Finite Group Databasebeta

← #22570140

Subgroups — #22570140

Order 8576.

414 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 133 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#16863907C211yesmaximal, characteristic
2#16863737C214yesmaximal, complemented
3#16863720C214yesmaximal, complemented
4#16863436C212yesmaximal, complemented
5#16863395C212yesmaximal, complemented
6#16863360C211yesmaximal, characteristic, complemented
7C2×#117×C67C211yesmaximal, characteristic, complemented
8C2×#11761628C2211yescharacteristic
9C2×#11761612C2212yes—
10#14524026—21no—
11#14524022C2212yes—
12#14524017C2212yes—
13#14524012—21no—
14C4×C22⋊D67—22no—
15#14524002C2211yescharacteristic
16#14523976C2214yescomplemented
17#14523974C2214yescomplemented
18#14523966C2214yescomplemented
19#14523963C2214yescomplemented
20C22⋊(C67⋊C8)C2214yes—
21C23×C4×C67C2211yescharacteristic
22#117×C67—22no—
23#117×C67C2214yes—
24C2×C4⋊C4×C67C2212yes—
25C2×C22⋊C4×C67—21no—
26C2×C42×C67—21no—
27#11761628—24no—
28C2×Dic134C2311yescharacteristic
29C2×D268C2311yescharacteristic
30C2×C22⋊D67—22no—
31C2×C22⋊D67—21no—
32C2×C4×D67—21no—
33#11761612—24no—
34C2×C67⋊C8C2314yes—
35C4⋊Dic67C2312yes—
36Dic67⋊C4—22no—
37C22⋊Dic67—21no—
38C22⋊Dic67—22no—
39(C2×D67)⋊C4—22no—
40C4×Dic67—21no—
41C24×C67D411yescharacteristic
42C22×C4×C67—24no—
43C22×C4×C67—22no—
44C22×C4×C67D411yescharacteristic
45C22×C4×C67D412yes—
46C22×C4×C67C2312yes—
47C22×C4×C67C2311yescharacteristic
48C4⋊C4×C67—22no—
49C4⋊C4×C67C2314yes—
50C22⋊C4×C67—24no—
51C42×C67—24no—
52C23×C67—22no—
53C23×C67—42no—
54C23×C67C2×D412yes—
55C23×C67C2×D411yescharacteristic
56C22×D67—21no—
57C2×C4×C67—22no—
58C2×C4×C67—24no—
59C2×C4×C67—24no—
60C2×C4×C67—24no—
61C2×C4×C67—44no—
62C2×C4×C67C2×D412yes—
63C2×C4×C67C2×D411yescharacteristic
64C2×C4×C67#4114yes—
65C2×C4×C67C2411yescharacteristic
66C22⋊D67—44no—
67C22⋊D67—42no—
68C2×Dic67—22no—
69C2×Dic67—21no—
70D268—22no—
71C4×D67—42no—
72Dic134—22no—
73C67⋊C8—24no—
74C22×C67—22no—
75C22×C67—22no—
76C22×C67—22no—
77C22×C67—22no—
78C22×C67—44no—
79C22×C67#12312yes—
80C22×C67C22×D411yescharacteristic
81C2×D67—41no—
82C2×D67—42no—
83C4×C67—22no—
84C4×C67—44no—
85C4×C67#12312yes—
86C4×C67C2×#4112yes—
87Dic67—41no—
88Dic67—42no—
89C2×C67—22no—
90C2×C67—42no—
91C2×C67C2×#12311yescharacteristic
92C2×C67#57512yes—
93D67—81no—
94#3093—671nomaximal, Sylow
95C67#309311yescharacteristic, complemented, Sylow
96#525—671no—
97C2×#117D6711yescharacteristic, complemented
98#485—674no—
99#482—674no—
100#428—672no—
101#417—672no—
102#407—671no—
103C2×#41—671no—
104C23×C4C2×D6711yescharacteristic
105C2×M16—672no—
106#123—1341no—
107C22⋊Q8—672no—
108#121—672no—
109#120—1341no—
110C4×D4—1342no—
111#117—22no—
112#117—671no—
113#117C2×D6714yes—
114C2×C4⋊C4C2×D6712yes—
115C2×C22⋊C4—21no—
116C2×C42—21no—
117#107—674no—
118#106—674no—
119#103—674no—
120#102—674no—
121C22⋊C8—674no—
122C24C22⋊D6711yescharacteristic
123#41—1344no—
124C2×Q8—671no—
125C2×D4—671no—
126C2×D4—1342no—
127C2×D4—1341no—
128C22×C4—24no—
129C22×C4—22no—
130C22×C4—1341no—
131C22×C4C22⋊D6711yescharacteristic
132C22×C4C22⋊D6712yes—
133C22×C4C22×D6712yes—
134C22×C4C22×D6711yescharacteristic
135M16—1344no—
136C2×C8—674no—
137C4⋊C4—22no—
138C4⋊C4—672no—
139C4⋊C4—1342no—
140C4⋊C4C22×D6714yes—
141C22⋊C4—24no—
142C22⋊C4—1341no—
143C22⋊C4—1342no—
144C22⋊C4—1342no—
145C42—24no—
146C42—1341no—
147C23—22no—
148C23—42no—
149C23—1341no—
150C23C2×C22⋊D6712yes—
151C23C2×C22⋊D6711yescharacteristic
152Q8—1342no—
153D4—2684no—
154D4—2682no—
155D4—1342no—
156C2×C4—22no—
157C2×C4—24no—
158C2×C4—24no—
159C2×C4—24no—
160C2×C4—44no—
161C2×C4—2682no—
162C2×C4—1342no—
163C2×C4—1341no—
164C2×C4C2×C22⋊D6712yes—
165C2×C4C2×C22⋊D6711yescharacteristic
166C2×C4#1176162814yes—
167C2×C4C23×D6711yescharacteristic
168C8—1344no—
169C22—22no—
170C22—22no—
171C22—22no—
172C22—22no—
173C22—44no—
174C22—2681no—
175C22—2682no—
176C22#1452402612yes—
177C22C22×C22⋊D6711yescharacteristic
178C4—22no—
179C4—44no—
180C4—2681no—
181C4—2682no—
182C4#1452402612yes—
183C4C2×#1176162812yes—
184C2—22no—
185C2—42no—
186C2—5361no—
187C2C2×#1452402611yescharacteristic
188C2#1686419412yes—
subgroupG/H
centerC22C22×C22⋊D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
FittingC2×#117×C67C2
socleC22×C67C22×D4
hypercenterC2×#117D67
hypocenterC67#3093
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. #16863737C2semidirect
3. #16863720C2semidirect
4. #16863436C2semidirect
5. #16863395C2semidirect
6. #16863360C2semidirect
7. C2×#117×C67C2semidirect
16. #14523976C22semidirect
17. #14523974C22semidirect
18. #14523966C22semidirect
19. #14523963C22semidirect
95. C67#3093semidirect
97. C2×#117D67semidirect