Finite Group Databasebeta

← #11737878

Subgroups — #11737878

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#90248C211yesmaximal, characteristic, complemented
3C7×#461C211yesmaximal, characteristic, complemented
4#89944C212yesmaximal, complemented
5#89940C212yesmaximal, complemented
6#89862C211yesmaximal, characteristic, complemented
7#89787C211yesmaximal, characteristic, complemented
8#89620C211yesmaximal, characteristic, complemented
9#89605C211yesmaximal, characteristic, complemented
10#89582C212yesmaximal, complemented
11#89572C212yesmaximal, complemented
12C2×#1161C2212yes—
13C2×C7×SD16C2212yes—
14C7×C4⋊Q8C2211yescharacteristic
15C4×C7×D4C2211yescharacteristic
16#6510C2212yes—
17C2×#1144C2212yes—
18#6501C2211yescharacteristic
19#6499C2211yescharacteristic
20C2×#1141C2212yes—
21C4×C22⋊D7—21no—
22#6489—21no—
23C2×#1132C2212yes—
24#6460C2211yescharacteristic
25#6458—21no—
26#6458—21no—
27#6456—21no—
28#6456—21no—
29#6453C2211yescharacteristic
30#6446—21no—
31#6442—21no—
32#6437C2211yescharacteristic
33C42×D7C2211yescharacteristic
34C4×Dic14C2211yescharacteristic
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×D4C2311yescharacteristic
46C22×C4×C7—21no—
47C2×C22⋊D7—21no—
48#1161—42no—
49#1161—42no—
50C2×D28—22no—
51C2×D28—21no—
52C2×D28—21no—
53C2×D28C2312yes—
54C2×C4×D7—22no—
55C2×C4×D7D411yescharacteristic
56C2×C4×D7D411yescharacteristic
57C2×C4×D7C2311yescharacteristic
58C2×Dic14C2311yescharacteristic
59C7×SD16—42no—
60C2×C7×C8C2312yes—
61C7×C4⋊C4—21no—
62C7×C4⋊C4—21no—
63C7×C4⋊C4C2311yescharacteristic
64C7×C22⋊C4—21no—
65C42×C7C2311yescharacteristic
66C22⋊Dic7—21no—
67C22⋊Dic7—21no—
68C22⋊Dic7—21no—
69#1144—42no—
70#1141—42no—
71(C2×D7)⋊C4—21no—
72(C2×D7)⋊C4—22no—
73(C2×D7)⋊C4—22no—
74C4⋊Dic7—21no—
75C4⋊Dic7—21no—
76C4⋊Dic7C2311yescharacteristic
77Dic7⋊C4—21no—
78Dic7⋊C4—21no—
79C4×Dic7—21no—
80C4×Dic7D411yescharacteristic
81C4×Dic7D411yescharacteristic
82C4×Dic7C2311yescharacteristic
83C2×C7⋊C8C2312yes—
84#1132—42no—
85#532—71no—
86#529—71no—
87#495—72no—
88#488—72no—
89#462—71no—
90#461D711yescharacteristic, complemented
91#460—71no—
92#459—71no—
93#452—72no—
94#449—72no—
95#431—71no—
96C23×C7—21no—
97C22×D7—22no—
98C22×D7C2×D411yescharacteristic
99C7×Q8—22no—
100C7×Q8—22no—
101C7×D4—21no—
102C7×D4—21no—
103C2×C4×C7—21no—
104C2×C4×C7—22no—
105C2×C4×C7—42no—
106C2×C4×C7C2×D411yescharacteristic
107C2×C4×C7C2×D411yescharacteristic
108C2×C4×C7C2411yescharacteristic
109C22⋊D7—42no—
110C2×Dic7—21no—
111C2×Dic7—21no—
112C2×Dic7C2×D411yescharacteristic
113C2×Dic7C2×D411yescharacteristic
114C2×Dic7C2×D411yescharacteristic
115D28—22no—
116D28—22no—
117D28—42no—
118D28—44no—
119D28—42no—
120C4×D7—22no—
121C4×D7—21no—
122C4×D7—21no—
123C4×D7—41no—
124C4×D7—44no—
125Dic14—21no—
126Dic14—21no—
127C7×C8—22no—
128C7⋊C8—22no—
129C2×#41—72no—
130C2×Q16—72no—
131C2×SD16—72no—
132C2×SD16C2×D712yes—
133C2×D8—72no—
134C4⋊Q8C2×D711yescharacteristic
135#127—71no—
136#126—141no—
137#125—71no—
138#124—72no—
139#124—71no—
140#123—141no—
141C22⋊Q8—71no—
142#121—71no—
143#121—141no—
144#121—141no—
145C4×Q8—71no—
146C4×D4—141no—
147C4×D4—141no—
148C4×D4—141no—
149C4×D4C2×D711yescharacteristic
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×C7—41no—
161C22×C7—41no—
162C22×C7—41no—
163C22×C7C22×D411yescharacteristic
164C2×D7—21no—
165C2×D7—21no—
166C2×D7—21no—
167C2×D7—42no—
168C2×D7—42no—
169C2×D7—42no—
170C4×C7—21no—
171C4×C7—42no—
172C4×C7—41no—
173C4×C7#13512yes—
174C4×C7H(5,𝔽2)11yescharacteristic
175C4×C7H(5,𝔽2)11yescharacteristic
176Dic7—21no—
177Dic7—21no—
178Dic7—21no—
179Dic7—41no—
180Dic7—41no—
181#41—282no—
182#41—282no—
183C2×Q8—71no—
184C2×Q8C22×D712yes—
185C2×D4—72no—
186C2×D4—142no—
187C2×D4—141no—
188C2×D4—141no—
189C2×D4—141no—
190C2×D4C22×D711yescharacteristic
191C22×C4—21no—
192C22×C4—71no—
193C22×C4—71no—
194C22×C4—71no—
195C22×C4—142no—
196Q16—282no—
197SD16—42no—
198SD16—282no—
199D8—282no—
200C2×C8—72no—
201C2×C8C22×D712yes—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—71no—
205C4⋊C4—141no—
206C4⋊C4—141no—
207C4⋊C4—141no—
208C4⋊C4—141no—
209C4⋊C4C22×D711yescharacteristic
210C22⋊C4—21no—
211C22⋊C4—142no—
212C22⋊C4—142no—
213C22⋊C4—141no—
214C22⋊C4—141no—
215C22⋊C4—141no—
216C22⋊C4—141no—
217C42—71no—
218C42—71no—
219C42—71no—
220C42—141no—
221C42C22×D711yescharacteristic
222C2×C7—81no—
223C2×C7#53511yescharacteristic
224C2×C7C2×#13511yescharacteristic
225C2×C7#57511yescharacteristic
226D7—41no—
227D7—82no—
228C23—21no—
229C23—71no—
230C23—142no—
231Q8—22no—
232Q8—22no—
233Q8—141no—
234Q8—141no—
235D4—21no—
236D4—21no—
237D4—142no—
238D4—142no—
239D4—282no—
240D4—284no—
241D4—282no—
242D4—282no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—42no—
246C2×C4—71no—
247C2×C4—71no—
248C2×C4—71no—
249C2×C4—142no—
250C2×C4—141no—
251C2×C4—141no—
252C2×C4—141no—
253C2×C4—141no—
254C2×C4—281no—
255C2×C4—284no—
256C2×C4D4×D711yescharacteristic
257C2×C4D4×D711yescharacteristic
258C2×C4C23×D711yescharacteristic
259C8—22no—
260C8—142no—
261C7#321211yescharacteristic, complemented, Sylow
262C22—41no—
263C22—41no—
264C22—41no—
265C22—141no—
266C22—141no—
267C22—141no—
268C22—282no—
269C22—282no—
270C22—282no—
271C22C2×D4×D711yescharacteristic
272C4—21no—
273C4—41no—
274C4—42no—
275C4—141no—
276C4—141no—
277C4—141no—
278C4—281no—
279C4—281no—
280C4#647912yes—
281C4#654811yescharacteristic
282C4#655311yescharacteristic
283C2—81no—
284C2—281no—
285C2—562no—
286C2#9029811yescharacteristic
287C2C2×#647911yescharacteristic
288C2#9046811yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#461C2
socleC22×C7C22×D4
hypercenter#461D7
hypocenterC7#3212
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #90416C2semidirect
2. #90248C2semidirect
3. C7×#461C2semidirect
4. #89944C2semidirect
5. #89940C2semidirect
6. #89862C2semidirect
7. #89787C2semidirect
8. #89620C2semidirect
9. #89605C2semidirect
10. #89582C2semidirect
11. #89572C2semidirect
90. #461D7semidirect
261. C7#3212semidirect

Maximal subgroups

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

subgroupindex
#904162
#902482
C7×#4612
#899442
#899402
#898622
#897872
#896202
#896052
#895822
#895722
#32127