Finite Group Databasebeta

← #11740177

Subgroups — #11740177

Order 896.

496 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×C22⋊Q8C211yesmaximal, characteristic, complemented
2C2×#6505C211yesmaximal, characteristic, complemented
3#90482C211yesmaximal, characteristic
4C22×#1136C211yesmaximal, characteristic, complemented
5#89997C211yesmaximal, characteristic, complemented
6#89824C212yesmaximal, complemented
7#89823C212yesmaximal, complemented
8#89821C212yesmaximal, complemented
9#89820C212yesmaximal, complemented
10#89738C211yesmaximal, characteristic, complemented
11#89736C211yesmaximal, characteristic, complemented
12C22×C7×Q8C2211yescharacteristic
13C23×C4×C7C2211yescharacteristic
14C2×#1157C2211yescharacteristic
15C7×C22⋊Q8—21no—
16C7×C22⋊Q8—21no—
17C7×C22⋊Q8C2212yes—
18C7×C22⋊Q8C2212yes—
19C2×C7×C4⋊C4—21no—
20C2×C7×C4⋊C4C2211yescharacteristic
21C2×C7×C22⋊C4—21no—
22#6516—21no—
23C2×#1144C2212yescomplemented
24#6505—24no—
25C2×#1143C2212yescomplemented
26#6493C2211yescharacteristic
27#6493C2211yescharacteristic
28#6492—21no—
29C4×C22⋊D7—21no—
30C4×C22⋊D7—21no—
31#6489C2211yescharacteristic
32#6487C2211yescharacteristic
33#6487C2211yescharacteristic
34C2×#1136—24no—
35C2×#1136C2211yescharacteristic
36C2×#1136C2212yes—
37C2×#1136C2211yescharacteristic
38C22×C7⋊C8C2212yes—
39#6409C2212yescomplemented
40#6409C2212yescomplemented
41#6384C2212yescomplemented
42#6383C2212yescomplemented
43#6382C2212yescomplemented
44#6381C2212yescomplemented
45#3056—71nomaximal, Sylow
46C24×C7D411yescharacteristic
47C2×C7×Q8—21no—
48C2×C7×Q8—22no—
49C2×C7×Q8—22no—
50C2×C7×Q8C2312yes—
51C22×C4×C7—22no—
52C22×C4×C7—22no—
53C22×C4×C7—21no—
54C22×C4×C7—21no—
55C22×C4×C7D411yescharacteristic
56C22×C4×C7D411yescharacteristic
57C22×C4×C7D411yescharacteristic
58C22×C4×C7C2311yescharacteristic
59C22×C4×C7C2311yescharacteristic
60C22×C4×C7C2311yescharacteristic
61C2×C22⋊D7—21no—
62C2×C22⋊D7—21no—
63C2×C22⋊D7—21no—
64#1157—22no—
65#1157—22no—
66C2×D28C2311yescharacteristic
67C2×C4×D7—21no—
68C2×Dic14C2311yescharacteristic
69C7×C4⋊C4—22no—
70C7×C4⋊C4—22no—
71C7×C4⋊C4—21no—
72C7×C4⋊C4C2312yes—
73C7×C22⋊C4—22no—
74C7×C22⋊C4—22no—
75C22⋊Dic7—21no—
76C22⋊Dic7—21no—
77C22⋊Dic7—21no—
78#1144—24no—
79#1143—24no—
80(C2×D7)⋊C4—21no—
81(C2×D7)⋊C4—21no—
82C4⋊Dic7C2311yescharacteristic
83C4⋊Dic7C2311yescharacteristic
84Dic7⋊C4—21no—
85Dic7⋊C4—21no—
86C4×Dic7—21no—
87#1136—22no—
88#1136—24no—
89#1136D414yescomplemented
90C2×C7⋊C8—24no—
91C2×C7⋊C8C2312yes—
92C2×C7⋊C8C2312yes—
93C2×#137—71no—
94C22×M16—71no—
95#525—71no—
96C2×C22⋊Q8D711yescharacteristic, complemented
97#470—72no—
98#468—72no—
99#467—72no—
100#465—72no—
101#428—71no—
102#419—71no—
103#417—71no—
104C23×C7—22no—
105C23×C7—22no—
106C23×C7—41no—
107C23×C7C2×D411yescharacteristic
108C23×C7C2×D411yescharacteristic
109C23×C7C2×D411yescharacteristic
110C22×D7—21no—
111C7×Q8—22no—
112C7×Q8—22no—
113C7×Q8—42no—
114C2×C4×C7—22no—
115C2×C4×C7—22no—
116C2×C4×C7—22no—
117C2×C4×C7—21no—
118C2×C4×C7—21no—
119C2×C4×C7—21no—
120C2×C4×C7—21no—
121C2×C4×C7—22no—
122C2×C4×C7—22no—
123C2×C4×C7—42no—
124C2×C4×C7—42no—
125C2×C4×C7C2×D412yes—
126C2×C4×C7C2×D411yescharacteristic
127C2×C4×C7C2×D411yescharacteristic
128C2×C4×C7C2×D411yescharacteristic
129C2×C4×C7#4112yes—
130C2×C4×C7C2411yescharacteristic
131C22⋊D7—42no—
132C22⋊D7—42no—
133C22⋊D7—42no—
134C2×Dic7—21no—
135C2×Dic7—21no—
136C2×Dic7—21no—
137D28—21no—
138D28—21no—
139C4×D7—42no—
140Dic14—21no—
141Dic14—21no—
142C7⋊C8—22no—
143C7⋊C8C2×D414yes—
144C2×#41—71no—
145C22×Q8C2×D711yescharacteristic
146C23×C4C2×D711yescharacteristic
147#137—144no—
148C2×Q16—72no—
149C2×SD16—72no—
150C2×M16—71no—
151C2×M16—71no—
152C2×M16—72no—
153C2×M16—144no—
154C22×C8—72no—
155#123—141no—
156C22⋊Q8—21no—
157C22⋊Q8—21no—
158C22⋊Q8—71no—
159C22⋊Q8—71no—
160C22⋊Q8C2×D712yes—
161C22⋊Q8C2×D712yes—
162#121—71no—
163#121—71no—
164#120—141no—
165C4×D4—141no—
166C4×D4—141no—
167#117—71no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4C2×D711yescharacteristic
170C2×C22⋊C4—21no—
171#107—72no—
172#106—72no—
173#103—72no—
174#103—72no—
175#103—72no—
176#102—72no—
177C22×C7—21no—
178C22×C7—21no—
179C22×C7—21no—
180C22×C7—21no—
181C22×C7—21no—
182C22×C7—21no—
183C22×C7—21no—
184C22×C7—44no—
185C22×C7#12112yes—
186C22×C7#13712yescomplemented
187C22×C7C22×D411yescharacteristic
188C2×D7—41no—
189C2×D7—41no—
190C2×D7—41no—
191C4×C7—21no—
192C4×C7—21no—
193C4×C7—42no—
194C4×C7—42no—
195C4×C7#12112yes—
196C4×C7C22×D411yescharacteristic
197C4×C7C2×#4111yescharacteristic
198Dic7—41no—
199Dic7—41no—
200Dic7—41no—
201C24C22⋊D711yescharacteristic
202#41—142no—
203#41—142no—
204C2×Q8—21no—
205C2×Q8—22no—
206C2×Q8—22no—
207C2×Q8—71no—
208C2×Q8C22×D712yes—
209C2×D4—71no—
210C2×D4—141no—
211C2×D4—141no—
212C2×D4—141no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—141no—
218C22×C4C22⋊D711yescharacteristic
219C22×C4C22⋊D711yescharacteristic
220C22×C4C22⋊D711yescharacteristic
221C22×C4C22×D711yescharacteristic
222C22×C4C22×D711yescharacteristic
223C22×C4C22×D711yescharacteristic
224Q16—144no—
225SD16—144no—
226M16—74no—
227M16—144no—
228M16—142no—
229C2×C8—72no—
230C2×C8—72no—
231C2×C8—144no—
232C4⋊C4—22no—
233C4⋊C4—22no—
234C4⋊C4—21no—
235C4⋊C4—71no—
236C4⋊C4—71no—
237C4⋊C4—141no—
238C4⋊C4—141no—
239C4⋊C4C22×D712yes—
240C22⋊C4—22no—
241C22⋊C4—22no—
242C22⋊C4—141no—
243C22⋊C4—141no—
244C22⋊C4—141no—
245C22⋊C4—141no—
246C22⋊C4—141no—
247C42—141no—
248C2×C7—22no—
249C2×C7—22no—
250C2×C7—41no—
251C2×C7C2×#12111yescharacteristic
252C2×C7C2×#13711yescharacteristic
253C2×C7#57511yescharacteristic
254D7—81no—
255C23—22no—
256C23—22no—
257C23—41no—
258C23—141no—
259C23C2×C22⋊D711yescharacteristic
260C23C2×C22⋊D711yescharacteristic
261C23C2×C22⋊D711yescharacteristic
262Q8—22no—
263Q8—22no—
264Q8—42no—
265Q8—141no—
266Q8—141no—
267D4—141no—
268D4—141no—
269D4—282no—
270D4—282no—
271D4—282no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—141no—
284C2×C4—141no—
285C2×C4—141no—
286C2×C4—282no—
287C2×C4D4×D712yes—
288C2×C4#115912yes—
289C2×C4C2×C22⋊D711yescharacteristic
290C2×C4C2×C22⋊D711yescharacteristic
291C2×C4C2×C22⋊D711yescharacteristic
292C2×C4C23×D711yescharacteristic
293C8—74no—
294C8—142no—
295C7#305611yescharacteristic, complemented, Sylow
296C22—21no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—44no—
304C22—281no—
305C22—281no—
306C22—281no—
307C22#650212yes—
308C22#650512yescomplemented
309C22C22×C22⋊D711yescharacteristic
310C4—21no—
311C4—21no—
312C4—42no—
313C4—42no—
314C4—281no—
315C4—281no—
316C4—281no—
317C4#650212yes—
318C4C2×D4×D711yescharacteristic
319C4C2×#115911yescharacteristic
320C2—22no—
321C2—22no—
322C2—41no—
323C2—561no—
324C2C2×#650211yescharacteristic
325C2C2×#650511yescharacteristic
326C2#9051311yescharacteristic
subgroupG/H
centerC22C22×C22⋊D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC2×C7×C22⋊Q8C2
socleC22×C7C22×D4
prime radical#90482C2
hypercenterC2×C22⋊Q8D7
hypocenterC7#3056
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×C7×C22⋊Q8C2semidirect
2. C2×#6505C2semidirect
4. C22×#1136C2semidirect
5. #89997C2semidirect
6. #89824C2semidirect
7. #89823C2semidirect
8. #89821C2semidirect
9. #89820C2semidirect
10. #89738C2semidirect
11. #89736C2semidirect
23. C2×#1144C22semidirect
25. C2×#1143C22semidirect
39. #6409C22semidirect
40. #6409C22semidirect
41. #6384C22semidirect
42. #6383C22semidirect
43. #6382C22semidirect
44. #6381C22semidirect
89. #1136D4semidirect
96. C2×C22⋊Q8D7semidirect
186. C22×C7#137semidirect
295. C7#3056semidirect
308. C22#6505semidirect

Maximal subgroups

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

subgroupindex
C2×C7×C22⋊Q82
C2×#65052
#904822
C22×#11362
#899972
#898242
#898232
#898212
#898202
#897382
#897362
#30567