Finite Group Databasebeta

← #11737635

Subgroups — #11737635

Order 896.

414 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 113 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×#6509C211yesmaximal, characteristic, complemented
2#90318C211yesmaximal, characteristic
3C7×#451C211yesmaximal, characteristic, complemented
4#89999C211yesmaximal, characteristic, complemented
5#89961C212yesmaximal, complemented
6#89959C212yesmaximal, complemented
7#89816C211yesmaximal, characteristic, complemented
8#89587C212yesmaximal, complemented
9#89584C212yesmaximal, complemented
10#89507C211yesmaximal, characteristic, complemented
11#89498C211yesmaximal, characteristic, complemented
12C22×C7×Q8C2211yescharacteristic
13C2×#1161C2211yescharacteristic
14C22×C4×D7C2211yescharacteristic
15C2×C7×Q16C2212yescomplemented
16C7×C22⋊Q8C2211yescharacteristic
17#6509—21no—
18#6509—21no—
19#6509C2212yes—
20#6509C2211yescharacteristic
21#6509C2212yes—
22C2×#1143C2212yescomplemented
23#6493C2211yescharacteristic
24C2×(C2×D7)⋊C4—21no—
25C2×C4⋊Dic7C2211yescharacteristic
26C2×Dic7⋊C4—21no—
27C2×#1131C2212yes—
28#6458C2211yescharacteristic
29#6457—21no—
30#6456—21no—
31#6455C2211yescharacteristic
32#6445—21no—
33(C22⋊D7)⋊C4—21no—
34C7×#103C2212yescomplemented
35C7×C22⋊C8C2211yescharacteristic
36#6409C2212yescomplemented
37#6409C2212yescomplemented
38C22⋊(C7⋊C8)C2211yescharacteristic
39#6395C2212yescomplemented
40(C2×D7)⋊C8C2212yes—
41#6391C2212yescomplemented
42#6381C2212yescomplemented
43#3196—71nomaximal, Sylow
44C23×D7D411yescharacteristic
45C2×C7×Q8—21no—
46C2×C7×Q8—42no—
47C2×C7×Q8C2311yescharacteristic
48C2×C7×Q8C2312yes—
49C22×C4×C7—21no—
50C22×C4×C7C2311yescharacteristic
51C2×C22⋊D7—21no—
52C22×Dic7—21no—
53C22×Dic7D411yescharacteristic
54#1161—42no—
55C2×D28—21no—
56C2×D28—21no—
57C2×D28C2311yescharacteristic
58C2×C4×D7—21no—
59C2×C4×D7—42no—
60C2×C4×D7D411yescharacteristic
61C2×C4×D7D411yescharacteristic
62C2×C4×D7C2311yescharacteristic
63C2×C4×D7C2311yescharacteristic
64C7×Q16—42no—
65C2×C7×C8C2312yes—
66C7×C4⋊C4—21no—
67C7×C4⋊C4C2311yescharacteristic
68C7×C22⋊C4—21no—
69#1143—42no—
70(C2×D7)⋊C4—21no—
71(C2×D7)⋊C4—21no—
72(C2×D7)⋊C4—22no—
73(C2×D7)⋊C4—22no—
74(C2×D7)⋊C4—21no—
75(C2×D7)⋊C4—21no—
76C4⋊Dic7—21no—
77C4⋊Dic7C2311yescharacteristic
78C4⋊Dic7C2312yes—
79Dic7⋊C4—21no—
80Dic7⋊C4—22no—
81Dic7⋊C4—22no—
82C4×Dic7—21no—
83C2×C7⋊C8C2312yes—
84#1131—42no—
85#525—71no—
86C2×C22⋊Q8—71no—
87#485—72no—
88#484—71no—
89#481—71no—
90#470—72no—
91#468—72no—
92#452—72no—
93#451D711yescharacteristic, complemented
94#448—71no—
95#407—71no—
96C23×C7C2×D411yescharacteristic
97C22×D7—22no—
98C22×D7—21no—
99C22×D7—41no—
100C22×D7—41no—
101C22×D7C2×D411yescharacteristic
102C22×D7C2×D411yescharacteristic
103C7×Q8—22no—
104C7×Q8—22no—
105C7×Q8—22no—
106C7×Q8—42no—
107C2×C4×C7—22no—
108C2×C4×C7—21no—
109C2×C4×C7—22no—
110C2×C4×C7—21no—
111C2×C4×C7—42no—
112C2×C4×C7C2×D411yescharacteristic
113C2×C4×C7C2411yescharacteristic
114C22⋊D7—42no—
115C2×Dic7—21no—
116C2×Dic7—22no—
117C2×Dic7—41no—
118C2×Dic7—42no—
119C2×Dic7C2×D411yescharacteristic
120C2×Dic7C2×D411yescharacteristic
121D28—22no—
122D28—42no—
123D28—42no—
124C4×D7—22no—
125C4×D7—22no—
126C4×D7—41no—
127C4×D7—41no—
128C4×D7—42no—
129C7×C8—22no—
130C7⋊C8—22no—
131C2×#41—71no—
132C22×Q8C2×D711yescharacteristic
133C23×C4—71no—
134C2×Q16C2×D712yescomplemented
135C2×SD16—72no—
136C2×M16—72no—
137#123—141no—
138C22⋊Q8—72no—
139C22⋊Q8—72no—
140C22⋊Q8—71no—
141C22⋊Q8—141no—
142C22⋊Q8—141no—
143C22⋊Q8C2×D711yescharacteristic
144#121—71no—
145#121—71no—
146#120—141no—
147C4×D4—141no—
148C4×D4—141no—
149#117—71no—
150C2×C4⋊C4—71no—
151C2×C4⋊C4—141no—
152C2×C22⋊C4—141no—
153#107—72no—
154#106—72no—
155#103—72no—
156#103—72no—
157#103C2×D712yescomplemented
158#102—72no—
159C22⋊C8—71no—
160C22⋊C8—72no—
161C22⋊C8C2×D711yescharacteristic
162C22×C7—21no—
163C22×C7—21no—
164C22×C7#13712yes—
165C22×C7C22×D411yescharacteristic
166C2×D7—21no—
167C2×D7—21no—
168C2×D7—21no—
169C2×D7—21no—
170C2×D7—21no—
171C2×D7—21no—
172C2×D7—44no—
173C2×D7—41no—
174C2×D7—41no—
175C2×D7—41no—
176C4×C7—21no—
177C4×C7—41no—
178C4×C7—42no—
179C4×C7—41no—
180C4×C7H(5,𝔽2)12yes—
181Dic7—21no—
182Dic7—21no—
183Dic7—41no—
184Dic7—42no—
185C24—71no—
186#41—282no—
187C2×Q8—21no—
188C2×Q8—42no—
189C2×Q8C22×D711yescharacteristic
190C2×Q8C22×D712yes—
191C2×D4—71no—
192C2×D4—141no—
193C2×D4—141no—
194C2×D4—141no—
195C22×C4—21no—
196C22×C4—71no—
197C22×C4—71no—
198C22×C4—71no—
199C22×C4—71no—
200C22×C4—71no—
201C22×C4—141no—
202C22×C4—141no—
203C22×C4—282no—
204C22×C4C22×D711yescharacteristic
205Q16—42no—
206SD16—282no—
207M16—282no—
208C2×C8—72no—
209C2×C8C22×D712yes—
210C4⋊C4—21no—
211C4⋊C4—72no—
212C4⋊C4—71no—
213C4⋊C4—142no—
214C4⋊C4—142no—
215C4⋊C4—141no—
216C4⋊C4—141no—
217C4⋊C4C22×D711yescharacteristic
218C22⋊C4—21no—
219C22⋊C4—141no—
220C22⋊C4—142no—
221C22⋊C4—142no—
222C22⋊C4—141no—
223C22⋊C4—141no—
224C22⋊C4—141no—
225C42—141no—
226C2×C7—22no—
227C2×C7#53411yescharacteristic
228C2×C7C2×#13711yescharacteristic
229C2×C7#57511yescharacteristic
230D7—41no—
231D7—41no—
232D7—81no—
233C23—71no—
234C23—71no—
235C23—142no—
236C23—141no—
237C23—281no—
238C23—281no—
239C23D4×D711yescharacteristic
240Q8—22no—
241Q8—22no—
242Q8—22no—
243Q8—42no—
244D4—142no—
245D4—282no—
246D4—282no—
247D4—282no—
248C2×C4—22no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—42no—
253C2×C4—71no—
254C2×C4—71no—
255C2×C4—142no—
256C2×C4—142no—
257C2×C4—141no—
258C2×C4—142no—
259C2×C4—281no—
260C2×C4—281no—
261C2×C4—282no—
262C2×C4—281no—
263C2×C4—282no—
264C2×C4D4×D711yescharacteristic
265C2×C4C23×D711yescharacteristic
266C8—22no—
267C8—142no—
268C7#319611yescharacteristic, complemented, Sylow
269C22—21no—
270C22—21no—
271C22—141no—
272C22—141no—
273C22—141no—
274C22—141no—
275C22—141no—
276C22—141no—
277C22—284no—
278C22—281no—
279C22—281no—
280C22—281no—
281C22#648112yes—
282C22C2×D4×D711yescharacteristic
283C4—21no—
284C4—41no—
285C4—42no—
286C4—41no—
287C4—141no—
288C4—141no—
289C4—281no—
290C4—282no—
291C4#654812yes—
292C2—22no—
293C2—281no—
294C2—281no—
295C2—561no—
296C2#9028011yescharacteristic
297C2C2×#648111yescharacteristic
298C2#9046811yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#451C2
socleC22×C7C22×D4
prime radical#90318C2
hypercenter#451D7
hypocenterC7#3196
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
1. C2×#6509C2semidirect
3. C7×#451C2semidirect
4. #89999C2semidirect
5. #89961C2semidirect
6. #89959C2semidirect
7. #89816C2semidirect
8. #89587C2semidirect
9. #89584C2semidirect
10. #89507C2semidirect
11. #89498C2semidirect
15. C2×C7×Q16C22semidirect
22. C2×#1143C22semidirect
34. C7×#103C22semidirect
36. #6409C22semidirect
37. #6409C22semidirect
39. #6395C22semidirect
41. #6391C22semidirect
42. #6381C22semidirect
93. #451D7semidirect
134. C2×Q16C2×D7semidirect
157. #103C2×D7semidirect
268. C7#3196semidirect

Maximal subgroups

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

subgroupindex
C2×#65092
#903182
C7×#4512
#899992
#899612
#899592
#898162
#895872
#895842
#895072
#894982
#31967