Finite Group Databasebeta

← #13058153

Subgroups — #13058153

Order 1344.

476 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
1#10621362C211yesmaximal, characteristic, complemented
2C7×#884C211yesmaximal, characteristic, complemented
3C3×#6449C211yesmaximal, characteristic, complemented
4#10621127C211yesmaximal, characteristic, complemented
5#67×Dic7C211yesmaximal, characteristic, complemented
6#67×Dic7C211yesmaximal, characteristic, complemented
7#10621111C211yesmaximal, characteristic, complemented
8#10621019C211yesmaximal, characteristic, complemented
9#10621015C211yesmaximal, characteristic, complemented
10#10621003C211yesmaximal, characteristic, complemented
11#10620997C211yesmaximal, characteristic, complemented
12D3×C4⋊Dic7C211yesmaximal, characteristic, complemented
13#10620956C211yesmaximal, characteristic, complemented
14#10620908C211yesmaximal, characteristic, complemented
15#10620903C211yesmaximal, characteristic, complemented
16#90309—31nomaximal, Hall
17C2×C7×#67C2211yescharacteristic
18C2×C7×#67C2211yescharacteristic
19C2×C7×D12C2211yescharacteristic
20C2×C4×D3×C7C2211yescharacteristic
21C2×C3×C22⋊D7C2211yescharacteristic
22C22×C3×Dic7C2211yescharacteristic
23C2×#4245C2211yescharacteristic
24C2×#4245C2211yescharacteristic
25C2×D3×Dic7C2211yescharacteristic
26C2×D3×Dic7C2211yescharacteristic
27C2×D3×Dic7C2211yescharacteristic
28C2×Dic3×D7C2211yescharacteristic
29C3×C7×C22⋊C4C2211yescharacteristic
30#66636C2211yescharacteristic
31#66636C2211yescharacteristic
32C4⋊Dic21C2211yescharacteristic
33#66629C2211yescharacteristic
34#66629C2211yescharacteristic
35C4×Dic21C2211yescharacteristic
36C7×(C2×D3)⋊C4C2211yescharacteristic
37C7×Dic3⋊C4C2211yescharacteristic
38C3×(C2×D7)⋊C4C2211yescharacteristic
39C3×(C2×D7)⋊C4C2211yescharacteristic
40C3×C4⋊Dic7C2211yescharacteristic
41C3×C4⋊Dic7C2211yescharacteristic
42Dic3⋊Dic7C2211yescharacteristic
43Dic3⋊Dic7C2211yescharacteristic
44Dic3⋊Dic7C2211yescharacteristic
45Dic7⋊Dic3C2211yescharacteristic
46(C2×D3)⋊Dic7C2211yescharacteristic
47(C2×D3)⋊Dic7C2211yescharacteristic
48(C2×D3)⋊Dic7C2211yescharacteristic
49(C2×D7)⋊Dic3C2211yescharacteristic
50Dic3×Dic7C2211yescharacteristic
51Dic3×Dic7C2211yescharacteristic
52C7×#121—31no—
53#6502—31no—
54#6502—31no—
55#6501—31no—
56D4×Dic7—31no—
57D4×Dic7—31no—
58D4×Dic7—31no—
59C4×C22⋊D7—31no—
60C2×C4⋊Dic7—31no—
61#6455—31no—
62#6453—31no—
63#6449—31no—
64#6449D311yescharacteristic, complemented
65#6442—31no—
66#6442—31no—
67#5661—71nomaximal, Hall
68C23×C3×C7C2311yescharacteristic
69C22×D3×C7C2311yescharacteristic
70C22×D3×C7C2311yescharacteristic
71C22×C3×D7C2311yescharacteristic
72C2×C3×C4×C7C2311yescharacteristic
73C2×C3×C4×C7C2311yescharacteristic
74C2×Dic21C2311yescharacteristic
75C2×Dic21C2311yescharacteristic
76C2×Dic21C2311yescharacteristic
77C2×Dic21C2311yescharacteristic
78C7×#67—22no—
79C7×#67—22no—
80C2×C7×Dic3C2311yescharacteristic
81C2×C7×Dic3C2311yescharacteristic
82C7×D12—22no—
83C4×D3×C7—22no—
84C3×C22⋊D7—41no—
85C2×C3×Dic7—41no—
86C2×C3×Dic7C2311yescharacteristic
87C2×C3×Dic7C2311yescharacteristic
88C2×C3×Dic7C2311yescharacteristic
89#4245—22no—
90#4245—41no—
91D3×Dic7—41no—
92D3×Dic7—41no—
93D3×Dic7—41no—
94Dic3×D7—22no—
95C2×C7×D4—31no—
96C2×C7×D4—31no—
97C2×C7×D4—31no—
98C22×C4×C7—31no—
99C2×C22⋊D7—31no—
100C2×C22⋊D7—31no—
101C2×C22⋊D7C2×D311yescharacteristic
102C22×Dic7—31no—
103C22×Dic7—31no—
104C22×Dic7—31no—
105C22×Dic7C2×D311yescharacteristic
106C2×C4×D7—31no—
107C7×C4⋊C4—31no—
108C7×C22⋊C4—31no—
109C7×C22⋊C4C2×D311yescharacteristic
110C22⋊Dic7—31no—
111C22⋊Dic7—31no—
112C22⋊Dic7—31no—
113C22⋊Dic7—31no—
114C22⋊Dic7—31no—
115(C2×D7)⋊C4—31no—
116(C2×D7)⋊C4C2×D311yescharacteristic
117(C2×D7)⋊C4C2×D311yescharacteristic
118C4⋊Dic7—31no—
119C4⋊Dic7—31no—
120C4⋊Dic7—31no—
121C4⋊Dic7—31no—
122C4⋊Dic7C2×D311yescharacteristic
123C4⋊Dic7C2×D311yescharacteristic
124Dic7⋊C4—31no—
125Dic7⋊C4—31no—
126Dic7⋊C4—31no—
127C4×Dic7—31no—
128C4×Dic7—31no—
129C4×Dic7—31no—
130C3×#123—71no—
131#939—71no—
132#938—71no—
133C4×#67—71no—
134C4×#67—71no—
135#898—71no—
136#893—71no—
137D3×C4⋊C4—71no—
138#889—71no—
139#884—71no—
140#884D711yescharacteristic, complemented
141#883—71no—
142#67⋊C4—71no—
143#879—71no—
144#877—71no—
145C22×C3×C7—21no—
146C22×C3×C7—21no—
147C22×C3×C7—21no—
148C22×C3×C7C2411yescharacteristic
149C2×D3×C7—21no—
150C2×D3×C7—21no—
151C2×D3×C7—21no—
152C2×D3×C7—21no—
153C2×D3×C7—21no—
154C2×D3×C7#4112yes—
155C2×C3×D7—21no—
156C2×C3×D7—21no—
157C2×C3×D7—21no—
158C3×C4×C7—21no—
159C3×C4×C7—21no—
160Dic21—21no—
161Dic21—21no—
162Dic21—21no—
163Dic21—21no—
164C3×Dic7—21no—
165C3×Dic7—21no—
166C3×Dic7—21no—
167C7×Dic3—21no—
168C7×Dic3#4112yes—
169#553—211noSylow
170C23×C7—31no—
171C23×C7—31no—
172C23×C7C22×D311yescharacteristic
173C22×D7C22×D311yescharacteristic
174C7×D4—62no—
175C7×D4—62no—
176C7×D4—62no—
177C2×C4×C7—31no—
178C2×C4×C7—31no—
179C2×C4×C7—62no—
180C2×C4×C7C22×D311yescharacteristic
181C2×C4×C7C22×D311yescharacteristic
182C22⋊D7—41no—
183C22⋊D7—62no—
184C22⋊D7—121no—
185C2×Dic7—31no—
186C2×Dic7—31no—
187C2×Dic7—31no—
188C2×Dic7—31no—
189C2×Dic7—41no—
190C2×Dic7—121no—
191C2×Dic7—121no—
192C2×Dic7—121no—
193C2×Dic7C22×D311yescharacteristic
194C2×Dic7C22×D311yescharacteristic
195C2×Dic7C22×D311yescharacteristic
196C4×D7—62no—
197C2×C3×D4—71no—
198C22×C3×C4—71no—
199C2×#67—71no—
200C2×#67—71no—
201C2×#67C2×D711yescharacteristic
202C2×#67C2×D711yescharacteristic
203C22×Dic3—71no—
204C2×D12C2×D711yescharacteristic
205C2×C4×D3—71no—
206C2×C4×D3—71no—
207C2×C4×D3—71no—
208C2×C4×D3C2×D711yescharacteristic
209C3×C4⋊C4—71no—
210C3×C4⋊C4—71no—
211C3×C22⋊C4—71no—
212C3×C22⋊C4—71no—
213C3×C22⋊C4C2×D711yescharacteristic
214#217—71no—
215#217—71no—
216#217—71no—
217(C2×D3)⋊C4—71no—
218(C2×D3)⋊C4—71no—
219(C2×D3)⋊C4—71no—
220(C2×D3)⋊C4C2×D711yescharacteristic
221C4⋊Dic3—71no—
222C4⋊Dic3—71no—
223Dic3⋊C4—71no—
224Dic3⋊C4—71no—
225Dic3⋊C4—71no—
226Dic3⋊C4—71no—
227Dic3⋊C4—71no—
228Dic3⋊C4C2×D711yescharacteristic
229C4×Dic3—71no—
230C4×Dic3—71no—
231C4×Dic3—71no—
232C2×C3×C7—41no—
233C2×C3×C7C2×#4111yescharacteristic
234C2×C3×C7C2×#4111yescharacteristic
235C2×C3×C7H(5,𝔽2)11yescharacteristic
236C3×D7—41no—
237D3×C7—22no—
238D3×C7—41no—
239#126—211no—
240#126—211no—
241#125—211no—
242#123—71no—
243#123—211no—
244#121—31no—
245#121—211no—
246#121—211no—
247#121—211no—
248C4×D4—211no—
249C4×D4—211no—
250C4×D4—211no—
251C4×D4—211no—
252#117—211no—
253C2×C4⋊C4—211no—
254C22×C7—21no—
255C22×C7—21no—
256C22×C7—21no—
257C22×C7—32no—
258C22×C7—61no—
259C22×C7—61no—
260C22×C7—61no—
261C22×C7—61no—
262C22×C7—61no—
263C22×C7C23×D311yescharacteristic
264C2×D7—21no—
265C2×D7—21no—
266C2×D7—21no—
267C4×C7—21no—
268C4×C7—21no—
269C4×C7—32no—
270C4×C7—61no—
271Dic7—21no—
272Dic7—21no—
273Dic7—21no—
274Dic7—61no—
275Dic7—61no—
276Dic7—61no—
277Dic7—61no—
278C23×C3—71no—
279C23×C3C22×D711yescharacteristic
280C22×D3C22×D711yescharacteristic
281C22×D3C22×D711yescharacteristic
282C3×D4—281no—
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—22no—
290#67—22no—
291#67—142no—
292#67—281no—
293C2×Dic3—71no—
294C2×Dic3—71no—
295C2×Dic3—71no—
296C2×Dic3—71no—
297C2×Dic3—142no—
298C2×Dic3C22×D711yescharacteristic
299C2×Dic3C22×D711yescharacteristic
300D12—22no—
301C4×D3—22no—
302C4×D3—281no—
303C4×D3—281no—
304C4×D3—281no—
305C3×C7#55311yescharacteristic, complemented, Hall
306C2×D4—31no—
307C2×D4—31no—
308C2×D4—31no—
309C2×D4—71no—
310C2×D4—211no—
311C2×D4—211no—
312C22×C4—31no—
313C22×C4—71no—
314C22×C4—211no—
315C22×C4—211no—
316C22×C4—211no—
317C22×C4—211no—
318C4⋊C4—31no—
319C4⋊C4—71no—
320C4⋊C4—71no—
321C4⋊C4—211no—
322C4⋊C4—211no—
323C4⋊C4—211no—
324C4⋊C4—211no—
325C4⋊C4—211no—
326C4⋊C4—211no—
327C4⋊C4—211no—
328C22⋊C4—31no—
329C22⋊C4—71no—
330C22⋊C4—71no—
331C22⋊C4—211no—
332C22⋊C4—211no—
333C22⋊C4—211no—
334C22⋊C4—211no—
335C22⋊C4—211no—
336C22⋊C4—211no—
337C22⋊C4D3×D711yescharacteristic
338C42—211no—
339C42—211no—
340C42—211no—
341C2×C7—41no—
342C2×C7—62no—
343C2×C7—121no—
344C2×C7#100411yescharacteristic
345C2×C7D3×#4111yescharacteristic
346C2×C7D3×#4111yescharacteristic
347D7—41no—
348C22×C3—21no—
349C22×C3—21no—
350C22×C3—21no—
351C22×C3—141no—
352C22×C3—141no—
353C22×C3—141no—
354C22×C3C23×D711yescharacteristic
355C2×D3—21no—
356C2×D3—21no—
357C2×D3—21no—
358C2×D3—21no—
359C2×D3—21no—
360C2×D3#115912yes—
361C3×C4—21no—
362C3×C4—21no—
363C3×C4—141no—
364C3×C4—141no—
365C3×C4—141no—
366Dic3—21no—
367Dic3—141no—
368Dic3—141no—
369Dic3—141no—
370Dic3—141no—
371Dic3#115912yes—
372C23—31no—
373C23—31no—
374C23—71no—
375C23C2×D3×D711yescharacteristic
376D4—62no—
377D4—62no—
378D4—62no—
379D4—281no—
380D4—422no—
381D4—841no—
382C2×C4—31no—
383C2×C4—31no—
384C2×C4—62no—
385C2×C4—71no—
386C2×C4—71no—
387C2×C4—71no—
388C2×C4—211no—
389C2×C4—211no—
390C2×C4—211no—
391C2×C4—211no—
392C2×C4—281no—
393C2×C4—422no—
394C2×C4—841no—
395C2×C4—841no—
396C2×C4—841no—
397C2×C4C2×D3×D711yescharacteristic
398C2×C4C2×D3×D711yescharacteristic
399C7#566111yescharacteristic, complemented, Sylow
400C2×C3—41no—
401C2×C3—281no—
402C2×C3C2×#115911yescharacteristic
403C2×C3C2×#115911yescharacteristic
404C2×C3#655311yescharacteristic
405D3—22no—
406D3—41no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—32no—
411C22—61no—
412C22—61no—
413C22—61no—
414C22—61no—
415C22—61no—
416C22—141no—
417C22—141no—
418C22—141no—
419C22C22×D3×D711yescharacteristic
420C4—21no—
421C4—21no—
422C4—32no—
423C4—61no—
424C4—141no—
425C4—141no—
426C4—141no—
427C4—421no—
428C4—421no—
429C4—421no—
430C4—421no—
431C3#9030911yescharacteristic, complemented, Sylow
432C2—41no—
433C2—62no—
434C2—121no—
435C2—281no—
436C2#1062156411yescharacteristic
437C2D3×#115911yescharacteristic
438C2D3×#115911yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC3×C7×C22⋊C4C22
hypercenterC22⋊C4D3×D7
hypocenterC3×C7#553
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. #10621362C2semidirect
2. C7×#884C2semidirect
3. C3×#6449C2semidirect
4. #10621127C2semidirect
5. #67×Dic7C2semidirect
6. #67×Dic7C2semidirect
7. #10621111C2semidirect
8. #10621019C2semidirect
9. #10621015C2semidirect
10. #10621003C2semidirect
11. #10620997C2semidirect
12. D3×C4⋊Dic7C2semidirect
13. #10620956C2semidirect
14. #10620908C2semidirect
15. #10620903C2semidirect
64. #6449D3semidirect
140. #884D7semidirect
305. C3×C7#553semidirect
399. C7#5661semidirect
431. C3#90309semidirect

Maximal subgroups

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

subgroupindex
#106213622
C7×#8842
C3×#64492
#106211272
#67×Dic72
#106211112
#106210192
#106210152
#106210032
#106209972
D3×C4⋊Dic72
#106209562
#106209082
#106209032
#903093
#56617