Finite Group Databasebeta

← #22571675

Subgroups — #22571675

Order 8576.

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#16864019C212yesmaximal, complemented
2#16863685C212yesmaximal, complemented
3#16863679C212yesmaximal, complemented
4#16863649C211yesmaximal, characteristic
5#16863627C212yesmaximal, complemented
6#16863620C212yesmaximal, complemented
7#16863614C212yesmaximal, complemented
8#16863424C211yesmaximal, characteristic, complemented
9#470×C67C211yesmaximal, characteristic, complemented
10C2×#11761615C2212yes—
11C2×D536C2211yescharacteristic
12C22×C67⋊C8C2211yescharacteristic
13#14524042C2212yes—
14#14524038—22no—
15#14524032C2212yes—
16#14524029—22no—
17#14524025—22no—
18#14524023—22no—
19#14524017C2212yes—
20Q8×Dic67C2212yes—
21#14524005C2212yes—
22#14524002C2211yescharacteristic
23#14523975C2211yescharacteristic
24#14523974C2212yes—
25Dic67⋊C8C2212yes—
26Dic67⋊C8C2212yes—
27#14523967C2212yes—
28#14523965C2212yes—
29#14523963C2212yes—
30C2×Q16×C67C2211yescharacteristic
31C2×M16×C67C2211yescharacteristic
32C22⋊Q8×C67C2212yes—
33#106×C67C2211yescharacteristic
34#103×C67C2212yes—
35C2×Dic134—22no—
36C2×D268C2312yes—
37C2×C22⋊D67—22no—
38C2×C4×D67—22no—
39#11761615—42no—
40D536—41no—
41C2×C67⋊C8—41no—
42C2×C67⋊C8C2312yes—
43C4⋊Dic67—22no—
44C4⋊Dic67D412yes—
45C4⋊Dic67C2312yes—
46Dic67⋊C4—22no—
47C22⋊Dic67D411yescharacteristic
48C22⋊Dic67D411yescharacteristic
49(C2×D67)⋊C4—22no—
50(C2×D67)⋊C4—22no—
51(C2×D67)⋊C4—22no—
52(C2×D67)⋊C4—22no—
53C4×Dic67—22no—
54C4×Dic67C2312yes—
55C2×Q8×C67C2312yes—
56C22×C4×C67C2311yescharacteristic
57Q16×C67—41no—
58M16×C67—41no—
59C2×C8×C67C2311yescharacteristic
60C2×C8×C67C2311yescharacteristic
61C4⋊C4×C67—22no—
62C4⋊C4×C67C2312yes—
63C22⋊C4×C67—22no—
64C23×C67C2×D411yescharacteristic
65C22×D67—22no—
66Q8×C67—22no—
67Q8×C67—22no—
68C2×C4×C67—21no—
69C2×C4×C67—21no—
70C2×C4×C67—22no—
71C2×C4×C67—22no—
72C2×C4×C67C2×D411yescharacteristic
73C2×C4×C67C2411yescharacteristic
74C22⋊D67—82no—
75C2×Dic67—22no—
76C2×Dic67C2×D412yes—
77C2×Dic67C2×D412yes—
78D268—22no—
79D268—22no—
80C4×D67—82no—
81Dic134—82no—
82C8×C67—21no—
83C8×C67—21no—
84C67⋊C8—22no—
85C22×C67—21no—
86C22×C67—22no—
87C22×C67C22×D411yescharacteristic
88C2×D67—42no—
89C2×D67—42no—
90C2×D67—42no—
91C4×C67—21no—
92C4×C67—42no—
93C4×C67—42no—
94C4×C67H(5,𝔽2)11yescharacteristic
95C4×C672−1+411yescharacteristic
96Dic67—22no—
97Dic67—22no—
98Dic67—42no—
99C2×C67—41no—
100C2×C67#53711yescharacteristic
101C2×C67#57712yes—
102D67—82no—
103#3263—671nomaximal, Sylow
104C67#326311yescharacteristic, complemented, Sylow
105#542—672no—
106#476—672no—
107#475—672no—
108#470D6711yescharacteristic, complemented
109#469—671no—
110#465—672no—
111#464—672no—
112#463—672no—
113#424—671no—
114C2×Q16C2×D6711yescharacteristic
115C2×SD16—672no—
116C2×D8—671no—
117C2×M16C2×D6711yescharacteristic
118C22×C8—671no—
119C4⋊Q8—672no—
120#126—1342no—
121#124—672no—
122#124—1342no—
123#123—1342no—
124C22⋊Q8—1342no—
125C22⋊Q8C2×D6712yes—
126#121—672no—
127C4×Q8—672no—
128C4×D4—672no—
129#117—671no—
130#106—671no—
131#106—672no—
132#106C2×D6711yescharacteristic
133C4⋊C8—672no—
134C4⋊C8—672no—
135#103—672no—
136#103C2×D6712yes—
137#102—672no—
138#102—672no—
139C2×Q8—1342no—
140C2×Q8C22×D6712yes—
141C2×D4—672no—
142C2×D4—1342no—
143C22×C4—1342no—
144C22×C4C22×D6711yescharacteristic
145Q16—41no—
146SD16—2682no—
147D8—2681no—
148M16—41no—
149C2×C8—2681no—
150C2×C8—672no—
151C2×C8C22×D6711yescharacteristic
152C2×C8C22×D6711yescharacteristic
153C4⋊C4—22no—
154C4⋊C4—672no—
155C4⋊C4—672no—
156C4⋊C4—1342no—
157C4⋊C4—1342no—
158C4⋊C4C22×D6712yes—
159C22⋊C4—22no—
160C22⋊C4—671no—
161C22⋊C4—671no—
162C22⋊C4—1342no—
163C22⋊C4—1342no—
164C22⋊C4—1342no—
165C22⋊C4—1342no—
166C42—672no—
167C42—1342no—
168C23—1342no—
169C23D4×D6711yescharacteristic
170Q8—22no—
171Q8—22no—
172Q8—5362no—
173D4—5362no—
174D4—1342no—
175D4—1342no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—5362no—
181C2×C4—672no—
182C2×C4—672no—
183C2×C4—1342no—
184C2×C4D4×D6711yescharacteristic
185C2×C4C23×D6711yescharacteristic
186C8—21no—
187C8—21no—
188C8—1342no—
189C22—22no—
190C22—21no—
191C22—2682no—
192C22—2682no—
193C22—2682no—
194C22C2×D4×D6711yescharacteristic
195C4—21no—
196C4—42no—
197C4—42no—
198C4—2682no—
199C4—1342no—
200C4—1342no—
201C4#1452408811yescharacteristic
202C4#1452409011yescharacteristic
203C2—41no—
204C2—5362no—
205C2#1686397311yescharacteristic
206C2#1686420512yes—
subgroupG/H
centerC22C2×D4×D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#470×C67C2
socleC22×C67C22×D4
hypercenter#470D67
hypocenterC67#3263
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. #16864019C2semidirect
2. #16863685C2semidirect
3. #16863679C2semidirect
5. #16863627C2semidirect
6. #16863620C2semidirect
7. #16863614C2semidirect
8. #16863424C2semidirect
9. #470×C67C2semidirect
104. C67#3263semidirect
108. #470D67semidirect