Finite Group Databasebeta

← #11739424

Subgroups — #11739424

Order 896.

366 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 111 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#90379C211yesmaximal, characteristic, complemented
3C7×#501C211yesmaximal, characteristic, complemented
4#89852C211yesmaximal, characteristic, complemented
5#89839C211yesmaximal, characteristic, complemented
6#89662C212yesmaximal, complemented
7#89657C212yesmaximal, complemented
8#89642C212yesmaximal, complemented
9#89638C212yesmaximal, complemented
10#89485C211yesmaximal, characteristic, complemented
11#89477C211yesmaximal, characteristic, complemented
12C2×#1161—21no—
13C7×C4⋊Q8C2211yescharacteristic
14C7×#125C2211yescharacteristic
15#6510—21no—
16#6459—21no—
17#6459—21no—
18#6458—21no—
19#6458C2212yes—
20#6456—21no—
21#6456C2212yes—
22#6455C2211yescharacteristic
23#6455C2211yescharacteristic
24#6451C2211yescharacteristic
25#6451C2211yescharacteristic
26#6450—21no—
27#6450—21no—
28#6437C2211yescharacteristic
29C42×D7C2211yescharacteristic
30#6433C2211yescharacteristic
31C7×#107C2212yes—
32C7×#106C2212yes—
33C7×#97C2211yescharacteristic
34#6395C2212yes—
35#6395C2212yes—
36(C2×D7)⋊C8C2212yes—
37Dic7⋊C8C2212yes—
38#6383C2212yes—
39#6383C2212yes—
40#6382C2212yes—
41#6381C2212yes—
42#6377C2211yescharacteristic
43#3235—71nomaximal, Sylow
44C2×C7×Q8—21no—
45#1161—44no—
46C2×D28—22no—
47C2×D28—21no—
48C2×D28—21no—
49C2×D28C2312yes—
50C2×C4×D7—22no—
51C2×C4×D7D411yescharacteristic
52C2×C4×D7D411yescharacteristic
53C2×C4×D7C2311yescharacteristic
54C2×Dic14—21no—
55C2×Dic14—21no—
56C2×C7×C8C2312yes—
57C7×C4⋊C4—21no—
58C7×C4⋊C4—21no—
59C7×C4⋊C4—21no—
60C7×C4⋊C4C2311yescharacteristic
61C7×C4⋊C4C2312yes—
62C7×C4⋊C4C2311yescharacteristic
63C42×C7C2311yescharacteristic
64(C2×D7)⋊C4—21no—
65(C2×D7)⋊C4—21no—
66(C2×D7)⋊C4—22no—
67(C2×D7)⋊C4—22no—
68C4⋊Dic7C2311yescharacteristic
69C4⋊Dic7C2311yescharacteristic
70Dic7⋊C4—21no—
71Dic7⋊C4—21no—
72Dic7⋊C4—21no—
73Dic7⋊C4—21no—
74C4×Dic7—21no—
75C4×Dic7—21no—
76C4×Dic7D411yescharacteristic
77C4×Dic7D411yescharacteristic
78C4×Dic7C2311yescharacteristic
79C2×C7⋊C8C2312yes—
80#533—71no—
81#532—71no—
82#501—71no—
83#501D711yescharacteristic, complemented
84#490—71no—
85#490—71no—
86#482—72no—
87#482—72no—
88#476—72no—
89#474—72no—
90#432—71no—
91C22×D7—22no—
92C22×D7C2×D411yescharacteristic
93C7×Q8—42no—
94C2×C4×C7—21no—
95C2×C4×C7—21no—
96C2×C4×C7—22no—
97C2×C4×C7C2×D411yescharacteristic
98C2×C4×C7C2×D411yescharacteristic
99C2×C4×C7C2411yescharacteristic
100C2×Dic7—21no—
101C2×Dic7—21no—
102C2×Dic7C2×D411yescharacteristic
103C2×Dic7C2×D411yescharacteristic
104C2×Dic7C2×D411yescharacteristic
105D28—24no—
106D28—42no—
107D28—42no—
108D28—44no—
109C4×D7—22no—
110C4×D7—22no—
111C4×D7—41no—
112C4×D7—44no—
113Dic14—42no—
114Dic14—42no—
115C7×C8—22no—
116C7⋊C8—22no—
117C2×#41—141no—
118C4⋊Q8—71no—
119C4⋊Q8—71no—
120C4⋊Q8C2×D711yescharacteristic
121#127—71no—
122#125—71no—
123#125C2×D711yescharacteristic
124#124—141no—
125C22⋊Q8—141no—
126C22⋊Q8—141no—
127#121—72no—
128#121—141no—
129C4×Q8—141no—
130C4×Q8—141no—
131C4×D4—72no—
132C4×D4—141no—
133#117—71no—
134#117—71no—
135C2×C42—71no—
136#107—72no—
137#107C2×D712yes—
138#106—72no—
139#106C2×D712yes—
140C4⋊C8—72no—
141#102—72no—
142#102—72no—
143#102—72no—
144#102—72no—
145C22⋊C8—72no—
146#97—71no—
147#97C2×D711yescharacteristic
148C22×C7C22×D411yescharacteristic
149C2×D7—21no—
150C2×D7—21no—
151C2×D7—21no—
152C2×D7—42no—
153C2×D7—42no—
154C2×D7—42no—
155C4×C7—21no—
156C4×C7—21no—
157C4×C7—41no—
158C4×C7—42no—
159C4×C7—41no—
160C4×C72−1+412yes—
161Dic7—21no—
162Dic7—21no—
163Dic7—41no—
164Dic7—41no—
165Dic7AGL(1,ℤ/8)12yes—
166#41—284no—
167C2×Q8—21no—
168C2×Q8—141no—
169C2×Q8—141no—
170C2×D4—72no—
171C2×D4—142no—
172C2×D4—141no—
173C2×D4—141no—
174C22×C4—71no—
175C22×C4—71no—
176C22×C4—71no—
177C22×C4—142no—
178C2×C8—72no—
179C2×C8C22×D712yes—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—71no—
184C4⋊C4—71no—
185C4⋊C4—141no—
186C4⋊C4—141no—
187C4⋊C4—141no—
188C4⋊C4—141no—
189C4⋊C4C22×D711yescharacteristic
190C4⋊C4C22×D711yescharacteristic
191C4⋊C4C22×D712yes—
192C22⋊C4—142no—
193C22⋊C4—142no—
194C22⋊C4—141no—
195C22⋊C4—141no—
196C42—71no—
197C42—71no—
198C42—71no—
199C42—141no—
200C42—141no—
201C42C22×D711yescharacteristic
202C2×C7#53611yescharacteristic
203C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
204C2×C7#57511yescharacteristic
205D7—41no—
206D7—82no—
207C23—71no—
208C23—142no—
209Q8—42no—
210Q8—282no—
211Q8—282no—
212D4—144no—
213D4—282no—
214D4—282no—
215D4—284no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—71no—
220C2×C4—71no—
221C2×C4—71no—
222C2×C4—142no—
223C2×C4—142no—
224C2×C4—141no—
225C2×C4—141no—
226C2×C4—281no—
227C2×C4—284no—
228C2×C4D4×D711yescharacteristic
229C2×C4D4×D711yescharacteristic
230C2×C4C23×D711yescharacteristic
231C8—22no—
232C8—142no—
233C7#323511yescharacteristic, complemented, Sylow
234C22—141no—
235C22—141no—
236C22—141no—
237C22—282no—
238C22—282no—
239C22—282no—
240C22C2×D4×D711yescharacteristic
241C4—21no—
242C4—21no—
243C4—42no—
244C4—41no—
245C4—41no—
246C4—72no—
247C4—141no—
248C4—141no—
249C4—281no—
250C4—281no—
251C4#655112yes—
252C2—281no—
253C2—562no—
254C2#9041511yescharacteristic
255C2D7×AGL(1,ℤ/8)11yescharacteristic
256C2#9046811yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#501C2
socleC22×C7C22×D4
prime radical#90416C2
hypercenter#501D7
hypocenterC7#3235
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #501

Complements

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

NH ≅ G/Ntype
2. #90379C2semidirect
3. C7×#501C2semidirect
4. #89852C2semidirect
5. #89839C2semidirect
6. #89662C2semidirect
7. #89657C2semidirect
8. #89642C2semidirect
9. #89638C2semidirect
10. #89485C2semidirect
11. #89477C2semidirect
83. #501D7semidirect
233. C7#3235semidirect

Maximal subgroups

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

subgroupindex
#904162
#903792
C7×#5012
#898522
#898392
#896622
#896572
#896422
#896382
#894852
#894772
#32357