Finite Group Databasebeta

← #11736331

Subgroups — #11736331

Order 896.

764 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 764 conjugacy classes of subgroups — too many to draw. Only its 149 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×#6553C211yesmaximal, characteristic, complemented
2C2×#6513C211yesmaximal, characteristic, complemented
3C2×#6471C211yesmaximal, characteristic, complemented
4#90201C211yesmaximal, characteristic, complemented
5C7×#417C211yesmaximal, characteristic, complemented
6#89901C211yesmaximal, characteristic, complemented
7C2×#6383C211yesmaximal, characteristic, complemented
8#89583C212yesmaximal, complemented
9#89582C212yesmaximal, complemented
10#89548C212yesmaximal, complemented
11#89545C212yesmaximal, complemented
12C2×C7×#41C2211yescharacteristic
13#6553—24no—
14#6553—24no—
15C2×#1161C2212yes—
16C2×D4×D7—22no—
17C2×D4×D7C2212yes—
18C2×#1157—21no—
19C2×#1157C2211yescharacteristic
20C2×#1157C2211yescharacteristic, complemented
21C22×D28—21no—
22C22×D28C2211yescharacteristic
23C2×C7×M16C2211yescharacteristic, complemented
24C2×C7×C4⋊C4C2211yescharacteristic, complemented
25#6513—24no—
26C2×#1144C2211yescharacteristic, complemented
27C2×#1143C2211yescharacteristic, complemented
28C2×#1142C2211yescharacteristic, complemented
29C2×#1141C2211yescharacteristic, complemented
30#6493—22no—
31C22×C7⋊C8C2211yescharacteristic, complemented
32#6471—24no—
33C2×D56C2212yescomplemented
34C2×#1132C2212yescomplemented
35#6460—21no—
36#6460C2212yescomplemented
37#6458—21no—
38#6458C2212yescomplemented
39C7×#103C2212yescomplemented
40C7×#102C2212yescomplemented
41(C2×D7)⋊C8C2212yescomplemented
42(C2×D7)⋊C8C2212yescomplemented
43#6383C2212yescomplemented
44#6383C2212yescomplemented
45#3012—71nomaximal, Sylow
46C23×D7—21no—
47C23×D7—22no—
48C7×#41—22no—
49C7×#41D414yescomplemented
50C2×C7×Q8C2311yescharacteristic
51C2×C7×D4—21no—
52C2×C7×D4C2311yescharacteristic
53C22×C4×C7—21no—
54C22×C4×C7—21no—
55C22×C4×C7C2311yescharacteristic
56C2×C22⋊D7—21no—
57C2×C22⋊D7—22no—
58C2×C22⋊D7D412yes—
59#1161—24no—
60#1161—24no—
61D4×D7—216no—
62D4×D7—24no—
63D4×D7—24no—
64#1157—28no—
65#1157—22no—
66#1157—22no—
67#1157—22no—
68#1157—22no—
69C2×D28—28no—
70C2×D28—22no—
71C2×D28—21no—
72C2×D28—22no—
73C2×D28—22no—
74C2×D28D411yescharacteristic
75C2×D28D414yescomplemented
76C2×D28C2311yescharacteristic
77C2×D28C2312yes—
78C2×C4×D7—22no—
79C2×C4×D7—21no—
80C2×C4×D7C2312yes—
81C2×Dic14D411yescharacteristic
82C2×Dic14C2311yescharacteristic
83C7×M16—22no—
84C2×C7×C8C2312yes—
85C7×C4⋊C4—21no—
86C7×C4⋊C4C2312yes—
87#1144—22no—
88#1143—22no—
89#1142—22no—
90#1141—22no—
91(C2×D7)⋊C4—22no—
92(C2×D7)⋊C4—22no—
93C4⋊Dic7—22no—
94C2×C7⋊C8—22no—
95C2×C7⋊C8C2311yescharacteristic
96C2×C7⋊C8C2311yescharacteristic
97D56—24no—
98#1132—24no—
99C2×H(5,𝔽2)—71no—
100C2×AGL(1,ℤ/8)—71no—
101C2×#135—71no—
102#537—71no—
103#452—72no—
104#450—72no—
105#449—72no—
106#447—72no—
107#417D711yescharacteristic, complemented
108C2×#102—71no—
109#408—71no—
110C23×C7—21no—
111C23×C7C2×D411yescharacteristic
112C22×D7—22no—
113C22×D7—28no—
114C22×D7—24no—
115C22×D7—21no—
116C22×D7—82no—
117C22×D7—82no—
118C22×D7—81no—
119C22×D7—81no—
120C22×D7C2×D412yes—
121C7×Q8—21no—
122C7×Q8C2×D412yes—
123C7×D4—24no—
124C7×D4—21no—
125C7×D4C2×D412yes—
126C2×C4×C7—21no—
127C2×C4×C7—24no—
128C2×C4×C7—22no—
129C2×C4×C7—81no—
130C2×C4×C7C2×D412yes—
131C2×C4×C7C2×D411yescharacteristic
132C2×C4×C7C2×D412yes—
133C2×C4×C7C2411yescharacteristic
134C22⋊D7—28no—
135C22⋊D7—24no—
136C22⋊D7—42no—
137C2×Dic7—21no—
138C2×Dic7C2×D412yes—
139D28—28no—
140D28—22no—
141D28—24no—
142D28—22no—
143D28—21no—
144D28—21no—
145D28—82no—
146D28C2×D414yes—
147C4×D7—28no—
148C4×D7—22no—
149C4×D7—22no—
150C4×D7—42no—
151Dic14—22no—
152Dic14—21no—
153Dic14—21no—
154C7×C8—22no—
155C7⋊C8—21no—
156C7⋊C8—21no—
157H(5,𝔽2)—144no—
158H(5,𝔽2)—144no—
159C2×#41—71no—
160C2×#41—72no—
161C2×#41—71no—
162C2×#41—141no—
163C2×#41C2×D711yescharacteristic
164C22×D4—72no—
165C22×D4—71no—
166C22×D4—141no—
167C22×D4—142no—
168AGL(1,ℤ/8)—144no—
169#135—144no—
170C2×Q16—71no—
171C2×SD16—71no—
172C2×SD16—71no—
173C2×SD16—72no—
174C2×D8—71no—
175C2×D8—72no—
176C2×M16C2×D711yescharacteristic, complemented
177C22×C8—71no—
178C22⋊Q8—72no—
179C22⋊Q8—141no—
180#121—72no—
181#121—141no—
182#121—142no—
183C2×C4⋊C4C2×D711yescharacteristic, complemented
184#103C2×D712yescomplemented
185#102—72no—
186#102—72no—
187#102C2×D712yescomplemented
188C22⋊C8—72no—
189C22⋊C8—72no—
190C22×C7—21no—
191C22×C7—21no—
192C22×C7—22no—
193C22×C7—41no—
194C22×C7—41no—
195C22×C7#12012yes—
196C22×C7C22×D411yescharacteristic
197C2×D7—22no—
198C2×D7—24no—
199C2×D7—22no—
200C2×D7—22no—
201C2×D7—42no—
202C2×D7—42no—
203C2×D7—41no—
204C2×D7—41no—
205C2×D7—41no—
206C2×D7—84no—
207C2×D7—88no—
208C4×C7—22no—
209C4×C7—42no—
210C4×C7#12012yes—
211C4×C7C22×D411yescharacteristic
212C4×C7C22×D411yescharacteristic
213Dic7—22no—
214Dic7—41no—
215C24—142no—
216C24—141no—
217#41—22no—
218#41—148no—
219#41—144no—
220#41—144no—
221#41—142no—
222#41—142no—
223#41—142no—
224#41—142no—
225#41D2814yescomplemented
226C2×Q8—71no—
227C2×Q8—71no—
228C2×Q8C22×D711yescharacteristic
229C2×D4—21no—
230C2×D4—72no—
231C2×D4—74no—
232C2×D4—71no—
233C2×D4—72no—
234C2×D4—71no—
235C2×D4—142no—
236C2×D4—148no—
237C2×D4—142no—
238C2×D4—142no—
239C2×D4—144no—
240C2×D4—1416no—
241C2×D4—144no—
242C2×D4—141no—
243C2×D4—142no—
244C2×D4—141no—
245C2×D4C22×D711yescharacteristic
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—72no—
249C22×C4—142no—
250C22×C4—141no—
251C22×C4C22×D711yescharacteristic
252Q16—142no—
253SD16—144no—
254SD16—142no—
255SD16—142no—
256D8—144no—
257D8—142no—
258M16—22no—
259C2×C8—71no—
260C2×C8—71no—
261C2×C8—142no—
262C2×C8C22×D712yes—
263C4⋊C4—21no—
264C4⋊C4—142no—
265C4⋊C4C22×D712yes—
266C22⋊C4—142no—
267C22⋊C4—142no—
268C2×C7—22no—
269C2×C7—42no—
270C2×C7C2×#12011yescharacteristic
271C2×C7#57611yescharacteristic
272C2×C7#57711yescharacteristic
273D7—42no—
274D7—44no—
275D7—81no—
276C23—21no—
277C23—72no—
278C23—148no—
279C23—142no—
280C23—144no—
281C23—141no—
282C23—561no—
283C23—561no—
284C23—562no—
285C23—562no—
286C23D4×D711yescharacteristic
287Q8—21no—
288Q8—142no—
289Q8—141no—
290Q8—141no—
291Q8C2×D2812yes—
292D4—24no—
293D4—21no—
294D4—74no—
295D4—148no—
296D4—148no—
297D4—144no—
298D4—144no—
299D4—142no—
300D4—142no—
301D4—141no—
302D4—141no—
303D4—282no—
304D4—562no—
305D4C2×D2812yes—
306C2×C4—21no—
307C2×C4—24no—
308C2×C4—22no—
309C2×C4—72no—
310C2×C4—81no—
311C2×C4—148no—
312C2×C4—142no—
313C2×C4—142no—
314C2×C4—141no—
315C2×C4—282no—
316C2×C4C2×D2812yes—
317C2×C4D4×D711yescharacteristic
318C2×C4D4×D712yes—
319C2×C4C23×D711yescharacteristic
320C8—22no—
321C8—141no—
322C8—141no—
323C7#301211yescharacteristic, complemented, Sylow
324C22—21no—
325C22—21no—
326C22—22no—
327C22—41no—
328C22—41no—
329C22—144no—
330C22—142no—
331C22—142no—
332C22—142no—
333C22—282no—
334C22—282no—
335C22—281no—
336C22—281no—
337C22—281no—
338C22—564no—
339C22—568no—
340C22#644512yes—
341C22C2×D4×D711yescharacteristic
342C4—22no—
343C4—42no—
344C4—142no—
345C4—281no—
346C4#644512yes—
347C4C22×D2811yescharacteristic
348C4C2×D4×D711yescharacteristic
349C2—22no—
350C2—42no—
351C2—282no—
352C2—284no—
353C2—561no—
354C2C2×#644511yescharacteristic
355C2#9046511yescharacteristic
356C2#9046911yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#417C2
socleC22×C7C22×D4
hypercenter#417D7
hypocenterC7#3012
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. C2×#6553C2semidirect
2. C2×#6513C2semidirect
3. C2×#6471C2semidirect
4. #90201C2semidirect
5. C7×#417C2semidirect
6. #89901C2semidirect
7. C2×#6383C2semidirect
8. #89583C2semidirect
9. #89582C2semidirect
10. #89548C2semidirect
11. #89545C2semidirect
20. C2×#1157C22semidirect
23. C2×C7×M16C22semidirect
24. C2×C7×C4⋊C4C22semidirect
26. C2×#1144C22semidirect
27. C2×#1143C22semidirect
28. C2×#1142C22semidirect
29. C2×#1141C22semidirect
31. C22×C7⋊C8C22semidirect
33. C2×D56C22semidirect
34. C2×#1132C22semidirect
36. #6460C22semidirect
38. #6458C22semidirect
39. C7×#103C22semidirect
40. C7×#102C22semidirect
41. (C2×D7)⋊C8C22semidirect
42. (C2×D7)⋊C8C22semidirect
43. #6383C22semidirect
44. #6383C22semidirect
49. C7×#41D4semidirect
75. C2×D28D4semidirect
107. #417D7semidirect
176. C2×M16C2×D7semidirect
183. C2×C4⋊C4C2×D7semidirect
184. #103C2×D7semidirect
187. #102C2×D7semidirect
225. #41D28semidirect
323. C7#3012semidirect

Maximal subgroups

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

subgroupindex
C2×#65532
C2×#65132
C2×#64712
#902012
C7×#4172
#899012
C2×#63832
#895832
#895822
#895482
#895452
#30127