Finite Group Databasebeta

← #13058714

Subgroups — #13058714

Order 1344.

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

normal subgroups only

This group has 472 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×#888C211yesmaximal, characteristic, complemented
3C3×#6458C211yesmaximal, characteristic, complemented
4#10621027C211yesmaximal, characteristic, complemented
5#10621017C212yesmaximal, complemented
6C4×#4246C211yesmaximal, characteristic, complemented
7Dic3×D28C211yesmaximal, characteristic, complemented
8#10620971C212yesmaximal, complemented
9#10620956C212yesmaximal, complemented
10#10620930C212yesmaximal, complemented
11#10620920C211yesmaximal, characteristic, complemented
12#90425—31nomaximal, Hall
13C2×C4×D21C2211yescharacteristic
14C2×C7×Dic6C2212yes—
15C2×C3×D28C2211yescharacteristic
16C2×C3×D28C2212yes—
17C2×C3×C4×D7C2211yescharacteristic
18C2×#4246C2211yescharacteristic
19C2×#4246C2212yes—
20C2×Dic3×D7C2212yes—
21C3×C7×C4⋊C4C2211yescharacteristic
22#66631C2212yes—
23C4⋊Dic21C2211yescharacteristic
24C4×Dic21C2212yes—
25C7×C4⋊Dic3C2211yescharacteristic
26C7×Dic3⋊C4C2212yes—
27C4×C7×Dic3C2211yescharacteristic
28C3×(C2×D7)⋊C4C2212yes—
29Dic3⋊Dic7C2211yescharacteristic
30#66575C2211yescharacteristic
31(C2×D7)⋊Dic3C2211yescharacteristic
32(C2×D7)⋊Dic3C2212yes—
33(C2×D7)⋊Dic3C2212yes—
34(C2×D7)⋊Dic3C2212yes—
35Dic3×Dic7C2212yes—
36C7×C4⋊Q8—31no—
37#6510—32no—
38#6510—32no—
39#6458—32no—
40#6458—31no—
41#6458D311yescharacteristic, complemented
42#6455—31no—
43#6455—31no—
44#6455—32no—
45C4×D28—31no—
46C4×D28—31no—
47#5616—71nomaximal, Hall
48C22×D21C2311yescharacteristic
49C22×C3×D7C2311yescharacteristic
50C22×C3×D7C2312yes—
51C2×C3×C4×C7C2312yes—
52C2×C3×C4×C7C2311yescharacteristic
53C2×Dic21C2312yes—
54C2×Dic21C2311yescharacteristic
55C4×D21—22no—
56C2×C7×Dic3C2312yes—
57C2×C7×Dic3C2311yescharacteristic
58C2×C7×Dic3C2311yescharacteristic
59C7×Dic6—42no—
60C2×C3×Dic7C2311yescharacteristic
61C3×D28—22no—
62C3×D28—42no—
63C3×C4×D7—22no—
64#4246—22no—
65#4246—42no—
66Dic3×D7—24no—
67C2×C7×Q8—32no—
68C2×D28—31no—
69C2×D28—32no—
70C2×D28C2×D311yescharacteristic
71C2×D28C2×D312yes—
72C2×C4×D7—32no—
73C2×C4×D7—31no—
74C2×C4×D7C2×D311yescharacteristic
75C7×C4⋊C4—32no—
76C7×C4⋊C4—31no—
77C7×C4⋊C4C2×D311yescharacteristic
78C42×C7—31no—
79(C2×D7)⋊C4—31no—
80(C2×D7)⋊C4—32no—
81(C2×D7)⋊C4—32no—
82(C2×D7)⋊C4—32no—
83(C2×D7)⋊C4—32no—
84(C2×D7)⋊C4—31no—
85(C2×D7)⋊C4C2×D312yes—
86C4⋊Dic7—31no—
87C4⋊Dic7—31no—
88C4×Dic7—32no—
89C4×Dic7—32no—
90C3×#121—71no—
91#939—72no—
92#938—71no—
93#936—72no—
94D4×Dic3—71no—
95C4×#67—71no—
96#926—71no—
97#892—71no—
98#888D711yescharacteristic, complemented
99#885—72no—
100#877—72no—
101C22×C3×C7C2411yescharacteristic
102C2×D21—21no—
103C2×D21—21no—
104C2×D21—21no—
105C2×C3×D7—22no—
106C2×C3×D7—21no—
107C2×C3×D7—21no—
108C2×C3×D7—22no—
109C2×C3×D7—22no—
110C2×C3×D7—21no—
111C3×C4×C7—22no—
112C3×C4×C7#4112yes—
113Dic21—22no—
114Dic21—21no—
115C3×Dic7—21no—
116C7×Dic3—22no—
117C7×Dic3—21no—
118C7×Dic3#4112yes—
119#555—211noSylow
120C22×D7—31no—
121C22×D7C22×D311yescharacteristic
122C22×D7C22×D312yes—
123C7×Q8—122no—
124C2×C4×C7—31no—
125C2×C4×C7—32no—
126C2×C4×C7—31no—
127C2×C4×C7C22×D312yes—
128C2×C4×C7C22×D311yescharacteristic
129C2×Dic7—31no—
130C2×Dic7—32no—
131C2×Dic7C22×D311yescharacteristic
132D28—22no—
133D28—42no—
134D28—62no—
135D28—122no—
136C4×D7—22no—
137C4×D7—64no—
138C4×D7—62no—
139C2×C3×D4—71no—
140C2×C3×D4—72no—
141C22×C3×C4—71no—
142C2×#67—71no—
143C2×#67—72no—
144C22×Dic3—72no—
145C2×C4×D3—71no—
146C2×Dic6C2×D712yes—
147C3×C4⋊C4C2×D711yescharacteristic
148C3×C22⋊C4—72no—
149#217—72no—
150#217—72no—
151#217—72no—
152#217—71no—
153(C2×D3)⋊C4—72no—
154(C2×D3)⋊C4—71no—
155C4⋊Dic3—71no—
156C4⋊Dic3C2×D711yescharacteristic
157Dic3⋊C4—71no—
158Dic3⋊C4C2×D712yes—
159C4×Dic3—72no—
160C4×Dic3—72no—
161C4×Dic3C2×D711yescharacteristic
162C2×C3×C7C2×#4111yescharacteristic
163C2×C3×C7C2×#4111yescharacteristic
164C2×C3×C7H(5,𝔽2)11yescharacteristic
165D21—41no—
166C3×D7—42no—
167C3×D7—41no—
168C4⋊Q8—31no—
169#124—212no—
170#124—212no—
171#121—71no—
172#121—211no—
173#121—212no—
174C4×D4—211no—
175C4×D4—211no—
176#117—212no—
177#117—211no—
178#117—211no—
179C22×C7C23×D311yescharacteristic
180C2×D7—21no—
181C2×D7—22no—
182C2×D7—21no—
183C2×D7—22no—
184C2×D7—22no—
185C2×D7—21no—
186C2×D7—61no—
187C2×D7—61no—
188C2×D7—61no—
189C4×C7—22no—
190C4×C7—32no—
191C4×C7—62no—
192C4×C7—61no—
193C4×C7#23712yes—
194Dic7—21no—
195Dic7—61no—
196Dic7—62no—
197C23×C3—71no—
198C23×C3—72no—
199C22×D3—71no—
200C3×D4—142no—
201C3×D4—282no—
202C2×C3×C4—71no—
203C2×C3×C4—142no—
204C2×C3×C4C22×D712yes—
205C2×C3×C4C22×D711yescharacteristic
206#67—142no—
207#67—282no—
208C2×Dic3—71no—
209C2×Dic3—72no—
210C2×Dic3—144no—
211C2×Dic3C22×D712yes—
212C2×Dic3C22×D711yescharacteristic
213C2×Dic3C22×D711yescharacteristic
214C4×D3—142no—
215Dic6—42no—
216C3×C7#55511yescharacteristic, complemented, Hall
217C2×Q8—32no—
218C2×D4—71no—
219C2×D4—72no—
220C2×D4—211no—
221C2×D4—212no—
222C22×C4—71no—
223C22×C4—212no—
224C22×C4—211no—
225C4⋊C4—32no—
226C4⋊C4—31no—
227C4⋊C4—211no—
228C4⋊C4—211no—
229C4⋊C4D3×D711yescharacteristic
230C22⋊C4—72no—
231C22⋊C4—211no—
232C22⋊C4—212no—
233C22⋊C4—212no—
234C22⋊C4—212no—
235C22⋊C4—211no—
236C22⋊C4—212no—
237C42—31no—
238C42—212no—
239C42—212no—
240C2×C7C2×#23711yescharacteristic
241C2×C7#100411yescharacteristic
242C2×C7D3×#4111yescharacteristic
243D7—41no—
244D7—42no—
245D7—121no—
246C22×C3—142no—
247C22×C3—141no—
248C22×C3—142no—
249C22×C3—141no—
250C22×C3—141no—
251C22×C3—142no—
252C22×C3C23×D711yescharacteristic
253C2×D3—141no—
254C2×D3—141no—
255C2×D3—141no—
256C3×C4—22no—
257C3×C4—141no—
258C3×C4#116112yes—
259Dic3—21no—
260Dic3—22no—
261Dic3—141no—
262Dic3—142no—
263Dic3#116112yes—
264C23—72no—
265C23—71no—
266C23—211no—
267Q8—122no—
268D4—142no—
269D4—282no—
270D4—422no—
271D4—842no—
272C2×C4—31no—
273C2×C4—31no—
274C2×C4—32no—
275C2×C4—71no—
276C2×C4—142no—
277C2×C4—211no—
278C2×C4—212no—
279C2×C4—424no—
280C2×C4—422no—
281C2×C4C2×D3×D712yes—
282C2×C4C2×D3×D711yescharacteristic
283C7#561611yescharacteristic, complemented, Sylow
284C2×C3—282no—
285C2×C3—281no—
286C2×C3#654811yescharacteristic
287C2×C3C2×#116111yescharacteristic
288C2×C3C2×#116111yescharacteristic
289D3—281no—
290C22—141no—
291C22—142no—
292C22—141no—
293C22—142no—
294C22—142no—
295C22—141no—
296C22—421no—
297C22—421no—
298C22—421no—
299C22C22×D3×D711yescharacteristic
300C4—22no—
301C4—32no—
302C4—62no—
303C4—61no—
304C4—141no—
305C4—421no—
306C4—422no—
307C4#6667012yes—
308C3#9042511yescharacteristic, complemented, Sylow
309C2—281no—
310C2—282no—
311C2—841no—
312C2C2×#6667011yescharacteristic
313C2#1062157211yescharacteristic
314C2D3×#116111yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC3×C7×C4⋊C4C22
hypercenterC4⋊C4D3×D7
hypocenterC3×C7#555
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×#888C2semidirect
3. C3×#6458C2semidirect
4. #10621027C2semidirect
5. #10621017C2semidirect
6. C4×#4246C2semidirect
7. Dic3×D28C2semidirect
8. #10620971C2semidirect
9. #10620956C2semidirect
10. #10620930C2semidirect
11. #10620920C2semidirect
41. #6458D3semidirect
98. #888D7semidirect
216. C3×C7#555semidirect
283. C7#5616semidirect
308. C3#90425semidirect

Maximal subgroups

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

subgroupindex
#106213752
C7×#8882
C3×#64582
#106210272
#106210172
C4×#42462
Dic3×D282
#106209712
#106209562
#106209302
#106209202
#904253
#56167