Finite Group Databasebeta

← #22598995

Subgroups — #22598995

Order 9088.

338 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 115 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#17986607C212yesmaximal
2#17986400C211yesmaximal, characteristic, complemented
3#17986377C211yesmaximal, characteristic, complemented
4#17986366C212yesmaximal, complemented
5#17986256C212yesmaximal
6#17986241C211yesmaximal, characteristic
7#17986237C211yesmaximal, characteristic
8#17986104C212yesmaximal, complemented
9C4×Dic284C211yesmaximal, characteristic, complemented
10#17985953C211yesmaximal, characteristic
11C4×Q16×C71C211yesmaximal, characteristic, complemented
12C2×Dic284C2211yescharacteristic
13C2×#11765855C2212yes—
14C2×#11765848C2211yescharacteristic
15C2×C8×D71C2211yescharacteristic
16#14556126C2212yes—
17#14556123—22no—
18#14556122—22no—
19#14556120—22no—
20#14556119C2212yes—
21#14556118—22no—
22#14556108C2212yes—
23#14556104C2212yes—
24C4×Dic142C2212yes—
25#14556085C2211yescharacteristic
26#14556062C2211yescharacteristic
27#14556059C2212yes—
28#14556053C2212yes—
29#14556052C2212yes—
30#14556051C2212yes—
31#14556039C2211yescharacteristic
32#14556038C2211yescharacteristic
33C8×Dic71C2211yescharacteristic
34C2×Q16×C71C2211yescharacteristic
35C4×Q8×C71C2212yes—
36#107×C71C2211yescharacteristic
37#103×C71C2212yes—
38C4×C8×C71C2211yescharacteristic
39C2×Dic142C2312yes—
40C2×C4×D71C2311yescharacteristic
41Dic284—22no—
42#11765855—24no—
43#11765848—22no—
44C2×C71⋊C8C2311yescharacteristic
45C2×C71⋊C8C2311yescharacteristic
46C8×D71—22no—
47C4⋊Dic71—22no—
48C4⋊Dic71—22no—
49C4⋊Dic71C2312yes—
50Dic71⋊C4—22no—
51Dic71⋊C4—22no—
52Dic71⋊C4—22no—
53Dic71⋊C4—22no—
54Dic71⋊C4—22no—
55Dic71⋊C4D411yescharacteristic
56Dic71⋊C4D411yescharacteristic
57(C2×D71)⋊C4—22no—
58(C2×D71)⋊C4—22no—
59(C2×D71)⋊C4D411yescharacteristic
60(C2×D71)⋊C4D411yescharacteristic
61C4×Dic71—22no—
62C4×Dic71C2311yescharacteristic
63C2×Q8×C71C2312yes—
64Q16×C71—22no—
65C2×C8×C71C2311yescharacteristic
66C2×C8×C71C2311yescharacteristic
67C4⋊C4×C71—22no—
68C4⋊C4×C71C2312yes—
69C42×C71—22no—
70C42×C71C2311yescharacteristic
71C22×D71C2×D411yescharacteristic
72Q8×C71—22no—
73Q8×C71—22no—
74C2×C4×C71—22no—
75C2×C4×C71—22no—
76C2×C4×C71C2×D411yescharacteristic
77C2×C4×C71C2×D411yescharacteristic
78C2×C4×C71C2411yescharacteristic
79C2×Dic71—22no—
80C2×Dic71—22no—
81C2×Dic71C2×D411yescharacteristic
82C2×Dic71C2×D411yescharacteristic
83C2×Dic71C2×D411yescharacteristic
84C4×D71—21no—
85C4×D71—21no—
86Dic142—22no—
87Dic142—22no—
88C8×C71—21no—
89C8×C71#4112yes—
90C71⋊C8—21no—
91C71⋊C8#4112yes—
92C22×C71C22×D411yescharacteristic
93C2×D71—21no—
94C2×D71—21no—
95C2×D71—21no—
96C4×C71—21no—
97C4×C71—21no—
98C4×C71—42no—
99C4×C71—42no—
100C4×C71C2×#4111yescharacteristic
101C4×C71C2×#4111yescharacteristic
102Dic71—21no—
103Dic71—21no—
104Dic71—21no—
105Dic71—42no—
106Dic71—42no—
107C2×C71#53211yescharacteristic
108C2×C71#57811yescharacteristic
109C2×C71#57811yescharacteristic
110D71—41no—
111#3177—711nomaximal, Sylow
112C71#317711yescharacteristic, complemented, Sylow
113#541—712no—
114#499—711no—
115#494—711no—
116#491—712no—
117#470—712no—
118#467—711no—
119#467—711no—
120#441—712no—
121C4×Q16—711no—
122C4×Q16D7111yescharacteristic, complemented
123#405—711no—
124C2×Q16—712no—
125C2×Q16—711no—
126C2×Q16C2×D7111yescharacteristic
127C2×M16—711no—
128C22×C8—711no—
129C4⋊Q8—712no—
130#126—1422no—
131#126—1422no—
132#125—712no—
133#125—1422no—
134#125—1422no—
135C22⋊Q8—712no—
136C22⋊Q8—712no—
137C4×Q8—712no—
138C4×Q8C2×D7112yes—
139#117—711no—
140#107—711no—
141#107C2×D7111yescharacteristic
142#106—712no—
143#103—712no—
144#103—712no—
145#103—712no—
146#103C2×D7112yes—
147#97—711no—
148#97—711no—
149C4×C8—711no—
150C4×C8C2×D7111yescharacteristic
151C2×Q8—712no—
152C2×Q8C22×D7112yes—
153C22×C4—711no—
154Q16—22no—
155Q16—1424no—
156Q16—1422no—
157M16—1422no—
158C2×C8—711no—
159C2×C8—711no—
160C2×C8—1422no—
161C2×C8C22×D7111yescharacteristic
162C2×C8C22×D7111yescharacteristic
163C4⋊C4—22no—
164C4⋊C4—711no—
165C4⋊C4—711no—
166C4⋊C4—712no—
167C4⋊C4—1422no—
168C4⋊C4—1422no—
169C4⋊C4—1422no—
170C4⋊C4—1422no—
171C4⋊C4—1422no—
172C4⋊C4—1422no—
173C4⋊C4—1422no—
174C4⋊C4C22×D7112yes—
175C22⋊C4—711no—
176C22⋊C4—711no—
177C22⋊C4—1422no—
178C22⋊C4—1422no—
179C42—22no—
180C42—711no—
181C42—1422no—
182C42C22×D7111yescharacteristic
183C23—711no—
184Q8—22no—
185Q8—22no—
186Q8—1422no—
187Q8—1422no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—711no—
191C2×C4—711no—
192C2×C4—711no—
193C2×C4—1422no—
194C2×C4—1422no—
195C2×C4—1421no—
196C2×C4—1421no—
197C2×C4D4×D7111yescharacteristic
198C2×C4D4×D7111yescharacteristic
199C2×C4C23×D7111yescharacteristic
200C8—21no—
201C8—712no—
202C8—1421no—
203C8#1176586512yes—
204C22—1421no—
205C22—1421no—
206C22—1421no—
207C22C2×D4×D7111yescharacteristic
208C4—21no—
209C4—21no—
210C4—42no—
211C4—42no—
212C4—2842no—
213C4—2842no—
214C4—1421no—
215C4—1421no—
216C4—1421no—
217C4#41×D7111yescharacteristic
218C4C2×#1176586511yescharacteristic
219C2—2841no—
220C2#1798654611yescharacteristic
221C2#1798681211yescharacteristic
222C2#1798681311yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
FittingC4×Q16×C71C2
socleC22×C71C22×D4
hypercenterC4×Q16D71
hypocenterC71#3177
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q16

Complements

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

NH ≅ G/Ntype
2. #17986400C2semidirect
3. #17986377C2semidirect
4. #17986366C2semidirect
8. #17986104C2semidirect
9. C4×Dic284C2semidirect
11. C4×Q16×C71C2semidirect
112. C71#3177semidirect
122. C4×Q16D71semidirect