Finite Group Databasebeta

← #22570953

Subgroups — #22570953

Order 8576.

358 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 121 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#16863951C211yesmaximal, characteristic
2#16863951C211yesmaximal, characteristic
3#16863792C211yesmaximal, characteristic, complemented
4#16863775C211yesmaximal, characteristic, complemented
5#16863747C211yesmaximal, characteristic, complemented
6#16863653C214yesmaximal
7#16863500C214yesmaximal, complemented
8C4×#11761611C211yesmaximal, characteristic
9#487×C67C211yesmaximal, characteristic, complemented
10C2×#11761618C2214yes—
11C2×#11761611C2212yes—
12#14524043C2211yescharacteristic
13#14524043C2211yescharacteristic
14#14524034—21no—
15#14524034—21no—
16#14524023—21no—
17#14524023C2212yes—
18#14524019—21no—
19#14524019C2212yes—
20Q8×Dic67—21no—
21Q8×Dic67C2212yes—
22#14524008—21no—
23#14524008C2212yes—
24#14524001—21no—
25#14524001—21no—
26C42×D67C2211yescharacteristic
27#14523974C2214yes—
28#14523968C2214yes—
29#14523953C2212yes—
30C4×C67⋊C8C2211yescharacteristic
31C4⋊Q8×C67C2211yescharacteristic
32C4⋊Q8×C67C2211yescharacteristic
33#103×C67C2214yes—
34C4×C8×C67C2211yescharacteristic
35C2×Dic134—21no—
36C2×Dic134C2312yes—
37C2×C4×D67D411yescharacteristic
38C2×C4×D67D411yescharacteristic
39C2×C4×D67C2311yescharacteristic
40#11761618—28no—
41#11761611—24no—
42C2×C67⋊C8C2312yes—
43C4⋊Dic67—21no—
44C4⋊Dic67—21no—
45C4⋊Dic67—21no—
46C4⋊Dic67—22no—
47C4⋊Dic67—22no—
48C4⋊Dic67C2312yes—
49Dic67⋊C4—22no—
50Dic67⋊C4—22no—
51(C2×D67)⋊C4—22no—
52(C2×D67)⋊C4—22no—
53C4×Dic67—22no—
54C4×Dic67—22no—
55C4×Dic67D411yescharacteristic
56C4×Dic67D411yescharacteristic
57C4×Dic67C2311yescharacteristic
58C2×Q8×C67—21no—
59C2×Q8×C67C2312yes—
60C2×C8×C67C2312yes—
61C4⋊C4×C67—21no—
62C4⋊C4×C67—21no—
63C4⋊C4×C67—21no—
64C4⋊C4×C67C2312yes—
65C42×C67C2311yescharacteristic
66C22×D67C2×D411yescharacteristic
67Q8×C67—24no—
68Q8×C67—42no—
69C2×C4×C67—22no—
70C2×C4×C67—22no—
71C2×C4×C67C2×D411yescharacteristic
72C2×C4×C67C2×D411yescharacteristic
73C2×C4×C67C2411yescharacteristic
74C2×Dic67—22no—
75C2×Dic67—22no—
76C2×Dic67C2×D411yescharacteristic
77C2×Dic67C2×D411yescharacteristic
78C2×Dic67C2×D411yescharacteristic
79C4×D67—22no—
80C4×D67—22no—
81C4×D67—22no—
82Dic134—24no—
83Dic134—42no—
84C8×C67—22no—
85C67⋊C8#4114yes—
86C22×C67C22×D411yescharacteristic
87C2×D67—21no—
88C2×D67—21no—
89C2×D67—21no—
90C4×C67—42no—
91C4×C67—42no—
92C4×C67#13712yes—
93C4×C67#13712yes—
94C4×C67C2×#4112yes—
95Dic67—21no—
96Dic67—21no—
97Dic67—21no—
98Dic67—42no—
99Dic67—42no—
100C2×C67#53211yescharacteristic
101C2×C67C2×#13711yescharacteristic
102C2×C67C2×#13711yescharacteristic
103D67—41no—
104#3185—671nomaximal, Sylow
105C67#318511yescharacteristic, complemented, Sylow
106#533—671no—
107#533—671no—
108#498—671no—
109#494—671no—
110#487—671no—
111#487D6711yescharacteristic, complemented
112#470—674no—
113#441—674no—
114C4×M16—671no—
115C2×Q16—674no—
116C2×M16—672no—
117C4⋊Q8—671no—
118C4⋊Q8—671no—
119C4⋊Q8C2×D6711yescharacteristic
120C4⋊Q8C2×D6711yescharacteristic
121#125—1341no—
122#125—1341no—
123C22⋊Q8—672no—
124C22⋊Q8—672no—
125C22⋊Q8—1341no—
126C22⋊Q8—1341no—
127C4×Q8—672no—
128C4×Q8—672no—
129C4×Q8—1341no—
130C4×Q8—1341no—
131#117—1341no—
132#117—1341no—
133C2×C42—671no—
134#106—674no—
135#103—674no—
136#103C2×D6714yes—
137#97—672no—
138C4×C8—671no—
139C4×C8C2×D6711yescharacteristic
140C2×Q8—21no—
141C2×Q8—672no—
142C2×Q8—1341no—
143C2×Q8C22×D6712yes—
144C22×C4—671no—
145C22×C4—671no—
146C22×C4—671no—
147Q16—1348no—
148M16—1344no—
149C2×C8—672no—
150C2×C8C22×D6712yes—
151C4⋊C4—21no—
152C4⋊C4—21no—
153C4⋊C4—21no—
154C4⋊C4—672no—
155C4⋊C4—1342no—
156C4⋊C4—1342no—
157C4⋊C4—1342no—
158C4⋊C4—1342no—
159C4⋊C4—1341no—
160C4⋊C4—1341no—
161C4⋊C4—1341no—
162C4⋊C4C22×D6712yes—
163C22⋊C4—1342no—
164C22⋊C4—1342no—
165C42—671no—
166C42—671no—
167C42—671no—
168C42—1342no—
169C42—1342no—
170C42C22×D6711yescharacteristic
171C23—671no—
172Q8—24no—
173Q8—42no—
174Q8—2682no—
175Q8—1344no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—671no—
179C2×C4—671no—
180C2×C4—671no—
181C2×C4—1342no—
182C2×C4—1342no—
183C2×C4—1342no—
184C2×C4—1342no—
185C2×C4—1342no—
186C2×C4D4×D6711yescharacteristic
187C2×C4D4×D6711yescharacteristic
188C2×C4C23×D6711yescharacteristic
189C8—22no—
190C8—674no—
191C22—1341no—
192C22—1341no—
193C22—1341no—
194C22C2×D4×D6711yescharacteristic
195C4—42no—
196C4—42no—
197C4—2682no—
198C4—2682no—
199C4—1341no—
200C4—1341no—
201C4—1341no—
202C4#1452407212yes—
203C4#1452407512yes—
204C4#41×D6712yes—
205C2—2681no—
206C2#1686394511yescharacteristic
207C2C2×#1452407211yescharacteristic
208C2C2×#1452407511yescharacteristic
subgroupG/H
centerC22C2×D4×D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#487×C67C2
socleC22×C67C22×D4
hypercenter#487D67
hypocenterC67#3185
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
3. #16863792C2semidirect
4. #16863775C2semidirect
5. #16863747C2semidirect
7. #16863500C2semidirect
9. #487×C67C2semidirect
105. C67#3185semidirect
111. #487D67semidirect