Finite Group Databasebeta

← #13059333

Subgroups — #13059333

Order 1344.

532 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 128 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
1#10621581C212yesmaximal, complemented
2Q16×D21C211yesmaximal, characteristic, complemented
3C7×#919C211yesmaximal, characteristic, complemented
4C3×#6482C211yesmaximal, characteristic, complemented
5#10621079C212yesmaximal, complemented
6#10621076C212yesmaximal, complemented
7#10621067C212yesmaximal, complemented
8#10620835C211yesmaximal, characteristic, complemented
9#10620832C211yesmaximal, characteristic, complemented
10#10620818C211yesmaximal, characteristic, complemented
11D3×C8×D7C211yesmaximal, characteristic, complemented
12D7×#135—31nomaximal, Hall
13Q8×D21C2212yes—
14C7×#239C2212yescomplemented
15C3×#1161C2212yescomplemented
16#66681—22no—
17#66681C2212yes—
18C4×D3×D7—22no—
19C4×D3×D7C2211yescharacteristic
20#66676—22no—
21#66676C2212yescomplemented
22#66669—22no—
23#66669C2212yescomplemented
24C3×C7×Q16C2211yescharacteristic, complemented
25#66635C2212yescomplemented
26Dic84C2211yescharacteristic, complemented
27C8×D21C2211yescharacteristic
28C7×#215C2212yescomplemented
29C7×D24C2211yescharacteristic, complemented
30D3×C7×C8C2211yescharacteristic, complemented
31C3×#1143C2212yescomplemented
32C3×D56C2211yescharacteristic, complemented
33C3×C8×D7C2211yescharacteristic, complemented
34#66567C2212yescomplemented
35#66565C2212yescomplemented
36#66560C2212yescomplemented
37D21⋊C8C2211yescharacteristic
38D3×C7⋊C8C2211yescharacteristic, complemented
39D7×C3⋊C8C2211yescharacteristic, complemented
40D7×#41—32no—
41C7×#135—31no—
42#6513—32no—
43#6482D311yescharacteristic, complemented
44D7×Q16—31no—
45#6479—32no—
46D7×SD16—32no—
47#6475—31no—
48D7×D8—31no—
49#6467—31no—
50C2×C8×D7—31no—
51D3×#135—71nomaximal, Hall
52C2×D3×D7—22no—
53C2×D3×D7D411yescharacteristic
54C3×C7×Q8C2312yes—
55C4×D21—22no—
56C4×D21C2311yescharacteristic
57Dic42—22no—
58Dic42C2312yes—
59C7×D12—22no—
60C7×D12C2312yes—
61C4×D3×C7—22no—
62C4×D3×C7C2311yescharacteristic
63C3×D28—22no—
64C3×D28C2312yes—
65C3×C4×D7—22no—
66C3×C4×D7C2311yescharacteristic
67#4245—22no—
68#4245—22no—
69#4245—22no—
70D21⋊C4D411yescharacteristic
71D3×Dic7—22no—
72D3×Dic7D411yescharacteristic
73Dic3×D7—22no—
74Dic3×D7D411yescharacteristic
75C3×C7×C8C2311yescharacteristic
76C3⋊(C7⋊C8)C2311yescharacteristic
77C3×C7⋊C8C2311yescharacteristic
78C7×C3⋊C8C2311yescharacteristic
79C7×#41—32no—
80#1161C2×D312yescomplemented
81Q8×D7—32no—
82#1159—32no—
83#1159—62no—
84D4×D7—32no—
85D4×D7—62no—
86#1157—32no—
87#1157—62no—
88C2×C4×D7—31no—
89C2×C4×D7—62no—
90C7×Q16C2×D311yescharacteristic, complemented
91C7×SD16—32no—
92C7×D8—31no—
93C2×C7×C8—31no—
94#1144—32no—
95#1143C2×D312yescomplemented
96#1142—32no—
97#1141—32no—
98C2×C7⋊C8—31no—
99Dic28—31no—
100D56C2×D311yescharacteristic, complemented
101#1132—32no—
102C8×D7—31no—
103C8×D7—31no—
104C8×D7—31no—
105C8×D7C2×D311yescharacteristic, complemented
106D3×#41—72no—
107C3×#135—71no—
108#950—72no—
109#919D711yescharacteristic, complemented
110D3×Q16—71no—
111#916—72no—
112D3×SD16—72no—
113#912—71no—
114D3×D8—71no—
115#904—71no—
116C2×D3×C8—71no—
117C2×D21C2×D411yescharacteristic
118C2×D3×C7—22no—
119C2×D3×C7C2×D411yescharacteristic
120C2×C3×D7—22no—
121C2×C3×D7C2×D411yescharacteristic
122D3×D7—22no—
123D3×D7—44no—
124C3×C4×C7—22no—
125C3×C4×C7C2411yescharacteristic
126Dic21—22no—
127Dic21C2×D411yescharacteristic
128C3×Dic7C2×D411yescharacteristic
129C7×Dic3C2×D411yescharacteristic
130C2×#135—211noSylow
131C22×D7—31no—
132C22×D7—62no—
133C7×Q8C22×D312yes—
134C7×D4—32no—
135C7×D4—62no—
136C2×C4×C7—31no—
137C2×C4×C7—62no—
138C22⋊D7—62no—
139C22⋊D7—62no—
140C22⋊D7—62no—
141C2×Dic7—31no—
142C2×Dic7—62no—
143D28—22no—
144D28C22×D312yes—
145C4×D7—22no—
146C4×D7—31no—
147C4×D7—31no—
148C4×D7—31no—
149C4×D7—62no—
150C4×D7—62no—
151C4×D7C22×D311yescharacteristic
152Dic14—32no—
153Dic14—62no—
154C7×C8—31no—
155C7×C8C22×D311yescharacteristic
156C7⋊C8—31no—
157C7⋊C8C22×D311yescharacteristic
158C3×#41—72no—
159#239C2×D712yescomplemented
160D3×Q8—72no—
161#237—72no—
162#237—142no—
163D3×D4—72no—
164D3×D4—142no—
165#235—72no—
166#235—142no—
167C2×C4×D3—71no—
168C2×C4×D3—142no—
169C3×Q16C2×D711yescharacteristic, complemented
170C3×SD16—72no—
171C3×D8—71no—
172C2×C3×C8—71no—
173#216—72no—
174#215C2×D712yescomplemented
175#214—72no—
176#213—72no—
177C2×C3⋊C8—71no—
178Dic12—71no—
179D24C2×D711yescharacteristic, complemented
180#204—72no—
181D3×C8—71no—
182D3×C8—71no—
183D3×C8—71no—
184D3×C8C2×D711yescharacteristic, complemented
185C2×C3×C7C22×D411yescharacteristic
186D21#13512yescomplemented
187C3×D7—21no—
188C3×D7—42no—
189D3×C7—21no—
190D3×C7—42no—
191C2×#41—212no—
192#135—31no—
193#135—71no—
194#135—211no—
195#135—211no—
196#135—212no—
197#135—212no—
198C2×Q16—211no—
199C2×SD16—212no—
200C2×D8—211no—
201C22×C8—211no—
202C22×C7—31no—
203C22×C7—62no—
204C2×D7—22no—
205C2×D7—31no—
206C2×D7—62no—
207C2×D7—124no—
208C2×D7D3×D411yescharacteristic
209C4×C7—22no—
210C4×C7—31no—
211C4×C7C23×D311yescharacteristic
212Dic7—31no—
213Dic7—62no—
214Dic7D3×D411yescharacteristic
215C22×D3—71no—
216C22×D3—142no—
217C3×Q8C22×D712yes—
218C3×D4—72no—
219C3×D4—142no—
220C2×C3×C4—71no—
221C2×C3×C4—142no—
222#67—142no—
223#67—142no—
224#67—142no—
225C2×Dic3—71no—
226C2×Dic3—142no—
227D12—22no—
228D12C22×D712yes—
229C4×D3—22no—
230C4×D3—71no—
231C4×D3—71no—
232C4×D3—71no—
233C4×D3—142no—
234C4×D3—142no—
235C4×D3C22×D711yescharacteristic
236Dic6—72no—
237Dic6—142no—
238C3×C8—71no—
239C3×C8C22×D711yescharacteristic
240C3⋊C8—71no—
241C3⋊C8C22×D711yescharacteristic
242C3×C7C2×#13511yescharacteristic, complemented, Hall
243#41—32no—
244#41—72no—
245#41—212no—
246#41—212no—
247#41—422no—
248#41—422no—
249C2×Q8—212no—
250C2×D4—212no—
251C2×D4—422no—
252C22×C4—211no—
253C22×C4—422no—
254Q16—212no—
255Q16—211no—
256Q16D3×D711yescharacteristic, complemented
257SD16—32no—
258SD16—72no—
259SD16—212no—
260SD16—212no—
261D8—31no—
262D8—71no—
263D8—212no—
264C2×C8—31no—
265C2×C8—71no—
266C2×C8—211no—
267C2×C8—211no—
268C2×C8—211no—
269C2×C8—211no—
270C2×C7—61no—
271C2×C7—122no—
272C2×C7C2×D3×D411yescharacteristic
273D7—21no—
274D7—32no—
275D7—42no—
276C22×C3—71no—
277C22×C3—142no—
278C2×D3—22no—
279C2×D3—71no—
280C2×D3—142no—
281C2×D3—284no—
282C2×D3D4×D711yescharacteristic
283C3×C4—22no—
284C3×C4—71no—
285C3×C4C23×D711yescharacteristic
286Dic3—71no—
287Dic3—142no—
288Dic3D4×D711yescharacteristic
289C23—211no—
290C23—422no—
291Q8—212no—
292Q8—422no—
293Q8C2×D3×D712yes—
294D4—32no—
295D4—62no—
296D4—72no—
297D4—142no—
298D4—422no—
299D4—422no—
300D4—422no—
301C2×C4—31no—
302C2×C4—62no—
303C2×C4—71no—
304C2×C4—142no—
305C2×C4—211no—
306C2×C4—211no—
307C2×C4—211no—
308C2×C4—211no—
309C2×C4—422no—
310C2×C4—422no—
311C2×C4—422no—
312C8—31no—
313C8—71no—
314C8—211no—
315C8C2×D3×D711yescharacteristic
316C7D3×#13511yescharacteristic, complemented, Sylow
317C2×C3—141no—
318C2×C3—282no—
319C2×C3C2×D4×D711yescharacteristic
320D3—21no—
321D3—42no—
322D3—72no—
323C22—31no—
324C22—62no—
325C22—71no—
326C22—142no—
327C22—211no—
328C22—422no—
329C22—844no—
330C4—22no—
331C4—31no—
332C4—71no—
333C4—211no—
334C4—422no—
335C4C22×D3×D711yescharacteristic
336C3D7×#13511yescharacteristic, complemented, Sylow
337C2—61no—
338C2—122no—
339C2—141no—
340C2—212no—
341C2—282no—
342C2D3×D4×D711yescharacteristic
subgroupG/H
centerC2D3×D4×D7
commutator, cosocleC3×C4×C7C24
FrattiniC4C22×D3×D7
FittingC3×C7×Q16C22
socleC2×C3×C7C22×D4
hypercenterQ16D3×D7
hypocenterC3×C7C2×#135
LCSG ▷ C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2 ◁ C4 ◁ Q16

Complements

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

NH ≅ G/Ntype
1. #10621581C2semidirect
2. Q16×D21C2semidirect
3. C7×#919C2semidirect
4. C3×#6482C2semidirect
5. #10621079C2semidirect
6. #10621076C2semidirect
7. #10621067C2semidirect
8. #10620835C2semidirect
9. #10620832C2semidirect
10. #10620818C2semidirect
11. D3×C8×D7C2semidirect
14. C7×#239C22semidirect
15. C3×#1161C22semidirect
21. #66676C22semidirect
23. #66669C22semidirect
24. C3×C7×Q16C22semidirect
25. #66635C22semidirect
26. Dic84C22semidirect
28. C7×#215C22semidirect
29. C7×D24C22semidirect
30. D3×C7×C8C22semidirect
31. C3×#1143C22semidirect
32. C3×D56C22semidirect
33. C3×C8×D7C22semidirect
34. #66567C22semidirect
35. #66565C22semidirect
36. #66560C22semidirect
38. D3×C7⋊C8C22semidirect
39. D7×C3⋊C8C22semidirect
43. #6482D3semidirect
80. #1161C2×D3semidirect
90. C7×Q16C2×D3semidirect
95. #1143C2×D3semidirect
100. D56C2×D3semidirect
105. C8×D7C2×D3semidirect
109. #919D7semidirect
159. #239C2×D7semidirect
169. C3×Q16C2×D7semidirect
174. #215C2×D7semidirect
179. D24C2×D7semidirect
184. D3×C8C2×D7semidirect
186. D21#135semidirect
242. C3×C7C2×#135semidirect
256. Q16D3×D7semidirect
316. C7D3×#135semidirect
336. C3D7×#135semidirect

Maximal subgroups

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

subgroupindex
#106215812
Q16×D212
C7×#9192
C3×#64822
#106210792
#106210762
#106210672
#106208352
#106208322
#106208182
D3×C8×D72
D7×#1353
D3×#1357