Finite Group Databasebeta

← #22599082

Subgroups — #22599082

Order 9088.

358 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 119 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#17986553C212yesmaximal
2#17986396C211yesmaximal, characteristic, complemented
3#17986377C211yesmaximal, characteristic, complemented
4#17986348C212yesmaximal, complemented
5#17986258C211yesmaximal, characteristic
6#17986256C212yesmaximal
7#17986254C211yesmaximal, characteristic
8#17986107C211yesmaximal, characteristic, complemented
9#17986104C212yesmaximal, complemented
10#17985951C211yesmaximal, characteristic
11#441×C71C211yesmaximal, characteristic, complemented
12C2×Dic284C2211yescharacteristic
13C2×#11765855C2212yes—
14C2×#11765848C2211yescharacteristic
15C2×#11765848C2211yescharacteristic
16#14556127C2212yes—
17#14556126C2212yes—
18#14556118—22no—
19#14556108C2212yes—
20#14556106—22no—
21#14556104C2212yes—
22C4×Dic142—22no—
23C4×Dic142C2212yes—
24#14556085—22no—
25C42×D71C2211yescharacteristic
26#14556060C2211yescharacteristic
27#14556059C2212yes—
28#14556053C2212yes—
29#14556052C2212yes—
30#14556051C2212yes—
31#14556039C2211yescharacteristic
32C8×Dic71C2211yescharacteristic
33C8×Dic71C2211yescharacteristic
34C2×Q16×C71C2211yescharacteristic
35C4×Q8×C71C2212yes—
36#106×C71C2211yescharacteristic
37#103×C71C2212yes—
38#97×C71C2211yescharacteristic
39C2×Dic142—22no—
40C2×Dic142C2312yes—
41C2×C4×D71D412yes—
42C2×C4×D71C2311yescharacteristic
43Dic284—22no—
44#11765855—24no—
45#11765848—22no—
46#11765848—22no—
47C2×C71⋊C8C2311yescharacteristic
48C2×C71⋊C8C2311yescharacteristic
49C4⋊Dic71—22no—
50C4⋊Dic71C2312yes—
51Dic71⋊C4—22no—
52Dic71⋊C4—22no—
53Dic71⋊C4—22no—
54Dic71⋊C4—22no—
55Dic71⋊C4—22no—
56(C2×D71)⋊C4—22no—
57(C2×D71)⋊C4—22no—
58C4×Dic71—22no—
59C4×Dic71D412yes—
60C4×Dic71C2311yescharacteristic
61C2×Q8×C71C2312yes—
62Q16×C71—22no—
63C2×C8×C71C2311yescharacteristic
64C2×C8×C71C2311yescharacteristic
65C4⋊C4×C71—22no—
66C4⋊C4×C71C2312yes—
67C42×C71—22no—
68C42×C71C2311yescharacteristic
69C22×D71C2×D411yescharacteristic
70Q8×C71—22no—
71Q8×C71—22no—
72C2×C4×C71—22no—
73C2×C4×C71—22no—
74C2×C4×C71C2×D412yes—
75C2×C4×C71C2411yescharacteristic
76C2×Dic71—22no—
77C2×Dic71—22no—
78C2×Dic71C2×D411yescharacteristic
79C2×Dic71C2×D412yes—
80C4×D71—21no—
81C4×D71—21no—
82C4×D71—24no—
83Dic142—22no—
84Dic142—22no—
85Dic142—44no—
86C8×C71—21no—
87C8×C71#4112yes—
88C71⋊C8—21no—
89C71⋊C8#4112yes—
90C22×C71C22×D411yescharacteristic
91C2×D71—21no—
92C2×D71—22no—
93C4×C71—22no—
94C4×C71—42no—
95C4×C71—42no—
96C4×C71C2×#4111yescharacteristic
97C4×C71C2×#4111yescharacteristic
98Dic71—21no—
99Dic71—42no—
100Dic71—42no—
101Dic71#13714yes—
102C2×C71#53211yescharacteristic
103C2×C71C2×#13712yes—
104D71—41no—
105#3185—711nomaximal, Sylow
106C71#318511yescharacteristic, complemented, Sylow
107#533—712no—
108#498—711no—
109#494—711no—
110#487—712no—
111#470—711no—
112#470—711no—
113#470—712no—
114#441—712no—
115#441—711no—
116#441D7111yescharacteristic, complemented
117C4×M16—711no—
118C2×Q16—712no—
119C2×Q16—711no—
120C2×Q16C2×D7111yescharacteristic
121C2×M16—711no—
122C2×M16—711no—
123C4⋊Q8—712no—
124C4⋊Q8—712no—
125#125—1422no—
126C22⋊Q8—712no—
127C22⋊Q8—712no—
128C22⋊Q8—1422no—
129C4×Q8—712no—
130C4×Q8—1422no—
131C4×Q8C2×D7112yes—
132#117—1422no—
133C2×C42—711no—
134#106—711no—
135#106—712no—
136#106C2×D7111yescharacteristic
137#103—712no—
138#103—712no—
139#103—712no—
140#103C2×D7112yes—
141#97—711no—
142#97C2×D7111yescharacteristic
143C4×C8—711no—
144C4×C8—711no—
145C2×Q8—712no—
146C2×Q8—1422no—
147C2×Q8C22×D7112yes—
148C22×C4—712no—
149C22×C4—711no—
150Q16—22no—
151Q16—1424no—
152Q16—1422no—
153M16—1422no—
154M16—1422no—
155C2×C8—711no—
156C2×C8—711no—
157C2×C8C22×D7111yescharacteristic
158C2×C8C22×D7111yescharacteristic
159C4⋊C4—22no—
160C4⋊C4—712no—
161C4⋊C4—1422no—
162C4⋊C4—1422no—
163C4⋊C4—1422no—
164C4⋊C4—1422no—
165C4⋊C4—1422no—
166C4⋊C4—1422no—
167C4⋊C4C22×D7112yes—
168C22⋊C4—1422no—
169C22⋊C4—1422no—
170C42—22no—
171C42—712no—
172C42—711no—
173C42—1422no—
174C42C22×D7111yescharacteristic
175C23—711no—
176Q8—22no—
177Q8—22no—
178Q8—2844no—
179Q8—1422no—
180Q8—1422no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—712no—
184C2×C4—711no—
185C2×C4—1424no—
186C2×C4—1422no—
187C2×C4—1422no—
188C2×C4—1421no—
189C2×C4—1421no—
190C2×C4D4×D7112yes—
191C2×C4C23×D7111yescharacteristic
192C8—21no—
193C8—712no—
194C8—1421no—
195C8#1176586512yes—
196C22—1422no—
197C22—1421no—
198C22C2×D4×D7111yescharacteristic
199C4—22no—
200C4—42no—
201C4—42no—
202C4—2842no—
203C4—2842no—
204C4—714no—
205C4—1421no—
206C4#41×D7111yescharacteristic
207C4C2×#1176586511yescharacteristic
208C2—2841no—
209C2#1798654611yescharacteristic
210C2#137×D7112yes—
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#441×C71C2
socleC22×C71C22×D4
hypercenter#441D71
hypocenterC71#3185
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #441

Complements

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

NH ≅ G/Ntype
2. #17986396C2semidirect
3. #17986377C2semidirect
4. #17986348C2semidirect
8. #17986107C2semidirect
9. #17986104C2semidirect
11. #441×C71C2semidirect
106. C71#3185semidirect
116. #441D71semidirect