Finite Group Databasebeta

← #11739386

Subgroups — #11739386

Order 896.

404 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 125 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#90416C211yesmaximal, characteristic
2D7×C4⋊Q8C211yesmaximal, characteristic, complemented
3C7×#500C211yesmaximal, characteristic, complemented
4#89860C211yesmaximal, characteristic, complemented
5#89850C211yesmaximal, characteristic, complemented
6#89662C214yesmaximal, complemented
7#89654C214yesmaximal, complemented
8#89466C211yesmaximal, characteristic, complemented
9#89457C211yesmaximal, characteristic, complemented
10C2×#1161—21no—
11C2×Q8×D7—21no—
12C7×C4⋊Q8C2211yescharacteristic
13C7×C4⋊Q8C2211yescharacteristic
14#6510—21no—
15#6507—21no—
16#6458—21no—
17#6458C2212yes—
18#6456—21no—
19#6456C2212yes—
20D7×C4⋊C4—21no—
21D7×C4⋊C4C2212yes—
22#6451—21no—
23#6451C2212yes—
24#6437C2211yescharacteristic
25C42×D7C2211yescharacteristic
26#6432C2211yescharacteristic
27C7×#107C2214yes—
28C4×C7×C8C2211yescharacteristic
29#6395C2214yes—
30(C2×D7)⋊C8C2212yes—
31Dic7⋊C8C2212yes—
32#6383C2214yes—
33#6381C2214yes—
34C4×C7⋊C8C2211yescharacteristic
35#3226—71nomaximal, Sylow
36C2×C7×Q8—21no—
37C2×C7×Q8—21no—
38#1161—44no—
39Q8×D7—44no—
40C2×D28—22no—
41C2×D28—21no—
42C2×D28—21no—
43C2×D28C2312yes—
44C2×C4×D7—22no—
45C2×C4×D7—22no—
46C2×C4×D7D411yescharacteristic
47C2×C4×D7D411yescharacteristic
48C2×C4×D7C2311yescharacteristic
49C2×Dic14—21no—
50C2×Dic14—22no—
51C2×C7×C8C2312yes—
52C7×C4⋊C4—21no—
53C7×C4⋊C4—21no—
54C7×C4⋊C4C2312yes—
55C7×C4⋊C4C2312yes—
56C42×C7C2311yescharacteristic
57(C2×D7)⋊C4—22no—
58(C2×D7)⋊C4—22no—
59C4⋊Dic7—21no—
60C4⋊Dic7C2312yes—
61Dic7⋊C4—22no—
62Dic7⋊C4—22no—
63C4×Dic7D411yescharacteristic
64C4×Dic7D411yescharacteristic
65C4×Dic7C2311yescharacteristic
66C2×C7⋊C8C2312yes—
67#532—71no—
68C2×C4⋊Q8—71no—
69#500—71no—
70#500D711yescharacteristic, complemented
71#486—71no—
72#486—71no—
73#480—74no—
74#474—74no—
75#431—71no—
76C22×D7—22no—
77C22×D7C2×D411yescharacteristic
78C7×Q8—42no—
79C7×Q8—42no—
80C2×C4×C7—22no—
81C2×C4×C7—22no—
82C2×C4×C7C2×D411yescharacteristic
83C2×C4×C7C2×D411yescharacteristic
84C2×C4×C7C2411yescharacteristic
85C2×Dic7—22no—
86C2×Dic7C2×D411yescharacteristic
87C2×Dic7C2×D411yescharacteristic
88C2×Dic7C2×D411yescharacteristic
89D28—24no—
90D28—44no—
91D28—42no—
92D28—42no—
93C4×D7—22no—
94C4×D7—22no—
95C4×D7—44no—
96C4×D7—44no—
97C4×D7D814yes—
98Dic14—42no—
99Dic14—44no—
100C7×C8—22no—
101C7⋊C8—22no—
102C2×#41—141no—
103C22×Q8—141no—
104C4⋊Q8—72no—
105C4⋊Q8—71no—
106C4⋊Q8—141no—
107C4⋊Q8—141no—
108C4⋊Q8C2×D711yescharacteristic
109C4⋊Q8C2×D711yescharacteristic
110#127—71no—
111#124—141no—
112#121—72no—
113#121—141no—
114C4×D4—72no—
115C4×D4—141no—
116C2×C4⋊C4—72no—
117C2×C4⋊C4—141no—
118C2×C42—71no—
119#107—74no—
120#107C2×D714yes—
121C4⋊C8—72no—
122#102—74no—
123#102—74no—
124C22⋊C8—72no—
125C4×C8—71no—
126C4×C8C2×D711yescharacteristic
127C22×C7C22×D411yescharacteristic
128C2×D7—21no—
129C2×D7—21no—
130C2×D7—42no—
131C2×D7—42no—
132C2×D7—42no—
133C2×D7C2×D812yes—
134C4×C7—42no—
135C4×C7—42no—
136C4×C7C2×D812yes—
137C4×C7#13712yes—
138C4×C72−1+412yes—
139Dic7—21no—
140Dic7—21no—
141Dic7—42no—
142Dic7C2×D812yes—
143#41—284no—
144C2×Q8—21no—
145C2×Q8—21no—
146C2×Q8—142no—
147C2×Q8—141no—
148C2×Q8—284no—
149C2×D4—72no—
150C2×D4—142no—
151C2×D4—141no—
152C2×D4—141no—
153C22×C4—71no—
154C22×C4—71no—
155C22×C4—71no—
156C22×C4—142no—
157C22×C4—142no—
158C2×C8—72no—
159C2×C8C22×D712yes—
160C4⋊C4—21no—
161C4⋊C4—21no—
162C4⋊C4—72no—
163C4⋊C4—142no—
164C4⋊C4—142no—
165C4⋊C4—141no—
166C4⋊C4C22×D712yes—
167C4⋊C4C22×D712yes—
168C22⋊C4—142no—
169C22⋊C4—142no—
170C42—71no—
171C42—71no—
172C42—71no—
173C42C22×D711yescharacteristic
174C2×C7#53611yescharacteristic
175C2×C7C22×D811yescharacteristic
176C2×C7C2×#13711yescharacteristic
177D7—22no—
178D7—82no—
179C23—71no—
180C23—142no—
181Q8—42no—
182Q8—42no—
183Q8—284no—
184Q8—282no—
185D4—144no—
186D4—284no—
187D4—282no—
188D4—282no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—71no—
192C2×C4—74no—
193C2×C4—71no—
194C2×C4—71no—
195C2×C4—142no—
196C2×C4—142no—
197C2×C4—142no—
198C2×C4—284no—
199C2×C4—284no—
200C2×C4D4×D711yescharacteristic
201C2×C4D4×D711yescharacteristic
202C2×C4C23×D711yescharacteristic
203C8—22no—
204C8—142no—
205C7#322611yescharacteristic, complemented, Sylow
206C22—72no—
207C22—141no—
208C22—141no—
209C22—282no—
210C22—282no—
211C22—282no—
212C22C2×D4×D711yescharacteristic
213C4—42no—
214C4—42no—
215C4—72no—
216C4—141no—
217C4—141no—
218C4—282no—
219C4D7×D812yes—
220C4#648112yes—
221C4#655112yes—
222C2—142no—
223C2—562no—
224C2#9041511yescharacteristic
225C2C2×D7×D811yescharacteristic
226C2C2×#648111yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#500C2
socleC22×C7C22×D4
prime radical#90416C2
hypercenter#500D7
hypocenterC7#3226
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #500

Complements

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

NH ≅ G/Ntype
2. D7×C4⋊Q8C2semidirect
3. C7×#500C2semidirect
4. #89860C2semidirect
5. #89850C2semidirect
6. #89662C2semidirect
7. #89654C2semidirect
8. #89466C2semidirect
9. #89457C2semidirect
70. #500D7semidirect
205. C7#3226semidirect

Maximal subgroups

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

subgroupindex
#904162
D7×C4⋊Q82
C7×#5002
#898602
#898502
#896622
#896542
#894662
#894572
#32267