Finite Group Databasebeta

← #22567115

Subgroups — #22567115

Order 8576.

448 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 141 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×#14524036C212yesmaximal
2#16863328C212yesmaximal, complemented
3#16863326C212yesmaximal, complemented
4#16863321C211yesmaximal, characteristic
5#16863321C211yesmaximal, characteristic
6#16863295C212yesmaximal, complemented
7#16863281C212yesmaximal, complemented
8#16863256C212yesmaximal
9#385×C67C211yesmaximal, characteristic
10#14524036—28no—
11C2×C4⋊Dic67C2212yes—
12C2×C4⋊Dic67C2212yes—
13C2×C4⋊Dic67C2212yes—
14C2×C4⋊Dic67C2211yescharacteristic
15C2×C4⋊Dic67C2211yescharacteristic
16C2×Dic67⋊C4C2212yes—
17C2×Dic67⋊C4C2212yes—
18C2×C22⋊Dic67C2212yes—
19C2×C22⋊Dic67C2212yes—
20C2×C4×Dic67C2212yes—
21C67⋊#95C2212yes—
22C67⋊#95C2212yes—
23C67⋊#95C2211yescharacteristic
24C67⋊#95C2212yes—
25C67⋊#95C2212yes—
26C67⋊#95C2211yescharacteristic
27C23×C4×C67C2211yescharacteristic
28C2×C4⋊C4×C67C2212yes—
29C2×C22⋊C4×C67C2211yescharacteristic
30C2×C22⋊C4×C67C2211yescharacteristic
31#95×C67C2212yes—
32C22×Dic67C2312yes—
33C22×Dic67C2312yes—
34C22×Dic67C2312yes—
35C22×Dic67C2312yes—
36C4⋊Dic67—24no—
37C4⋊Dic67—24no—
38C4⋊Dic67—42no—
39C4⋊Dic67—41no—
40C4⋊Dic67—41no—
41Dic67⋊C4—24no—
42Dic67⋊C4—24no—
43C22⋊Dic67—24no—
44C22⋊Dic67—24no—
45C4×Dic67—24no—
46C24×C67C2311yescharacteristic
47C22×C4×C67—24no—
48C22×C4×C67—41no—
49C22×C4×C67C2311yescharacteristic
50C22×C4×C67C2311yescharacteristic
51C22×C4×C67C2312yes—
52C22×C4×C67C2312yes—
53C4⋊C4×C67—24no—
54C22⋊C4×C67—41no—
55C22⋊C4×C67D414yes—
56C23×C67—22no—
57C23×C67—21no—
58C23×C67—22no—
59C23×C67—21no—
60C23×C67C2×D412yes—
61C23×C67C2411yescharacteristic
62C2×C4×C67—22no—
63C2×C4×C67—22no—
64C2×C4×C67—22no—
65C2×C4×C67—22no—
66C2×C4×C67—22no—
67C2×C4×C67—21no—
68C2×C4×C67—21no—
69C2×C4×C67—22no—
70C2×C4×C67—44no—
71C2×C4×C67C2×D414yes—
72C2×C4×C67#4114yes—
73C2×Dic67—22no—
74C2×Dic67—22no—
75C2×Dic67—22no—
76C2×Dic67—22no—
77C2×Dic67—22no—
78C2×Dic67—22no—
79C2×Dic67—22no—
80C2×Dic67—22no—
81C2×Dic67—22no—
82C2×Dic67—22no—
83C2×Dic67—22no—
84C2×Dic67—22no—
85C22×C67—22no—
86C22×C67—24no—
87C22×C67—42no—
88C22×C67—41no—
89C22×C67—41no—
90C22×C67C22×D411yescharacteristic
91C22×C67C2×#4112yes—
92C22×C67H(5,𝔽2)11yescharacteristic
93C22×C672−1+412yes—
94C22×C672−1+411yescharacteristic
95C4×C67—24no—
96C4×C67—24no—
97C4×C67—41no—
98C4×C67—41no—
99Dic67—42no—
100Dic67—42no—
101Dic67—42no—
102Dic67—42no—
103C2×C67—42no—
104C2×C67#53711yescharacteristic
105C2×C67#54112yes—
106C2×C67#54712yes—
107C2×C67#55111yescharacteristic
108C2×C67#56311yescharacteristic
109#2722—671nomaximal, Sylow
110C67#272211yescharacteristic, complemented, Sylow
111C2×#126—672no—
112#398—672no—
113#398—672no—
114#397—671no—
115#397—671no—
116#389—672no—
117#387—672no—
118#385D6711yescharacteristic
119#380—672no—
120C23×C4C2×D6711yescharacteristic
121#126—1348no—
122C2×C4⋊C4—672no—
123C2×C4⋊C4—671no—
124C2×C4⋊C4—671no—
125C2×C4⋊C4—672no—
126C2×C4⋊C4—672no—
127C2×C4⋊C4—672no—
128C2×C4⋊C4—672no—
129C2×C4⋊C4C2×D6712yes—
130C2×C22⋊C4—672no—
131C2×C22⋊C4—672no—
132C2×C22⋊C4C2×D6711yescharacteristic
133C2×C22⋊C4C2×D6711yescharacteristic
134C2×C42—672no—
135#95—672no—
136#95—672no—
137#95—671no—
138#95—672no—
139#95—672no—
140#95—671no—
141#95C2×D6712yes—
142C24C22×D6711yescharacteristic
143C22×C4—24no—
144C22×C4—41no—
145C22×C4—672no—
146C22×C4—672no—
147C22×C4—672no—
148C22×C4—672no—
149C22×C4C22×D6711yescharacteristic
150C22×C4C22×D6711yescharacteristic
151C22×C4C22×D6712yes—
152C22×C4C22×D6712yes—
153C4⋊C4—24no—
154C4⋊C4—2682no—
155C4⋊C4—2681no—
156C4⋊C4—2681no—
157C4⋊C4—1344no—
158C4⋊C4—1344no—
159C4⋊C4—1344no—
160C4⋊C4—1344no—
161C22⋊C4—41no—
162C22⋊C4—1344no—
163C22⋊C4—1344no—
164C22⋊C4D26814yes—
165C42—1344no—
166C23—22no—
167C23—21no—
168C23—22no—
169C23—21no—
170C23C2×D26812yes—
171C23C23×D6711yescharacteristic
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—22no—
180C2×C4—44no—
181C2×C4—1342no—
182C2×C4—1342no—
183C2×C4—1342no—
184C2×C4—1342no—
185C2×C4—1342no—
186C2×C4—1342no—
187C2×C4—1342no—
188C2×C4—1342no—
189C2×C4—1342no—
190C2×C4—1342no—
191C2×C4—1342no—
192C2×C4—1342no—
193C2×C4C2×D26814yes—
194C2×C4#1176162714yes—
195C22—24no—
196C22—22no—
197C22—42no—
198C22—41no—
199C22—41no—
200C22C22×D26811yescharacteristic
201C22C2×#1176162712yes—
202C22#1452408811yescharacteristic
203C22#1452409012yes—
204C22#1452409111yescharacteristic
205C4—24no—
206C4—24no—
207C4—41no—
208C4—41no—
209C4—2682no—
210C4—2682no—
211C4—2682no—
212C4—2682no—
213C2—42no—
214C2#1686397711yescharacteristic
215C2#1686400612yes—
216C2#1686405012yes—
217C2#1686407711yescharacteristic
218C2#1686414911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D67
commutator, socleC23×C67C24
Fitting#385×C67C2
hypercenter#385D67
hypocenterC67#2722
LCSG ▷ C23×C67 ▷ C67
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
2. #16863328C2semidirect
3. #16863326C2semidirect
6. #16863295C2semidirect
7. #16863281C2semidirect
110. C67#2722semidirect