Finite Group Databasebeta

← #13058057

Subgroups — #13058057

Order 1344.

660 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 660 conjugacy classes of subgroups — too many to draw. Only its 148 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×D21⋊C4C211yesmaximal, characteristic, complemented
2C2×#66686C211yesmaximal, characteristic, complemented
3#10621365C211yesmaximal, characteristic, complemented
4C7×#886C211yesmaximal, characteristic, complemented
5C3×#6440C211yesmaximal, characteristic, complemented
6#10621135C211yesmaximal, characteristic, complemented
7#10621104C211yesmaximal, characteristic, complemented
8#10620982C212yesmaximal, complemented
9#10620981C212yesmaximal, complemented
10#10620980C212yesmaximal, complemented
11#10620967C212yesmaximal, complemented
12#90347—31nomaximal, Hall
13C23×D21C2211yescharacteristic
14C2×#4268C2211yescharacteristic
15C2×D84C2212yes—
16C2×C7×#67C2211yescharacteristic
17C22×C7×Dic3C2211yescharacteristic
18C22×C3×Dic7C2211yescharacteristic
19C2×#4248C2211yescharacteristic, complemented
20C2×#4247C2212yes—
21C2×#4247C2211yescharacteristic, complemented
22C2×D21⋊C4—24no—
23C2×D21⋊C4C2212yes—
24C2×D21⋊C4C2212yes—
25C2×D21⋊C4C2211yescharacteristic, complemented
26#66686—24no—
27C2×D3×Dic7C2211yescharacteristic, complemented
28C3×C7×C22⋊C4C2211yescharacteristic
29#66631C2212yes—
30C7×(C2×D3)⋊C4C2212yescomplemented
31C7×C4⋊Dic3C2212yescomplemented
32C3×C22⋊Dic7C2211yescharacteristic
33C3×Dic7⋊C4C2212yescomplemented
34C3×C4×Dic7C2212yescomplemented
35Dic7⋊Dic3C2212yescomplemented
36#66575C2212yes—
37#66575C2212yescomplemented
38C2×#1159—31no—
39C22×C4×D7—31no—
40C7×#123—31no—
41#6500—31no—
42#6492—31no—
43#6459—32no—
44#6456—32no—
45#6447—32no—
46#6445—31no—
47(C22⋊D7)⋊C4—32no—
48#6440D311yescharacteristic, complemented
49#5524—71nomaximal, Hall
50C23×C3×C7C2311yescharacteristic
51C22×D21—22no—
52C22×D21—24no—
53C22×D21C2311yescharacteristic
54C22×D21C2312yes—
55C22×D3×C7C2311yescharacteristic
56C2×C3×C4×C7C2312yes—
57#4268—22no—
58C2×Dic21C2311yescharacteristic
59D84—24no—
60C7×#67—22no—
61C2×C7×Dic3—22no—
62C2×C7×Dic3C2311yescharacteristic
63C2×C7×Dic3C2312yes—
64C2×C3×Dic7D414yescomplemented
65C2×C3×Dic7C2312yes—
66C2×C3×Dic7C2311yescharacteristic
67C2×C3×Dic7C2311yescharacteristic
68#4248—22no—
69#4247—24no—
70#4247—22no—
71D21⋊C4—22no—
72D21⋊C4—24no—
73D21⋊C4—24no—
74D3×Dic7—22no—
75C23×D7—31no—
76C2×C7×D4—31no—
77C22×C4×C7—31no—
78C2×C22⋊D7—31no—
79C2×C22⋊D7—32no—
80C2×C22⋊D7—31no—
81C22×Dic7—31no—
82C22×Dic7C2×D311yescharacteristic
83#1159—64no—
84C2×D28—32no—
85C2×C4×D7—32no—
86C2×C4×D7—32no—
87C2×C4×D7—31no—
88C2×C4×D7—64no—
89C2×Dic14—31no—
90C7×C4⋊C4—32no—
91C7×C22⋊C4—32no—
92C7×C22⋊C4C2×D311yescharacteristic
93C22⋊Dic7C2×D311yescharacteristic
94(C2×D7)⋊C4—32no—
95(C2×D7)⋊C4—32no—
96(C2×D7)⋊C4—32no—
97Dic7⋊C4—32no—
98Dic7⋊C4C2×D312yescomplemented
99C4×Dic7C2×D312yescomplemented
100C2×#235—71no—
101C22×C4×D3—71no—
102C3×#117—71no—
103#897—72no—
104#895—72no—
105#886—71no—
106#886D711yescharacteristic, complemented
107#882—71no—
108#882—71no—
109C4×D12—72no—
110C4×D12—72no—
111C22×C3×C7—22no—
112C22×C3×C7C2×D412yes—
113C22×C3×C7C2411yescharacteristic
114C2×D21—28no—
115C2×D21—22no—
116C2×D21—22no—
117C2×D21—21no—
118C2×D21—22no—
119C2×D21#4114yescomplemented
120C2×D3×C7—21no—
121C2×D3×C7—22no—
122C3×C4×C7—22no—
123Dic21—21no—
124C3×Dic7—22no—
125C3×Dic7C2×D412yes—
126C3×Dic7C2×D412yes—
127C7×Dic3—22no—
128C7×Dic3—21no—
129#525—211noSylow
130C23×C7—31no—
131C23×C7C22×D311yescharacteristic
132C22×D7—32no—
133C22×D7—31no—
134C22×D7—64no—
135C22×D7—62no—
136C7×D4—62no—
137C2×C4×C7—32no—
138C2×C4×C7—31no—
139C2×C4×C7—62no—
140C2×C4×C7C22×D312yes—
141C22⋊D7—64no—
142C22⋊D7—62no—
143C22⋊D7—62no—
144C2×Dic7—31no—
145C2×Dic7—62no—
146C2×Dic7D1214yescomplemented
147C2×Dic7C22×D312yes—
148C2×Dic7C22×D311yescharacteristic
149C2×Dic7C22×D311yescharacteristic
150D28—64no—
151C4×D7—64no—
152C4×D7—64no—
153C4×D7—62no—
154Dic14—62no—
155C23×D3—71no—
156C22×C3×C4—71no—
157C2×#67—71no—
158C2×#67C2×D711yescharacteristic
159C22×Dic3C2×D711yescharacteristic
160#235—144no—
161C2×D12—72no—
162C2×D12—71no—
163C2×D12—72no—
164C2×C4×D3—71no—
165C2×C4×D3—72no—
166C2×C4×D3—71no—
167C2×C4×D3—72no—
168C2×C4×D3—144no—
169C2×Dic6—71no—
170C3×C4⋊C4—72no—
171C3×C22⋊C4—71no—
172C3×C22⋊C4C2×D711yescharacteristic
173C3×C42—72no—
174(C2×D3)⋊C4—72no—
175(C2×D3)⋊C4—72no—
176(C2×D3)⋊C4—72no—
177(C2×D3)⋊C4C2×D712yescomplemented
178C4⋊Dic3—72no—
179C4⋊Dic3C2×D712yescomplemented
180C2×C3×C7—22no—
181C2×C3×C7C22×D411yescharacteristic
182C2×C3×C7C2×#4112yes—
183D21—24no—
184D21—41no—
185D3×C7—41no—
186C2×#41—211no—
187C23×C4—211no—
188#123—31no—
189#123—211no—
190C22⋊Q8—212no—
191#121—212no—
192#120—211no—
193#120—211no—
194C4×D4—212no—
195C4×D4—212no—
196#117—71no—
197C22×C7—22no—
198C22×C7—61no—
199C22×C7—62no—
200C22×C7C2×D1212yes—
201C22×C7C23×D311yescharacteristic
202C2×D7—34no—
203C2×D7—68no—
204C2×D7—62no—
205C2×D7—62no—
206C2×D7—61no—
207C2×D7—62no—
208C4×C7—22no—
209C4×C7—61no—
210C4×C7—62no—
211Dic7—22no—
212Dic7—61no—
213Dic7C2×D1212yes—
214Dic7C2×D1212yes—
215C23×C3C22×D711yescharacteristic
216C22×D3—72no—
217C22×D3—71no—
218C22×D3—144no—
219C22×D3—142no—
220C22×D3C22×D711yescharacteristic
221C2×C3×C4—74no—
222C2×C3×C4—71no—
223C2×C3×C4—71no—
224C2×C3×C4—72no—
225C2×C3×C4C22×D712yes—
226#67—22no—
227#67—142no—
228C2×Dic3—22no—
229C2×Dic3—71no—
230C2×Dic3C22×D712yes—
231C2×Dic3C22×D711yescharacteristic
232D12—144no—
233D12—142no—
234D12—144no—
235C4×D3—144no—
236C4×D3—144no—
237C4×D3—142no—
238C4×D3—142no—
239Dic6—142no—
240C3×C7#52511yescharacteristic, complemented, Hall
241C24—211no—
242#41—424no—
243C2×Q8—211no—
244C2×D4—31no—
245C2×D4—212no—
246C2×D4—211no—
247C2×D4—211no—
248C2×D4—212no—
249C22×C4—31no—
250C22×C4—71no—
251C22×C4—212no—
252C22×C4—211no—
253C22×C4—212no—
254C22×C4—211no—
255C22×C4—424no—
256C4⋊C4—32no—
257C4⋊C4—72no—
258C4⋊C4—212no—
259C22⋊C4—32no—
260C22⋊C4—71no—
261C22⋊C4—212no—
262C22⋊C4—212no—
263C22⋊C4—212no—
264C22⋊C4D3×D711yescharacteristic
265C42—72no—
266C2×C7—22no—
267C2×C7—121no—
268C2×C7C22×D1211yescharacteristic
269C2×C7D3×#4112yes—
270D7—64no—
271D7—121no—
272C22×C3—22no—
273C22×C3D4×D712yes—
274C22×C3C23×D711yescharacteristic
275C2×D3—21no—
276C2×D3—22no—
277C2×D3—74no—
278C2×D3—148no—
279C2×D3—142no—
280C2×D3—142no—
281C2×D3—141no—
282C2×D3—142no—
283C3×C4—22no—
284C3×C4—72no—
285C3×C4—72no—
286C3×C4—142no—
287Dic3—21no—
288Dic3—22no—
289Dic3—141no—
290C23—31no—
291C23—212no—
292C23—211no—
293C23—424no—
294C23—422no—
295C23C2×D3×D711yescharacteristic
296Q8—422no—
297D4—62no—
298D4—424no—
299D4—422no—
300D4—424no—
301D4—422no—
302C2×C4—32no—
303C2×C4—31no—
304C2×C4—62no—
305C2×C4—74no—
306C2×C4—71no—
307C2×C4—71no—
308C2×C4—72no—
309C2×C4—211no—
310C2×C4—424no—
311C2×C4—424no—
312C2×C4—422no—
313C2×C4—422no—
314C2×C4C2×D3×D712yes—
315C7#552411yescharacteristic, complemented, Sylow
316C2×C3—22no—
317C2×C3C2×D4×D711yescharacteristic
318C2×C3D7×#4112yes—
319D3—41no—
320D3—144no—
321D3—281no—
322C22—22no—
323C22—61no—
324C22—62no—
325C22—214no—
326C22—428no—
327C22—422no—
328C22—422no—
329C22—421no—
330C22—422no—
331C22D7×D1212yes—
332C22C22×D3×D711yescharacteristic
333C4—22no—
334C4—62no—
335C4—61no—
336C4—72no—
337C4—72no—
338C4—142no—
339C4—421no—
340C3#9034711yescharacteristic, complemented, Sylow
341C2—22no—
342C2—121no—
343C2—424no—
344C2—841no—
345C2C2×D7×D1211yescharacteristic
346C2#1062157112yes—
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×D21⋊C4C2semidirect
2. C2×#66686C2semidirect
3. #10621365C2semidirect
4. C7×#886C2semidirect
5. C3×#6440C2semidirect
6. #10621135C2semidirect
7. #10621104C2semidirect
8. #10620982C2semidirect
9. #10620981C2semidirect
10. #10620980C2semidirect
11. #10620967C2semidirect
19. C2×#4248C22semidirect
21. C2×#4247C22semidirect
25. C2×D21⋊C4C22semidirect
27. C2×D3×Dic7C22semidirect
30. C7×(C2×D3)⋊C4C22semidirect
31. C7×C4⋊Dic3C22semidirect
33. C3×Dic7⋊C4C22semidirect
34. C3×C4×Dic7C22semidirect
35. Dic7⋊Dic3C22semidirect
37. #66575C22semidirect
48. #6440D3semidirect
64. C2×C3×Dic7D4semidirect
98. Dic7⋊C4C2×D3semidirect
99. C4×Dic7C2×D3semidirect
106. #886D7semidirect
119. C2×D21#41semidirect
146. C2×Dic7D12semidirect
177. (C2×D3)⋊C4C2×D7semidirect
179. C4⋊Dic3C2×D7semidirect
240. C3×C7#525semidirect
315. C7#5524semidirect
340. C3#90347semidirect

Maximal subgroups

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

subgroupindex
C22×D21⋊C42
C2×#666862
#106213652
C7×#8862
C3×#64402
#106211352
#106211042
#106209822
#106209812
#106209802
#106209672
#903473
#55247