Finite Group Databasebeta

← #22564653

Subgroups — #22564653

Order 8576.

546 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 157 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
1C2×#14524026C211yesmaximal, characteristic
2C2×#14524002C211yesmaximal, characteristic, complemented
3#16863336C211yesmaximal, characteristic, complemented
4#16863333C211yesmaximal, characteristic, complemented
5#16863328C211yesmaximal, characteristic, complemented
6#16863323C211yesmaximal, characteristic, complemented
7#16863288C212yesmaximal, complemented
8#16863281C212yesmaximal, complemented
9#16863275C212yesmaximal
10#16863256C212yesmaximal, complemented
11C2×#95×C67C211yesmaximal, characteristic, complemented
12C22×C22⋊D67C2211yescharacteristic
13#14524026—24no—
14#14524002—24no—
15C2×C4⋊Dic67C2212yes—
16C2×C4⋊Dic67C2211yescharacteristic
17C2×C4⋊Dic67C2211yescharacteristic
18C2×C4⋊Dic67C2212yes—
19C2×Dic67⋊C4C2211yescharacteristic
20C2×Dic67⋊C4C2211yescharacteristic
21C2×C22⋊Dic67C2211yescharacteristic
22C2×C22⋊Dic67C2212yes—
23C2×(C2×D67)⋊C4C2212yes—
24C2×(C2×D67)⋊C4C2211yescharacteristic
25C2×(C2×D67)⋊C4C2211yescharacteristic
26C2×(C2×D67)⋊C4C2212yes—
27C2×C4×Dic67C2212yes—
28C67⋊#95C2212yes—
29C67⋊#95C2212yes—
30C67⋊#95C2212yes—
31C67⋊#95C2212yes—
32C23×C4×C67C2212yes—
33C23×C4×C67C2211yescharacteristic
34#95×C67C2212yes—
35#95×C67C2211yescharacteristic
36#95×C67C2211yescharacteristic
37C23×D67C2311yescharacteristic
38C2×C22⋊D67—22no—
39C2×C22⋊D67—24no—
40C22×Dic67C2311yescharacteristic
41C22×Dic67C2312yes—
42C22×Dic67C2312yes—
43C22×Dic67C2311yescharacteristic
44C22×Dic67C2311yescharacteristic
45C4⋊Dic67—22no—
46C4⋊Dic67—22no—
47C4⋊Dic67—24no—
48C4⋊Dic67—42no—
49Dic67⋊C4—22no—
50Dic67⋊C4—22no—
51C22⋊Dic67—24no—
52C22⋊Dic67—22no—
53(C2×D67)⋊C4—22no—
54(C2×D67)⋊C4—22no—
55(C2×D67)⋊C4—24no—
56(C2×D67)⋊C4—42no—
57C4×Dic67—24no—
58C24×C67C2311yescharacteristic
59C22×C4×C67—24no—
60C22×C4×C67—24no—
61C22×C4×C67—24no—
62C22×C4×C67—42no—
63C22×C4×C67D414yes—
64C22×C4×C67C2312yes—
65C22×C4×C67C2311yescharacteristic
66C22×C4×C67C2311yescharacteristic
67C22×C4×C67C2312yes—
68C23×C67—22no—
69C23×C67—21no—
70C23×C67—21no—
71C23×C67C2×D412yes—
72C23×C67#4114yes—
73C23×C67C2411yescharacteristic
74C22×D67—21no—
75C22×D67—22no—
76C22×D67—22no—
77C22×D67—21no—
78C22×D67—21no—
79C2×C4×C67—28no—
80C2×C4×C67—22no—
81C2×C4×C67—22no—
82C2×C4×C67—22no—
83C2×C4×C67—22no—
84C2×C4×C67—22no—
85C2×C4×C67—22no—
86C2×C4×C67—22no—
87C2×C4×C67—48no—
88C2×C4×C67C2×D412yes—
89C2×C4×C67C2×D412yes—
90C2×C4×C67#4114yes—
91C22⋊D67—44no—
92C2×Dic67—22no—
93C2×Dic67—22no—
94C2×Dic67—21no—
95C2×Dic67—21no—
96C2×Dic67—21no—
97C2×Dic67—22no—
98C2×Dic67—22no—
99C2×Dic67—22no—
100C2×Dic67—22no—
101C2×Dic67—22no—
102C2×Dic67—21no—
103C2×Dic67—22no—
104C2×Dic67—21no—
105C22×C67—22no—
106C22×C67—22no—
107C22×C67—24no—
108C22×C67—24no—
109C22×C67#12314yes—
110C22×C67C22×D411yescharacteristic
111C22×C67C2×#4112yes—
112C22×C67C2×#4112yes—
113C22×C67H(5,𝔽2)11yescharacteristic
114C22×C672−1+411yescharacteristic
115C2×D67—41no—
116C2×D67—41no—
117C2×D67—41no—
118C2×D67—42no—
119C2×D67—42no—
120C4×C67—22no—
121C4×C67—22no—
122C4×C67—24no—
123C4×C67—42no—
124Dic67—41no—
125Dic67—42no—
126Dic67—41no—
127Dic67—42no—
128Dic67—41no—
129C2×C67—24no—
130C2×C67C2×#12311yescharacteristic
131C2×C67#53211yescharacteristic
132C2×C67#53711yescharacteristic
133C2×C67#55112yes—
134C2×C67#55212yes—
135D67—81no—
136#2498—671nomaximal, Sylow
137C67#249811yescharacteristic, complemented, Sylow
138C2×#123—671no—
139C2×#117—671no—
140#400—671no—
141#399—671no—
142#398—671no—
143#397—671no—
144#388—672no—
145#387—672no—
146#386—672no—
147#380—672no—
148C2×#95D6711yescharacteristic, complemented
149C22×D4—671no—
150C23×C4C2×D6712yes—
151C23×C4C2×D6711yescharacteristic
152#123—1344no—
153#117—1344no—
154C2×C4⋊C4—672no—
155C2×C4⋊C4—671no—
156C2×C4⋊C4—671no—
157C2×C4⋊C4—671no—
158C2×C4⋊C4—671no—
159C2×C4⋊C4—672no—
160C2×C22⋊C4—672no—
161C2×C22⋊C4—672no—
162C2×C22⋊C4—671no—
163C2×C22⋊C4—671no—
164C2×C22⋊C4—671no—
165C2×C22⋊C4—672no—
166C2×C42—672no—
167#95—672no—
168#95—672no—
169#95—672no—
170#95—672no—
171#95C2×D6712yes—
172#95C2×D6711yescharacteristic
173#95C2×D6711yescharacteristic
174C24—671no—
175C24C22×D6711yescharacteristic
176C2×D4—1342no—
177C2×D4—1344no—
178C22×C4—24no—
179C22×C4—24no—
180C22×C4—24no—
181C22×C4—42no—
182C22×C4—671no—
183C22×C4—672no—
184C22×C4—672no—
185C22×C4—671no—
186C22×C4—671no—
187C22×C4D26814yes—
188C22×C4C22×D6712yes—
189C22×C4C22×D6711yescharacteristic
190C22×C4C22×D6711yescharacteristic
191C22×C4C22×D6712yes—
192C4⋊C4—2682no—
193C4⋊C4—1342no—
194C4⋊C4—1342no—
195C4⋊C4—1342no—
196C4⋊C4—1342no—
197C4⋊C4—1344no—
198C22⋊C4—2682no—
199C22⋊C4—1344no—
200C22⋊C4—1342no—
201C22⋊C4—1342no—
202C22⋊C4—1342no—
203C22⋊C4—1344no—
204C42—1344no—
205C23—22no—
206C23—21no—
207C23—21no—
208C23—1341no—
209C23—1342no—
210C23—1342no—
211C23—1341no—
212C23—1341no—
213C23C2×D26812yes—
214C23#1176162714yes—
215C23C23×D6711yescharacteristic
216D4—2684no—
217C2×C4—28no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—48no—
226C2×C4—1342no—
227C2×C4—1342no—
228C2×C4—1341no—
229C2×C4—1341no—
230C2×C4—1341no—
231C2×C4—1342no—
232C2×C4—1342no—
233C2×C4—1342no—
234C2×C4—1342no—
235C2×C4—1342no—
236C2×C4—1341no—
237C2×C4—1342no—
238C2×C4—1341no—
239C2×C4C2×D26812yes—
240C2×C4C2×D26812yes—
241C2×C4#1176162814yes—
242C22—24no—
243C22—24no—
244C22—22no—
245C22—22no—
246C22—2682no—
247C22—2682no—
248C22—2681no—
249C22—2681no—
250C22—2681no—
251C22#1452402714yes—
252C22C22×D26811yescharacteristic
253C22C2×#1176162712yes—
254C22C2×#1176162812yes—
255C22#1452408811yescharacteristic
256C22#1452409111yescharacteristic
257C4—24no—
258C4—22no—
259C4—22no—
260C4—42no—
261C4—2681no—
262C4—2682no—
263C4—2681no—
264C4—2682no—
265C4—2681no—
266C2—24no—
267C2—5361no—
268C2C2×#1452402711yescharacteristic
269C2#1686394611yescharacteristic
270C2#1686397711yescharacteristic
271C2#1686407212yes—
272C2#1686408112yes—
subgroupG/H
center, FrattiniC23C23×D67
commutator, socleC23×C67C24
FittingC2×#95×C67C2
hypercenterC2×#95D67
hypocenterC67#2498
LCSG ▷ C23×C67 ▷ C67
UCSC1 ◁ C23 ◁ C2×#95

Complements

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

NH ≅ G/Ntype
2. C2×#14524002C2semidirect
3. #16863336C2semidirect
4. #16863333C2semidirect
5. #16863328C2semidirect
6. #16863323C2semidirect
7. #16863288C2semidirect
8. #16863281C2semidirect
10. #16863256C2semidirect
11. C2×#95×C67C2semidirect
137. C67#2498semidirect
148. C2×#95D67semidirect