Finite Group Databasebeta

← #13060037

Subgroups — #13060037

Order 1344.

480 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10621423C211yesmaximal, characteristic, complemented
2C7×#935C211yesmaximal, characteristic, complemented
3C3×#6502C211yesmaximal, characteristic, complemented
4#10621131C211yesmaximal, characteristic, complemented
5#10621124C211yesmaximal, characteristic, complemented
6#10621118C211yesmaximal, characteristic, complemented
7#10621114C211yesmaximal, characteristic, complemented
8#10621112C211yesmaximal, characteristic, complemented
9#10621112C211yesmaximal, characteristic, complemented
10#10621108C211yesmaximal, characteristic, complemented
11#10621095C211yesmaximal, characteristic, complemented
12#10621021C211yesmaximal, characteristic, complemented
13#10620995C211yesmaximal, characteristic, complemented
14#10620956C211yesmaximal, characteristic, complemented
15#10620944C211yesmaximal, characteristic, complemented
16#90354—31nomaximal, Hall
17C2×C3×C7×D4C2211yescharacteristic
18C2×#4268C2211yescharacteristic
19C2×#4268C2211yescharacteristic
20C2×#4268C2211yescharacteristic
21C2×#4268C2211yescharacteristic
22C2×D84C2211yescharacteristic
23C22×C7×Dic3C2211yescharacteristic
24C2×C3×C22⋊D7C2211yescharacteristic
25C2×C3×C22⋊D7C2211yescharacteristic
26C22×C3×Dic7C2211yescharacteristic
27C2×#4246C2211yescharacteristic
28C2×#4246C2211yescharacteristic
29C2×D21⋊C4C2211yescharacteristic
30C2×D21⋊C4C2211yescharacteristic
31C2×Dic3×D7C2211yescharacteristic
32C4×Dic21C2211yescharacteristic
33C7×#217C2211yescharacteristic
34C7×#217C2211yescharacteristic
35C7×#217C2211yescharacteristic
36C7×Dic3⋊C4C2211yescharacteristic
37C7×Dic3⋊C4C2211yescharacteristic
38C3×C22⋊Dic7C2211yescharacteristic
39C3×(C2×D7)⋊C4C2211yescharacteristic
40C3×Dic7⋊C4C2211yescharacteristic
41Dic3⋊Dic7C2211yescharacteristic
42Dic3⋊Dic7C2211yescharacteristic
43#66577C2211yescharacteristic
44#66577C2211yescharacteristic
45Dic7⋊Dic3C2211yescharacteristic
46#66575C2211yescharacteristic
47#66575C2211yescharacteristic
48#66575C2211yescharacteristic
49#66575C2211yescharacteristic
50(C2×D7)⋊Dic3C2211yescharacteristic
51Dic3×Dic7C2211yescharacteristic
52C7×#123—31no—
53#6503—31no—
54#6502D311yescharacteristic, complemented
55#6493—31no—
56C4×C22⋊D7—31no—
57#6458—31no—
58#6457—31no—
59#6455—31no—
60#6452—31no—
61#6449—31no—
62#6447—31no—
63#6447—31no—
64#6447—31no—
65#6446—31no—
66(C22⋊D7)⋊C4—31no—
67#5615—71nomaximal, Hall
68C23×C3×C7C2311yescharacteristic
69C23×C3×C7C2311yescharacteristic
70C22×D21C2311yescharacteristic
71C22×D21C2311yescharacteristic
72C22×C3×D7C2311yescharacteristic
73C3×C7×D4—41no—
74C2×C3×C4×C7C2311yescharacteristic
75#4268—22no—
76#4268—22no—
77#4268—41no—
78#4268—41no—
79C2×Dic21C2311yescharacteristic
80C2×Dic21C2311yescharacteristic
81D84—41no—
82C2×C7×Dic3—41no—
83C2×C7×Dic3C2311yescharacteristic
84C2×C7×Dic3C2311yescharacteristic
85C2×C7×Dic3C2311yescharacteristic
86C2×C7×Dic3C2311yescharacteristic
87C3×C22⋊D7—41no—
88C3×C22⋊D7—41no—
89C2×C3×Dic7—41no—
90C2×C3×Dic7C2311yescharacteristic
91C2×C3×Dic7C2311yescharacteristic
92C2×C3×Dic7C2311yescharacteristic
93#4246—41no—
94#4246—41no—
95D21⋊C4—41no—
96D21⋊C4—41no—
97Dic3×D7—22no—
98C2×C7×D4C2×D311yescharacteristic
99C22×C4×C7—31no—
100C2×C22⋊D7—31no—
101C2×C22⋊D7—31no—
102C2×C22⋊D7—31no—
103C2×C22⋊D7—31no—
104C2×C22⋊D7C2×D311yescharacteristic
105C2×C22⋊D7C2×D311yescharacteristic
106C22×Dic7C2×D311yescharacteristic
107C2×D28—31no—
108C2×D28—31no—
109C2×D28—31no—
110C2×C4×D7—31no—
111C2×C4×D7—31no—
112C2×C4×D7—31no—
113C7×C4⋊C4—31no—
114C7×C4⋊C4—31no—
115C7×C22⋊C4—31no—
116C7×C22⋊C4—31no—
117C7×C22⋊C4—31no—
118C22⋊Dic7C2×D311yescharacteristic
119(C2×D7)⋊C4—31no—
120(C2×D7)⋊C4—31no—
121(C2×D7)⋊C4—31no—
122(C2×D7)⋊C4—31no—
123(C2×D7)⋊C4—31no—
124(C2×D7)⋊C4C2×D311yescharacteristic
125C4⋊Dic7—31no—
126C4⋊Dic7—31no—
127Dic7⋊C4—31no—
128Dic7⋊C4—31no—
129Dic7⋊C4—31no—
130Dic7⋊C4C2×D311yescharacteristic
131C4×Dic7—31no—
132C4×Dic7—31no—
133C3×#121—71no—
134#940—71no—
135#939—71no—
136#939—71no—
137#939—71no—
138#938—71no—
139#935D711yescharacteristic, complemented
140#929—71no—
141C4×#67—71no—
142#894—71no—
143#889—71no—
144#884—71no—
145#883—71no—
146#67⋊C4—71no—
147#877—71no—
148C22×C3×C7—21no—
149C22×C3×C7—21no—
150C22×C3×C7—21no—
151C22×C3×C7—21no—
152C22×C3×C7—21no—
153C22×C3×C7—21no—
154C22×C3×C7C2411yescharacteristic
155C2×D21—21no—
156C2×D21—21no—
157C2×D21—21no—
158C2×D21—21no—
159C2×D21—21no—
160C2×D21—21no—
161C2×C3×D7—21no—
162C2×C3×D7—21no—
163C2×C3×D7—21no—
164C3×C4×C7—21no—
165Dic21—21no—
166Dic21#4112yes—
167C3×Dic7—21no—
168C3×Dic7—21no—
169C3×Dic7—21no—
170C7×Dic3—21no—
171C7×Dic3—21no—
172C7×Dic3—21no—
173C7×Dic3—21no—
174#540—211noSylow
175C23×C7C22×D311yescharacteristic
176C23×C7C22×D311yescharacteristic
177C22×D7—31no—
178C22×D7—31no—
179C22×D7C22×D311yescharacteristic
180C7×D4—41no—
181C2×C4×C7—31no—
182C2×C4×C7—31no—
183C2×C4×C7—31no—
184C2×C4×C7—31no—
185C2×C4×C7—121no—
186C2×C4×C7C22×D311yescharacteristic
187C22⋊D7—41no—
188C22⋊D7—41no—
189C22⋊D7—62no—
190C22⋊D7—62no—
191C22⋊D7—121no—
192C22⋊D7—121no—
193C2×Dic7—31no—
194C2×Dic7—31no—
195C2×Dic7—41no—
196C2×Dic7C22×D311yescharacteristic
197C2×Dic7C22×D311yescharacteristic
198C2×Dic7C22×D311yescharacteristic
199D28—121no—
200D28—121no—
201D28—121no—
202C4×D7—62no—
203C4×D7—121no—
204C4×D7—121no—
205C2×C3×D4—71no—
206C2×C3×D4—71no—
207C2×C3×D4C2×D711yescharacteristic
208C22×C3×C4—71no—
209C2×#67—71no—
210C2×#67—71no—
211C2×#67—71no—
212C2×#67—71no—
213C2×#67—71no—
214C2×#67—71no—
215C22×Dic3—71no—
216C22×Dic3C2×D711yescharacteristic
217C2×D12—71no—
218C2×C4×D3—71no—
219C2×C4×D3—71no—
220C3×C4⋊C4—71no—
221C3×C22⋊C4—71no—
222C3×C22⋊C4—71no—
223#217—71no—
224#217C2×D711yescharacteristic
225#217C2×D711yescharacteristic
226#217C2×D711yescharacteristic
227(C2×D3)⋊C4—71no—
228(C2×D3)⋊C4—71no—
229(C2×D3)⋊C4—71no—
230(C2×D3)⋊C4—71no—
231C4⋊Dic3—71no—
232Dic3⋊C4—71no—
233Dic3⋊C4—71no—
234Dic3⋊C4—71no—
235Dic3⋊C4—71no—
236Dic3⋊C4C2×D711yescharacteristic
237Dic3⋊C4C2×D711yescharacteristic
238C4×Dic3—71no—
239C4×Dic3—71no—
240C2×C3×C7—41no—
241C2×C3×C7—41no—
242C2×C3×C7C2×#4111yescharacteristic
243C2×C3×C7H(5,𝔽2)11yescharacteristic
244C2×C3×C7H(5,𝔽2)11yescharacteristic
245D21—41no—
246D21—41no—
247C3×D7—41no—
248#127—211no—
249#125—211no—
250#123—31no—
251#123—211no—
252#123—211no—
253#123—211no—
254#121—71no—
255#121—211no—
256#121—211no—
257#121—211no—
258#121—211no—
259#121—211no—
260C4×D4—211no—
261C4×D4—211no—
262#117—211no—
263C22×C7—21no—
264C22×C7—21no—
265C22×C7—21no—
266C22×C7—21no—
267C22×C7—21no—
268C22×C7—21no—
269C22×C7C23×D311yescharacteristic
270C2×D7—21no—
271C2×D7—21no—
272C2×D7—21no—
273C2×D7—61no—
274C2×D7—61no—
275C2×D7—61no—
276C2×D7—61no—
277C2×D7—61no—
278C2×D7—61no—
279C4×C7—21no—
280C4×C7—61no—
281C4×C7—61no—
282C4×C7—61no—
283C4×C7—61no—
284Dic7—21no—
285Dic7—21no—
286Dic7—21no—
287Dic7—32no—
288Dic7—61no—
289C23×C3—71no—
290C23×C3C22×D711yescharacteristic
291C23×C3C22×D711yescharacteristic
292C22×D3—71no—
293C22×D3—71no—
294C3×D4—41no—
295C3×D4—281no—
296C3×D4—281no—
297C2×C3×C4—71no—
298C2×C3×C4—71no—
299C2×C3×C4—71no—
300C2×C3×C4—281no—
301C2×C3×C4C22×D711yescharacteristic
302#67—142no—
303#67—142no—
304#67—281no—
305#67—281no—
306#67—281no—
307#67—281no—
308C2×Dic3—41no—
309C2×Dic3—71no—
310C2×Dic3—71no—
311C2×Dic3—142no—
312C2×Dic3C22×D711yescharacteristic
313C2×Dic3C22×D711yescharacteristic
314C2×Dic3C22×D711yescharacteristic
315C2×Dic3C22×D711yescharacteristic
316D12—281no—
317C4×D3—281no—
318C4×D3—281no—
319C3×C7#54011yescharacteristic, complemented, Hall
320C2×D4—71no—
321C2×D4—71no—
322C2×D4—211no—
323C2×D4—211no—
324C2×D4—211no—
325C2×D4—211no—
326C2×D4—211no—
327C2×D4—211no—
328C2×D4—211no—
329C2×D4D3×D711yescharacteristic
330C22×C4—31no—
331C22×C4—71no—
332C22×C4—211no—
333C22×C4—211no—
334C22×C4—211no—
335C4⋊C4—31no—
336C4⋊C4—31no—
337C4⋊C4—71no—
338C4⋊C4—211no—
339C4⋊C4—211no—
340C4⋊C4—211no—
341C4⋊C4—211no—
342C4⋊C4—211no—
343C22⋊C4—31no—
344C22⋊C4—31no—
345C22⋊C4—31no—
346C22⋊C4—71no—
347C22⋊C4—71no—
348C22⋊C4—211no—
349C22⋊C4—211no—
350C22⋊C4—211no—
351C22⋊C4—211no—
352C22⋊C4—211no—
353C42—211no—
354C42—211no—
355C2×C7—41no—
356C2×C7—41no—
357C2×C7#100411yescharacteristic
358C2×C7#100411yescharacteristic
359C2×C7D3×#4111yescharacteristic
360D7—41no—
361D7—121no—
362D7—121no—
363C22×C3—21no—
364C22×C3—21no—
365C22×C3—21no—
366C22×C3—21no—
367C22×C3—21no—
368C22×C3—21no—
369C22×C3—141no—
370C22×C3—141no—
371C22×C3—141no—
372C22×C3C23×D711yescharacteristic
373C2×D3—141no—
374C2×D3—141no—
375C2×D3—141no—
376C2×D3—141no—
377C2×D3—141no—
378C2×D3—141no—
379C3×C4—21no—
380C3×C4—141no—
381C3×C4—141no—
382C3×C4—141no—
383Dic3—21no—
384Dic3—21no—
385Dic3—21no—
386Dic3—21no—
387Dic3—72no—
388Dic3—141no—
389C23—71no—
390C23—211no—
391C23—211no—
392C23C2×D3×D711yescharacteristic
393C23C2×D3×D711yescharacteristic
394D4—41no—
395D4—281no—
396D4—281no—
397D4—422no—
398D4—422no—
399D4—841no—
400D4—841no—
401D4—841no—
402D4—841no—
403D4—841no—
404C2×C4—31no—
405C2×C4—31no—
406C2×C4—31no—
407C2×C4—31no—
408C2×C4—71no—
409C2×C4—71no—
410C2×C4—71no—
411C2×C4—121no—
412C2×C4—211no—
413C2×C4—211no—
414C2×C4—281no—
415C2×C4—422no—
416C2×C4—841no—
417C2×C4—841no—
418C2×C4C2×D3×D711yescharacteristic
419C7#561511yescharacteristic, complemented, Sylow
420C2×C3—41no—
421C2×C3—41no—
422C2×C3—281no—
423C2×C3#654811yescharacteristic
424C2×C3D7×#4111yescharacteristic
425C2×C3#655311yescharacteristic
426D3—281no—
427D3—281no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—141no—
435C22—141no—
436C22—141no—
437C22—421no—
438C22—421no—
439C22—421no—
440C22—421no—
441C22—421no—
442C22—421no—
443C22C22×D3×D711yescharacteristic
444C4—21no—
445C4—61no—
446C4—61no—
447C4—61no—
448C4—61no—
449C4—141no—
450C4—141no—
451C4—141no—
452C4—212no—
453C4—421no—
454C3#9035411yescharacteristic, complemented, Sylow
455C2—41no—
456C2—41no—
457C2—281no—
458C2—841no—
459C2—841no—
460C2#1062157111yescharacteristic
461C2#1062157311yescharacteristic
462C2#1062159611yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC2×C3×C7×D4C22
hypercenterC2×D4D3×D7
hypocenterC3×C7#540
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. #10621423C2semidirect
2. C7×#935C2semidirect
3. C3×#6502C2semidirect
4. #10621131C2semidirect
5. #10621124C2semidirect
6. #10621118C2semidirect
7. #10621114C2semidirect
8. #10621112C2semidirect
9. #10621112C2semidirect
10. #10621108C2semidirect
11. #10621095C2semidirect
12. #10621021C2semidirect
13. #10620995C2semidirect
14. #10620956C2semidirect
15. #10620944C2semidirect
54. #6502D3semidirect
139. #935D7semidirect
319. C3×C7#540semidirect
419. C7#5615semidirect
454. C3#90354semidirect

Maximal subgroups

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

subgroupindex
#106214232
C7×#9352
C3×#65022
#106211312
#106211242
#106211182
#106211142
#106211122
#106211082
#106210952
#106210212
#106209952
#106209562
#106209442
#903543
#56157