Finite Group Databasebeta

← #22600944

Subgroups — #22600944

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#17986702C212yesmaximal, complemented
2#17986326C212yesmaximal, complemented
3#17986314C212yesmaximal, complemented
4#17986296C212yesmaximal, complemented
5#17986253C212yesmaximal, complemented
6C4×D568C211yesmaximal, characteristic, complemented
7#17986078C211yesmaximal, characteristic, complemented
8#17986038C212yesmaximal, complemented
9#499×C71C211yesmaximal, characteristic, complemented
10C2×D568C2211yescharacteristic
11C2×#11765848C2212yes—
12#14556123—22no—
13#14556113—22no—
14#14556103—22no—
15#14556103C2212yes—
16#14556103C2212yes—
17C4×D284C2212yes—
18C4×D284C2211yescharacteristic
19#14556090—22no—
20#14556086C2212yes—
21C4⋊C4×D71C2212yes—
22#14556062C2211yescharacteristic
23#14556061C2212yes—
24#14556059C2212yes—
25C4⋊(C71⋊C8)C2211yescharacteristic
26#14556050C2212yes—
27#14556048C2212yes—
28#14556048C2212yes—
29(C2×D71)⋊C8C2212yes—
30#125×C71C2212yes—
31#107×C71C2211yescharacteristic
32#107×C71C2211yescharacteristic
33#106×C71C2212yes—
34C4×C8×C71C2211yescharacteristic
35C2×D284—22no—
36C2×D284—22no—
37C2×D284D411yescharacteristic
38C2×D284C2312yes—
39C2×C4×D71—22no—
40C2×C4×D71—22no—
41C2×C4×D71C2312yes—
42D568—22no—
43#11765848—24no—
44C2×C71⋊C8C2312yes—
45C4⋊Dic71D411yescharacteristic
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—
52(C2×D71)⋊C4D412yes—
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×D71—22no—
62C22×D71C2×D412yes—
63C2×C4×C71—22no—
64C2×C4×C71—22no—
65C2×C4×C71C2×D411yescharacteristic
66C2×C4×C71C2×D411yescharacteristic
67C2×C4×C71C2411yescharacteristic
68C2×Dic71—22no—
69C2×Dic71C2×D412yes—
70D284—22no—
71D284—22no—
72D284—41no—
73D284—44no—
74D284—82no—
75C4×D71—22no—
76C4×D71—22no—
77C4×D71—82no—
78C4×D71—82no—
79C8×C71—21no—
80C8×C71#4112yes—
81C71⋊C8—22no—
82C22×C71C22×D411yescharacteristic
83C2×D71—22no—
84C2×D71—22no—
85C2×D71—22no—
86C2×D71—42no—
87C2×D71—42no—
88C2×D71—42no—
89C4×C71—21no—
90C4×C71—21no—
91C4×C71—42no—
92C4×C71—42no—
93C4×C71C2×#4111yescharacteristic
94C4×C712−1+411yescharacteristic
95Dic71—22no—
96Dic71—42no—
97C2×C71#54711yescharacteristic
98C2×C71#57511yescharacteristic
99C2×C71#57611yescharacteristic
100D71—42no—
101D71—82no—
102#3355—711nomaximal, Sylow
103C71#335511yescharacteristic, complemented, Sylow
104#553—712no—
105#499D7111yescharacteristic, complemented
106#482—712no—
107#480—712no—
108#478—712no—
109#469—712no—
110C4×D8—711no—
111#436—711no—
112#428—712no—
113C2×D8—711no—
114C2×M16—712no—
115#126—1422no—
116#125C2×D7112yes—
117#123—1422no—
118#121—712no—
119#121—712no—
120#121—1422no—
121C4×D4—711no—
122C4×D4—712no—
123C4×D4—1422no—
124#117—712no—
125C2×C4⋊C4—712no—
126#107—711no—
127#107—712no—
128#107C2×D7111yescharacteristic
129#107C2×D7111yescharacteristic
130#106—712no—
131#106C2×D7112yes—
132C4⋊C8—711no—
133#102—712no—
134#102—712no—
135#102—712no—
136C22⋊C8—712no—
137C4×C8C2×D7111yescharacteristic
138C2×D4—711no—
139C2×D4—712no—
140C2×D4—1422no—
141C2×D4—1422no—
142C22×C4—712no—
143C22×C4—1422no—
144C22×C4—1422no—
145D8—1422no—
146M16—1424no—
147C2×C8—712no—
148C2×C8C22×D7111yescharacteristic
149C2×C8C22×D7111yescharacteristic
150C4⋊C4—22no—
151C4⋊C4—22no—
152C4⋊C4—711no—
153C4⋊C4—712no—
154C4⋊C4—1422no—
155C4⋊C4C22×D7112yes—
156C4⋊C4C22×D7112yes—
157C22⋊C4—712no—
158C22⋊C4—1422no—
159C22⋊C4—1422no—
160C22⋊C4—1422no—
161C22⋊C4—1422no—
162C42—1422no—
163C42C22×D7111yescharacteristic
164C23—712no—
165C23—1422no—
166D4—2841no—
167D4—2844no—
168D4—5682no—
169D4—1422no—
170D4—1422no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—5682no—
174C2×C4—5682no—
175C2×C4—712no—
176C2×C4—1422no—
177C2×C4—1422no—
178C2×C4—1422no—
179C2×C4D4×D7111yescharacteristic
180C2×C4D4×D7111yescharacteristic
181C2×C4C23×D7111yescharacteristic
182C8—21no—
183C8—1422no—
184C8#1176586612yes—
185C22—2842no—
186C22—2842no—
187C22—2842no—
188C22—1422no—
189C22—1422no—
190C22—1422no—
191C22C2×D4×D7111yescharacteristic
192C4—21no—
193C4—21no—
194C4—42no—
195C4—42no—
196C4—2842no—
197C4—1422no—
198C4C2×#1176586611yescharacteristic
199C4#1455617611yescharacteristic
200C2—2842no—
201C2—5682no—
202C2#1798665211yescharacteristic
203C2#1798679611yescharacteristic
204C2#1798680111yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#499×C71C2
socleC22×C71C22×D4
hypercenter#499D71
hypocenterC71#3355
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
1. #17986702C2semidirect
2. #17986326C2semidirect
3. #17986314C2semidirect
4. #17986296C2semidirect
5. #17986253C2semidirect
6. C4×D568C2semidirect
7. #17986078C2semidirect
8. #17986038C2semidirect
9. #499×C71C2semidirect
103. C71#3355semidirect
105. #499D71semidirect