Finite Group Databasebeta

← #13059899

Subgroups — #13059899

Order 1344.

668 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 136 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#66728C211yesmaximal, characteristic, complemented
2C2×#66681C211yesmaximal, characteristic, complemented
3C7×#930C211yesmaximal, characteristic, complemented
4C3×#6493C211yesmaximal, characteristic, complemented
5#10621139C212yesmaximal, complemented
6#10621129C212yesmaximal, complemented
7#10621038C212yesmaximal, complemented
8#10621028C211yesmaximal, characteristic, complemented
9#10621027C211yesmaximal, characteristic, complemented
10#10621024C212yesmaximal, complemented
11#10620909C211yesmaximal, characteristic, complemented
12#90298—31nomaximal, Hall
13C22×D3×D7C2212yes—
14C22×C3×C4×C7C2211yescharacteristic, complemented
15C2×#4268C2211yescharacteristic
16C2×#4268C2211yescharacteristic, complemented
17#66728—24no—
18C2×D84C2211yescharacteristic
19C2×C4×D21C2211yescharacteristic
20C2×C4×D21C2211yescharacteristic, complemented
21C2×Dic42C2211yescharacteristic, complemented
22C2×C7×#67C2212yescomplemented
23C2×C7×D12C2211yescharacteristic
24C2×C3×C22⋊D7C2212yescomplemented
25C2×C3×D28C2211yescharacteristic
26C2×#4247C2212yescomplemented
27C2×#4246C2212yescomplemented
28C2×#4245C2212yes—
29#66681—24no—
30C7×(C2×D3)⋊C4C2212yescomplemented
31C7×C4⋊Dic3C2211yescharacteristic, complemented
32C3×(C2×D7)⋊C4C2212yescomplemented
33C3×C4⋊Dic7C2211yescharacteristic, complemented
34#66575C2212yes—
35(C2×D3)⋊Dic7C2212yescomplemented
36(C2×D7)⋊Dic3C2212yescomplemented
37Dic3×Dic7C2212yescomplemented
38C2×D4×D7—31no—
39C2×#1157—31no—
40C7×#121—31no—
41#6503—32no—
42#6501—31no—
43#6500—32no—
44#6493D311yescharacteristic, complemented
45#6458—31no—
46#6455—31no—
47#6448—32no—
48#6445—32no—
49#5607—71nomaximal, Hall
50C23×C3×C7C2311yescharacteristic
51C22×D21C2311yescharacteristic
52C22×D21C2311yescharacteristic
53C22×D3×C7C2312yes—
54C22×C3×D7C2312yes—
55C2×D3×D7—24no—
56C2×D3×D7—42no—
57C2×D3×D7—42no—
58C2×C3×C4×C7—22no—
59C2×C3×C4×C7C2311yescharacteristic
60C2×C3×C4×C7C2311yescharacteristic
61#4268—22no—
62#4268—22no—
63C2×Dic21C2311yescharacteristic
64C2×Dic21C2311yescharacteristic
65D84—22no—
66C4×D21—22no—
67C4×D21D414yescomplemented
68Dic42—22no—
69C7×#67—42no—
70C2×C7×Dic3C2312yes—
71C7×D12—22no—
72C3×C22⋊D7—42no—
73C2×C3×Dic7C2312yes—
74C3×D28—22no—
75#4247—42no—
76#4246—42no—
77#4245—24no—
78C23×D7—32no—
79C2×C7×D4—32no—
80C2×C7×D4—31no—
81C22×C4×C7C2×D311yescharacteristic, complemented
82C2×C22⋊D7—31no—
83C2×C22⋊D7—31no—
84C2×C22⋊D7—32no—
85C2×C22⋊D7—32no—
86C2×C22⋊D7C2×D312yescomplemented
87D4×D7—64no—
88#1157—64no—
89C2×D28—31no—
90C2×D28—32no—
91C2×D28C2×D311yescharacteristic
92C2×C4×D7—31no—
93C2×C4×D7—31no—
94C2×Dic14—31no—
95C7×C4⋊C4—31no—
96C7×C22⋊C4—32no—
97C22⋊Dic7—32no—
98(C2×D7)⋊C4—32no—
99(C2×D7)⋊C4—32no—
100(C2×D7)⋊C4C2×D312yescomplemented
101C4⋊Dic7C2×D311yescharacteristic, complemented
102C4×Dic7—32no—
103C2×D3×D4—71no—
104C2×#235—71no—
105C3×#121—71no—
106#940—72no—
107#938—71no—
108#937—72no—
109#930D711yescharacteristic, complemented
110#895—71no—
111#892—71no—
112#885—72no—
113#882—72no—
114C22×C3×C7—21no—
115C22×C3×C7—21no—
116C22×C3×C7—21no—
117C22×C3×C7C2411yescharacteristic
118C2×D21—21no—
119C2×D21—21no—
120C2×D21—21no—
121C2×D21—21no—
122C2×D21—21no—
123C2×D21C2×D412yes—
124C2×D3×C7—22no—
125C2×D3×C7—22no—
126C2×D3×C7—22no—
127C2×C3×D7—22no—
128C2×C3×D7—22no—
129C2×C3×D7—22no—
130D3×D7—48no—
131C3×C4×C7—21no—
132C3×C4×C7C2×D412yes—
133Dic21—21no—
134Dic21C2×D412yes—
135C3×Dic7—22no—
136C7×Dic3—22no—
137#535—211noSylow
138C23×C7—32no—
139C23×C7C22×D311yescharacteristic
140C22×D7—31no—
141C22×D7—31no—
142C22×D7—64no—
143C22×D7—122no—
144C22×D7—122no—
145C22×D7C22×D312yes—
146C7×D4—62no—
147C7×D4—122no—
148C2×C4×C7—22no—
149C2×C4×C7—32no—
150C2×C4×C7C22×D311yescharacteristic
151C2×C4×C7C22×D311yescharacteristic
152C22⋊D7—42no—
153C22⋊D7—64no—
154C22⋊D7—62no—
155C22⋊D7—62no—
156C22⋊D7—122no—
157C2×Dic7—31no—
158C2×Dic7—31no—
159C2×Dic7C22×D312yes—
160D28—22no—
161D28—62no—
162D28—122no—
163C4×D7—34no—
164C4×D7—62no—
165Dic14—62no—
166C23×D3—72no—
167C2×C3×D4—72no—
168C2×C3×D4—71no—
169C22×C3×C4C2×D711yescharacteristic, complemented
170C2×#67—71no—
171C2×#67—71no—
172C2×#67—72no—
173C2×#67—72no—
174C2×#67C2×D712yescomplemented
175D3×D4—144no—
176#235—144no—
177C2×D12—71no—
178C2×D12—72no—
179C2×D12C2×D711yescharacteristic
180C2×C4×D3—71no—
181C2×C4×D3—71no—
182C2×Dic6—71no—
183C3×C4⋊C4—71no—
184C3×C22⋊C4—72no—
185#217—72no—
186(C2×D3)⋊C4—72no—
187(C2×D3)⋊C4—72no—
188(C2×D3)⋊C4C2×D712yescomplemented
189C4⋊Dic3C2×D711yescharacteristic, complemented
190C4×Dic3—72no—
191C2×C3×C7—41no—
192C2×C3×C7C22×D411yescharacteristic
193C2×C3×C7H(5,𝔽2)11yescharacteristic
194C2×C3×C7H(5,𝔽2)11yescharacteristic
195D21—22no—
196D21—41no—
197C3×D7—42no—
198D3×C7—42no—
199C2×#41—211no—
200C22×D4—211no—
201#127—212no—
202#124—212no—
203#121—31no—
204#121—71no—
205#121—211no—
206#121—211no—
207#120—212no—
208#120—212no—
209#117—211no—
210C22×C7—21no—
211C22×C7—21no—
212C22×C7—21no—
213C22×C7—62no—
214C22×C7—62no—
215C22×C7—62no—
216C22×C7C23×D311yescharacteristic
217C2×D7—22no—
218C2×D7—22no—
219C2×D7—22no—
220C2×D7—32no—
221C2×D7—61no—
222C2×D7—61no—
223C2×D7—61no—
224C2×D7—61no—
225C2×D7—61no—
226C2×D7—128no—
227C4×C7—21no—
228C4×C7—62no—
229C4×C7D3×D412yes—
230Dic7—22no—
231Dic7—32no—
232Dic7—61no—
233C23×C3—72no—
234C23×C3C22×D711yescharacteristic
235C22×D3—71no—
236C22×D3—71no—
237C22×D3—144no—
238C22×D3—282no—
239C22×D3—282no—
240C22×D3C22×D712yes—
241C3×D4—142no—
242C3×D4—282no—
243C2×C3×C4—22no—
244C2×C3×C4—72no—
245C2×C3×C4C22×D711yescharacteristic
246C2×C3×C4C22×D711yescharacteristic
247#67—42no—
248#67—144no—
249#67—142no—
250#67—142no—
251#67—282no—
252C2×Dic3—71no—
253C2×Dic3—71no—
254C2×Dic3C22×D712yes—
255D12—22no—
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—32no—
266C2×D4—31no—
267C2×D4—72no—
268C2×D4—71no—
269C2×D4—212no—
270C2×D4—212no—
271C2×D4—212no—
272C2×D4—211no—
273C2×D4—211no—
274C2×D4—211no—
275C2×D4—424no—
276C22×C4—211no—
277C22×C4—211no—
278C22×C4D3×D711yescharacteristic, complemented
279C4⋊C4—31no—
280C4⋊C4—71no—
281C22⋊C4—32no—
282C22⋊C4—72no—
283C22⋊C4—212no—
284C22⋊C4—212no—
285C22⋊C4—212no—
286C42—212no—
287C2×C7—41no—
288C2×C7—122no—
289C2×C7C2×D3×D411yescharacteristic
290C2×C7#100411yescharacteristic
291C2×C7#100911yescharacteristic
292D7—42no—
293D7—62no—
294D7—121no—
295C22×C3—21no—
296C22×C3—21no—
297C22×C3—21no—
298C22×C3—142no—
299C22×C3—142no—
300C22×C3—142no—
301C22×C3C23×D711yescharacteristic
302C2×D3—22no—
303C2×D3—22no—
304C2×D3—22no—
305C2×D3—72no—
306C2×D3—141no—
307C2×D3—141no—
308C2×D3—141no—
309C2×D3—141no—
310C2×D3—141no—
311C2×D3—288no—
312C3×C4—21no—
313C3×C4—142no—
314C3×C4D4×D712yes—
315Dic3—22no—
316Dic3—72no—
317Dic3—141no—
318C23—32no—
319C23—72no—
320C23—211no—
321C23—211no—
322C23—424no—
323C23—842no—
324C23—842no—
325C23C2×D3×D711yescharacteristic
326Q8—422no—
327D4—62no—
328D4—122no—
329D4—142no—
330D4—282no—
331D4—422no—
332D4—422no—
333D4—424no—
334D4—422no—
335D4—842no—
336D4—842no—
337C2×C4—22no—
338C2×C4—32no—
339C2×C4—72no—
340C2×C4—211no—
341C2×C4—214no—
342C2×C4—211no—
343C2×C4—422no—
344C2×C4C2×D3×D711yescharacteristic
345C2×C4C2×D3×D711yescharacteristic
346C7#560711yescharacteristic, complemented, Sylow
347C2×C3—41no—
348C2×C3—282no—
349C2×C3C2×D4×D711yescharacteristic
350C2×C3#654811yescharacteristic
351C2×C3#655311yescharacteristic
352D3—42no—
353D3—142no—
354D3—281no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—62no—
359C22—62no—
360C22—62no—
361C22—142no—
362C22—142no—
363C22—142no—
364C22—212no—
365C22—421no—
366C22—421no—
367C22—421no—
368C22—421no—
369C22—421no—
370C22—848no—
371C22C22×D3×D711yescharacteristic
372C4—21no—
373C4—62no—
374C4—142no—
375C4—212no—
376C4—421no—
377C4#6669412yes—
378C3#9029811yescharacteristic, complemented, Sylow
379C2—41no—
380C2—122no—
381C2—282no—
382C2—422no—
383C2—841no—
384C2#1062156411yescharacteristic
385C2#1062156511yescharacteristic
386C2C2×#6669411yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC22×C3×C4×C7C22
hypercenterC22×C4D3×D7
hypocenterC3×C7#535
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. C2×#66728C2semidirect
2. C2×#66681C2semidirect
3. C7×#930C2semidirect
4. C3×#6493C2semidirect
5. #10621139C2semidirect
6. #10621129C2semidirect
7. #10621038C2semidirect
8. #10621028C2semidirect
9. #10621027C2semidirect
10. #10621024C2semidirect
11. #10620909C2semidirect
14. C22×C3×C4×C7C22semidirect
16. C2×#4268C22semidirect
20. C2×C4×D21C22semidirect
21. C2×Dic42C22semidirect
22. C2×C7×#67C22semidirect
24. C2×C3×C22⋊D7C22semidirect
26. C2×#4247C22semidirect
27. C2×#4246C22semidirect
30. C7×(C2×D3)⋊C4C22semidirect
31. C7×C4⋊Dic3C22semidirect
32. C3×(C2×D7)⋊C4C22semidirect
33. C3×C4⋊Dic7C22semidirect
35. (C2×D3)⋊Dic7C22semidirect
36. (C2×D7)⋊Dic3C22semidirect
37. Dic3×Dic7C22semidirect
44. #6493D3semidirect
67. C4×D21D4semidirect
81. C22×C4×C7C2×D3semidirect
86. C2×C22⋊D7C2×D3semidirect
100. (C2×D7)⋊C4C2×D3semidirect
101. C4⋊Dic7C2×D3semidirect
109. #930D7semidirect
169. C22×C3×C4C2×D7semidirect
174. C2×#67C2×D7semidirect
188. (C2×D3)⋊C4C2×D7semidirect
189. C4⋊Dic3C2×D7semidirect
261. C3×C7#535semidirect
278. C22×C4D3×D7semidirect
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×#667282
C2×#666812
C7×#9302
C3×#64932
#106211392
#106211292
#106210382
#106210282
#106210272
#106210242
#106209092
#902983
#56077