Finite Group Databasebeta

← #13058032

Subgroups — #13058032

Order 1344.

620 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#66686C211yesmaximal, characteristic, complemented
2C2×#66676C211yesmaximal, characteristic, complemented
3#10621368C211yesmaximal, characteristic, complemented
4C7×#884C211yesmaximal, characteristic, complemented
5C3×(C22⋊D7)⋊C4C211yesmaximal, characteristic, complemented
6#10621129C211yesmaximal, characteristic, complemented
7#10621127C211yesmaximal, characteristic, complemented
8#67×Dic7C211yesmaximal, characteristic, complemented
9#10621026C211yesmaximal, characteristic, complemented
10#10621016C211yesmaximal, characteristic, complemented
11#10621006C211yesmaximal, characteristic, complemented
12#10620978C211yesmaximal, characteristic, complemented
13#10620967C211yesmaximal, characteristic, complemented
14C4×D3×Dic7C211yesmaximal, characteristic, complemented
15#10620890C211yesmaximal, characteristic, complemented
16#90287—31nomaximal, Hall
17C2×#4268C2211yescharacteristic
18C2×Dic42C2211yescharacteristic
19C2×C7×#67C2211yescharacteristic
20C2×C7×#67C2211yescharacteristic
21C2×C7×D12C2211yescharacteristic
22C2×C4×D3×C7C2211yescharacteristic
23C2×C3×C22⋊D7C2211yescharacteristic
24C22×C3×Dic7C2211yescharacteristic
25C2×C3×C4×D7C2211yescharacteristic
26C2×#4248C2211yescharacteristic
27C2×#4247C2211yescharacteristic
28C2×#4247C2211yescharacteristic
29C2×#4246C2211yescharacteristic
30C2×#4245C2211yescharacteristic
31C2×#4245C2211yescharacteristic
32C2×D21⋊C4C2211yescharacteristic
33#66686—24no—
34C2×D3×Dic7C2211yescharacteristic
35C2×D3×Dic7C2211yescharacteristic
36C2×D3×Dic7C2211yescharacteristic
37#66676—24no—
38C3×C7×C22⋊C4C2211yescharacteristic
39#66636C2211yescharacteristic
40#66631C2211yescharacteristic
41#66631C2211yescharacteristic
42C4×Dic21C2211yescharacteristic
43C7×(C2×D3)⋊C4C2211yescharacteristic
44C7×Dic3⋊C4C2211yescharacteristic
45C3×(C2×D7)⋊C4C2211yescharacteristic
46C3×Dic7⋊C4C2211yescharacteristic
47C3×C4×Dic7C2211yescharacteristic
48Dic3⋊Dic7C2211yescharacteristic
49Dic7⋊Dic3C2211yescharacteristic
50(C2×D3)⋊Dic7C2211yescharacteristic
51(C2×D7)⋊Dic3C2211yescharacteristic
52Dic3×Dic7C2211yescharacteristic
53Dic3×Dic7C2211yescharacteristic
54C2×#1159—31no—
55C2×#1159—31no—
56C7×#121—31no—
57#6503—31no—
58#6502—31no—
59#6502—31no—
60D4×Dic7—31no—
61#6493—31no—
62C2×C4×Dic7—31no—
63#6456—31no—
64#6451—31no—
65#6448—31no—
66#6448—31no—
67(C22⋊D7)⋊C4—31no—
68(C22⋊D7)⋊C4D311yescharacteristic, complemented
69#5548—71nomaximal, Hall
70C23×C3×C7C2311yescharacteristic
71C22×D21C2311yescharacteristic
72C22×D3×C7C2311yescharacteristic
73C22×D3×C7C2311yescharacteristic
74C22×C3×D7C2311yescharacteristic
75C2×C3×C4×C7C2311yescharacteristic
76C2×C3×C4×C7C2311yescharacteristic
77#4268—22no—
78C2×Dic21C2311yescharacteristic
79C2×Dic21C2311yescharacteristic
80C2×Dic21C2311yescharacteristic
81Dic42—22no—
82C7×#67—22no—
83C7×#67—22no—
84C2×C7×Dic3C2311yescharacteristic
85C2×C7×Dic3C2311yescharacteristic
86C7×D12—22no—
87C4×D3×C7—22no—
88C3×C22⋊D7—22no—
89C2×C3×Dic7—22no—
90C2×C3×Dic7C2311yescharacteristic
91C2×C3×Dic7C2311yescharacteristic
92C2×C3×Dic7C2311yescharacteristic
93C3×C4×D7—22no—
94#4248—22no—
95#4247—22no—
96#4247—22no—
97#4246—22no—
98#4245—22no—
99#4245—22no—
100D21⋊C4—22no—
101D3×Dic7—22no—
102D3×Dic7—22no—
103D3×Dic7D414yescomplemented
104C2×C7×D4—31no—
105C2×C7×D4—31no—
106C2×C7×D4—31no—
107C22×C4×C7—31no—
108C2×C22⋊D7—31no—
109C2×C22⋊D7—31no—
110C2×C22⋊D7—31no—
111C2×C22⋊D7—31no—
112C2×C22⋊D7—31no—
113C2×C22⋊D7C2×D311yescharacteristic
114C22×Dic7—31no—
115C22×Dic7—31no—
116C22×Dic7—31no—
117C22×Dic7C2×D311yescharacteristic
118#1159—64no—
119#1159—64no—
120C2×D28—31no—
121C2×C4×D7—31no—
122C2×C4×D7C2×D311yescharacteristic
123C2×Dic14—31no—
124C2×Dic14—31no—
125C7×C4⋊C4—31no—
126C7×C22⋊C4—31no—
127C7×C22⋊C4C2×D311yescharacteristic
128C22⋊Dic7—31no—
129C22⋊Dic7—31no—
130(C2×D7)⋊C4—31no—
131(C2×D7)⋊C4—31no—
132(C2×D7)⋊C4—31no—
133(C2×D7)⋊C4C2×D311yescharacteristic
134C4⋊Dic7—31no—
135Dic7⋊C4—31no—
136Dic7⋊C4C2×D311yescharacteristic
137C4×Dic7—31no—
138C4×Dic7—31no—
139C4×Dic7—31no—
140C4×Dic7C2×D311yescharacteristic
141C2×#235—71no—
142C2×#235—71no—
143C3×C4×D4—71no—
144#940—71no—
145#938—71no—
146C4×#67—71no—
147C4×#67—71no—
148#895—71no—
149#888—71no—
150#885—71no—
151#885—71no—
152#884—71no—
153#884D711yescharacteristic, complemented
154C4×D12—71no—
155C42×D3—71no—
156C22×C3×C7—21no—
157C22×C3×C7—21no—
158C22×C3×C7—21no—
159C22×C3×C7C2411yescharacteristic
160C2×D21—21no—
161C2×D21—21no—
162C2×D21—21no—
163C2×D3×C7—21no—
164C2×D3×C7—21no—
165C2×D3×C7—21no—
166C2×D3×C7—21no—
167C2×D3×C7—21no—
168C2×D3×C7C2×D412yes—
169C2×C3×D7—21no—
170C2×C3×D7—21no—
171C2×C3×D7—21no—
172C3×C4×C7—21no—
173C3×C4×C7—21no—
174Dic21—21no—
175Dic21—21no—
176Dic21C2×D412yes—
177C3×Dic7—21no—
178C3×Dic7C2×D412yes—
179C3×Dic7#4112yescomplemented
180C7×Dic3—21no—
181C7×Dic3#4112yescomplemented
182#532—211noSylow
183C23×C7—31no—
184C23×C7—31no—
185C23×C7C22×D311yescharacteristic
186C22×D7—31no—
187C22×D7C22×D311yescharacteristic
188C7×D4—62no—
189C7×D4—62no—
190C7×D4—62no—
191C2×C4×C7—31no—
192C2×C4×C7—31no—
193C2×C4×C7—62no—
194C2×C4×C7C22×D311yescharacteristic
195C2×C4×C7C22×D311yescharacteristic
196C22⋊D7—22no—
197C22⋊D7—62no—
198C22⋊D7—62no—
199C22⋊D7—62no—
200C22⋊D7—62no—
201C22⋊D7—62no—
202C2×Dic7—22no—
203C2×Dic7—31no—
204C2×Dic7—34no—
205C2×Dic7—31no—
206C2×Dic7—31no—
207C2×Dic7—62no—
208C2×Dic7—62no—
209C2×Dic7C22×D311yescharacteristic
210C2×Dic7C22×D311yescharacteristic
211C2×Dic7C22×D311yescharacteristic
212D28—62no—
213C4×D7—22no—
214C4×D7—62no—
215Dic14—62no—
216Dic14—62no—
217C2×C3×D4—71no—
218C22×C3×C4—71no—
219C22×C3×C4—71no—
220C2×#67—71no—
221C2×#67—71no—
222C2×#67—71no—
223C2×#67—71no—
224C2×#67C2×D711yescharacteristic
225C2×#67C2×D711yescharacteristic
226#235—144no—
227#235—144no—
228C2×D12—71no—
229C2×D12—71no—
230C2×D12C2×D711yescharacteristic
231C2×C4×D3—71no—
232C2×C4×D3—71no—
233C2×C4×D3—71no—
234C2×C4×D3—71no—
235C2×C4×D3C2×D711yescharacteristic
236C2×Dic6—71no—
237C2×Dic6—71no—
238C3×C4⋊C4—71no—
239C3×C22⋊C4—71no—
240C3×C22⋊C4C2×D711yescharacteristic
241C3×C42—71no—
242#217—71no—
243#217—71no—
244(C2×D3)⋊C4—71no—
245(C2×D3)⋊C4—71no—
246(C2×D3)⋊C4—71no—
247(C2×D3)⋊C4C2×D711yescharacteristic
248C4⋊Dic3—71no—
249Dic3⋊C4—71no—
250Dic3⋊C4C2×D711yescharacteristic
251C4×Dic3—71no—
252C4×Dic3—71no—
253C4×Dic3—71no—
254C2×C3×C7—41no—
255C2×C3×C7C22×D411yescharacteristic
256C2×C3×C7C2×#4111yescharacteristic
257C2×C3×C7C2×#4111yescharacteristic
258D21—41no—
259C3×D7—41no—
260D3×C7—22no—
261D3×C7—41no—
262C2×#41—211no—
263C2×#41—211no—
264C4⋊Q8—211no—
265#127—211no—
266#124—211no—
267#124—211no—
268#121—31no—
269#121—211no—
270#121—211no—
271#121—211no—
272C4×D4—71no—
273C4×D4—211no—
274C4×D4—211no—
275C4×D4—211no—
276C2×C42—211no—
277C22×C7—21no—
278C22×C7—21no—
279C22×C7—21no—
280C22×C7—32no—
281C22×C7—61no—
282C22×C7—61no—
283C22×C7—61no—
284C22×C7—61no—
285C22×C7—61no—
286C22×C7C23×D311yescharacteristic
287C2×D7—21no—
288C2×D7—21no—
289C2×D7—21no—
290C2×D7—61no—
291C2×D7—61no—
292C2×D7—61no—
293C4×C7—21no—
294C4×C7—21no—
295C4×C7—32no—
296C4×C7—61no—
297Dic7—21no—
298Dic7—32no—
299Dic7—61no—
300Dic7—61no—
301Dic7#23512yescomplemented
302Dic7D3×D412yes—
303C23×C3—71no—
304C23×C3C22×D711yescharacteristic
305C22×D3—71no—
306C22×D3C22×D711yescharacteristic
307C22×D3C22×D711yescharacteristic
308C3×D4—142no—
309C2×C3×C4—71no—
310C2×C3×C4—71no—
311C2×C3×C4—71no—
312C2×C3×C4—142no—
313C2×C3×C4—142no—
314C2×C3×C4C22×D711yescharacteristic
315C2×C3×C4C22×D711yescharacteristic
316#67—22no—
317#67—22no—
318#67—142no—
319#67—142no—
320#67—142no—
321#67—142no—
322C2×Dic3—71no—
323C2×Dic3—71no—
324C2×Dic3—71no—
325C2×Dic3C22×D711yescharacteristic
326C2×Dic3C22×D711yescharacteristic
327D12—22no—
328D12—142no—
329D12—142no—
330C4×D3—22no—
331C4×D3—74no—
332C4×D3—142no—
333C4×D3—142no—
334C4×D3—142no—
335Dic6—142no—
336Dic6—142no—
337C3×C7#53211yescharacteristic, complemented, Hall
338#41—424no—
339#41—424no—
340C2×Q8—211no—
341C2×Q8—211no—
342C2×D4—31no—
343C2×D4—31no—
344C2×D4—31no—
345C2×D4—71no—
346C2×D4—211no—
347C2×D4—211no—
348C2×D4—211no—
349C2×D4—211no—
350C2×D4—211no—
351C2×D4—211no—
352C22×C4—31no—
353C22×C4—71no—
354C22×C4—71no—
355C22×C4—211no—
356C22×C4—211no—
357C22×C4—211no—
358C22×C4—211no—
359C4⋊C4—31no—
360C4⋊C4—71no—
361C4⋊C4—211no—
362C4⋊C4—211no—
363C22⋊C4—31no—
364C22⋊C4—71no—
365C22⋊C4—211no—
366C22⋊C4—211no—
367C22⋊C4—211no—
368C22⋊C4—211no—
369C22⋊C4—211no—
370C22⋊C4D3×D711yescharacteristic
371C42—71no—
372C42—211no—
373C42—211no—
374C42—211no—
375C2×C7—41no—
376C2×C7—62no—
377C2×C7—121no—
378C2×C7C2×#23511yescharacteristic
379C2×C7C2×D3×D411yescharacteristic
380C2×C7D3×#4111yescharacteristic
381D7—41no—
382D7—121no—
383C22×C3—21no—
384C22×C3—21no—
385C22×C3—21no—
386C22×C3—141no—
387C22×C3—141no—
388C22×C3—141no—
389C22×C3C23×D711yescharacteristic
390C2×D3—21no—
391C2×D3—21no—
392C2×D3—21no—
393C2×D3—21no—
394C2×D3—21no—
395C2×D3—141no—
396C2×D3—141no—
397C2×D3—141no—
398C2×D3D4×D712yes—
399C3×C4—21no—
400C3×C4—21no—
401C3×C4—72no—
402C3×C4—72no—
403C3×C4—141no—
404Dic3—21no—
405Dic3—72no—
406Dic3—141no—
407Dic3—141no—
408Dic3#115912yescomplemented
409C23—31no—
410C23—31no—
411C23—71no—
412C23—211no—
413C23C2×D3×D711yescharacteristic
414Q8—422no—
415Q8—422no—
416D4—62no—
417D4—62no—
418D4—62no—
419D4—142no—
420D4—422no—
421D4—422no—
422D4—422no—
423D4—422no—
424D4—422no—
425D4—422no—
426C2×C4—31no—
427C2×C4—31no—
428C2×C4—62no—
429C2×C4—71no—
430C2×C4—71no—
431C2×C4—71no—
432C2×C4—142no—
433C2×C4—142no—
434C2×C4—214no—
435C2×C4—211no—
436C2×C4—211no—
437C2×C4—211no—
438C2×C4—422no—
439C2×C4—422no—
440C2×C4—422no—
441C2×C4C2×D3×D711yescharacteristic
442C2×C4C2×D3×D711yescharacteristic
443C7#554811yescharacteristic, complemented, Sylow
444C2×C3—41no—
445C2×C3—281no—
446C2×C3C2×D4×D711yescharacteristic
447C2×C3C2×#115911yescharacteristic
448C2×C3D7×#4111yescharacteristic
449D3—22no—
450D3—41no—
451D3—281no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—32no—
456C22—61no—
457C22—61no—
458C22—61no—
459C22—61no—
460C22—61no—
461C22—141no—
462C22—141no—
463C22—141no—
464C22—421no—
465C22—421no—
466C22—421no—
467C22C22×D3×D711yescharacteristic
468C4—21no—
469C4—21no—
470C4—32no—
471C4—61no—
472C4—72no—
473C4—72no—
474C4—141no—
475C4—212no—
476C4—421no—
477C4—421no—
478C3#9028711yescharacteristic, complemented, Sylow
479C2—41no—
480C2—62no—
481C2—121no—
482C2—281no—
483C2—841no—
484C2D7×#23511yescharacteristic
485C2D3×D4×D711yescharacteristic
486C2D3×#115911yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC3×C7×C22⋊C4C22
hypercenterC22⋊C4D3×D7
hypocenterC3×C7#532
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#66686C2semidirect
2. C2×#66676C2semidirect
3. #10621368C2semidirect
4. C7×#884C2semidirect
5. C3×(C22⋊D7)⋊C4C2semidirect
6. #10621129C2semidirect
7. #10621127C2semidirect
8. #67×Dic7C2semidirect
9. #10621026C2semidirect
10. #10621016C2semidirect
11. #10621006C2semidirect
12. #10620978C2semidirect
13. #10620967C2semidirect
14. C4×D3×Dic7C2semidirect
15. #10620890C2semidirect
68. (C22⋊D7)⋊C4D3semidirect
103. D3×Dic7D4semidirect
153. #884D7semidirect
179. C3×Dic7#41semidirect
181. C7×Dic3#41semidirect
301. Dic7#235semidirect
337. C3×C7#532semidirect
408. Dic3#1159semidirect
443. C7#5548semidirect
478. C3#90287semidirect

Maximal subgroups

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

subgroupindex
C2×#666862
C2×#666762
#106213682
C7×#8842
C3×(C22⋊D7)⋊C42
#106211292
#106211272
#67×Dic72
#106210262
#106210162
#106210062
#106209782
#106209672
C4×D3×Dic72
#106208902
#902873
#55487