Finite Group Databasebeta

← #13060030

Subgroups — #13060030

Order 1344.

468 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 134 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#10621421C211yesmaximal, characteristic
2C7×D4×Dic3C211yesmaximal, characteristic, complemented
3C3×#6499C211yesmaximal, characteristic, complemented
4#10621124C212yesmaximal, complemented
5#10621118C212yesmaximal, complemented
6#10621093C212yesmaximal, complemented
7#10621091C212yesmaximal, complemented
8C4×D21⋊C4C211yesmaximal, characteristic, complemented
9#10620952C211yesmaximal, characteristic, complemented
10#10620950C211yesmaximal, characteristic, complemented
11Dic3×Dic14C211yesmaximal, characteristic, complemented
12#90248—31nomaximal, Hall
13C2×C3×C7×D4C2211yescharacteristic
14C2×#4268C2212yes—
15C2×C4×D21C2211yescharacteristic
16C22×C7×Dic3C2212yes—
17C2×C3×Dic14C2211yescharacteristic
18C2×D21⋊C4C2211yescharacteristic
19C2×D21⋊C4C2211yescharacteristic
20#66636C2212yes—
21C4⋊Dic21C2211yescharacteristic
22C7×#217C2212yes—
23C7×C4⋊Dic3C2211yescharacteristic
24C4×C7×Dic3C2211yescharacteristic
25C3×C22⋊Dic7C2212yes—
26C3×C22⋊Dic7C2212yes—
27C3×C4×Dic7C2211yescharacteristic
28Dic3⋊Dic7C2212yes—
29Dic3⋊Dic7C2212yes—
30#66577C2212yes—
31Dic7⋊Dic3C2212yes—
32#66575C2212yes—
33Dic3×Dic7C2211yescharacteristic
34Dic3×Dic7C2211yescharacteristic
35Dic3×Dic7C2212yes—
36C4×C7×D4—31no—
37#6501—31no—
38#6499D311yescharacteristic, complemented
39C4×C22⋊D7—32no—
40#6489—32no—
41#6460—31no—
42#6453—31no—
43#6446—32no—
44#6442—32no—
45C42×D7—31no—
46C4×Dic14—31no—
47#5680—71nomaximal, Hall
48C23×C3×C7C2312yes—
49C22×D21C2311yescharacteristic
50C3×C7×D4—22no—
51C2×C3×C4×C7C2311yescharacteristic
52#4268—24no—
53C2×Dic21C2311yescharacteristic
54C2×Dic21C2312yes—
55C4×D21—22no—
56C2×C7×Dic3—24no—
57C2×C7×Dic3C2311yescharacteristic
58C2×C7×Dic3C2312yes—
59C2×C7×Dic3C2311yescharacteristic
60C2×C3×Dic7C2312yes—
61C2×C3×Dic7C2311yescharacteristic
62C2×C3×Dic7C2311yescharacteristic
63C3×Dic14—22no—
64D21⋊C4—22no—
65D21⋊C4—22no—
66C2×C7×D4C2×D311yescharacteristic
67C22×C4×C7—32no—
68C2×C22⋊D7—32no—
69C2×C4×D7—31no—
70C2×C4×D7—31no—
71C2×C4×D7—31no—
72C2×Dic14C2×D311yescharacteristic
73C7×C4⋊C4—31no—
74C7×C22⋊C4—32no—
75C42×C7—31no—
76C22⋊Dic7—32no—
77C22⋊Dic7C2×D312yes—
78C22⋊Dic7C2×D312yes—
79(C2×D7)⋊C4—32no—
80C4⋊Dic7—31no—
81C4⋊Dic7—32no—
82C4⋊Dic7—32no—
83Dic7⋊C4—32no—
84Dic7⋊C4—32no—
85C4×Dic7—32no—
86C4×Dic7—31no—
87C4×Dic7—31no—
88C4×Dic7C2×D311yescharacteristic
89C3×#124—71no—
90#946—71no—
91Q8×Dic3—71no—
92#938—71no—
93D4×Dic3D711yescharacteristic, complemented
94#883—72no—
95#67⋊C4—72no—
96#879—72no—
97#877—72no—
98C42×D3—71no—
99#870—71no—
100C22×C3×C7—22no—
101C22×C3×C7—22no—
102C22×C3×C7—22no—
103C22×C3×C7C2411yescharacteristic
104C2×D21—21no—
105C2×D21—21no—
106C2×D21#4112yes—
107C3×C4×C7#4112yes—
108Dic21—21no—
109Dic21—22no—
110C3×Dic7—21no—
111C3×Dic7—21no—
112C3×Dic7—22no—
113C7×Dic3—21no—
114C7×Dic3—22no—
115C7×Dic3#4112yes—
116#529—211noSylow
117C23×C7C22×D312yes—
118C22×D7—31no—
119C7×D4—22no—
120C2×C4×C7—31no—
121C2×C4×C7—31no—
122C2×C4×C7—32no—
123C2×C4×C7—64no—
124C2×C4×C7C22×D311yescharacteristic
125C22⋊D7—64no—
126C2×Dic7—31no—
127C2×Dic7—32no—
128C2×Dic7C22×D311yescharacteristic
129C2×Dic7C22×D311yescharacteristic
130C2×Dic7C22×D312yes—
131C4×D7—62no—
132C4×D7—62no—
133C4×D7—62no—
134Dic14—22no—
135C2×C3×Q8—71no—
136C2×C3×D4C2×D711yescharacteristic
137C2×#67—72no—
138C22×Dic3C2×D712yes—
139C2×C4×D3—71no—
140C2×C4×D3—71no—
141C2×C4×D3—71no—
142C3×C22⋊C4—72no—
143C3×C22⋊C4—72no—
144C3×C42—71no—
145#217—72no—
146#217C2×D712yes—
147(C2×D3)⋊C4—72no—
148C4⋊Dic3—71no—
149C4⋊Dic3—72no—
150C4⋊Dic3C2×D711yescharacteristic
151Dic3⋊C4—72no—
152Dic3⋊C4—72no—
153Dic3⋊C4—72no—
154C4×Dic3—71no—
155C4×Dic3—71no—
156C4×Dic3—72no—
157C4×Dic3C2×D711yescharacteristic
158C2×C3×C7—42no—
159C2×C3×C7C2×#4111yescharacteristic
160C2×C3×C7C2×#4111yescharacteristic
161C2×C3×C7C2×#4111yescharacteristic
162D21—22no—
163#126—212no—
164#125—211no—
165#124—71no—
166#123—212no—
167C22⋊Q8—211no—
168#121—211no—
169C4×Q8—211no—
170C4×D4—31no—
171C4×D4—212no—
172#117—212no—
173C2×C42—211no—
174C22×C7—22no—
175C22×C7—22no—
176C22×C7—22no—
177C22×C7C23×D311yescharacteristic
178C2×D7—32no—
179C2×D7—61no—
180C2×D7—61no—
181C4×C7—32no—
182C4×C7—61no—
183C4×C7—62no—
184C4×C7#23712yes—
185Dic7—22no—
186Dic7—21no—
187Dic7—21no—
188Dic7—61no—
189Dic7—62no—
190C23×C3C22×D712yes—
191C22×D3—71no—
192C3×Q8—142no—
193C3×D4—22no—
194C2×C3×C4—71no—
195C2×C3×C4—71no—
196C2×C3×C4—72no—
197C2×C3×C4C22×D711yescharacteristic
198#67—144no—
199C2×Dic3—24no—
200C2×Dic3—71no—
201C2×Dic3—72no—
202C2×Dic3C22×D711yescharacteristic
203C2×Dic3C22×D712yes—
204C2×Dic3C22×D711yescharacteristic
205C4×D3—142no—
206C4×D3—142no—
207C4×D3—142no—
208C3×C7#52911yescharacteristic, complemented, Hall
209C2×Q8—71no—
210C2×D4—212no—
211C2×D4D3×D711yescharacteristic
212C22×C4—32no—
213C22×C4—211no—
214C22×C4—211no—
215C22×C4—211no—
216C4⋊C4—31no—
217C4⋊C4—212no—
218C4⋊C4—212no—
219C4⋊C4—212no—
220C4⋊C4—212no—
221C4⋊C4—211no—
222C22⋊C4—32no—
223C22⋊C4—72no—
224C22⋊C4—72no—
225C22⋊C4—212no—
226C22⋊C4—212no—
227C42—31no—
228C42—71no—
229C42—211no—
230C42—211no—
231C42—212no—
232C2×C7—42no—
233C2×C7C2×#23711yescharacteristic
234C2×C7D3×#4111yescharacteristic
235C2×C7D3×#4111yescharacteristic
236D7—62no—
237C22×C3—22no—
238C22×C3—22no—
239C22×C3—22no—
240C22×C3C23×D711yescharacteristic
241C2×D3—72no—
242C2×D3—141no—
243C2×D3—141no—
244C3×C4—142no—
245C3×C4—141no—
246C3×C4—141no—
247C3×C4#115712yes—
248Dic3—22no—
249Dic3—21no—
250Dic3—141no—
251Dic3—142no—
252Dic3#115912yes—
253C23—211no—
254C23C2×D3×D712yes—
255Q8—142no—
256D4—22no—
257D4—424no—
258C2×C4—31no—
259C2×C4—31no—
260C2×C4—32no—
261C2×C4—64no—
262C2×C4—71no—
263C2×C4—71no—
264C2×C4—72no—
265C2×C4—211no—
266C2×C4—212no—
267C2×C4—422no—
268C2×C4—422no—
269C2×C4—422no—
270C2×C4C2×D3×D711yescharacteristic
271C7#568011yescharacteristic, complemented, Sylow
272C2×C3—42no—
273C2×C3C2×#115711yescharacteristic
274C2×C3C2×#115911yescharacteristic
275C2×C3D7×#4111yescharacteristic
276D3—142no—
277C22—22no—
278C22—22no—
279C22—22no—
280C22—212no—
281C22—421no—
282C22—421no—
283C22C22×D3×D711yescharacteristic
284C4—32no—
285C4—62no—
286C4—61no—
287C4—141no—
288C4—141no—
289C4—142no—
290C4—422no—
291C4—421no—
292C4#6668312yes—
293C3#9024811yescharacteristic, complemented, Sylow
294C2—42no—
295C2—422no—
296C2D3×#115911yescharacteristic
297C2#1062157111yescharacteristic
298C2C2×#6668311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC2×C3×C7×D4C22
prime radical#10621421C2
hypercenterC2×D4D3×D7
hypocenterC3×C7#529
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C7×D4×Dic3C2semidirect
3. C3×#6499C2semidirect
4. #10621124C2semidirect
5. #10621118C2semidirect
6. #10621093C2semidirect
7. #10621091C2semidirect
8. C4×D21⋊C4C2semidirect
9. #10620952C2semidirect
10. #10620950C2semidirect
11. Dic3×Dic14C2semidirect
38. #6499D3semidirect
93. D4×Dic3D7semidirect
208. C3×C7#529semidirect
271. C7#5680semidirect
293. C3#90248semidirect

Maximal subgroups

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

subgroupindex
#106214212
C7×D4×Dic32
C3×#64992
#106211242
#106211182
#106210932
#106210912
C4×D21⋊C42
#106209522
#106209502
Dic3×Dic142
#902483
#56807