Finite Group Databasebeta

← #11736792

Subgroups — #11736792

Order 896.

430 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 125 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#90168C211yesmaximal, characteristic, complemented
2C4×#1157C211yesmaximal, characteristic, complemented
3C7×#432C211yesmaximal, characteristic, complemented
4#89623C211yesmaximal, characteristic, complemented
5#89620C211yesmaximal, characteristic, complemented
6#89618C211yesmaximal, characteristic, complemented
7#89615C211yesmaximal, characteristic, complemented
8#89509C212yesmaximal, complemented
9#89506C212yesmaximal, complemented
10#89483C212yesmaximal, complemented
11#89482C212yesmaximal, complemented
12C2×#1157—21no—
13C2×#1157C2211yescharacteristic
14C2×C42×C7C2211yescharacteristic
15#6493—21no—
16#6493C2212yes—
17C4×C22⋊D7—21no—
18C4×C22⋊D7—21no—
19#6489C2211yescharacteristic
20C2×D56C2212yes—
21C2×#1132C2212yes—
22#6438—21no—
23#6437C2211yescharacteristic
24C4×D28—21no—
25C4×D28C2211yescharacteristic
26C4×D28C2211yescharacteristic
27C4×D28C2212yes—
28#6435—21no—
29C42×D7—21no—
30#6432C2211yescharacteristic
31C4×Dic14C2211yescharacteristic
32C4×Dic14C2211yescharacteristic
33C7×C4⋊C8C2211yescharacteristic
34C7×C4⋊C8C2211yescharacteristic
35C7×C22⋊C8C2212yes—
36C7×#97C2212yes—
37#6395C2212yes—
38#6395C2212yes—
39#6395C2212yes—
40#6392C2212yes—
41#6391C2212yes—
42#6390C2212yes—
43#3115—71nomaximal, Sylow
44C22×C4×C7D411yescharacteristic
45C22×C4×C7D411yescharacteristic
46C22×C4×C7C2311yescharacteristic
47C2×C22⋊D7—21no—
48C2×C22⋊D7—22no—
49#1157—22no—
50#1157—22no—
51#1157—44no—
52C2×D28—21no—
53C2×D28—21no—
54C2×D28C2312yes—
55C2×D28C2311yescharacteristic
56C2×C4×D7—21no—
57C2×C4×D7—21no—
58C2×C4×D7—21no—
59C2×C4×D7—22no—
60C2×Dic14—21no—
61C2×Dic14C2311yescharacteristic
62C2×C7×C8C2312yes—
63C2×C7×C8C2312yes—
64C42×C7D411yescharacteristic
65C42×C7D411yescharacteristic
66C42×C7C2311yescharacteristic
67C42×C7C2311yescharacteristic
68C22⋊Dic7—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
75C4⋊Dic7C2311yescharacteristic
76C4⋊Dic7C2312yes—
77Dic7⋊C4—21no—
78Dic7⋊C4—21no—
79C4×Dic7—21no—
80C4×Dic7—21no—
81C4×Dic7—21no—
82D56—24no—
83#1132—24no—
84#532—71no—
85C4×#41—71no—
86#482—72no—
87#475—71no—
88#474—71no—
89#460—71no—
90#459—71no—
91#449—72no—
92#442—72no—
93#440—72no—
94#432D711yescharacteristic, complemented
95C23×C7C2×D411yescharacteristic
96C22×D7—21no—
97C22×D7—22no—
98C2×C4×C7—22no—
99C2×C4×C7—21no—
100C2×C4×C7—21no—
101C2×C4×C7—41no—
102C2×C4×C7C2×D411yescharacteristic
103C2×C4×C7C2×D411yescharacteristic
104C2×C4×C7C2×D411yescharacteristic
105C2×C4×C7C2×D411yescharacteristic
106C2×C4×C7C2×D411yescharacteristic
107C2×C4×C7C2411yescharacteristic
108C22⋊D7—24no—
109C22⋊D7—44no—
110C2×Dic7—21no—
111C2×Dic7—21no—
112C2×Dic7—21no—
113C2×Dic7—22no—
114D28—21no—
115D28—22no—
116D28—22no—
117D28—42no—
118D28—42no—
119D28#4112yescomplemented
120C4×D7—24no—
121C4×D7—42no—
122C4×D7—44no—
123C4×D7—42no—
124Dic14—21no—
125Dic14—42no—
126Dic14#4112yescomplemented
127C7×C8—22no—
128C7×C8—22no—
129C2×#41—71no—
130C2×#41—141no—
131C2×SD16—72no—
132C2×D8—72no—
133C4⋊Q8—71no—
134#127—71no—
135#124—141no—
136#121—72no—
137#121—141no—
138C4×Q8—71no—
139C4×Q8—71no—
140C4×D4—71no—
141C4×D4—71no—
142C4×D4—72no—
143C4×D4—141no—
144C4×D4—141no—
145C4×D4—141no—
146#117—71no—
147#117—141no—
148C2×C42—141no—
149C2×C42C2×D711yescharacteristic
150#107—72no—
151#106—72no—
152C4⋊C8C2×D711yescharacteristic
153C4⋊C8C2×D711yescharacteristic
154#103—72no—
155#102—72no—
156#102—72no—
157#102—72no—
158C22⋊C8C2×D712yes—
159#97C2×D712yes—
160C22×C7—21no—
161C22×C7—21no—
162C22×C7—21no—
163C22×C7C22×D411yescharacteristic
164C2×D7—22no—
165C2×D7—41no—
166C2×D7—41no—
167C2×D7—42no—
168C2×D7—42no—
169C2×D7—42no—
170C4×C7—21no—
171C4×C7—21no—
172C4×C7—21no—
173C4×C7—21no—
174C4×C7AGL(1,ℤ/8)12yes—
175C4×C7C2×#4111yescharacteristic
176C4×C7C2×#4111yescharacteristic
177Dic7—22no—
178Dic7—42no—
179Dic7—41no—
180Dic7—41no—
181#41—142no—
182#41—142no—
183#41—284no—
184C2×Q8—71no—
185C2×Q8—141no—
186C2×D4—71no—
187C2×D4—72no—
188C2×D4—141no—
189C2×D4—142no—
190C2×D4—141no—
191C2×D4—141no—
192C22×C4—141no—
193C22×C4—141no—
194C22×C4—141no—
195C22×C4—142no—
196C22×C4D2811yescharacteristic
197C22×C4D2811yescharacteristic
198C22×C4C22×D711yescharacteristic
199SD16—144no—
200D8—144no—
201C2×C8C22×D712yes—
202C2×C8C22×D712yes—
203C4⋊C4—71no—
204C4⋊C4—71no—
205C4⋊C4—72no—
206C4⋊C4—141no—
207C4⋊C4—141no—
208C4⋊C4—141no—
209C22⋊C4—141no—
210C22⋊C4—141no—
211C22⋊C4—141no—
212C22⋊C4—142no—
213C22⋊C4—142no—
214C42—141no—
215C42—141no—
216C42—141no—
217C42D2811yescharacteristic
218C42D2811yescharacteristic
219C42C22×D711yescharacteristic
220C42C22×D711yescharacteristic
221C2×C7—41no—
222C2×C7#52511yescharacteristic
223C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
224C2×C7#57511yescharacteristic
225D7—42no—
226D7—82no—
227C23—141no—
228C23—142no—
229C23C2×D2811yescharacteristic
230Q8—72no—
231Q8—141no—
232Q8—282no—
233D4—72no—
234D4—144no—
235D4—141no—
236D4—142no—
237D4—142no—
238D4—282no—
239D4—284no—
240D4—282no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—41no—
245C2×C4—144no—
246C2×C4—141no—
247C2×C4—141no—
248C2×C4—141no—
249C2×C4—142no—
250C2×C4—282no—
251C2×C4—284no—
252C2×C4—282no—
253C2×C4C2×D2811yescharacteristic
254C2×C4C2×D2811yescharacteristic
255C2×C4C2×D2811yescharacteristic
256C2×C4C2×D2811yescharacteristic
257C2×C4C2×D2811yescharacteristic
258C2×C4C23×D711yescharacteristic
259C8—22no—
260C8—22no—
261C7#311511yescharacteristic, complemented, Sylow
262C22—21no—
263C22—21no—
264C22—21no—
265C22—142no—
266C22—281no—
267C22—281no—
268C22—282no—
269C22—282no—
270C22—282no—
271C22C22×D2811yescharacteristic
272C4—21no—
273C4—21no—
274C4—21no—
275C4—21no—
276C4—142no—
277C4—282no—
278C4—281no—
279C4—281no—
280C4#647112yes—
281C4D7×#4111yescharacteristic
282C4D7×#4111yescharacteristic
283C2—41no—
284C2—282no—
285C2—562no—
286C2#9021511yescharacteristic
287C2C2×#647111yescharacteristic
288C2#9043911yescharacteristic
subgroupG/H
centerC22C22×D28
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#432C2
socleC22×C7C22×D4
hypercenter#432D7
hypocenterC7#3115
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #432

Complements

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

NH ≅ G/Ntype
1. #90168C2semidirect
2. C4×#1157C2semidirect
3. C7×#432C2semidirect
4. #89623C2semidirect
5. #89620C2semidirect
6. #89618C2semidirect
7. #89615C2semidirect
8. #89509C2semidirect
9. #89506C2semidirect
10. #89483C2semidirect
11. #89482C2semidirect
94. #432D7semidirect
119. D28#41semidirect
126. Dic14#41semidirect
261. C7#3115semidirect

Maximal subgroups

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

subgroupindex
#901682
C4×#11572
C7×#4322
#896232
#896202
#896182
#896152
#895092
#895062
#894832
#894822
#31157