Finite Group Databasebeta

← #22598965

Subgroups — #22598965

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#17986610C211yesmaximal, characteristic, complemented
2#17986603C211yesmaximal, characteristic
3#17986401C211yesmaximal, characteristic, complemented
4#17986367C211yesmaximal, characteristic, complemented
5#17986363C211yesmaximal, characteristic, complemented
6#17986242C212yesmaximal, complemented
7#17986225C212yesmaximal, complemented
8#17986100C212yesmaximal, complemented
9SD16×Dic71C212yesmaximal, complemented
10#17985953C211yesmaximal, characteristic, complemented
11#492×C71C211yesmaximal, characteristic, complemented
12C2×#11765848C2211yescharacteristic
13C2×C8×D71C2211yescharacteristic
14#14556123—22no—
15#14556120C2211yescharacteristic
16#14556120C2211yescharacteristic
17#14556118—22no—
18#14556109—22no—
19#14556104C2212yes—
20#14556100—22no—
21#14556099C2212yes—
22Q8×Dic71C2212yes—
23D4×Dic71C2212yes—
24#14556085C2211yescharacteristic
25#14556062C2212yes—
26#14556060C2212yes—
27#14556052C2212yes—
28#14556052C2212yes—
29#14556049C2212yes—
30#14556049C2212yes—
31#14556039C2211yescharacteristic
32#14556038C2211yescharacteristic
33C8×Dic71C2211yescharacteristic
34C2×SD16×C71C2212yes—
35C2×SD16×C71C2212yes—
36C4⋊Q8×C71C2211yescharacteristic
37#127×C71C2211yescharacteristic
38C4×C8×C71C2211yescharacteristic
39C2×C22⋊D71—22no—
40C22×Dic71—22no—
41C2×C4×D71C2311yescharacteristic
42#11765848—22no—
43C2×C71⋊C8C2311yescharacteristic
44C2×C71⋊C8C2311yescharacteristic
45C8×D71—22no—
46C4⋊Dic71—22no—
47C4⋊Dic71C2312yes—
48C4⋊Dic71C2312yes—
49Dic71⋊C4—22no—
50Dic71⋊C4—22no—
51Dic71⋊C4—22no—
52Dic71⋊C4D412yes—
53C22⋊Dic71—22no—
54C22⋊Dic71—22no—
55(C2×D71)⋊C4—22no—
56(C2×D71)⋊C4D412yes—
57C4×Dic71—22no—
58C4×Dic71C2311yescharacteristic
59C2×Q8×C71C2312yes—
60C2×D4×C71—22no—
61C2×D4×C71C2312yes—
62SD16×C71—24no—
63SD16×C71—24no—
64C2×C8×C71C2311yescharacteristic
65C2×C8×C71C2311yescharacteristic
66C4⋊C4×C71—22no—
67C42×C71C2311yescharacteristic
68C23×C71—22no—
69C22×D71C2×D411yescharacteristic
70Q8×C71—22no—
71Q8×C71—22no—
72D4×C71—22no—
73D4×C71—22no—
74D4×C71—82no—
75C2×C4×C71—22no—
76C2×C4×C71C2×D412yes—
77C2×C4×C71C2411yescharacteristic
78C22⋊D71—82no—
79C2×Dic71—22no—
80C2×Dic71—22no—
81C2×Dic71—82no—
82C2×Dic71C2×D412yes—
83C2×Dic71C2×D411yescharacteristic
84C4×D71—21no—
85C4×D71—21no—
86C8×C71#4112yes—
87C8×C71#4112yes—
88C71⋊C8—21no—
89C71⋊C8—21no—
90C22×C71—42no—
91C22×C71—42no—
92C22×C71—42no—
93C22×C71C22×D411yescharacteristic
94C2×D71—21no—
95C2×D71—22no—
96C4×C71—22no—
97C4×C71—42no—
98C4×C71C2×#4111yescharacteristic
99C4×C71C2×#4111yescharacteristic
100Dic71—21no—
101Dic71—22no—
102Dic71—42no—
103Dic71—42no—
104C2×C71—82no—
105C2×C71#53211yescharacteristic
106C2×C71#57712yes—
107D71—41no—
108#3174—711nomaximal, Sylow
109C71#317411yescharacteristic, complemented, Sylow
110#541—711no—
111#540—711no—
112#499—711no—
113#492D7111yescharacteristic, complemented
114#491—711no—
115#490—711no—
116#468—712no—
117#465—712no—
118#440—712no—
119C4×SD16—712no—
120#405—711no—
121C2×SD16C2×D7112yes—
122C2×SD16C2×D7112yes—
123C2×M16—711no—
124C22×C8—711no—
125C4⋊Q8C2×D7111yescharacteristic
126#127C2×D7111yescharacteristic
127#126—1422no—
128#125—711no—
129#125—711no—
130#125—1422no—
131#123—1422no—
132C22⋊Q8—712no—
133#121—712no—
134#121—1422no—
135C4×Q8—712no—
136C4×D4—712no—
137#117—711no—
138#107—712no—
139#106—712no—
140#103—712no—
141#103—712no—
142#102—712no—
143#102—712no—
144#97—711no—
145#97—711no—
146C4×C8—711no—
147C4×C8C2×D7111yescharacteristic
148C2×Q8C22×D7112yes—
149C2×D4—22no—
150C2×D4—1422no—
151C2×D4C22×D7112yes—
152C22×C4—711no—
153C22×C4—1422no—
154SD16—24no—
155SD16—24no—
156M16—1422no—
157C2×C8—711no—
158C2×C8—711no—
159C2×C8—1422no—
160C2×C8C22×D7111yescharacteristic
161C2×C8C22×D7111yescharacteristic
162C4⋊C4—22no—
163C4⋊C4—712no—
164C4⋊C4—712no—
165C4⋊C4—712no—
166C4⋊C4—1422no—
167C4⋊C4—1422no—
168C4⋊C4—1422no—
169C4⋊C4—1422no—
170C22⋊C4—712no—
171C22⋊C4—1422no—
172C22⋊C4—1422no—
173C22⋊C4—1422no—
174C42—711no—
175C42—1422no—
176C42C22×D7111yescharacteristic
177C23—22no—
178C23—711no—
179Q8—22no—
180Q8—22no—
181D4—22no—
182D4—22no—
183D4—82no—
184D4—5682no—
185C2×C4—22no—
186C2×C4—5682no—
187C2×C4—712no—
188C2×C4—711no—
189C2×C4—1422no—
190C2×C4—1422no—
191C2×C4—1421no—
192C2×C4—1421no—
193C2×C4D4×D7112yes—
194C2×C4C23×D7111yescharacteristic
195C8—1421no—
196C8—1421no—
197C8#1176586412yes—
198C8#1176586412yes—
199C22—42no—
200C22—42no—
201C22—42no—
202C22—1421no—
203C22—1422no—
204C22C2×D4×D7111yescharacteristic
205C4—22no—
206C4—42no—
207C4—2842no—
208C4—2842no—
209C4—1421no—
210C4—1422no—
211C4C2×#1176586411yescharacteristic
212C4C2×#1176586411yescharacteristic
213C2—82no—
214C2—2841no—
215C2#1798654711yescharacteristic
216C2#1798680512yes—
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#492×C71C2
socleC22×C71C22×D4
hypercenter#492D71
hypocenterC71#3174
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #492

Complements

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

NH ≅ G/Ntype
1. #17986610C2semidirect
3. #17986401C2semidirect
4. #17986367C2semidirect
5. #17986363C2semidirect
6. #17986242C2semidirect
7. #17986225C2semidirect
8. #17986100C2semidirect
9. SD16×Dic71C2semidirect
10. #17985953C2semidirect
11. #492×C71C2semidirect
109. C71#3174semidirect
113. #492D71semidirect