Finite Group Databasebeta

← #13059606

Subgroups — #13059606

Order 1344.

456 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10621407C211yesmaximal, characteristic, complemented
2C2×C7×Dic3⋊C4C211yesmaximal, characteristic, complemented
3C3×C4×C22⋊D7C211yesmaximal, characteristic
4#10621137C211yesmaximal, characteristic, complemented
5#10621137C211yesmaximal, characteristic, complemented
6Dic3×C22⋊D7C211yesmaximal, characteristic
7Dic3×C22⋊D7C211yesmaximal, characteristic
8#10620993C211yesmaximal, characteristic, complemented
9#10620984C211yesmaximal, characteristic, complemented
10D7×Dic3⋊C4C211yesmaximal, characteristic, complemented
11#10620940C211yesmaximal, characteristic, complemented
12#10620911C211yesmaximal, characteristic, complemented
13#10620896C211yesmaximal, characteristic, complemented
14#10620890C211yesmaximal, characteristic, complemented
15#10620885C211yesmaximal, characteristic, complemented
16#90202—31nomaximal, Hall
17C22×C3×C4×C7C2211yescharacteristic
18C2×Dic42C2211yescharacteristic
19C22×C7×Dic3C2211yescharacteristic
20C22×C7×Dic3C2211yescharacteristic
21C2×C3×C22⋊D7C2211yescharacteristic
22C2×C3×C4×D7C2211yescharacteristic
23C2×#4248C2211yescharacteristic
24C2×#4248C2211yescharacteristic
25C2×Dic3×D7C2211yescharacteristic
26C2×Dic3×D7C2211yescharacteristic
27#66636C2211yescharacteristic
28#66636C2211yescharacteristic
29C4⋊Dic21C2211yescharacteristic
30#66629C2211yescharacteristic
31#66629C2211yescharacteristic
32C7×Dic3⋊C4C2211yescharacteristic
33C7×Dic3⋊C4C2211yescharacteristic
34C7×Dic3⋊C4C2211yescharacteristic
35C7×Dic3⋊C4C2211yescharacteristic
36C3×C22⋊Dic7C2211yescharacteristic
37C3×(C2×D7)⋊C4C2211yescharacteristic
38C3×Dic7⋊C4C2211yescharacteristic
39C3×C4×Dic7C2211yescharacteristic
40Dic3⋊Dic7C2211yescharacteristic
41Dic3⋊Dic7C2211yescharacteristic
42#66577C2211yescharacteristic
43#66577C2211yescharacteristic
44Dic7⋊Dic3C2211yescharacteristic
45Dic7⋊Dic3C2211yescharacteristic
46Dic7⋊Dic3C2211yescharacteristic
47Dic7⋊Dic3C2211yescharacteristic
48(C2×D7)⋊Dic3C2211yescharacteristic
49(C2×D7)⋊Dic3C2211yescharacteristic
50Dic3×Dic7C2211yescharacteristic
51Dic3×Dic7C2211yescharacteristic
52C2×C7×C4⋊C4—31no—
53C4×C22⋊D7—31no—
54C4×C22⋊D7—31no—
55C4×C22⋊D7D311yescharacteristic
56#6487—31no—
57#6487—31no—
58#6487—31no—
59#6460—31no—
60#6459—31no—
61#6459—31no—
62D7×C4⋊C4—31no—
63#6452—31no—
64#6452—31no—
65#6451—31no—
66#6450—31no—
67#5539—71nomaximal, Hall
68C23×C3×C7C2311yescharacteristic
69C22×C3×D7C2311yescharacteristic
70C2×C3×C4×C7—41no—
71C2×C3×C4×C7C2311yescharacteristic
72C2×C3×C4×C7C2311yescharacteristic
73C2×Dic21C2311yescharacteristic
74C2×Dic21C2311yescharacteristic
75C2×Dic21C2311yescharacteristic
76C2×Dic21C2311yescharacteristic
77Dic42—41no—
78C2×C7×Dic3—22no—
79C2×C7×Dic3—41no—
80C2×C7×Dic3C2311yescharacteristic
81C2×C7×Dic3C2311yescharacteristic
82C2×C7×Dic3C2311yescharacteristic
83C2×C7×Dic3C2311yescharacteristic
84C3×C22⋊D7Q814yescomplemented
85C2×C3×Dic7C2311yescharacteristic
86C2×C3×Dic7C2311yescharacteristic
87C2×C3×Dic7C2311yescharacteristic
88C3×C4×D7—41no—
89#4248—22no—
90#4248—41no—
91Dic3×D7—22no—
92Dic3×D7—41no—
93C22×C4×C7—31no—
94C22×C4×C7—31no—
95C22×C4×C7C2×D311yescharacteristic
96C2×C22⋊D7C2×D311yescharacteristic
97C2×C4×D7—31no—
98C2×C4×D7—31no—
99C2×C4×D7C2×D311yescharacteristic
100C2×Dic14—31no—
101C2×Dic14—31no—
102C2×Dic14—31no—
103C7×C4⋊C4—31no—
104C7×C4⋊C4—31no—
105C7×C4⋊C4—31no—
106C7×C4⋊C4—31no—
107C22⋊Dic7—31no—
108C22⋊Dic7—31no—
109C22⋊Dic7C2×D311yescharacteristic
110(C2×D7)⋊C4—31no—
111(C2×D7)⋊C4—31no—
112(C2×D7)⋊C4C2×D311yescharacteristic
113C4⋊Dic7—31no—
114C4⋊Dic7—31no—
115C4⋊Dic7—31no—
116Dic7⋊C4—31no—
117Dic7⋊C4—31no—
118Dic7⋊C4—31no—
119Dic7⋊C4—31no—
120Dic7⋊C4—31no—
121Dic7⋊C4—31no—
122Dic7⋊C4—31no—
123Dic7⋊C4—31no—
124Dic7⋊C4C2×D311yescharacteristic
125C4×Dic7—31no—
126C4×Dic7—31no—
127C4×Dic7C2×D311yescharacteristic
128C3×C4×D4—71no—
129D4×Dic3—71no—
130D4×Dic3—71no—
131#924—71no—
132#924—71no—
133C2×Dic3⋊C4—71no—
134C2×Dic3⋊C4D711yescharacteristic, complemented
135#890—71no—
136#888—71no—
137#878—71no—
138#878—71no—
139#878—71no—
140#878—71no—
141#870—71no—
142C4×Dic6—71no—
143C22×C3×C7—21no—
144C22×C3×C7—21no—
145C22×C3×C7C2×Q812yes—
146C22×C3×C7C2411yescharacteristic
147C2×C3×D7—21no—
148C2×C3×D7—21no—
149C2×C3×D7C2×Q812yes—
150C3×C4×C7—21no—
151C3×C4×C7—21no—
152Dic21—21no—
153Dic21—21no—
154Dic21—21no—
155Dic21—21no—
156C3×Dic7—21no—
157C3×Dic7—21no—
158C3×Dic7C2×Q812yes—
159C7×Dic3—21no—
160C7×Dic3—21no—
161C7×Dic3—21no—
162C7×Dic3#4112yes—
163#554—211noSylow
164C23×C7C22×D311yescharacteristic
165C22×D7C22×D311yescharacteristic
166C2×C4×C7—31no—
167C2×C4×C7—31no—
168C2×C4×C7—31no—
169C2×C4×C7—31no—
170C2×C4×C7—41no—
171C2×C4×C7—62no—
172C2×C4×C7—121no—
173C2×C4×C7C22×D311yescharacteristic
174C2×C4×C7C22×D311yescharacteristic
175C22⋊D7Dic614yescomplemented
176C2×Dic7—31no—
177C2×Dic7—31no—
178C2×Dic7—31no—
179C2×Dic7—31no—
180C2×Dic7C22×D311yescharacteristic
181C2×Dic7C22×D311yescharacteristic
182C2×Dic7C22×D311yescharacteristic
183C4×D7—41no—
184C4×D7—62no—
185C4×D7—121no—
186Dic14—62no—
187Dic14—121no—
188Dic14—121no—
189C2×C3×D4—71no—
190C22×C3×C4—71no—
191C22×C3×C4C2×D711yescharacteristic
192C22×Dic3—71no—
193C22×Dic3—71no—
194C22×Dic3C2×D711yescharacteristic
195C22×Dic3C2×D711yescharacteristic
196C2×Dic6—71no—
197C2×Dic6—71no—
198C2×Dic6—71no—
199C3×C4⋊C4—71no—
200C3×C22⋊C4—71no—
201C3×C22⋊C4—71no—
202C3×C42—71no—
203#217—71no—
204#217—71no—
205#217—71no—
206#217—71no—
207C4⋊Dic3—71no—
208C4⋊Dic3—71no—
209C4⋊Dic3—71no—
210C4⋊Dic3—71no—
211C4⋊Dic3—71no—
212Dic3⋊C4—71no—
213Dic3⋊C4—71no—
214Dic3⋊C4—71no—
215Dic3⋊C4—71no—
216Dic3⋊C4—71no—
217Dic3⋊C4—71no—
218Dic3⋊C4C2×D711yescharacteristic
219Dic3⋊C4C2×D711yescharacteristic
220Dic3⋊C4C2×D711yescharacteristic
221Dic3⋊C4C2×D711yescharacteristic
222C4×Dic3—71no—
223C4×Dic3—71no—
224C2×C3×C7—22no—
225C2×C3×C7C22×Q811yescharacteristic
226C2×C3×C7C2×#4111yescharacteristic
227C2×C3×C7H(5,𝔽2)11yescharacteristic
228C3×D7—22no—
229C4⋊Q8—211no—
230#125—211no—
231#125—211no—
232C22⋊Q8—211no—
233C22⋊Q8—211no—
234C22⋊Q8—211no—
235C22⋊Q8—211no—
236C22⋊Q8—211no—
237C22⋊Q8—211no—
238C4×Q8—211no—
239C4×D4—71no—
240C4×D4—211no—
241C4×D4—211no—
242C2×C4⋊C4—31no—
243C2×C4⋊C4—211no—
244C22×C7—21no—
245C22×C7—21no—
246C22×C7C2×Dic612yes—
247C22×C7C23×D311yescharacteristic
248C2×D7—21no—
249C2×D7—21no—
250C2×D7C2×Dic612yes—
251C4×C7—21no—
252C4×C7—21no—
253C4×C7—32no—
254C4×C7—61no—
255C4×C7—61no—
256C4×C7—61no—
257Dic7—21no—
258Dic7—21no—
259Dic7—61no—
260Dic7—61no—
261Dic7—61no—
262Dic7—61no—
263Dic7C2×Dic612yes—
264C23×C3—71no—
265C23×C3C22×D711yescharacteristic
266C3×D4—74no—
267C2×C3×C4—41no—
268C2×C3×C4—71no—
269C2×C3×C4—71no—
270C2×C3×C4—71no—
271C2×C3×C4—281no—
272C2×C3×C4C22×D711yescharacteristic
273C2×C3×C4C22×D711yescharacteristic
274C2×Dic3—22no—
275C2×Dic3—41no—
276C2×Dic3—71no—
277C2×Dic3—71no—
278C2×Dic3—71no—
279C2×Dic3—71no—
280C2×Dic3—142no—
281C2×Dic3—281no—
282C2×Dic3C22×D711yescharacteristic
283C2×Dic3C22×D711yescharacteristic
284C2×Dic3C22×D711yescharacteristic
285C2×Dic3C22×D711yescharacteristic
286Dic6—142no—
287Dic6—281no—
288Dic6—281no—
289C3×C7#55411yescharacteristic, complemented, Hall
290C2×Q8—211no—
291C2×Q8—211no—
292C2×Q8—211no—
293C2×D4—71no—
294C22×C4—31no—
295C22×C4—31no—
296C22×C4—71no—
297C22×C4—211no—
298C22×C4—211no—
299C22×C4D3×D711yescharacteristic
300C4⋊C4—31no—
301C4⋊C4—31no—
302C4⋊C4—31no—
303C4⋊C4—31no—
304C4⋊C4—71no—
305C4⋊C4—211no—
306C4⋊C4—211no—
307C4⋊C4—211no—
308C4⋊C4—211no—
309C4⋊C4—211no—
310C4⋊C4—211no—
311C4⋊C4—211no—
312C4⋊C4—211no—
313C4⋊C4—211no—
314C4⋊C4—211no—
315C4⋊C4—211no—
316C22⋊C4—71no—
317C22⋊C4—71no—
318C22⋊C4—211no—
319C22⋊C4—211no—
320C22⋊C4—211no—
321C22⋊C4—211no—
322C42—71no—
323C42—211no—
324C42—211no—
325C2×C7—22no—
326C2×C7C22×Dic611yescharacteristic
327C2×C7#100411yescharacteristic
328C2×C7D3×#4111yescharacteristic
329D7—22no—
330C22×C3—21no—
331C22×C3—21no—
332C22×C3—72no—
333C22×C3—141no—
334C22×C3—141no—
335C22×C3Q8×D712yes—
336C22×C3C23×D711yescharacteristic
337C3×C4—21no—
338C3×C4—21no—
339C3×C4—72no—
340C3×C4—141no—
341C3×C4—141no—
342Dic3—21no—
343Dic3—21no—
344Dic3—21no—
345Dic3—141no—
346Dic3—141no—
347Dic3—141no—
348Dic3—141no—
349Dic3#115712yes—
350C23—71no—
351C23C2×D3×D711yescharacteristic
352Q8—422no—
353Q8—841no—
354Q8—841no—
355D4—74no—
356C2×C4—31no—
357C2×C4—31no—
358C2×C4—31no—
359C2×C4—31no—
360C2×C4—41no—
361C2×C4—62no—
362C2×C4—71no—
363C2×C4—71no—
364C2×C4—71no—
365C2×C4—121no—
366C2×C4—211no—
367C2×C4—211no—
368C2×C4—211no—
369C2×C4—211no—
370C2×C4—281no—
371C2×C4—422no—
372C2×C4—841no—
373C2×C4C2×D3×D711yescharacteristic
374C2×C4C2×D3×D711yescharacteristic
375C7#553911yescharacteristic, complemented, Sylow
376C2×C3—22no—
377C2×C3—142no—
378C2×C3C2×#115711yescharacteristic
379C2×C3#654811yescharacteristic
380C2×C3C2×Q8×D711yescharacteristic
381C22—21no—
382C22—21no—
383C22—72no—
384C22—141no—
385C22—141no—
386C22D7×Dic612yes—
387C22C22×D3×D711yescharacteristic
388C4—21no—
389C4—21no—
390C4—32no—
391C4—61no—
392C4—61no—
393C4—61no—
394C4—72no—
395C4—141no—
396C4—141no—
397C4—421no—
398C4—421no—
399C4—421no—
400C4—421no—
401C3#9020211yescharacteristic, complemented, Sylow
402C2—22no—
403C2—142no—
404C2C2×D7×Dic611yescharacteristic
405C2D3×#115711yescharacteristic
406C2#1062159611yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC22×C3×C4×C7C22
prime radicalC2×C3×C22⋊D7C22
hypercenterC22×C4D3×D7
hypocenterC3×C7#554
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. #10621407C2semidirect
2. C2×C7×Dic3⋊C4C2semidirect
4. #10621137C2semidirect
5. #10621137C2semidirect
8. #10620993C2semidirect
9. #10620984C2semidirect
10. D7×Dic3⋊C4C2semidirect
11. #10620940C2semidirect
12. #10620911C2semidirect
13. #10620896C2semidirect
14. #10620890C2semidirect
15. #10620885C2semidirect
84. C3×C22⋊D7Q8semidirect
134. C2×Dic3⋊C4D7semidirect
175. C22⋊D7Dic6semidirect
289. C3×C7#554semidirect
375. C7#5539semidirect
401. C3#90202semidirect

Maximal subgroups

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

subgroupindex
#106214072
C2×C7×Dic3⋊C42
C3×C4×C22⋊D72
#106211372
Dic3×C22⋊D72
#106209932
#106209842
D7×Dic3⋊C42
#106209402
#106209112
#106208962
#106208902
#106208852
#902023
#55397