Finite Group Databasebeta

← #11736799

Subgroups — #11736799

Order 896.

396 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 117 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#90170C211yesmaximal, characteristic, complemented
2#90168C211yesmaximal, characteristic, complemented
3C7×#432C211yesmaximal, characteristic, complemented
4#89617C212yesmaximal, complemented
5#89616C212yesmaximal, complemented
6#89506C212yesmaximal, complemented
7#89505C212yesmaximal, complemented
8#89487C211yesmaximal, characteristic, complemented
9#89485C211yesmaximal, characteristic, complemented
10#89484C211yesmaximal, characteristic, complemented
11#89475C211yesmaximal, characteristic, complemented
12C2×#1157C2212yes—
13C2×C42×C7C2211yescharacteristic
14#6493—21no—
15#6493—21no—
16#6493C2211yescharacteristic
17#6492—21no—
18C4×C22⋊D7—21no—
19#6487C2211yescharacteristic
20C2×Dic28C2212yes—
21C2×D56C2212yes—
22C2×#1132C2212yes—
23C2×#1132C2212yes—
24#6439—21no—
25#6438C2212yes—
26#6438C2211yescharacteristic
27#6437C2211yescharacteristic
28C4×D28—21no—
29C4×D28—21no—
30C4×D28C2211yescharacteristic
31#6435—21no—
32#6433C2211yescharacteristic
33#6432C2211yescharacteristic
34C4×Dic14C2211yescharacteristic
35C7×C4⋊C8C2212yes—
36C7×C22⋊C8C2212yes—
37C7×#97C2211yescharacteristic
38C7×#97C2211yescharacteristic
39#6395C2212yes—
40#6395C2212yes—
41#6390C2212yes—
42#6390C2212yes—
43#3219—71nomaximal, Sylow
44C22×C4×C7D411yescharacteristic
45C22×C4×C7D411yescharacteristic
46C22×C4×C7C2311yescharacteristic
47C2×C22⋊D7—21no—
48C2×C22⋊D7—22no—
49#1157—42no—
50#1157—42no—
51C2×D28—21no—
52C2×D28—21no—
53C2×D28C2311yescharacteristic
54C2×D28C2312yes—
55C2×C4×D7—21no—
56C2×C4×D7—22no—
57C2×Dic14C2311yescharacteristic
58C2×Dic14C2312yes—
59C2×C7×C8C2312yes—
60C2×C7×C8C2312yes—
61C42×C7D411yescharacteristic
62C42×C7D411yescharacteristic
63C42×C7C2311yescharacteristic
64C42×C7C2311yescharacteristic
65C22⋊Dic7—21no—
66(C2×D7)⋊C4—21no—
67(C2×D7)⋊C4—21no—
68(C2×D7)⋊C4—22no—
69(C2×D7)⋊C4—22no—
70(C2×D7)⋊C4—21no—
71(C2×D7)⋊C4—21no—
72C4⋊Dic7—21no—
73C4⋊Dic7—21no—
74C4⋊Dic7C2311yescharacteristic
75C4⋊Dic7C2311yescharacteristic
76Dic7⋊C4—21no—
77Dic7⋊C4—21no—
78Dic7⋊C4—21no—
79Dic7⋊C4—21no—
80C4×Dic7—21no—
81Dic28—42no—
82D56—42no—
83#1132—42no—
84#1132—42no—
85#532—71no—
86#529—71no—
87#497—71no—
88#496—71no—
89#491—71no—
90#490—71no—
91#464—72no—
92#463—72no—
93#452—72no—
94#449—72no—
95#432D711yescharacteristic, complemented
96C23×C7C2×D411yescharacteristic
97C22×D7—21no—
98C22×D7—22no—
99C2×C4×C7—22no—
100C2×C4×C7—41no—
101C2×C4×C7—41no—
102C2×C4×C7C2×D411yescharacteristic
103C2×C4×C7C2×D411yescharacteristic
104C2×C4×C7C2×D411yescharacteristic
105C2×C4×C7C2×D411yescharacteristic
106C2×C4×C7C2×D411yescharacteristic
107C2×C4×C7C2411yescharacteristic
108C22⋊D7—44no—
109C22⋊D7—42no—
110C2×Dic7—21no—
111C2×Dic7—21no—
112C2×Dic7—21no—
113C2×Dic7—22no—
114D28—22no—
115D28—22no—
116D28—22no—
117D28—42no—
118D28—42no—
119C4×D7—42no—
120C4×D7—44no—
121Dic14—22no—
122Dic14—22no—
123Dic14—22no—
124C7×C8—22no—
125C7×C8—22no—
126C2×#41—72no—
127C2×Q16—72no—
128C2×SD16—72no—
129C2×SD16—72no—
130C2×D8—72no—
131C4⋊Q8—71no—
132#127—71no—
133#126—141no—
134#125—71no—
135#124—71no—
136#124—72no—
137#123—141no—
138C22⋊Q8—71no—
139#121—71no—
140#121—141no—
141#121—141no—
142C4×Q8—71no—
143C4×D4—71no—
144C4×D4—141no—
145C4×D4—141no—
146C4×D4—141no—
147#117—141no—
148C2×C42C2×D711yescharacteristic
149C4⋊C8C2×D712yes—
150#103—72no—
151#103—72no—
152#102—72no—
153#102—72no—
154C22⋊C8C2×D712yes—
155#97C2×D711yescharacteristic
156#97C2×D711yescharacteristic
157C22×C7—21no—
158C22×C7—21no—
159C22×C7—21no—
160C22×C7C22×D411yescharacteristic
161C2×D7—42no—
162C2×D7—41no—
163C2×D7—41no—
164C2×D7—42no—
165C2×D7—42no—
166C2×D7—41no—
167C4×C7—21no—
168C4×C7—21no—
169C4×C7—21no—
170C4×C7—21no—
171C4×C7—21no—
172C4×C7H(5,𝔽2)12yes—
173Dic7—42no—
174Dic7—41no—
175Dic7—41no—
176Dic7—41no—
177#41—282no—
178#41—282no—
179C2×Q8—72no—
180C2×Q8—71no—
181C2×D4—72no—
182C2×D4—71no—
183C2×D4—141no—
184C2×D4—142no—
185C2×D4—141no—
186C2×D4—141no—
187C22×C4—141no—
188C22×C4—142no—
189C22×C4D2811yescharacteristic
190C22×C4D2811yescharacteristic
191C22×C4C22×D711yescharacteristic
192Q16—282no—
193SD16—282no—
194SD16—282no—
195D8—282no—
196C2×C8C22×D712yes—
197C2×C8C22×D712yes—
198C4⋊C4—71no—
199C4⋊C4—71no—
200C4⋊C4—141no—
201C4⋊C4—141no—
202C4⋊C4—141no—
203C4⋊C4—141no—
204C4⋊C4—141no—
205C4⋊C4—141no—
206C22⋊C4—141no—
207C22⋊C4—141no—
208C22⋊C4—141no—
209C22⋊C4—141no—
210C22⋊C4—142no—
211C22⋊C4—142no—
212C22⋊C4—141no—
213C42—141no—
214C42D2811yescharacteristic
215C42D2811yescharacteristic
216C42C22×D711yescharacteristic
217C42C22×D711yescharacteristic
218C2×C7—41no—
219C2×C7#53511yescharacteristic
220C2×C7#57511yescharacteristic
221C2×C7#57511yescharacteristic
222D7—82no—
223D7—81no—
224C23—141no—
225C23—142no—
226C23C2×D2811yescharacteristic
227Q8—142no—
228Q8—142no—
229Q8—142no—
230D4—142no—
231D4—142no—
232D4—142no—
233D4—282no—
234D4—282no—
235D4—282no—
236D4—284no—
237C2×C4—22no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—141no—
241C2×C4—141no—
242C2×C4—141no—
243C2×C4—142no—
244C2×C4—282no—
245C2×C4—284no—
246C2×C4C2×D2811yescharacteristic
247C2×C4C2×D2811yescharacteristic
248C2×C4C2×D2811yescharacteristic
249C2×C4C2×D2811yescharacteristic
250C2×C4C2×D2811yescharacteristic
251C2×C4C23×D711yescharacteristic
252C8—22no—
253C8—22no—
254C7#321911yescharacteristic, complemented, Sylow
255C22—21no—
256C22—21no—
257C22—21no—
258C22—281no—
259C22—282no—
260C22—282no—
261C22—281no—
262C22—282no—
263C22—281no—
264C22C22×D2811yescharacteristic
265C4—21no—
266C4—21no—
267C4—21no—
268C4—21no—
269C4—21no—
270C4—281no—
271C4—281no—
272C4—282no—
273C4—281no—
274C4#655312yes—
275C2—41no—
276C2—561no—
277C2—562no—
278C2#9021611yescharacteristic
279C2#9043911yescharacteristic
280C2#9043911yescharacteristic
subgroupG/H
centerC22C22×D28
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#432C2
socleC22×C7C22×D4
hypercenter#432D7
hypocenterC7#3219
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #432

Complements

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

NH ≅ G/Ntype
1. #90170C2semidirect
2. #90168C2semidirect
3. C7×#432C2semidirect
4. #89617C2semidirect
5. #89616C2semidirect
6. #89506C2semidirect
7. #89505C2semidirect
8. #89487C2semidirect
9. #89485C2semidirect
10. #89484C2semidirect
11. #89475C2semidirect
95. #432D7semidirect
254. C7#3219semidirect

Maximal subgroups

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

subgroupindex
#901702
#901682
C7×#4322
#896172
#896162
#895062
#895052
#894872
#894852
#894842
#894752
#32197