Finite Group Databasebeta

← #11736543

Subgroups — #11736543

Order 896.

346 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 117 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#90225C212yesmaximal, complemented
2C7×#424C211yesmaximal, characteristic, complemented
3#89909C211yesmaximal, characteristic, complemented
4#89907C211yesmaximal, characteristic
5#89886C212yesmaximal, complemented
6#89623C212yesmaximal, complemented
7#89617C212yesmaximal, complemented
8#89616C212yesmaximal, complemented
9#89614C212yesmaximal, complemented
10C2×C7×M16C2211yescharacteristic
11C22×C7×C8C2211yescharacteristic
12C7×#117C2211yescharacteristic
13#6493C2212yes—
14#6487C2212yes—
15C2×Dic28C2211yescharacteristic
16C2×D56C2211yescharacteristic
17C2×#1132C2212yes—
18#6461—22no—
19#6459—22no—
20#6448—22no—
21#6446—22no—
22#6438C2212yes—
23C4×D28C2212yes—
24#6432C2212yes—
25C4×Dic14C2212yes—
26C7×C4⋊C8C2212yes—
27C7×C4⋊C8C2212yes—
28#6395C2212yes—
29#6395C2212yes—
30#6391C2211yescharacteristic
31#6391C2212yes—
32#6391C2211yescharacteristic
33#6390C2212yes—
34#6390C2212yes—
35#3263—71nomaximal, Sylow
36C22×C4×C7C2311yescharacteristic
37C2×C22⋊D7—22no—
38C2×D28C2312yes—
39C2×C4×D7—22no—
40C2×Dic14—22no—
41C2×Dic14C2312yes—
42C7×M16—41no—
43C2×C7×C8—41no—
44C2×C7×C8C2312yes—
45C2×C7×C8C2311yescharacteristic
46C2×C7×C8C2311yescharacteristic
47C7×C4⋊C4D412yes—
48C7×C22⋊C4D411yescharacteristic
49C7×C22⋊C4D411yescharacteristic
50C42×C7C2312yes—
51C22⋊Dic7—22no—
52(C2×D7)⋊C4—22no—
53(C2×D7)⋊C4—22no—
54(C2×D7)⋊C4—22no—
55(C2×D7)⋊C4—22no—
56C4⋊Dic7—22no—
57C4⋊Dic7C2312yes—
58C4⋊Dic7C2312yes—
59Dic7⋊C4—22no—
60Dic7⋊C4—22no—
61C4×Dic7—22no—
62Dic28—41no—
63D56—41no—
64#1132—42no—
65#542—72no—
66#476—72no—
67#475—72no—
68#470—71no—
69#469—71no—
70#465—72no—
71#464—72no—
72#463—72no—
73#424D711yescharacteristic, complemented
74C23×C7C2×D411yescharacteristic
75C22×D7—22no—
76C2×C4×C7—21no—
77C2×C4×C7—21no—
78C2×C4×C7C2×D412yes—
79C2×C4×C7C2×D411yescharacteristic
80C2×C4×C7C2×D412yes—
81C2×C4×C7C2411yescharacteristic
82C22⋊D7—82no—
83C2×Dic7—22no—
84C2×Dic7—22no—
85C2×Dic7—22no—
86D28—22no—
87D28—22no—
88C4×D7—82no—
89Dic14—22no—
90Dic14—22no—
91Dic14—82no—
92C7×C8—22no—
93C7×C8—21no—
94C7×C8—21no—
95C2×Q16—71no—
96C2×SD16—72no—
97C2×D8—71no—
98C2×M16C2×D711yescharacteristic
99C22×C8C2×D711yescharacteristic
100C4⋊Q8—72no—
101#126—142no—
102#124—72no—
103#124—142no—
104#123—142no—
105C22⋊Q8—72no—
106C22⋊Q8—142no—
107#121—72no—
108C4×Q8—72no—
109C4×D4—72no—
110#117C2×D711yescharacteristic
111#106—72no—
112#106—71no—
113#106—71no—
114C4⋊C8C2×D712yes—
115C4⋊C8C2×D712yes—
116#103—72no—
117#103—72no—
118#102—72no—
119#102—72no—
120C22×C7—21no—
121C22×C7—22no—
122C22×C7C22×D411yescharacteristic
123C2×D7—42no—
124C2×D7—42no—
125C2×D7—42no—
126C4×C7—21no—
127C4×C7—22no—
128C4×C7—22no—
129C4×C7H(5,𝔽2)11yescharacteristic
130C4×C72−1+411yescharacteristic
131Dic7—42no—
132Dic7—42no—
133Dic7—42no—
134C2×Q8—72no—
135C2×Q8—142no—
136C2×D4—72no—
137C2×D4—142no—
138C22×C4—142no—
139C22×C4C22×D711yescharacteristic
140Q16—281no—
141SD16—282no—
142D8—281no—
143M16—41no—
144C2×C8—41no—
145C2×C8C22×D711yescharacteristic
146C2×C8C22×D712yes—
147C2×C8C22×D711yescharacteristic
148C4⋊C4—72no—
149C4⋊C4—72no—
150C4⋊C4—142no—
151C4⋊C4—142no—
152C4⋊C4—142no—
153C4⋊C4D2812yes—
154C22⋊C4—142no—
155C22⋊C4—142no—
156C22⋊C4—142no—
157C22⋊C4—142no—
158C22⋊C4—142no—
159C22⋊C4D2811yescharacteristic
160C22⋊C4D2811yescharacteristic
161C42—142no—
162C42C22×D712yes—
163C2×C7—41no—
164C2×C7#53711yescharacteristic
165C2×C7#57712yes—
166D7—82no—
167C23—142no—
168C23C2×D2811yescharacteristic
169Q8—142no—
170Q8—142no—
171Q8—562no—
172D4—142no—
173D4—142no—
174D4—562no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—142no—
178C2×C4—142no—
179C2×C4—142no—
180C2×C4—562no—
181C2×C4C2×D2812yes—
182C2×C4C2×D2812yes—
183C2×C4C2×D2811yescharacteristic
184C2×C4C23×D711yescharacteristic
185C8—22no—
186C8—21no—
187C8—21no—
188C7#326311yescharacteristic, complemented, Sylow
189C22—22no—
190C22—21no—
191C22—282no—
192C22—282no—
193C22—282no—
194C22C22×D2811yescharacteristic
195C4—22no—
196C4—22no—
197C4—21no—
198C4—282no—
199C4—282no—
200C4—282no—
201C4#654811yescharacteristic
202C4#655111yescharacteristic
203C2—41no—
204C2—562no—
205C2#9020111yescharacteristic
206C2#9044212yes—
subgroupG/H
centerC22C22×D28
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#424C2
socleC22×C7C22×D4
prime radical#89907C2
hypercenter#424D7
hypocenterC7#3263
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ #117 ◁ #424

Complements

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

NH ≅ G/Ntype
1. #90225C2semidirect
2. C7×#424C2semidirect
3. #89909C2semidirect
5. #89886C2semidirect
6. #89623C2semidirect
7. #89617C2semidirect
8. #89616C2semidirect
9. #89614C2semidirect
73. #424D7semidirect
188. C7#3263semidirect

Maximal subgroups

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

subgroupindex
#902252
C7×#4242
#899092
#899072
#898862
#896232
#896172
#896162
#896142
#32637