×#433—54no—
82C2×#432—51no—
83C2×#432—51no—
84C23×D5C22×C411yescharacteristic
85C2×C5×Q8—44no—
86C2×C5×Q8C22×C418yes—
87C22×C4×C5C22×C411yescharacteristic
88C22×C4×C5C22×C411yescharacteristic
89C22×C4×C5C22×C411yescharacteristic
90C22×C4×C5C22×C414yes—
91C22×Dic5C22×C414yes—
92C22×Dic5C22×C411yescharacteristic
93C22×Dic5C22×C411yescharacteristic
94C22×Dic5C2411yescharacteristic
95C2×C4×D5—24no—
96C2×C4×D5—42no—
97C2×C4×D5—42no—
98C2×C4×D5C22×C412yes—
99C2×C4×D5C22×C414yes—
100C2×C4×D5C22×C414yes—
101C2×C4×D5C2412yes—
102C2×Dic10—44no—
103C2×Dic10C22×C418yes—
104C2×#170—28no—
105C2×#170—28no—
106C2×#170C2418yes—
107C2×#170C2418yes—
108C5×C4⋊C4C22×C418yes—
109C5×C4⋊C4C22×C418yes—
110C42×C5C22×C414yes—
111#685C22×C4116yes—
112C4⋊Dic5C22×C418yes—
113C4⋊Dic5C22×C418yes—
114C4⋊Dic5C22×C4116yes—
115#683C22×C4116yes—
116C4×Dic5C22×C412yes—
117C4×Dic5C22×C412yes—
118C4×Dic5C22×C414yes—
119C4×Dic5C22×C418yes—
120C4×Dic5C2418yes—
121C4×Dic5C2412yes—
122C4×Dic5C2412yes—
123#533—58no—
124#533—58no—
125C2×C4⋊Q8—51no—
126C2×C4⋊Q8AGL(1,𝔽5)11yescharacteristic
127C2×#125—52no—
128C2×C22⋊Q8—52no—
129C2×C22⋊Q8—52no—
130C2×C4×Q8—52no—
131C2×C4×Q8—52no—
132C2×#117—52no—
133C22×C42—51no—
134#446—532no—
135#446—532no—
136#433—532no—
137#432—58no—
138#432—58no—
139C2×C4⋊C8—52no—
140C2×C4⋊C8—52no—
141C2×C4⋊C8—54no—
142C2×C4⋊C8—54no—
143C2×C22⋊C8—52no—
144C2×C22⋊C8—52no—
145C2×#97—52no—
146C2×#97—52no—
147C2×#97—54no—
148C2×C4×C8—54no—
149C23×C5C23×C411yescharacteristic
150C22×D5—22no—
151C22×D5—22no—
152C22×D5—22no—
153C22×D5C23×C412yes—
154C5×Q8—48no—
155C2×C4×C5—22no—
156C2×C4×C5—22no—
157C2×C4×C5—22no—
158C2×C4×C5—28no—
159C2×C4×C5C23×C412yes—
160C2×C4×C5C23×C418yes—
161C2×C4×C5C23×C412yes—
162C2×C4×C5C23×C412yes—
163C2×Dic5—28no—
164C2×Dic5—22no—
165C2×Dic5—22no—
166C2×Dic5C23×C412yes—
167C2×Dic5C23×C412yes—
168C2×Dic5C23×C418yes—
169C2×Dic5C23×C411yescharacteristic
170C2×Dic52−1+414yes—
171C2×Dic5C2511yescharacteristic
172C4×D5—24no—
173C4×D5—24no—
174C4×D5—44no—
175C4×D5—44no—
176Dic10—48no—
177#170—28no—
178#170—28no—
179C22×Q8—52no—
180C22×Q8C2×AGL(1,𝔽5)12yes—
181C23×C4—51no—
182C23×C4—51no—
183C23×C4—51no—
184C22×C8—54no—
185C22×C8—54no—
186C4⋊Q8—58no—
187C4⋊Q8C2×AGL(1,𝔽5)18yes—
188#125—516no—
189C22⋊Q8—516no—
190C22⋊Q8—516no—
191C4×Q8—58no—
192C4×Q8—58no—
193C4×Q8—58no—
194C4×Q8—58no—
195#117—58no—
196#117—58no—
197C2×C4⋊C4—52no—
198C2×C4⋊C4—54no—
199C2×C4⋊C4—54no—
200C2×C4⋊C4—52no—
201C2×C4⋊C4C2×AGL(1,𝔽5)12yes—
202C2×C4⋊C4C2×AGL(1,𝔽5)12yes—
203C2×C22⋊C4—54no—
204C2×C42—54no—
205C2×C42—51no—
206C2×C42—51no—
207C2×C42—51no—
208C2×C42—54no—
209C2×C42—54no—
210C2×C42C2×AGL(1,𝔽5)11yescharacteristic
211C4⋊C8—516no—
212C4⋊C8—58no—
213C4⋊C8—58no—
214C4⋊C8—516no—
215C22⋊C8—58no—
216C22⋊C8—58no—
217#97—58no—
218#97—58no—
219#97—516no—
220C4×C8—516no—
221C22×C5#51912yes—
222C22×C5#56812yes—
223C22×C5#56812yes—
224C22×C5C24×C411yescharacteristic
225C2×D5—22no—
226C2×D5—22no—
227C2×D5—22no—
228C2×D5—44no—
229C4×C5—22no—
230C4×C5—22no—
231C4×C5—22no—
232C4×C5—28no—
233Dic5—28no—
234Dic5—22no—
235Dic5—22no—
236Dic5#51912yes—
237Dic5C2×2−1+412yes—
238C24—51no—
239C2×Q8—44no—
240C2×Q8—58no—
241C2×Q8—204no—
242C2×Q8C22×AGL(1,𝔽5)18yes—
243C22×C4—51no—
244C22×C4—51no—
245C22×C4—51no—
246C22×C4—54no—
247C22×C4—52no—
248C22×C4—52no—
249C22×C4—54no—
250C22×C4—54no—
251C22×C4—104no—
252C22×C4—202no—
253C22×C4—202no—
254C22×C4C22×AGL(1,𝔽5)11yescharacteristic
255C22×C4C22×AGL(1,𝔽5)11yescharacteristic
256C22×C4C22×AGL(1,𝔽5)11yescharacteristic
257C22×C4C22×AGL(1,𝔽5)14yes—
258C2×C8—58no—
259C2×C8—58no—
260C2×C8—108no—
261C2×C8—108no—
262C4⋊C4—58no—
263C4⋊C4—516no—
264C4⋊C4—516no—
265C4⋊C4—58no—
266C4⋊C4C22×AGL(1,𝔽5)18yes—
267C4⋊C4C22×AGL(1,𝔽5)18yes—
268C22⋊C4—516no—
269C42—58no—
270C42—52no—
271C42—52no—
272C42—52no—
273C42—52no—
274C42—54no—
275C42—58no—
276C42C22×AGL(1,𝔽5)14yes—
277C2×C5#298714yescomplemented
278C2×C5C2×#51911yescharacteristic
279C2×C5C2×#56811yescharacteristic
280C2×C5C2×#56811yescharacteristic
281D5—42no—
282C23—52no—
283C23—102no—
284C23—102no—
285C23—102no—
286C23C23×AGL(1,𝔽5)11yescharacteristic
287Q8—48no—
288Q8—208no—
289C2×C4—22no—
290C2×C4—28no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—54no—
294C2×C4—51no—
295C2×C4—51no—
296C2×C4—52no—
297C2×C4—52no—
298C2×C4—58no—
299C2×C4—104no—
300C2×C4—104no—
301C2×C4—102no—
302C2×C4—102no—
303C2×C4—108no—
304C2×C4—204no—
305C2×C4—204no—
306C2×C4C23×AGL(1,𝔽5)12yes—
307C2×C4C23×AGL(1,𝔽5)12yes—
308C2×C4C23×AGL(1,𝔽5)12yes—
309C2×C4C23×AGL(1,𝔽5)18yes—
310C8—108no—
311C8—108no—
312C5C2×#298711yescharacteristic, complemented, Sylow
313C22—102no—
314C22—102no—
315C22—102no—
316C22—204no—
317C22#6625612yes—
318C22#6626012yes—
319C22#6626212yes—
320C22C24×AGL(1,𝔽5)11yescharacteristic
321C4—22no—
322C4—28no—
323C4—22no—
324C4—22no—
325C4—52no—
326C4—52no—
327C4—102no—
328C4—102no—
329C4—108no—
330C2—202no—
331C2#1061738414yesdirect product factor
332C2C2×#6625611yescharacteristic
333C2C2×#6626011yescharacteristic
334C2C2×#6626211yescharacteristic
subgroupG/H
centerC23C23×AGL(1,𝔽5)
commutatorC22×C5C24×C4
FrattiniC22C24×AGL(1,𝔽5)
FittingC2×C5×C4⋊Q8C4
socleC23×C5C23×C4
prime radicalC23×D5C22×C4
cosocleC2×Dic5C25
hypercenterC2×C4⋊Q8AGL(1,𝔽5)
hypocenterC5C2×#2987
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#65788C2semidirect
3. #10617384C2direct (N × H)
4. C2×#65707C2semidirect
19. #65788C22semidirect
30. #65707C22semidirect
277. C2×C5#2987semidirect
312. C5C2×#2987semidirect
331. C2#10617384direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#660622
C2×#657882
#106173842
C2×#657072
C2×#657062
C2×#656842
C2×#29875