Finite Group Databasebeta

← #22597825

Subgroups — #22597825

Order 9088.

480 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 145 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
1C2×#14556158C211yesmaximal, characteristic
2C2×#14556126C211yesmaximal, characteristic, complemented
3#41×Dic71C211yesmaximal, characteristic
4#17986207C214yesmaximal, complemented
5#17986202C214yesmaximal, complemented
6#17986022C211yesmaximal, characteristic, complemented
7#17986009C211yesmaximal, characteristic, complemented
8#17986008C211yesmaximal, characteristic, complemented
9C2×#137×C71C211yesmaximal, characteristic
10C22×Dic142C2211yescharacteristic
11#14556158—24no—
12C2×#11765855C2212yes—
13C2×#11765853C2212yes—
14C2×#11765849C2211yescharacteristic
15#14556126—21no—
16#14556126—21no—
17#14556126C2214yes—
18Q8×Dic71C2212yes—
19D4×Dic71—22no—
20D4×Dic71C2212yes—
21#14556087—21no—
22#14556087C2211yescharacteristic
23C2×Dic71⋊C4—22no—
24C2×C4×Dic71—21no—
25C2×C4×Dic71C2211yescharacteristic
26Dic71⋊C8C2214yes—
27#14556053C2214yes—
28#14556052C2214yes—
29C2×#41×C71C2211yescharacteristic
30C22×Q8×C71C2211yescharacteristic
31#137×C71—24no—
32C2×Q16×C71C2212yes—
33C2×SD16×C71C2212yes—
34C2×M16×C71C2211yescharacteristic
35C2×Dic142—22no—
36C2×Dic142—22no—
37C2×Dic142—21no—
38C2×Dic142C2312yes—
39C22×Dic71—21no—
40C22×Dic71—22no—
41C22×Dic71D412yes—
42#11765855—24no—
43#11765853—24no—
44#11765849—22no—
45C2×C71⋊C8C2312yes—
46C4⋊Dic71—22no—
47C4⋊Dic71C2312yes—
48Dic71⋊C4—24no—
49Dic71⋊C4—24no—
50C22⋊Dic71—21no—
51C22⋊Dic71—22no—
52C4×Dic71—21no—
53C4×Dic71—22no—
54C4×Dic71D411yescharacteristic
55C4×Dic71D411yescharacteristic
56C4×Dic71C2312yes—
57#41×C71—22no—
58#41×C71D414yes—
59C2×Q8×C71—21no—
60C2×Q8×C71—22no—
61C2×Q8×C71—22no—
62C2×Q8×C71C2312yes—
63C2×Q8×C71C2311yescharacteristic
64C2×D4×C71—21no—
65C2×D4×C71C2311yescharacteristic
66C22×C4×C71—21no—
67C22×C4×C71—21no—
68C22×C4×C71C2311yescharacteristic
69Q16×C71—24no—
70SD16×C71—24no—
71M16×C71—22no—
72C2×C8×C71C2312yes—
73C23×C71—21no—
74C23×C71C2×D411yescharacteristic
75Q8×C71—21no—
76Q8×C71—22no—
77Q8×C71—22no—
78Q8×C71—42no—
79Q8×C71C2×D412yes—
80D4×C71—24no—
81D4×C71—21no—
82D4×C71C2×D412yes—
83C2×C4×C71—21no—
84C2×C4×C71—24no—
85C2×C4×C71—22no—
86C2×C4×C71—42no—
87C2×C4×C71C2×D411yescharacteristic
88C2×C4×C71C2×D412yes—
89C2×C4×C71#4112yes—
90C2×C4×C71C2411yescharacteristic
91C2×Dic71—24no—
92C2×Dic71—22no—
93C2×Dic71—22no—
94C2×Dic71—42no—
95C2×Dic71—44no—
96C2×Dic71C2×D412yes—
97C2×Dic71C2×D412yes—
98Dic142—22no—
99Dic142—22no—
100Dic142—42no—
101C8×C71—22no—
102C71⋊C8—22no—
103C22×C71—22no—
104C22×C71—22no—
105C22×C71—42no—
106C22×C71#12112yes—
107C22×C71C22×D411yescharacteristic
108C4×C71—22no—
109C4×C71—42no—
110C4×C71#12112yes—
111C4×C71C22×D411yescharacteristic
112C4×C71C2×#4111yescharacteristic
113Dic71—22no—
114Dic71—42no—
115Dic71—42no—
116Dic71#13714yescomplemented
117C2×C71—22no—
118C2×C71—42no—
119C2×C71C2×#12111yescharacteristic
120C2×C71C2×#13712yes—
121#3039—711nomaximal, Sylow
122C71#303911yescharacteristic, complemented, Sylow
123C2×#137—711no—
124C2×#137D7111yescharacteristic
125C2×C4⋊Q8—711no—
126C4×#41—711no—
127#462—714no—
128#461—714no—
129#423—711no—
130#419—711no—
131#419—711no—
132C2×#41C2×D7111yescharacteristic
133C22×Q8—711no—
134C22×Q8C2×D7111yescharacteristic
135#137—24no—
136#137—1424no—
137C2×Q16—712no—
138C2×Q16C2×D7112yes—
139C2×SD16—712no—
140C2×SD16C2×D7112yes—
141C2×M16—711no—
142C2×M16C2×D7111yescharacteristic
143C4⋊Q8—714no—
144C4⋊Q8—1421no—
145C4⋊Q8—1421no—
146C4×Q8—712no—
147C4×D4—712no—
148C4×D4—1422no—
149#117—711no—
150#117—1421no—
151C2×C4⋊C4—1422no—
152C2×C42—711no—
153C2×C42—1421no—
154C4⋊C8—714no—
155#103—714no—
156#103—714no—
157#41—22no—
158#41C22⋊D7114yes—
159C2×Q8—21no—
160C2×Q8—22no—
161C2×Q8—22no—
162C2×Q8—712no—
163C2×Q8—1422no—
164C2×Q8—1422no—
165C2×Q8—1421no—
166C2×Q8C22×D7112yes—
167C2×Q8C22×D7111yescharacteristic
168C2×D4—21no—
169C2×D4C22×D7111yescharacteristic
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—712no—
173C22×C4—1422no—
174C22×C4—1421no—
175C22×C4C22×D7111yescharacteristic
176Q16—24no—
177Q16—1424no—
178SD16—24no—
179SD16—1424no—
180M16—22no—
181M16—1422no—
182C2×C8—712no—
183C2×C8C22×D7112yes—
184C4⋊C4—712no—
185C4⋊C4—1424no—
186C4⋊C4—1424no—
187C4⋊C4—1422no—
188C22⋊C4—1422no—
189C22⋊C4—1421no—
190C42—711no—
191C42—711no—
192C42—712no—
193C42—1422no—
194C42—1421no—
195C23—21no—
196C23D4×D7111yescharacteristic
197Q8—21no—
198Q8—22no—
199Q8—22no—
200Q8—42no—
201Q8—2842no—
202Q8—1422no—
203Q8—1422no—
204Q8C2×C22⋊D7112yes—
205D4—24no—
206D4—21no—
207D4C2×C22⋊D7112yes—
208C2×C4—21no—
209C2×C4—24no—
210C2×C4—22no—
211C2×C4—42no—
212C2×C4—2842no—
213C2×C4—2844no—
214C2×C4—712no—
215C2×C4—712no—
216C2×C4—1422no—
217C2×C4—1422no—
218C2×C4—1424no—
219C2×C4D4×D7111yescharacteristic
220C2×C4C2×C22⋊D7112yes—
221C2×C4#1176586412yes—
222C2×C4C23×D7111yescharacteristic
223C8—22no—
224C8—1422no—
225C22—22no—
226C22—22no—
227C22—42no—
228C22#1455610012yes—
229C22C2×D4×D7111yescharacteristic
230C4—22no—
231C4—42no—
232C4—2842no—
233C4—2842no—
234C4—714no—
235C4—1422no—
236C4#1455610012yes—
237C4C22×C22⋊D7111yescharacteristic
238C4C2×#1176586411yescharacteristic
239C2—22no—
240C2—42no—
241C2C2×#1455610011yescharacteristic
242C2#137×D7112yes—
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
FittingC2×#137×C71C2
socleC22×C71C22×D4
hypercenterC2×#137D71
hypocenterC71#3039
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
2. C2×#14556126C2semidirect
4. #17986207C2semidirect
5. #17986202C2semidirect
6. #17986022C2semidirect
7. #17986009C2semidirect
8. #17986008C2semidirect
116. Dic71#137semidirect
122. C71#3039semidirect