Finite Group Databasebeta

← #22566172

Subgroups — #22566172

Order 8576.

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

normal subgroups only

This group has 420 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#16863325C212yesmaximal, complemented
2#16863323C212yesmaximal, complemented
3#16863322C212yesmaximal, complemented
4#16863318C212yesmaximal, complemented
5#16863298C211yesmaximal, characteristic
6#16863297C211yesmaximal, characteristic
7#16863297C211yesmaximal, characteristic
8#16863295C211yesmaximal, characteristic, complemented
9#16863292C211yesmaximal, characteristic, complemented
10#16863242C211yesmaximal, characteristic, complemented
11#389×C67C211yesmaximal, characteristic, complemented
12C22×D268C2211yescharacteristic
13C2×C4⋊Dic67C2211yescharacteristic
14C2×C4⋊Dic67C2212yes—
15C2×Dic67⋊C4C2212yes—
16C2×(C2×D67)⋊C4C2212yes—
17C2×(C2×D67)⋊C4C2212yes—
18C2×(C2×D67)⋊C4C2212yes—
19C2×(C2×D67)⋊C4C2212yes—
20C2×(C2×D67)⋊C4C2212yes—
21C2×(C2×D67)⋊C4C2212yes—
22C2×C4×Dic67C2211yescharacteristic
23C2×C4×Dic67C2211yescharacteristic
24C67⋊#95C2211yescharacteristic
25C67⋊#95C2211yescharacteristic
26C67⋊#95C2211yescharacteristic
27C67⋊#95C2211yescharacteristic
28C67⋊#95C2212yes—
29C67⋊#95C2212yes—
30C2×C4⋊C4×C67C2211yescharacteristic
31C2×C4⋊C4×C67C2211yescharacteristic
32C2×C4⋊C4×C67C2212yes—
33C2×C42×C67C2211yescharacteristic
34#95×C67C2212yes—
35C23×D67C2312yes—
36C2×D268—22no—
37C2×D268—22no—
38C2×D268—22no—
39C22×Dic67C2311yescharacteristic
40C22×Dic67C2311yescharacteristic
41C22×Dic67C2311yescharacteristic
42C22×Dic67C2311yescharacteristic
43C22×Dic67C2312yes—
44C4⋊Dic67—22no—
45C4⋊Dic67—42no—
46Dic67⋊C4—42no—
47(C2×D67)⋊C4—42no—
48(C2×D67)⋊C4—42no—
49(C2×D67)⋊C4—42no—
50(C2×D67)⋊C4—42no—
51(C2×D67)⋊C4—42no—
52(C2×D67)⋊C4—42no—
53C4×Dic67—22no—
54C4×Dic67—22no—
55C22×C4×C67C2311yescharacteristic
56C22×C4×C67C2311yescharacteristic
57C22×C4×C67C2312yes—
58C22×C4×C67C2312yes—
59C22×C4×C67C2311yescharacteristic
60C4⋊C4×C67—22no—
61C4⋊C4×C67—22no—
62C4⋊C4×C67—42no—
63C42×C67—22no—
64C23×C67C2411yescharacteristic
65C22×D67—22no—
66C22×D67—22no—
67C22×D67—22no—
68C22×D67—22no—
69C22×D67—22no—
70C22×D67—22no—
71C22×D67—22no—
72C2×C4×C67—21no—
73C2×C4×C67—21no—
74C2×C4×C67—22no—
75C2×C4×C67—22no—
76C2×C4×C67—21no—
77C2×C4×C67—22no—
78C2×C4×C67—22no—
79C2×C4×C67—21no—
80C2×C4×C67—21no—
81C2×C4×C67—22no—
82C2×C4×C67—22no—
83C2×C4×C67—21no—
84C2×C4×C67#4112yes—
85C2×C4×C67#4112yes—
86C2×C4×C67#4112yes—
87C2×Dic67—21no—
88C2×Dic67—21no—
89C2×Dic67—21no—
90C2×Dic67—21no—
91C2×Dic67—22no—
92C2×Dic67—21no—
93C2×Dic67—21no—
94C2×Dic67—21no—
95C2×Dic67—21no—
96C2×Dic67—22no—
97C2×Dic67—22no—
98C2×Dic67—21no—
99C2×Dic67—21no—
100C2×Dic67—21no—
101C2×Dic67—21no—
102D268—44no—
103C22×C67C2×#4111yescharacteristic
104C22×C67C2×#4111yescharacteristic
105C22×C67C2×#4111yescharacteristic
106C22×C67H(5,𝔽2)11yescharacteristic
107C22×C67H(5,𝔽2)11yescharacteristic
108C22×C67H(5,𝔽2)11yescharacteristic
109C22×C672−1+411yescharacteristic
110C2×D67—42no—
111C2×D67—42no—
112C2×D67—42no—
113C2×D67—42no—
114C2×D67—42no—
115C2×D67—42no—
116C2×D67—42no—
117C4×C67—42no—
118C4×C67—41no—
119C4×C67—42no—
120C4×C67—41no—
121C4×C67#12614yes—
122Dic67—41no—
123Dic67—42no—
124Dic67—41no—
125Dic67—41no—
126Dic67—41no—
127C2×C67C2×#12611yescharacteristic
128C2×C67#54011yescharacteristic
129C2×C67#54011yescharacteristic
130C2×C67#54011yescharacteristic
131C2×C67#54211yescharacteristic
132C2×C67#54211yescharacteristic
133C2×C67#54211yescharacteristic
134D67—82no—
135#2644—671nomaximal, Sylow
136C67#264411yescharacteristic, complemented, Sylow
137#397—672no—
138#397—672no—
139#397—672no—
140#396—672no—
141#390—671no—
142#390—671no—
143#390—671no—
144#389—671no—
145#389—671no—
146#389D6711yescharacteristic, complemented
147#376—671no—
148C22×D4—671no—
149C2×C4⋊C4—672no—
150C2×C4⋊C4—672no—
151C2×C4⋊C4—671no—
152C2×C4⋊C4C2×D6711yescharacteristic
153C2×C4⋊C4C2×D6711yescharacteristic
154C2×C4⋊C4C2×D6712yes—
155C2×C22⋊C4—672no—
156C2×C22⋊C4—672no—
157C2×C22⋊C4—672no—
158C2×C22⋊C4—672no—
159C2×C22⋊C4—672no—
160C2×C22⋊C4—672no—
161C2×C42—671no—
162C2×C42—671no—
163C2×C42C2×D6711yescharacteristic
164#95—671no—
165#95—671no—
166#95—671no—
167#95—671no—
168#95—672no—
169#95—672no—
170#95C2×D6712yes—
171C24—672no—
172C2×D4—1342no—
173C2×D4—1342no—
174C2×D4—1342no—
175C22×C4—671no—
176C22×C4—671no—
177C22×C4—671no—
178C22×C4—671no—
179C22×C4—672no—
180C22×C4C22×D6711yescharacteristic
181C22×C4C22×D6711yescharacteristic
182C22×C4C22×D6712yes—
183C22×C4C22×D6712yes—
184C22×C4C22×D6711yescharacteristic
185C4⋊C4—22no—
186C4⋊C4—22no—
187C4⋊C4—42no—
188C4⋊C4—2682no—
189C4⋊C4—2682no—
190C4⋊C4—1342no—
191C22⋊C4—2682no—
192C22⋊C4—2682no—
193C22⋊C4—2682no—
194C22⋊C4—2682no—
195C22⋊C4—2682no—
196C22⋊C4—2682no—
197C42—22no—
198C42—1342no—
199C42—1342no—
200C23—1342no—
201C23—1342no—
202C23—1342no—
203C23—1342no—
204C23—1342no—
205C23—1342no—
206C23—1342no—
207C23C23×D6711yescharacteristic
208D4—2684no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—1341no—
222C2×C4—1341no—
223C2×C4—1341no—
224C2×C4—1341no—
225C2×C4—1342no—
226C2×C4—1341no—
227C2×C4—1341no—
228C2×C4—1341no—
229C2×C4—1341no—
230C2×C4—1342no—
231C2×C4—1342no—
232C2×C4—1341no—
233C2×C4—1341no—
234C2×C4—1341no—
235C2×C4—1341no—
236C2×C4#1176162712yes—
237C2×C4#1176162812yes—
238C2×C4#1176162912yes—
239C22—2682no—
240C22—2682no—
241C22—2682no—
242C22—2682no—
243C22—2682no—
244C22—2682no—
245C22—2682no—
246C22C2×#1176162711yescharacteristic
247C22C2×#1176162811yescharacteristic
248C22C2×#1176162911yescharacteristic
249C22#1452408811yescharacteristic
250C22#1452408911yescharacteristic
251C22#1452408911yescharacteristic
252C22#1452409111yescharacteristic
253C4—41no—
254C4—42no—
255C4—41no—
256C4—42no—
257C4—2681no—
258C4—2682no—
259C4—2681no—
260C4—2681no—
261C4—2681no—
262C4#1452403814yes—
263C2—5362no—
264C2C2×#1452403811yescharacteristic
265C2#1686399711yescharacteristic
266C2#1686399911yescharacteristic
267C2#1686400211yescharacteristic
268C2#1686401811yescharacteristic
269C2#1686401911yescharacteristic
270C2#1686402011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D67
commutator, socleC23×C67C24
Fitting#389×C67C2
hypercenter#389D67
hypocenterC67#2644
LCSG ▷ C23×C67 ▷ C67
UCSC1 ◁ C23 ◁ #389

Complements

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

NH ≅ G/Ntype
1. #16863325C2semidirect
2. #16863323C2semidirect
3. #16863322C2semidirect
4. #16863318C2semidirect
8. #16863295C2semidirect
9. #16863292C2semidirect
10. #16863242C2semidirect
11. #389×C67C2semidirect
136. C67#2644semidirect
146. #389D67semidirect