Finite Group Databasebeta

← #11737847

Subgroups — #11737847

Order 896.

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

normal subgroups only

This group has 392 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
1#90407C211yesmaximal, characteristic, complemented
2#90271C211yesmaximal, characteristic, complemented
3C7×#460C211yesmaximal, characteristic, complemented
4#89946C212yesmaximal, complemented
5#89937C212yesmaximal, complemented
6#89844C211yesmaximal, characteristic, complemented
7#89792C211yesmaximal, characteristic, complemented
8#89618C211yesmaximal, characteristic, complemented
9#89609C211yesmaximal, characteristic, complemented
10#89554C212yesmaximal, complemented
11#89538C212yesmaximal, complemented
12C2×#1159—21no—
13C2×C7×SD16C2212yes—
14C7×#127C2211yescharacteristic
15C4×C7×Q8C2211yescharacteristic
16#6510C2211yescharacteristic
17#6509C2211yescharacteristic
18#6501—21no—
19#6501C2212yes—
20#6499—21no—
21D4×Dic7—21no—
22D4×Dic7C2212yes—
23C2×#1141C2212yes—
24C2×#1131C2212yes—
25#6461—21no—
26#6458C2211yescharacteristic
27#6457—21no—
28#6453C2211yescharacteristic
29C4×D28—21no—
30C4×D28C2211yescharacteristic
31#6435—21no—
32C42×D7C2211yescharacteristic
33#6432C2211yescharacteristic
34C7×C4⋊C8C2211yescharacteristic
35C7×#102C2212yes—
36#6409C2212yes—
37#6406C2212yes—
38#6395C2212yes—
39#6392C2212yes—
40Dic7⋊C8C2212yes—
41#6382C2212yes—
42C4⋊(C7⋊C8)C2211yescharacteristic
43#3251—71nomaximal, Sylow
44C2×C7×Q8C2311yescharacteristic
45C2×C7×D4—21no—
46C2×C7×D4—21no—
47C2×C7×D4C2312yes—
48C2×C22⋊D7—22no—
49C22×Dic7—22no—
50#1159—44no—
51C2×D28—21no—
52C2×D28C2311yescharacteristic
53C2×C4×D7—21no—
54C2×C4×D7D411yescharacteristic
55C2×C4×D7D411yescharacteristic
56C2×C4×D7C2311yescharacteristic
57C2×Dic14—21no—
58C7×SD16—42no—
59C2×C7×C8C2312yes—
60C7×C4⋊C4—21no—
61C7×C4⋊C4C2311yescharacteristic
62C42×C7—21no—
63C42×C7C2311yescharacteristic
64C22⋊Dic7—22no—
65C22⋊Dic7—22no—
66#1141—42no—
67(C2×D7)⋊C4—21no—
68(C2×D7)⋊C4—21no—
69(C2×D7)⋊C4—21no—
70(C2×D7)⋊C4—21no—
71(C2×D7)⋊C4—21no—
72C4⋊Dic7—21no—
73C4⋊Dic7—21no—
74C4⋊Dic7C2311yescharacteristic
75C4⋊Dic7C2312yes—
76Dic7⋊C4—21no—
77Dic7⋊C4—21no—
78C4×Dic7D411yescharacteristic
79C4×Dic7D411yescharacteristic
80C4×Dic7C2311yescharacteristic
81C2×C7⋊C8C2312yes—
82#1131—42no—
83#532—71no—
84#529—71no—
85#478—72no—
86#475—71no—
87#474—71no—
88#469—72no—
89#468—72no—
90#463—72no—
91#460D711yescharacteristic, complemented
92#459—71no—
93#423—71no—
94C23×C7—22no—
95C22×D7—21no—
96C22×D7C2×D411yescharacteristic
97C7×Q8—21no—
98C7×Q8—21no—
99C7×D4—22no—
100C7×D4—22no—
101C7×D4—42no—
102C7×D4—42no—
103C2×C4×C7—21no—
104C2×C4×C7—21no—
105C2×C4×C7C2×D411yescharacteristic
106C2×C4×C7C2×D411yescharacteristic
107C2×C4×C7C2411yescharacteristic
108C22⋊D7—44no—
109C2×Dic7—21no—
110C2×Dic7—22no—
111C2×Dic7—44no—
112C2×Dic7C2×D411yescharacteristic
113C2×Dic7C2×D411yescharacteristic
114C2×Dic7C2×D411yescharacteristic
115D28—21no—
116D28—21no—
117D28—42no—
118C4×D7—22no—
119C4×D7—21no—
120C4×D7—21no—
121C4×D7—41no—
122C4×D7—42no—
123Dic14—42no—
124C7×C8—22no—
125C7⋊C8—22no—
126C2×#41—141no—
127C2×SD16C2×D712yes—
128C2×D8—72no—
129C2×M16—72no—
130C4⋊Q8—71no—
131#127C2×D711yescharacteristic
132#126—141no—
133#125—71no—
134#124—71no—
135#124—141no—
136#123—141no—
137C22⋊Q8—71no—
138#121—72no—
139#121—71no—
140#121—141no—
141C4×Q8C2×D711yescharacteristic
142C4×D4—72no—
143C4×D4—71no—
144C4×D4—141no—
145C4×D4—141no—
146#117—141no—
147C2×C42—71no—
148#107—72no—
149#106—72no—
150C4⋊C8—72no—
151C4⋊C8—71no—
152C4⋊C8C2×D711yescharacteristic
153#103—72no—
154#102—72no—
155#102—72no—
156#102C2×D712yes—
157C22×C7—42no—
158C22×C7—42no—
159C22×C7—42no—
160C22×C7C22×D411yescharacteristic
161C2×D7—21no—
162C2×D7—21no—
163C2×D7—21no—
164C2×D7—41no—
165C2×D7—41no—
166C2×D7—41no—
167C4×C7—21no—
168C4×C7—41no—
169C4×C7—41no—
170C4×C7AGL(1,ℤ/8)12yes—
171C4×C7H(5,𝔽2)11yescharacteristic
172C4×C72−1+411yescharacteristic
173Dic7—21no—
174Dic7—21no—
175Dic7—21no—
176Dic7—42no—
177Dic7—41no—
178#41—284no—
179C2×Q8—141no—
180C2×Q8C22×D711yescharacteristic
181C2×D4—21no—
182C2×D4—21no—
183C2×D4—71no—
184C2×D4—141no—
185C2×D4—142no—
186C2×D4C22×D712yes—
187C22×C4—71no—
188C22×C4—71no—
189C22×C4—71no—
190C22×C4—142no—
191C22×C4—141no—
192SD16—42no—
193D8—282no—
194M16—282no—
195C2×C8—72no—
196C2×C8C22×D712yes—
197C4⋊C4—21no—
198C4⋊C4—71no—
199C4⋊C4—72no—
200C4⋊C4—141no—
201C4⋊C4—141no—
202C4⋊C4—141no—
203C4⋊C4—141no—
204C4⋊C4C22×D711yescharacteristic
205C22⋊C4—141no—
206C22⋊C4—141no—
207C22⋊C4—141no—
208C22⋊C4—141no—
209C22⋊C4—142no—
210C22⋊C4—142no—
211C22⋊C4—141no—
212C42—21no—
213C42—71no—
214C42—71no—
215C42—71no—
216C42C22×D711yescharacteristic
217C2×C7—82no—
218C2×C7#53711yescharacteristic
219C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
220C2×C7#57511yescharacteristic
221D7—41no—
222D7—81no—
223C23—22no—
224C23—71no—
225C23—141no—
226Q8—21no—
227Q8—21no—
228Q8—282no—
229D4—22no—
230D4—22no—
231D4—42no—
232D4—42no—
233D4—141no—
234D4—141no—
235D4—282no—
236D4—284no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—71no—
240C2×C4—71no—
241C2×C4—71no—
242C2×C4—142no—
243C2×C4—141no—
244C2×C4—141no—
245C2×C4—141no—
246C2×C4—142no—
247C2×C4—281no—
248C2×C4—284no—
249C2×C4—282no—
250C2×C4D4×D711yescharacteristic
251C2×C4D4×D711yescharacteristic
252C2×C4C23×D711yescharacteristic
253C8—22no—
254C8—142no—
255C7#325111yescharacteristic, complemented, Sylow
256C22—42no—
257C22—42no—
258C22—42no—
259C22—141no—
260C22—141no—
261C22—141no—
262C22—281no—
263C22—281no—
264C22—281no—
265C22C2×D4×D711yescharacteristic
266C4—21no—
267C4—41no—
268C4—41no—
269C4—141no—
270C4—141no—
271C4—141no—
272C4—282no—
273C4—281no—
274C4#647712yes—
275C4#654811yescharacteristic
276C4#655411yescharacteristic
277C2—82no—
278C2—281no—
279C2—561no—
280C2#9029911yescharacteristic
281C2C2×#647711yescharacteristic
282C2#9046411yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#460C2
socleC22×C7C22×D4
hypercenter#460D7
hypocenterC7#3251
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. #90407C2semidirect
2. #90271C2semidirect
3. C7×#460C2semidirect
4. #89946C2semidirect
5. #89937C2semidirect
6. #89844C2semidirect
7. #89792C2semidirect
8. #89618C2semidirect
9. #89609C2semidirect
10. #89554C2semidirect
11. #89538C2semidirect
91. #460D7semidirect
255. C7#3251semidirect

Maximal subgroups

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

subgroupindex
#904072
#902712
C7×#4602
#899462
#899372
#898442
#897922
#896182
#896092
#895542
#895382
#32517