Finite Group Databasebeta

← #22561433

Subgroups — #22561433

Order 8576.

306 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 69 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#16863977C211yesmaximal, characteristic
2C2×#14523971C212yesmaximal, complemented
3#16863395C212yesmaximal, complemented
4#16863165C211yesmaximal, characteristic, complemented
5#384×C67C211yesmaximal, characteristic, complemented
6C2×#11761628C2212yes—
7C2×#11761612C2212yes—
8#14524023—22no—
9#14524018—22no—
10#14524017—22no—
11#14523971—24no—
12#14523966—22no—
13#14523963—22no—
14C2×C4⋊C4×C67—22no—
15C2×C4⋊C4×C67C2211yescharacteristic
16C2×C4⋊C4×C67C2211yescharacteristic
17C2×C42×C67C2211yescharacteristic
18#11761628—22no—
19#11761628—22no—
20#11761628—22no—
21#11761628—22no—
22C2×Dic134D412yes—
23C2×D268—22no—
24C2×D268D412yes—
25C2×C22⋊D67—22no—
26C2×C4×D67—22no—
27#11761612—22no—
28#11761612—22no—
29C2×C67⋊C8—22no—
30C4⋊Dic67—22no—
31(C2×D67)⋊C4—24no—
32C22×C4×C67—21no—
33C22×C4×C67—21no—
34C22×C4×C67D412yes—
35C22×C4×C67C2311yescharacteristic
36C4⋊C4×C67—22no—
37C4⋊C4×C67—22no—
38C4⋊C4×C67—44no—
39C42×C67—21no—
40C42×C67D412yescomplemented
41C23×C67C2×D411yescharacteristic
42C22×D67—22no—
43C2×C4×C67—22no—
44C2×C4×C67—22no—
45C2×C4×C67—22no—
46C2×C4×C67—22no—
47C2×C4×C67—42no—
48C2×C4×C67—42no—
49C2×C4×C67—41no—
50C2×C4×C67C2×D412yes—
51C2×C4×C67C2×D411yescharacteristic
52C2×C4×C67#4111yescharacteristic
53C2×C4×C67#4112yes—
54C22⋊D67—42no—
55C22⋊D67—42no—
56C2×Dic67—22no—
57D268—22no—
58D268—22no—
59D268—82no—
60C4×D67—42no—
61C4×D67—42no—
62Dic134—22no—
63Dic134—22no—
64C67⋊C8—42no—
65C22×C67—21no—
66C22×C67—21no—
67C22×C67#12011yescharacteristic
68C22×C67#12111yescharacteristic
69C22×C67#12411yescharacteristic
70C2×D67—42no—
71C2×D67—42no—
72C2×D67—42no—
73C4×C67—24no—
74C4×C67—42no—
75C4×C67—81no—
76C4×C67#12112yes—
77C4×C67#12312yes—
78Dic67—42no—
79C2×C67—21no—
80C2×C67—21no—
81C2×C67#39411yescharacteristic
82C2×C67#45311yescharacteristic
83C2×C67#45611yescharacteristic
84D67—82no—
85#2039—671nomaximal, Sylow
86C67#203911yescharacteristic, complemented, Sylow
87#537—671no—
88C2×#104—672no—
89#417—672no—
90#384D6711yescharacteristic, complemented
91#343—671no—
92C2×#41—672no—
93C2×M16—672no—
94C22⋊Q8—1342no—
95#121—1342no—
96#121—1342no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4C2×D6711yescharacteristic
99C2×C4⋊C4C2×D6711yescharacteristic
100C2×C42C2×D6711yescharacteristic
101#104—1344no—
102#103—1342no—
103#102—1342no—
104#41—1342no—
105#41—1342no—
106#41—1342no—
107#41—1342no—
108C2×Q8—672no—
109C2×D4—672no—
110C2×D4—1342no—
111C2×D4—1342no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—1342no—
115C22×C4C22⋊D6712yes—
116C22×C4C22×D6711yescharacteristic
117M16—1342no—
118M16—1342no—
119C2×C8—1342no—
120C4⋊C4—22no—
121C4⋊C4—22no—
122C4⋊C4—44no—
123C4⋊C4—1342no—
124C22⋊C4—1344no—
125C42—21no—
126C42D26812yescomplemented
127C23—1342no—
128C23C2×C22⋊D6711yescharacteristic
129Q8—1342no—
130Q8—1342no—
131D4—2682no—
132D4—2682no—
133D4—5362no—
134D4—1342no—
135D4—1342no—
136C2×C4—22no—
137C2×C4—22no—
138C2×C4—22no—
139C2×C4—22no—
140C2×C4—42no—
141C2×C4—42no—
142C2×C4—41no—
143C2×C4—2682no—
144C2×C4—2682no—
145C2×C4—1342no—
146C2×C4D4×D6712yes—
147C2×C4C2×D26811yescharacteristic
148C2×C4#1176162811yescharacteristic
149C2×C4#1176162912yes—
150C8—2682no—
151C22—21no—
152C22—21no—
153C22—2682no—
154C22—2682no—
155C22—2682no—
156C22#1452401111yescharacteristic
157C22#1452401711yescharacteristic
158C22#1452403211yescharacteristic
159C4—24no—
160C4—42no—
161C4—81no—
162C4—2682no—
163C4#1452401812yes—
164C4#1452402812yes—
165C2—21no—
166C2—21no—
167C2—5362no—
168C2#1686331311yescharacteristic
169C2#1686356311yescharacteristic
170C2#1686357811yescharacteristic
subgroupG/H
centerC22#14524011
commutatorC22×C4×C67C23
FrattiniC22×C4C22×D67
Fitting#384×C67C2
socleC22×C67#120
hypercenter#384D67
hypocenterC67#2039
LCSG ▷ C22×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ #384

Complements

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

NH ≅ G/Ntype
2. C2×#14523971C2semidirect
3. #16863395C2semidirect
4. #16863165C2semidirect
5. #384×C67C2semidirect
40. C42×C67D4semidirect
86. C67#2039semidirect
90. #384D67semidirect
126. C42D268semidirect