Finite Group Databasebeta

← #22600913

Subgroups — #22600913

Order 9088.

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

normal subgroups only

This group has 336 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#17986687C212yesmaximal
2#17986342C212yesmaximal, complemented
3#17986336C212yesmaximal, complemented
4#17986303C212yesmaximal, complemented
5#17986258C212yesmaximal, complemented
6C4×Dic284C211yesmaximal, characteristic, complemented
7#17986078C211yesmaximal, characteristic
8#17986038C212yesmaximal, complemented
9#499×C71C211yesmaximal, characteristic, complemented
10C2×Dic284C2211yescharacteristic
11C2×#11765848C2212yes—
12#14556123—22no—
13#14556119—22no—
14#14556113—22no—
15#14556108C2212yes—
16#14556108C2212yes—
17C4×Dic142C2212yes—
18C4×D284C2211yescharacteristic
19#14556086—22no—
20#14556086C2212yes—
21C4⋊C4×D71C2212yes—
22#14556062C2211yescharacteristic
23#14556061C2212yes—
24#14556059C2212yes—
25C4⋊(C71⋊C8)C2211yescharacteristic
26#14556053C2212yes—
27#14556051C2212yes—
28#14556051C2212yes—
29(C2×D71)⋊C8C2212yes—
30#125×C71C2212yes—
31#107×C71C2211yescharacteristic
32#107×C71C2211yescharacteristic
33#106×C71C2212yes—
34C4×C8×C71C2211yescharacteristic
35C2×Dic142C2312yes—
36C2×D284D411yescharacteristic
37C2×C4×D71—22no—
38C2×C4×D71C2312yes—
39Dic284—22no—
40#11765848—24no—
41C2×C71⋊C8C2312yes—
42C4⋊Dic71—22no—
43C4⋊Dic71—22no—
44C4⋊Dic71D411yescharacteristic
45C4⋊Dic71C2312yes—
46Dic71⋊C4—22no—
47Dic71⋊C4—22no—
48(C2×D71)⋊C4—22no—
49(C2×D71)⋊C4—22no—
50(C2×D71)⋊C4—22no—
51(C2×D71)⋊C4D412yes—
52C4×Dic71—22no—
53C4×Dic71—22no—
54C2×C8×C71C2311yescharacteristic
55C2×C8×C71C2311yescharacteristic
56C4⋊C4×C71—22no—
57C4⋊C4×C71—22no—
58C4⋊C4×C71C2312yes—
59C4⋊C4×C71C2312yes—
60C42×C71C2311yescharacteristic
61C22×D71C2×D412yes—
62C2×C4×C71—22no—
63C2×C4×C71—22no—
64C2×C4×C71C2×D411yescharacteristic
65C2×C4×C71C2×D411yescharacteristic
66C2×C4×C71C2411yescharacteristic
67C2×Dic71—22no—
68C2×Dic71—22no—
69C2×Dic71C2×D412yes—
70D284—41no—
71C4×D71—22no—
72C4×D71—22no—
73C4×D71—44no—
74Dic142—22no—
75Dic142—22no—
76C8×C71—21no—
77C8×C71#4112yes—
78C71⋊C8—22no—
79C22×C71C22×D411yescharacteristic
80C2×D71—22no—
81C2×D71—22no—
82C2×D71—22no—
83C4×C71—21no—
84C4×C71—21no—
85C4×C71—42no—
86C4×C71—42no—
87C4×C71C2×#4111yescharacteristic
88C4×C712−1+411yescharacteristic
89Dic71—22no—
90Dic71—42no—
91Dic71—42no—
92C2×C71#54711yescharacteristic
93C2×C71#57511yescharacteristic
94C2×C71#57811yescharacteristic
95D71—42no—
96#3352—711nomaximal, Sylow
97C71#335211yescharacteristic, complemented, Sylow
98#552—712no—
99#499D7111yescharacteristic, complemented
100#485—712no—
101#484—712no—
102#479—712no—
103#470—712no—
104C4×Q16—711no—
105#436—711no—
106#428—712no—
107C2×Q16—711no—
108C2×M16—712no—
109#126—1422no—
110#125—1422no—
111#125C2×D7112yes—
112#123—1422no—
113C22⋊Q8—712no—
114C22⋊Q8—712no—
115C4×Q8—712no—
116C4×D4—711no—
117#117—712no—
118#117—1422no—
119C2×C4⋊C4—712no—
120#107—711no—
121#107—712no—
122#107C2×D7111yescharacteristic
123#107C2×D7111yescharacteristic
124#106—712no—
125#106C2×D7112yes—
126C4⋊C8—711no—
127#103—712no—
128#103—712no—
129#103—712no—
130C22⋊C8—712no—
131C4×C8C2×D7111yescharacteristic
132C2×Q8—712no—
133C2×D4—711no—
134C22×C4—712no—
135C22×C4—1422no—
136Q16—1422no—
137M16—1424no—
138C2×C8—712no—
139C2×C8C22×D7111yescharacteristic
140C2×C8C22×D7111yescharacteristic
141C4⋊C4—22no—
142C4⋊C4—22no—
143C4⋊C4—711no—
144C4⋊C4—712no—
145C4⋊C4—1422no—
146C4⋊C4—1422no—
147C4⋊C4—1422no—
148C4⋊C4—1422no—
149C4⋊C4C22×D7112yes—
150C4⋊C4C22×D7112yes—
151C22⋊C4—712no—
152C22⋊C4—1422no—
153C22⋊C4—1422no—
154C22⋊C4—1422no—
155C42—1422no—
156C42—1422no—
157C42C22×D7111yescharacteristic
158C23—712no—
159Q8—1422no—
160Q8—1422no—
161D4—2841no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—2844no—
165C2×C4—712no—
166C2×C4—1422no—
167C2×C4—1422no—
168C2×C4—1422no—
169C2×C4—1422no—
170C2×C4D4×D7111yescharacteristic
171C2×C4D4×D7111yescharacteristic
172C2×C4C23×D7111yescharacteristic
173C8—21no—
174C8—1422no—
175C8#1176586612yes—
176C22—1422no—
177C22—1422no—
178C22—1422no—
179C22C2×D4×D7111yescharacteristic
180C4—21no—
181C4—21no—
182C4—42no—
183C4—42no—
184C4—2842no—
185C4—2842no—
186C4—1422no—
187C4C2×#1176586611yescharacteristic
188C4#1455617611yescharacteristic
189C2—2842no—
190C2#1798665211yescharacteristic
191C2#1798679611yescharacteristic
192C2#1798681311yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#499×C71C2
socleC22×C71C22×D4
hypercenter#499D71
hypocenterC71#3352
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #499

Complements

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

NH ≅ G/Ntype
2. #17986342C2semidirect
3. #17986336C2semidirect
4. #17986303C2semidirect
5. #17986258C2semidirect
6. C4×Dic284C2semidirect
8. #17986038C2semidirect
9. #499×C71C2semidirect
97. C71#3352semidirect
99. #499D71semidirect