Finite Group Databasebeta

← #22598959

Subgroups — #22598959

Order 9088.

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

normal subgroups only

This group has 362 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#17986623C212yesmaximal, complemented
2#17986383C211yesmaximal, characteristic, complemented
3#17986361C212yesmaximal, complemented
4#17986258C211yesmaximal, characteristic, complemented
5#17986253C211yesmaximal, characteristic
6#17986229C212yesmaximal, complemented
7#17986110C211yesmaximal, characteristic, complemented
8#17986105C211yesmaximal, characteristic, complemented
9#17986086C212yesmaximal, complemented
10#17985953C211yesmaximal, characteristic, complemented
11#498×C71C211yesmaximal, characteristic, complemented
12C2×Dic284C2211yescharacteristic
13C2×#11765854C2212yes—
14C2×D568C2211yescharacteristic
15C2×#11765848C2211yescharacteristic
16C2×C8×D71C2211yescharacteristic
17#14556123—22no—
18#14556117—22no—
19#14556116C2212yes—
20#14556113—22no—
21#14556108C2212yes—
22#14556104—22no—
23#14556103C2212yes—
24#14556093C2212yes—
25#14556090C2212yes—
26#14556085C2211yescharacteristic
27#14556051C2212yes—
28#14556051C2212yes—
29#14556048C2212yes—
30#14556048C2212yes—
31#14556039C2211yescharacteristic
32#14556038C2211yescharacteristic
33C8×Dic71C2211yescharacteristic
34C4⋊Q8×C71C2212yes—
35#106×C71C2212yes—
36#106×C71C2211yescharacteristic
37#106×C71C2211yescharacteristic
38C4×C8×C71C2211yescharacteristic
39C2×Dic142C2312yes—
40C2×D284—22no—
41C2×D284C2312yes—
42C2×C4×D71—22no—
43C2×C4×D71C2311yescharacteristic
44Dic284—22no—
45#11765854—24no—
46D568—22no—
47#11765848—22no—
48C2×C71⋊C8C2311yescharacteristic
49C2×C71⋊C8C2311yescharacteristic
50C8×D71—22no—
51C4⋊Dic71—22no—
52Dic71⋊C4—22no—
53Dic71⋊C4D412yes—
54(C2×D71)⋊C4—22no—
55(C2×D71)⋊C4—22no—
56(C2×D71)⋊C4—22no—
57(C2×D71)⋊C4—22no—
58(C2×D71)⋊C4—22no—
59(C2×D71)⋊C4D412yes—
60C4×Dic71—22no—
61C4×Dic71C2311yescharacteristic
62C2×Q8×C71—22no—
63C2×C8×C71C2311yescharacteristic
64C2×C8×C71C2311yescharacteristic
65C4⋊C4×C71—22no—
66C4⋊C4×C71C2312yes—
67C4⋊C4×C71C2312yes—
68C42×C71C2311yescharacteristic
69C22×D71—22no—
70C22×D71C2×D411yescharacteristic
71Q8×C71—82no—
72C2×C4×C71—22no—
73C2×C4×C71—22no—
74C2×C4×C71C2×D412yes—
75C2×C4×C71C2411yescharacteristic
76C2×Dic71—22no—
77C2×Dic71C2×D412yes—
78C2×Dic71C2×D411yescharacteristic
79D284—22no—
80D284—22no—
81D284—82no—
82C4×D71—21no—
83C4×D71—21no—
84C4×D71—82no—
85Dic142—22no—
86Dic142—22no—
87C8×C71#4112yes—
88C8×C71#4112yes—
89C71⋊C8—21no—
90C71⋊C8—21no—
91C22×C71C22×D411yescharacteristic
92C2×D71—21no—
93C2×D71—22no—
94C2×D71—42no—
95C2×D71—42no—
96C2×D71—42no—
97C4×C71—22no—
98C4×C71—42no—
99C4×C71—42no—
100C4×C71C2×#4111yescharacteristic
101C4×C71C2×#4111yescharacteristic
102Dic71—21no—
103Dic71—22no—
104Dic71—42no—
105C2×C71#53211yescharacteristic
106C2×C71#57712yes—
107D71—41no—
108D71—82no—
109#3173—711nomaximal, Sylow
110C71#317311yescharacteristic, complemented, Sylow
111#542—712no—
112#498D7111yescharacteristic, complemented
113#495—711no—
114#489—712no—
115#470—711no—
116#469—711no—
117#465—712no—
118#442—711no—
119#441—711no—
120C4×SD16—712no—
121#405—711no—
122C2×Q16—711no—
123C2×SD16—712no—
124C2×D8—711no—
125C2×M16—711no—
126C22×C8—711no—
127C4⋊Q8C2×D7112yes—
128#126—1422no—
129#124—712no—
130#124—1422no—
131#123—1422no—
132C22⋊Q8—712no—
133C22⋊Q8—1422no—
134#121—712no—
135C4×Q8—712no—
136C4×D4—712no—
137#117—711no—
138#106C2×D7112yes—
139#106C2×D7111yescharacteristic
140#106C2×D7111yescharacteristic
141#103—712no—
142#103—712no—
143#102—712no—
144#102—712no—
145#97—711no—
146#97—711no—
147C4×C8—711no—
148C4×C8C2×D7111yescharacteristic
149C2×Q8—22no—
150C2×Q8—712no—
151C2×D4—712no—
152C2×D4—1422no—
153C22×C4—711no—
154C22×C4—1422no—
155Q16—1422no—
156SD16—1424no—
157D8—1422no—
158M16—1422no—
159C2×C8—711no—
160C2×C8—711no—
161C2×C8—1422no—
162C2×C8C22×D7111yescharacteristic
163C2×C8C22×D7111yescharacteristic
164C4⋊C4—22no—
165C4⋊C4—712no—
166C4⋊C4—1422no—
167C4⋊C4—1422no—
168C4⋊C4C22×D7112yes—
169C4⋊C4C22×D7112yes—
170C22⋊C4—712no—
171C22⋊C4—1422no—
172C22⋊C4—1422no—
173C22⋊C4—1422no—
174C22⋊C4—1422no—
175C22⋊C4—1422no—
176C42—711no—
177C42—1422no—
178C42C22×D7111yescharacteristic
179C23—711no—
180C23—1422no—
181Q8—82no—
182Q8—1422no—
183Q8—1422no—
184D4—5682no—
185D4—1422no—
186D4—1422no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—5682no—
190C2×C4—712no—
191C2×C4—711no—
192C2×C4—1422no—
193C2×C4—1421no—
194C2×C4—1421no—
195C2×C4D4×D7112yes—
196C2×C4C23×D7111yescharacteristic
197C8—1421no—
198C8—1421no—
199C8#1176586612yes—
200C8#1176586612yes—
201C22—2842no—
202C22—2842no—
203C22—2842no—
204C22—1422no—
205C22—1421no—
206C22C2×D4×D7111yescharacteristic
207C4—22no—
208C4—42no—
209C4—42no—
210C4—2842no—
211C4—1422no—
212C4—1421no—
213C4C2×#1176586611yescharacteristic
214C4C2×#1176586611yescharacteristic
215C2—2841no—
216C2—5682no—
217C2#1798655011yescharacteristic
218C2#1798680512yes—
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#498×C71C2
socleC22×C71C22×D4
hypercenter#498D71
hypocenterC71#3173
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #498

Complements

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

NH ≅ G/Ntype
1. #17986623C2semidirect
2. #17986383C2semidirect
3. #17986361C2semidirect
4. #17986258C2semidirect
6. #17986229C2semidirect
7. #17986110C2semidirect
8. #17986105C2semidirect
9. #17986086C2semidirect
10. #17985953C2semidirect
11. #498×C71C2semidirect
110. C71#3173semidirect
112. #498D71semidirect