Finite Group Databasebeta

← #22598144

Subgroups — #22598144

Order 9088.

376 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 135 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#17986552C211yesmaximal, characteristic
2C2×#14556093C211yesmaximal, characteristic
3#17986309C212yesmaximal, complemented
4#17986305C212yesmaximal, complemented
5#17986293C212yesmaximal, complemented
6#17986283C212yesmaximal, complemented
7#17986039C211yesmaximal, characteristic
8#17986022C211yesmaximal, characteristic
9#17986009C211yesmaximal, characteristic
10#17986007C211yesmaximal, characteristic
11#428×C71C211yesmaximal, characteristic
12C22×Dic142C2211yescharacteristic
13C2×#11765849C2211yescharacteristic
14#14556127C2211yescharacteristic
15#14556127C2211yescharacteristic
16#14556119C2211yescharacteristic
17#14556119C2211yescharacteristic
18#14556105—21no—
19#14556105—21no—
20C4×Dic142—21no—
21C4×Dic142—21no—
22#14556093—21no—
23#14556093—21no—
24#14556093C2212yes—
25#14556093C2212yes—
26#14556087C2211yescharacteristic
27C2×Dic71⋊C4—21no—
28C2×C4×Dic71—21no—
29C2×C4×Dic71C2211yescharacteristic
30#14556061C2212yes—
31#14556059C2212yes—
32Dic71⋊C8C2212yes—
33Dic71⋊C8C2212yes—
34#14556053C2212yes—
35#14556053C2212yes—
36#14556051C2212yes—
37#14556051C2212yes—
38C2×M16×C71C2211yescharacteristic
39#117×C71C2211yescharacteristic
40C2×C4⋊C4×C71C2211yescharacteristic
41#107×C71C2212yes—
42#106×C71C2212yes—
43C2×Dic142—21no—
44C2×Dic142—22no—
45C2×Dic142—21no—
46C2×Dic142—21no—
47C2×Dic142Q814yescomplemented
48C2×Dic142C2312yes—
49C22×Dic71—21no—
50C22×Dic71D411yescharacteristic
51C22×Dic71D411yescharacteristic
52#11765849—22no—
53C2×C71⋊C8C2312yes—
54C4⋊Dic71—21no—
55C4⋊Dic71—21no—
56C4⋊Dic71C2311yescharacteristic
57C4⋊Dic71C2311yescharacteristic
58Dic71⋊C4—22no—
59Dic71⋊C4—22no—
60Dic71⋊C4—21no—
61Dic71⋊C4—21no—
62Dic71⋊C4—21no—
63Dic71⋊C4—21no—
64C22⋊Dic71—21no—
65C4×Dic71—21no—
66C4×Dic71—22no—
67C4×Dic71—22no—
68C4×Dic71D411yescharacteristic
69C4×Dic71D411yescharacteristic
70C4×Dic71C2311yescharacteristic
71C4×Dic71C2311yescharacteristic
72C22×C4×C71—21no—
73C22×C4×C71C2311yescharacteristic
74M16×C71—22no—
75C2×C8×C71C2312yes—
76C4⋊C4×C71—21no—
77C4⋊C4×C71C2311yescharacteristic
78C4⋊C4×C71C2312yes—
79C4⋊C4×C71C2311yescharacteristic
80C22⋊C4×C71—21no—
81C42×C71—21no—
82C23×C71C2×D411yescharacteristic
83C2×C4×C71—21no—
84C2×C4×C71—21no—
85C2×C4×C71—22no—
86C2×C4×C71—42no—
87C2×C4×C71C2×D411yescharacteristic
88C2×C4×C71C2×Q812yes—
89C2×C4×C71#4112yes—
90C2×C4×C71C2411yescharacteristic
91C2×Dic71—24no—
92C2×Dic71—22no—
93C2×Dic71—22no—
94C2×Dic71—21no—
95C2×Dic71—21no—
96C2×Dic71—41no—
97C2×Dic71C2×D411yescharacteristic
98C2×Dic71C2×D411yescharacteristic
99C2×Dic71C2×D411yescharacteristic
100C2×Dic71C2×D411yescharacteristic
101Dic142—24no—
102Dic142—22no—
103Dic142—42no—
104Dic142—42no—
105Dic142C2×Q814yes—
106C8×C71—22no—
107C71⋊C8—22no—
108C22×C71—21no—
109C22×C71—21no—
110C22×C71C22⋊Q812yes—
111C22×C71C22×D411yescharacteristic
112C4×C71—41no—
113C4×C71—41no—
114C4×C71—42no—
115C4×C71C22⋊Q812yes—
116C4×C71C22×Q811yescharacteristic
117C4×C71C2×#4111yescharacteristic
118Dic71—24no—
119Dic71—21no—
120Dic71—21no—
121Dic71—21no—
122Dic71—41no—
123Dic71—41no—
124Dic71#13712yes—
125C2×C71—22no—
126C2×C71C2×C22⋊Q811yescharacteristic
127C2×C71C2×#13711yescharacteristic
128C2×C71#57511yescharacteristic
129#3078—711nomaximal, Sylow
130C71#307811yescharacteristic, complemented, Sylow
131#533—711no—
132C2×C4×Q8—711no—
133#479—712no—
134#479—712no—
135#477—712no—
136#475—712no—
137#428—711no—
138#428D7111yescharacteristic
139#423—711no—
140#419—711no—
141#419—711no—
142C22×Q8—711no—
143C2×M16—711no—
144C2×M16C2×D7111yescharacteristic
145C4⋊Q8—711no—
146C4⋊Q8—711no—
147#125—711no—
148#125—711no—
149C22⋊Q8—1421no—
150C22⋊Q8—1421no—
151C4×Q8—712no—
152C4×Q8—712no—
153C4×Q8—1421no—
154C4×Q8—1421no—
155C4×Q8—1421no—
156C4×Q8—1421no—
157#117—711no—
158#117C2×D7111yescharacteristic
159C2×C4⋊C4—1421no—
160C2×C4⋊C4C2×D7111yescharacteristic
161C2×C42—711no—
162C2×C42—1421no—
163#107—712no—
164#107C2×D7112yes—
165#106—712no—
166#106C2×D7112yes—
167C4⋊C8—712no—
168C4⋊C8—712no—
169#103—712no—
170#103—712no—
171#103—712no—
172#103—712no—
173C2×Q8—714no—
174C2×Q8—712no—
175C2×Q8—1421no—
176C2×Q8—1421no—
177C2×Q8—1421no—
178C2×Q8—1422no—
179C22×C4—21no—
180C22×C4—711no—
181C22×C4—711no—
182C22×C4—1421no—
183C22×C4C22×D7111yescharacteristic
184M16—22no—
185M16—1422no—
186C2×C8—712no—
187C2×C8C22×D7112yes—
188C4⋊C4—21no—
189C4⋊C4—711no—
190C4⋊C4—711no—
191C4⋊C4—1422no—
192C4⋊C4—1422no—
193C4⋊C4—1421no—
194C4⋊C4—1421no—
195C4⋊C4—1421no—
196C4⋊C4—1421no—
197C4⋊C4—1421no—
198C4⋊C4—1421no—
199C4⋊C4C22×D7111yescharacteristic
200C4⋊C4C22×D7111yescharacteristic
201C4⋊C4C22×D7112yes—
202C22⋊C4—21no—
203C22⋊C4—1421no—
204C42—21no—
205C42—711no—
206C42—711no—
207C42—711no—
208C42—711no—
209C42—1422no—
210C42—1422no—
211C42—1421no—
212C23D4×D7111yescharacteristic
213Q8—2842no—
214Q8—2842no—
215Q8—714no—
216Q8—1422no—
217Q8—1424no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—42no—
222C2×C4—2841no—
223C2×C4—711no—
224C2×C4—711no—
225C2×C4—711no—
226C2×C4—711no—
227C2×C4—1421no—
228C2×C4—1421no—
229C2×C4—1422no—
230C2×C4—1422no—
231C2×C4—1424no—
232C2×C4D4×D7111yescharacteristic
233C2×C4Q8×D7112yes—
234C2×C4#1176586512yes—
235C2×C4C23×D7111yescharacteristic
236C8—22no—
237C8—1422no—
238C22—21no—
239C22—21no—
240C22#1455610612yes—
241C22C2×D4×D7111yescharacteristic
242C4—41no—
243C4—42no—
244C4—41no—
245C4—2841no—
246C4—2841no—
247C4—712no—
248C4—1424no—
249C4—1421no—
250C4—1421no—
251C4—1421no—
252C4#1455610612yes—
253C4C2×Q8×D7111yescharacteristic
254C4C2×#1176586511yescharacteristic
255C2—22no—
256C2C2×#1455610611yescharacteristic
257C2#137×D7111yescharacteristic
258C2#1798679511yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#428×C71C2
socleC22×C71C22×D4
hypercenter#428D71
hypocenterC71#3078
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
3. #17986309C2semidirect
4. #17986305C2semidirect
5. #17986293C2semidirect
6. #17986283C2semidirect
47. C2×Dic142Q8semidirect
130. C71#3078semidirect