Finite Group Databasebeta

← #13058635

Subgroups — #13058635

Order 1344.

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

normal subgroups only

This group has 468 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
1#10621373C211yesmaximal, characteristic, complemented
2C7×#887C211yesmaximal, characteristic, complemented
3C3×#6458C211yesmaximal, characteristic
4Dic3×D28C212yesmaximal, complemented
5Dic3×D28C211yesmaximal, characteristic, complemented
6#10620969C212yesmaximal, complemented
7#10620955C212yesmaximal, complemented
8C4×Dic3×D7C211yesmaximal, characteristic, complemented
9#10620930C211yesmaximal, characteristic, complemented
10#10620924C212yesmaximal, complemented
11#10620921C211yesmaximal, characteristic, complemented
12#90271—31nomaximal, Hall
13C2×C7×Dic6C2211yescharacteristic
14C2×C3×D28C2211yescharacteristic
15C2×C3×D28C2212yes—
16C2×C3×C4×D7C2211yescharacteristic
17C2×Dic3×D7C2211yescharacteristic
18C2×Dic3×D7C2211yescharacteristic
19C2×Dic3×D7C2212yes—
20C3×C7×C4⋊C4C2211yescharacteristic
21C4⋊Dic21C2211yescharacteristic
22C4⋊Dic21C2212yes—
23#66629C2212yes—
24C4×Dic21C2211yescharacteristic
25C7×Dic3⋊C4C2212yes—
26C4×C7×Dic3C2211yescharacteristic
27C4×C7×Dic3C2212yes—
28C3×(C2×D7)⋊C4C2212yes—
29#66577C2212yes—
30(C2×D7)⋊Dic3C2212yes—
31(C2×D7)⋊Dic3C2212yes—
32(C2×D7)⋊Dic3C2212yes—
33(C2×D7)⋊Dic3C2212yes—
34Dic3×Dic7C2211yescharacteristic
35Dic3×Dic7C2211yescharacteristic
36C4×C7×Q8—31no—
37#6510—31no—
38#6509—31no—
39#6461—32no—
40#6458D311yescharacteristic
41#6457—32no—
42#6453—31no—
43C4×D28—32no—
44C4×D28—31no—
45#6435—32no—
46C42×D7—31no—
47#5600—71nomaximal, Hall
48C22×C3×D7C2312yes—
49C22×C3×D7C2311yescharacteristic
50C2×C3×C4×C7C2312yes—
51C2×C3×C4×C7C2311yescharacteristic
52C2×Dic21C2311yescharacteristic
53C2×Dic21C2311yescharacteristic
54C2×Dic21C2312yes—
55C2×C7×Dic3C2311yescharacteristic
56C2×C7×Dic3C2312yes—
57C2×C7×Dic3C2311yescharacteristic
58C7×Dic6—22no—
59C2×C3×Dic7C2311yescharacteristic
60C3×D28—24no—
61C3×D28—22no—
62C3×C4×D7—22no—
63Dic3×D7—22no—
64Dic3×D7—22no—
65Dic3×D7—24no—
66C2×C7×Q8—31no—
67C2×D28C2×D312yes—
68C2×D28C2×D311yescharacteristic
69C2×C4×D7—31no—
70C2×C4×D7—31no—
71C2×C4×D7—32no—
72C2×C4×D7C2×D311yescharacteristic
73C7×C4⋊C4—32no—
74C7×C4⋊C4C2×D311yescharacteristic
75C42×C7—31no—
76C42×C7—32no—
77(C2×D7)⋊C4—32no—
78(C2×D7)⋊C4—32no—
79(C2×D7)⋊C4—32no—
80(C2×D7)⋊C4—32no—
81(C2×D7)⋊C4C2×D312yes—
82C4⋊Dic7—32no—
83C4⋊Dic7—31no—
84Dic7⋊C4—32no—
85Dic7⋊C4—32no—
86C4×Dic7—31no—
87C4×Dic7—31no—
88C4×Dic7—31no—
89C3×#121—71no—
90#936—71no—
91#935—72no—
92D4×Dic3—72no—
93D4×Dic3—71no—
94#924—71no—
95C2×C4×Dic3—71no—
96#890—71no—
97#887D711yescharacteristic, complemented
98#879—72no—
99#877—72no—
100C22×C3×C7C2411yescharacteristic
101C2×C3×D7—21no—
102C2×C3×D7—21no—
103C2×C3×D7—22no—
104C2×C3×D7—22no—
105C2×C3×D7—22no—
106C2×C3×D7#4112yes—
107C3×C4×C7—22no—
108C3×C4×C7#4112yes—
109Dic21—21no—
110Dic21—22no—
111Dic21—21no—
112C3×Dic7—21no—
113C7×Dic3—21no—
114C7×Dic3—22no—
115C7×Dic3#4112yes—
116#529—211noSylow
117C22×D7C22×D312yes—
118C22×D7C22×D311yescharacteristic
119C7×Q8—62no—
120C2×C4×C7—31no—
121C2×C4×C7—31no—
122C2×C4×C7—32no—
123C2×C4×C7C22×D312yes—
124C2×C4×C7C22×D311yescharacteristic
125C2×Dic7—31no—
126C2×Dic7—31no—
127C2×Dic7—32no—
128C2×Dic7C22×D311yescharacteristic
129D28—24no—
130D28—22no—
131C4×D7—22no—
132C4×D7—62no—
133C4×D7—64no—
134C4×D7—62no—
135C2×C3×D4—72no—
136C2×C3×D4—71no—
137C22×C3×C4—71no—
138C22×Dic3—72no—
139C22×Dic3—71no—
140C22×Dic3—71no—
141C2×Dic6C2×D711yescharacteristic
142C3×C4⋊C4C2×D711yescharacteristic
143C3×C22⋊C4—72no—
144#217—72no—
145#217—72no—
146#217—72no—
147#217—72no—
148C4⋊Dic3—72no—
149C4⋊Dic3—71no—
150Dic3⋊C4—72no—
151Dic3⋊C4—72no—
152Dic3⋊C4C2×D712yes—
153C4×Dic3—71no—
154C4×Dic3—71no—
155C4×Dic3—71no—
156C4×Dic3C2×D712yes—
157C4×Dic3C2×D711yescharacteristic
158C2×C3×C7C2×#4111yescharacteristic
159C2×C3×C7C2×#4111yescharacteristic
160C2×C3×C7C2×#4111yescharacteristic
161C3×D7—22no—
162C3×D7—42no—
163#126—212no—
164#125—211no—
165#124—211no—
166#123—212no—
167C22⋊Q8—211no—
168#121—71no—
169C4×Q8—31no—
170C4×D4—211no—
171C4×D4—212no—
172#117—212no—
173C2×C42—211no—
174C22×C7C23×D311yescharacteristic
175C2×D7—21no—
176C2×D7—21no—
177C2×D7—22no—
178C2×D7—22no—
179C2×D7—22no—
180C2×D7#23712yes—
181C4×C7—22no—
182C4×C7—32no—
183C4×C7—61no—
184C4×C7—62no—
185C4×C7#23712yes—
186Dic7—21no—
187Dic7—61no—
188Dic7—61no—
189Dic7—62no—
190C23×C3—71no—
191C23×C3—72no—
192C3×D4—144no—
193C3×D4—142no—
194C2×C3×C4—71no—
195C2×C3×C4—142no—
196C2×C3×C4C22×D712yes—
197C2×C3×C4C22×D711yescharacteristic
198C2×Dic3—71no—
199C2×Dic3—71no—
200C2×Dic3—72no—
201C2×Dic3—142no—
202C2×Dic3—142no—
203C2×Dic3—144no—
204C2×Dic3C22×D711yescharacteristic
205C2×Dic3C22×D712yes—
206C2×Dic3C22×D711yescharacteristic
207Dic6—22no—
208C3×C7#52911yescharacteristic, complemented, Hall
209C2×Q8—31no—
210C2×D4—72no—
211C2×D4—71no—
212C22×C4—71no—
213C22×C4—211no—
214C22×C4—211no—
215C22×C4—212no—
216C4⋊C4—32no—
217C4⋊C4—212no—
218C4⋊C4—212no—
219C4⋊C4—211no—
220C4⋊C4—212no—
221C4⋊C4D3×D711yescharacteristic
222C22⋊C4—72no—
223C22⋊C4—212no—
224C22⋊C4—212no—
225C22⋊C4—212no—
226C22⋊C4—212no—
227C42—31no—
228C42—32no—
229C42—211no—
230C42—211no—
231C42—211no—
232C2×C7C2×#23711yescharacteristic
233C2×C7C2×#23711yescharacteristic
234C2×C7D3×#4111yescharacteristic
235D7—22no—
236D7—42no—
237C22×C3—72no—
238C22×C3—141no—
239C22×C3—141no—
240C22×C3—142no—
241C22×C3—142no—
242C22×C3—142no—
243C22×C3C23×D711yescharacteristic
244C3×C4—22no—
245C3×C4—141no—
246C3×C4#115712yes—
247Dic3—22no—
248Dic3—21no—
249Dic3—141no—
250Dic3—142no—
251Dic3—141no—
252Dic3#116112yes—
253C23—71no—
254C23—72no—
255Q8—62no—
256D4—144no—
257D4—142no—
258C2×C4—31no—
259C2×C4—31no—
260C2×C4—32no—
261C2×C4—71no—
262C2×C4—142no—
263C2×C4—211no—
264C2×C4—211no—
265C2×C4—212no—
266C2×C4—422no—
267C2×C4—424no—
268C2×C4—422no—
269C2×C4C2×D3×D712yes—
270C2×C4C2×D3×D711yescharacteristic
271C7#560011yescharacteristic, complemented, Sylow
272C2×C3—142no—
273C2×C3—282no—
274C2×C3C2×#115711yescharacteristic
275C2×C3C2×#116111yescharacteristic
276C2×C3D7×#4111yescharacteristic
277C22—72no—
278C22—141no—
279C22—141no—
280C22—142no—
281C22—142no—
282C22—142no—
283C22C22×D3×D711yescharacteristic
284C4—22no—
285C4—32no—
286C4—62no—
287C4—61no—
288C4—141no—
289C4—421no—
290C4—421no—
291C4—422no—
292C4#6666912yes—
293C3#9027111yescharacteristic, complemented, Sylow
294C2—142no—
295C2—282no—
296C2C2×#6666911yescharacteristic
297C2D7×#23711yescharacteristic
298C2D3×#116111yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC3×C7×C4⋊C4C22
prime radicalC3×#6458C2
hypercenterC4⋊C4D3×D7
hypocenterC3×C7#529
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. #10621373C2semidirect
2. C7×#887C2semidirect
4. Dic3×D28C2semidirect
5. Dic3×D28C2semidirect
6. #10620969C2semidirect
7. #10620955C2semidirect
8. C4×Dic3×D7C2semidirect
9. #10620930C2semidirect
10. #10620924C2semidirect
11. #10620921C2semidirect
97. #887D7semidirect
208. C3×C7#529semidirect
271. C7#5600semidirect
293. C3#90271semidirect

Maximal subgroups

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

subgroupindex
#106213732
C7×#8872
C3×#64582
Dic3×D282
#106209692
#106209552
C4×Dic3×D72
#106209302
#106209242
#106209212
#902713
#56007