Finite Group Databasebeta

← #11738447

Subgroups — #11738447

Order 896.

392 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 121 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×#475C211yesmaximal, characteristic, complemented
4#89866C211yesmaximal, characteristic, complemented
5#89799C211yesmaximal, characteristic, complemented
6#89642C212yesmaximal, complemented
7#89636C212yesmaximal, complemented
8#89620C211yesmaximal, characteristic, complemented
9D7×C4⋊C8C211yesmaximal, characteristic, complemented
10#89590C212yesmaximal, complemented
11#89579C212yesmaximal, complemented
12C2×#1161—21no—
13C7×C4⋊Q8C2211yescharacteristic
14C4×C7×Q8C2211yescharacteristic
15#6510—21no—
16#6509C2211yescharacteristic
17#6507C2211yescharacteristic
18C2×#1143C2212yes—
19C2×#1132C2212yes—
20C2×C8×D7C2212yes—
21#6460C2211yescharacteristic
22#6459—21no—
23#6458—21no—
24#6458C2212yes—
25#6456—21no—
26#6456C2212yes—
27#6452—21no—
28#6451C2211yescharacteristic
29#6437C2211yescharacteristic
30#6435—21no—
31C42×D7C2211yescharacteristic
32C4×Dic14—21no—
33C4×Dic14C2211yescharacteristic
34C7×#106C2212yes—
35C7×C4⋊C8C2211yescharacteristic
36C7×#103C2212yes—
37#6395C2212yes—
38Dic7⋊C8C2212yes—
39#6384C2212yes—
40#6383C2212yes—
41#6382C2212yes—
42C4⋊(C7⋊C8)C2211yescharacteristic
43#3246—71nomaximal, Sylow
44C2×C7×Q8—21no—
45C2×C7×Q8C2311yescharacteristic
46#1161—44no—
47C2×D28—22no—
48C2×D28—21no—
49C2×D28—21no—
50C2×D28C2312yes—
51C2×C4×D7—22no—
52C2×C4×D7D411yescharacteristic
53C2×C4×D7D411yescharacteristic
54C2×C4×D7C2311yescharacteristic
55C2×Dic14—21no—
56C2×Dic14C2311yescharacteristic
57C2×C7×C8C2312yes—
58C7×C4⋊C4—21no—
59C7×C4⋊C4—21no—
60C7×C4⋊C4C2312yes—
61C7×C4⋊C4C2311yescharacteristic
62C42×C7—21no—
63C42×C7C2311yescharacteristic
64#1143—42no—
65(C2×D7)⋊C4—21no—
66(C2×D7)⋊C4—21no—
67(C2×D7)⋊C4—22no—
68(C2×D7)⋊C4—22no—
69C4⋊Dic7—21no—
70C4⋊Dic7C2311yescharacteristic
71Dic7⋊C4—21no—
72Dic7⋊C4—21no—
73Dic7⋊C4—21no—
74Dic7⋊C4—21no—
75Dic7⋊C4—21no—
76C4×Dic7—21no—
77C4×Dic7D411yescharacteristic
78C4×Dic7D411yescharacteristic
79C4×Dic7C2311yescharacteristic
80C2×C7⋊C8C2312yes—
81#1132—42no—
82C8×D7—42no—
83#533—71no—
84#532—71no—
85#476—72no—
86#475—71no—
87#475D711yescharacteristic, complemented
88#465—72no—
89#465—72no—
90#461—72no—
91#460—71no—
92#460—71no—
93C2×C4⋊C8—71no—
94C22×D7—22no—
95C22×D7C2×D411yescharacteristic
96C7×Q8—21no—
97C7×Q8—21no—
98C7×Q8—42no—
99C2×C4×C7—21no—
100C2×C4×C7—21no—
101C2×C4×C7—22no—
102C2×C4×C7C2×D411yescharacteristic
103C2×C4×C7C2×D411yescharacteristic
104C2×C4×C7C2411yescharacteristic
105C2×Dic7—21no—
106C2×Dic7—21no—
107C2×Dic7C2×D411yescharacteristic
108C2×Dic7C2×D411yescharacteristic
109C2×Dic7C2×D411yescharacteristic
110D28—22no—
111D28—22no—
112D28—42no—
113D28—44no—
114D28—42no—
115C4×D7—21no—
116C4×D7—21no—
117C4×D7—41no—
118C4×D7—44no—
119C4×D7SD1614yescomplemented
120Dic14—21no—
121Dic14—21no—
122Dic14—42no—
123C7×C8—22no—
124C7⋊C8—22no—
125C2×#41—141no—
126C2×SD16—72no—
127C2×SD16—72no—
128C22×C8—72no—
129C4⋊Q8—71no—
130C4⋊Q8—71no—
131C4⋊Q8C2×D711yescharacteristic
132#127—71no—
133#125—141no—
134#124—141no—
135C22⋊Q8—71no—
136C22⋊Q8—71no—
137C22⋊Q8—141no—
138#121—72no—
139#121—141no—
140C4×Q8—71no—
141C4×Q8—141no—
142C4×Q8C2×D711yescharacteristic
143C4×D4—72no—
144C4×D4—141no—
145#117—141no—
146C2×C42—71no—
147#106—72no—
148#106C2×D712yes—
149C4⋊C8—72no—
150C4⋊C8—71no—
151C4⋊C8C2×D711yescharacteristic
152#103—72no—
153#103C2×D712yes—
154#102—72no—
155#102—72no—
156C22×C7C22×D411yescharacteristic
157C2×D7—21no—
158C2×D7—21no—
159C2×D7—42no—
160C2×D7—42no—
161C2×D7—42no—
162C2×D7C2×SD1612yes—
163C4×C7—21no—
164C4×C7—41no—
165C4×C7—42no—
166C4×C7—41no—
167C4×C7C2×SD1612yes—
168C4×C7H(5,𝔽2)11yescharacteristic
169C4×C72−1+411yescharacteristic
170Dic7—21no—
171Dic7—21no—
172Dic7—41no—
173Dic7—41no—
174Dic7C2×SD1612yes—
175#41—284no—
176C2×Q8—21no—
177C2×Q8—71no—
178C2×Q8—141no—
179C2×Q8C22×D711yescharacteristic
180C2×D4—72no—
181C2×D4—142no—
182C2×D4—141no—
183C2×D4—141no—
184C22×C4—71no—
185C22×C4—71no—
186C22×C4—71no—
187C22×C4—142no—
188SD16—282no—
189SD16—282no—
190C2×C8—72no—
191C2×C8—282no—
192C2×C8C22×D712yes—
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—71no—
196C4⋊C4—141no—
197C4⋊C4—141no—
198C4⋊C4—141no—
199C4⋊C4—141no—
200C4⋊C4—141no—
201C4⋊C4—141no—
202C4⋊C4C22×D711yescharacteristic
203C4⋊C4C22×D712yes—
204C22⋊C4—142no—
205C22⋊C4—142no—
206C22⋊C4—141no—
207C22⋊C4—141no—
208C42—21no—
209C42—71no—
210C42—71no—
211C42—71no—
212C42—141no—
213C42C22×D711yescharacteristic
214C2×C7#53711yescharacteristic
215C2×C7C22×SD1611yescharacteristic
216C2×C7#57511yescharacteristic
217D7—22no—
218D7—82no—
219C23—71no—
220C23—142no—
221Q8—21no—
222Q8—21no—
223Q8—42no—
224Q8—141no—
225Q8—141no—
226Q8—282no—
227D4—142no—
228D4—142no—
229D4—282no—
230D4—284no—
231D4—282no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—71no—
236C2×C4—74no—
237C2×C4—71no—
238C2×C4—71no—
239C2×C4—141no—
240C2×C4—141no—
241C2×C4—141no—
242C2×C4—141no—
243C2×C4—281no—
244C2×C4—284no—
245C2×C4D4×D711yescharacteristic
246C2×C4D4×D711yescharacteristic
247C2×C4C23×D711yescharacteristic
248C8—22no—
249C8—142no—
250C7#324611yescharacteristic, complemented, Sylow
251C22—72no—
252C22—141no—
253C22—141no—
254C22—282no—
255C22—282no—
256C22—282no—
257C22C2×D4×D711yescharacteristic
258C4—21no—
259C4—42no—
260C4—41no—
261C4—41no—
262C4—72no—
263C4—141no—
264C4—141no—
265C4—281no—
266C4—281no—
267C4D7×SD1612yes—
268C4#655111yescharacteristic
269C4#655311yescharacteristic
270C2—142no—
271C2—562no—
272C2#9032011yescharacteristic
273C2C2×D7×SD1611yescharacteristic
274C2#9046811yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#475C2
socleC22×C7C22×D4
prime radical#90416C2
hypercenter#475D7
hypocenterC7#3246
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
2. #90269C2semidirect
3. C7×#475C2semidirect
4. #89866C2semidirect
5. #89799C2semidirect
6. #89642C2semidirect
7. #89636C2semidirect
8. #89620C2semidirect
9. D7×C4⋊C8C2semidirect
10. #89590C2semidirect
11. #89579C2semidirect
87. #475D7semidirect
119. C4×D7SD16semidirect
250. C7#3246semidirect

Maximal subgroups

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

subgroupindex
#904162
#902692
C7×#4752
#898662
#897992
#896422
#896362
#896202
D7×C4⋊C82
#895902
#895792
#32467