Finite Group Databasebeta

← #11738576

Subgroups — #11738576

Order 896.

398 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 115 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×#477C211yesmaximal, characteristic, complemented
4#89856C211yesmaximal, characteristic, complemented
5#89799C211yesmaximal, characteristic, complemented
6#89663C212yesmaximal, complemented
7#89655C212yesmaximal, complemented
8#89615C211yesmaximal, characteristic, complemented
9D7×C4⋊C8C211yesmaximal, characteristic, complemented
10#89591C212yesmaximal, complemented
11#89579C212yesmaximal, complemented
12C2×#1161—21no—
13C7×C4⋊Q8C2211yescharacteristic
14C4×C7×Q8C2211yescharacteristic
15#6510—21no—
16#6510C2211yescharacteristic
17#6509C2211yescharacteristic
18C2×#1143C2212yes—
19C2×D56C2212yes—
20C2×C8×D7C2212yes—
21#6461—21no—
22#6458—21no—
23#6458C2212yes—
24#6458C2211yescharacteristic
25#6457—21no—
26#6456—21no—
27#6456C2212yes—
28#6453C2211yescharacteristic
29#6437C2211yescharacteristic
30C4×D28—21no—
31C4×D28C2211yescharacteristic
32#6435—21no—
33C42×D7C2211yescharacteristic
34C7×#107C2212yes—
35C7×C4⋊C8C2211yescharacteristic
36C7×#103C2212yes—
37#6395C2212yes—
38Dic7⋊C8C2212yes—
39#6383C2212yes—
40#6383C2212yes—
41#6382C2212yes—
42C4⋊(C7⋊C8)C2211yescharacteristic
43#3247—71nomaximal, Sylow
44C2×C7×Q8—21no—
45C2×C7×Q8C2311yescharacteristic
46#1161—44no—
47C2×D28—21no—
48C2×D28—22no—
49C2×D28—21no—
50C2×D28—21no—
51C2×D28C2311yescharacteristic
52C2×D28C2312yes—
53C2×C4×D7—21no—
54C2×C4×D7—22no—
55C2×C4×D7D411yescharacteristic
56C2×C4×D7D411yescharacteristic
57C2×C4×D7C2311yescharacteristic
58C2×C7×C8C2312yes—
59C7×C4⋊C4—21no—
60C7×C4⋊C4—21no—
61C7×C4⋊C4C2312yes—
62C7×C4⋊C4C2311yescharacteristic
63C42×C7—21no—
64C42×C7C2311yescharacteristic
65#1143—42no—
66(C2×D7)⋊C4—21no—
67(C2×D7)⋊C4—21no—
68(C2×D7)⋊C4—21no—
69(C2×D7)⋊C4—21no—
70(C2×D7)⋊C4—21no—
71(C2×D7)⋊C4—22no—
72(C2×D7)⋊C4—22no—
73C4⋊Dic7—21no—
74C4⋊Dic7C2311yescharacteristic
75Dic7⋊C4—21no—
76Dic7⋊C4—21no—
77C4×Dic7D411yescharacteristic
78C4×Dic7D411yescharacteristic
79C4×Dic7C2311yescharacteristic
80C2×C7⋊C8C2312yes—
81D56—42no—
82C8×D7—42no—
83#532—71no—
84#529—71no—
85#478—72no—
86#477D711yescharacteristic, complemented
87#476—71no—
88#466—72no—
89#465—72no—
90#463—72no—
91#460—71no—
92#459—71no—
93C2×C4⋊C8—71no—
94C22×D7—21no—
95C22×D7—22no—
96C22×D7C2×D411yescharacteristic
97C7×Q8—21no—
98C7×Q8—21no—
99C7×Q8—42no—
100C2×C4×C7—21no—
101C2×C4×C7—21no—
102C2×C4×C7—22no—
103C2×C4×C7C2×D411yescharacteristic
104C2×C4×C7C2×D411yescharacteristic
105C2×C4×C7C2411yescharacteristic
106C2×Dic7—21no—
107C2×Dic7C2×D411yescharacteristic
108C2×Dic7C2×D411yescharacteristic
109C2×Dic7C2×D411yescharacteristic
110D28—21no—
111D28—22no—
112D28—21no—
113D28—22no—
114D28—42no—
115D28—42no—
116D28—44no—
117D28—42no—
118C4×D7—22no—
119C4×D7—22no—
120C4×D7—21no—
121C4×D7—21no—
122C4×D7—44no—
123C4×D7—42no—
124C7×C8—22no—
125C7⋊C8—22no—
126C2×#41—141no—
127C2×SD16—72no—
128C2×D8—72no—
129C22×C8—72no—
130C4⋊Q8C2×D711yescharacteristic
131#127—71no—
132#126—141no—
133#125—71no—
134#124—71no—
135#124—141no—
136#123—141no—
137C22⋊Q8—71no—
138#121—71no—
139#121—72no—
140#121—141no—
141C4×Q8C2×D711yescharacteristic
142C4×D4—72no—
143C4×D4—71no—
144C4×D4—141no—
145C4×D4—141no—
146#117—141no—
147C2×C42—71no—
148#107C2×D712yes—
149#106—72no—
150C4⋊C8—72no—
151C4⋊C8—71no—
152C4⋊C8C2×D711yescharacteristic
153#103C2×D712yes—
154#102—72no—
155#102—72no—
156#102—72no—
157C22×C7C22×D411yescharacteristic
158C2×D7—21no—
159C2×D7—21no—
160C2×D7—41no—
161C2×D7—42no—
162C2×D7—41no—
163C2×D7—42no—
164C2×D7—42no—
165C2×D7—41no—
166C2×D7AGL(1,ℤ/8)12yes—
167C4×C7—21no—
168C4×C7—41no—
169C4×C7—42no—
170C4×C7—41no—
171C4×C7#13512yes—
172C4×C7H(5,𝔽2)11yescharacteristic
173C4×C72−1+411yescharacteristic
174Dic7—21no—
175Dic7—21no—
176Dic7—21no—
177Dic7—41no—
178#41—284no—
179C2×Q8—21no—
180C2×Q8C22×D711yescharacteristic
181C2×D4—71no—
182C2×D4—72no—
183C2×D4—141no—
184C2×D4—142no—
185C2×D4—141no—
186C2×D4—141no—
187C22×C4—71no—
188C22×C4—71no—
189C22×C4—71no—
190C22×C4—142no—
191C22×C4—141no—
192SD16—282no—
193D8—282no—
194C2×C8—72no—
195C2×C8—282no—
196C2×C8C22×D712yes—
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4—71no—
200C4⋊C4—141no—
201C4⋊C4—141no—
202C4⋊C4—141no—
203C4⋊C4C22×D711yescharacteristic
204C4⋊C4C22×D712yes—
205C22⋊C4—141no—
206C22⋊C4—141no—
207C22⋊C4—142no—
208C22⋊C4—142no—
209C22⋊C4—141no—
210C22⋊C4—141no—
211C22⋊C4—141no—
212C42—21no—
213C42—71no—
214C42—71no—
215C42—71no—
216C42C22×D711yescharacteristic
217C2×C7#53711yescharacteristic
218C2×C7C2×#13511yescharacteristic
219C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
220D7—22no—
221D7—81no—
222D7—82no—
223C23—71no—
224C23—141no—
225C23—142no—
226Q8—21no—
227Q8—21no—
228Q8—42no—
229D4—141no—
230D4—142no—
231D4—141no—
232D4—142no—
233D4—282no—
234D4—282no—
235D4—284no—
236D4—282no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—71no—
241C2×C4—71no—
242C2×C4—71no—
243C2×C4—142no—
244C2×C4—142no—
245C2×C4—141no—
246C2×C4—141no—
247C2×C4—141no—
248C2×C4—284no—
249C2×C4—282no—
250C2×C4D4×D711yescharacteristic
251C2×C4D4×D711yescharacteristic
252C2×C4C23×D711yescharacteristic
253C8—22no—
254C8—142no—
255C7#324711yescharacteristic, complemented, Sylow
256C22—72no—
257C22—141no—
258C22—141no—
259C22—281no—
260C22—282no—
261C22—282no—
262C22—281no—
263C22—281no—
264C22—282no—
265C22C2×D4×D711yescharacteristic
266C4—21no—
267C4—42no—
268C4—41no—
269C4—41no—
270C4—141no—
271C4—141no—
272C4—141no—
273C4—281no—
274C4#648212yes—
275C4#655111yescharacteristic
276C4#655311yescharacteristic
277C2—142no—
278C2—561no—
279C2—562no—
280C2#9032011yescharacteristic
281C2C2×#648211yescharacteristic
282C2D7×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#477C2
socleC22×C7C22×D4
hypercenter#477D7
hypocenterC7#3247
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #477

Complements

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

NH ≅ G/Ntype
1. #90416C2semidirect
2. #90271C2semidirect
3. C7×#477C2semidirect
4. #89856C2semidirect
5. #89799C2semidirect
6. #89663C2semidirect
7. #89655C2semidirect
8. #89615C2semidirect
9. D7×C4⋊C8C2semidirect
10. #89591C2semidirect
11. #89579C2semidirect
86. #477D7semidirect
255. C7#3247semidirect

Maximal subgroups

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

subgroupindex
#904162
#902712
C7×#4772
#898562
#897992
#896632
#896552
#896152
D7×C4⋊C82
#895912
#895792
#32477