Finite Group Databasebeta

← #11736631

Subgroups — #11736631

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#90220C212yesmaximal
2C7×#427C211yesmaximal, characteristic, complemented
3#89888C211yesmaximal, characteristic, complemented
4C8×C22⋊D7C211yesmaximal, characteristic
5#89768C212yesmaximal, complemented
6#89659C211yesmaximal, characteristic, complemented
7D7×#107C211yesmaximal, characteristic, complemented
8#89650C211yesmaximal, characteristic, complemented
9#89645C211yesmaximal, characteristic, complemented
10#89634C212yesmaximal, complemented
11#89629C212yesmaximal, complemented
12C22×C7×C8C2211yescharacteristic
13C7×#117C2212yes—
14C4×C22⋊D7C2211yescharacteristic
15#6487C2212yes—
16C2×Dic28C2211yescharacteristic
17C2×C8×D7C2211yescharacteristic
18#6460C2212yes—
19D7×C4⋊C4C2212yes—
20#6453C2212yes—
21#6450C2212yes—
22#6446—22no—
23#6442—22no—
24#6435—22no—
25#6433—22no—
26C7×#107C2211yescharacteristic
27C7×#107C2211yescharacteristic
28C7×#106C2212yes—
29C22⋊(C7⋊C8)C2211yescharacteristic
30(C2×D7)⋊C8C2211yescharacteristic
31#6392C2211yescharacteristic
32#6390C2212yes—
33Dic7⋊C8C2211yescharacteristic
34C8×Dic7C2211yescharacteristic
35#6384C2212yes—
36#6384C2212yes—
37#6382C2212yes—
38#6381C2212yes—
39#3336—71nomaximal, Sylow
40C22×C4×C7C2311yescharacteristic
41C2×C22⋊D7D411yescharacteristic
42C2×C4×D7—22no—
43C2×C4×D7C2311yescharacteristic
44C2×Dic14C2312yes—
45C2×C7×C8—22no—
46C2×C7×C8C2311yescharacteristic
47C2×C7×C8C2311yescharacteristic
48C7×C4⋊C4C2312yes—
49C7×C4⋊C4C2312yes—
50C7×C22⋊C4—22no—
51C42×C7—22no—
52C22⋊Dic7—22no—
53C22⋊Dic7D411yescharacteristic
54(C2×D7)⋊C4—22no—
55(C2×D7)⋊C4D411yescharacteristic
56C4⋊Dic7—22no—
57C4⋊Dic7—22no—
58C4⋊Dic7C2312yes—
59Dic7⋊C4—22no—
60Dic7⋊C4—22no—
61Dic7⋊C4—22no—
62Dic7⋊C4—22no—
63Dic7⋊C4D411yescharacteristic
64C4×Dic7—22no—
65C4×Dic7C2311yescharacteristic
66C2×C7⋊C8C2311yescharacteristic
67C2×C7⋊C8C2311yescharacteristic
68Dic28—22no—
69C8×D7—22no—
70#552—72no—
71#499—71no—
72#485—72no—
73#483—72no—
74#479—72no—
75#467—71no—
76#467—71no—
77C4×Q16—71no—
78C8×D4—71no—
79#427D711yescharacteristic, complemented
80C2×#107—71no—
81C23×C7C2×D411yescharacteristic
82C22×D7C2×D411yescharacteristic
83C2×C4×C7—21no—
84C2×C4×C7—21no—
85C2×C4×C7—22no—
86C2×C4×C7—22no—
87C2×C4×C7C2×D411yescharacteristic
88C2×C4×C7C2411yescharacteristic
89C22⋊D7—22no—
90C2×Dic7—22no—
91C2×Dic7—22no—
92C2×Dic7C2×D411yescharacteristic
93C2×Dic7C2×D411yescharacteristic
94C2×Dic7C2×D411yescharacteristic
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×C8C2×D711yescharacteristic
107#126—142no—
108#125—72no—
109#125—142no—
110#123—142no—
111C22⋊Q8—72no—
112C22⋊Q8—72no—
113C4×Q8—72no—
114C4×D4—71no—
115#117—142no—
116#117C2×D712yes—
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×C8—71no—
131C22×C7—21no—
132C22×C7—21no—
133C22×C7—21no—
134C22×C7C22×D411yescharacteristic
135C2×D7—21no—
136C2×D7—21no—
137C2×D7#13512yes—
138C4×C7—21no—
139C4×C7—42no—
140C4×C7—42no—
141C4×C7C2×#4111yescharacteristic
142C4×C72−1+411yescharacteristic
143Dic7—21no—
144Dic7—21no—
145Dic7—21no—
146Dic7—42no—
147Dic7—42no—
148C2×Q8—72no—
149C2×D4—71no—
150C22×C4—71no—
151C22×C4—142no—
152C22×C4C22×D711yescharacteristic
153Q16—142no—
154C2×C8—22no—
155C2×C8—71no—
156C2×C8—71no—
157C2×C8—142no—
158C2×C8C22×D711yescharacteristic
159C2×C8C22×D711yescharacteristic
160C4⋊C4—71no—
161C4⋊C4—72no—
162C4⋊C4—142no—
163C4⋊C4—142no—
164C4⋊C4—142no—
165C4⋊C4—142no—
166C4⋊C4—142no—
167C4⋊C4—142no—
168C4⋊C4C22×D712yes—
169C4⋊C4C22×D712yes—
170C22⋊C4—22no—
171C22⋊C4—71no—
172C22⋊C4—71no—
173C22⋊C4—142no—
174C22⋊C4—142no—
175C42—22no—
176C42—71no—
177C42—142no—
178C2×C7—41no—
179C2×C7#54711yescharacteristic
180C2×C7C2×#13511yescharacteristic
181C2×C7#57811yescharacteristic
182D7—22no—
183C23—71no—
184C23D4×D711yescharacteristic
185Q8—142no—
186Q8—142no—
187D4—142no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—71no—
193C2×C4—71no—
194C2×C4—71no—
195C2×C4—141no—
196C2×C4—141no—
197C2×C4—142no—
198C2×C4—142no—
199C2×C4—284no—
200C2×C4D4×D711yescharacteristic
201C2×C4C23×D711yescharacteristic
202C8—21no—
203C8—141no—
204C8—141no—
205C8#115712yes—
206C7#333611yescharacteristic, complemented, Sylow
207C22—21no—
208C22—21no—
209C22—21no—
210C22—72no—
211C22—141no—
212C22—141no—
213C22C2×D4×D711yescharacteristic
214C4—21no—
215C4—42no—
216C4—42no—
217C4—141no—
218C4—141no—
219C4—141no—
220C4—282no—
221C4—282no—
222C4C2×#115711yescharacteristic
223C4#655111yescharacteristic
224C2—41no—
225C2—142no—
226C2#9019811yescharacteristic
227C2D7×#13511yescharacteristic
228C2#9046211yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#427C2
socleC22×C7C22×D4
prime radicalC2×C22⋊D7D4
hypercenter#427D7
hypocenterC7#3336
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #427

Complements

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

NH ≅ G/Ntype
2. C7×#427C2semidirect
3. #89888C2semidirect
5. #89768C2semidirect
6. #89659C2semidirect
7. D7×#107C2semidirect
8. #89650C2semidirect
9. #89645C2semidirect
10. #89634C2semidirect
11. #89629C2semidirect
79. #427D7semidirect
206. C7#3336semidirect

Maximal subgroups

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

subgroupindex
#902202
C7×#4272
#898882
C8×C22⋊D72
#897682
#896592
D7×#1072
#896502
#896452
#896342
#896292
#33367