Finite Group Databasebeta

← #22599693

Subgroups — #22599693

Order 9088.

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

normal subgroups only

This group has 360 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
1#17986535C212yesmaximal, complemented
2#17986308C212yesmaximal, complemented
3#17986302C212yesmaximal, complemented
4#17986235C211yesmaximal, characteristic
5#17986230C212yesmaximal, complemented
6#17986223C212yesmaximal, complemented
7#17986217C212yesmaximal, complemented
8C2×Dic71⋊C8C211yesmaximal, characteristic, complemented
9#467×C71C211yesmaximal, characteristic, complemented
10C2×#11765852C2212yes—
11C2×D568C2211yescharacteristic
12C22×C71⋊C8C2211yescharacteristic
13#14556123—22no—
14#14556119C2212yes—
15#14556117C2212yes—
16#14556112—22no—
17#14556102C2212yes—
18Q8×Dic71C2212yes—
19#14556090C2212yes—
20(C22⋊D71)⋊C4—22no—
21#14556086—22no—
22C2×C4×Dic71C2211yescharacteristic
23#14556062C2211yescharacteristic
24#14556059C2212yes—
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×D284C2312yes—
36C2×C22⋊D71—22no—
37C22×Dic71D411yescharacteristic
38C22×Dic71D411yescharacteristic
39C2×C4×D71—22no—
40#11765852—42no—
41D568—41no—
42C2×C71⋊C8—41no—
43C2×C71⋊C8C2312yes—
44C4⋊Dic71—22no—
45C4⋊Dic71—22no—
46C4⋊Dic71C2312yes—
47Dic71⋊C4—22no—
48(C2×D71)⋊C4—22no—
49(C2×D71)⋊C4—22no—
50(C2×D71)⋊C4—22no—
51(C2×D71)⋊C4—22no—
52C4×Dic71—22no—
53C4×Dic71D412yes—
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—44no—
75C2×Dic71—22no—
76C2×Dic71—22no—
77C2×Dic71—41no—
78C2×Dic71C2×D412yes—
79C2×Dic71C2×D412yes—
80D284—22no—
81D284—22no—
82C4×D71—44no—
83C8×C71—21no—
84C8×C71—21no—
85C71⋊C8—22no—
86C22×C71—21no—
87C22×C71—21no—
88C22×C71#13512yes—
89C22×C71C22×D411yescharacteristic
90C2×D71—42no—
91C2×D71—42no—
92C2×D71—42no—
93C4×C71—21no—
94C4×C71—42no—
95C4×C71—42no—
96C4×C71H(5,𝔽2)11yescharacteristic
97C4×C712−1+411yescharacteristic
98Dic71—22no—
99Dic71—22no—
100Dic71—42no—
101C2×C71—22no—
102C2×C71#53711yescharacteristic
103C2×C71C2×#13511yescharacteristic
104C2×C71#57511yescharacteristic
105D71—82no—
106#3250—711nomaximal, Sylow
107C71#325011yescharacteristic, complemented, Sylow
108#529—712no—
109#479—712no—
110#478—712no—
111#467D7111yescharacteristic, complemented
112#466—711no—
113#465—712no—
114#464—712no—
115#463—712no—
116C2×C4⋊C8—711no—
117C2×Q16C2×D7111yescharacteristic
118C2×SD16—712no—
119C2×D8—711no—
120C22×C8—711no—
121C22×C8C2×D7111yescharacteristic
122#126—1422no—
123#125—712no—
124#124—712no—
125#123—1422no—
126C22⋊Q8C2×D7112yes—
127#121—712no—
128C4×Q8—712no—
129C4×D4—712no—
130C4×D4—1422no—
131#117—1422no—
132C2×C42—711no—
133#107—711no—
134#107C2×D7111yescharacteristic
135#106—712no—
136C4⋊C8—712no—
137C4⋊C8—712no—
138#103—712no—
139#103C2×D7112yes—
140#102—712no—
141#102—712no—
142C2×Q8C22×D7112yes—
143C2×D4—712no—
144C2×D4—1422no—
145C22×C4—711no—
146C22×C4—711no—
147C22×C4—1422no—
148C22×C4C22×D7111yescharacteristic
149Q16—41no—
150SD16—2842no—
151D8—2841no—
152C2×C8—41no—
153C2×C8—2841no—
154C2×C8—712no—
155C2×C8C22×D7111yescharacteristic
156C2×C8C22×D7111yescharacteristic
157C4⋊C4—22no—
158C4⋊C4—712no—
159C4⋊C4—1422no—
160C4⋊C4—1422no—
161C4⋊C4—1422no—
162C4⋊C4C22×D7112yes—
163C22⋊C4—22no—
164C22⋊C4—1422no—
165C22⋊C4—1422no—
166C22⋊C4—1422no—
167C22⋊C4—1422no—
168C42—712no—
169C42—712no—
170C42—1422no—
171C23—1422no—
172C23D4×D7111yescharacteristic
173Q8—22no—
174Q8—22no—
175D4—2844no—
176D4—1422no—
177D4—1422no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—2844no—
183C2×C4—2841no—
184C2×C4—712no—
185C2×C4—712no—
186C2×C4—1422no—
187C2×C4—1422no—
188C2×C4D4×D7111yescharacteristic
189C2×C4C23×D7111yescharacteristic
190C8—21no—
191C8—21no—
192C8—1422no—
193C22—21no—
194C22—21no—
195C22—2842no—
196C22—2842no—
197C22—2842no—
198C22#1455615112yes—
199C22C2×D4×D7111yescharacteristic
200C4—21no—
201C4—42no—
202C4—42no—
203C4—2842no—
204C4—1422no—
205C4—1422no—
206C4#1455617311yescharacteristic
207C4#1455617511yescharacteristic
208C2—22no—
209C2—5682no—
210C2#1798657611yescharacteristic
211C2C2×#1455615111yescharacteristic
212C2#1798679611yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#467×C71C2
socleC22×C71C22×D4
hypercenter#467D71
hypocenterC71#3250
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. #17986535C2semidirect
2. #17986308C2semidirect
3. #17986302C2semidirect
5. #17986230C2semidirect
6. #17986223C2semidirect
7. #17986217C2semidirect
8. C2×Dic71⋊C8C2semidirect
9. #467×C71C2semidirect
107. C71#3250semidirect
111. #467D71semidirect