Finite Group Databasebeta

← #11739342

Subgroups — #11739342

Order 896.

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

normal subgroups only

This group has 326 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#90419C211yesmaximal, characteristic
2#90419C211yesmaximal, characteristic
3C7×#498C211yesmaximal, characteristic, complemented
4#89865C211yesmaximal, characteristic, complemented
5#89850C211yesmaximal, characteristic, complemented
6#89639C214yesmaximal
7#89629C214yesmaximal, complemented
8#89457C211yesmaximal, characteristic
9#89453C211yesmaximal, characteristic, complemented
10C7×C4⋊Q8C2211yescharacteristic
11C7×C4⋊Q8C2211yescharacteristic
12#6509—21no—
13#6509—21no—
14Q8×Dic7—21no—
15Q8×Dic7—21no—
16#6460—21no—
17#6460C2212yes—
18#6455—21no—
19#6455C2212yes—
20#6453—21no—
21#6453C2212yes—
22#6450—21no—
23#6450C2212yes—
24C42×D7C2211yescharacteristic
25#6432C2211yescharacteristic
26#6432C2211yescharacteristic
27C7×#106C2214yes—
28C4×C7×C8C2211yescharacteristic
29(C2×D7)⋊C8C2212yes—
30#6390C2214yes—
31Dic7⋊C8C2212yes—
32#6384C2214yes—
33#6381C2214yes—
34C4×C7⋊C8C2211yescharacteristic
35#3231—71nomaximal, Sylow
36C2×C7×Q8—21no—
37C2×C7×Q8—21no—
38C2×C4×D7D411yescharacteristic
39C2×C4×D7D411yescharacteristic
40C2×C4×D7C2311yescharacteristic
41C2×Dic14—21no—
42C2×Dic14C2312yes—
43C2×C7×C8C2312yes—
44C7×C4⋊C4—21no—
45C7×C4⋊C4—21no—
46C7×C4⋊C4C2312yes—
47C7×C4⋊C4C2312yes—
48C42×C7C2311yescharacteristic
49(C2×D7)⋊C4—22no—
50(C2×D7)⋊C4—22no—
51C4⋊Dic7—21no—
52C4⋊Dic7—21no—
53C4⋊Dic7—21no—
54C4⋊Dic7—22no—
55C4⋊Dic7—22no—
56C4⋊Dic7C2312yes—
57Dic7⋊C4—22no—
58Dic7⋊C4—22no—
59C4×Dic7—22no—
60C4×Dic7—22no—
61C4×Dic7D411yescharacteristic
62C4×Dic7D411yescharacteristic
63C4×Dic7C2311yescharacteristic
64C2×C7⋊C8C2312yes—
65#533—71no—
66#533—71no—
67#500—71no—
68#498D711yescharacteristic, complemented
69#487—71no—
70#487—71no—
71#485—74no—
72#479—74no—
73#431—71no—
74C22×D7C2×D411yescharacteristic
75C7×Q8—42no—
76C7×Q8—42no—
77C2×C4×C7—22no—
78C2×C4×C7—22no—
79C2×C4×C7C2×D411yescharacteristic
80C2×C4×C7C2×D411yescharacteristic
81C2×C4×C7C2411yescharacteristic
82C2×Dic7—22no—
83C2×Dic7—22no—
84C2×Dic7C2×D411yescharacteristic
85C2×Dic7C2×D411yescharacteristic
86C2×Dic7C2×D411yescharacteristic
87C4×D7—22no—
88C4×D7—22no—
89C4×D7—22no—
90Dic14—24no—
91Dic14—42no—
92C7×C8—22no—
93C7⋊C8—22no—
94C4⋊Q8—71no—
95C4⋊Q8—71no—
96C4⋊Q8C2×D711yescharacteristic
97C4⋊Q8C2×D711yescharacteristic
98#125—72no—
99#125—141no—
100C22⋊Q8—72no—
101C22⋊Q8—141no—
102C22⋊Q8—141no—
103C22⋊Q8—141no—
104C4×Q8—72no—
105C4×Q8—141no—
106C4×Q8—141no—
107C4×Q8—141no—
108#117—72no—
109#117—141no—
110C2×C42—71no—
111#107—74no—
112#106C2×D714yes—
113C4⋊C8—72no—
114#103—74no—
115#103—74no—
116C22⋊C8—72no—
117C4×C8—71no—
118C4×C8C2×D711yescharacteristic
119C22×C7C22×D411yescharacteristic
120C2×D7—21no—
121C2×D7—21no—
122C2×D7—21no—
123C4×C7—42no—
124C4×C7—42no—
125C4×C7#13512yes—
126C4×C7#13712yes—
127C4×C72−1+412yes—
128Dic7—21no—
129Dic7—21no—
130Dic7—21no—
131Dic7—42no—
132Dic7—42no—
133C2×Q8—21no—
134C2×Q8—21no—
135C2×Q8—72no—
136C2×Q8—141no—
137C22×C4—71no—
138C22×C4—71no—
139C22×C4—71no—
140C2×C8—72no—
141C2×C8C22×D712yes—
142C4⋊C4—21no—
143C4⋊C4—21no—
144C4⋊C4—72no—
145C4⋊C4—142no—
146C4⋊C4—142no—
147C4⋊C4—142no—
148C4⋊C4—142no—
149C4⋊C4—141no—
150C4⋊C4—141no—
151C4⋊C4—141no—
152C4⋊C4C22×D712yes—
153C4⋊C4C22×D712yes—
154C22⋊C4—142no—
155C22⋊C4—142no—
156C42—71no—
157C42—71no—
158C42—71no—
159C42—142no—
160C42—142no—
161C42C22×D711yescharacteristic
162C2×C7#53611yescharacteristic
163C2×C7C2×#13511yescharacteristic
164C2×C7C2×#13711yescharacteristic
165D7—41no—
166C23—71no—
167Q8—42no—
168Q8—42no—
169Q8—144no—
170Q8—282no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—71no—
174C2×C4—71no—
175C2×C4—71no—
176C2×C4—142no—
177C2×C4—142no—
178C2×C4—142no—
179C2×C4—142no—
180C2×C4—142no—
181C2×C4D4×D711yescharacteristic
182C2×C4D4×D711yescharacteristic
183C2×C4C23×D711yescharacteristic
184C8—22no—
185C8—142no—
186C7#323111yescharacteristic, complemented, Sylow
187C22—141no—
188C22—141no—
189C22—141no—
190C22C2×D4×D711yescharacteristic
191C4—42no—
192C4—42no—
193C4—141no—
194C4—141no—
195C4—141no—
196C4—282no—
197C4—282no—
198C4#647812yes—
199C4#647912yes—
200C4#655112yes—
201C2—281no—
202C2#9041511yescharacteristic
203C2C2×#647811yescharacteristic
204C2C2×#647911yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#498C2
socleC22×C7C22×D4
prime radicalC22×D7C2×D4
hypercenter#498D7
hypocenterC7#3231
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #498

Complements

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

NH ≅ G/Ntype
3. C7×#498C2semidirect
4. #89865C2semidirect
5. #89850C2semidirect
7. #89629C2semidirect
9. #89453C2semidirect
68. #498D7semidirect
186. C7#3231semidirect

Maximal subgroups

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

subgroupindex
#904192
C7×#4982
#898652
#898502
#896392
#896292
#894572
#894532
#32317