Finite Group Databasebeta

← #11737926

Subgroups — #11737926

Order 896.

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

normal subgroups only

This group has 400 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, complemented
2#90271C211yesmaximal, characteristic, complemented
3C7×#462C211yesmaximal, characteristic, complemented
4#89963C212yesmaximal, complemented
5#89959C212yesmaximal, complemented
6#89855C211yesmaximal, characteristic, complemented
7#89799C211yesmaximal, characteristic, complemented
8#89615C211yesmaximal, characteristic, complemented
9#89605C211yesmaximal, characteristic, complemented
10#89583C212yesmaximal, complemented
11#89572C212yesmaximal, complemented
12C2×#1161C2212yes—
13C2×C7×Q16C2212yes—
14C7×C4⋊Q8C2211yescharacteristic
15C4×C7×Q8C2211yescharacteristic
16#6510C2212yes—
17#6510C2211yescharacteristic
18#6509C2211yescharacteristic
19C2×#1143C2212yes—
20C2×#1143C2212yes—
21C2×D56C2212yes—
22#6461—21no—
23#6458—21no—
24#6458—21no—
25#6458C2211yescharacteristic
26#6457—21no—
27#6456—21no—
28#6456—21no—
29#6453C2211yescharacteristic
30#6437C2211yescharacteristic
31C4×D28—21no—
32C4×D28C2211yescharacteristic
33#6435—21no—
34C42×D7C2211yescharacteristic
35C7×C4⋊C8C2211yescharacteristic
36C7×#103C2212yes—
37#6409C2212yes—
38#6395C2212yes—
39(C2×D7)⋊C8C2212yes—
40C8×Dic7C2212yes—
41#6383C2212yes—
42C4⋊(C7⋊C8)C2211yescharacteristic
43#3212—71nomaximal, Sylow
44C2×C7×Q8C2312yes—
45C2×C7×Q8C2311yescharacteristic
46#1161—42no—
47#1161—42no—
48C2×D28—21no—
49C2×D28—22no—
50C2×D28—21no—
51C2×D28—21no—
52C2×D28C2311yescharacteristic
53C2×D28C2312yes—
54C2×C4×D7—21no—
55C2×C4×D7—22no—
56C2×C4×D7D411yescharacteristic
57C2×C4×D7D411yescharacteristic
58C2×C4×D7C2311yescharacteristic
59C7×Q16—42no—
60C2×C7×C8C2312yes—
61C7×C4⋊C4—21no—
62C7×C4⋊C4—21no—
63C7×C4⋊C4—21no—
64C7×C4⋊C4C2311yescharacteristic
65C42×C7—21no—
66C42×C7C2311yescharacteristic
67#1143—42no—
68#1143—42no—
69(C2×D7)⋊C4—21no—
70(C2×D7)⋊C4—21no—
71(C2×D7)⋊C4—21no—
72(C2×D7)⋊C4—21no—
73(C2×D7)⋊C4—21no—
74(C2×D7)⋊C4—22no—
75(C2×D7)⋊C4—22no—
76C4⋊Dic7—21no—
77C4⋊Dic7C2311yescharacteristic
78Dic7⋊C4—21no—
79Dic7⋊C4—21no—
80C4×Dic7D411yescharacteristic
81C4×Dic7D411yescharacteristic
82C4×Dic7C2311yescharacteristic
83C2×C7⋊C8C2312yes—
84D56—42no—
85#532—71no—
86#529—71no—
87#495—72no—
88#488—72no—
89#462D711yescharacteristic, complemented
90#461—71no—
91#460—71no—
92#459—71no—
93#452—72no—
94#449—72no—
95#431—71no—
96C22×D7—21no—
97C22×D7—22no—
98C22×D7C2×D411yescharacteristic
99C7×Q8—21no—
100C7×Q8—22no—
101C7×Q8—21no—
102C7×Q8—22no—
103C2×C4×C7—21no—
104C2×C4×C7—21no—
105C2×C4×C7—22no—
106C2×C4×C7C2×D411yescharacteristic
107C2×C4×C7C2×D411yescharacteristic
108C2×C4×C7C2411yescharacteristic
109C2×Dic7—21no—
110C2×Dic7C2×D411yescharacteristic
111C2×Dic7C2×D411yescharacteristic
112C2×Dic7C2×D411yescharacteristic
113D28—21no—
114D28—22no—
115D28—21no—
116D28—22no—
117D28—42no—
118D28—42no—
119D28—44no—
120D28—42no—
121C4×D7—22no—
122C4×D7—21no—
123C4×D7—21no—
124C4×D7—41no—
125C4×D7—44no—
126C4×D7—42no—
127C7×C8—22no—
128C7⋊C8—22no—
129C2×#41—72no—
130C2×Q16C2×D712yes—
131C2×SD16—72no—
132C2×SD16—72no—
133C2×D8—72no—
134C4⋊Q8C2×D711yescharacteristic
135#127—71no—
136#126—141no—
137#125—71no—
138#124—71no—
139#124—72no—
140#123—141no—
141C22⋊Q8—71no—
142#121—71no—
143#121—141no—
144#121—141no—
145C4×Q8C2×D711yescharacteristic
146C4×D4—71no—
147C4×D4—141no—
148C4×D4—141no—
149C4×D4—141no—
150#117—141no—
151C2×C42—71no—
152C4⋊C8—71no—
153C4⋊C8C2×D711yescharacteristic
154#103—72no—
155#103C2×D712yes—
156#102—72no—
157#102—72no—
158C22⋊C8—72no—
159C4×C8—72no—
160C22×C7C22×D411yescharacteristic
161C2×D7—21no—
162C2×D7—21no—
163C2×D7—21no—
164C2×D7—42no—
165C2×D7—41no—
166C2×D7—41no—
167C2×D7—42no—
168C2×D7—42no—
169C2×D7—41no—
170C4×C7—21no—
171C4×C7—42no—
172C4×C7—41no—
173C4×C7—41no—
174C4×C7#13512yes—
175C4×C7H(5,𝔽2)11yescharacteristic
176C4×C7H(5,𝔽2)11yescharacteristic
177Dic7—21no—
178Dic7—21no—
179Dic7—21no—
180Dic7—41no—
181#41—282no—
182#41—282no—
183C2×Q8C22×D712yes—
184C2×Q8C22×D711yescharacteristic
185C2×D4—72no—
186C2×D4—71no—
187C2×D4—141no—
188C2×D4—142no—
189C2×D4—141no—
190C2×D4—141no—
191C22×C4—71no—
192C22×C4—71no—
193C22×C4—71no—
194C22×C4—142no—
195C22×C4—141no—
196Q16—42no—
197SD16—282no—
198SD16—282no—
199D8—282no—
200C2×C8—72no—
201C2×C8C22×D712yes—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—21no—
205C4⋊C4—71no—
206C4⋊C4—141no—
207C4⋊C4—141no—
208C4⋊C4—141no—
209C4⋊C4C22×D711yescharacteristic
210C22⋊C4—141no—
211C22⋊C4—141no—
212C22⋊C4—142no—
213C22⋊C4—142no—
214C22⋊C4—141no—
215C22⋊C4—141no—
216C22⋊C4—141no—
217C42—21no—
218C42—71no—
219C42—71no—
220C42—71no—
221C42C22×D711yescharacteristic
222C2×C7#53511yescharacteristic
223C2×C7C2×#13511yescharacteristic
224C2×C7#57511yescharacteristic
225D7—41no—
226D7—82no—
227D7—81no—
228C23—71no—
229C23—141no—
230C23—142no—
231Q8—21no—
232Q8—22no—
233Q8—21no—
234Q8—22no—
235D4—141no—
236D4—142no—
237D4—141no—
238D4—142no—
239D4—282no—
240D4—282no—
241D4—284no—
242D4—282no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—71no—
247C2×C4—71no—
248C2×C4—71no—
249C2×C4—142no—
250C2×C4—141no—
251C2×C4—141no—
252C2×C4—141no—
253C2×C4—281no—
254C2×C4—284no—
255C2×C4—282no—
256C2×C4D4×D711yescharacteristic
257C2×C4D4×D711yescharacteristic
258C2×C4C23×D711yescharacteristic
259C8—22no—
260C8—142no—
261C7#321211yescharacteristic, complemented, Sylow
262C22—141no—
263C22—141no—
264C22—141no—
265C22—281no—
266C22—282no—
267C22—282no—
268C22—281no—
269C22—282no—
270C22—281no—
271C22C2×D4×D711yescharacteristic
272C4—21no—
273C4—41no—
274C4—42no—
275C4—41no—
276C4—141no—
277C4—141no—
278C4—141no—
279C4—281no—
280C4#648212yes—
281C4#654811yescharacteristic
282C4#655311yescharacteristic
283C2—281no—
284C2—561no—
285C2—562no—
286C2#9029811yescharacteristic
287C2C2×#648211yescharacteristic
288C2#9046811yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#462C2
socleC22×C7C22×D4
hypercenter#462D7
hypocenterC7#3212
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
1. #90416C2semidirect
2. #90271C2semidirect
3. C7×#462C2semidirect
4. #89963C2semidirect
5. #89959C2semidirect
6. #89855C2semidirect
7. #89799C2semidirect
8. #89615C2semidirect
9. #89605C2semidirect
10. #89583C2semidirect
11. #89572C2semidirect
89. #462D7semidirect
261. C7#3212semidirect

Maximal subgroups

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

subgroupindex
#904162
#902712
C7×#4622
#899632
#899592
#898552
#897992
#896152
#896052
#895832
#895722
#32127