Finite Group Databasebeta

← #72221335

Subgroups — #72221335

Order 3200.

1434 conjugacy classes of subgroups; only the 251 normal subgroups are shown.

normal subgroups only

This group has 1434 conjugacy classes of subgroups — too many to draw. Only its 251 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
1C2×#11717353C212yesmaximal
2#13104978C211yesmaximal, characteristic
3#13104826C211yesmaximal, characteristic
4C42×#1043C211yesmaximal, characteristic
5#13104001C214yesmaximal
6#13103978C214yesmaximal
7#13103880C212yesmaximal
8C2×#87872C2212yes—
9C2×C4⋊#1044C2212yes—
10C2×C4⋊#1044C2214yes—
11C2×C4×#1043C2211yescharacteristic
12C2×C4×#1043C2212yes—
13C2×C4×#1043C2211yescharacteristic
14C2×C4×#1043C2212yes—
15C2×#87832C2214yes—
16C2×#87830C2214yes—
17#11717353—28no—
18C42×C5⋊D5C2211yescharacteristic
19C5×#4015C412yescomplemented
20#11717037C414yescomplemented
21#11717018C2214yes—
22#11716985C412yescomplemented
23#11716776C2218yes—
24AGL(1,𝔽5)×C4⋊Q8—52nomaximal
25C22×#1044C2314yes—
26C22×#1043C2311yescharacteristic
27C22×#1043C2312yes—
28C22×#1043C2311yescharacteristic
29C2×C4×C5⋊D5C2311yescharacteristic
30C2×C4×C5⋊D5C2312yes—
31C2×C5×Dic10C2×C414yes—
32C2×#6039C2×C414yes—
33C2×#6036C2314yes—
34#87872—24no—
35#87872—24no—
36C4⋊#1044—24no—
37C4⋊#1044—28no—
38C4×#1043—22no—
39C4×#1043—24no—
40C4×#1043D414yes—
41C4×#1043Q818yescomplemented
42#87832—28no—
43#87830—28no—
44C42×C52C2×C411yescharacteristic
45C4×#1039C2×C411yescharacteristic
46C4×#1039C2×C412yes—
47C5×C4⋊Dic5C2×C418yes—
48#87781C2×C418yes—
49C2×Q8×AGL(1,𝔽5)—54no—
50D5×C4⋊Q8—52no—
51(C2×AGL(1,𝔽5))⋊Q8—54no—
52#65712—58no—
53C4⋊C4×AGL(1,𝔽5)—58no—
54#65688—52no—
55C42×AGL(1,𝔽5)—52no—
56C42×AGL(1,𝔽5)—102no—
57C22×C5⋊D5C2411yescharacteristic
58C2×#1044—24no—
59C2×#1044—28no—
60C2×#1043—22no—
61C2×#1043—22no—
62C2×#1043—22no—
63C2×#1043C2×D412yes—
64C2×#1043C2×D412yes—
65C2×#1043C2×Q814yes—
66C2×#1043C2×Q814yes—
67C2×C4×C52C22×C411yescharacteristic
68C2×C4×C52C22×C412yes—
69C2×#1039C22×C411yescharacteristic
70C2×#1039C22×C412yes—
71C4×C5⋊D5C2×D412yes—
72C4×C5⋊D5C2×Q814yes—
73C4×C5⋊D5#4112yes—
74C4×C5⋊D5#4114yes—
75C2×C5×Dic5C22×C418yes—
76C5×Dic10—28no—
77#6039—28no—
78#6036—24no—
79#6036—24no—
80C2×Q8×D5—54no—
81Q8×AGL(1,𝔽5)—1016no—
82C2×C4⋊AGL(1,𝔽5)—58no—
83C2×C4⋊AGL(1,𝔽5)—54no—
84C2×C4⋊AGL(1,𝔽5)—58no—
85C2×C4×AGL(1,𝔽5)—54no—
86C2×C4×AGL(1,𝔽5)—52no—
87C2×C4×AGL(1,𝔽5)—54no—
88C2×C4×AGL(1,𝔽5)—52no—
89C2×C4×AGL(1,𝔽5)—58no—
90C2×C4×AGL(1,𝔽5)—102no—
91C2×C4×AGL(1,𝔽5)—104no—
92C2×C4×AGL(1,𝔽5)—104no—
93C2×C4×AGL(1,𝔽5)—102no—
94C5×C4⋊Q8—52no—
95#4090—54no—
96D5×C4⋊C4—58no—
97#4034—58no—
98C42×D5—52no—
99C42×D5—102no—
100#4015AGL(1,𝔽5)12yescomplemented
101(C2×AGL(1,𝔽5))⋊C4—516no—
102C4×C4⋊Q8—251noSylow
103C22×C52C23×C411yescharacteristic
104C2×C5⋊D5C22×D411yescharacteristic
105C2×C5⋊D5C22×Q812yes—
106C2×C5⋊D5C2×#4112yes—
107C2×C5⋊D5C2×#4111yescharacteristic
108#1044—44no—
109#1043—22no—
110#1043—24no—
111#1043C4⋊Q814yescomplemented
112C4×C52C4×D412yes—
113C4×C52C4×Q814yescomplemented
114#1039C4×D412yes—
115#1039C4×Q814yescomplemented
116C5×Dic5—28no—
117C22×AGL(1,𝔽5)—52no—
118C22×AGL(1,𝔽5)—52no—
119C22×AGL(1,𝔽5)—54no—
120C22×AGL(1,𝔽5)—58no—
121C22×AGL(1,𝔽5)—102no—
122C22×AGL(1,𝔽5)—104no—
123C22×AGL(1,𝔽5)—102no—
124C2×C5×Q8—54no—
125Q8×D5—108no—
126Q8×D5—108no—
127C2×C4×D5—52no—
128C2×C4×D5—54no—
129C2×C4×D5—58no—
130C2×C4×D5—104no—
131C2×C4×D5—102no—
132C2×Dic10—58no—
133C2×Dic10C2×AGL(1,𝔽5)14yes—
134C4⋊AGL(1,𝔽5)—108no—
135C4⋊AGL(1,𝔽5)—1016no—
136C4⋊AGL(1,𝔽5)—1016no—
137C4×AGL(1,𝔽5)—58no—
138C4×AGL(1,𝔽5)—516no—
139C4×AGL(1,𝔽5)—108no—
140C4×AGL(1,𝔽5)—1016no—
141C4×AGL(1,𝔽5)—104no—
142C4×AGL(1,𝔽5)—108no—
143C4×AGL(1,𝔽5)—1016no—
144C4×AGL(1,𝔽5)—1016no—
145C4×AGL(1,𝔽5)—108no—
146C5×C4⋊C4—58no—
147C42×C5—22no—
148C42×C5C2×AGL(1,𝔽5)12yes—
149C4⋊Dic5C2×AGL(1,𝔽5)18yes—
150#683—516no—
151C4×Dic5—54no—
152C4×Dic5—52no—
153C4×Dic5—102no—
154C4×Dic5—104no—
155C2×C4⋊Q8—251no—
156C2×C4×Q8—252no—
157#391—252no—
158#389—254no—
159#384—251no—
160C4×C4⋊C4—254no—
161C43—251no—
162C2×C52C2×C4×D411yescharacteristic
163C2×C52C2×C4×Q812yes—
164C5⋊D5C2×C4⋊Q811yescharacteristic
165C5⋊D5#53211yescharacteristic
166C5⋊D5#53312yes—
167C22×D5—52no—
168C22×D5—102no—
169C2×AGL(1,𝔽5)—54no—
170C2×AGL(1,𝔽5)—58no—
171C2×AGL(1,𝔽5)—54no—
172C2×AGL(1,𝔽5)—58no—
173C2×AGL(1,𝔽5)—104no—
174C2×AGL(1,𝔽5)—104no—
175C2×AGL(1,𝔽5)—108no—
176C2×AGL(1,𝔽5)—108no—
177C2×AGL(1,𝔽5)—104no—
178C2×AGL(1,𝔽5)—104no—
179C2×AGL(1,𝔽5)—104no—
180C2×AGL(1,𝔽5)—108no—
181C2×AGL(1,𝔽5)—108no—
182C2×AGL(1,𝔽5)—108no—
183C2×AGL(1,𝔽5)—108no—
184C2×AGL(1,𝔽5)—1016no—
185C5×Q8—108no—
186C2×C4×C5—24no—
187C2×C4×C5—22no—
188C2×C4×C5—58no—
189C2×C4×C5C22×AGL(1,𝔽5)12yes—
190C2×C4×C5C22×AGL(1,𝔽5)14yes—
191C2×Dic5—54no—
192C2×Dic5—52no—
193C2×Dic5—104no—
194C2×Dic5—102no—
195C2×Dic5C22×AGL(1,𝔽5)18yes—
196C4×D5—54no—
197C4×D5—54no—
198C4×D5—58no—
199C4×D5—58no—
200C4×D5—104no—
201C4×D5—108no—
202C4×D5—108no—
203C4×D5—104no—
204C4×D5—108no—
205C4×D5—108no—
206Dic10—28no—
207Dic10—1016no—
208C22×Q8—252no—
209C4⋊Q8—52no—
210C4⋊Q8—254no—
211C4⋊Q8—252no—
212C4×Q8—508no—
213C2×C4⋊C4—254no—
214C2×C4⋊C4—252no—
215C2×C4⋊C4—254no—
216C2×C4⋊C4—254no—
217C2×C42—251no—
218C2×C42—252no—
219C2×C42—251no—
220C2×C42—254no—
221C2×C42—252no—
222C2×C42—251no—
223#95—258no—
224C52C4×C4⋊Q811yescharacteristic, complemented, Sylow
225C22×C5—22no—
226C22×C5C23×AGL(1,𝔽5)12yes—
227C2×D5—52no—
228C2×D5—52no—
229C2×D5—54no—
230C2×D5—54no—
231C2×D5—102no—
232C2×D5—104no—
233C2×D5—104no—
234C2×D5—102no—
235AGL(1,𝔽5)—58no—
236AGL(1,𝔽5)—108no—
237AGL(1,𝔽5)—104no—
238AGL(1,𝔽5)—108no—
239AGL(1,𝔽5)—108no—
240AGL(1,𝔽5)—1016no—
241AGL(1,𝔽5)—208no—
242C4×C5—28no—
243C4×C5—24no—
244C4×C5—108no—
245C4×C5D4×AGL(1,𝔽5)14yes—
246C4×C5Q8×AGL(1,𝔽5)18yescomplemented
247Dic5—28no—
248Dic5—54no—
249Dic5—58no—
250Dic5—104no—
251Dic5—108no—
252C2×Q8—54no—
253C2×Q8—254no—
254C2×Q8—504no—
255C2×Q8—504no—
256C22×C4—254no—
257C22×C4—251no—
258C22×C4—252no—
259C22×C4—251no—
260C22×C4—252no—
261C22×C4—251no—
262C22×C4—254no—
263C4⋊C4—58no—
264C4⋊C4—258no—
265C4⋊C4—504no—
266C4⋊C4—508no—
267C4⋊C4—508no—
268C42—254no—
269C42—258no—
270C42—251no—
271C42—252no—
272C42—508no—
273C42—504no—
274C42—502no—
275C42D5⋊AGL(1,𝔽5)11yescharacteristic
276C2×C5—24no—
277C2×C5—22no—
278C2×C5C2×D4×AGL(1,𝔽5)12yes—
279C2×C5C2×Q8×AGL(1,𝔽5)14yes—
280D5—52no—
281D5—52no—
282D5—54no—
283D5—102no—
284D5—102no—
285D5—104no—
286C23—251no—
287Q8—108no—
288Q8—508no—
289C2×C4—58no—
290C2×C4—252no—
291C2×C4—254no—
292C2×C4—252no—
293C2×C4—251no—
294C2×C4—254no—
295C2×C4—254no—
296C2×C4—252no—
297C2×C4—252no—
298C2×C4—252no—
299C2×C4—254no—
300C2×C4—504no—
301C2×C4—504no—
302C2×C4—502no—
303C2×C4—502no—
304C2×C4—502no—
305C2×C4—504no—
306C2×C4—508no—
307C2×C4C2×D5⋊AGL(1,𝔽5)12yes—
308C2×C4C2×D5⋊AGL(1,𝔽5)11yescharacteristic
309C5—22no—
310C5AGL(1,𝔽5)×C4⋊Q812yescomplemented
311C22—252no—
312C22—252no—
313C22—251no—
314C22—251no—
315C22C22×D5⋊AGL(1,𝔽5)11yescharacteristic
316C4—108no—
317C4—254no—
318C4—254no—
319C4—252no—
320C4—502no—
321C4—504no—
322C4—1004no—
323C4#1171735314yescomplemented
324C4#1171736312yes—
325C2—251no—
326C2—251no—
327C2—252no—
328C2C2×#1171735312yes—
329C2C2×#1171736311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D5⋊AGL(1,𝔽5)
commutator, socleC22×C52C23×C4
FittingC42×C52C2×C4
prime radical, cosocleC22×C5⋊D5C24
hypercenterC42D5⋊AGL(1,𝔽5)
hypocenterC52C4×C4⋊Q8
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
19. C5×#4015C4semidirect
20. #11717037C4semidirect
22. #11716985C4semidirect
41. C4×#1043Q8semidirect
100. #4015AGL(1,𝔽5)semidirect
111. #1043C4⋊Q8semidirect
113. C4×C52C4×Q8semidirect
115. #1039C4×Q8semidirect
224. C52C4×C4⋊Q8semidirect
246. C4×C5Q8×AGL(1,𝔽5)semidirect
310. C5AGL(1,𝔽5)×C4⋊Q8semidirect
323. C4#11717353semidirect

Maximal subgroups

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

subgroupindex
C2×#117173532
#131049782
#131048262
C42×#10432
#131040012
#131039782
#131038802
AGL(1,𝔽5)×C4⋊Q85