Finite Group Databasebeta

← #22560592

Subgroups — #22560592

Order 8576.

332 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 89 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#16863942C211yesmaximal, characteristic
2#16863412C212yesmaximal, complemented
3#16863370C211yesmaximal, characteristic, complemented
4#16863157C212yesmaximal, complemented
5#568×C67C211yesmaximal, characteristic, complemented
6C2×#11761628C2211yescharacteristic
7#14524041C411yescharacteristic, complemented
8#14524039C411yescharacteristic, complemented
9#14524032C411yescharacteristic, complemented
10#14524030C411yescharacteristic, complemented
11#14524015—22no—
12C4×C22⋊D67—22no—
13#14524002C2212yes—
14C2×C4×Dic67C2211yescharacteristic
15#14523969—24no—
16#14523958—22no—
17C2×#41×C67C2211yescharacteristic
18#131×C67—24no—
19C2×M16×C67—22no—
20C2×M16×C67C2212yes—
21#11761628—22no—
22#11761628—22no—
23C2×Dic134C2×C411yescharacteristic
24C2×D268C2×C411yescharacteristic
25C2×C22⋊D67—22no—
26C2×C22⋊D67—21no—
27C22×Dic67—21no—
28C2×C4×D67—21no—
29C4⋊Dic67—22no—
30Dic67⋊C4—22no—
31C22⋊Dic67—22no—
32C22⋊Dic67—22no—
33(C2×D67)⋊C4—22no—
34C4×Dic67—21no—
35C4×Dic67—22no—
36C4×Dic67C2×C411yescharacteristic
37C4×Dic67C2×C411yescharacteristic
38#41×C67—22no—
39#41×C67—22no—
40C2×Q8×C67C2×C411yescharacteristic
41C2×D4×C67C2×C411yescharacteristic
42C2×D4×C67D412yes—
43C22×C4×C67D412yes—
44C22×C4×C67C2311yescharacteristic
45M16×C67—24no—
46M16×C67—22no—
47M16×C67D414yescomplemented
48C2×C8×C67—24no—
49C2×C8×C67—22no—
50C23×C67C22⋊C412yes—
51C23×C67C2×D411yescharacteristic
52C22×D67—21no—
53Q8×C67—22no—
54D4×C67—24no—
55D4×C67—22no—
56C2×C4×C67—22no—
57C2×C4×C67—22no—
58C2×C4×C67C22⋊C412yes—
59C2×C4×C67C22×C411yescharacteristic
60C2×C4×C67C2×D412yes—
61C2×C4×C67C2×D411yescharacteristic
62C2×C4×C67#4112yes—
63C22⋊D67—44no—
64C22⋊D67—42no—
65C2×Dic67—21no—
66C2×Dic67—21no—
67C2×Dic67—21no—
68C2×Dic67—42no—
69C2×Dic67—42no—
70D268—22no—
71C4×D67—41no—
72C4×D67—41no—
73Dic134—22no—
74C8×C67—24no—
75C22×C67—22no—
76C22×C67—41no—
77C22×C67—42no—
78C22×C67C2×C22⋊C411yescharacteristic
79C22×C67#12112yes—
80C2×D67—42no—
81C2×D67—41no—
82C4×C67—22no—
83C4×C67C4×D412yes—
84C4×C67#12011yescharacteristic
85C4×C67#12311yescharacteristic
86Dic67—22no—
87Dic67—41no—
88Dic67—41no—
89Dic67—82no—
90C2×C67—21no—
91C2×C67—22no—
92C2×C67—42no—
93C2×C67#38611yescharacteristic
94D67—81no—
95#1884—671nomaximal, Sylow
96C67#188411yescharacteristic, complemented, Sylow
97#568D6711yescharacteristic, complemented
98#532—671no—
99#421—672no—
100#410—671no—
101#339—672no—
102C2×#41—671no—
103C2×#41C2×D6711yescharacteristic
104#131—24no—
105C2×M16—22no—
106C2×M16C2×D6712yes—
107C4⋊Q8—671no—
108#127—671no—
109#124—671no—
110#124—671no—
111#121—1342no—
112C4×D4—1342no—
113#117—672no—
114C2×C42—671no—
115#104—1344no—
116#99—1342no—
117#41—22no—
118#41—22no—
119#41—1342no—
120#41—1342no—
121C2×Q8—671no—
122C2×Q8C4×D6711yescharacteristic
123C2×D4—671no—
124C2×D4—1342no—
125C2×D4—1341no—
126C2×D4C4×D6711yescharacteristic
127C2×D4D26812yes—
128C22×C4—1341no—
129C22×C4—1341no—
130C22×C4C22⋊D6712yes—
131C22×C4C22×D6711yescharacteristic
132M16—24no—
133M16—22no—
134M16C22⋊D6714yescomplemented
135C2×C8—24no—
136C2×C8—22no—
137C4⋊C4—1342no—
138C4⋊C4—1342no—
139C22⋊C4—1342no—
140C22⋊C4—1342no—
141C22⋊C4—1342no—
142C42—671no—
143C42—671no—
144C42—1342no—
145C42—1341no—
146C23—1341no—
147C23(C2×D67)⋊C412yes—
148C23C2×D26811yescharacteristic
149Q8—22no—
150Q8—1342no—
151D4—24no—
152D4—22no—
153D4—2682no—
154D4—2684no—
155D4—1342no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—2682no—
159C2×C4—2681no—
160C2×C4—2681no—
161C2×C4—2682no—
162C2×C4—1341no—
163C2×C4—1341no—
164C2×C4—1341no—
165C2×C4(C2×D67)⋊C412yes—
166C2×C4C2×C4×D6711yescharacteristic
167C2×C4C2×C22⋊D6712yes—
168C2×C4C2×C22⋊D6711yescharacteristic
169C2×C4#1176162812yes—
170C8—24no—
171C22—22no—
172C22—41no—
173C22—42no—
174C22—2682no—
175C22—2681no—
176C22C2×(C2×D67)⋊C411yescharacteristic
177C22#1452401712yes—
178C4—22no—
179C4—2681no—
180C4—2681no—
181C4—5362no—
182C4—1342no—
183C4C4×C22⋊D6712yes—
184C4#1452401211yescharacteristic
185C4#1452402611yescharacteristic
186C2—21no—
187C2—22no—
188C2—42no—
189C2—5361no—
190C2#1686327511yescharacteristic
subgroupG/H
centerC4#14524012
commutatorC2×C4×C67C22×C4
FrattiniC22×C4C22×D67
Fitting#568×C67C2
socleC2×C67#386
hypercenter#568D67
hypocenterC67#1884
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C4 ◁ C22×C4 ◁ #568

Complements

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

NH ≅ G/Ntype
2. #16863412C2semidirect
3. #16863370C2semidirect
4. #16863157C2semidirect
5. #568×C67C2semidirect
7. #14524041C4semidirect
8. #14524039C4semidirect
9. #14524032C4semidirect
10. #14524030C4semidirect
47. M16×C67D4semidirect
96. C67#1884semidirect
97. #568D67semidirect
134. M16C22⋊D67semidirect