Finite Group Databasebeta

← C2×#11717004 #13106066

Subgroups — C2×#11717004 #13106066

Order 1600.

944 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 944 conjugacy classes of subgroups — too many to draw. Only its 184 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C5×D20C211yesmaximal, characteristic, complemented
2C22×#6037C211yesmaximal, characteristic, complemented
3C22×#6037C211yesmaximal, characteristic, complemented
4C22×D5×Dic5C211yesmaximal, characteristic, complemented
5C2×#87806C211yesmaximal, characteristic, complemented
6C2×C5×#685C211yesmaximal, characteristic, complemented
7C2×#87778C211yesmaximal, characteristic, complemented
8#11717004C218yesmaximal, direct product factor
9C23×C5×D5C2211yescharacteristic
10C23×C5×D5C2211yescharacteristic
11C23×C5×D5C2211yescharacteristic
12C22×C4×C52C2211yescharacteristic
13C22×#1039C2211yescharacteristic
14C22×#1039C2211yescharacteristic
15C22×C5×Dic5C2211yescharacteristic
16C2×C5×D20—24no—
17C2×C5×D20C2214yescomplemented
18C2×#6037—24no—
19C2×#6037—24no—
20C2×#6037C2214yescomplemented
21C2×#6037C2214yescomplemented
22C2×D5×Dic5—24no—
23C2×D5×Dic5C2214yescomplemented
24#87806C2214yescomplemented
25C5×#685C2214yescomplemented
26#87778C2214yescomplemented
27C2×#4085—51nomaximal
28C2×#4030—51nomaximal
29C23×C52C2311yescharacteristic
30C22×C5×D5—22no—
31C22×C5×D5—22no—
32C22×C5×D5—22no—
33C22×C5×D5—22no—
34C22×C5×D5—22no—
35C22×C5×D5—22no—
36C22×C5×D5—22no—
37C22×C5×D5—22no—
38C22×C5×D5D414yescomplemented
39C22×C5×D5C2312yes—
40C22×C5×D5C2312yes—
41C22×C5×D5C2312yes—
42C2×C4×C52—22no—
43C2×C4×C52C2312yes—
44C2×#1039—22no—
45C2×#1039D414yescomplemented
46C2×#1039C2312yes—
47C2×#1039C2312yes—
48C2×C5×Dic5—22no—
49C2×C5×Dic5C2312yes—
50C5×D20—28no—
51#6037—28no—
52#6037—28no—
53D5×Dic5—28no—
54C22×C5×D4—51no—
55C22×C22⋊D5—51no—
56C22×C22⋊D5—51no—
57C22×C22⋊D5—51no—
58C22×C22⋊D5—51no—
59C23×Dic5—51no—
60C22×D20D511yescharacteristic, complemented
61C22×C4×D5—51no—
62C2×C5×C22⋊C4—51no—
63C2×C22⋊Dic5—51no—
64#4085—58no—
65C2×#685—51no—
66C2×#685D511yescharacteristic, complemented
67C2×#683—51no—
68C2×#683—51no—
69C2×#683—102no—
70#4030—58no—
71C22×C52C2×D412yes—
72C22×C52C2×D412yes—
73C22×C52#4112yes—
74C22×C52C2411yescharacteristic
75C2×C5×D5—28no—
76C2×C5×D5—22no—
77C2×C5×D5—22no—
78C2×C5×D5—22no—
79C2×C5×D5—22no—
80C2×C5×D5—22no—
81C2×C5×D5—22no—
82C2×C5×D5—22no—
83C2×C5×D5—22no—
84C2×C5×D5—44no—
85C2×C5×D5—44no—
86C2×C5×D5C2×D414yescomplemented
87C4×C52—22no—
88#1039—22no—
89#1039C2×D414yescomplemented
90C5×Dic5—22no—
91C24×C5—51no—
92C24×C5—51no—
93C24×C5—51no—
94C23×D5C2×D511yescharacteristic
95C23×D5C2×D511yescharacteristic
96C23×D5C2×D511yescharacteristic
97C2×C5×D4—54no—
98C2×C5×D4—104no—
99C22×C4×C5—22no—
100C22×C4×C5—51no—
101C22×C4×C5C2×D511yescharacteristic
102C22×C4×C5C2×D511yescharacteristic
103C2×C22⋊D5—54no—
104C2×C22⋊D5—54no—
105C2×C22⋊D5—54no—
106C2×C22⋊D5—54no—
107C2×C22⋊D5—104no—
108C2×C22⋊D5—104no—
109C2×C22⋊D5—104no—
110C2×C22⋊D5—104no—
111C22×Dic5—51no—
112C22×Dic5—51no—
113C22×Dic5—51no—
114C22×Dic5—51no—
115C22×Dic5—54no—
116C22×Dic5—102no—
117C22×Dic5—102no—
118C22×Dic5—104no—
119C22×Dic5C2×D511yescharacteristic
120C2×D20—24no—
121C2×D20C2×D514yescomplemented
122C2×C4×D5—54no—
123C2×C4×D5—104no—
124C5×C22⋊C4—54no—
125C22⋊Dic5—54no—
126#685—54no—
127#685C2×D514yescomplemented
128#683—54no—
129#683—54no—
130#683—108no—
131C2×#121—251noSylow
132C2×C52#12114yescomplemented
133C2×C52C22×D411yescharacteristic
134C2×C52C22×D411yescharacteristic
135C2×C52C2×#4111yescharacteristic
136C5×D5—24no—
137C5×D5—42no—
138C5×D5—42no—
139C23×C5—22no—
140C23×C5—54no—
141C23×C5—52no—
142C23×C5—52no—
143C23×C5—52no—
144C23×C5—102no—
145C23×C5—102no—
146C23×C5—102no—
147C23×C5—102no—
148C23×C5—102no—
149C23×C5—102no—
150C23×C5—102no—
151C23×C5—102no—
152C23×C5C22×D511yescharacteristic
153C23×C5C22×D511yescharacteristic
154C22×D5—22no—
155C22×D5—22no—
156C22×D5—22no—
157C22×D5—22no—
158C22×D5—22no—
159C22×D5—22no—
160C22×D5—22no—
161C22×D5—22no—
162C22×D5C22⋊D514yescomplemented
163C22×D5C22×D512yes—
164C22×D5C22×D512yes—
165C22×D5C22×D512yes—
166C5×D4—108no—
167C2×C4×C5—22no—
168C2×C4×C5—22no—
169C2×C4×C5—24no—
170C2×C4×C5—44no—
171C2×C4×C5—52no—
172C2×C4×C5—102no—
173C2×C4×C5C22×D512yes—
174C2×C4×C5C22×D512yes—
175C22⋊D5—108no—
176C22⋊D5—108no—
177C22⋊D5—108no—
178C22⋊D5—108no—
179C2×Dic5—22no—
180C2×Dic5—54no—
181C2×Dic5—54no—
182C2×Dic5—52no—
183C2×Dic5—52no—
184C2×Dic5—52no—
185C2×Dic5—52no—
186C2×Dic5—108no—
187C2×Dic5—104no—
188C2×Dic5—104no—
189C2×Dic5—108no—
190C2×Dic5—102no—
191C2×Dic5—102no—
192C2×Dic5—204no—
193C2×Dic5C22×D512yes—
194D20—28no—
195C4×D5—108no—
196C22×D4—51no—
197C22×D4—251no—
198C22×D4—251no—
199C23×C4—251no—
200#121—258no—
201C2×C4⋊C4—251no—
202C2×C22⋊C4—51no—
203C2×C22⋊C4—251no—
204C52C2×#12111yescharacteristic, complemented, Sylow
205C22×C5—24no—
206C22×C5—24no—
207C22×C5—24no—
208C22×C5—22no—
209C22×C5—54no—
210C22×C5—102no—
211C22×C5—102no—
212C22×C5—108no—
213C22×C5—102no—
214C22×C5—102no—
215C22×C5—102no—
216C22×C5—102no—
217C22×C5—102no—
218C22×C5—102no—
219C22×C5—204no—
220C22×C5—204no—
221C22×C5#70912yes—
222C22×C5D4×D512yes—
223C22×C5D4×D512yes—
224C22×C5D4×D512yes—
225C22×C5#71112yes—
226C22×C5C2×C22⋊D512yes—
227C22×C5C23×D511yescharacteristic
228C22×C5C23×D511yescharacteristic
229C2×D5—22no—
230C2×D5—22no—
231C2×D5—28no—
232C2×D5—22no—
233C2×D5—22no—
234C2×D5—22no—
235C2×D5—22no—
236C2×D5—22no—
237C2×D5—22no—
238C2×D5—44no—
239C2×D5—44no—
240C2×D5C2×C22⋊D514yescomplemented
241C4×C5—22no—
242C4×C5—22no—
243C4×C5—44no—
244C4×C5—102no—
245Dic5—22no—
246Dic5—54no—
247Dic5—54no—
248Dic5—108no—
249Dic5—102no—
250Dic5—102no—
251Dic5—204no—
252C24—51no—
253C24—51no—
254C24—51no—
255C2×D4—54no—
256C2×D4—104no—
257C2×D4—254no—
258C2×D4—254no—
259C2×D4—504no—
260C2×D4—504no—
261C22×C4—51no—
262C22×C4—254no—
263C22×C4—251no—
264C22×C4—251no—
265C22×C4—504no—
266C22×C4D5211yescharacteristic
267C4⋊C4—254no—
268C22⋊C4—54no—
269C22⋊C4—254no—
270C2×C5—22no—
271C2×C5—22no—
272C2×C5—22no—
273C2×C5—28no—
274C2×C5—104no—
275C2×C5—202no—
276C2×C5—202no—
277C2×C5#403014yescomplemented
278C2×C5#408514yescomplemented
279C2×C5C2×#70911yescharacteristic
280C2×C5C2×D4×D511yescharacteristic
281C2×C5C2×D4×D511yescharacteristic
282C2×C5C2×D4×D511yescharacteristic
283C2×C5C2×#71111yescharacteristic
284C2×C5C22×C22⋊D511yescharacteristic
285D5—24no—
286D5—42no—
287D5—42no—
288C23—54no—
289C23—52no—
290C23—52no—
291C23—52no—
292C23—102no—
293C23—102no—
294C23—102no—
295C23—102no—
296C23—102no—
297C23—102no—
298C23—102no—
299C23—102no—
300C23C2×D5211yescharacteristic
301D4—108no—
302D4—508no—
303D4—508no—
304C2×C4—22no—
305C2×C4—52no—
306C2×C4—102no—
307C2×C4—254no—
308C2×C4—252no—
309C2×C4—252no—
310C2×C4—508no—
311C2×C4—502no—
312C2×C4C2×D5212yes—
313C5—22no—
314C5C2×#403011yescharacteristic, complemented
315C5C2×#408511yescharacteristic, complemented
316C22—54no—
317C22—102no—
318C22—102no—
319C22—108no—
320C22—102no—
321C22—102no—
322C22—102no—
323C22—102no—
324C22—102no—
325C22—102no—
326C22—204no—
327C22—204no—
328C22#8787012yes—
329C22#8787712yes—
330C22D5×C22⋊D512yes—
331C22C22×D5211yescharacteristic
332C4—22no—
333C4—102no—
334C4—254no—
335C4—502no—
336C2—104no—
337C2—202no—
338C2—202no—
339C2#1171700414yesdirect product factor
340C2C2×#8787011yescharacteristic
341C2C2×#8787711yescharacteristic
342C2C2×D5×C22⋊D511yescharacteristic
subgroupG/H
centerC23C2×D52
commutator, cosocleC22×C52C24
FrattiniC22C22×D52
FittingC22×C4×C52C22
socleC23×C52C23
hypercenterC22×C4D52
hypocenterC52C2×#121
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×C5×D20C2semidirect
2. C22×#6037C2semidirect
3. C22×#6037C2semidirect
4. C22×D5×Dic5C2semidirect
5. C2×#87806C2semidirect
6. C2×C5×#685C2semidirect
7. C2×#87778C2semidirect
8. #11717004C2direct (N × H)
17. C2×C5×D20C22semidirect
20. C2×#6037C22semidirect
21. C2×#6037C22semidirect
23. C2×D5×Dic5C22semidirect
24. #87806C22semidirect
25. C5×#685C22semidirect
26. #87778C22semidirect
38. C22×C5×D5D4semidirect
45. C2×#1039D4semidirect
60. C22×D20D5semidirect
66. C2×#685D5semidirect
86. C2×C5×D5C2×D4semidirect
89. #1039C2×D4semidirect
121. C2×D20C2×D5semidirect
127. #685C2×D5semidirect
132. C2×C52#121semidirect
162. C22×D5C22⋊D5semidirect
204. C52C2×#121semidirect
240. C2×D5C2×C22⋊D5semidirect
277. C2×C5#4030semidirect
278. C2×C5#4085semidirect
314. C5C2×#4030semidirect
315. C5C2×#4085semidirect
339. C2#11717004direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C5×D202
C22×#60372
C22×D5×Dic52
C2×#878062
C2×C5×#6852
C2×#877782
#117170042
C2×#40855
C2×#40305