Finite Group Databasebeta

← #13057773

Subgroups — #13057773

Order 1344.

444 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×Dic42C211yesmaximal, characteristic, complemented
2C7×#872C211yesmaximal, characteristic
3C3×C4×Dic14C211yesmaximal, characteristic, complemented
4C4×#4248C211yesmaximal, characteristic, complemented
5C4×#4248C211yesmaximal, characteristic, complemented
6#10620998C211yesmaximal, characteristic
7#10620994C211yesmaximal, characteristic
8#10620979C211yesmaximal, characteristic
9#10620972C211yesmaximal, characteristic
10C4×D3×Dic7C211yesmaximal, characteristic
11#10620933C211yesmaximal, characteristic
12#10620908C211yesmaximal, characteristic, complemented
13#10620905C211yesmaximal, characteristic, complemented
14#10620890C211yesmaximal, characteristic, complemented
15#10620889C211yesmaximal, characteristic, complemented
16#90208—31nomaximal, Hall
17C2×Dic42C2211yescharacteristic
18C2×C4×D3×C7C2211yescharacteristic
19C2×C3×Dic14C2211yescharacteristic
20C2×#4248C2211yescharacteristic
21C2×#4248C2211yescharacteristic
22C2×D3×Dic7C2211yescharacteristic
23C2×D3×Dic7C2211yescharacteristic
24C3×C42×C7C2211yescharacteristic
25C4⋊Dic21C2211yescharacteristic
26#66629C2211yescharacteristic
27#66629C2211yescharacteristic
28C4×Dic21C2211yescharacteristic
29C4×Dic21C2211yescharacteristic
30C7×(C2×D3)⋊C4C2211yescharacteristic
31C7×(C2×D3)⋊C4C2211yescharacteristic
32C7×Dic3⋊C4C2211yescharacteristic
33C7×Dic3⋊C4C2211yescharacteristic
34C4×C7×Dic3C2211yescharacteristic
35C3×C4⋊Dic7C2211yescharacteristic
36C3×Dic7⋊C4C2211yescharacteristic
37C3×Dic7⋊C4C2211yescharacteristic
38C3×C4×Dic7C2211yescharacteristic
39C3×C4×Dic7C2211yescharacteristic
40Dic3⋊Dic7C2211yescharacteristic
41Dic3⋊Dic7C2211yescharacteristic
42Dic3⋊Dic7C2211yescharacteristic
43Dic3⋊Dic7C2211yescharacteristic
44#66577C2211yescharacteristic
45#66577C2211yescharacteristic
46Dic7⋊Dic3C2211yescharacteristic
47Dic7⋊Dic3C2211yescharacteristic
48(C2×D3)⋊Dic7C2211yescharacteristic
49(C2×D3)⋊Dic7C2211yescharacteristic
50Dic3×Dic7C2211yescharacteristic
51Dic3×Dic7C2211yescharacteristic
52C7×#117—31no—
53#6489—31no—
54C2×C4×Dic7—31no—
55#6453—31no—
56#6453—31no—
57#6451—31no—
58#6451—31no—
59#6441—31no—
60#6441—31no—
61#6441—31no—
62#6441—31no—
63C4×Dic14—31no—
64C4×Dic14—31no—
65C4×Dic14—31no—
66C4×Dic14D311yescharacteristic, complemented
67#5578—71nomaximal, Hall
68C22×D3×C7C2311yescharacteristic
69C2×C3×C4×C7C2311yescharacteristic
70C2×C3×C4×C7C2311yescharacteristic
71C2×C3×C4×C7C2311yescharacteristic
72C2×Dic21C2311yescharacteristic
73C2×Dic21C2311yescharacteristic
74C2×Dic21C2311yescharacteristic
75C2×Dic21C2311yescharacteristic
76Dic42—22no—
77C2×C7×Dic3C2311yescharacteristic
78C2×C7×Dic3C2311yescharacteristic
79C2×C7×Dic3C2311yescharacteristic
80C4×D3×C7Q814yescomplemented
81C2×C3×Dic7C2311yescharacteristic
82C2×C3×Dic7C2311yescharacteristic
83C2×C3×Dic7C2311yescharacteristic
84C2×C3×Dic7C2311yescharacteristic
85C3×Dic14—22no—
86#4248—22no—
87#4248—22no—
88D3×Dic7—22no—
89D3×Dic7—22no—
90C22×C4×C7—31no—
91C22×Dic7—31no—
92C22×Dic7—31no—
93C2×Dic14—31no—
94C2×Dic14—31no—
95C2×Dic14—31no—
96C2×Dic14C2×D311yescharacteristic
97C7×C4⋊C4—31no—
98C7×C4⋊C4—31no—
99C7×C22⋊C4—31no—
100C7×C22⋊C4—31no—
101C42×C7—31no—
102C42×C7C2×D311yescharacteristic
103C22⋊Dic7—31no—
104C22⋊Dic7—31no—
105C4⋊Dic7—31no—
106C4⋊Dic7—31no—
107C4⋊Dic7—31no—
108C4⋊Dic7—31no—
109C4⋊Dic7—31no—
110C4⋊Dic7C2×D311yescharacteristic
111Dic7⋊C4—31no—
112Dic7⋊C4—31no—
113Dic7⋊C4—31no—
114Dic7⋊C4—31no—
115Dic7⋊C4—31no—
116Dic7⋊C4—31no—
117Dic7⋊C4C2×D311yescharacteristic
118Dic7⋊C4C2×D311yescharacteristic
119C4×Dic7—31no—
120C4×Dic7—31no—
121C4×Dic7—31no—
122C4×Dic7—31no—
123C4×Dic7C2×D311yescharacteristic
124C4×Dic7C2×D311yescharacteristic
125C3×C4×Q8—71no—
126#946—71no—
127#944—71no—
128#897—71no—
129#896—71no—
130#896—71no—
131#889—71no—
132#889—71no—
133#888—71no—
134#872—71no—
135#872D711yescharacteristic
136C42×D3—71no—
137C4×Dic6—71no—
138C4×Dic6—71no—
139C4×Dic6—71no—
140C22×C3×C7C2411yescharacteristic
141C2×D3×C7—21no—
142C2×D3×C7—21no—
143C2×D3×C7C2×Q812yes—
144C3×C4×C7—21no—
145C3×C4×C7—21no—
146C3×C4×C7C2×Q812yes—
147Dic21—21no—
148Dic21—21no—
149Dic21—21no—
150Dic21#4112yes—
151C3×Dic7—21no—
152C3×Dic7—21no—
153C3×Dic7—21no—
154C3×Dic7#4112yes—
155C7×Dic3—21no—
156C7×Dic3—21no—
157C7×Dic3C2×Q812yes—
158#533—211noSylow
159C23×C7—31no—
160C2×C4×C7—31no—
161C2×C4×C7—34no—
162C2×C4×C7—31no—
163C2×C4×C7—31no—
164C2×C4×C7C22×D311yescharacteristic
165C2×C4×C7C22×D311yescharacteristic
166C2×C4×C7C22×D311yescharacteristic
167C2×Dic7—31no—
168C2×Dic7—31no—
169C2×Dic7—31no—
170C2×Dic7—31no—
171C2×Dic7—62no—
172C2×Dic7—62no—
173C2×Dic7C22×D311yescharacteristic
174C2×Dic7C22×D311yescharacteristic
175C2×Dic7C22×D311yescharacteristic
176C2×Dic7C22×D311yescharacteristic
177Dic14—22no—
178Dic14—62no—
179Dic14—62no—
180Dic14—62no—
181C2×C3×Q8—71no—
182C2×C4×D3—71no—
183C2×C4×D3—71no—
184C2×C4×D3C2×D711yescharacteristic
185C2×Dic6—71no—
186C2×Dic6—71no—
187C2×Dic6—71no—
188C3×C4⋊C4—71no—
189C3×C4⋊C4—71no—
190C3×C4⋊C4—71no—
191C3×C42—71no—
192C3×C42—71no—
193C3×C42C2×D711yescharacteristic
194(C2×D3)⋊C4—71no—
195(C2×D3)⋊C4—71no—
196(C2×D3)⋊C4C2×D711yescharacteristic
197(C2×D3)⋊C4C2×D711yescharacteristic
198C4⋊Dic3—71no—
199C4⋊Dic3—71no—
200C4⋊Dic3—71no—
201Dic3⋊C4—71no—
202Dic3⋊C4—71no—
203Dic3⋊C4—71no—
204Dic3⋊C4—71no—
205Dic3⋊C4—71no—
206Dic3⋊C4—71no—
207Dic3⋊C4—71no—
208Dic3⋊C4—71no—
209Dic3⋊C4C2×D711yescharacteristic
210Dic3⋊C4C2×D711yescharacteristic
211C4×Dic3—71no—
212C4×Dic3—71no—
213C4×Dic3—71no—
214C4×Dic3—71no—
215C4×Dic3C2×D711yescharacteristic
216C2×C3×C7C22×Q811yescharacteristic
217C2×C3×C7C2×#4111yescharacteristic
218C2×C3×C7C2×#4111yescharacteristic
219D3×C7—22no—
220C4⋊Q8—211no—
221C4⋊Q8—211no—
222#125—211no—
223#125—211no—
224C22⋊Q8—211no—
225C22⋊Q8—211no—
226C22⋊Q8—211no—
227C22⋊Q8—211no—
228C4×Q8—71no—
229C4×Q8—211no—
230C4×Q8—211no—
231C4×Q8—211no—
232#117—31no—
233#117—211no—
234C2×C42—211no—
235C22×C7—32no—
236C22×C7—61no—
237C22×C7—61no—
238C22×C7C23×D311yescharacteristic
239C4×C7—21no—
240C4×C7—21no—
241C4×C7—32no—
242C4×C7—61no—
243C4×C7—61no—
244C4×C7D3×Q812yes—
245Dic7—21no—
246Dic7—21no—
247Dic7—21no—
248Dic7—32no—
249Dic7—61no—
250Dic7—61no—
251Dic7—61no—
252Dic7#23512yes—
253C22×D3C22×D711yescharacteristic
254C3×Q8—142no—
255C2×C3×C4—71no—
256C2×C3×C4—71no—
257C2×C3×C4—71no—
258C2×C3×C4—71no—
259C2×C3×C4C22×D711yescharacteristic
260C2×C3×C4C22×D711yescharacteristic
261C2×C3×C4C22×D711yescharacteristic
262C2×Dic3—71no—
263C2×Dic3—71no—
264C2×Dic3—71no—
265C2×Dic3—71no—
266C2×Dic3C22×D711yescharacteristic
267C2×Dic3C22×D711yescharacteristic
268C2×Dic3C22×D711yescharacteristic
269C4×D3—142no—
270C4×D3—142no—
271C4×D3Dic1414yescomplemented
272Dic6—142no—
273Dic6—142no—
274Dic6—142no—
275C3×C7#53311yescharacteristic, complemented, Hall
276C2×Q8—71no—
277C2×Q8—211no—
278C2×Q8—211no—
279C2×Q8—211no—
280C22×C4—31no—
281C22×C4—211no—
282C22×C4—211no—
283C4⋊C4—31no—
284C4⋊C4—31no—
285C4⋊C4—71no—
286C4⋊C4—71no—
287C4⋊C4—71no—
288C4⋊C4—211no—
289C4⋊C4—211no—
290C4⋊C4—211no—
291C4⋊C4—211no—
292C4⋊C4—211no—
293C4⋊C4—211no—
294C4⋊C4—211no—
295C4⋊C4—211no—
296C4⋊C4—211no—
297C4⋊C4—211no—
298C4⋊C4—211no—
299C22⋊C4—31no—
300C22⋊C4—31no—
301C22⋊C4—211no—
302C22⋊C4—211no—
303C42—31no—
304C42—71no—
305C42—71no—
306C42—211no—
307C42—211no—
308C42—211no—
309C42—211no—
310C42D3×D711yescharacteristic
311C2×C7—62no—
312C2×C7C2×#23511yescharacteristic
313C2×C7C2×D3×Q811yescharacteristic
314C2×C7D3×#4111yescharacteristic
315C22×C3C23×D711yescharacteristic
316C2×D3—21no—
317C2×D3—21no—
318C2×D3C2×Dic1412yes—
319C3×C4—21no—
320C3×C4—21no—
321C3×C4—72no—
322C3×C4—141no—
323C3×C4—141no—
324C3×C4—141no—
325C3×C4C2×Dic1412yes—
326Dic3—21no—
327Dic3—21no—
328Dic3—72no—
329Dic3—141no—
330Dic3—141no—
331Dic3—141no—
332Dic3C2×Dic1412yes—
333C23—31no—
334Q8—142no—
335Q8—422no—
336Q8—422no—
337Q8—422no—
338C2×C4—31no—
339C2×C4—34no—
340C2×C4—31no—
341C2×C4—31no—
342C2×C4—71no—
343C2×C4—71no—
344C2×C4—71no—
345C2×C4—71no—
346C2×C4—211no—
347C2×C4—211no—
348C2×C4—211no—
349C2×C4—211no—
350C2×C4—422no—
351C2×C4—422no—
352C2×C4C2×D3×D711yescharacteristic
353C2×C4C2×D3×D711yescharacteristic
354C2×C4C2×D3×D711yescharacteristic
355C7#557811yescharacteristic, complemented, Sylow
356C2×C3C22×Dic1411yescharacteristic
357C2×C3D7×#4111yescharacteristic
358C2×C3D7×#4111yescharacteristic
359D3—22no—
360C22—32no—
361C22—61no—
362C22—61no—
363C22C22×D3×D711yescharacteristic
364C4—21no—
365C4—21no—
366C4—32no—
367C4—61no—
368C4—61no—
369C4—72no—
370C4—141no—
371C4—141no—
372C4—141no—
373C4—212no—
374C4—421no—
375C4—421no—
376C4—421no—
377C4D3×Dic1412yes—
378C3#9020811yescharacteristic, complemented, Sylow
379C2—62no—
380C2C2×D3×Dic1411yescharacteristic
381C2D7×#23511yescharacteristic
382C2#1062156311yescharacteristic
subgroupG/H
centerC2×C4C2×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FrattiniC22C22×D3×D7
FittingC3×C42×C7C22
prime radicalC22×D3×C7C23
hypercenterC42D3×D7
hypocenterC3×C7#533
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C4×Dic42C2semidirect
3. C3×C4×Dic14C2semidirect
4. C4×#4248C2semidirect
5. C4×#4248C2semidirect
12. #10620908C2semidirect
13. #10620905C2semidirect
14. #10620890C2semidirect
15. #10620889C2semidirect
66. C4×Dic14D3semidirect
80. C4×D3×C7Q8semidirect
271. C4×D3Dic14semidirect
275. C3×C7#533semidirect
355. C7#5578semidirect
378. C3#90208semidirect

Maximal subgroups

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

subgroupindex
C4×Dic422
C7×#8722
C3×C4×Dic142
C4×#42482
#106209982
#106209942
#106209792
#106209722
C4×D3×Dic72
#106209332
#106209082
#106209052
#106208902
#106208892
#902083
#55787