Finite Group Databasebeta

← #22571347

Subgroups — #22571347

Order 8576.

370 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 125 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#16863953C211yesmaximal, characteristic, complemented
2#16863946C211yesmaximal, characteristic
3#16863801C211yesmaximal, characteristic, complemented
4#16863794C211yesmaximal, characteristic, complemented
5#16863743C211yesmaximal, characteristic, complemented
6#16863743C211yesmaximal, characteristic, complemented
7#16863721C214yesmaximal, complemented
8#16863697C214yesmaximal, complemented
9#431×C67C211yesmaximal, characteristic, complemented
10C2×#11761628—21no—
11#14524043C2211yescharacteristic
12#14524043C2211yescharacteristic
13#14524043C2211yescharacteristic
14#14524040C2211yescharacteristic
15#14524035C2212yes—
16#14524031—21no—
17#14524022—21no—
18#14524022—21no—
19#14524017—21no—
20#14524017C2212yes—
21C4×Dic134—21no—
22C4×Dic134—21no—
23C4×D268—21no—
24C4×D268C2212yes—
25#14524002C2212yes—
26#14523977C2214yes—
27#14523975C2214yes—
28#14523965C2214yes—
29#14523965C2214yes—
30C2×C42×C67C2211yescharacteristic
31C4⋊C8×C67C2212yes—
32C22⋊C8×C67C2212yes—
33C4×C8×C67C2211yescharacteristic
34C4×C8×C67C2211yescharacteristic
35#11761628—44no—
36C2×Dic134—21no—
37C2×Dic134—21no—
38C2×Dic134—21no—
39C2×D268—21no—
40C2×D268—21no—
41C2×D268C2312yes—
42C2×C22⋊D67—22no—
43C2×C4×D67—22no—
44C4⋊Dic67—21no—
45C4⋊Dic67—21no—
46C4⋊Dic67—21no—
47C4⋊Dic67C2312yes—
48C4⋊Dic67C2312yes—
49C4⋊Dic67C2312yes—
50Dic67⋊C4—22no—
51Dic67⋊C4—22no—
52C22⋊Dic67—22no—
53(C2×D67)⋊C4—22no—
54(C2×D67)⋊C4—22no—
55C4×Dic67—22no—
56C22×C4×C67D411yescharacteristic
57C22×C4×C67D411yescharacteristic
58C22×C4×C67C2311yescharacteristic
59C2×C8×C67C2312yes—
60C2×C8×C67C2312yes—
61C42×C67D411yescharacteristic
62C42×C67D411yescharacteristic
63C42×C67C2311yescharacteristic
64C42×C67C2311yescharacteristic
65C23×C67C2×D411yescharacteristic
66C22×D67—22no—
67C2×C4×C67—22no—
68C2×C4×C67—22no—
69C2×C4×C67—22no—
70C2×C4×C67C2×D411yescharacteristic
71C2×C4×C67C2×D411yescharacteristic
72C2×C4×C67C2×D411yescharacteristic
73C2×C4×C67C2×D411yescharacteristic
74C2×C4×C67C2×D411yescharacteristic
75C2×C4×C67C2411yescharacteristic
76C22⋊D67—44no—
77C2×Dic67—22no—
78C2×Dic67—22no—
79C2×Dic67—22no—
80D268—24no—
81D268—42no—
82D268—42no—
83C4×D67—44no—
84Dic134—42no—
85Dic134—42no—
86Dic134—42no—
87C8×C67—22no—
88C8×C67—22no—
89C22×C67—21no—
90C22×C67—21no—
91C22×C67—21no—
92C22×C67C22×D411yescharacteristic
93C2×D67—42no—
94C2×D67—42no—
95C2×D67—42no—
96C4×C67—21no—
97C4×C67—21no—
98C4×C67—21no—
99C4×C67#13512yes—
100C4×C67AGL(1,ℤ/8)12yes—
101C4×C672−1+412yes—
102Dic67—42no—
103Dic67—42no—
104Dic67—42no—
105C2×C67—41no—
106C2×C67#53611yescharacteristic
107C2×C67C2×#13511yescharacteristic
108C2×C67C2×AGL(1,ℤ/8)11yescharacteristic
109D67—82no—
110#3228—671nomaximal, Sylow
111C67#322811yescharacteristic, complemented, Sylow
112#533—671no—
113#532—671no—
114#500—671no—
115#498—671no—
116#486—671no—
117#486—671no—
118#482—674no—
119#478—674no—
120#431D6711yescharacteristic, complemented
121C2×#41—1341no—
122C4⋊Q8—671no—
123C4⋊Q8—671no—
124C4⋊Q8—671no—
125#127—671no—
126#125—672no—
127#124—1341no—
128C22⋊Q8—1341no—
129C22⋊Q8—1341no—
130#121—672no—
131#121—1341no—
132C4×Q8—1341no—
133C4×Q8—1341no—
134C4×D4—672no—
135C4×D4—1341no—
136#117—672no—
137C2×C42C2×D6711yescharacteristic
138#107—674no—
139#106—674no—
140C4⋊C8C2×D6712yes—
141#102—674no—
142#102—674no—
143C22⋊C8C2×D6712yes—
144C4×C8C2×D6711yescharacteristic
145C4×C8C2×D6711yescharacteristic
146#41—2684no—
147C2×Q8—1341no—
148C2×Q8—1341no—
149C2×Q8—1341no—
150C2×D4—672no—
151C2×D4—1342no—
152C2×D4—1341no—
153C2×D4—1341no—
154C22×C4—1342no—
155C22×C4D26811yescharacteristic
156C22×C4D26811yescharacteristic
157C22×C4C22×D6711yescharacteristic
158C2×C8C22×D6712yes—
159C2×C8C22×D6712yes—
160C4⋊C4—672no—
161C4⋊C4—672no—
162C4⋊C4—672no—
163C4⋊C4—1341no—
164C4⋊C4—1342no—
165C4⋊C4—1342no—
166C4⋊C4—1341no—
167C4⋊C4—1341no—
168C22⋊C4—1342no—
169C22⋊C4—1342no—
170C22⋊C4—1342no—
171C42—1342no—
172C42D26811yescharacteristic
173C42D26811yescharacteristic
174C42C22×D6711yescharacteristic
175C42C22×D6711yescharacteristic
176C23—1342no—
177C23C2×D26811yescharacteristic
178Q8—2682no—
179Q8—2682no—
180Q8—2682no—
181D4—2684no—
182D4—2682no—
183D4—2682no—
184D4—1344no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—2684no—
189C2×C4—1342no—
190C2×C4—1342no—
191C2×C4—1342no—
192C2×C4C2×D26811yescharacteristic
193C2×C4C2×D26811yescharacteristic
194C2×C4C2×D26811yescharacteristic
195C2×C4C2×D26811yescharacteristic
196C2×C4C2×D26811yescharacteristic
197C2×C4C23×D6711yescharacteristic
198C8—22no—
199C8—22no—
200C22—21no—
201C22—21no—
202C22—21no—
203C22—2682no—
204C22—2682no—
205C22—2682no—
206C22C22×D26811yescharacteristic
207C4—21no—
208C4—21no—
209C4—21no—
210C4—2682no—
211C4—2682no—
212C4—2682no—
213C4#1452406512yes—
214C4#1452407112yes—
215C4#1452409012yes—
216C2—41no—
217C2—5362no—
218C2#1686397211yescharacteristic
219C2C2×#1452406511yescharacteristic
220C2C2×#1452407111yescharacteristic
subgroupG/H
centerC22C22×D268
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#431×C67C2
socleC22×C67C22×D4
hypercenter#431D67
hypocenterC67#3228
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C2×C42 ◁ #431

Complements

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

NH ≅ G/Ntype
1. #16863953C2semidirect
3. #16863801C2semidirect
4. #16863794C2semidirect
5. #16863743C2semidirect
6. #16863743C2semidirect
7. #16863721C2semidirect
8. #16863697C2semidirect
9. #431×C67C2semidirect
111. C67#3228semidirect
120. #431D67semidirect