Finite Group Databasebeta

← #13058660

Subgroups — #13058660

Order 1344.

428 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 132 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
1#10621375C211yesmaximal, characteristic, complemented
2C7×#890C211yesmaximal, characteristic, complemented
3C3×#6460C211yesmaximal, characteristic, complemented
4C4×#4246C211yesmaximal, characteristic
5#10620975C212yesmaximal
6D7×C4⋊Dic3C211yesmaximal, characteristic, complemented
7#10620956C212yesmaximal, complemented
8#10620952C211yesmaximal, characteristic, complemented
9#10620946C212yesmaximal, complemented
10#10620907C212yesmaximal, complemented
11Dic3×Dic14C211yesmaximal, characteristic, complemented
12#90385—31nomaximal, Hall
13C2×C4×D21C2211yescharacteristic
14C2×C3×C4×D7C2211yescharacteristic
15C2×C3×Dic14C2211yescharacteristic
16C2×#4246C2211yescharacteristic
17C2×Dic3×D7C2212yes—
18C3×C7×C4⋊C4C2211yescharacteristic
19#66631C2212yes—
20C4⋊Dic21C2211yescharacteristic
21C4×Dic21C2212yes—
22C7×C4⋊Dic3C2212yes—
23C7×C4⋊Dic3C2211yescharacteristic
24C7×Dic3⋊C4C2212yes—
25C4×C7×Dic3C2211yescharacteristic
26C3×(C2×D7)⋊C4C2212yes—
27C3×C4⋊Dic7C2212yes—
28Dic3⋊Dic7C2211yescharacteristic
29Dic3⋊Dic7C2212yes—
30Dic7⋊Dic3C2212yes—
31Dic7⋊Dic3C2212yes—
32#66575C2212yes—
33#66575C2211yescharacteristic
34(C2×D7)⋊Dic3C2211yescharacteristic
35Dic3×Dic7C2212yes—
36C7×#125—31no—
37#6461—32no—
38#6460—31no—
39#6460D311yescharacteristic, complemented
40#6457—32no—
41#6455—31no—
42#6455—32no—
43D7×C4⋊C4—31no—
44#6453—32no—
45C4×D28—31no—
46C4×Dic14—31no—
47#5642—71nomaximal, Hall
48C22×D21C2311yescharacteristic
49C22×C3×D7C2311yescharacteristic
50C2×C3×C4×C7C2312yes—
51C2×C3×C4×C7C2311yescharacteristic
52C2×Dic21C2312yes—
53C2×Dic21C2311yescharacteristic
54C4×D21—22no—
55C2×C7×Dic3C2311yescharacteristic
56C2×C7×Dic3C2312yes—
57C2×C7×Dic3C2311yescharacteristic
58C2×C3×Dic7C2312yes—
59C2×C3×Dic7C2311yescharacteristic
60C3×C4×D7—22no—
61C3×Dic14—22no—
62#4246—22no—
63Dic3×D7—24no—
64C2×D28—31no—
65C2×C4×D7—32no—
66C2×C4×D7—31no—
67C2×C4×D7C2×D311yescharacteristic
68C2×Dic14C2×D311yescharacteristic
69C7×C4⋊C4—32no—
70C7×C4⋊C4—32no—
71C7×C4⋊C4—31no—
72C7×C4⋊C4C2×D311yescharacteristic
73C42×C7—31no—
74(C2×D7)⋊C4—31no—
75(C2×D7)⋊C4—32no—
76(C2×D7)⋊C4—31no—
77(C2×D7)⋊C4—32no—
78(C2×D7)⋊C4C2×D312yes—
79C4⋊Dic7—31no—
80C4⋊Dic7—31no—
81C4⋊Dic7—32no—
82C4⋊Dic7C2×D312yes—
83Dic7⋊C4—32no—
84Dic7⋊C4—32no—
85C4×Dic7—32no—
86C4×Dic7—32no—
87C3×C22⋊Q8—71no—
88#946—71no—
89Q8×Dic3—71no—
90C4×#67—71no—
91C2×C4⋊Dic3—71no—
92#898—72no—
93#892—71no—
94#890—72no—
95#890D711yescharacteristic, complemented
96#886—72no—
97#877—72no—
98C22×C3×C7C2411yescharacteristic
99C2×D21—21no—
100C2×D21—21no—
101C2×D21—21no—
102C2×C3×D7—21no—
103C2×C3×D7—21no—
104C2×C3×D7#4112yes—
105C3×C4×C7—22no—
106C3×C4×C7#4112yes—
107Dic21—22no—
108Dic21—21no—
109C3×Dic7—22no—
110C3×Dic7—21no—
111C7×Dic3—22no—
112C7×Dic3—21no—
113C7×Dic3—21no—
114#552—211noSylow
115C22×D7—31no—
116C22×D7C22×D311yescharacteristic
117C2×C4×C7—31no—
118C2×C4×C7—31no—
119C2×C4×C7—32no—
120C2×C4×C7C22×D312yes—
121C2×C4×C7C22×D311yescharacteristic
122C2×Dic7—32no—
123C2×Dic7—31no—
124C2×Dic7C22×D312yes—
125C2×Dic7C22×D311yescharacteristic
126D28—62no—
127C4×D7—22no—
128C4×D7—64no—
129C4×D7—62no—
130Dic14—22no—
131C2×C3×Q8—71no—
132C22×C3×C4—71no—
133C2×#67—71no—
134C22×Dic3—72no—
135C2×C4×D3—71no—
136C3×C4⋊C4—72no—
137C3×C4⋊C4C2×D711yescharacteristic
138C3×C22⋊C4—72no—
139#217—71no—
140(C2×D3)⋊C4—72no—
141(C2×D3)⋊C4—72no—
142(C2×D3)⋊C4—71no—
143C4⋊Dic3—71no—
144C4⋊Dic3—72no—
145C4⋊Dic3—72no—
146C4⋊Dic3C2×D712yes—
147C4⋊Dic3C2×D711yescharacteristic
148Dic3⋊C4—72no—
149Dic3⋊C4—71no—
150Dic3⋊C4C2×D712yes—
151C4×Dic3—72no—
152C4×Dic3—72no—
153C4×Dic3C2×D711yescharacteristic
154C2×C3×C7C2×#4111yescharacteristic
155C2×C3×C7C2×#4111yescharacteristic
156C2×C3×C72−1+411yescharacteristic
157D21—41no—
158C3×D7—22no—
159#126—212no—
160#125—31no—
161#125—212no—
162#123—212no—
163C22⋊Q8—71no—
164C22⋊Q8—211no—
165C4×Q8—211no—
166C4×D4—211no—
167#117—212no—
168#117—211no—
169C2×C4⋊C4—211no—
170C22×C7C23×D311yescharacteristic
171C2×D7—21no—
172C2×D7—21no—
173C2×D7—61no—
174C2×D7—61no—
175C2×D7—61no—
176C2×D7#23912yes—
177C4×C7—22no—
178C4×C7—62no—
179C4×C7—61no—
180C4×C7—61no—
181C4×C7#23712yes—
182Dic7—22no—
183Dic7—21no—
184Dic7—61no—
185Dic7—62no—
186C23×C3—71no—
187C22×D3—71no—
188C3×Q8—142no—
189C2×C3×C4—72no—
190C2×C3×C4—71no—
191C2×C3×C4—142no—
192C2×C3×C4C22×D712yes—
193C2×C3×C4C22×D711yescharacteristic
194#67—142no—
195C2×Dic3—71no—
196C2×Dic3—72no—
197C2×Dic3—144no—
198C2×Dic3C22×D711yescharacteristic
199C2×Dic3C22×D711yescharacteristic
200C2×Dic3C22×D712yes—
201C4×D3—142no—
202C3×C7#55211yescharacteristic, complemented, Hall
203C2×Q8—71no—
204C2×D4—211no—
205C22×C4—71no—
206C22×C4—212no—
207C22×C4—211no—
208C4⋊C4—31no—
209C4⋊C4—32no—
210C4⋊C4—32no—
211C4⋊C4—72no—
212C4⋊C4—212no—
213C4⋊C4—212no—
214C4⋊C4—212no—
215C4⋊C4—211no—
216C4⋊C4—211no—
217C4⋊C4D3×D711yescharacteristic
218C22⋊C4—72no—
219C22⋊C4—211no—
220C22⋊C4—212no—
221C22⋊C4—211no—
222C22⋊C4—212no—
223C42—31no—
224C42—212no—
225C42—212no—
226C2×C7C2×#23711yescharacteristic
227C2×C7C2×#23911yescharacteristic
228C2×C7#101011yescharacteristic
229D7—22no—
230D7—121no—
231C22×C3—72no—
232C22×C3—141no—
233C22×C3—141no—
234C22×C3C23×D711yescharacteristic
235C2×D3—141no—
236C2×D3—141no—
237C2×D3—141no—
238C3×C4—22no—
239C3×C4—141no—
240C3×C4—142no—
241C3×C4#116112yes—
242Dic3—22no—
243Dic3—21no—
244Dic3—21no—
245Dic3—141no—
246Dic3—142no—
247C23—71no—
248C23—211no—
249Q8—142no—
250D4—422no—
251C2×C4—31no—
252C2×C4—31no—
253C2×C4—32no—
254C2×C4—72no—
255C2×C4—71no—
256C2×C4—142no—
257C2×C4—211no—
258C2×C4—212no—
259C2×C4—424no—
260C2×C4—422no—
261C2×C4C2×D3×D712yes—
262C2×C4C2×D3×D711yescharacteristic
263C7#564211yescharacteristic, complemented, Sylow
264C2×C3—142no—
265C2×C3C2×#116111yescharacteristic
266C2×C3D7×#4111yescharacteristic
267C2×C3#655411yescharacteristic
268D3—281no—
269C22—72no—
270C22—141no—
271C22—141no—
272C22—421no—
273C22—421no—
274C22—421no—
275C22C22×D3×D711yescharacteristic
276C4—22no—
277C4—61no—
278C4—62no—
279C4—61no—
280C4—141no—
281C4—142no—
282C4—421no—
283C4—422no—
284C4#6667012yes—
285C3#9038511yescharacteristic, complemented, Sylow
286C2—142no—
287C2—841no—
288C2C2×#6667011yescharacteristic
289C2#1062156711yescharacteristic
290C2D7×#23911yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC3×C7×C4⋊C4C22
prime radicalC2×#4246C22
hypercenterC4⋊C4D3×D7
hypocenterC3×C7#552
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #10621375C2semidirect
2. C7×#890C2semidirect
3. C3×#6460C2semidirect
6. D7×C4⋊Dic3C2semidirect
7. #10620956C2semidirect
8. #10620952C2semidirect
9. #10620946C2semidirect
10. #10620907C2semidirect
11. Dic3×Dic14C2semidirect
39. #6460D3semidirect
95. #890D7semidirect
202. C3×C7#552semidirect
263. C7#5642semidirect
285. C3#90385semidirect

Maximal subgroups

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

subgroupindex
#106213752
C7×#8902
C3×#64602
C4×#42462
#106209752
D7×C4⋊Dic32
#106209562
#106209522
#106209462
#106209072
Dic3×Dic142
#903853
#56427