Finite Group Databasebeta

← #11736334

Subgroups — #11736334

Order 896.

496 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 135 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#90519C211yesmaximal, characteristic
2C2×#6514C211yesmaximal, characteristic, complemented
3C2×D7×M16C211yesmaximal, characteristic, complemented
4C2×#6460C211yesmaximal, characteristic, complemented
5C7×#417C211yesmaximal, characteristic, complemented
6#89892C211yesmaximal, characteristic, complemented
7#89736C211yesmaximal, characteristic, complemented
8#89588C212yesmaximal, complemented
9#89585C212yesmaximal, complemented
10#89552C212yesmaximal, complemented
11#89550C212yesmaximal, complemented
12C2×C7×#41C2211yescharacteristic
13C22×C4×D7C2211yescharacteristic
14C22×Dic14C2211yescharacteristic
15C2×C7×M16C2211yescharacteristic
16C2×C7×C4⋊C4C2211yescharacteristic
17#6514—24no—
18#6509C2211yescharacteristic
19#6509C2211yescharacteristic
20C2×#1144C2212yescomplemented
21#6501C2211yescharacteristic
22#6501C2211yescharacteristic
23#6500—21no—
24#6498—21no—
25C2×#1142C2212yescomplemented
26C4×C22⋊D7—21no—
27C4×C22⋊D7—21no—
28C2×(C2×D7)⋊C4—21no—
29#6489C2211yescharacteristic
30C2×C4⋊Dic7—21no—
31C2×#1136C2211yescharacteristic
32D7×M16—24no—
33C2×#1131C2212yes—
34C2×C8×D7C2212yes—
35#6460—21no—
36#6460—21no—
37#6460C2212yes—
38#6460C2212yes—
39C7×#103C2212yescomplemented
40C7×#102C2212yescomplemented
41#6390C2212yescomplemented
42#6390C2212yescomplemented
43#6382C2212yescomplemented
44#6381C2212yescomplemented
45#3056—71nomaximal, Sylow
46C23×D7D411yescharacteristic
47C7×#41—22no—
48C7×#41—22no—
49C2×C7×Q8C2311yescharacteristic
50C2×C7×D4—21no—
51C2×C7×D4C2311yescharacteristic
52C22×C4×C7—21no—
53C22×C4×C7—21no—
54C22×C4×C7C2311yescharacteristic
55C2×C22⋊D7—21no—
56C2×C22⋊D7—21no—
57C22×Dic7—21no—
58C22×Dic7D411yescharacteristic
59C2×C4×D7—22no—
60C2×C4×D7—22no—
61C2×C4×D7D411yescharacteristic
62C2×C4×D7D411yescharacteristic
63C2×C4×D7C2311yescharacteristic
64C2×C4×D7C2311yescharacteristic
65C2×Dic14—22no—
66C2×Dic14—22no—
67C2×Dic14—21no—
68C2×Dic14C2312yes—
69C7×M16—22no—
70C2×C7×C8C2312yes—
71C7×C4⋊C4—21no—
72C7×C4⋊C4C2312yes—
73C22⋊Dic7—21no—
74C22⋊Dic7—21no—
75C22⋊Dic7—21no—
76#1144—24no—
77#1142—24no—
78(C2×D7)⋊C4—22no—
79(C2×D7)⋊C4—22no—
80(C2×D7)⋊C4—21no—
81(C2×D7)⋊C4—21no—
82C4⋊Dic7—22no—
83C4⋊Dic7—22no—
84C4⋊Dic7C2311yescharacteristic
85C4⋊Dic7C2311yescharacteristic
86Dic7⋊C4—21no—
87Dic7⋊C4—21no—
88C4×Dic7—21no—
89#1136D414yescomplemented
90C2×C7⋊C8C2312yes—
91#1131—24no—
92C8×D7—24no—
93C2×#137—71no—
94C22×M16—71no—
95#525—71no—
96C2×C22⋊Q8—71no—
97#470—72no—
98#468—72no—
99#467—72no—
100#465—72no—
101#428—71no—
102#419—71no—
103#417D711yescharacteristic, complemented
104C23×C7—21no—
105C23×C7C2×D411yescharacteristic
106C22×D7—22no—
107C22×D7—22no—
108C22×D7—41no—
109C22×D7C2×D411yescharacteristic
110C22×D7C2×D411yescharacteristic
111C7×Q8—21no—
112C7×Q8—21no—
113C7×D4—21no—
114C7×D4—21no—
115C7×D4—42no—
116C2×C4×C7—21no—
117C2×C4×C7—22no—
118C2×C4×C7—42no—
119C2×C4×C7—42no—
120C2×C4×C7C2×D411yescharacteristic
121C2×C4×C7C2×D412yes—
122C2×C4×C7#4112yes—
123C2×C4×C7C2411yescharacteristic
124C22⋊D7—42no—
125C22⋊D7—42no—
126C2×Dic7—22no—
127C2×Dic7—21no—
128C2×Dic7—21no—
129C2×Dic7—22no—
130C2×Dic7—42no—
131C2×Dic7C2×D411yescharacteristic
132C2×Dic7C2×D411yescharacteristic
133C4×D7—22no—
134C4×D7—22no—
135C4×D7—21no—
136C4×D7—21no—
137C4×D7—21no—
138C4×D7—21no—
139Dic14—22no—
140Dic14—22no—
141Dic14—42no—
142C7×C8—22no—
143C7⋊C8C2×D414yes—
144C2×#41C2×D711yescharacteristic
145C22×Q8—71no—
146C23×C4—71no—
147#137—144no—
148C2×Q16—72no—
149C2×SD16—72no—
150C2×M16—71no—
151C2×M16—72no—
152C2×M16—144no—
153C2×M16C2×D711yescharacteristic
154C22×C8—72no—
155#123—141no—
156C22⋊Q8—71no—
157C22⋊Q8—71no—
158C22⋊Q8—72no—
159C22⋊Q8—72no—
160C22⋊Q8—141no—
161C22⋊Q8—141no—
162#121—71no—
163#121—71no—
164#120—141no—
165C4×D4—141no—
166C4×D4—141no—
167#117—71no—
168C2×C4⋊C4—141no—
169C2×C4⋊C4C2×D711yescharacteristic
170C2×C22⋊C4—141no—
171#107—72no—
172#106—72no—
173#103—72no—
174#103—72no—
175#103C2×D712yescomplemented
176#102C2×D712yescomplemented
177C22×C7—21no—
178C22×C7—21no—
179C22×C7—41no—
180C22×C7—41no—
181C22×C7—41no—
182C22×C7#12112yes—
183C22×C7C22×D411yescharacteristic
184C2×D7—21no—
185C2×D7—21no—
186C2×D7—21no—
187C2×D7—21no—
188C2×D7—21no—
189C2×D7—44no—
190C2×D7#13712yescomplemented
191C4×C7—41no—
192C4×C7—42no—
193C4×C7#12112yes—
194C4×C7C22×D411yescharacteristic
195C4×C7C2×#4111yescharacteristic
196Dic7—21no—
197Dic7—21no—
198Dic7—42no—
199Dic7—41no—
200Dic7—41no—
201C24—71no—
202#41—22no—
203#41—22no—
204C2×Q8—72no—
205C2×Q8—141no—
206C2×Q8—142no—
207C2×Q8—142no—
208C2×Q8C22×D711yescharacteristic
209C2×D4—21no—
210C2×D4—141no—
211C2×D4—141no—
212C2×D4C22×D711yescharacteristic
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—71no—
216C22×C4—71no—
217C22×C4—71no—
218C22×C4—71no—
219C22×C4—71no—
220C22×C4—142no—
221C22×C4—142no—
222C22×C4—141no—
223C22×C4C22×D711yescharacteristic
224Q16—144no—
225SD16—144no—
226M16—22no—
227M16—74no—
228M16—144no—
229C2×C8—72no—
230C2×C8—144no—
231C2×C8C22×D712yes—
232C4⋊C4—21no—
233C4⋊C4—71no—
234C4⋊C4—71no—
235C4⋊C4—141no—
236C4⋊C4—141no—
237C4⋊C4—142no—
238C4⋊C4—142no—
239C4⋊C4C22×D712yes—
240C22⋊C4—142no—
241C22⋊C4—142no—
242C22⋊C4—141no—
243C22⋊C4—141no—
244C22⋊C4—141no—
245C22⋊C4—141no—
246C22⋊C4—141no—
247C42—141no—
248C2×C7—22no—
249C2×C7—81no—
250C2×C7C2×#12111yescharacteristic
251C2×C7C2×#13711yescharacteristic
252C2×C7#57511yescharacteristic
253D7—22no—
254D7—41no—
255C23—21no—
256C23—71no—
257C23—71no—
258C23—142no—
259C23—142no—
260C23—281no—
261C23D4×D711yescharacteristic
262Q8—21no—
263Q8—21no—
264Q8—142no—
265Q8—142no—
266Q8—282no—
267D4—21no—
268D4—21no—
269D4—42no—
270D4—282no—
271D4—282no—
272C2×C4—22no—
273C2×C4—21no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—71no—
277C2×C4—71no—
278C2×C4—142no—
279C2×C4—142no—
280C2×C4—142no—
281C2×C4—141no—
282C2×C4—141no—
283C2×C4—141no—
284C2×C4—141no—
285C2×C4—141no—
286C2×C4—141no—
287C2×C4—142no—
288C2×C4—282no—
289C2×C4#115712yes—
290C2×C4D4×D711yescharacteristic
291C2×C4D4×D712yes—
292C2×C4C23×D711yescharacteristic
293C8—22no—
294C8—74no—
295C7#305611yescharacteristic, complemented, Sylow
296C22—21no—
297C22—21no—
298C22—41no—
299C22—41no—
300C22—41no—
301C22—72no—
302C22—141no—
303C22—141no—
304C22—141no—
305C22—141no—
306C22—141no—
307C22—284no—
308C22#644712yes—
309C22C2×D4×D711yescharacteristic
310C4—41no—
311C4—42no—
312C4—141no—
313C4—141no—
314C4—282no—
315C4—281no—
316C4—281no—
317C4#644712yes—
318C4C2×#115711yescharacteristic
319C4C2×D4×D711yescharacteristic
320C2—22no—
321C2—81no—
322C2—142no—
323C2—281no—
324C2C2×#644711yescharacteristic
325C2#9046411yescharacteristic
326C2D7×#13711yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#417C2
socleC22×C7C22×D4
prime radical#90519C2
hypercenter#417D7
hypocenterC7#3056
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
2. C2×#6514C2semidirect
3. C2×D7×M16C2semidirect
4. C2×#6460C2semidirect
5. C7×#417C2semidirect
6. #89892C2semidirect
7. #89736C2semidirect
8. #89588C2semidirect
9. #89585C2semidirect
10. #89552C2semidirect
11. #89550C2semidirect
20. C2×#1144C22semidirect
25. C2×#1142C22semidirect
39. C7×#103C22semidirect
40. C7×#102C22semidirect
41. #6390C22semidirect
42. #6390C22semidirect
43. #6382C22semidirect
44. #6381C22semidirect
89. #1136D4semidirect
103. #417D7semidirect
175. #103C2×D7semidirect
176. #102C2×D7semidirect
190. C2×D7#137semidirect
295. C7#3056semidirect

Maximal subgroups

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

subgroupindex
#905192
C2×#65142
C2×D7×M162
C2×#64602
C7×#4172
#898922
#897362
#895882
#895852
#895522
#895502
#30567