Finite Group Databasebeta

← #13061690

Subgroups — #13061690

Order 1344.

468 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C7×#121C211yesmaximal, characteristic
2#10621419C211yesmaximal, characteristic, complemented
3#10621418C212yesmaximal, complemented
4D4×Dic21C212yesmaximal, complemented
5D4×Dic21C211yesmaximal, characteristic, complemented
6#10621407C211yesmaximal, characteristic, complemented
7C2×C4×Dic21C211yesmaximal, characteristic, complemented
8#10621373C211yesmaximal, characteristic, complemented
9#10621370C211yesmaximal, characteristic, complemented
10#10621362C212yesmaximal, complemented
11#10621360C212yesmaximal, complemented
12#90291—31nomaximal, Hall
13C2×C3×C7×D4C2212yes—
14C2×C3×C7×D4C2211yescharacteristic
15C22×C3×C4×C7C2211yescharacteristic
16C22×Dic21C2212yes—
17C22×Dic21C2211yescharacteristic
18C22×Dic21C2211yescharacteristic
19C2×Dic42C2211yescharacteristic
20C3×C7×C4⋊C4C2211yescharacteristic
21C3×C7×C22⋊C4C2212yes—
22#66636C2212yes—
23#66636C2212yes—
24#66636C2212yes—
25#66636C2212yes—
26C4⋊Dic21C2212yes—
27C4⋊Dic21C2211yescharacteristic
28#66629C2212yes—
29#66629C2212yes—
30#66629C2212yes—
31C4×Dic21C2211yescharacteristic
32C4×Dic21C2212yes—
33C4×Dic21C2211yescharacteristic
34C4×Dic21C2211yescharacteristic
35C4×Dic21C2211yescharacteristic
36C7×#121D311yescharacteristic
37#6499—31no—
38#6498—32no—
39D4×Dic7—32no—
40D4×Dic7—31no—
41#6487—31no—
42C2×C4×Dic7—31no—
43#6453—31no—
44#6450—31no—
45#6442—32no—
46#6440—32no—
47#5600—71nomaximal, Hall
48C23×C3×C7C2312yes—
49C23×C3×C7C2311yescharacteristic
50C3×C7×D4—24no—
51C3×C7×D4—22no—
52C2×C3×C4×C7—22no—
53C2×C3×C4×C7C2311yescharacteristic
54C2×C3×C4×C7C2312yes—
55C2×C3×C4×C7C2311yescharacteristic
56C2×Dic21—22no—
57C2×Dic21—22no—
58C2×Dic21—24no—
59C2×Dic21C2311yescharacteristic
60C2×Dic21C2311yescharacteristic
61C2×Dic21C2311yescharacteristic
62C2×Dic21C2312yes—
63C2×Dic21C2312yes—
64C2×Dic21C2311yescharacteristic
65Dic42—22no—
66C2×C7×D4C2×D312yes—
67C2×C7×D4C2×D311yescharacteristic
68C22×C4×C7C2×D311yescharacteristic
69C22×Dic7—32no—
70C22×Dic7—31no—
71C22×Dic7—31no—
72C2×Dic14—31no—
73C7×C4⋊C4C2×D311yescharacteristic
74C7×C22⋊C4C2×D312yes—
75C22⋊Dic7—32no—
76C22⋊Dic7—32no—
77C22⋊Dic7—32no—
78C22⋊Dic7—32no—
79C4⋊Dic7—32no—
80C4⋊Dic7—31no—
81Dic7⋊C4—32no—
82Dic7⋊C4—32no—
83Dic7⋊C4—32no—
84C4×Dic7—31no—
85C4×Dic7—32no—
86C4×Dic7—31no—
87C4×Dic7—31no—
88C4×Dic7—31no—
89C3×#121D711yescharacteristic
90#936—71no—
91#935—72no—
92D4×Dic3—72no—
93D4×Dic3—71no—
94#924—71no—
95C2×C4×Dic3—71no—
96#890—71no—
97#887—71no—
98#879—72no—
99#877—72no—
100C22×C3×C7—21no—
101C22×C3×C7—21no—
102C22×C3×C7—22no—
103C22×C3×C7—22no—
104C22×C3×C7—22no—
105C22×C3×C7#4112yes—
106C22×C3×C7C2411yescharacteristic
107C3×C4×C7—21no—
108C3×C4×C7—22no—
109C3×C4×C7#4112yes—
110Dic21—22no—
111Dic21—21no—
112Dic21—21no—
113Dic21—21no—
114Dic21—22no—
115Dic21#4112yes—
116#529—211noSylow
117C23×C7C22×D312yes—
118C23×C7C22×D311yescharacteristic
119C7×D4—24no—
120C7×D4—22no—
121C2×C4×C7—22no—
122C2×C4×C7C22×D311yescharacteristic
123C2×C4×C7C22×D312yes—
124C2×C4×C7C22×D311yescharacteristic
125C2×Dic7—31no—
126C2×Dic7—31no—
127C2×Dic7—31no—
128C2×Dic7—31no—
129C2×Dic7—32no—
130C2×Dic7—32no—
131C2×Dic7—62no—
132C2×Dic7—64no—
133C2×Dic7—62no—
134Dic14—62no—
135C2×C3×D4C2×D712yes—
136C2×C3×D4C2×D711yescharacteristic
137C22×C3×C4C2×D711yescharacteristic
138C22×Dic3—72no—
139C22×Dic3—71no—
140C22×Dic3—71no—
141C2×Dic6—71no—
142C3×C4⋊C4C2×D711yescharacteristic
143C3×C22⋊C4C2×D712yes—
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⋊C4—72no—
153C4×Dic3—71no—
154C4×Dic3—72no—
155C4×Dic3—71no—
156C4×Dic3—71no—
157C4×Dic3—71no—
158C2×C3×C7—22no—
159C2×C3×C7—42no—
160C2×C3×C7C2×#4111yescharacteristic
161C2×C3×C7C2×#4111yescharacteristic
162C2×C3×C7C2×#4111yescharacteristic
163#126—212no—
164#125—211no—
165#124—211no—
166#123—212no—
167C22⋊Q8—211no—
168#121D2111yescharacteristic
169C4×Q8—211no—
170C4×D4—211no—
171C4×D4—212no—
172#117—212no—
173C2×C42—211no—
174C22×C7—21no—
175C22×C7—21no—
176C22×C7—22no—
177C22×C7—22no—
178C22×C7—22no—
179C22×C7#23712yes—
180C22×C7C23×D311yescharacteristic
181C4×C7—21no—
182C4×C7—22no—
183C4×C7#23712yes—
184Dic7—32no—
185Dic7—62no—
186Dic7—61no—
187Dic7—61no—
188Dic7—61no—
189Dic7—62no—
190C23×C3C22×D712yes—
191C23×C3C22×D711yescharacteristic
192C3×D4—24no—
193C3×D4—22no—
194C2×C3×C4—22no—
195C2×C3×C4C22×D711yescharacteristic
196C2×C3×C4C22×D712yes—
197C2×C3×C4C22×D711yescharacteristic
198C2×Dic3—71no—
199C2×Dic3—71no—
200C2×Dic3—71no—
201C2×Dic3—71no—
202C2×Dic3—72no—
203C2×Dic3—72no—
204C2×Dic3—142no—
205C2×Dic3—144no—
206C2×Dic3—142no—
207Dic6—142no—
208C3×C7#52911yescharacteristic, complemented, Hall
209C2×Q8—211no—
210C2×D4C2×D2112yes—
211C2×D4C2×D2111yescharacteristic
212C22×C4—211no—
213C22×C4—211no—
214C22×C4—212no—
215C22×C4C2×D2111yescharacteristic
216C4⋊C4—212no—
217C4⋊C4—212no—
218C4⋊C4—212no—
219C4⋊C4—211no—
220C4⋊C4—212no—
221C4⋊C4C2×D2111yescharacteristic
222C22⋊C4—212no—
223C22⋊C4—212no—
224C22⋊C4—212no—
225C22⋊C4—212no—
226C22⋊C4C2×D2112yes—
227C42—211no—
228C42—211no—
229C42—211no—
230C42—211no—
231C42—212no—
232C2×C7—22no—
233C2×C7—42no—
234C2×C7C2×#23711yescharacteristic
235C2×C7C2×#23711yescharacteristic
236C2×C7D3×#4111yescharacteristic
237C22×C3—21no—
238C22×C3—21no—
239C22×C3—22no—
240C22×C3—22no—
241C22×C3—22no—
242C22×C3#115912yes—
243C22×C3C23×D711yescharacteristic
244C3×C4—21no—
245C3×C4—22no—
246C3×C4#115912yes—
247Dic3—72no—
248Dic3—142no—
249Dic3—141no—
250Dic3—141no—
251Dic3—141no—
252Dic3—142no—
253C23C22×D2112yes—
254C23C22×D2111yescharacteristic
255Q8—422no—
256D4—24no—
257D4—22no—
258C2×C4—22no—
259C2×C4—211no—
260C2×C4—211no—
261C2×C4—211no—
262C2×C4—211no—
263C2×C4—212no—
264C2×C4—212no—
265C2×C4—422no—
266C2×C4—424no—
267C2×C4—422no—
268C2×C4C22×D2111yescharacteristic
269C2×C4C22×D2112yes—
270C2×C4C22×D2111yescharacteristic
271C7#560011yescharacteristic, complemented, Sylow
272C2×C3—22no—
273C2×C3—42no—
274C2×C3C2×#115911yescharacteristic
275C2×C3C2×#115911yescharacteristic
276C2×C3D7×#4111yescharacteristic
277C22—21no—
278C22—21no—
279C22—22no—
280C22—22no—
281C22—22no—
282C22#6673012yes—
283C22C23×D2111yescharacteristic
284C4—21no—
285C4—22no—
286C4—212no—
287C4—421no—
288C4—421no—
289C4—422no—
290C4—421no—
291C4—422no—
292C4#6673012yes—
293C3#9029111yescharacteristic, complemented, Sylow
294C2—22no—
295C2—42no—
296C2C2×#6673011yescharacteristic
297C2C2×#6673011yescharacteristic
298C2#41×D2111yescharacteristic
subgroupG/H
center, FrattiniC22C23×D21
commutator, socle, cosocleC22×C3×C7C24
Fitting, prime radicalC3×C7×#121C2
hypercenter#121D21
hypocenterC3×C7#529
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
2. #10621419C2semidirect
3. #10621418C2semidirect
4. D4×Dic21C2semidirect
5. D4×Dic21C2semidirect
6. #10621407C2semidirect
7. C2×C4×Dic21C2semidirect
8. #10621373C2semidirect
9. #10621370C2semidirect
10. #10621362C2semidirect
11. #10621360C2semidirect
208. C3×C7#529semidirect
271. C7#5600semidirect
293. C3#90291semidirect

Maximal subgroups

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

subgroupindex
C3×C7×#1212
#106214192
#106214182
D4×Dic212
#106214072
C2×C4×Dic212
#106213732
#106213702
#106213622
#106213602
#902913
#56007