Finite Group Databasebeta

← #11736650

Subgroups — #11736650

Order 896.

340 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 149 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#90210C211yesmaximal, characteristic
2#90191C211yesmaximal, characteristic
3C7×#428C211yesmaximal, characteristic
4#89894C211yesmaximal, characteristic
5#89893C211yesmaximal, characteristic
6#89770C211yesmaximal, characteristic
7C2×#6382C211yesmaximal, characteristic
8#89650C212yesmaximal, complemented
9#89648C212yesmaximal, complemented
10#89628C212yesmaximal, complemented
11#89627C212yesmaximal, complemented
12C2×C7×M16C2211yescharacteristic
13C7×#117C2211yescharacteristic
14C2×C7×C4⋊C4C2211yescharacteristic
15#6489C2211yescharacteristic
16#6489C2211yescharacteristic
17C2×C4⋊Dic7C2211yescharacteristic
18#6487—22no—
19C22×C7⋊C8C2211yescharacteristic
20#6453—21no—
21#6453C2212yes—
22#6453C2212yes—
23#6451—21no—
24#6451C2212yes—
25#6451C2212yes—
26#6441—22no—
27#6433—21no—
28#6432—21no—
29C7×#107C2212yes—
30C7×#106C2212yes—
31#6392C2212yes—
32#6391C2212yes—
33#6389C2212yes—
34C8×Dic7C2212yes—
35#6382C2211yescharacteristic
36#6382C2212yes—
37#6382C2212yes—
38#6382C2211yescharacteristic
39#6381C2211yescharacteristic
40#6381C2211yescharacteristic
41#3163—71nomaximal, Sylow
42C22×C4×C7—21no—
43C22×C4×C7C2311yescharacteristic
44C22×Dic7—21no—
45C2×Dic14—22no—
46C2×Dic14—22no—
47C7×M16Q814yescomplemented
48C2×C7×C8C2312yes—
49C7×C4⋊C4—21no—
50C7×C4⋊C4C2311yescharacteristic
51C7×C4⋊C4C2312yes—
52C7×C4⋊C4C2311yescharacteristic
53C7×C22⋊C4—21no—
54C42×C7—21no—
55C22⋊Dic7—21no—
56C22⋊Dic7D412yes—
57C4⋊Dic7—21no—
58C4⋊Dic7—22no—
59C4⋊Dic7—22no—
60C4⋊Dic7D411yescharacteristic
61C4⋊Dic7D411yescharacteristic
62C4⋊Dic7C2311yescharacteristic
63C4⋊Dic7C2312yes—
64C4⋊Dic7C2311yescharacteristic
65Dic7⋊C4—22no—
66Dic7⋊C4—22no—
67Dic7⋊C4—22no—
68Dic7⋊C4—22no—
69C4×Dic7—21no—
70C4×Dic7C2312yes—
71C2×C7⋊C8Q814yescomplemented
72C2×C7⋊C8C2311yescharacteristic
73C2×C7⋊C8C2311yescharacteristic
74#544—71no—
75#544—71no—
76#501—72no—
77#501—72no—
78#499—72no—
79#498—72no—
80#428—71no—
81#428D711yescharacteristic
82#427—71no—
83C2×#106—71no—
84#405—71no—
85C23×C7C2×D411yescharacteristic
86C2×C4×C7—21no—
87C2×C4×C7—22no—
88C2×C4×C7—21no—
89C2×C4×C7—81no—
90C2×C4×C7C2×D411yescharacteristic
91C2×C4×C7C2×Q812yes—
92C2×C4×C7C2×Q812yes—
93C2×C4×C7C2411yescharacteristic
94C2×Dic7—21no—
95C2×Dic7—21no—
96C2×Dic7—22no—
97C2×Dic7—81no—
98C2×Dic7C2×D412yes—
99C2×Dic7C2×D412yes—
100Dic14—82no—
101Dic14—82no—
102C7×C8C2×Q814yes—
103C7⋊C8C2×Q812yes—
104C7⋊C8C2×Q812yes—
105C2×M16C2×D711yescharacteristic
106C22×C8—71no—
107C4⋊Q8—72no—
108C4⋊Q8—72no—
109C4⋊Q8—141no—
110C4⋊Q8—141no—
111#125—72no—
112#125—72no—
113#125—141no—
114#125—141no—
115C22⋊Q8—142no—
116C22⋊Q8—142no—
117#117—71no—
118#117—71no—
119#117C2×D711yescharacteristic
120C2×C4⋊C4—71no—
121C2×C4⋊C4C2×D711yescharacteristic
122#107—72no—
123#107—71no—
124#107—71no—
125#107C2×D712yes—
126#106—72no—
127#106—72no—
128#106—72no—
129#106—71no—
130#106—71no—
131#106C2×D712yes—
132#97—72no—
133C4×C8—72no—
134C22×C7—21no—
135C22×C7—21no—
136C22×C7C4⋊Q812yes—
137C22×C7C22×D411yescharacteristic
138C4×C7—41no—
139C4×C7—42no—
140C4×C7—41no—
141C4×C7C4⋊Q812yes—
142C4×C7C22×Q811yescharacteristic
143C4×C7C22×Q811yescharacteristic
144Dic7—22no—
145Dic7—22no—
146Dic7—41no—
147Dic7—42no—
148Dic7—41no—
149C2×Q8—142no—
150C2×Q8—142no—
151C22×C4—21no—
152C22×C4—141no—
153C22×C4C22×D711yescharacteristic
154M16Dic1414yescomplemented
155C2×C8—74no—
156C2×C8—71no—
157C2×C8—71no—
158C2×C8C22×D712yes—
159C4⋊C4—21no—
160C4⋊C4—71no—
161C4⋊C4—71no—
162C4⋊C4—72no—
163C4⋊C4—71no—
164C4⋊C4—71no—
165C4⋊C4—142no—
166C4⋊C4—142no—
167C4⋊C4—142no—
168C4⋊C4—142no—
169C4⋊C4—142no—
170C4⋊C4—142no—
171C4⋊C4—141no—
172C4⋊C4C22×D712yes—
173C4⋊C4C22×D711yescharacteristic
174C4⋊C4C22×D711yescharacteristic
175C22⋊C4—21no—
176C22⋊C4—72no—
177C22⋊C4—141no—
178C42—21no—
179C42—72no—
180C42—141no—
181C2×C7—22no—
182C2×C7C2×C4⋊Q811yescharacteristic
183C2×C7#57711yescharacteristic
184C2×C7#57711yescharacteristic
185C23D4×D711yescharacteristic
186Q8—562no—
187Q8—562no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—72no—
192C2×C4—72no—
193C2×C4—81no—
194C2×C4—141no—
195C2×C4—141no—
196C2×C4—142no—
197C2×C4—561no—
198C2×C4C2×Dic1412yes—
199C2×C4D4×D711yescharacteristic
200C2×C4Q8×D712yes—
201C2×C4C23×D711yescharacteristic
202C8—72no—
203C8—72no—
204C8C2×Dic1414yes—
205C7#316311yescharacteristic, complemented, Sylow
206C22—21no—
207C22—21no—
208C22#645112yes—
209C22C2×D4×D711yescharacteristic
210C4—42no—
211C4—41no—
212C4—41no—
213C4—142no—
214C4—142no—
215C4—281no—
216C4—282no—
217C4—281no—
218C4#645112yes—
219C4C22×Dic1411yescharacteristic
220C4C2×Q8×D711yescharacteristic
221C2—22no—
222C2C2×#645111yescharacteristic
223C2#9046611yescharacteristic
224C2#9046911yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#428C2
socleC22×C7C22×D4
prime radicalC23×C7C2×D4
hypercenter#428D7
hypocenterC7#3163
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
8. #89650C2semidirect
9. #89648C2semidirect
10. #89628C2semidirect
11. #89627C2semidirect
47. C7×M16Q8semidirect
71. C2×C7⋊C8Q8semidirect
154. M16Dic14semidirect
205. C7#3163semidirect

Maximal subgroups

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

subgroupindex
#902102
#901912
C7×#4282
#898942
#898932
#897702
C2×#63822
#896502
#896482
#896282
#896272
#31637