Finite Group Databasebeta

← #13061699

Subgroups — #13061699

Order 1344.

668 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×D21C211yesmaximal, characteristic, complemented
2C2×#66728C211yesmaximal, characteristic, complemented
3C3×C7×#121C211yesmaximal, characteristic, complemented
4#10621423C212yesmaximal, complemented
5#10621421C211yesmaximal, characteristic, complemented
6#10621420C212yesmaximal, complemented
7#10621413C211yesmaximal, characteristic, complemented
8#10621378C211yesmaximal, characteristic, complemented
9#10621375C211yesmaximal, characteristic, complemented
10#10621368C212yesmaximal, complemented
11#10621365C212yesmaximal, complemented
12#90298—31nomaximal, Hall
13C23×D21C2212yes—
14C2×C3×C7×D4C2211yescharacteristic
15C2×C3×C7×D4C2212yescomplemented
16C22×C3×C4×C7C2211yescharacteristic, complemented
17C2×#4268C2212yes—
18C2×#4268C2211yescharacteristic
19C2×#4268C2211yescharacteristic, complemented
20C2×#4268C2212yescomplemented
21C2×#4268C2212yescomplemented
22D4×D21—24no—
23#66728—24no—
24C2×D84C2211yescharacteristic
25C2×D84C2211yescharacteristic
26C2×D84C2212yescomplemented
27C2×C4×D21C2211yescharacteristic
28C2×C4×D21C2211yescharacteristic, complemented
29C2×Dic42C2211yescharacteristic, complemented
30C3×C7×C4⋊C4C2211yescharacteristic, complemented
31C3×C7×C22⋊C4C2212yescomplemented
32#66636C2212yescomplemented
33#66631C2212yes—
34#66631C2212yescomplemented
35#66631C2212yescomplemented
36C4⋊Dic21C2211yescharacteristic, complemented
37C4×Dic21C2212yescomplemented
38C2×D4×D7—31no—
39C2×#1157—31no—
40C7×#121D311yescharacteristic, complemented
41#6503—32no—
42#6501—31no—
43#6500—32no—
44#6493—31no—
45#6458—31no—
46#6455—31no—
47#6448—32no—
48#6445—32no—
49#5607—71nomaximal, Hall
50C23×C3×C7C2312yes—
51C23×C3×C7C2311yescharacteristic
52C22×D21—24no—
53C22×D21—42no—
54C22×D21—42no—
55C22×D21C2312yes—
56C22×D21C2311yescharacteristic
57C22×D21C2311yescharacteristic
58C3×C7×D4—22no—
59C3×C7×D4—42no—
60C2×C3×C4×C7—22no—
61C2×C3×C4×C7C2312yes—
62C2×C3×C4×C7C2311yescharacteristic
63C2×C3×C4×C7C2311yescharacteristic
64#4268—22no—
65#4268—22no—
66#4268—24no—
67#4268—42no—
68#4268—42no—
69C2×Dic21C2312yes—
70C2×Dic21C2311yescharacteristic
71C2×Dic21C2311yescharacteristic
72D84—22no—
73D84—22no—
74D84—42no—
75C4×D21—22no—
76C4×D21D414yescomplemented
77Dic42—22no—
78C23×D7—32no—
79C2×C7×D4C2×D311yescharacteristic
80C2×C7×D4C2×D312yescomplemented
81C22×C4×C7C2×D311yescharacteristic, complemented
82C2×C22⋊D7—32no—
83C2×C22⋊D7—31no—
84C2×C22⋊D7—31no—
85C2×C22⋊D7—32no—
86C2×C22⋊D7—32no—
87D4×D7—64no—
88#1157—64no—
89C2×D28—31no—
90C2×D28—31no—
91C2×D28—32no—
92C2×C4×D7—31no—
93C2×C4×D7—31no—
94C2×Dic14—31no—
95C7×C4⋊C4C2×D311yescharacteristic, complemented
96C7×C22⋊C4C2×D312yescomplemented
97C22⋊Dic7—32no—
98(C2×D7)⋊C4—32no—
99(C2×D7)⋊C4—32no—
100(C2×D7)⋊C4—32no—
101C4⋊Dic7—31no—
102C4×Dic7—32no—
103C2×D3×D4—71no—
104C2×#235—71no—
105C3×#121D711yescharacteristic, complemented
106#940—72no—
107#938—71no—
108#937—72no—
109#930—71no—
110#895—71no—
111#892—71no—
112#885—72no—
113#882—72no—
114C22×C3×C7—21no—
115C22×C3×C7—22no—
116C22×C3×C7—21no—
117C22×C3×C7—22no—
118C22×C3×C7—21no—
119C22×C3×C7—22no—
120C22×C3×C7C2411yescharacteristic
121C2×D21—21no—
122C2×D21—21no—
123C2×D21—21no—
124C2×D21—22no—
125C2×D21—21no—
126C2×D21—22no—
127C2×D21—21no—
128C2×D21—22no—
129C2×D21—48no—
130C2×D21C2×D412yes—
131C3×C4×C7—21no—
132C3×C4×C7—22no—
133C3×C4×C7C2×D412yes—
134Dic21—21no—
135Dic21—22no—
136Dic21C2×D412yes—
137#535—211noSylow
138C23×C7C22×D312yes—
139C23×C7C22×D311yescharacteristic
140C22×D7—31no—
141C22×D7—32no—
142C22×D7—31no—
143C22×D7—64no—
144C22×D7—122no—
145C22×D7—122no—
146C7×D4—22no—
147C7×D4—42no—
148C2×C4×C7—22no—
149C2×C4×C7C22×D312yes—
150C2×C4×C7C22×D311yescharacteristic
151C2×C4×C7C22×D311yescharacteristic
152C22⋊D7—62no—
153C22⋊D7—64no—
154C22⋊D7—62no—
155C22⋊D7—122no—
156C22⋊D7—122no—
157C2×Dic7—31no—
158C2×Dic7—32no—
159C2×Dic7—31no—
160D28—62no—
161D28—62no—
162D28—122no—
163C4×D7—34no—
164C4×D7—62no—
165Dic14—62no—
166C23×D3—72no—
167C2×C3×D4C2×D711yescharacteristic
168C2×C3×D4C2×D712yescomplemented
169C22×C3×C4C2×D711yescharacteristic, complemented
170C2×#67—72no—
171C2×#67—71no—
172C2×#67—71no—
173C2×#67—72no—
174C2×#67—72no—
175D3×D4—144no—
176#235—144no—
177C2×D12—71no—
178C2×D12—71no—
179C2×D12—72no—
180C2×C4×D3—71no—
181C2×C4×D3—71no—
182C2×Dic6—71no—
183C3×C4⋊C4C2×D711yescharacteristic, complemented
184C3×C22⋊C4C2×D712yescomplemented
185#217—72no—
186(C2×D3)⋊C4—72no—
187(C2×D3)⋊C4—72no—
188(C2×D3)⋊C4—72no—
189C4⋊Dic3—71no—
190C4×Dic3—72no—
191C2×C3×C7—42no—
192C2×C3×C7—41no—
193C2×C3×C7C22×D411yescharacteristic
194C2×C3×C7H(5,𝔽2)11yescharacteristic
195C2×C3×C7H(5,𝔽2)11yescharacteristic
196D21—22no—
197D21—41no—
198D21—42no—
199C2×#41—211no—
200C22×D4—211no—
201#127—212no—
202#124—212no—
203#121—211no—
204#121—211no—
205#121—211no—
206#121D2111yescharacteristic, complemented
207#120—212no—
208#120—212no—
209#117—211no—
210C22×C7—21no—
211C22×C7—22no—
212C22×C7—21no—
213C22×C7—22no—
214C22×C7—21no—
215C22×C7—22no—
216C22×C7C23×D311yescharacteristic
217C2×D7—32no—
218C2×D7—61no—
219C2×D7—61no—
220C2×D7—61no—
221C2×D7—62no—
222C2×D7—61no—
223C2×D7—62no—
224C2×D7—61no—
225C2×D7—62no—
226C2×D7—128no—
227C4×C7—21no—
228C4×C7—22no—
229C4×C7D3×D412yes—
230Dic7—32no—
231Dic7—61no—
232Dic7—62no—
233C23×C3C22×D712yes—
234C23×C3C22×D711yescharacteristic
235C22×D3—71no—
236C22×D3—72no—
237C22×D3—71no—
238C22×D3—144no—
239C22×D3—282no—
240C22×D3—282no—
241C3×D4—22no—
242C3×D4—42no—
243C2×C3×C4—22no—
244C2×C3×C4C22×D712yes—
245C2×C3×C4C22×D711yescharacteristic
246C2×C3×C4C22×D711yescharacteristic
247#67—142no—
248#67—144no—
249#67—142no—
250#67—282no—
251#67—282no—
252C2×Dic3—71no—
253C2×Dic3—72no—
254C2×Dic3—71no—
255D12—142no—
256D12—142no—
257D12—282no—
258C4×D3—74no—
259C4×D3—142no—
260Dic6—142no—
261C3×C7#53511yescharacteristic, complemented, Hall
262C24—212no—
263#41—424no—
264C2×Q8—211no—
265C2×D4—211no—
266C2×D4—212no—
267C2×D4—211no—
268C2×D4—212no—
269C2×D4—212no—
270C2×D4—211no—
271C2×D4—211no—
272C2×D4—212no—
273C2×D4—424no—
274C2×D4C2×D2111yescharacteristic
275C2×D4C2×D2112yescomplemented
276C22×C4—211no—
277C22×C4—211no—
278C22×C4C2×D2111yescharacteristic, complemented
279C4⋊C4—211no—
280C4⋊C4C2×D2111yescharacteristic, complemented
281C22⋊C4—212no—
282C22⋊C4—212no—
283C22⋊C4—212no—
284C22⋊C4—212no—
285C22⋊C4C2×D2112yescomplemented
286C42—212no—
287C2×C7—42no—
288C2×C7—41no—
289C2×C7C2×D3×D411yescharacteristic
290C2×C7#100411yescharacteristic
291C2×C7#100911yescharacteristic
292D7—62no—
293D7—122no—
294D7—121no—
295C22×C3—21no—
296C22×C3—22no—
297C22×C3—21no—
298C22×C3—22no—
299C22×C3—21no—
300C22×C3—22no—
301C22×C3C23×D711yescharacteristic
302C2×D3—72no—
303C2×D3—141no—
304C2×D3—141no—
305C2×D3—141no—
306C2×D3—142no—
307C2×D3—141no—
308C2×D3—142no—
309C2×D3—141no—
310C2×D3—142no—
311C2×D3—288no—
312C3×C4—21no—
313C3×C4—22no—
314C3×C4D4×D712yes—
315Dic3—72no—
316Dic3—141no—
317Dic3—142no—
318C23—211no—
319C23—212no—
320C23—211no—
321C23—424no—
322C23—842no—
323C23—842no—
324C23C22×D2112yes—
325C23C22×D2111yescharacteristic
326Q8—422no—
327D4—22no—
328D4—42no—
329D4—422no—
330D4—424no—
331D4—422no—
332D4—422no—
333D4—422no—
334D4—842no—
335D4—842no—
336D4—842no—
337C2×C4—22no—
338C2×C4—211no—
339C2×C4—214no—
340C2×C4—212no—
341C2×C4—211no—
342C2×C4—422no—
343C2×C4C22×D2112yes—
344C2×C4C22×D2111yescharacteristic
345C2×C4C22×D2111yescharacteristic
346C7#560711yescharacteristic, complemented, Sylow
347C2×C3—42no—
348C2×C3—41no—
349C2×C3C2×D4×D711yescharacteristic
350C2×C3#654811yescharacteristic
351C2×C3#655311yescharacteristic
352D3—142no—
353D3—282no—
354D3—281no—
355C22—21no—
356C22—22no—
357C22—21no—
358C22—22no—
359C22—21no—
360C22—22no—
361C22—212no—
362C22—421no—
363C22—421no—
364C22—421no—
365C22—422no—
366C22—421no—
367C22—422no—
368C22—421no—
369C22—422no—
370C22—848no—
371C22C23×D2111yescharacteristic
372C4—21no—
373C4—22no—
374C4—212no—
375C4—421no—
376C4—422no—
377C4D4×D2112yes—
378C3#9029811yescharacteristic, complemented, Sylow
379C2—42no—
380C2—41no—
381C2—422no—
382C2—842no—
383C2—841no—
384C2C2×D4×D2111yescharacteristic
385C2#1062165711yescharacteristic
386C2#1062166211yescharacteristic
subgroupG/H
center, FrattiniC22C23×D21
commutator, socle, cosocleC22×C3×C7C24
FittingC3×C7×#121C2
hypercenter#121D21
hypocenterC3×C7#535
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×D4×D21C2semidirect
2. C2×#66728C2semidirect
3. C3×C7×#121C2semidirect
4. #10621423C2semidirect
5. #10621421C2semidirect
6. #10621420C2semidirect
7. #10621413C2semidirect
8. #10621378C2semidirect
9. #10621375C2semidirect
10. #10621368C2semidirect
11. #10621365C2semidirect
15. C2×C3×C7×D4C22semidirect
16. C22×C3×C4×C7C22semidirect
19. C2×#4268C22semidirect
20. C2×#4268C22semidirect
21. C2×#4268C22semidirect
26. C2×D84C22semidirect
28. C2×C4×D21C22semidirect
29. C2×Dic42C22semidirect
30. C3×C7×C4⋊C4C22semidirect
31. C3×C7×C22⋊C4C22semidirect
32. #66636C22semidirect
34. #66631C22semidirect
35. #66631C22semidirect
36. C4⋊Dic21C22semidirect
37. C4×Dic21C22semidirect
40. C7×#121D3semidirect
76. C4×D21D4semidirect
80. C2×C7×D4C2×D3semidirect
81. C22×C4×C7C2×D3semidirect
95. C7×C4⋊C4C2×D3semidirect
96. C7×C22⋊C4C2×D3semidirect
105. C3×#121D7semidirect
168. C2×C3×D4C2×D7semidirect
169. C22×C3×C4C2×D7semidirect
183. C3×C4⋊C4C2×D7semidirect
184. C3×C22⋊C4C2×D7semidirect
206. #121D21semidirect
261. C3×C7#535semidirect
275. C2×D4C2×D21semidirect
278. C22×C4C2×D21semidirect
280. C4⋊C4C2×D21semidirect
285. C22⋊C4C2×D21semidirect
346. C7#5607semidirect
378. C3#90298semidirect

Maximal subgroups

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

subgroupindex
C2×D4×D212
C2×#667282
C3×C7×#1212
#106214232
#106214212
#106214202
#106214132
#106213782
#106213752
#106213682
#106213652
#902983
#56077