Finite Group Databasebeta

← #13058074

Subgroups — #13058074

Order 1344.

436 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10621362C211yesmaximal, characteristic, complemented
2C7×#883C211yesmaximal, characteristic, complemented
3C3×(C22⋊D7)⋊C4C211yesmaximal, characteristic, complemented
4#10621136C211yesmaximal, characteristic, complemented
5#10621111C211yesmaximal, characteristic
6#10621110C211yesmaximal, characteristic, complemented
7#10621101C211yesmaximal, characteristic, complemented
8C4×#4245C211yesmaximal, characteristic, complemented
9#10620992C211yesmaximal, characteristic, complemented
10#10620976C211yesmaximal, characteristic, complemented
11D7×Dic3⋊C4C211yesmaximal, characteristic, complemented
12#10620950C211yesmaximal, characteristic, complemented
13#10620944C211yesmaximal, characteristic, complemented
14#10620934C211yesmaximal, characteristic, complemented
15#10620914C211yesmaximal, characteristic, complemented
16#90351—31nomaximal, Hall
17C2×C7×#67C2211yescharacteristic
18C2×C4×D3×C7C2211yescharacteristic
19C2×C3×C22⋊D7C2211yescharacteristic
20C22×C3×Dic7C2211yescharacteristic
21C2×C3×C4×D7C2211yescharacteristic
22C2×#4248C2211yescharacteristic
23C2×#4245C2211yescharacteristic
24C2×Dic3×D7C2211yescharacteristic
25C2×Dic3×D7C2211yescharacteristic
26C3×C7×C22⋊C4C2211yescharacteristic
27#66636C2211yescharacteristic
28#66636C2211yescharacteristic
29C4⋊Dic21C2211yescharacteristic
30#66629C2211yescharacteristic
31#66629C2211yescharacteristic
32C4×Dic21C2211yescharacteristic
33C7×#217C2211yescharacteristic
34C7×(C2×D3)⋊C4C2211yescharacteristic
35C7×C4⋊Dic3C2211yescharacteristic
36C7×Dic3⋊C4C2211yescharacteristic
37C3×(C2×D7)⋊C4C2211yescharacteristic
38C3×Dic7⋊C4C2211yescharacteristic
39C3×C4×Dic7C2211yescharacteristic
40Dic3⋊Dic7C2211yescharacteristic
41Dic3⋊Dic7C2211yescharacteristic
42Dic3⋊Dic7C2211yescharacteristic
43#66577C2211yescharacteristic
44#66577C2211yescharacteristic
45#66577C2211yescharacteristic
46#66577C2211yescharacteristic
47Dic7⋊Dic3C2211yescharacteristic
48(C2×D3)⋊Dic7C2211yescharacteristic
49(C2×D3)⋊Dic7C2211yescharacteristic
50(C2×D3)⋊Dic7C2211yescharacteristic
51(C2×D7)⋊Dic3C2211yescharacteristic
52C7×#123—31no—
53#6501—31no—
54#6498—31no—
55C4×C22⋊D7—31no—
56#6487—31no—
57#6459—31no—
58D7×C4⋊C4—31no—
59#6453—31no—
60#6452—31no—
61#6446—31no—
62#6446—31no—
63(C22⋊D7)⋊C4D311yescharacteristic, complemented
64#6442—31no—
65#6442—31no—
66#6441—31no—
67#5564—71nomaximal, Hall
68C23×C3×C7C2311yescharacteristic
69C22×D3×C7C2311yescharacteristic
70C22×C3×D7C2311yescharacteristic
71C2×C3×C4×C7C2311yescharacteristic
72C2×C3×C4×C7C2311yescharacteristic
73C2×Dic21C2311yescharacteristic
74C2×Dic21C2311yescharacteristic
75C2×Dic21C2311yescharacteristic
76C2×Dic21C2311yescharacteristic
77C7×#67—41no—
78C2×C7×Dic3C2311yescharacteristic
79C2×C7×Dic3C2311yescharacteristic
80C2×C7×Dic3C2311yescharacteristic
81C4×D3×C7—41no—
82C3×C22⋊D7—22no—
83C2×C3×Dic7—41no—
84C2×C3×Dic7C2311yescharacteristic
85C2×C3×Dic7C2311yescharacteristic
86C2×C3×Dic7C2311yescharacteristic
87C3×C4×D7—41no—
88#4248—41no—
89#4245—22no—
90Dic3×D7—22no—
91Dic3×D7—41no—
92C2×C7×D4—31no—
93C22×C4×C7—31no—
94C2×C22⋊D7—31no—
95C2×C22⋊D7C2×D311yescharacteristic
96C22×Dic7C2×D311yescharacteristic
97C2×C4×D7—31no—
98C2×C4×D7—31no—
99C2×C4×D7C2×D311yescharacteristic
100C2×Dic14—31no—
101C7×C4⋊C4—31no—
102C7×C4⋊C4—31no—
103C7×C22⋊C4—31no—
104C7×C22⋊C4—31no—
105C7×C22⋊C4C2×D311yescharacteristic
106C22⋊Dic7—31no—
107C22⋊Dic7—31no—
108C22⋊Dic7—31no—
109C22⋊Dic7—31no—
110C22⋊Dic7—31no—
111(C2×D7)⋊C4—31no—
112(C2×D7)⋊C4C2×D311yescharacteristic
113C4⋊Dic7—31no—
114C4⋊Dic7—31no—
115C4⋊Dic7—31no—
116C4⋊Dic7—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×Dic7C2×D311yescharacteristic
127C3×C4×D4—71no—
128#939—71no—
129#935—71no—
130#929—71no—
131C4×#67—71no—
132#924—71no—
133C2×Dic3⋊C4—71no—
134#896—71no—
135#889—71no—
136#886—71no—
137#883D711yescharacteristic, complemented
138#879—71no—
139#878—71no—
140#876—71no—
141#870—71no—
142C22×C3×C7—21no—
143C22×C3×C7—21no—
144C22×C3×C7—21no—
145C22×C3×C7C2411yescharacteristic
146C2×D3×C7—21no—
147C2×D3×C7—21no—
148C2×D3×C7—21no—
149C2×C3×D7—21no—
150C2×C3×D7—21no—
151C2×C3×D7#4112yes—
152C3×C4×C7—21no—
153C3×C4×C7—21no—
154Dic21—21no—
155Dic21—21no—
156Dic21—21no—
157Dic21—21no—
158C3×Dic7—21no—
159C3×Dic7—21no—
160C3×Dic7—21no—
161C7×Dic3—21no—
162C7×Dic3—21no—
163C7×Dic3—21no—
164#539—211noSylow
165C23×C7—31no—
166C23×C7C22×D311yescharacteristic
167C22×D7C22×D311yescharacteristic
168C7×D4—121no—
169C2×C4×C7—31no—
170C2×C4×C7—31no—
171C2×C4×C7—31no—
172C2×C4×C7—121no—
173C2×C4×C7C22×D311yescharacteristic
174C2×C4×C7C22×D311yescharacteristic
175C22⋊D7—22no—
176C22⋊D7—62no—
177C2×Dic7—31no—
178C2×Dic7—31no—
179C2×Dic7—31no—
180C2×Dic7—31no—
181C2×Dic7—41no—
182C2×Dic7C22×D311yescharacteristic
183C2×Dic7C22×D311yescharacteristic
184C2×Dic7C22×D311yescharacteristic
185C4×D7—41no—
186C4×D7—62no—
187C4×D7—121no—
188Dic14—121no—
189C2×C3×D4—71no—
190C22×C3×C4—71no—
191C22×C3×C4—71no—
192C2×#67—71no—
193C2×#67C2×D711yescharacteristic
194C22×Dic3—71no—
195C22×Dic3—71no—
196C2×C4×D3C2×D711yescharacteristic
197C2×Dic6—71no—
198C3×C4⋊C4—71no—
199C3×C22⋊C4—71no—
200C3×C22⋊C4C2×D711yescharacteristic
201C3×C42—71no—
202#217—71no—
203#217—71no—
204#217—71no—
205#217C2×D711yescharacteristic
206(C2×D3)⋊C4—71no—
207(C2×D3)⋊C4—71no—
208(C2×D3)⋊C4—71no—
209(C2×D3)⋊C4C2×D711yescharacteristic
210C4⋊Dic3—71no—
211C4⋊Dic3—71no—
212C4⋊Dic3C2×D711yescharacteristic
213Dic3⋊C4—71no—
214Dic3⋊C4—71no—
215Dic3⋊C4—71no—
216Dic3⋊C4—71no—
217Dic3⋊C4—71no—
218Dic3⋊C4—71no—
219Dic3⋊C4—71no—
220Dic3⋊C4—71no—
221Dic3⋊C4—71no—
222Dic3⋊C4C2×D711yescharacteristic
223C4×Dic3—71no—
224C2×C3×C7—41no—
225C2×C3×C7C2×#4111yescharacteristic
226C2×C3×C7H(5,𝔽2)11yescharacteristic
227C2×C3×C72−1+411yescharacteristic
228C3×D7—22no—
229D3×C7—41no—
230#126—211no—
231#126—211no—
232#125—211no—
233#125—211no—
234#123—31no—
235#123—211no—
236#123—211no—
237#123—211no—
238C22⋊Q8—211no—
239C22⋊Q8—211no—
240C22⋊Q8—211no—
241#121—211no—
242C4×D4—71no—
243C4×D4—211no—
244C2×C4⋊C4—211no—
245C22×C7—21no—
246C22×C7—21no—
247C22×C7—21no—
248C22×C7—61no—
249C22×C7—61no—
250C22×C7—61no—
251C22×C7C23×D311yescharacteristic
252C2×D7—21no—
253C2×D7—21no—
254C2×D7#23512yes—
255C4×C7—21no—
256C4×C7—21no—
257C4×C7—61no—
258C4×C7—61no—
259C4×C7—61no—
260Dic7—21no—
261Dic7—21no—
262Dic7—21no—
263Dic7—61no—
264Dic7—61no—
265Dic7—61no—
266Dic7—61no—
267C23×C3—71no—
268C23×C3C22×D711yescharacteristic
269C22×D3C22×D711yescharacteristic
270C3×D4—142no—
271C2×C3×C4—71no—
272C2×C3×C4—71no—
273C2×C3×C4—71no—
274C2×C3×C4—281no—
275C2×C3×C4—281no—
276C2×C3×C4C22×D711yescharacteristic
277C2×C3×C4C22×D711yescharacteristic
278#67—41no—
279#67—142no—
280C2×Dic3—71no—
281C2×Dic3—71no—
282C2×Dic3—71no—
283C2×Dic3—71no—
284C2×Dic3—142no—
285C2×Dic3—281no—
286C2×Dic3C22×D711yescharacteristic
287C2×Dic3C22×D711yescharacteristic
288C2×Dic3C22×D711yescharacteristic
289C4×D3—41no—
290Dic6—281no—
291C3×C7#53911yescharacteristic, complemented, Hall
292C2×Q8—211no—
293C2×D4—31no—
294C2×D4—71no—
295C2×D4—211no—
296C22×C4—31no—
297C22×C4—71no—
298C22×C4—71no—
299C22×C4—211no—
300C22×C4—211no—
301C4⋊C4—31no—
302C4⋊C4—31no—
303C4⋊C4—71no—
304C4⋊C4—211no—
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—
315C22⋊C4—31no—
316C22⋊C4—31no—
317C22⋊C4—71no—
318C22⋊C4—211no—
319C22⋊C4—211no—
320C22⋊C4—211no—
321C22⋊C4—211no—
322C22⋊C4—211no—
323C22⋊C4—211no—
324C22⋊C4D3×D711yescharacteristic
325C42—71no—
326C42—211no—
327C2×C7—41no—
328C2×C7—121no—
329C2×C7C2×#23511yescharacteristic
330C2×C7#100411yescharacteristic
331C2×C7#101011yescharacteristic
332D7—22no—
333C22×C3—21no—
334C22×C3—21no—
335C22×C3—21no—
336C22×C3—72no—
337C22×C3—141no—
338C22×C3—141no—
339C22×C3C23×D711yescharacteristic
340C2×D3—21no—
341C2×D3—21no—
342C2×D3—21no—
343C3×C4—21no—
344C3×C4—21no—
345C3×C4—141no—
346C3×C4—141no—
347C3×C4—141no—
348Dic3—21no—
349Dic3—21no—
350Dic3—21no—
351Dic3—141no—
352Dic3—141no—
353Dic3—141no—
354Dic3—141no—
355C23—31no—
356C23—71no—
357C23C2×D3×D711yescharacteristic
358Q8—841no—
359D4—121no—
360D4—142no—
361D4—422no—
362C2×C4—31no—
363C2×C4—31no—
364C2×C4—31no—
365C2×C4—71no—
366C2×C4—71no—
367C2×C4—71no—
368C2×C4—121no—
369C2×C4—211no—
370C2×C4—211no—
371C2×C4—211no—
372C2×C4—211no—
373C2×C4—281no—
374C2×C4—281no—
375C2×C4—422no—
376C2×C4—841no—
377C2×C4C2×D3×D711yescharacteristic
378C2×C4C2×D3×D711yescharacteristic
379C7#556411yescharacteristic, complemented, Sylow
380C2×C3—41no—
381C2×C3—142no—
382C2×C3#654811yescharacteristic
383C2×C3D7×#4111yescharacteristic
384C2×C3#655411yescharacteristic
385D3—41no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—61no—
390C22—61no—
391C22—61no—
392C22—72no—
393C22—141no—
394C22—141no—
395C22C22×D3×D711yescharacteristic
396C4—21no—
397C4—21no—
398C4—61no—
399C4—61no—
400C4—61no—
401C4—141no—
402C4—141no—
403C4—141no—
404C4—421no—
405C4—421no—
406C4—421no—
407C4—421no—
408C3#9035111yescharacteristic, complemented, Sylow
409C2—41no—
410C2—121no—
411C2—142no—
412C2D7×#23511yescharacteristic
413C2#1062156711yescharacteristic
414C2#1062157211yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC3×C7×C22⋊C4C22
prime radical#10621111C2
hypercenterC22⋊C4D3×D7
hypocenterC3×C7#539
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×#883C2semidirect
3. C3×(C22⋊D7)⋊C4C2semidirect
4. #10621136C2semidirect
6. #10621110C2semidirect
7. #10621101C2semidirect
8. C4×#4245C2semidirect
9. #10620992C2semidirect
10. #10620976C2semidirect
11. D7×Dic3⋊C4C2semidirect
12. #10620950C2semidirect
13. #10620944C2semidirect
14. #10620934C2semidirect
15. #10620914C2semidirect
63. (C22⋊D7)⋊C4D3semidirect
137. #883D7semidirect
291. C3×C7#539semidirect
379. C7#5564semidirect
408. C3#90351semidirect

Maximal subgroups

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

subgroupindex
#106213622
C7×#8832
C3×(C22⋊D7)⋊C42
#106211362
#106211112
#106211102
#106211012
C4×#42452
#106209922
#106209762
D7×Dic3⋊C42
#106209502
#106209442
#106209342
#106209142
#903513
#55647