Finite Group Databasebeta

← #22599802

Subgroups — #22599802

Order 9088.

346 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 109 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#17986622C212yesmaximal, complemented
2#17986292C212yesmaximal, complemented
3#17986278C212yesmaximal, complemented
4#17986247C212yesmaximal, complemented
5#17986235C211yesmaximal, characteristic
6#17986223C212yesmaximal, complemented
7#17986217C212yesmaximal, complemented
8#17986025C211yesmaximal, characteristic, complemented
9#467×C71C211yesmaximal, characteristic, complemented
10C2×#11765852C2212yes—
11C2×D568C2211yescharacteristic
12C2×#11765849C2211yescharacteristic
13#14556127C2212yes—
14#14556123—22no—
15#14556117C2212yes—
16#14556114—22no—
17#14556110—22no—
18#14556108—22no—
19#14556102C2212yes—
20Q8×Dic71C2212yes—
21#14556090C2212yes—
22#14556087C2211yescharacteristic
23#14556062C2211yescharacteristic
24#14556061C2212yes—
25Dic71⋊C8C2212yes—
26Dic71⋊C8C2212yes—
27#14556052C2212yes—
28#14556050C2212yes—
29#14556048C2212yes—
30C2×Q16×C71C2211yescharacteristic
31C22×C8×C71C2211yescharacteristic
32C22⋊Q8×C71C2212yes—
33#107×C71C2211yescharacteristic
34#103×C71C2212yes—
35C2×Dic142—22no—
36C2×D284C2312yes—
37C2×C22⋊D71—22no—
38C2×C4×D71—22no—
39#11765852—42no—
40D568—41no—
41#11765849—41no—
42C2×C71⋊C8C2312yes—
43C4⋊Dic71—22no—
44C4⋊Dic71D411yescharacteristic
45C4⋊Dic71D411yescharacteristic
46C4⋊Dic71C2312yes—
47Dic71⋊C4—22no—
48C22⋊Dic71D412yes—
49(C2×D71)⋊C4—22no—
50(C2×D71)⋊C4—22no—
51(C2×D71)⋊C4—22no—
52(C2×D71)⋊C4—22no—
53C4×Dic71—22no—
54C4×Dic71C2312yes—
55C2×Q8×C71C2312yes—
56C22×C4×C71C2311yescharacteristic
57Q16×C71—41no—
58C2×C8×C71—41no—
59C2×C8×C71C2311yescharacteristic
60C2×C8×C71C2311yescharacteristic
61C4⋊C4×C71—22no—
62C4⋊C4×C71C2312yes—
63C22⋊C4×C71—22no—
64C23×C71C2×D411yescharacteristic
65C22×D71—22no—
66Q8×C71—22no—
67Q8×C71—22no—
68C2×C4×C71—21no—
69C2×C4×C71—21no—
70C2×C4×C71—22no—
71C2×C4×C71—22no—
72C2×C4×C71C2×D411yescharacteristic
73C2×C4×C71C2411yescharacteristic
74C22⋊D71—82no—
75C2×Dic71—22no—
76C2×Dic71C2×D412yes—
77C2×Dic71C2×D412yes—
78D284—22no—
79D284—22no—
80C4×D71—82no—
81Dic142—82no—
82C8×C71—21no—
83C8×C71—21no—
84C71⋊C8—22no—
85C22×C71—21no—
86C22×C71—21no—
87C22×C71—21no—
88C22×C71C22×D411yescharacteristic
89C2×D71—42no—
90C2×D71—42no—
91C2×D71—42no—
92C4×C71—21no—
93C4×C71—42no—
94C4×C71—42no—
95C4×C71H(5,𝔽2)11yescharacteristic
96C4×C712−1+411yescharacteristic
97Dic71—22no—
98Dic71—22no—
99Dic71—42no—
100C2×C71—41no—
101C2×C71#53711yescharacteristic
102C2×C71#57611yescharacteristic
103C2×C71#57811yescharacteristic
104D71—82no—
105#3262—711nomaximal, Sylow
106C71#326211yescharacteristic, complemented, Sylow
107#542—712no—
108#477—712no—
109#474—712no—
110#468—712no—
111#467D7111yescharacteristic, complemented
112#466—711no—
113#464—712no—
114#463—712no—
115#424—711no—
116C2×Q16C2×D7111yescharacteristic
117C2×SD16—712no—
118C2×D8—711no—
119C2×M16—711no—
120C22×C8C2×D7111yescharacteristic
121C4⋊Q8—712no—
122#126—1422no—
123#124—712no—
124#124—1422no—
125#123—1422no—
126C22⋊Q8—1422no—
127C22⋊Q8C2×D7112yes—
128#121—712no—
129C4×Q8—712no—
130C4×D4—712no—
131#117—711no—
132#107—711no—
133#107—712no—
134#107C2×D7111yescharacteristic
135C4⋊C8—712no—
136C4⋊C8—712no—
137#103—712no—
138#103C2×D7112yes—
139#102—712no—
140#102—712no—
141C2×Q8—1422no—
142C2×Q8C22×D7112yes—
143C2×D4—712no—
144C2×D4—1422no—
145C22×C4—1422no—
146C22×C4C22×D7111yescharacteristic
147Q16—41no—
148SD16—2842no—
149D8—2841no—
150M16—2841no—
151C2×C8—41no—
152C2×C8—712no—
153C2×C8C22×D7111yescharacteristic
154C2×C8C22×D7111yescharacteristic
155C4⋊C4—22no—
156C4⋊C4—711no—
157C4⋊C4—711no—
158C4⋊C4—712no—
159C4⋊C4—1422no—
160C4⋊C4—1422no—
161C4⋊C4C22×D7112yes—
162C22⋊C4—22no—
163C22⋊C4—712no—
164C22⋊C4—1422no—
165C22⋊C4—1422no—
166C22⋊C4—1422no—
167C22⋊C4—1422no—
168C42—712no—
169C42—1422no—
170C23—1422no—
171C23D4×D7111yescharacteristic
172Q8—22no—
173Q8—22no—
174Q8—5682no—
175D4—5682no—
176D4—1422no—
177D4—1422no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—5682no—
183C2×C4—712no—
184C2×C4—712no—
185C2×C4—1422no—
186C2×C4D4×D7111yescharacteristic
187C2×C4C23×D7111yescharacteristic
188C8—21no—
189C8—21no—
190C8—1422no—
191C22—21no—
192C22—21no—
193C22—21no—
194C22—2842no—
195C22—2842no—
196C22—2842no—
197C22C2×D4×D7111yescharacteristic
198C4—21no—
199C4—42no—
200C4—42no—
201C4—2842no—
202C4—1422no—
203C4—1422no—
204C4#1455617311yescharacteristic
205C4#1455617511yescharacteristic
206C2—41no—
207C2—5682no—
208C2#1798657611yescharacteristic
209C2#1798680111yescharacteristic
210C2#1798681211yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#467×C71C2
socleC22×C71C22×D4
hypercenter#467D71
hypocenterC71#3262
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
1. #17986622C2semidirect
2. #17986292C2semidirect
3. #17986278C2semidirect
4. #17986247C2semidirect
6. #17986223C2semidirect
7. #17986217C2semidirect
8. #17986025C2semidirect
9. #467×C71C2semidirect
106. C71#3262semidirect
111. #467D71semidirect