Finite Group Databasebeta

← #13058069

Subgroups — #13058069

Order 1344.

660 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 660 conjugacy classes of subgroups — too many to draw. Only its 144 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
1C22×D3×Dic7C211yesmaximal, characteristic, complemented
2C2×#66684C211yesmaximal, characteristic, complemented
3#10621360C211yesmaximal, characteristic, complemented
4C7×#882C211yesmaximal, characteristic, complemented
5C3×#6449C211yesmaximal, characteristic, complemented
6#10621130C211yesmaximal, characteristic, complemented
7#10621101C211yesmaximal, characteristic, complemented
8#10621015C212yesmaximal, complemented
9#10621004C212yesmaximal, complemented
10#10620998C212yesmaximal, complemented
11#10620997C212yesmaximal, complemented
12#90277—31nomaximal, Hall
13C23×D3×C7C2211yescharacteristic
14C22×Dic21C2211yescharacteristic
15C2×C7×#67C2211yescharacteristic
16C2×C7×D12C2212yes—
17C2×C3×C22⋊D7C2211yescharacteristic
18C22×C3×Dic7C2211yescharacteristic
19C2×#4248C2211yescharacteristic, complemented
20C2×#4245C2212yes—
21C2×#4245C2211yescharacteristic, complemented
22C2×D3×Dic7—24no—
23C2×D3×Dic7C2212yes—
24C2×D3×Dic7C2212yes—
25C2×D3×Dic7C2211yescharacteristic, complemented
26#66684—24no—
27C2×Dic3×D7C2211yescharacteristic, complemented
28C3×C7×C22⋊C4C2211yescharacteristic
29#66636C2211yescharacteristic
30#66629C2212yescomplemented
31C4×Dic21C2212yescomplemented
32C7×(C2×D3)⋊C4C2212yes—
33C3×(C2×D7)⋊C4C2212yescomplemented
34C3×C4⋊Dic7C2212yescomplemented
35#66577C2212yescomplemented
36(C2×D3)⋊Dic7C2212yes—
37(C2×D3)⋊Dic7C2212yescomplemented
38C23×Dic7—31no—
39C2×#1159—31no—
40C7×#120—31no—
41#6516—31no—
42#6502—32no—
43#6498—31no—
44D4×Dic7—32no—
45#6449D311yescharacteristic, complemented
46(C22⋊D7)⋊C4—32no—
47#6441—32no—
48#6440—31no—
49#5656—71nomaximal, Hall
50C23×C3×C7C2311yescharacteristic
51C22×D3×C7—22no—
52C22×D3×C7—24no—
53C22×D3×C7C2311yescharacteristic
54C22×D3×C7C2312yes—
55C22×C3×D7C2311yescharacteristic
56C2×C3×C4×C7C2312yes—
57C2×Dic21D414yescomplemented
58C2×Dic21C2312yes—
59C2×Dic21C2311yescharacteristic
60C2×Dic21C2311yescharacteristic
61C7×#67—22no—
62C2×C7×Dic3C2311yescharacteristic
63C7×D12—24no—
64C3×C22⋊D7—22no—
65C2×C3×Dic7—22no—
66C2×C3×Dic7C2311yescharacteristic
67C2×C3×Dic7C2312yes—
68#4248—22no—
69#4245—24no—
70#4245—22no—
71D3×Dic7—22no—
72D3×Dic7—24no—
73D3×Dic7—24no—
74Dic3×D7—22no—
75C24×C7—31no—
76C2×C7×D4—31no—
77C2×C7×D4—32no—
78C2×C22⋊D7—32no—
79C2×C22⋊D7—31no—
80C2×C22⋊D7C2×D311yescharacteristic
81C22×Dic7—31no—
82C22×Dic7—32no—
83C22×Dic7—32no—
84C22×Dic7—31no—
85C22×Dic7—64no—
86C22×Dic7C2×D311yescharacteristic
87#1159—64no—
88C2×C4×D7—31no—
89C2×Dic14—31no—
90C7×C22⋊C4—32no—
91C7×C22⋊C4C2×D311yescharacteristic
92C22⋊Dic7—31no—
93C22⋊Dic7—32no—
94C22⋊Dic7—32no—
95(C2×D7)⋊C4C2×D312yescomplemented
96C4⋊Dic7C2×D312yescomplemented
97Dic7⋊C4—32no—
98Dic7⋊C4—32no—
99C4×Dic7—32no—
100C2×#237—71no—
101C22×C4×D3—71no—
102C3×#123—71no—
103#937—71no—
104#929—71no—
105#896—72no—
106#893—72no—
107#884—72no—
108#882D711yescharacteristic, complemented
109#67⋊C4—72no—
110#877—71no—
111C22×C3×C7—22no—
112C22×C3×C7C2×D412yes—
113C22×C3×C7C2411yescharacteristic
114C2×D3×C7—28no—
115C2×D3×C7—22no—
116C2×D3×C7—22no—
117C2×D3×C7—21no—
118C2×D3×C7—22no—
119C2×D3×C7#4114yescomplemented
120C2×C3×D7—21no—
121C2×C3×D7—22no—
122C3×C4×C7—22no—
123Dic21—22no—
124Dic21C2×D412yes—
125Dic21C2×D412yes—
126C3×Dic7—22no—
127C3×Dic7—21no—
128C7×Dic3—21no—
129#525—211noSylow
130C23×C7—32no—
131C23×C7—31no—
132C23×C7—64no—
133C23×C7—62no—
134C23×C7C22×D311yescharacteristic
135C22×D7C22×D311yescharacteristic
136C7×D4—64no—
137C7×D4—62no—
138C2×C4×C7—31no—
139C2×C4×C7C22×D312yes—
140C22⋊D7—22no—
141C22⋊D7—62no—
142C22⋊D7—64no—
143C2×Dic7—22no—
144C2×Dic7—34no—
145C2×Dic7—31no—
146C2×Dic7—31no—
147C2×Dic7—32no—
148C2×Dic7—62no—
149C2×Dic7—64no—
150C2×Dic7—64no—
151C2×Dic7C22×D312yes—
152C2×Dic7C22×D311yescharacteristic
153C4×D7—62no—
154Dic14—62no—
155C23×D3C2×D711yescharacteristic
156C2×C3×D4—71no—
157C22×C3×C4—71no—
158C2×#67—72no—
159C2×#67—71no—
160C2×#67C2×D711yescharacteristic
161C22×Dic3—71no—
162C22×Dic3—71no—
163#237—144no—
164C2×D12C2×D712yes—
165C2×C4×D3—72no—
166C2×C4×D3—72no—
167C2×C4×D3—71no—
168C2×C4×D3—144no—
169C2×Dic6—71no—
170C3×C4⋊C4—72no—
171C3×C22⋊C4—72no—
172C3×C22⋊C4C2×D711yescharacteristic
173#217—71no—
174(C2×D3)⋊C4—72no—
175(C2×D3)⋊C4—72no—
176(C2×D3)⋊C4C2×D712yes—
177Dic3⋊C4—72no—
178Dic3⋊C4—72no—
179C4×Dic3—72no—
180C2×C3×C7—22no—
181C2×C3×C7C22×D411yescharacteristic
182C2×C3×C7C2×#4112yes—
183C3×D7—41no—
184D3×C7—24no—
185D3×C7—41no—
186C2×#41—211no—
187C23×C4—211no—
188#123—71no—
189#123—211no—
190C22⋊Q8—212no—
191#121—212no—
192#120—31no—
193#120—211no—
194C4×D4—212no—
195C4×D4—212no—
196#117—211no—
197C22×C7—22no—
198C22×C7—34no—
199C22×C7—68no—
200C22×C7—62no—
201C22×C7—62no—
202C22×C7—61no—
203C22×C7—62no—
204C22×C7D3×D412yes—
205C22×C7C23×D311yescharacteristic
206C2×D7—21no—
207C2×D7—22no—
208C4×C7—22no—
209C4×C7—61no—
210Dic7—21no—
211Dic7—22no—
212Dic7—32no—
213Dic7—32no—
214Dic7—62no—
215C23×C3—71no—
216C23×C3C22×D711yescharacteristic
217C22×D3—22no—
218C22×D3—24no—
219C22×D3C22×D711yescharacteristic
220C22×D3C22×D712yes—
221C3×D4—142no—
222C2×C3×C4—72no—
223C2×C3×C4—71no—
224C2×C3×C4—142no—
225C2×C3×C4C22×D712yes—
226#67—22no—
227#67—142no—
228#67—144no—
229C2×Dic3—74no—
230C2×Dic3—71no—
231C2×Dic3—71no—
232C2×Dic3—72no—
233C2×Dic3—142no—
234C2×Dic3C22×D711yescharacteristic
235D12—24no—
236C4×D3—144no—
237C4×D3—144no—
238C4×D3—142no—
239Dic6—142no—
240C3×C7#52511yescharacteristic, complemented, Hall
241C24—31no—
242#41—424no—
243C2×Q8—211no—
244C2×D4—31no—
245C2×D4—32no—
246C2×D4—71no—
247C2×D4—212no—
248C2×D4—211no—
249C22×C4—71no—
250C22×C4—211no—
251C22×C4—212no—
252C22×C4—211no—
253C22×C4—212no—
254C22×C4—211no—
255C22×C4—424no—
256C4⋊C4—72no—
257C4⋊C4—212no—
258C4⋊C4—212no—
259C22⋊C4—32no—
260C22⋊C4—72no—
261C22⋊C4—212no—
262C22⋊C4—212no—
263C22⋊C4—211no—
264C22⋊C4D3×D711yescharacteristic
265C42—212no—
266C2×C7—22no—
267C2×C7—64no—
268C2×C7—121no—
269C2×C7C2×D3×D411yescharacteristic
270C2×C7D3×#4112yes—
271D7—41no—
272C22×C3—22no—
273C22×C3—141no—
274C22×C3—142no—
275C22×C3D4×D712yes—
276C22×C3C23×D711yescharacteristic
277C2×D3—28no—
278C2×D3—22no—
279C2×D3—22no—
280C2×D3—21no—
281C2×D3—22no—
282C2×D3#115914yescomplemented
283C3×C4—22no—
284C3×C4—141no—
285C3×C4—142no—
286Dic3—21no—
287Dic3—72no—
288Dic3—72no—
289Dic3—142no—
290C23—32no—
291C23—31no—
292C23—64no—
293C23—62no—
294C23—71no—
295C23C2×D3×D711yescharacteristic
296Q8—422no—
297D4—64no—
298D4—62no—
299D4—142no—
300D4—422no—
301D4—424no—
302C2×C4—31no—
303C2×C4—72no—
304C2×C4—71no—
305C2×C4—142no—
306C2×C4—214no—
307C2×C4—211no—
308C2×C4—211no—
309C2×C4—212no—
310C2×C4—422no—
311C2×C4—424no—
312C2×C4—424no—
313C2×C4—422no—
314C2×C4C2×D3×D712yes—
315C7#565611yescharacteristic, complemented, Sylow
316C2×C3—22no—
317C2×C3—281no—
318C2×C3C2×D4×D711yescharacteristic
319C2×C3C2×#115912yes—
320D3—24no—
321D3—41no—
322C22—22no—
323C22—34no—
324C22—68no—
325C22—62no—
326C22—62no—
327C22—61no—
328C22—62no—
329C22—141no—
330C22—142no—
331C22#6668112yes—
332C22C22×D3×D711yescharacteristic
333C4—22no—
334C4—61no—
335C4—142no—
336C4—141no—
337C4—212no—
338C4—212no—
339C4—422no—
340C3#9027711yescharacteristic, complemented, Sylow
341C2—22no—
342C2—64no—
343C2—121no—
344C2—281no—
345C2C2×#6668111yescharacteristic
346C2D3×#115912yes—
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC3×C7×C22⋊C4C22
hypercenterC22⋊C4D3×D7
hypocenterC3×C7#525
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. C22×D3×Dic7C2semidirect
2. C2×#66684C2semidirect
3. #10621360C2semidirect
4. C7×#882C2semidirect
5. C3×#6449C2semidirect
6. #10621130C2semidirect
7. #10621101C2semidirect
8. #10621015C2semidirect
9. #10621004C2semidirect
10. #10620998C2semidirect
11. #10620997C2semidirect
19. C2×#4248C22semidirect
21. C2×#4245C22semidirect
25. C2×D3×Dic7C22semidirect
27. C2×Dic3×D7C22semidirect
30. #66629C22semidirect
31. C4×Dic21C22semidirect
33. C3×(C2×D7)⋊C4C22semidirect
34. C3×C4⋊Dic7C22semidirect
35. #66577C22semidirect
37. (C2×D3)⋊Dic7C22semidirect
45. #6449D3semidirect
57. C2×Dic21D4semidirect
95. (C2×D7)⋊C4C2×D3semidirect
96. C4⋊Dic7C2×D3semidirect
108. #882D7semidirect
119. C2×D3×C7#41semidirect
240. C3×C7#525semidirect
282. C2×D3#1159semidirect
315. C7#5656semidirect
340. C3#90277semidirect

Maximal subgroups

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

subgroupindex
C22×D3×Dic72
C2×#666842
#106213602
C7×#8822
C3×#64492
#106211302
#106211012
#106210152
#106210042
#106209982
#106209972
#902773
#56567