Finite Group Databasebeta

← #11739371

Subgroups — #11739371

Order 896.

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

normal subgroups only

This group has 366 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#90386C212yesmaximal, complemented
2C7×#499C211yesmaximal, characteristic, complemented
3#89838C212yesmaximal, complemented
4#89655C211yesmaximal, characteristic, complemented
5#89655C211yesmaximal, characteristic, complemented
6#89652C211yesmaximal, characteristic, complemented
7D7×#107C211yesmaximal, characteristic, complemented
8#89638C212yesmaximal, complemented
9#89635C212yesmaximal, complemented
10C4×D56C211yesmaximal, characteristic, complemented
11C8×D28C211yesmaximal, characteristic, complemented
12C7×#125C2212yes—
13C2×D56C2211yescharacteristic
14C2×C8×D7C2211yescharacteristic
15C2×C8×D7C2211yescharacteristic
16#6461—22no—
17#6458—22no—
18#6458C2212yes—
19#6458C2212yes—
20#6457—22no—
21#6456—22no—
22#6455C2212yes—
23D7×C4⋊C4C2212yes—
24C4×D28C2211yescharacteristic
25C4×D28C2212yes—
26C7×#107C2211yescharacteristic
27C7×#107C2211yescharacteristic
28C7×#106C2212yes—
29C4×C7×C8C2211yescharacteristic
30#6395C2212yes—
31(C2×D7)⋊C8C2211yescharacteristic
32(C2×D7)⋊C8C2211yescharacteristic
33#6392C2211yescharacteristic
34#6383C2212yes—
35#6383C2212yes—
36#6382C2212yes—
37#6381C2212yes—
38C4⋊(C7⋊C8)C2211yescharacteristic
39#3339—71nomaximal, Sylow
40C2×D28—22no—
41C2×D28—22no—
42C2×D28D411yescharacteristic
43C2×D28C2312yes—
44C2×C4×D7—22no—
45C2×C4×D7—22no—
46C2×C4×D7C2311yescharacteristic
47C2×C4×D7C2311yescharacteristic
48C2×C7×C8C2311yescharacteristic
49C2×C7×C8C2311yescharacteristic
50C7×C4⋊C4—22no—
51C7×C4⋊C4—22no—
52C7×C4⋊C4C2312yes—
53C7×C4⋊C4C2312yes—
54C42×C7C2311yescharacteristic
55(C2×D7)⋊C4—22no—
56(C2×D7)⋊C4—22no—
57(C2×D7)⋊C4—22no—
58(C2×D7)⋊C4—22no—
59(C2×D7)⋊C4D411yescharacteristic
60(C2×D7)⋊C4D411yescharacteristic
61C4⋊Dic7D411yescharacteristic
62C4⋊Dic7C2312yes—
63Dic7⋊C4—22no—
64C4×Dic7—22no—
65C2×C7⋊C8C2311yescharacteristic
66C2×C7⋊C8C2311yescharacteristic
67D56—22no—
68C8×D7—22no—
69C8×D7—22no—
70#553—72no—
71#499D711yescharacteristic, complemented
72#482—72no—
73#481—72no—
74#478—72no—
75#466—71no—
76#466—71no—
77C4×D8—71no—
78C8×D4—71no—
79#427—71no—
80C2×#107—71no—
81C22×D7—22no—
82C22×D7C2×D411yescharacteristic
83C22×D7C2×D411yescharacteristic
84C2×C4×C7—22no—
85C2×C4×C7—22no—
86C2×C4×C7C2×D411yescharacteristic
87C2×C4×C7C2×D411yescharacteristic
88C2×C4×C7C2411yescharacteristic
89C2×Dic7—22no—
90C2×Dic7C2×D411yescharacteristic
91C2×Dic7C2×D411yescharacteristic
92D28—22no—
93D28—22no—
94D28—22no—
95D28—44no—
96D28—82no—
97C4×D7—21no—
98C4×D7—21no—
99C4×D7—21no—
100C4×D7—21no—
101C4×D7—82no—
102C4×D7—82no—
103C7×C8—21no—
104C7×C8#4112yes—
105C7⋊C8—21no—
106C7⋊C8—21no—
107C2×D8—71no—
108C22×C8—71no—
109C22×C8—71no—
110#126—142no—
111#125C2×D712yes—
112#123—142no—
113#121—72no—
114#121—72no—
115#121—142no—
116C4×D4—71no—
117C4×D4—72no—
118C4×D4—142no—
119#117—72no—
120C2×C4⋊C4—72no—
121#107—71no—
122#107—72no—
123#107C2×D711yescharacteristic
124#107C2×D711yescharacteristic
125#106—72no—
126#106C2×D712yes—
127C4⋊C8—71no—
128#102—72no—
129#102—72no—
130#102—72no—
131C22⋊C8—71no—
132C22⋊C8—71no—
133C4×C8C2×D711yescharacteristic
134C22×C7C22×D411yescharacteristic
135C2×D7—21no—
136C2×D7—21no—
137C2×D7—21no—
138C2×D7—21no—
139C2×D7—21no—
140C2×D7—42no—
141C2×D7—42no—
142C2×D7—42no—
143C2×D7#13512yes—
144C4×C7—21no—
145C4×C7—21no—
146C4×C7—42no—
147C4×C7—42no—
148C4×C7C2×#4111yescharacteristic
149C4×C72−1+411yescharacteristic
150Dic7—21no—
151Dic7—21no—
152Dic7—42no—
153C2×D4—71no—
154C2×D4—72no—
155C2×D4—142no—
156C2×D4—142no—
157C22×C4—71no—
158C22×C4—71no—
159C22×C4—142no—
160C22×C4—142no—
161D8—142no—
162C2×C8—71no—
163C2×C8—71no—
164C2×C8—142no—
165C2×C8—142no—
166C2×C8C22×D711yescharacteristic
167C2×C8C22×D711yescharacteristic
168C4⋊C4—22no—
169C4⋊C4—22no—
170C4⋊C4—71no—
171C4⋊C4—72no—
172C4⋊C4—142no—
173C4⋊C4C22×D712yes—
174C4⋊C4C22×D712yes—
175C22⋊C4—71no—
176C22⋊C4—71no—
177C22⋊C4—142no—
178C22⋊C4—142no—
179C22⋊C4—142no—
180C22⋊C4—142no—
181C42—142no—
182C42C22×D711yescharacteristic
183C2×C7#54711yescharacteristic
184C2×C7C2×#13511yescharacteristic
185C2×C7#57611yescharacteristic
186D7—22no—
187D7—41no—
188D7—82no—
189C23—71no—
190C23—71no—
191C23—142no—
192D4—142no—
193D4—142no—
194D4—142no—
195D4—284no—
196D4—562no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—71no—
200C2×C4—71no—
201C2×C4—141no—
202C2×C4—141no—
203C2×C4—141no—
204C2×C4—141no—
205C2×C4—142no—
206C2×C4—562no—
207C2×C4—562no—
208C2×C4D4×D711yescharacteristic
209C2×C4D4×D711yescharacteristic
210C2×C4C23×D711yescharacteristic
211C8—21no—
212C8—141no—
213C8—141no—
214C8#116112yes—
215C7#333911yescharacteristic, complemented, Sylow
216C22—72no—
217C22—141no—
218C22—141no—
219C22—141no—
220C22—141no—
221C22—141no—
222C22—282no—
223C22—282no—
224C22—282no—
225C22C2×D4×D711yescharacteristic
226C4—21no—
227C4—21no—
228C4—42no—
229C4—42no—
230C4—141no—
231C4—141no—
232C4—282no—
233C4C2×#116111yescharacteristic
234C4#655111yescharacteristic
235C2—142no—
236C2—281no—
237C2—562no—
238C2#9041711yescharacteristic
239C2D7×#13511yescharacteristic
240C2#9046011yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#499C2
socleC22×C7C22×D4
hypercenter#499D7
hypocenterC7#3339
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #499

Complements

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

NH ≅ G/Ntype
1. #90386C2semidirect
2. C7×#499C2semidirect
3. #89838C2semidirect
4. #89655C2semidirect
5. #89655C2semidirect
6. #89652C2semidirect
7. D7×#107C2semidirect
8. #89638C2semidirect
9. #89635C2semidirect
10. C4×D56C2semidirect
11. C8×D28C2semidirect
71. #499D7semidirect
215. C7#3339semidirect

Maximal subgroups

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

subgroupindex
#903862
C7×#4992
#898382
#896552
#896522
D7×#1072
#896382
#896352
C4×D562
C8×D282
#33397