Finite Group Databasebeta

← #22564808

Subgroups — #22564808

Order 8576.

440 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 137 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#16863299C211yesmaximal, characteristic
4#16863299C211yesmaximal, characteristic
5#16863292C211yesmaximal, characteristic, complemented
6#16863290C212yesmaximal, complemented
7#16863284C212yesmaximal, complemented
8#16863262C211yesmaximal, characteristic, complemented
9C4⋊C4×Dic67C211yesmaximal, characteristic, complemented
10C4×Dic67⋊C4C211yesmaximal, characteristic, complemented
11#390×C67C211yesmaximal, characteristic
12C2×C4⋊Dic67C2211yescharacteristic
13C2×C4⋊Dic67C2211yescharacteristic
14C2×Dic67⋊C4C2211yescharacteristic
15C2×Dic67⋊C4C2211yescharacteristic
16C2×Dic67⋊C4C2212yes—
17C2×C22⋊Dic67C2212yes—
18C2×C22⋊Dic67C2212yes—
19C2×C22⋊Dic67C2212yes—
20C2×C22⋊Dic67C2212yes—
21C2×C4×Dic67C2211yescharacteristic
22C2×C4×Dic67C2211yescharacteristic
23C2×C4×Dic67C2212yes—
24C67⋊#95C2211yescharacteristic
25C67⋊#95C2211yescharacteristic
26C67⋊#95C2212yes—
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×Dic67C2312yes—
37C22×Dic67C2312yes—
38C22×Dic67C2311yescharacteristic
39C22×Dic67C2311yescharacteristic
40C22×Dic67C2311yescharacteristic
41C4⋊Dic67—22no—
42C4⋊Dic67—22no—
43Dic67⋊C4—22no—
44Dic67⋊C4—22no—
45Dic67⋊C4—42no—
46C22⋊Dic67—24no—
47C22⋊Dic67—24no—
48C22⋊Dic67—42no—
49C22⋊Dic67—42no—
50C4×Dic67—22no—
51C4×Dic67—22no—
52C4×Dic67—42no—
53C24×C67C2312yes—
54C2×D4×C67—22no—
55C2×D4×C67—22no—
56C2×D4×C67—22no—
57C22×C4×C67C2311yescharacteristic
58C22×C4×C67C2311yescharacteristic
59C22×C4×C67C2312yes—
60C22×C4×C67C2311yescharacteristic
61C4⋊C4×C67—22no—
62C22⋊C4×C67—42no—
63C22⋊C4×C67—42no—
64C42×C67—22no—
65C23×C67—22no—
66C23×C67—22no—
67C23×C67—22no—
68C23×C67—22no—
69C23×C67—22no—
70C23×C67—22no—
71C23×C67—22no—
72C23×C67C2411yescharacteristic
73D4×C67—44no—
74C2×C4×C67—21no—
75C2×C4×C67—21no—
76C2×C4×C67—22no—
77C2×C4×C67—21no—
78C2×C4×C67—21no—
79C2×C4×C67—21no—
80C2×C4×C67—21no—
81C2×C4×C67—22no—
82C2×C4×C67—22no—
83C2×C4×C67#4112yes—
84C2×C4×C67#4112yes—
85C2×C4×C67#4112yes—
86C2×Dic67—21no—
87C2×Dic67—21no—
88C2×Dic67—21no—
89C2×Dic67—21no—
90C2×Dic67—21no—
91C2×Dic67—21no—
92C2×Dic67—22no—
93C2×Dic67—22no—
94C2×Dic67—21no—
95C2×Dic67—22no—
96C2×Dic67—22no—
97C2×Dic67—22no—
98C2×Dic67—21no—
99C2×Dic67—21no—
100C2×Dic67—22no—
101C2×Dic67—21no—
102C2×Dic67#4112yes—
103C2×Dic67#4112yes—
104C22×C67—42no—
105C22×C67—42no—
106C22×C67—42no—
107C22×C67—42no—
108C22×C67—42no—
109C22×C67—42no—
110C22×C67—42no—
111C22×C67C2×#4111yescharacteristic
112C22×C67C2×#4111yescharacteristic
113C22×C67C2×#4111yescharacteristic
114C22×C67C2×#4111yescharacteristic
115C22×C67C2×#4111yescharacteristic
116C22×C67H(5,𝔽2)11yescharacteristic
117C22×C67H(5,𝔽2)11yescharacteristic
118C4×C67—42no—
119C4×C67—41no—
120C4×C67—41no—
121C4×C67#12614yes—
122Dic67—22no—
123Dic67—22no—
124Dic67—41no—
125Dic67—41no—
126Dic67—42no—
127Dic67—42no—
128C2×C67—82no—
129C2×C67C2×#12611yescharacteristic
130C2×C67#52911yescharacteristic
131C2×C67#54011yescharacteristic
132C2×C67#55311yescharacteristic
133C2×C67#55511yescharacteristic
134C2×C67#55911yescharacteristic
135C2×C67#55911yescharacteristic
136#2513—671nomaximal, Sylow
137C67#251311yescharacteristic, complemented, Sylow
138#397—672no—
139#397—672no—
140#390—671no—
141#390—671no—
142#390D6711yescharacteristic
143#389—671no—
144#388—672no—
145#388—672no—
146#382—671no—
147C4×C4⋊C4—671no—
148C4×C4⋊C4—671no—
149C22×D4C2×D6711yescharacteristic
150C2×C4⋊C4—672no—
151C2×C4⋊C4—671no—
152C2×C4⋊C4—671no—
153C2×C4⋊C4—671no—
154C2×C4⋊C4—671no—
155C2×C4⋊C4C2×D6711yescharacteristic
156C2×C22⋊C4—672no—
157C2×C22⋊C4—672no—
158C2×C22⋊C4—672no—
159C2×C22⋊C4—672no—
160C2×C22⋊C4C2×D6712yes—
161C2×C22⋊C4C2×D6712yes—
162C2×C42—672no—
163C2×C42—671no—
164C2×C42—671no—
165C2×C42C2×D6711yescharacteristic
166#95—671no—
167#95—672no—
168#95—671no—
169#95—672no—
170#95—672no—
171#95—672no—
172C24C22×D6712yes—
173C2×D4—22no—
174C2×D4—22no—
175C2×D4—22no—
176C22×C4—671no—
177C22×C4—671no—
178C22×C4—672no—
179C22×C4—671no—
180C22×C4—672no—
181C22×C4—671no—
182C22×C4C22×D6711yescharacteristic
183C22×C4C22×D6711yescharacteristic
184C22×C4C22×D6712yes—
185C22×C4C22×D6711yescharacteristic
186C4⋊C4—22no—
187C4⋊C4—2682no—
188C4⋊C4—1342no—
189C4⋊C4—1342no—
190C4⋊C4—1342no—
191C4⋊C4—1342no—
192C22⋊C4—42no—
193C22⋊C4—42no—
194C22⋊C4—2682no—
195C22⋊C4—2682no—
196C22⋊C4—1344no—
197C22⋊C4—1344no—
198C42—22no—
199C42—2682no—
200C42—1342no—
201C42—1342no—
202C23—22no—
203C23—22no—
204C23—22no—
205C23—22no—
206C23—22no—
207C23—22no—
208C23—22no—
209C23C23×D6711yescharacteristic
210D4—44no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—672no—
221C2×C4—672no—
222C2×C4—1342no—
223C2×C4—1342no—
224C2×C4—1341no—
225C2×C4—1341no—
226C2×C4—1341no—
227C2×C4—1342no—
228C2×C4—1342no—
229C2×C4—1342no—
230C2×C4—1341no—
231C2×C4—1341no—
232C2×C4—1342no—
233C2×C4—1341no—
234C2×C4—1341no—
235C2×C4—1341no—
236C2×C4—1341no—
237C2×C4—1341no—
238C2×C4#1176162712yes—
239C2×C4#1176162712yes—
240C2×C4#1176162812yes—
241C22—42no—
242C22—42no—
243C22—42no—
244C22—42no—
245C22—42no—
246C22—42no—
247C22—42no—
248C22#41×D6711yescharacteristic
249C22#41×D6711yescharacteristic
250C22C2×#1176162711yescharacteristic
251C22C2×#1176162711yescharacteristic
252C22C2×#1176162811yescharacteristic
253C22#1452408811yescharacteristic
254C22#1452408811yescharacteristic
255C4—42no—
256C4—41no—
257C4—41no—
258C4—2681no—
259C4—2681no—
260C4—2682no—
261C4—2682no—
262C4—1342no—
263C4—1342no—
264C4#1452403614yes—
265C2—82no—
266C2C2×#1452403611yescharacteristic
267C2#1686392511yescharacteristic
268C2#1686400011yescharacteristic
269C2#1686409311yescharacteristic
270C2#1686411011yescharacteristic
271C2#1686412911yescharacteristic
272C2#1686412911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D67
commutator, socleC23×C67C24
Fitting#390×C67C2
hypercenter#390D67
hypocenterC67#2513
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
5. #16863292C2semidirect
6. #16863290C2semidirect
7. #16863284C2semidirect
8. #16863262C2semidirect
9. C4⋊C4×Dic67C2semidirect
10. C4×Dic67⋊C4C2semidirect
137. C67#2513semidirect