Finite Group Databasebeta

← #13058022

Subgroups — #13058022

Order 1344.

540 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#66684C211yesmaximal, characteristic
2C2×D7×Dic6C211yesmaximal, characteristic, complemented
3#10621361C211yesmaximal, characteristic, complemented
4C7×#885C211yesmaximal, characteristic, complemented
5C3×#6446C211yesmaximal, characteristic, complemented
6#10621138C211yesmaximal, characteristic, complemented
7#10621110C211yesmaximal, characteristic, complemented
8#10621087C211yesmaximal, characteristic, complemented
9#10621031C211yesmaximal, characteristic, complemented
10#10621009C211yesmaximal, characteristic, complemented
11#10620998C211yesmaximal, characteristic, complemented
12#10620994C211yesmaximal, characteristic, complemented
13#10620976C211yesmaximal, characteristic, complemented
14#10620955C211yesmaximal, characteristic, complemented
15#10620891C211yesmaximal, characteristic, complemented
16#90366—31nomaximal, Hall
17C22×Dic21C2211yescharacteristic
18C2×Dic42C2211yescharacteristic
19C2×C7×#67C2211yescharacteristic
20C2×C7×Dic6C2211yescharacteristic
21C2×C3×C22⋊D7C2211yescharacteristic
22C2×C3×C4×D7C2211yescharacteristic
23C2×#4248C2211yescharacteristic
24C2×#4248C2211yescharacteristic
25C2×#4248C2211yescharacteristic
26C2×#4245C2211yescharacteristic
27C2×D3×Dic7C2211yescharacteristic
28#66684—24no—
29C2×Dic3×D7C2211yescharacteristic
30C2×Dic3×D7C2211yescharacteristic
31D7×Dic6—24no—
32C3×C7×C22⋊C4C2211yescharacteristic
33#66636C2211yescharacteristic
34C4⋊Dic21C2211yescharacteristic
35#66629C2211yescharacteristic
36#66629C2211yescharacteristic
37C7×#217C2211yescharacteristic
38C7×(C2×D3)⋊C4C2211yescharacteristic
39C7×(C2×D3)⋊C4C2211yescharacteristic
40C4×C7×Dic3C2211yescharacteristic
41C3×C22⋊Dic7C2211yescharacteristic
42C3×(C2×D7)⋊C4C2211yescharacteristic
43C3×C4⋊Dic7C2211yescharacteristic
44C3×Dic7⋊C4C2211yescharacteristic
45Dic3⋊Dic7C2211yescharacteristic
46Dic3⋊Dic7C2211yescharacteristic
47#66577C2211yescharacteristic
48#66577C2211yescharacteristic
49Dic7⋊Dic3C2211yescharacteristic
50(C2×D3)⋊Dic7C2211yescharacteristic
51(C2×D3)⋊Dic7C2211yescharacteristic
52(C2×D7)⋊Dic3C2211yescharacteristic
53Dic3×Dic7C2211yescharacteristic
54C2×Q8×D7—31no—
55C2×#1159—31no—
56C7×#124—31no—
57#6507—31no—
58#6499—31no—
59#6446—31no—
60#6446—31no—
61#6446—31no—
62#6446D311yescharacteristic, complemented
63#6441—31no—
64#6441—31no—
65#6441—31no—
66#6441—31no—
67#6435—31no—
68#6432—31no—
69#5648—71nomaximal, Hall
70C23×C3×C7C2311yescharacteristic
71C22×D3×C7C2311yescharacteristic
72C22×C3×D7C2311yescharacteristic
73C2×C3×C4×C7C2311yescharacteristic
74C2×C3×C4×C7C2311yescharacteristic
75C2×Dic21—22no—
76C2×Dic21C2311yescharacteristic
77C2×Dic21C2311yescharacteristic
78C2×Dic21C2311yescharacteristic
79C2×Dic21C2311yescharacteristic
80Dic42—22no—
81C7×#67—22no—
82C2×C7×Dic3C2311yescharacteristic
83C2×C7×Dic3C2311yescharacteristic
84C2×C7×Dic3C2311yescharacteristic
85C7×Dic6—22no—
86C3×C22⋊D7—22no—
87C2×C3×Dic7C2311yescharacteristic
88C2×C3×Dic7C2311yescharacteristic
89C2×C3×Dic7C2311yescharacteristic
90C3×C4×D7—22no—
91#4248—22no—
92#4248—22no—
93#4248—22no—
94#4245—22no—
95D3×Dic7—22no—
96Dic3×D7—22no—
97Dic3×D7D414yes—
98C2×C7×Q8—31no—
99C2×C7×D4—31no—
100C2×C22⋊D7—31no—
101C2×C22⋊D7C2×D311yescharacteristic
102C22×Dic7—31no—
103C22×Dic7—31no—
104Q8×D7—64no—
105#1159—64no—
106C2×C4×D7—31no—
107C2×C4×D7—31no—
108C2×C4×D7C2×D311yescharacteristic
109C2×Dic14—31no—
110C2×Dic14—31no—
111C2×Dic14—31no—
112C2×Dic14—31no—
113C7×C22⋊C4—31no—
114C7×C22⋊C4—31no—
115C7×C22⋊C4—31no—
116C7×C22⋊C4C2×D311yescharacteristic
117C42×C7—31no—
118C22⋊Dic7—31no—
119C22⋊Dic7—31no—
120C22⋊Dic7—31no—
121C22⋊Dic7C2×D311yescharacteristic
122(C2×D7)⋊C4—31no—
123(C2×D7)⋊C4C2×D311yescharacteristic
124C4⋊Dic7—31no—
125C4⋊Dic7—31no—
126C4⋊Dic7—31no—
127C4⋊Dic7C2×D311yescharacteristic
128Dic7⋊C4—31no—
129Dic7⋊C4—31no—
130Dic7⋊C4—31no—
131Dic7⋊C4—31no—
132Dic7⋊C4—31no—
133Dic7⋊C4C2×D311yescharacteristic
134C4×Dic7—31no—
135C2×#237—71no—
136C22×Dic6—71no—
137C3×#123—71no—
138#935—71no—
139#929—71no—
140#896—71no—
141#896—71no—
142#888—71no—
143#888—71no—
144#886—71no—
145#885—71no—
146#885D711yescharacteristic, complemented
147#878—71no—
148#878—71no—
149#877—71no—
150C22×C3×C7—21no—
151C22×C3×C7—21no—
152C22×C3×C7—21no—
153C22×C3×C7C2411yescharacteristic
154C2×D3×C7—21no—
155C2×D3×C7—21no—
156C2×D3×C7—21no—
157C2×C3×D7—21no—
158C2×C3×D7—21no—
159C2×C3×D7C2×D412yes—
160C3×C4×C7—21no—
161C3×C4×C7—21no—
162Dic21—21no—
163Dic21—21no—
164Dic21—21no—
165Dic21C2×D412yes—
166C3×Dic7—21no—
167C3×Dic7—21no—
168C3×Dic7—21no—
169C7×Dic3—21no—
170C7×Dic3—21no—
171C7×Dic3C2×D412yes—
172#536—211noSylow
173C23×C7—31no—
174C23×C7C22×D311yescharacteristic
175C22×D7C22×D311yescharacteristic
176C7×Q8—62no—
177C7×D4—62no—
178C2×C4×C7—31no—
179C2×C4×C7—31no—
180C2×C4×C7—31no—
181C2×C4×C7C22×D311yescharacteristic
182C2×C4×C7C22×D311yescharacteristic
183C22⋊D7—22no—
184C22⋊D7—62no—
185C2×Dic7—31no—
186C2×Dic7—31no—
187C2×Dic7—31no—
188C2×Dic7—31no—
189C2×Dic7—62no—
190C2×Dic7—62no—
191C2×Dic7C22×D311yescharacteristic
192C2×Dic7C22×D311yescharacteristic
193C2×Dic7C22×D311yescharacteristic
194C4×D7—22no—
195C4×D7—34no—
196C4×D7—62no—
197Dic14—62no—
198Dic14—62no—
199Dic14—62no—
200Dic14—62no—
201C2×C3×D4—71no—
202C22×C3×C4—71no—
203C2×#67—71no—
204C2×#67C2×D711yescharacteristic
205C22×Dic3—71no—
206C22×Dic3—71no—
207C22×Dic3—71no—
208#237—144no—
209C2×C4×D3—71no—
210C2×Dic6—71no—
211C2×Dic6—71no—
212C2×Dic6—71no—
213C2×Dic6—71no—
214C2×Dic6—144no—
215C2×Dic6C2×D711yescharacteristic
216C3×C4⋊C4—71no—
217C3×C4⋊C4—71no—
218C3×C22⋊C4—71no—
219C3×C22⋊C4—71no—
220C3×C22⋊C4C2×D711yescharacteristic
221#217—71no—
222#217—71no—
223#217C2×D711yescharacteristic
224(C2×D3)⋊C4—71no—
225(C2×D3)⋊C4—71no—
226(C2×D3)⋊C4C2×D711yescharacteristic
227(C2×D3)⋊C4C2×D711yescharacteristic
228C4⋊Dic3—71no—
229C4⋊Dic3—71no—
230Dic3⋊C4—71no—
231Dic3⋊C4—71no—
232Dic3⋊C4—71no—
233Dic3⋊C4—71no—
234Dic3⋊C4—71no—
235Dic3⋊C4—71no—
236C4×Dic3—71no—
237C4×Dic3C2×D711yescharacteristic
238C2×C3×C7—41no—
239C2×C3×C7C22×D411yescharacteristic
240C2×C3×C72−1+411yescharacteristic
241C2×C3×C72−1+411yescharacteristic
242C3×D7—22no—
243D3×C7—41no—
244C2×#41—211no—
245C22×Q8—211no—
246C4⋊Q8—211no—
247C4⋊Q8—211no—
248#124—31no—
249#124—211no—
250#123—71no—
251#123—211no—
252#123—211no—
253#123—211no—
254C22⋊Q8—211no—
255C22⋊Q8—211no—
256C22⋊Q8—211no—
257C22⋊Q8—211no—
258#117—211no—
259C22×C7—21no—
260C22×C7—21no—
261C22×C7—21no—
262C22×C7—61no—
263C22×C7—61no—
264C22×C7—61no—
265C22×C7C23×D311yescharacteristic
266C2×D7—21no—
267C2×D7—21no—
268C2×D7D3×D412yes—
269C4×C7—21no—
270C4×C7—21no—
271C4×C7—32no—
272C4×C7—61no—
273C4×C7—61no—
274Dic7—21no—
275Dic7—21no—
276Dic7—21no—
277Dic7—32no—
278Dic7—61no—
279Dic7—61no—
280Dic7—61no—
281C23×C3—71no—
282C23×C3C22×D711yescharacteristic
283C22×D3C22×D711yescharacteristic
284C3×D4—142no—
285C2×C3×C4—71no—
286C2×C3×C4—71no—
287C2×C3×C4—71no—
288C2×C3×C4—142no—
289C2×C3×C4C22×D711yescharacteristic
290C2×C3×C4C22×D711yescharacteristic
291#67—22no—
292#67—142no—
293C2×Dic3—71no—
294C2×Dic3—74no—
295C2×Dic3—71no—
296C2×Dic3—71no—
297C2×Dic3—71no—
298C2×Dic3—142no—
299C2×Dic3—142no—
300C2×Dic3C22×D711yescharacteristic
301C2×Dic3C22×D711yescharacteristic
302C2×Dic3C22×D711yescharacteristic
303C4×D3—142no—
304Dic6—22no—
305Dic6—142no—
306Dic6—142no—
307Dic6—142no—
308Dic6—142no—
309C3×C7#53611yescharacteristic, complemented, Hall
310#41—424no—
311C2×Q8—31no—
312C2×Q8—211no—
313C2×Q8—211no—
314C2×Q8—211no—
315C2×Q8—211no—
316C2×Q8—424no—
317C2×D4—31no—
318C2×D4—71no—
319C2×D4—211no—
320C22×C4—71no—
321C22×C4—211no—
322C22×C4—211no—
323C22×C4—211no—
324C22×C4—211no—
325C4⋊C4—71no—
326C4⋊C4—71no—
327C4⋊C4—211no—
328C4⋊C4—211no—
329C4⋊C4—211no—
330C4⋊C4—211no—
331C4⋊C4—211no—
332C4⋊C4—211no—
333C4⋊C4—211no—
334C4⋊C4—211no—
335C22⋊C4—31no—
336C22⋊C4—31no—
337C22⋊C4—31no—
338C22⋊C4—71no—
339C22⋊C4—71no—
340C22⋊C4—211no—
341C22⋊C4—211no—
342C22⋊C4—211no—
343C22⋊C4—211no—
344C22⋊C4D3×D711yescharacteristic
345C42—31no—
346C42—211no—
347C2×C7—41no—
348C2×C7—121no—
349C2×C7C2×D3×D411yescharacteristic
350C2×C7#101011yescharacteristic
351C2×C7#101011yescharacteristic
352D7—22no—
353C22×C3—21no—
354C22×C3—21no—
355C22×C3—21no—
356C22×C3—72no—
357C22×C3—141no—
358C22×C3—141no—
359C22×C3C23×D711yescharacteristic
360C2×D3—21no—
361C2×D3—21no—
362C2×D3—21no—
363C3×C4—21no—
364C3×C4—21no—
365C3×C4—141no—
366C3×C4—141no—
367C3×C4—141no—
368Dic3—21no—
369Dic3—21no—
370Dic3—72no—
371Dic3—141no—
372Dic3—141no—
373Dic3—141no—
374Dic3D4×D712yes—
375C23—31no—
376C23—71no—
377C23C2×D3×D711yescharacteristic
378Q8—62no—
379Q8—422no—
380Q8—422no—
381Q8—422no—
382Q8—422no—
383D4—62no—
384D4—142no—
385D4—422no—
386C2×C4—31no—
387C2×C4—31no—
388C2×C4—31no—
389C2×C4—71no—
390C2×C4—71no—
391C2×C4—71no—
392C2×C4—142no—
393C2×C4—214no—
394C2×C4—211no—
395C2×C4—211no—
396C2×C4—211no—
397C2×C4—211no—
398C2×C4—422no—
399C2×C4—422no—
400C2×C4—422no—
401C2×C4C2×D3×D711yescharacteristic
402C2×C4C2×D3×D711yescharacteristic
403C7#564811yescharacteristic, complemented, Sylow
404C2×C3—41no—
405C2×C3—142no—
406C2×C3C2×D4×D711yescharacteristic
407C2×C3#655411yescharacteristic
408C2×C3#655411yescharacteristic
409D3—41no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—61no—
414C22—61no—
415C22—61no—
416C22—72no—
417C22—141no—
418C22—141no—
419C22C22×D3×D711yescharacteristic
420C4—21no—
421C4—21no—
422C4—32no—
423C4—61no—
424C4—61no—
425C4—141no—
426C4—141no—
427C4—141no—
428C4—212no—
429C4—421no—
430C4—421no—
431C4—421no—
432C3#9036611yescharacteristic, complemented, Sylow
433C2—41no—
434C2—121no—
435C2—142no—
436C2#1062154811yescharacteristic
437C2#1062156711yescharacteristic
438C2D3×D4×D711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC3×C7×C22⋊C4C22
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
2. C2×D7×Dic6C2semidirect
3. #10621361C2semidirect
4. C7×#885C2semidirect
5. C3×#6446C2semidirect
6. #10621138C2semidirect
7. #10621110C2semidirect
8. #10621087C2semidirect
9. #10621031C2semidirect
10. #10621009C2semidirect
11. #10620998C2semidirect
12. #10620994C2semidirect
13. #10620976C2semidirect
14. #10620955C2semidirect
15. #10620891C2semidirect
62. #6446D3semidirect
146. #885D7semidirect
309. C3×C7#536semidirect
403. C7#5648semidirect
432. C3#90366semidirect

Maximal subgroups

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

subgroupindex
C2×#666842
C2×D7×Dic62
#106213612
C7×#8852
C3×#64462
#106211382
#106211102
#106210872
#106210312
#106210092
#106209982
#106209942
#106209762
#106209552
#106208912
#903663
#56487