Finite Group Databasebeta

← #13058719

Subgroups — #13058719

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×#897C211yesmaximal, characteristic, complemented
3C3×#6453C211yesmaximal, characteristic, complemented
4C4×#4247C211yesmaximal, characteristic
5#10620989C212yesmaximal
6D3×C4⋊Dic7C211yesmaximal, characteristic, complemented
7#10620962C212yesmaximal, complemented
8#10620946C212yesmaximal, complemented
9#10620945C211yesmaximal, characteristic, complemented
10#10620899C212yesmaximal, complemented
11Dic6×Dic7C211yesmaximal, characteristic, complemented
12#90333—31nomaximal, Hall
13C2×C4×D21C2211yescharacteristic
14C2×C4×D3×C7C2211yescharacteristic
15C2×C7×Dic6C2211yescharacteristic
16C2×#4247C2211yescharacteristic
17C2×D3×Dic7C2212yes—
18C3×C7×C4⋊C4C2211yescharacteristic
19#66631C2212yes—
20C4⋊Dic21C2211yescharacteristic
21C4×Dic21C2212yes—
22C7×(C2×D3)⋊C4C2212yes—
23C7×C4⋊Dic3C2212yes—
24C3×C4⋊Dic7C2211yescharacteristic
25C3×C4⋊Dic7C2212yes—
26C3×Dic7⋊C4C2212yes—
27C3×C4×Dic7C2211yescharacteristic
28Dic3⋊Dic7C2212yes—
29Dic3⋊Dic7C2212yes—
30Dic7⋊Dic3C2212yes—
31Dic7⋊Dic3C2211yescharacteristic
32#66575C2212yes—
33#66575C2211yescharacteristic
34(C2×D3)⋊Dic7C2211yescharacteristic
35Dic3×Dic7C2212yes—
36C7×C22⋊Q8—31no—
37#6509—31no—
38Q8×Dic7—31no—
39C4×C22⋊D7—31no—
40C2×C4⋊Dic7—31no—
41#6461—32no—
42#6455—31no—
43#6453—32no—
44#6453D311yescharacteristic, complemented
45#6449—32no—
46#6440—32no—
47#5694—71nomaximal, Hall
48C22×D21C2311yescharacteristic
49C22×D3×C7C2311yescharacteristic
50C2×C3×C4×C7C2312yes—
51C2×C3×C4×C7C2311yescharacteristic
52C2×Dic21C2312yes—
53C2×Dic21C2311yescharacteristic
54C4×D21—22no—
55C2×C7×Dic3C2312yes—
56C2×C7×Dic3C2311yescharacteristic
57C4×D3×C7—22no—
58C7×Dic6—22no—
59C2×C3×Dic7C2311yescharacteristic
60C2×C3×Dic7C2312yes—
61C2×C3×Dic7C2311yescharacteristic
62#4247—22no—
63D3×Dic7—24no—
64C2×C7×Q8—31no—
65C22×C4×C7—31no—
66C2×C22⋊D7—31no—
67C22×Dic7—32no—
68C2×C4×D7—31no—
69C7×C4⋊C4—32no—
70C7×C4⋊C4C2×D311yescharacteristic
71C7×C22⋊C4—32no—
72C22⋊Dic7—31no—
73(C2×D7)⋊C4—32no—
74(C2×D7)⋊C4—32no—
75(C2×D7)⋊C4—31no—
76C4⋊Dic7—31no—
77C4⋊Dic7—32no—
78C4⋊Dic7—32no—
79C4⋊Dic7C2×D312yes—
80C4⋊Dic7C2×D311yescharacteristic
81Dic7⋊C4—32no—
82Dic7⋊C4—31no—
83Dic7⋊C4C2×D312yes—
84C4×Dic7—32no—
85C4×Dic7—32no—
86C4×Dic7C2×D311yescharacteristic
87C3×#125—71no—
88#898—72no—
89#897—71no—
90#897D711yescharacteristic, complemented
91#894—72no—
92#892—71no—
93#892—72no—
94D3×C4⋊C4—71no—
95#890—72no—
96C4×D12—71no—
97C4×Dic6—71no—
98C22×C3×C7C2411yescharacteristic
99C2×D21—21no—
100C2×D21—21no—
101C2×D21—21no—
102C2×D3×C7—21no—
103C2×D3×C7—21no—
104C2×D3×C7#4112yes—
105C3×C4×C7—22no—
106C3×C4×C7#4112yes—
107Dic21—22no—
108Dic21—21no—
109C3×Dic7—22no—
110C3×Dic7—21no—
111C3×Dic7—21no—
112C7×Dic3—22no—
113C7×Dic3—21no—
114#552—211noSylow
115C23×C7—31no—
116C22×D7—31no—
117C7×Q8—62no—
118C2×C4×C7—32no—
119C2×C4×C7—31no—
120C2×C4×C7—62no—
121C2×C4×C7C22×D312yes—
122C2×C4×C7C22×D311yescharacteristic
123C22⋊D7—62no—
124C2×Dic7—31no—
125C2×Dic7—32no—
126C2×Dic7—64no—
127C2×Dic7C22×D311yescharacteristic
128C2×Dic7C22×D311yescharacteristic
129C2×Dic7C22×D312yes—
130C4×D7—62no—
131C2×D12—71no—
132C2×C4×D3—72no—
133C2×C4×D3—71no—
134C2×C4×D3C2×D711yescharacteristic
135C2×Dic6C2×D711yescharacteristic
136C3×C4⋊C4—72no—
137C3×C4⋊C4—72no—
138C3×C4⋊C4—71no—
139C3×C4⋊C4C2×D711yescharacteristic
140C3×C42—71no—
141(C2×D3)⋊C4—71no—
142(C2×D3)⋊C4—72no—
143(C2×D3)⋊C4—71no—
144(C2×D3)⋊C4—72no—
145(C2×D3)⋊C4C2×D712yes—
146C4⋊Dic3—71no—
147C4⋊Dic3—71no—
148C4⋊Dic3—72no—
149C4⋊Dic3C2×D712yes—
150Dic3⋊C4—72no—
151Dic3⋊C4—72no—
152C4×Dic3—72no—
153C4×Dic3—72no—
154C2×C3×C7C2×#4111yescharacteristic
155C2×C3×C7C2×#4111yescharacteristic
156C2×C3×C72−1+411yescharacteristic
157D21—41no—
158D3×C7—22no—
159#126—212no—
160#125—71no—
161#125—212no—
162#123—212no—
163C22⋊Q8—31no—
164C22⋊Q8—211no—
165C4×Q8—211no—
166C4×D4—211no—
167#117—212no—
168#117—211no—
169C2×C4⋊C4—211no—
170C22×C7—32no—
171C22×C7—61no—
172C22×C7—61no—
173C22×C7C23×D311yescharacteristic
174C2×D7—61no—
175C2×D7—61no—
176C2×D7—61no—
177C4×C7—22no—
178C4×C7—61no—
179C4×C7—62no—
180C4×C7#23912yes—
181Dic7—22no—
182Dic7—21no—
183Dic7—21no—
184Dic7—61no—
185Dic7—62no—
186C22×D3—71no—
187C22×D3C22×D711yescharacteristic
188C2×C3×C4—71no—
189C2×C3×C4—71no—
190C2×C3×C4—72no—
191C2×C3×C4C22×D712yes—
192C2×C3×C4C22×D711yescharacteristic
193C2×Dic3—72no—
194C2×Dic3—71no—
195C2×Dic3C22×D712yes—
196C2×Dic3C22×D711yescharacteristic
197D12—142no—
198C4×D3—22no—
199C4×D3—144no—
200C4×D3—142no—
201Dic6—22no—
202C3×C7#55211yescharacteristic, complemented, Hall
203C2×Q8—31no—
204C2×D4—211no—
205C22×C4—31no—
206C22×C4—212no—
207C22×C4—211no—
208C4⋊C4—32no—
209C4⋊C4—72no—
210C4⋊C4—71no—
211C4⋊C4—72no—
212C4⋊C4—212no—
213C4⋊C4—211no—
214C4⋊C4—212no—
215C4⋊C4—212no—
216C4⋊C4—211no—
217C4⋊C4D3×D711yescharacteristic
218C22⋊C4—32no—
219C22⋊C4—211no—
220C22⋊C4—212no—
221C22⋊C4—211no—
222C22⋊C4—212no—
223C42—71no—
224C42—212no—
225C42—212no—
226C2×C7—62no—
227C2×C7C2×#23911yescharacteristic
228C2×C7D3×#4111yescharacteristic
229C2×C7#101011yescharacteristic
230D7—121no—
231C22×C3C23×D711yescharacteristic
232C2×D3—21no—
233C2×D3—21no—
234C2×D3—141no—
235C2×D3—141no—
236C2×D3—141no—
237C2×D3#116112yes—
238C3×C4—22no—
239C3×C4—142no—
240C3×C4—141no—
241C3×C4—141no—
242C3×C4#115912yes—
243Dic3—22no—
244Dic3—21no—
245Dic3—141no—
246Dic3—142no—
247C23—31no—
248C23—211no—
249Q8—62no—
250D4—422no—
251C2×C4—32no—
252C2×C4—31no—
253C2×C4—62no—
254C2×C4—71no—
255C2×C4—71no—
256C2×C4—72no—
257C2×C4—211no—
258C2×C4—212no—
259C2×C4—424no—
260C2×C4—422no—
261C2×C4C2×D3×D712yes—
262C2×C4C2×D3×D711yescharacteristic
263C7#569411yescharacteristic, complemented, Sylow
264C2×C3C2×#115911yescharacteristic
265C2×C3C2×#116111yescharacteristic
266C2×C3#655411yescharacteristic
267D3—22no—
268D3—281no—
269C22—32no—
270C22—61no—
271C22—61no—
272C22—421no—
273C22—421no—
274C22—421no—
275C22C22×D3×D711yescharacteristic
276C4—22no—
277C4—61no—
278C4—62no—
279C4—141no—
280C4—142no—
281C4—141no—
282C4—421no—
283C4—422no—
284C4#6667212yes—
285C3#9033311yescharacteristic, complemented, Sylow
286C2—62no—
287C2—841no—
288C2C2×#6667211yescharacteristic
289C2#1062156711yescharacteristic
290C2D3×#116111yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC3×C7×C4⋊C4C22
prime radicalC2×#4247C22
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×#897C2semidirect
3. C3×#6453C2semidirect
6. D3×C4⋊Dic7C2semidirect
7. #10620962C2semidirect
8. #10620946C2semidirect
9. #10620945C2semidirect
10. #10620899C2semidirect
11. Dic6×Dic7C2semidirect
44. #6453D3semidirect
90. #897D7semidirect
202. C3×C7#552semidirect
263. C7#5694semidirect
285. C3#90333semidirect

Maximal subgroups

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

subgroupindex
#106213752
C7×#8972
C3×#64532
C4×#42472
#106209892
D3×C4⋊Dic72
#106209622
#106209462
#106209452
#106208992
Dic6×Dic72
#903333
#56947