Finite Group Databasebeta

← #11737440

Subgroups — #11737440

Order 896.

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

normal subgroups only

This group has 410 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
1C2×#6499C211yesmaximal, characteristic, complemented
2#90291C211yesmaximal, characteristic, complemented
3C7×#447C211yesmaximal, characteristic
4#89990C211yesmaximal, characteristic, complemented
5#89927C212yesmaximal, complemented
6#89925C212yesmaximal, complemented
7#89812C211yesmaximal, characteristic, complemented
8#89540C212yesmaximal, complemented
9#89539C212yesmaximal, complemented
10#89511C211yesmaximal, characteristic, complemented
11#89490C211yesmaximal, characteristic, complemented
12C22×C7×D4C2211yescharacteristic
13C22×Dic14C2211yescharacteristic
14C2×C7×D8C2212yes—
15C7×#121C2211yescharacteristic
16C2×C22⋊Dic7—21no—
17C2×C22⋊Dic7—21no—
18#6499—21no—
19#6499—21no—
20#6499C2212yes—
21#6499C2212yes—
22#6499C2211yescharacteristic, complemented
23#6498—21no—
24D4×Dic7—21no—
25D4×Dic7C2211yescharacteristic, complemented
26C2×#1142C2212yes—
27C2×#1142C2212yescomplemented
28#6487C2211yescharacteristic
29C2×C4×Dic7C2211yescharacteristic
30C2×Dic28C2212yescomplemented
31#6453C2211yescharacteristic, complemented
32#6450C2211yescharacteristic, complemented
33#6442—21no—
34#6440—21no—
35C7×#102C2212yes—
36C7×C22⋊C8C2211yescharacteristic
37#6406C2212yes—
38C22⋊(C7⋊C8)C2211yescharacteristic
39#6390C2212yescomplemented
40Dic7⋊C8C2212yescomplemented
41C8×Dic7C2212yescomplemented
42#6384C2212yescomplemented
43#3211—71nomaximal, Sylow
44C24×C7—21no—
45C2×C7×D4—21no—
46C2×C7×D4—21no—
47C2×C7×D4—42no—
48C2×C7×D4C2312yes—
49C2×C7×D4C2311yescharacteristic
50C22×C4×C7C2311yescharacteristic
51C22×Dic7—21no—
52C22×Dic7—21no—
53C22×Dic7D411yescharacteristic
54C22×Dic7D411yescharacteristic
55C2×Dic14—21no—
56C2×Dic14—42no—
57C2×Dic14C2311yescharacteristic
58C2×Dic14C2312yes—
59C7×D8—42no—
60C2×C7×C8C2312yes—
61C7×C4⋊C4C2311yescharacteristic
62C7×C22⋊C4—21no—
63C22⋊Dic7—21no—
64C22⋊Dic7—21no—
65C22⋊Dic7—21no—
66C22⋊Dic7—21no—
67C22⋊Dic7—22no—
68C22⋊Dic7—22no—
69C22⋊Dic7—22no—
70C22⋊Dic7—22no—
71#1142—42no—
72#1142—42no—
73C4⋊Dic7—21no—
74C4⋊Dic7C2311yescharacteristic
75Dic7⋊C4—21no—
76Dic7⋊C4—21no—
77Dic7⋊C4—21no—
78C4×Dic7—21no—
79C4×Dic7D411yescharacteristic
80C4×Dic7D411yescharacteristic
81C4×Dic7C2311yescharacteristic
82C4×Dic7C2311yescharacteristic
83C2×C7⋊C8C2312yes—
84Dic28—42no—
85#529—71no—
86C2×#124—71no—
87#495—72no—
88#488—72no—
89#464—72no—
90#463—72no—
91#451—71no—
92#450—71no—
93#448—71no—
94#447D711yescharacteristic
95#431—71no—
96C23×C7—22no—
97C23×C7—21no—
98C23×C7—42no—
99C23×C7—42no—
100C23×C7—42no—
101C23×C7C2×D411yescharacteristic
102C7×D4—22no—
103C7×D4—22no—
104C7×D4—22no—
105C7×D4—42no—
106C7×D4—42no—
107C2×C4×C7—22no—
108C2×C4×C7—21no—
109C2×C4×C7C2×D411yescharacteristic
110C2×C4×C7C2411yescharacteristic
111C2×Dic7—22no—
112C2×Dic7—21no—
113C2×Dic7—21no—
114C2×Dic7—22no—
115C2×Dic7—41no—
116C2×Dic7—42no—
117C2×Dic7—42no—
118C2×Dic7C2×D411yescharacteristic
119C2×Dic7C2×D411yescharacteristic
120C2×Dic7C2×D411yescharacteristic
121C2×Dic7C2×D411yescharacteristic
122Dic14—22no—
123Dic14—22no—
124Dic14—22no—
125Dic14—42no—
126C7×C8—22no—
127C7⋊C8—22no—
128C22×Q8—71no—
129C22×D4C2×D711yescharacteristic
130C2×Q16—72no—
131C2×SD16—72no—
132C2×SD16—72no—
133C2×D8C2×D712yes—
134#126—141no—
135#125—71no—
136#124—72no—
137#124—72no—
138#124—71no—
139#124—141no—
140#124—141no—
141#123—141no—
142C22⋊Q8—71no—
143#121C2×D711yescharacteristic
144C4×Q8—71no—
145C4×D4—71no—
146C4×D4—141no—
147#117—141no—
148C2×C22⋊C4—141no—
149C2×C22⋊C4—141no—
150C2×C42—71no—
151C4⋊C8—72no—
152#103—72no—
153#103—72no—
154#102—72no—
155#102C2×D712yes—
156C22⋊C8—71no—
157C22⋊C8C2×D711yescharacteristic
158C4×C8—72no—
159C22×C7—21no—
160C22×C7—21no—
161C22×C7—42no—
162C22×C7—41no—
163C22×C7—41no—
164C22×C7—42no—
165C22×C7—41no—
166C22×C7—42no—
167C22×C7—84no—
168C22×C7#13512yes—
169C22×C7C22×D411yescharacteristic
170C4×C7—21no—
171C4×C7—41no—
172C4×C7H(5,𝔽2)12yes—
173Dic7—21no—
174Dic7—21no—
175Dic7—21no—
176Dic7—21no—
177Dic7—42no—
178Dic7—41no—
179Dic7—41no—
180C24—21no—
181C2×Q8—72no—
182C2×Q8—71no—
183C2×Q8—141no—
184C2×Q8—282no—
185C2×D4—21no—
186C2×D4—21no—
187C2×D4—42no—
188C2×D4C22×D712yes—
189C2×D4C22×D711yescharacteristic
190C22×C4—71no—
191C22×C4—71no—
192C22×C4—141no—
193C22×C4—141no—
194C22×C4C22×D711yescharacteristic
195Q16—282no—
196SD16—282no—
197SD16—282no—
198D8—42no—
199C2×C8—72no—
200C2×C8C22×D712yes—
201C4⋊C4—71no—
202C4⋊C4—141no—
203C4⋊C4—141no—
204C4⋊C4—141no—
205C4⋊C4—141no—
206C4⋊C4C22×D711yescharacteristic
207C22⋊C4—21no—
208C22⋊C4—142no—
209C22⋊C4—142no—
210C22⋊C4—142no—
211C22⋊C4—142no—
212C22⋊C4—141no—
213C22⋊C4—141no—
214C22⋊C4—141no—
215C22⋊C4—141no—
216C42—71no—
217C42—71no—
218C42—71no—
219C42—71no—
220C42—141no—
221C2×C7—22no—
222C2×C7—81no—
223C2×C7—82no—
224C2×C7#53511yescharacteristic
225C2×C7C2×#13511yescharacteristic
226C2×C7#57511yescharacteristic
227C23—21no—
228C23—22no—
229C23—42no—
230C23—42no—
231C23—42no—
232C23D4×D711yescharacteristic
233Q8—142no—
234Q8—142no—
235Q8—142no—
236Q8—282no—
237D4—22no—
238D4—22no—
239D4—22no—
240D4—42no—
241D4—42no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—71no—
245C2×C4—71no—
246C2×C4—71no—
247C2×C4—71no—
248C2×C4—142no—
249C2×C4—141no—
250C2×C4—142no—
251C2×C4—141no—
252C2×C4—281no—
253C2×C4—282no—
254C2×C4—282no—
255C2×C4D4×D711yescharacteristic
256C2×C4C23×D711yescharacteristic
257C8—22no—
258C8—142no—
259C7#321111yescharacteristic, complemented, Sylow
260C22—21no—
261C22—21no—
262C22—41no—
263C22—42no—
264C22—41no—
265C22—42no—
266C22—42no—
267C22—41no—
268C22—84no—
269C22#647512yes—
270C22C2×D4×D711yescharacteristic
271C4—21no—
272C4—41no—
273C4—141no—
274C4—141no—
275C4—141no—
276C4—141no—
277C4—281no—
278C4—282no—
279C4—281no—
280C4#654812yes—
281C2—22no—
282C2—82no—
283C2—81no—
284C2#9028311yescharacteristic
285C2C2×#647511yescharacteristic
286C2#9046411yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
Fitting, prime radicalC7×#447C2
socleC22×C7C22×D4
hypercenter#447D7
hypocenterC7#3211
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. C2×#6499C2semidirect
2. #90291C2semidirect
4. #89990C2semidirect
5. #89927C2semidirect
6. #89925C2semidirect
7. #89812C2semidirect
8. #89540C2semidirect
9. #89539C2semidirect
10. #89511C2semidirect
11. #89490C2semidirect
22. #6499C22semidirect
25. D4×Dic7C22semidirect
27. C2×#1142C22semidirect
30. C2×Dic28C22semidirect
31. #6453C22semidirect
32. #6450C22semidirect
39. #6390C22semidirect
40. Dic7⋊C8C22semidirect
41. C8×Dic7C22semidirect
42. #6384C22semidirect
259. C7#3211semidirect

Maximal subgroups

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

subgroupindex
C2×#64992
#902912
C7×#4472
#899902
#899272
#899252
#898122
#895402
#895392
#895112
#894902
#32117