Finite Group Databasebeta

← #22601153

Subgroups — #22601153

Order 9088.

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

normal subgroups only

This group has 350 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#17986734C211yesmaximal, characteristic
2#17986729C211yesmaximal, characteristic, complemented
3#17986360C212yesmaximal, complemented
4#17986347C211yesmaximal, characteristic
5#17986278C212yesmaximal, complemented
6#17986224C212yesmaximal, complemented
7#17986200C211yesmaximal, characteristic
8#17986126C211yesmaximal, characteristic, complemented
9#17986098C212yesmaximal, complemented
10C4×#11765855C211yesmaximal, characteristic, complemented
11#477×C71C211yesmaximal, characteristic, complemented
12C2×#11765855C2211yescharacteristic
13C2×#11765854C2212yes—
14#14556127—21no—
15#14556127C2212yes—
16#14556126—22no—
17#14556125C2211yescharacteristic
18#14556117C2212yes—
19#14556116—21no—
20#14556113—22no—
21C4×Dic142C2211yescharacteristic
22C4×Dic142C2211yescharacteristic
23#14556093—21no—
24#14556093C2212yes—
25C4×D284—21no—
26#14556090C2212yes—
27#14556061C2211yescharacteristic
28Dic71⋊C8C2212yes—
29#14556052C2211yescharacteristic
30#14556051C2211yescharacteristic
31#14556050C2212yes—
32#14556048C2212yes—
33#14556048C2212yes—
34#14556038C2212yes—
35C4×C71⋊C8C2211yescharacteristic
36C4⋊Q8×C71C2211yescharacteristic
37C4×Q8×C71C2211yescharacteristic
38#107×C71C2212yes—
39C4⋊C8×C71C2211yescharacteristic
40#103×C71C2212yes—
41C2×Dic142—22no—
42C2×Dic142D411yescharacteristic
43C2×Dic142C2311yescharacteristic
44C2×D284—21no—
45C2×D284—21no—
46C2×D284C2312yes—
47C2×C4×D71—22no—
48#11765855—22no—
49#11765854—24no—
50C2×C71⋊C8C2311yescharacteristic
51C2×C71⋊C8C2311yescharacteristic
52C4⋊Dic71D411yescharacteristic
53C4⋊Dic71C2311yescharacteristic
54Dic71⋊C4—21no—
55Dic71⋊C4—22no—
56Dic71⋊C4—22no—
57Dic71⋊C4D412yes—
58(C2×D71)⋊C4—22no—
59(C2×D71)⋊C4—22no—
60(C2×D71)⋊C4—22no—
61C4×Dic71—21no—
62C4×Dic71—22no—
63C4×Dic71C2312yes—
64C2×Q8×C71—21no—
65C2×Q8×C71C2311yescharacteristic
66C2×C8×C71C2312yes—
67C4⋊C4×C71—21no—
68C4⋊C4×C71—21no—
69C4⋊C4×C71C2312yes—
70C4⋊C4×C71C2311yescharacteristic
71C42×C71—21no—
72C42×C71C2311yescharacteristic
73C22×D71—22no—
74Q8×C71—21no—
75Q8×C71—21no—
76Q8×C71—81no—
77C2×C4×C71—21no—
78C2×C4×C71—21no—
79C2×C4×C71—22no—
80C2×C4×C71C2×D411yescharacteristic
81C2×C4×C71C2×D411yescharacteristic
82C2×C4×C71C2411yescharacteristic
83C2×Dic71—21no—
84C2×Dic71—21no—
85C2×Dic71C2×D412yes—
86C2×Dic71C2×D412yes—
87D284—22no—
88D284—22no—
89D284—42no—
90D284—81no—
91C4×D71—82no—
92Dic142—22no—
93Dic142—21no—
94Dic142—82no—
95Dic142#4112yes—
96C8×C71—22no—
97C71⋊C8—21no—
98C71⋊C8—21no—
99C22×C71C22×D411yescharacteristic
100C2×D71—42no—
101C2×D71—42no—
102C2×D71—42no—
103C4×C71—21no—
104C4×C71—41no—
105C4×C71—42no—
106C4×C71—41no—
107C4×C71#13712yes—
108C4×C71C2×#4111yescharacteristic
109C4×C712−1+411yescharacteristic
110Dic71—22no—
111Dic71—22no—
112Dic71—22no—
113Dic71—41no—
114C2×C71#54811yescharacteristic
115C2×C71C2×#13711yescharacteristic
116C2×C71#57611yescharacteristic
117D71—82no—
118#3375—711nomaximal, Sylow
119C71#337511yescharacteristic, complemented, Sylow
120#559—711no—
121Q82—711no—
122#489—712no—
123#486—711no—
124#477D7111yescharacteristic, complemented
125#474—712no—
126#464—712no—
127#460—711no—
128#446—711no—
129#440—712no—
130C4×Q16—711no—
131C2×Q16—711no—
132C2×SD16—712no—
133C4⋊Q8—712no—
134C4⋊Q8—1421no—
135C4⋊Q8—1422no—
136C4⋊Q8C2×D7111yescharacteristic
137#127—711no—
138#124—712no—
139#124—1421no—
140#123—1422no—
141C4×Q8—711no—
142C4×Q8—712no—
143C4×Q8—711no—
144C4×Q8—1421no—
145C4×Q8C2×D7111yescharacteristic
146C4×D4—712no—
147C4×D4—1421no—
148#107—711no—
149#107C2×D7112yes—
150C4⋊C8—712no—
151C4⋊C8C2×D7111yescharacteristic
152#103—711no—
153#103—711no—
154#103C2×D7112yes—
155#102—712no—
156#102—712no—
157#102—712no—
158#97—712no—
159C4×C8—711no—
160C2×Q8—21no—
161C2×Q8—711no—
162C2×Q8—711no—
163C2×Q8—1422no—
164C2×Q8C22×D7111yescharacteristic
165C2×D4—712no—
166C2×D4—1421no—
167C2×D4—1421no—
168C22×C4—1422no—
169Q16—1422no—
170SD16—1424no—
171C2×C8—711no—
172C2×C8—711no—
173C2×C8C22×D7112yes—
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—712no—
177C4⋊C4—711no—
178C4⋊C4—711no—
179C4⋊C4—1422no—
180C4⋊C4—1422no—
181C4⋊C4—1421no—
182C4⋊C4C22×D7112yes—
183C4⋊C4C22×D7111yescharacteristic
184C22⋊C4—1422no—
185C22⋊C4—1422no—
186C22⋊C4—1422no—
187C42—21no—
188C42—712no—
189C42—1422no—
190C42—1421no—
191C42C22×D7111yescharacteristic
192C23—1422no—
193Q8—21no—
194Q8—21no—
195Q8—81no—
196Q8—5682no—
197Q8—712no—
198Q8—1422no—
199Q8—1421no—
200D4—2842no—
201D4—5681no—
202D4—1422no—
203D4—1422no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—5682no—
208C2×C4—712no—
209C2×C4—712no—
210C2×C4—1421no—
211C2×C4—1421no—
212C2×C4D4×D7111yescharacteristic
213C2×C4D4×D7111yescharacteristic
214C2×C4C23×D7111yescharacteristic
215C8—22no—
216C8—1421no—
217C8—1421no—
218C22—2842no—
219C22—2842no—
220C22—2842no—
221C22C2×D4×D7111yescharacteristic
222C4—21no—
223C4—41no—
224C4—41no—
225C4—42no—
226C4—2841no—
227C4—1422no—
228C4—1422no—
229C4—1422no—
230C4#1455616012yes—
231C4#41×D7111yescharacteristic
232C4#1455617611yescharacteristic
233C2—5682no—
234C2#1798665911yescharacteristic
235C2C2×#1455616011yescharacteristic
236C2#1798680311yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#477×C71C2
socleC22×C71C22×D4
hypercenter#477D71
hypocenterC71#3375
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #477

Complements

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

NH ≅ G/Ntype
2. #17986729C2semidirect
3. #17986360C2semidirect
5. #17986278C2semidirect
6. #17986224C2semidirect
8. #17986126C2semidirect
9. #17986098C2semidirect
10. C4×#11765855C2semidirect
11. #477×C71C2semidirect
119. C71#3375semidirect
124. #477D71semidirect