Finite Group Databasebeta

← #22571167

Subgroups — #22571167

Order 8576.

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

normal subgroups only

This group has 394 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#16863776C211yesmaximal, characteristic, complemented
4#16863769C211yesmaximal, characteristic, complemented
5#16863747C211yesmaximal, characteristic, complemented
6#16863743C211yesmaximal, characteristic, complemented
7#16863619C214yesmaximal, complemented
8#16863560C214yesmaximal, complemented
9#431×C67C211yesmaximal, characteristic, complemented
10C2×#11761628C2212yes—
11C2×Dic268C2214yes—
12C2×#11761617C2214yes—
13#14524043C2211yescharacteristic
14#14524043C2211yescharacteristic
15#14524043C2211yescharacteristic
16#14524040C2211yescharacteristic
17#14524035—21no—
18#14524031C2212yes—
19#14524022—21no—
20#14524022C2212yes—
21#14524017—21no—
22#14524017—21no—
23C4×Dic134—21no—
24C4×Dic134C2212yes—
25C4×D268—21no—
26C4×D268—21no—
27#14524002—21no—
28#14523968C2214yes—
29#14523965C2214yes—
30C2×C42×C67C2211yescharacteristic
31C4⋊C8×C67C2212yes—
32C22⋊C8×C67C2212yes—
33C4×C8×C67C2211yescharacteristic
34C4×C8×C67C2211yescharacteristic
35#11761628—44no—
36C2×Dic134—21no—
37C2×Dic134C2312yes—
38C2×Dic134C2312yes—
39C2×D268—21no—
40C2×D268—21no—
41C2×D268C2312yes—
42C2×C22⋊D67—22no—
43C2×C4×D67—22no—
44Dic268—44no—
45#11761617—44no—
46C4⋊Dic67—21no—
47C4⋊Dic67—21no—
48C4⋊Dic67—21no—
49C4⋊Dic67—21no—
50C4⋊Dic67—21no—
51C4⋊Dic67C2312yes—
52Dic67⋊C4—22no—
53Dic67⋊C4—22no—
54C22⋊Dic67—22no—
55(C2×D67)⋊C4—22no—
56(C2×D67)⋊C4—22no—
57C4×Dic67—22no—
58C22×C4×C67D411yescharacteristic
59C22×C4×C67D411yescharacteristic
60C22×C4×C67C2311yescharacteristic
61C2×C8×C67C2312yes—
62C2×C8×C67C2312yes—
63C42×C67D411yescharacteristic
64C42×C67D411yescharacteristic
65C42×C67C2311yescharacteristic
66C42×C67C2311yescharacteristic
67C23×C67C2×D411yescharacteristic
68C22×D67—22no—
69C2×C4×C67—22no—
70C2×C4×C67—22no—
71C2×C4×C67—22no—
72C2×C4×C67C2×D411yescharacteristic
73C2×C4×C67C2×D411yescharacteristic
74C2×C4×C67C2×D411yescharacteristic
75C2×C4×C67C2×D411yescharacteristic
76C2×C4×C67C2×D411yescharacteristic
77C2×C4×C67C2411yescharacteristic
78C22⋊D67—44no—
79C2×Dic67—22no—
80C2×Dic67—22no—
81C2×Dic67—22no—
82D268—24no—
83D268—42no—
84D268—42no—
85C4×D67—44no—
86Dic134—24no—
87Dic134—24no—
88Dic134—42no—
89C8×C67—22no—
90C8×C67—22no—
91C22×C67—21no—
92C22×C67—21no—
93C22×C67—21no—
94C22×C67C22×D411yescharacteristic
95C2×D67—42no—
96C2×D67—42no—
97C2×D67—42no—
98C4×C67—21no—
99C4×C67—21no—
100C4×C67—21no—
101C4×C67#13512yes—
102C4×C67#13712yes—
103C4×C67H(5,𝔽2)12yes—
104Dic67—42no—
105Dic67—42no—
106Dic67—42no—
107C2×C67—41no—
108C2×C67#53511yescharacteristic
109C2×C67C2×#13511yescharacteristic
110C2×C67C2×#13711yescharacteristic
111D67—82no—
112#3210—671nomaximal, Sylow
113C67#321011yescharacteristic, complemented, Sylow
114#533—671no—
115#532—671no—
116#494—671no—
117#492—671no—
118#487—671no—
119#486—671no—
120#464—674no—
121#452—674no—
122#431D6711yescharacteristic, complemented
123C2×#41—672no—
124C2×Q16—674no—
125C2×SD16—674no—
126C4⋊Q8—671no—
127C4⋊Q8—671no—
128C4⋊Q8—671no—
129#127—671no—
130#125—1341no—
131#124—672no—
132C22⋊Q8—672no—
133C22⋊Q8—1341no—
134#121—1341no—
135#121—1341no—
136C4×Q8—672no—
137C4×Q8—1341no—
138C4×D4—1341no—
139C4×D4—1341no—
140#117—1341no—
141C2×C42C2×D6711yescharacteristic
142C4⋊C8C2×D6712yes—
143#103—674no—
144#102—674no—
145C22⋊C8C2×D6712yes—
146C4×C8C2×D6711yescharacteristic
147C4×C8C2×D6711yescharacteristic
148#41—2684no—
149C2×Q8—672no—
150C2×Q8—672no—
151C2×Q8—1341no—
152C2×D4—672no—
153C2×D4—1342no—
154C2×D4—1341no—
155C2×D4—1341no—
156C22×C4—1342no—
157C22×C4D26811yescharacteristic
158C22×C4D26811yescharacteristic
159C22×C4C22×D6711yescharacteristic
160Q16—2684no—
161SD16—2684no—
162C2×C8C22×D6712yes—
163C2×C8C22×D6712yes—
164C4⋊C4—672no—
165C4⋊C4—1341no—
166C4⋊C4—1341no—
167C4⋊C4—1341no—
168C4⋊C4—1341no—
169C4⋊C4—1342no—
170C4⋊C4—1342no—
171C4⋊C4—1341no—
172C22⋊C4—1342no—
173C22⋊C4—1342no—
174C22⋊C4—1342no—
175C42—1342no—
176C42D26811yescharacteristic
177C42D26811yescharacteristic
178C42C22×D6711yescharacteristic
179C42C22×D6711yescharacteristic
180C23—1342no—
181C23C2×D26811yescharacteristic
182Q8—2682no—
183Q8—1344no—
184Q8—1344no—
185D4—2684no—
186D4—2682no—
187D4—2682no—
188D4—1344no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—2684no—
193C2×C4—1342no—
194C2×C4—1342no—
195C2×C4—1342no—
196C2×C4C2×D26811yescharacteristic
197C2×C4C2×D26811yescharacteristic
198C2×C4C2×D26811yescharacteristic
199C2×C4C2×D26811yescharacteristic
200C2×C4C2×D26811yescharacteristic
201C2×C4C23×D6711yescharacteristic
202C8—22no—
203C8—22no—
204C22—21no—
205C22—21no—
206C22—21no—
207C22—2682no—
208C22—2682no—
209C22—2682no—
210C22C22×D26811yescharacteristic
211C4—21no—
212C4—21no—
213C4—21no—
214C4—2682no—
215C4—2682no—
216C4—2682no—
217C4#1452406512yes—
218C4#1452407612yes—
219C4#1452408912yes—
220C2—41no—
221C2—5362no—
222C2#1686396511yescharacteristic
223C2C2×#1452406511yescharacteristic
224C2C2×#1452407611yescharacteristic
subgroupG/H
centerC22C22×D268
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#431×C67C2
socleC22×C67C22×D4
hypercenter#431D67
hypocenterC67#3210
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. #16863776C2semidirect
4. #16863769C2semidirect
5. #16863747C2semidirect
6. #16863743C2semidirect
7. #16863619C2semidirect
8. #16863560C2semidirect
9. #431×C67C2semidirect
113. C67#3210semidirect
122. #431D67semidirect