Finite Group Databasebeta

← #22599571

Subgroups — #22599571

Order 9088.

362 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 113 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
1C4⋊Q8×D71C211yesmaximal, characteristic, complemented
2#17986534C211yesmaximal, characteristic
3#17986409C211yesmaximal, characteristic, complemented
4#17986361C211yesmaximal, characteristic, complemented
5#17986359C211yesmaximal, characteristic, complemented
6#17986336C212yesmaximal, complemented
7#17986319C212yesmaximal
8#17986283C212yesmaximal, complemented
9#17986278C212yesmaximal, complemented
10#17986054C211yesmaximal, characteristic, complemented
11#501×C71C211yesmaximal, characteristic, complemented
12C2×Q8×D71—21no—
13#14556128C2211yescharacteristic
14#14556127—21no—
15#14556127C2212yes—
16#14556126—21no—
17#14556123—21no—
18#14556116C2211yescharacteristic
19#14556113—21no—
20#14556108C2211yescharacteristic
21#14556103C2211yescharacteristic
22#14556093C2211yescharacteristic
23#14556090—21no—
24#14556090C2211yescharacteristic
25#14556086—21no—
26C4⋊C4×D71—21no—
27C4⋊C4×D71C2212yes—
28C42×D71C2211yescharacteristic
29#14556061C2212yes—
30#14556059C2212yes—
31Dic71⋊C8C2212yes—
32#14556053C2212yes—
33#14556051C2212yes—
34#14556050C2212yes—
35#14556048C2212yes—
36(C2×D71)⋊C8C2212yes—
37#14556039C2211yescharacteristic
38C4⋊Q8×C71C2211yescharacteristic
39#125×C71C2211yescharacteristic
40#107×C71C2212yes—
41#106×C71C2212yes—
42#97×C71C2211yescharacteristic
43C2×Dic142—21no—
44C2×Dic142—22no—
45C2×Dic142C2311yescharacteristic
46Q8×D71—44no—
47C2×D284—21no—
48C2×D284C2311yescharacteristic
49C2×C4×D71—22no—
50C2×C4×D71—21no—
51C2×C4×D71D411yescharacteristic
52C2×C4×D71D411yescharacteristic
53C2×C4×D71C2311yescharacteristic
54C2×C71⋊C8C2312yes—
55C4⋊Dic71—21no—
56C4⋊Dic71—21no—
57C4⋊Dic71C2312yes—
58Dic71⋊C4—22no—
59Dic71⋊C4—22no—
60Dic71⋊C4—21no—
61(C2×D71)⋊C4—21no—
62(C2×D71)⋊C4—21no—
63(C2×D71)⋊C4—21no—
64(C2×D71)⋊C4—21no—
65(C2×D71)⋊C4—21no—
66C4×Dic71—21no—
67C4×Dic71D411yescharacteristic
68C4×Dic71D411yescharacteristic
69C4×Dic71C2311yescharacteristic
70C2×Q8×C71—21no—
71C2×C8×C71C2312yes—
72C4⋊C4×C71—21no—
73C4⋊C4×C71—21no—
74C4⋊C4×C71—21no—
75C4⋊C4×C71C2311yescharacteristic
76C4⋊C4×C71C2312yes—
77C4⋊C4×C71C2311yescharacteristic
78C42×C71C2311yescharacteristic
79C22×D71—21no—
80C22×D71C2×D411yescharacteristic
81Q8×C71—42no—
82C2×C4×C71—21no—
83C2×C4×C71—21no—
84C2×C4×C71—22no—
85C2×C4×C71C2×D411yescharacteristic
86C2×C4×C71C2×D411yescharacteristic
87C2×C4×C71C2411yescharacteristic
88C2×Dic71—21no—
89C2×Dic71—22no—
90C2×Dic71C2×D411yescharacteristic
91C2×Dic71C2×D411yescharacteristic
92C2×Dic71C2×D411yescharacteristic
93D284—22no—
94D284—42no—
95C4×D71—22no—
96C4×D71—22no—
97C4×D71—22no—
98C4×D71—44no—
99C4×D71—42no—
100Dic142—22no—
101Dic142—42no—
102Dic142—44no—
103C8×C71—22no—
104C71⋊C8—22no—
105C22×C71C22×D411yescharacteristic
106C2×D71—21no—
107C2×D71—21no—
108C2×D71—41no—
109C2×D71—41no—
110C2×D71—41no—
111C2×D71#13712yes—
112C4×C71—21no—
113C4×C71—21no—
114C4×C71—42no—
115C4×C71—41no—
116C4×C71—41no—
117C4×C712−1+412yes—
118Dic71—21no—
119Dic71—21no—
120Dic71—41no—
121Dic71—42no—
122Dic71AGL(1,ℤ/8)12yes—
123C2×C71#53611yescharacteristic
124C2×C71C2×AGL(1,ℤ/8)11yescharacteristic
125C2×C71C2×#13711yescharacteristic
126D71—22no—
127D71—81no—
128#3239—711nomaximal, Sylow
129C71#323911yescharacteristic, complemented, Sylow
130C2×C4⋊Q8—711no—
131#529—711no—
132#501—711no—
133#501D7111yescharacteristic, complemented
134#489—711no—
135#489—711no—
136#484—712no—
137#481—712no—
138#475—712no—
139#474—712no—
140#432—711no—
141C22×Q8—1421no—
142C4⋊Q8—712no—
143C4⋊Q8—711no—
144C4⋊Q8—1421no—
145C4⋊Q8—1421no—
146C4⋊Q8C2×D7111yescharacteristic
147#126—1421no—
148#125C2×D7111yescharacteristic
149#124—711no—
150#123—1421no—
151C22⋊Q8—711no—
152#121—711no—
153C4×Q8—711no—
154C4×D4—711no—
155C4×D4—1421no—
156#117—1421no—
157C2×C4⋊C4—712no—
158C2×C4⋊C4—1421no—
159C2×C42—711no—
160#107—712no—
161#107C2×D7112yes—
162#106—712no—
163#106C2×D7112yes—
164C4⋊C8—712no—
165#103—712no—
166#103—712no—
167#102—712no—
168#102—712no—
169C22⋊C8—712no—
170#97—711no—
171#97C2×D7111yescharacteristic
172C2×Q8—21no—
173C2×Q8—2844no—
174C2×Q8—711no—
175C2×Q8—1422no—
176C2×Q8—1421no—
177C2×D4—711no—
178C2×D4—1421no—
179C22×C4—711no—
180C22×C4—711no—
181C22×C4—711no—
182C22×C4—1421no—
183C22×C4—1422no—
184C2×C8—712no—
185C2×C8C22×D7112yes—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—712no—
190C4⋊C4—1422no—
191C4⋊C4—1422no—
192C4⋊C4—1421no—
193C4⋊C4—1421no—
194C4⋊C4—1421no—
195C4⋊C4C22×D7112yes—
196C4⋊C4C22×D7111yescharacteristic
197C4⋊C4C22×D7111yescharacteristic
198C22⋊C4—1421no—
199C22⋊C4—1421no—
200C22⋊C4—1421no—
201C22⋊C4—1421no—
202C22⋊C4—1421no—
203C42—711no—
204C42—711no—
205C42—711no—
206C42—1421no—
207C42C22×D7111yescharacteristic
208C23—711no—
209C23—1421no—
210Q8—42no—
211Q8—2842no—
212Q8—2844no—
213Q8—1422no—
214D4—2842no—
215D4—1422no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—2844no—
220C2×C4—2842no—
221C2×C4—711no—
222C2×C4—711no—
223C2×C4—711no—
224C2×C4—1422no—
225C2×C4—1422no—
226C2×C4—1421no—
227C2×C4—1422no—
228C2×C4—1422no—
229C2×C4D4×D7111yescharacteristic
230C2×C4D4×D7111yescharacteristic
231C2×C4C23×D7111yescharacteristic
232C8—22no—
233C8—1422no—
234C22—2841no—
235C22—2841no—
236C22—2841no—
237C22—712no—
238C22—1421no—
239C22—1421no—
240C22C2×D4×D7111yescharacteristic
241C4—21no—
242C4—21no—
243C4—42no—
244C4—41no—
245C4—41no—
246C4—2841no—
247C4—2842no—
248C4—712no—
249C4—1421no—
250C4—1421no—
251C4#1455617612yes—
252C2—5681no—
253C2—1422no—
254C2#1798657411yescharacteristic
255C2AGL(1,ℤ/8)×D7111yescharacteristic
256C2#137×D7111yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#501×C71C2
socleC22×C71C22×D4
hypercenter#501D71
hypocenterC71#3239
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #501

Complements

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

NH ≅ G/Ntype
1. C4⋊Q8×D71C2semidirect
3. #17986409C2semidirect
4. #17986361C2semidirect
5. #17986359C2semidirect
6. #17986336C2semidirect
8. #17986283C2semidirect
9. #17986278C2semidirect
10. #17986054C2semidirect
11. #501×C71C2semidirect
129. C71#3239semidirect
133. #501D71semidirect