Finite Group Databasebeta

← #13059550

Subgroups — #13059550

Order 1344.

540 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 136 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×Dic42C211yesmaximal, characteristic, complemented
2C2×#66684C211yesmaximal, characteristic
3C7×#929C211yesmaximal, characteristic, complemented
4C3×#6492C211yesmaximal, characteristic, complemented
5#10621105C211yesmaximal, characteristic, complemented
6#10621099C211yesmaximal, characteristic, complemented
7#10621094C211yesmaximal, characteristic, complemented
8#10621024C212yesmaximal, complemented
9#10620994C212yesmaximal, complemented
10#10620993C212yesmaximal, complemented
11#10620890C212yesmaximal, complemented
12#90339—31nomaximal, Hall
13C22×C3×C4×C7C2211yescharacteristic
14C22×Dic21C2211yescharacteristic
15C22×Dic21C2211yescharacteristic
16C2×Dic42—24no—
17C2×Dic42C2212yes—
18C2×Dic42C2212yes—
19C2×C7×#67C2211yescharacteristic
20C2×C3×C22⋊D7C2211yescharacteristic
21C2×#4248C2211yescharacteristic
22C2×#4245C2211yescharacteristic
23C2×D3×Dic7C2211yescharacteristic
24#66684—24no—
25C2×Dic3×D7C2211yescharacteristic
26C7×#217C2211yescharacteristic
27C7×(C2×D3)⋊C4C2212yes—
28C7×Dic3⋊C4C2212yes—
29C3×C22⋊Dic7C2211yescharacteristic
30C3×(C2×D7)⋊C4C2212yes—
31C3×Dic7⋊C4C2212yes—
32Dic3⋊Dic7C2212yes—
33#66577C2212yes—
34Dic7⋊Dic3C2212yes—
35(C2×D3)⋊Dic7C2212yes—
36(C2×D7)⋊Dic3C2212yes—
37Dic3×Dic7C2212yes—
38C2×#1159—31no—
39C22×Dic14—31no—
40C7×#123—31no—
41#6498—31no—
42#6492D311yescharacteristic, complemented
43#6459—32no—
44#6451—32no—
45#6449—31no—
46#6448—32no—
47#6441—32no—
48#6440—31no—
49#5648—71nomaximal, Hall
50C23×C3×C7C2311yescharacteristic
51C22×D3×C7C2311yescharacteristic
52C22×C3×D7C2311yescharacteristic
53C2×C3×C4×C7—22no—
54C2×C3×C4×C7C2312yes—
55C2×Dic21—22no—
56C2×Dic21D414yes—
57C2×Dic21C2312yes—
58C2×Dic21C2311yescharacteristic
59C2×Dic21C2311yescharacteristic
60Dic42—24no—
61Dic42—24no—
62C7×#67—22no—
63C2×C7×Dic3C2312yes—
64C2×C7×Dic3C2311yescharacteristic
65C3×C22⋊D7—22no—
66C2×C3×Dic7C2312yes—
67C2×C3×Dic7C2311yescharacteristic
68#4248—22no—
69#4245—22no—
70D3×Dic7—22no—
71Dic3×D7—22no—
72C2×C7×D4—31no—
73C22×C4×C7C2×D311yescharacteristic
74C2×C22⋊D7—31no—
75C2×C22⋊D7C2×D311yescharacteristic
76C22×Dic7—31no—
77C22×Dic7—31no—
78C22×Dic7—31no—
79#1159—64no—
80C2×C4×D7—31no—
81C2×Dic14—32no—
82C2×Dic14—32no—
83C2×Dic14—31no—
84C2×Dic14—64no—
85C7×C4⋊C4—32no—
86C7×C22⋊C4—31no—
87C7×C22⋊C4—32no—
88C22⋊Dic7—32no—
89C22⋊Dic7C2×D311yescharacteristic
90(C2×D7)⋊C4—32no—
91(C2×D7)⋊C4C2×D312yes—
92C4⋊Dic7—32no—
93Dic7⋊C4—32no—
94Dic7⋊C4—32no—
95Dic7⋊C4C2×D312yes—
96C4×Dic7—32no—
97C2×#237—71no—
98C22×Dic6—71no—
99C3×#123—71no—
100#935—71no—
101#929D711yescharacteristic, complemented
102#896—72no—
103#888—72no—
104#886—71no—
105#885—72no—
106#878—72no—
107#877—71no—
108C22×C3×C7—22no—
109C22×C3×C7C2×D412yes—
110C22×C3×C7C2411yescharacteristic
111C2×D3×C7—21no—
112C2×D3×C7—22no—
113C2×C3×D7—21no—
114C2×C3×D7—22no—
115C3×C4×C7—22no—
116Dic21—22no—
117Dic21C2×D412yes—
118Dic21C2×D412yes—
119C3×Dic7—21no—
120C3×Dic7—22no—
121C7×Dic3—21no—
122C7×Dic3—22no—
123#536—211noSylow
124C23×C7—31no—
125C23×C7C22×D311yescharacteristic
126C22×D7C22×D311yescharacteristic
127C7×D4—62no—
128C2×C4×C7—22no—
129C2×C4×C7—32no—
130C2×C4×C7—31no—
131C2×C4×C7C22×D312yes—
132C22⋊D7—22no—
133C22⋊D7—62no—
134C2×Dic7—34no—
135C2×Dic7—31no—
136C2×Dic7—31no—
137C2×Dic7—32no—
138C2×Dic7—62no—
139C2×Dic7—62no—
140C2×Dic7C22×D312yes—
141C2×Dic7C22×D311yescharacteristic
142C4×D7—62no—
143Dic14—62no—
144Dic14—64no—
145Dic14—64no—
146C2×C3×D4—71no—
147C22×C3×C4C2×D711yescharacteristic
148C2×#67—71no—
149C2×#67C2×D711yescharacteristic
150C22×Dic3—71no—
151C22×Dic3—71no—
152C22×Dic3—71no—
153#237—144no—
154C2×C4×D3—71no—
155C2×Dic6—72no—
156C2×Dic6—72no—
157C2×Dic6—71no—
158C2×Dic6—144no—
159C3×C4⋊C4—72no—
160C3×C22⋊C4—71no—
161C3×C22⋊C4—72no—
162#217—72no—
163#217C2×D711yescharacteristic
164(C2×D3)⋊C4—72no—
165(C2×D3)⋊C4C2×D712yes—
166C4⋊Dic3—72no—
167Dic3⋊C4—72no—
168Dic3⋊C4—72no—
169Dic3⋊C4C2×D712yes—
170C4×Dic3—72no—
171C2×C3×C7—22no—
172C2×C3×C7C22×D411yescharacteristic
173C2×C3×C72−1+412yes—
174C3×D7—41no—
175D3×C7—41no—
176C2×#41—211no—
177C22×Q8—211no—
178C4⋊Q8—212no—
179#124—212no—
180#123—31no—
181#123—71no—
182#123—211no—
183#123—211no—
184C22⋊Q8—212no—
185C22⋊Q8—212no—
186#117—211no—
187C22×C7—22no—
188C22×C7—61no—
189C22×C7—62no—
190C22×C7D3×D412yes—
191C22×C7C23×D311yescharacteristic
192C2×D7—21no—
193C2×D7—22no—
194C4×C7—22no—
195C4×C7—61no—
196C4×C7—62no—
197Dic7—21no—
198Dic7—22no—
199Dic7—32no—
200Dic7—32no—
201Dic7—62no—
202C23×C3—71no—
203C23×C3C22×D711yescharacteristic
204C22×D3C22×D711yescharacteristic
205C3×D4—142no—
206C2×C3×C4—22no—
207C2×C3×C4—72no—
208C2×C3×C4—71no—
209C2×C3×C4C22×D712yes—
210#67—22no—
211#67—142no—
212C2×Dic3—74no—
213C2×Dic3—71no—
214C2×Dic3—71no—
215C2×Dic3—72no—
216C2×Dic3—142no—
217C2×Dic3—142no—
218C2×Dic3C22×D712yes—
219C2×Dic3C22×D711yescharacteristic
220C4×D3—142no—
221Dic6—142no—
222Dic6—144no—
223Dic6—144no—
224C3×C7#53611yescharacteristic, complemented, Hall
225#41—424no—
226C2×Q8—211no—
227C2×Q8—212no—
228C2×Q8—212no—
229C2×Q8—424no—
230C2×D4—31no—
231C2×D4—71no—
232C2×D4—211no—
233C22×C4—211no—
234C22×C4—211no—
235C22×C4—211no—
236C22×C4—211no—
237C22×C4D3×D711yescharacteristic
238C4⋊C4—32no—
239C4⋊C4—72no—
240C4⋊C4—212no—
241C4⋊C4—212no—
242C4⋊C4—212no—
243C22⋊C4—31no—
244C22⋊C4—32no—
245C22⋊C4—71no—
246C22⋊C4—72no—
247C22⋊C4—212no—
248C22⋊C4—212no—
249C42—212no—
250C2×C7—22no—
251C2×C7—121no—
252C2×C7C2×D3×D411yescharacteristic
253C2×C7#101012yes—
254D7—41no—
255C22×C3—22no—
256C22×C3—141no—
257C22×C3—142no—
258C22×C3D4×D712yes—
259C22×C3C23×D711yescharacteristic
260C2×D3—21no—
261C2×D3—22no—
262C3×C4—22no—
263C3×C4—141no—
264C3×C4—142no—
265Dic3—21no—
266Dic3—22no—
267Dic3—72no—
268Dic3—72no—
269Dic3—142no—
270C23—31no—
271C23—71no—
272C23C2×D3×D711yescharacteristic
273Q8—424no—
274Q8—424no—
275Q8—422no—
276D4—62no—
277D4—142no—
278D4—422no—
279C2×C4—22no—
280C2×C4—32no—
281C2×C4—31no—
282C2×C4—72no—
283C2×C4—71no—
284C2×C4—214no—
285C2×C4—211no—
286C2×C4—211no—
287C2×C4—212no—
288C2×C4—422no—
289C2×C4—422no—
290C2×C4—422no—
291C2×C4C2×D3×D712yes—
292C7#564811yescharacteristic, complemented, Sylow
293C2×C3—22no—
294C2×C3—281no—
295C2×C3C2×D4×D711yescharacteristic
296C2×C3#655412yes—
297D3—41no—
298C22—22no—
299C22—62no—
300C22—61no—
301C22—142no—
302C22—141no—
303C22#6669412yes—
304C22C22×D3×D711yescharacteristic
305C4—22no—
306C4—62no—
307C4—61no—
308C4—142no—
309C4—141no—
310C4—212no—
311C4—212no—
312C4—422no—
313C3#9033911yescharacteristic, complemented, Sylow
314C2—22no—
315C2—121no—
316C2—281no—
317C2#1062154812yes—
318C2C2×#6669411yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC22×C3×C4×C7C22
prime radicalC2×#66684C2
hypercenterC22×C4D3×D7
hypocenterC3×C7#536
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×Dic42C2semidirect
3. C7×#929C2semidirect
4. C3×#6492C2semidirect
5. #10621105C2semidirect
6. #10621099C2semidirect
7. #10621094C2semidirect
8. #10621024C2semidirect
9. #10620994C2semidirect
10. #10620993C2semidirect
11. #10620890C2semidirect
42. #6492D3semidirect
101. #929D7semidirect
224. C3×C7#536semidirect
292. C7#5648semidirect
313. C3#90339semidirect

Maximal subgroups

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

subgroupindex
C22×Dic422
C2×#666842
C7×#9292
C3×#64922
#106211052
#106210992
#106210942
#106210242
#106209942
#106209932
#106208902
#903393
#56487