Finite Group Databasebeta

← #13059466

Subgroups — #13059466

Order 1344.

588 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#66683C211yesmaximal, characteristic
2C4×#4268C211yesmaximal, characteristic, complemented
3C2×C7×Dic3⋊C4C211yesmaximal, characteristic, complemented
4C3×C4×C22⋊D7C211yesmaximal, characteristic, complemented
5#10621118C211yesmaximal, characteristic, complemented
6Dic3×C22⋊D7C211yesmaximal, characteristic, complemented
7#10621091C211yesmaximal, characteristic, complemented
8#10621006C211yesmaximal, characteristic, complemented
9#10620966C211yesmaximal, characteristic, complemented
10#10620964C211yesmaximal, characteristic, complemented
11#10620958C211yesmaximal, characteristic, complemented
12#10620956C211yesmaximal, characteristic, complemented
13D7×Dic3⋊C4C211yesmaximal, characteristic, complemented
14#10620887C211yesmaximal, characteristic, complemented
15#10620885C211yesmaximal, characteristic, complemented
16#90195—31nomaximal, Hall
17C22×C3×C4×C7C2211yescharacteristic
18C2×#4268C2211yescharacteristic
19C2×C4×D21C2211yescharacteristic
20C22×C7×Dic3C2211yescharacteristic
21C22×C7×Dic3C2211yescharacteristic
22C2×C3×C22⋊D7C2211yescharacteristic
23C2×C3×C4×D7C2211yescharacteristic
24C2×#4248C2211yescharacteristic
25C2×#4246C2211yescharacteristic
26C2×D21⋊C4C2211yescharacteristic
27C2×D21⋊C4C2211yescharacteristic
28#66683C418yescomplemented
29C2×Dic3×D7C2211yescharacteristic
30C2×Dic3×D7C2211yescharacteristic
31#66636C2211yescharacteristic
32#66631C2211yescharacteristic
33#66629C2211yescharacteristic
34C4×Dic21C2211yescharacteristic
35C7×Dic3⋊C4C2211yescharacteristic
36C7×Dic3⋊C4C2211yescharacteristic
37C7×Dic3⋊C4C2211yescharacteristic
38C7×Dic3⋊C4C2211yescharacteristic
39C3×C22⋊Dic7C2211yescharacteristic
40C3×(C2×D7)⋊C4C2211yescharacteristic
41C3×Dic7⋊C4C2211yescharacteristic
42C3×C4×Dic7C2211yescharacteristic
43Dic3⋊Dic7C2211yescharacteristic
44Dic3⋊Dic7C2211yescharacteristic
45#66577C2211yescharacteristic
46Dic7⋊Dic3C2211yescharacteristic
47#66575C2211yescharacteristic
48(C2×D7)⋊Dic3C2211yescharacteristic
49Dic3×Dic7C2211yescharacteristic
50Dic3×Dic7C2211yescharacteristic
51Dic3×Dic7C2211yescharacteristic
52Dic3×Dic7C2211yescharacteristic
53C2×#1157—31no—
54C2×C7×C4⋊C4—31no—
55C4×C22⋊D7—31no—
56C4×C22⋊D7—31no—
57C4×C22⋊D7—31no—
58C4×C22⋊D7D311yescharacteristic, complemented
59#6489—31no—
60#6456—31no—
61#6456—31no—
62#6455—31no—
63#6455—31no—
64D7×C4⋊C4—31no—
65D7×C4⋊C4—31no—
66#6450—31no—
67#6450—31no—
68#5546—71nomaximal, Hall
69C23×C3×C7C2311yescharacteristic
70C22×D21C2311yescharacteristic
71C22×C3×D7C2311yescharacteristic
72C2×C3×C4×C7—41no—
73C2×C3×C4×C7C2311yescharacteristic
74C2×C3×C4×C7C2311yescharacteristic
75#4268C2×C414yescomplemented
76C2×Dic21C2311yescharacteristic
77C2×Dic21C2311yescharacteristic
78C2×Dic21C2311yescharacteristic
79C4×D21—41no—
80C2×C7×Dic3—41no—
81C2×C7×Dic3C2×C414yescomplemented
82C2×C7×Dic3C2311yescharacteristic
83C2×C7×Dic3C2311yescharacteristic
84C2×C7×Dic3C2311yescharacteristic
85C2×C7×Dic3C2311yescharacteristic
86C3×C22⋊D7C2×C414yescomplemented
87C2×C3×Dic7C2311yescharacteristic
88C2×C3×Dic7C2311yescharacteristic
89C2×C3×Dic7C2311yescharacteristic
90C3×C4×D7—41no—
91#4248C2×C414yescomplemented
92#4246C2×C414yescomplemented
93D21⋊C4—41no—
94D21⋊C4C2×C414yescomplemented
95Dic3×D7—41no—
96Dic3×D7C2×C414yescomplemented
97C22×C4×C7—31no—
98C22×C4×C7—31no—
99C22×C4×C7C2×D311yescharacteristic
100C2×C22⋊D7—31no—
101C2×C22⋊D7C2×D311yescharacteristic
102#1157—38no—
103C2×D28—31no—
104C2×C4×D7—31no—
105C2×C4×D7—31no—
106C2×C4×D7—31no—
107C2×C4×D7—31no—
108C2×C4×D7—31no—
109C2×C4×D7C2×D311yescharacteristic
110C2×Dic14—31no—
111C7×C4⋊C4—31no—
112C7×C4⋊C4—31no—
113C7×C4⋊C4—31no—
114C7×C4⋊C4—31no—
115C22⋊Dic7—31no—
116C22⋊Dic7C2×D311yescharacteristic
117(C2×D7)⋊C4—31no—
118(C2×D7)⋊C4—31no—
119(C2×D7)⋊C4—31no—
120(C2×D7)⋊C4C2×D311yescharacteristic
121C4⋊Dic7—31no—
122C4⋊Dic7—31no—
123Dic7⋊C4—31no—
124Dic7⋊C4—31no—
125Dic7⋊C4—31no—
126Dic7⋊C4C2×D311yescharacteristic
127C4×Dic7—31no—
128C4×Dic7—31no—
129C4×Dic7—31no—
130C4×Dic7—31no—
131C4×Dic7—31no—
132C4×Dic7C2×D311yescharacteristic
133C2×#237—71no—
134C3×C4×D4—71no—
135D4×Dic3—71no—
136C4×#67—71no—
137C4×#67—71no—
138C2×Dic3⋊C4—71no—
139C2×Dic3⋊C4D711yescharacteristic, complemented
140D3×C4⋊C4—71no—
141#887—71no—
142#67⋊C4—71no—
143#67⋊C4—71no—
144#877—71no—
145#877—71no—
146#872—71no—
147C4×Dic6—71no—
148C22×C3×C7—21no—
149C22×C3×C7—21no—
150C22×C3×C7C22×C412yes—
151C22×C3×C7C2411yescharacteristic
152C2×D21—21no—
153C2×D21—21no—
154C2×D21C22×C412yes—
155C2×C3×D7—21no—
156C2×C3×D7—21no—
157C2×C3×D7C22×C412yes—
158C3×C4×C7—21no—
159C3×C4×C7—21no—
160Dic21—21no—
161Dic21—21no—
162Dic21C22×C412yes—
163C3×Dic7—21no—
164C3×Dic7—21no—
165C3×Dic7C22×C412yes—
166C7×Dic3—21no—
167C7×Dic3—21no—
168C7×Dic3C22×C412yes—
169C7×Dic3C22×C412yes—
170#520—211noSylow
171C23×C7C22×D311yescharacteristic
172C22×D7—31no—
173C22×D7C22×D311yescharacteristic
174C2×C4×C7—34no—
175C2×C4×C7—31no—
176C2×C4×C7—31no—
177C2×C4×C7—31no—
178C2×C4×C7—31no—
179C2×C4×C7—41no—
180C2×C4×C7—121no—
181C2×C4×C7C22×D311yescharacteristic
182C2×C4×C7C22×D311yescharacteristic
183C22⋊D7—34no—
184C22⋊D7C4×D314yescomplemented
185C2×Dic7—31no—
186C2×Dic7—31no—
187C2×Dic7—31no—
188C2×Dic7C22×D311yescharacteristic
189C2×Dic7C22×D311yescharacteristic
190C2×Dic7C22×D311yescharacteristic
191D28—34no—
192C4×D7—34no—
193C4×D7—34no—
194C4×D7—41no—
195C4×D7—121no—
196C4×D7—121no—
197C4×D7—121no—
198Dic14—34no—
199C2×C3×D4—71no—
200C22×C3×C4—71no—
201C22×C3×C4C2×D711yescharacteristic
202C2×#67—71no—
203C2×#67—71no—
204C22×Dic3—71no—
205C22×Dic3—71no—
206C22×Dic3C2×D711yescharacteristic
207C22×Dic3C2×D711yescharacteristic
208#237—78no—
209C2×C4×D3—71no—
210C2×C4×D3—71no—
211C2×C4×D3—71no—
212C2×Dic6—71no—
213C3×C4⋊C4—71no—
214C3×C22⋊C4—71no—
215C3×C22⋊C4—71no—
216C3×C42—71no—
217#217—71no—
218#217—71no—
219(C2×D3)⋊C4—71no—
220(C2×D3)⋊C4—71no—
221C4⋊Dic3—71no—
222Dic3⋊C4—71no—
223Dic3⋊C4—71no—
224Dic3⋊C4—71no—
225Dic3⋊C4—71no—
226Dic3⋊C4C2×D711yescharacteristic
227Dic3⋊C4C2×D711yescharacteristic
228Dic3⋊C4C2×D711yescharacteristic
229Dic3⋊C4C2×D711yescharacteristic
230C4×Dic3—71no—
231C4×Dic3—71no—
232C4×Dic3—71no—
233C4×Dic3—71no—
234C4×Dic3—71no—
235C2×C3×C7—22no—
236C2×C3×C7C23×C411yescharacteristic
237C2×C3×C7H(5,𝔽2)11yescharacteristic
238C2×C3×C72−1+411yescharacteristic
239D21—22no—
240C3×D7—22no—
241C2×#41—211no—
242C4×Q8—211no—
243C4×Q8—211no—
244C4×D4—71no—
245C4×D4—211no—
246C4×D4—211no—
247C4×D4—211no—
248C4×D4—211no—
249C4×D4—211no—
250#117—211no—
251#117—211no—
252#117—211no—
253C2×C4⋊C4—31no—
254C2×C4⋊C4—211no—
255C2×C4⋊C4—211no—
256C22×C7—21no—
257C22×C7—21no—
258C22×C7C2×C4×D312yes—
259C22×C7C23×D311yescharacteristic
260C2×D7—21no—
261C2×D7—21no—
262C2×D7—32no—
263C2×D7—61no—
264C2×D7—61no—
265C2×D7C2×C4×D312yes—
266C4×C7—21no—
267C4×C7—21no—
268C4×C7—32no—
269C4×C7—32no—
270C4×C7—61no—
271C4×C7—61no—
272Dic7—21no—
273Dic7—21no—
274Dic7—32no—
275Dic7—61no—
276Dic7—61no—
277Dic7C2×C4×D312yes—
278C23×C3—71no—
279C23×C3C22×D711yescharacteristic
280C22×D3—71no—
281C3×D4—74no—
282C2×C3×C4—41no—
283C2×C3×C4—71no—
284C2×C3×C4—71no—
285C2×C3×C4—71no—
286C2×C3×C4—281no—
287C2×C3×C4C22×D711yescharacteristic
288C2×C3×C4C22×D711yescharacteristic
289#67—74no—
290#67—74no—
291C2×Dic3—41no—
292C2×Dic3—71no—
293C2×Dic3—74no—
294C2×Dic3—71no—
295C2×Dic3—71no—
296C2×Dic3—281no—
297C2×Dic3C4×D714yescomplemented
298C2×Dic3C22×D711yescharacteristic
299C2×Dic3C22×D711yescharacteristic
300C2×Dic3C22×D711yescharacteristic
301C2×Dic3C22×D711yescharacteristic
302C4×D3—74no—
303C4×D3—281no—
304C4×D3—281no—
305Dic6—74no—
306C3×C7#52011yescharacteristic, complemented, Hall
307#41—218no—
308C2×Q8—211no—
309C2×D4—71no—
310C2×D4—211no—
311C2×D4—211no—
312C22×C4—31no—
313C22×C4—31no—
314C22×C4—71no—
315C22×C4—211no—
316C22×C4—211no—
317C22×C4—211no—
318C22×C4—211no—
319C22×C4—211no—
320C22×C4D3×D711yescharacteristic
321C4⋊C4—31no—
322C4⋊C4—31no—
323C4⋊C4—31no—
324C4⋊C4—31no—
325C4⋊C4—71no—
326C4⋊C4—211no—
327C4⋊C4—211no—
328C4⋊C4—211no—
329C4⋊C4—211no—
330C4⋊C4—211no—
331C22⋊C4—71no—
332C22⋊C4—71no—
333C22⋊C4—211no—
334C22⋊C4—211no—
335C22⋊C4—211no—
336C22⋊C4—211no—
337C42—71no—
338C42—211no—
339C42—211no—
340C42—211no—
341C42—211no—
342C42—211no—
343C2×C7—22no—
344C2×C7C22×C4×D311yescharacteristic
345C2×C7#100411yescharacteristic
346C2×C7#101011yescharacteristic
347D7—22no—
348D7—62no—
349C22×C3—21no—
350C22×C3—21no—
351C22×C3—72no—
352C22×C3—141no—
353C22×C3—141no—
354C22×C3C2×C4×D712yes—
355C22×C3C23×D711yescharacteristic
356C2×D3—72no—
357C2×D3—141no—
358C2×D3—141no—
359C3×C4—21no—
360C3×C4—21no—
361C3×C4—72no—
362C3×C4—141no—
363C3×C4—141no—
364Dic3—21no—
365Dic3—21no—
366Dic3—72no—
367Dic3—141no—
368Dic3—141no—
369Dic3C2×C4×D712yes—
370Dic3C2×C4×D712yes—
371C23—71no—
372C23—211no—
373C23C2×D3×D711yescharacteristic
374Q8—214no—
375D4—74no—
376D4—214no—
377D4—214no—
378C2×C4—34no—
379C2×C4—31no—
380C2×C4—31no—
381C2×C4—31no—
382C2×C4—31no—
383C2×C4—41no—
384C2×C4—71no—
385C2×C4—71no—
386C2×C4—71no—
387C2×C4—121no—
388C2×C4—214no—
389C2×C4—214no—
390C2×C4—211no—
391C2×C4—211no—
392C2×C4—211no—
393C2×C4—281no—
394C2×C4—841no—
395C2×C4—841no—
396C2×C4—841no—
397C2×C4C2×D3×D711yescharacteristic
398C2×C4C2×D3×D711yescharacteristic
399C7#554611yescharacteristic, complemented, Sylow
400C2×C3—22no—
401C2×C3—142no—
402C2×C3C22×C4×D711yescharacteristic
403C2×C3#654811yescharacteristic
404C2×C3#655111yescharacteristic
405D3—142no—
406C22—21no—
407C22—21no—
408C22—72no—
409C22—141no—
410C22—141no—
411C22—212no—
412C22—421no—
413C22—421no—
414C22C4×D3×D712yes—
415C22C22×D3×D711yescharacteristic
416C4—21no—
417C4—21no—
418C4—32no—
419C4—32no—
420C4—61no—
421C4—61no—
422C4—72no—
423C4—141no—
424C4—141no—
425C4—212no—
426C4—421no—
427C4—421no—
428C3#9019511yescharacteristic, complemented, Sylow
429C2—22no—
430C2—142no—
431C2—422no—
432C2#1062154711yescharacteristic
433C2C2×C4×D3×D711yescharacteristic
434C2#1062159611yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D7
commutatorC2×C3×C7C23×C4
FittingC22×C3×C4×C7C22
socle, cosocleC22×C3×C7C24
prime radicalC2×#66683C2
hypercenterC22×C4D3×D7
hypocenterC3×C7#520
LCSG ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C4×#4268C2semidirect
3. C2×C7×Dic3⋊C4C2semidirect
4. C3×C4×C22⋊D7C2semidirect
5. #10621118C2semidirect
6. Dic3×C22⋊D7C2semidirect
7. #10621091C2semidirect
8. #10621006C2semidirect
9. #10620966C2semidirect
10. #10620964C2semidirect
11. #10620958C2semidirect
12. #10620956C2semidirect
13. D7×Dic3⋊C4C2semidirect
14. #10620887C2semidirect
15. #10620885C2semidirect
28. #66683C4semidirect
58. C4×C22⋊D7D3semidirect
75. #4268C2×C4semidirect
81. C2×C7×Dic3C2×C4semidirect
86. C3×C22⋊D7C2×C4semidirect
91. #4248C2×C4semidirect
92. #4246C2×C4semidirect
94. D21⋊C4C2×C4semidirect
96. Dic3×D7C2×C4semidirect
139. C2×Dic3⋊C4D7semidirect
184. C22⋊D7C4×D3semidirect
297. C2×Dic3C4×D7semidirect
306. C3×C7#520semidirect
399. C7#5546semidirect
428. C3#90195semidirect

Maximal subgroups

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

subgroupindex
C2×#666832
C4×#42682
C2×C7×Dic3⋊C42
C3×C4×C22⋊D72
#106211182
Dic3×C22⋊D72
#106210912
#106210062
#106209662
#106209642
#106209582
#106209562
D7×Dic3⋊C42
#106208872
#106208852
#901953
#55467