Finite Group Databasebeta

← #11737927

Subgroups — #11737927

Order 896.

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

normal subgroups only

This group has 390 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#90416C211yesmaximal, characteristic
2#90269C211yesmaximal, characteristic, complemented
3C7×#462C211yesmaximal, characteristic, complemented
4#89962C212yesmaximal, complemented
5#89959C212yesmaximal, complemented
6#89862C211yesmaximal, characteristic, complemented
7#89799C211yesmaximal, characteristic, complemented
8#89620C211yesmaximal, characteristic, complemented
9#89607C211yesmaximal, characteristic, complemented
10#89582C212yesmaximal, complemented
11#89567C212yesmaximal, complemented
12C2×#1161C2212yes—
13C2×C7×Q16C2212yes—
14C7×C4⋊Q8C2211yescharacteristic
15C4×C7×Q8C2211yescharacteristic
16#6510C2212yes—
17#6509C2211yescharacteristic
18#6507C2211yescharacteristic
19C2×#1144C2212yes—
20C2×#1143C2212yes—
21C2×#1132C2212yes—
22#6460C2211yescharacteristic
23#6459—21no—
24#6458—21no—
25#6458—21no—
26#6456—21no—
27#6456—21no—
28#6452—21no—
29#6451C2211yescharacteristic
30#6437C2211yescharacteristic
31#6435—21no—
32C42×D7C2211yescharacteristic
33C4×Dic14—21no—
34C4×Dic14C2211yescharacteristic
35C7×C4⋊C8C2211yescharacteristic
36C7×#103C2212yes—
37#6409C2212yes—
38#6395C2212yes—
39(C2×D7)⋊C8C2212yes—
40#6389C2212yes—
41#6383C2212yes—
42C4⋊(C7⋊C8)C2211yescharacteristic
43#3216—71nomaximal, Sylow
44C2×C7×Q8C2312yes—
45C2×C7×Q8C2311yescharacteristic
46#1161—42no—
47#1161—42no—
48C2×D28—22no—
49C2×D28—21no—
50C2×D28—21no—
51C2×D28C2312yes—
52C2×C4×D7—22no—
53C2×C4×D7D411yescharacteristic
54C2×C4×D7D411yescharacteristic
55C2×C4×D7C2311yescharacteristic
56C2×Dic14—21no—
57C2×Dic14C2311yescharacteristic
58C7×Q16—42no—
59C2×C7×C8C2312yes—
60C7×C4⋊C4—21no—
61C7×C4⋊C4—21no—
62C7×C4⋊C4—21no—
63C7×C4⋊C4C2311yescharacteristic
64C42×C7—21no—
65C42×C7C2311yescharacteristic
66#1144—42no—
67#1143—42no—
68(C2×D7)⋊C4—21no—
69(C2×D7)⋊C4—21no—
70(C2×D7)⋊C4—22no—
71(C2×D7)⋊C4—22no—
72C4⋊Dic7—21no—
73C4⋊Dic7C2311yescharacteristic
74Dic7⋊C4—21no—
75Dic7⋊C4—21no—
76Dic7⋊C4—21no—
77Dic7⋊C4—21no—
78Dic7⋊C4—21no—
79C4×Dic7—21no—
80C4×Dic7D411yescharacteristic
81C4×Dic7D411yescharacteristic
82C4×Dic7C2311yescharacteristic
83C2×C7⋊C8C2312yes—
84#1132—42no—
85#533—71no—
86#532—71no—
87#497—72no—
88#489—72no—
89#462—71no—
90#462D711yescharacteristic, complemented
91#460—71no—
92#460—71no—
93#452—72no—
94#452—72no—
95#432—71no—
96C22×D7—22no—
97C22×D7C2×D411yescharacteristic
98C7×Q8—21no—
99C7×Q8—22no—
100C7×Q8—21no—
101C7×Q8—22no—
102C2×C4×C7—21no—
103C2×C4×C7—21no—
104C2×C4×C7—22no—
105C2×C4×C7C2×D411yescharacteristic
106C2×C4×C7C2×D411yescharacteristic
107C2×C4×C7C2411yescharacteristic
108C2×Dic7—21no—
109C2×Dic7—21no—
110C2×Dic7C2×D411yescharacteristic
111C2×Dic7C2×D411yescharacteristic
112C2×Dic7C2×D411yescharacteristic
113D28—22no—
114D28—22no—
115D28—42no—
116D28—44no—
117D28—42no—
118C4×D7—22no—
119C4×D7—21no—
120C4×D7—21no—
121C4×D7—41no—
122C4×D7—44no—
123Dic14—21no—
124Dic14—21no—
125Dic14—42no—
126C7×C8—22no—
127C7⋊C8—22no—
128C2×#41—72no—
129C2×Q16—72no—
130C2×Q16C2×D712yes—
131C2×SD16—72no—
132C2×SD16—72no—
133C4⋊Q8—71no—
134C4⋊Q8—71no—
135C4⋊Q8C2×D711yescharacteristic
136#127—71no—
137#125—141no—
138#124—72no—
139C22⋊Q8—71no—
140C22⋊Q8—71no—
141C22⋊Q8—141no—
142#121—141no—
143#121—141no—
144C4×Q8—71no—
145C4×Q8—141no—
146C4×Q8C2×D711yescharacteristic
147C4×D4—141no—
148C4×D4—141no—
149#117—141no—
150C2×C42—71no—
151C4⋊C8—71no—
152C4⋊C8C2×D711yescharacteristic
153#103—72no—
154#103C2×D712yes—
155#102—72no—
156#102—72no—
157C22⋊C8—72no—
158#97—72no—
159C22×C7C22×D411yescharacteristic
160C2×D7—21no—
161C2×D7—21no—
162C2×D7—21no—
163C2×D7—42no—
164C2×D7—42no—
165C2×D7—42no—
166C4×C7—21no—
167C4×C7—42no—
168C4×C7—41no—
169C4×C7—41no—
170C4×C7#13712yes—
171C4×C7H(5,𝔽2)11yescharacteristic
172C4×C7H(5,𝔽2)11yescharacteristic
173Dic7—21no—
174Dic7—21no—
175Dic7—21no—
176Dic7—41no—
177Dic7—41no—
178#41—282no—
179#41—282no—
180C2×Q8—71no—
181C2×Q8—141no—
182C2×Q8C22×D712yes—
183C2×Q8C22×D711yescharacteristic
184C2×D4—72no—
185C2×D4—142no—
186C2×D4—141no—
187C2×D4—141no—
188C22×C4—71no—
189C22×C4—71no—
190C22×C4—71no—
191C22×C4—142no—
192Q16—42no—
193Q16—282no—
194SD16—282no—
195SD16—282no—
196C2×C8—72no—
197C2×C8C22×D712yes—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—71no—
202C4⋊C4—141no—
203C4⋊C4—141no—
204C4⋊C4—141no—
205C4⋊C4—141no—
206C4⋊C4—141no—
207C4⋊C4—141no—
208C4⋊C4C22×D711yescharacteristic
209C22⋊C4—142no—
210C22⋊C4—142no—
211C22⋊C4—141no—
212C22⋊C4—141no—
213C42—21no—
214C42—71no—
215C42—71no—
216C42—71no—
217C42—141no—
218C42C22×D711yescharacteristic
219C2×C7#53511yescharacteristic
220C2×C7C2×#13711yescharacteristic
221C2×C7#57511yescharacteristic
222D7—41no—
223D7—82no—
224C23—71no—
225C23—142no—
226Q8—21no—
227Q8—22no—
228Q8—21no—
229Q8—22no—
230Q8—141no—
231Q8—141no—
232Q8—282no—
233D4—142no—
234D4—142no—
235D4—282no—
236D4—284no—
237D4—282no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—71no—
242C2×C4—71no—
243C2×C4—71no—
244C2×C4—142no—
245C2×C4—141no—
246C2×C4—141no—
247C2×C4—141no—
248C2×C4—141no—
249C2×C4—281no—
250C2×C4—284no—
251C2×C4D4×D711yescharacteristic
252C2×C4D4×D711yescharacteristic
253C2×C4C23×D711yescharacteristic
254C8—22no—
255C8—142no—
256C7#321611yescharacteristic, complemented, Sylow
257C22—141no—
258C22—141no—
259C22—141no—
260C22—282no—
261C22—282no—
262C22—282no—
263C22C2×D4×D711yescharacteristic
264C4—21no—
265C4—41no—
266C4—42no—
267C4—41no—
268C4—141no—
269C4—141no—
270C4—141no—
271C4—281no—
272C4—281no—
273C4#648112yes—
274C4#654811yescharacteristic
275C4#655311yescharacteristic
276C2—281no—
277C2—562no—
278C2#9029811yescharacteristic
279C2C2×#648111yescharacteristic
280C2#9046811yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#462C2
socleC22×C7C22×D4
prime radical#90416C2
hypercenter#462D7
hypocenterC7#3216
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

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

NH ≅ G/Ntype
2. #90269C2semidirect
3. C7×#462C2semidirect
4. #89962C2semidirect
5. #89959C2semidirect
6. #89862C2semidirect
7. #89799C2semidirect
8. #89620C2semidirect
9. #89607C2semidirect
10. #89582C2semidirect
11. #89567C2semidirect
90. #462D7semidirect
256. C7#3216semidirect

Maximal subgroups

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

subgroupindex
#904162
#902692
C7×#4622
#899622
#899592
#898622
#897992
#896202
#896072
#895822
#895672
#32167