Finite Group Databasebeta

← #11739369

Subgroups — #11739369

Order 896.

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

normal subgroups only

This group has 342 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#90385C212yesmaximal
2C7×#499C211yesmaximal, characteristic, complemented
3#89835C212yesmaximal, complemented
4#89659C211yesmaximal, characteristic, complemented
5#89659C211yesmaximal, characteristic, complemented
6#89652C211yesmaximal, characteristic, complemented
7D7×#107C211yesmaximal, characteristic, complemented
8#89639C212yesmaximal, complemented
9#89634C212yesmaximal, complemented
10C4×Dic28C211yesmaximal, characteristic, complemented
11C8×D28C211yesmaximal, characteristic
12C7×#125C2212yes—
13C2×Dic28C2211yescharacteristic
14C2×C8×D7C2211yescharacteristic
15C2×C8×D7C2211yescharacteristic
16#6461—22no—
17#6460C2212yes—
18#6460C2212yes—
19#6457—22no—
20#6455—22no—
21#6455C2212yes—
22D7×C4⋊C4C2212yes—
23#6453—22no—
24C4×D28C2211yescharacteristic
25C4×Dic14C2212yes—
26C7×#107C2211yescharacteristic
27C7×#107C2211yescharacteristic
28C7×#106C2212yes—
29C4×C7×C8C2211yescharacteristic
30(C2×D7)⋊C8C2211yescharacteristic
31(C2×D7)⋊C8C2211yescharacteristic
32#6392C2211yescharacteristic
33#6390C2212yes—
34#6384C2212yes—
35#6384C2212yes—
36#6382C2212yes—
37#6381C2212yes—
38C4⋊(C7⋊C8)C2211yescharacteristic
39#3336—71nomaximal, Sylow
40C2×D28D411yescharacteristic
41C2×C4×D7—22no—
42C2×C4×D7C2311yescharacteristic
43C2×C4×D7C2311yescharacteristic
44C2×Dic14C2312yes—
45C2×C7×C8C2311yescharacteristic
46C2×C7×C8C2311yescharacteristic
47C7×C4⋊C4—22no—
48C7×C4⋊C4—22no—
49C7×C4⋊C4C2312yes—
50C7×C4⋊C4C2312yes—
51C42×C7C2311yescharacteristic
52(C2×D7)⋊C4—22no—
53(C2×D7)⋊C4—22no—
54(C2×D7)⋊C4—22no—
55(C2×D7)⋊C4D411yescharacteristic
56(C2×D7)⋊C4D411yescharacteristic
57C4⋊Dic7—22no—
58C4⋊Dic7—22no—
59C4⋊Dic7D411yescharacteristic
60C4⋊Dic7C2312yes—
61Dic7⋊C4—22no—
62Dic7⋊C4—22no—
63C4×Dic7—22no—
64C4×Dic7—22no—
65C2×C7⋊C8C2311yescharacteristic
66C2×C7⋊C8C2311yescharacteristic
67Dic28—22no—
68C8×D7—22no—
69C8×D7—22no—
70#552—72no—
71#499D711yescharacteristic, complemented
72#485—72no—
73#483—72no—
74#479—72no—
75#467—71no—
76#467—71no—
77C4×Q16—71no—
78C8×D4—71no—
79#427—71no—
80C2×#107—71no—
81C22×D7C2×D411yescharacteristic
82C22×D7C2×D411yescharacteristic
83C2×C4×C7—22no—
84C2×C4×C7—22no—
85C2×C4×C7C2×D411yescharacteristic
86C2×C4×C7C2×D411yescharacteristic
87C2×C4×C7C2411yescharacteristic
88C2×Dic7—22no—
89C2×Dic7—22no—
90C2×Dic7C2×D411yescharacteristic
91C2×Dic7C2×D411yescharacteristic
92D28—22no—
93C4×D7—21no—
94C4×D7—21no—
95C4×D7—21no—
96C4×D7—21no—
97C4×D7—44no—
98Dic14—22no—
99Dic14—22no—
100C7×C8—21no—
101C7×C8#4112yes—
102C7⋊C8—21no—
103C7⋊C8—21no—
104C2×Q16—71no—
105C22×C8—71no—
106C22×C8—71no—
107#126—142no—
108#125—142no—
109#125C2×D712yes—
110#123—142no—
111C22⋊Q8—72no—
112C22⋊Q8—72no—
113C4×Q8—72no—
114C4×D4—71no—
115#117—72no—
116#117—142no—
117C2×C4⋊C4—72no—
118#107—71no—
119#107—72no—
120#107C2×D711yescharacteristic
121#107C2×D711yescharacteristic
122#106—72no—
123#106C2×D712yes—
124C4⋊C8—71no—
125#103—72no—
126#103—72no—
127#103—72no—
128C22⋊C8—71no—
129C22⋊C8—71no—
130C4×C8C2×D711yescharacteristic
131C22×C7C22×D411yescharacteristic
132C2×D7—21no—
133C2×D7—21no—
134C2×D7—21no—
135C2×D7—21no—
136C2×D7—21no—
137C2×D7#13512yes—
138C4×C7—21no—
139C4×C7—21no—
140C4×C7—42no—
141C4×C7—42no—
142C4×C7C2×#4111yescharacteristic
143C4×C72−1+411yescharacteristic
144Dic7—21no—
145Dic7—21no—
146Dic7—42no—
147Dic7—42no—
148C2×Q8—72no—
149C2×D4—71no—
150C22×C4—71no—
151C22×C4—71no—
152C22×C4—142no—
153Q16—142no—
154C2×C8—71no—
155C2×C8—71no—
156C2×C8—142no—
157C2×C8—142no—
158C2×C8C22×D711yescharacteristic
159C2×C8C22×D711yescharacteristic
160C4⋊C4—22no—
161C4⋊C4—22no—
162C4⋊C4—71no—
163C4⋊C4—72no—
164C4⋊C4—142no—
165C4⋊C4—142no—
166C4⋊C4—142no—
167C4⋊C4—142no—
168C4⋊C4C22×D712yes—
169C4⋊C4C22×D712yes—
170C22⋊C4—71no—
171C22⋊C4—71no—
172C22⋊C4—142no—
173C22⋊C4—142no—
174C22⋊C4—142no—
175C42—142no—
176C42—142no—
177C42C22×D711yescharacteristic
178C2×C7#54711yescharacteristic
179C2×C7C2×#13511yescharacteristic
180C2×C7#57811yescharacteristic
181D7—22no—
182D7—41no—
183C23—71no—
184C23—71no—
185Q8—142no—
186Q8—142no—
187D4—142no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—71no—
191C2×C4—71no—
192C2×C4—141no—
193C2×C4—141no—
194C2×C4—141no—
195C2×C4—141no—
196C2×C4—142no—
197C2×C4—142no—
198C2×C4—284no—
199C2×C4D4×D711yescharacteristic
200C2×C4D4×D711yescharacteristic
201C2×C4C23×D711yescharacteristic
202C8—21no—
203C8—141no—
204C8—141no—
205C8#116112yes—
206C7#333611yescharacteristic, complemented, Sylow
207C22—72no—
208C22—141no—
209C22—141no—
210C22—141no—
211C22—141no—
212C22—141no—
213C22C2×D4×D711yescharacteristic
214C4—21no—
215C4—21no—
216C4—42no—
217C4—42no—
218C4—141no—
219C4—141no—
220C4—282no—
221C4—282no—
222C4C2×#116111yescharacteristic
223C4#655111yescharacteristic
224C2—142no—
225C2—281no—
226C2#9041711yescharacteristic
227C2D7×#13511yescharacteristic
228C2#9046211yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#499C2
socleC22×C7C22×D4
prime radicalC2×D28D4
hypercenter#499D7
hypocenterC7#3336
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
2. C7×#499C2semidirect
3. #89835C2semidirect
4. #89659C2semidirect
5. #89659C2semidirect
6. #89652C2semidirect
7. D7×#107C2semidirect
8. #89639C2semidirect
9. #89634C2semidirect
10. C4×Dic28C2semidirect
71. #499D7semidirect
206. C7#3336semidirect

Maximal subgroups

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

subgroupindex
#903852
C7×#4992
#898352
#896592
#896522
D7×#1072
#896392
#896342
C4×Dic282
C8×D282
#33367