Finite Group Databasebeta

← #22598717

Subgroups — #22598717

Order 9088.

380 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 125 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#17986554C211yesmaximal, characteristic
2C4⋊Q8×D71C211yesmaximal, characteristic
3#17986366C214yesmaximal, complemented
4#17986351C211yesmaximal, characteristic, complemented
5#17986350C211yesmaximal, characteristic, complemented
6#17986349C211yesmaximal, characteristic, complemented
7#17986006C214yesmaximal
8C4×#11765848C211yesmaximal, characteristic
9#487×C71C211yesmaximal, characteristic, complemented
10C2×Q8×D71C2212yes—
11C2×#11765848C2212yes—
12#14556128C2211yescharacteristic
13#14556128C2211yescharacteristic
14#14556127—21no—
15#14556127—21no—
16#14556126C2212yes—
17#14556119C2212yes—
18#14556108—21no—
19#14556104—21no—
20Q8×Dic71—21no—
21#14556093—21no—
22#14556086C2212yes—
23C4⋊C4×D71—21no—
24C4⋊C4×D71—21no—
25C42×D71C2211yescharacteristic
26#14556053C2214yes—
27#14556052C2214yes—
28#14556051C2214yes—
29#14556038C2212yes—
30C4×C71⋊C8C2211yescharacteristic
31C4⋊Q8×C71C2211yescharacteristic
32C4⋊Q8×C71C2211yescharacteristic
33#103×C71C2214yes—
34C4×C8×C71C2211yescharacteristic
35C2×Dic142—21no—
36C2×Dic142—22no—
37C2×Dic142C2312yes—
38Q8×D71—28no—
39C2×C4×D71—22no—
40C2×C4×D71D411yescharacteristic
41C2×C4×D71D411yescharacteristic
42C2×C4×D71C2311yescharacteristic
43#11765848—24no—
44C2×C71⋊C8C2312yes—
45C4⋊Dic71—21no—
46C4⋊Dic71—21no—
47C4⋊Dic71—21no—
48C4⋊Dic71—22no—
49C4⋊Dic71C2312yes—
50Dic71⋊C4—22no—
51Dic71⋊C4—22no—
52Dic71⋊C4—22no—
53(C2×D71)⋊C4—22no—
54C4×Dic71—22no—
55C4×Dic71D411yescharacteristic
56C4×Dic71D411yescharacteristic
57C4×Dic71C2311yescharacteristic
58C2×Q8×C71—21no—
59C2×Q8×C71C2312yes—
60C2×C8×C71C2312yes—
61C4⋊C4×C71—21no—
62C4⋊C4×C71—21no—
63C4⋊C4×C71—21no—
64C4⋊C4×C71C2312yes—
65C42×C71C2311yescharacteristic
66C22×D71C2×D411yescharacteristic
67Q8×C71—24no—
68Q8×C71—42no—
69C2×C4×C71—22no—
70C2×C4×C71—22no—
71C2×C4×C71C2×D411yescharacteristic
72C2×C4×C71C2×D411yescharacteristic
73C2×C4×C71C2411yescharacteristic
74C2×Dic71—22no—
75C2×Dic71—22no—
76C2×Dic71C2×D411yescharacteristic
77C2×Dic71C2×D411yescharacteristic
78C2×Dic71C2×D411yescharacteristic
79C4×D71—22no—
80C4×D71—22no—
81C4×D71—44no—
82C4×D71#4114yes—
83Dic142—24no—
84Dic142—42no—
85Dic142—44no—
86C8×C71—22no—
87C71⋊C8—22no—
88C22×C71C22×D411yescharacteristic
89C2×D71—21no—
90C2×D71—21no—
91C2×D71#12412yes—
92C4×C71—42no—
93C4×C71—42no—
94C4×C71#13712yes—
95C4×C71#13712yes—
96C4×C71C2×#4112yes—
97Dic71—21no—
98Dic71—21no—
99Dic71—42no—
100Dic71—42no—
101Dic71#12412yes—
102C2×C71C2×#12411yescharacteristic
103C2×C71C2×#13711yescharacteristic
104C2×C71C2×#13711yescharacteristic
105D71—22no—
106#3139—711nomaximal, Sylow
107C71#313911yescharacteristic, complemented, Sylow
108#533—711no—
109C2×C4⋊Q8—711no—
110#491—714no—
111#487—711no—
112#487—711no—
113#487—711no—
114#487D7111yescharacteristic, complemented
115#419—714no—
116C4×M16—711no—
117C22×Q8—712no—
118C2×M16—712no—
119C4⋊Q8—712no—
120C4⋊Q8—711no—
121C4⋊Q8—711no—
122C4⋊Q8—1421no—
123C4⋊Q8—1421no—
124C4⋊Q8C2×D7111yescharacteristic
125C4⋊Q8C2×D7111yescharacteristic
126#125—712no—
127C22⋊Q8—1421no—
128C22⋊Q8—1421no—
129C4×Q8—1421no—
130C4×Q8—1421no—
131#117—712no—
132C2×C4⋊C4—1421no—
133C2×C4⋊C4—1421no—
134C2×C42—711no—
135#103—714no—
136#103—714no—
137#103—714no—
138#103C2×D7114yes—
139#97—712no—
140C4×C8—711no—
141C4×C8C2×D7111yescharacteristic
142C2×Q8—21no—
143C2×Q8—712no—
144C2×Q8—1422no—
145C2×Q8—1428no—
146C2×Q8—1421no—
147C2×Q8C22×D7112yes—
148C22×C4—711no—
149C22×C4—711no—
150C22×C4—711no—
151C22×C4—1422no—
152M16—1424no—
153C2×C8—712no—
154C2×C8C22×D7112yes—
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—21no—
158C4⋊C4—712no—
159C4⋊C4—1422no—
160C4⋊C4—1422no—
161C4⋊C4—1422no—
162C4⋊C4—1422no—
163C4⋊C4—1421no—
164C4⋊C4—1421no—
165C4⋊C4—1421no—
166C4⋊C4C22×D7112yes—
167C22⋊C4—1422no—
168C42—711no—
169C42—711no—
170C42—711no—
171C42—1422no—
172C42C22×D7111yescharacteristic
173C23—711no—
174Q8—24no—
175Q8—42no—
176Q8—2844no—
177Q8—2842no—
178Q8—1424no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—2844no—
182C2×C4—711no—
183C2×C4—714no—
184C2×C4—711no—
185C2×C4—711no—
186C2×C4—1422no—
187C2×C4—1422no—
188C2×C4—1422no—
189C2×C4—1422no—
190C2×C4D4×D7111yescharacteristic
191C2×C4D4×D7111yescharacteristic
192C2×C4C23×D7111yescharacteristic
193C8—22no—
194C8—1422no—
195C22—712no—
196C22—1421no—
197C22—1421no—
198C22C2×D4×D7111yescharacteristic
199C4—42no—
200C4—42no—
201C4—2842no—
202C4—2842no—
203C4—712no—
204C4—1421no—
205C4—1421no—
206C4#1455615712yes—
207C4#1455616012yes—
208C4#41×D7112yes—
209C2—1422no—
210C2#124×D7111yescharacteristic
211C2C2×#1455615711yescharacteristic
212C2C2×#1455616011yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#487×C71C2
socleC22×C71C22×D4
hypercenter#487D71
hypocenterC71#3139
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
3. #17986366C2semidirect
4. #17986351C2semidirect
5. #17986350C2semidirect
6. #17986349C2semidirect
9. #487×C71C2semidirect
107. C71#3139semidirect
114. #487D71semidirect