Finite Group Databasebeta

← #22566170

Subgroups — #22566170

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#16863324C212yesmaximal, complemented
2#16863324C212yesmaximal, complemented
3#16863321C212yesmaximal, complemented
4#16863319C212yesmaximal, complemented
5#16863299C211yesmaximal, characteristic
6#16863299C211yesmaximal, characteristic
7#16863295C211yesmaximal, characteristic, complemented
8#16863294C211yesmaximal, characteristic, complemented
9#16863292C211yesmaximal, characteristic, complemented
10#16863242C211yesmaximal, characteristic, complemented
11#390×C67C211yesmaximal, characteristic
12C2×C4⋊Dic67C2211yescharacteristic
13C2×C4⋊Dic67C2211yescharacteristic
14C2×C4⋊Dic67C2212yes—
15C2×Dic67⋊C4C2212yes—
16C2×Dic67⋊C4C2212yes—
17C2×C22⋊Dic67C2212yes—
18C2×C22⋊Dic67C2212yes—
19C2×C22⋊Dic67C2212yes—
20C2×C22⋊Dic67C2212yes—
21C2×C4×Dic67C2211yescharacteristic
22C2×C4×Dic67C2211yescharacteristic
23C67⋊#95C2211yescharacteristic
24C67⋊#95C2211yescharacteristic
25C67⋊#95C2211yescharacteristic
26C67⋊#95C2211yescharacteristic
27C67⋊#95C2212yes—
28C67⋊#95C2212yes—
29C67⋊#95C2212yes—
30C22×D4×C67C2211yescharacteristic
31C2×C4⋊C4×C67C2211yescharacteristic
32C2×C22⋊C4×C67C2212yes—
33C2×C22⋊C4×C67C2212yes—
34C2×C42×C67C2211yescharacteristic
35C22×Dic67C2311yescharacteristic
36C22×Dic67C2311yescharacteristic
37C22×Dic67C2311yescharacteristic
38C22×Dic67C2311yescharacteristic
39C22×Dic67C2312yes—
40C22×Dic67C2312yes—
41C4⋊Dic67—22no—
42C4⋊Dic67—22no—
43C4⋊Dic67—42no—
44Dic67⋊C4—42no—
45Dic67⋊C4—42no—
46C22⋊Dic67—42no—
47C22⋊Dic67—42no—
48C22⋊Dic67—42no—
49C22⋊Dic67—42no—
50C4×Dic67—22no—
51C4×Dic67—22no—
52C24×C67C2312yes—
53C2×D4×C67—22no—
54C2×D4×C67—22no—
55C2×D4×C67—22no—
56C22×C4×C67C2311yescharacteristic
57C22×C4×C67C2311yescharacteristic
58C22×C4×C67C2312yes—
59C22×C4×C67C2311yescharacteristic
60C4⋊C4×C67—22no—
61C22⋊C4×C67—42no—
62C22⋊C4×C67—42no—
63C42×C67—22no—
64C23×C67—22no—
65C23×C67—22no—
66C23×C67—22no—
67C23×C67—22no—
68C23×C67—22no—
69C23×C67—22no—
70C23×C67—22no—
71C23×C67C2411yescharacteristic
72D4×C67—44no—
73C2×C4×C67—21no—
74C2×C4×C67—21no—
75C2×C4×C67—22no—
76C2×C4×C67—21no—
77C2×C4×C67—21no—
78C2×C4×C67—22no—
79C2×C4×C67—22no—
80C2×C4×C67—21no—
81C2×C4×C67—21no—
82C2×C4×C67#4112yes—
83C2×C4×C67#4112yes—
84C2×C4×C67#4112yes—
85C2×Dic67—21no—
86C2×Dic67—21no—
87C2×Dic67—21no—
88C2×Dic67—21no—
89C2×Dic67—22no—
90C2×Dic67—22no—
91C2×Dic67—21no—
92C2×Dic67—21no—
93C2×Dic67—21no—
94C2×Dic67—21no—
95C2×Dic67—22no—
96C2×Dic67—22no—
97C2×Dic67—22no—
98C2×Dic67—22no—
99C2×Dic67—21no—
100C2×Dic67—21no—
101C2×Dic67—21no—
102C2×Dic67—21no—
103C22×C67—42no—
104C22×C67—42no—
105C22×C67—42no—
106C22×C67—42no—
107C22×C67—42no—
108C22×C67—42no—
109C22×C67—42no—
110C22×C67C2×#4111yescharacteristic
111C22×C67C2×#4111yescharacteristic
112C22×C67C2×#4111yescharacteristic
113C22×C67H(5,𝔽2)11yescharacteristic
114C22×C67H(5,𝔽2)11yescharacteristic
115C22×C67H(5,𝔽2)11yescharacteristic
116C22×C672−1+411yescharacteristic
117C4×C67—42no—
118C4×C67—41no—
119C4×C67—41no—
120C4×C67#12614yes—
121Dic67—41no—
122Dic67—41no—
123Dic67—42no—
124Dic67—42no—
125Dic67—41no—
126Dic67—41no—
127C2×C67—82no—
128C2×C67C2×#12611yescharacteristic
129C2×C67#54011yescharacteristic
130C2×C67#54011yescharacteristic
131C2×C67#54011yescharacteristic
132C2×C67#54211yescharacteristic
133C2×C67#54211yescharacteristic
134C2×C67#54211yescharacteristic
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#390D6711yescharacteristic
144#389—671no—
145#389—671no—
146#389—671no—
147#376—671no—
148C22×D4C2×D6711yescharacteristic
149C2×C4⋊C4—672no—
150C2×C4⋊C4—672no—
151C2×C4⋊C4—672no—
152C2×C4⋊C4—671no—
153C2×C4⋊C4—671no—
154C2×C4⋊C4C2×D6711yescharacteristic
155C2×C22⋊C4—672no—
156C2×C22⋊C4—672no—
157C2×C22⋊C4—672no—
158C2×C22⋊C4—672no—
159C2×C22⋊C4C2×D6712yes—
160C2×C22⋊C4C2×D6712yes—
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#95—672no—
171C24C22×D6712yes—
172C2×D4—22no—
173C2×D4—22no—
174C2×D4—22no—
175C22×C4—671no—
176C22×C4—671no—
177C22×C4—671no—
178C22×C4—671no—
179C22×C4—672no—
180C22×C4—672no—
181C22×C4C22×D6711yescharacteristic
182C22×C4C22×D6711yescharacteristic
183C22×C4C22×D6712yes—
184C22×C4C22×D6711yescharacteristic
185C4⋊C4—22no—
186C4⋊C4—2682no—
187C4⋊C4—2682no—
188C4⋊C4—2682no—
189C4⋊C4—1342no—
190C4⋊C4—1342no—
191C22⋊C4—42no—
192C22⋊C4—42no—
193C22⋊C4—2682no—
194C22⋊C4—2682no—
195C22⋊C4—2682no—
196C22⋊C4—2682no—
197C42—22no—
198C42—1342no—
199C42—1342no—
200C23—22no—
201C23—22no—
202C23—22no—
203C23—22no—
204C23—22no—
205C23—22no—
206C23—22no—
207C23C23×D6711yescharacteristic
208D4—44no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—1341no—
219C2×C4—1341no—
220C2×C4—1342no—
221C2×C4—1341no—
222C2×C4—1341no—
223C2×C4—1342no—
224C2×C4—1341no—
225C2×C4—1341no—
226C2×C4—1342no—
227C2×C4—1342no—
228C2×C4—1341no—
229C2×C4—1341no—
230C2×C4—1341no—
231C2×C4—1341no—
232C2×C4—1342no—
233C2×C4—1342no—
234C2×C4—1341no—
235C2×C4—1341no—
236C2×C4#1176162712yes—
237C2×C4#1176162712yes—
238C2×C4#1176162812yes—
239C22—42no—
240C22—42no—
241C22—42no—
242C22—42no—
243C22—42no—
244C22—42no—
245C22—42no—
246C22C2×#1176162711yescharacteristic
247C22C2×#1176162711yescharacteristic
248C22C2×#1176162811yescharacteristic
249C22#1452408811yescharacteristic
250C22#1452408811yescharacteristic
251C22#1452408911yescharacteristic
252C22#1452409011yescharacteristic
253C4—42no—
254C4—41no—
255C4—41no—
256C4—2681no—
257C4—2681no—
258C4—2681no—
259C4—2681no—
260C4—2682no—
261C4—2682no—
262C4#1452403614yes—
263C2—82no—
264C2C2×#1452403611yescharacteristic
265C2#1686399711yescharacteristic
266C2#1686399911yescharacteristic
267C2#1686400011yescharacteristic
268C2#1686401111yescharacteristic
269C2#1686401311yescharacteristic
270C2#1686401511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D67
commutator, socleC23×C67C24
Fitting#390×C67C2
hypercenter#390D67
hypocenterC67#2644
LCSG ▷ C23×C67 ▷ C67
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
1. #16863324C2semidirect
2. #16863324C2semidirect
3. #16863321C2semidirect
4. #16863319C2semidirect
7. #16863295C2semidirect
8. #16863294C2semidirect
9. #16863292C2semidirect
10. #16863242C2semidirect
136. C67#2644semidirect