Finite Group Databasebeta

← #11735945

Subgroups — #11735945

Order 896.

410 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 129 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
3C2×C7×#97C211yesmaximal, characteristic, complemented
4#89887C212yesmaximal, complemented
5#89886C212yesmaximal, complemented
6#89486C211yesmaximal, characteristic, complemented
7#89485C211yesmaximal, characteristic, complemented
8#89484C211yesmaximal, characteristic, complemented
9#89483C212yesmaximal, complemented
10#89482C212yesmaximal, complemented
11#89474C211yesmaximal, characteristic, complemented
12C2×#1157—21no—
13C22×C7×C8C2212yes—
14C2×C42×C7C2211yescharacteristic
15#6493—21no—
16#6493C2212yes—
17#6493C2211yescharacteristic
18#6492—21no—
19C4×C22⋊D7—21no—
20#6487C2211yescharacteristic
21C2×D56C2212yes—
22C2×#1132C2212yes—
23#6439—21no—
24#6438—21no—
25#6438C2211yescharacteristic
26#6437C2211yescharacteristic
27C4×D28—21no—
28C4×D28C2212yes—
29C4×D28C2211yescharacteristic
30#6435—21no—
31#6433C2211yescharacteristic
32#6432C2211yescharacteristic
33C4×Dic14C2211yescharacteristic
34C7×#97C2211yescharacteristic
35C7×#97C2212yes—
36C7×#97C2211yescharacteristic
37#6395C2212yes—
38#6395C2212yes—
39#6395C2212yes—
40#6392C2212yes—
41#6391C2212yes—
42#6390C2212yes—
43#3180—71nomaximal, Sylow
44C22×C4×C7D411yescharacteristic
45C22×C4×C7D411yescharacteristic
46C22×C4×C7C2311yescharacteristic
47C2×C22⋊D7—21no—
48C2×C22⋊D7—22no—
49#1157—44no—
50C2×D28—21no—
51C2×D28—21no—
52C2×D28C2312yes—
53C2×D28C2311yescharacteristic
54C2×C4×D7—21no—
55C2×C4×D7—22no—
56C2×Dic14—21no—
57C2×Dic14C2311yescharacteristic
58C2×C7×C8—24no—
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⋊Dic7C2311yescharacteristic
74C4⋊Dic7C2312yes—
75C4⋊Dic7C2311yescharacteristic
76Dic7⋊C4—21no—
77Dic7⋊C4—21no—
78Dic7⋊C4—21no—
79Dic7⋊C4—21no—
80C4×Dic7—21no—
81D56—24no—
82#1132—24no—
83#532—71no—
84#529—71no—
85#501—71no—
86#496—71no—
87#490—71no—
88#489—71no—
89#466—72no—
90#465—72no—
91#442—72no—
92#440—72no—
93C2×#97D711yescharacteristic, complemented
94C23×C7C2×D411yescharacteristic
95C22×D7—21no—
96C22×D7—22no—
97C2×C4×C7—22no—
98C2×C4×C7—22no—
99C2×C4×C7—41no—
100C2×C4×C7C2×D411yescharacteristic
101C2×C4×C7C2×D411yescharacteristic
102C2×C4×C7C2×D411yescharacteristic
103C2×C4×C7C2×D411yescharacteristic
104C2×C4×C7C2×D411yescharacteristic
105C2×C4×C7C2411yescharacteristic
106C22⋊D7—44no—
107C22⋊D7—42no—
108C2×Dic7—21no—
109C2×Dic7—21no—
110C2×Dic7—21no—
111C2×Dic7—22no—
112D28—22no—
113D28—22no—
114D28—22no—
115D28—42no—
116D28—42no—
117C4×D7—42no—
118C4×D7—44no—
119Dic14—22no—
120Dic14—42no—
121C7×C8—22no—
122C7×C8#4114yescomplemented
123C2×#41—141no—
124C2×SD16—72no—
125C2×D8—72no—
126C22×C8C2×D712yes—
127C4⋊Q8—71no—
128#127—71no—
129#126—141no—
130#125—71no—
131#124—71no—
132#124—141no—
133#123—141no—
134C22⋊Q8—71no—
135#121—71no—
136#121—72no—
137#121—141no—
138C4×Q8—71no—
139C4×D4—71no—
140C4×D4—72no—
141C4×D4—141no—
142C4×D4—141no—
143#117—141no—
144C2×C42C2×D711yescharacteristic
145#107—72no—
146#106—72no—
147#103—72no—
148#102—72no—
149#102—72no—
150#102—72no—
151#97C2×D712yes—
152#97C2×D711yescharacteristic
153#97C2×D711yescharacteristic
154C22×C7—21no—
155C22×C7—21no—
156C22×C7AGL(1,ℤ/8)12yes—
157C22×C7C22×D411yescharacteristic
158C2×D7—42no—
159C2×D7—41no—
160C2×D7—42no—
161C2×D7—41no—
162C2×D7—41no—
163C2×D7—42no—
164C4×C7—21no—
165C4×C7—21no—
166C4×C7—21no—
167C4×C7—21no—
168C4×C7—21no—
169C4×C7C2×#4112yes—
170Dic7—41no—
171Dic7—42no—
172Dic7—41no—
173Dic7—41no—
174#41—284no—
175C2×Q8—71no—
176C2×Q8—141no—
177C2×D4—71no—
178C2×D4—72no—
179C2×D4—141no—
180C2×D4—142no—
181C2×D4—141no—
182C2×D4—141no—
183C22×C4—141no—
184C22×C4—142no—
185C22×C4D2811yescharacteristic
186C22×C4D2811yescharacteristic
187C22×C4C22×D711yescharacteristic
188SD16—144no—
189D8—144no—
190C2×C8—24no—
191C2×C8C22×D712yes—
192C2×C8C22×D712yes—
193C4⋊C4—71no—
194C4⋊C4—71no—
195C4⋊C4—72no—
196C4⋊C4—141no—
197C4⋊C4—141no—
198C4⋊C4—141no—
199C4⋊C4—141no—
200C4⋊C4—141no—
201C22⋊C4—141no—
202C22⋊C4—141no—
203C22⋊C4—141no—
204C22⋊C4—142no—
205C22⋊C4—141no—
206C22⋊C4—142no—
207C22⋊C4—141no—
208C42—141no—
209C42D2811yescharacteristic
210C42D2811yescharacteristic
211C42C22×D711yescharacteristic
212C42C22×D711yescharacteristic
213C2×C7—22no—
214C2×C7#53211yescharacteristic
215C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
216C2×C7#57511yescharacteristic
217D7—82no—
218D7—81no—
219C23—141no—
220C23—142no—
221C23C2×D2811yescharacteristic
222Q8—142no—
223Q8—282no—
224D4—142no—
225D4—142no—
226D4—142no—
227D4—282no—
228D4—282no—
229D4—282no—
230D4—284no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—41no—
234C2×C4—141no—
235C2×C4—141no—
236C2×C4—141no—
237C2×C4—142no—
238C2×C4—282no—
239C2×C4—284no—
240C2×C4C2×D2811yescharacteristic
241C2×C4C2×D2811yescharacteristic
242C2×C4C2×D2811yescharacteristic
243C2×C4C2×D2811yescharacteristic
244C2×C4C2×D2811yescharacteristic
245C2×C4C23×D711yescharacteristic
246C8—22no—
247C8#115714yescomplemented
248C7#318011yescharacteristic, complemented, Sylow
249C22—21no—
250C22—21no—
251C22—281no—
252C22—282no—
253C22—282no—
254C22—281no—
255C22—281no—
256C22—282no—
257C22#647112yes—
258C22C22×D2811yescharacteristic
259C4—21no—
260C4—21no—
261C4—21no—
262C4—21no—
263C4—21no—
264C4—281no—
265C4—282no—
266C4—281no—
267C4—281no—
268C4C2×#115712yes—
269C2—22no—
270C2—562no—
271C2—561no—
272C2#9016811yescharacteristic
273C2C2×#647111yescharacteristic
274C2#9043911yescharacteristic
subgroupG/H
centerC22C22×D28
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC2×C7×#97C2
socleC22×C7C22×D4
hypercenterC2×#97D7
hypocenterC7#3180
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#97

Complements

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

NH ≅ G/Ntype
1. #90170C2semidirect
2. #90168C2semidirect
3. C2×C7×#97C2semidirect
4. #89887C2semidirect
5. #89886C2semidirect
6. #89486C2semidirect
7. #89485C2semidirect
8. #89484C2semidirect
9. #89483C2semidirect
10. #89482C2semidirect
11. #89474C2semidirect
93. C2×#97D7semidirect
122. C7×C8#41semidirect
247. C8#1157semidirect
248. C7#3180semidirect

Maximal subgroups

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

subgroupindex
#901702
#901682
C2×C7×#972
#898872
#898862
#894862
#894852
#894842
#894832
#894822
#894742
#31807