Finite Group Databasebeta

← #22565992

Subgroups — #22565992

Order 8576.

526 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 149 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
2C22×C4⋊Dic67C211yesmaximal, characteristic, complemented
3#16863336C212yesmaximal, complemented
4#16863333C212yesmaximal, complemented
5#16863328C212yesmaximal, complemented
6#16863323C212yesmaximal, complemented
7#16863273C212yesmaximal
8#16863271C212yesmaximal, complemented
9#381×C67C211yesmaximal, characteristic, complemented
10C22×C22⋊D67C2211yescharacteristic
11C23×Dic67C2212yes—
12#14524026—24no—
13C2×C4⋊Dic67—24no—
14C2×C4⋊Dic67C2212yes—
15C2×C4⋊Dic67C2212yes—
16C2×C4⋊Dic67C2214yes—
17C2×Dic67⋊C4C2211yescharacteristic
18C2×Dic67⋊C4C2211yescharacteristic
19C2×C22⋊Dic67C2211yescharacteristic
20C2×(C2×D67)⋊C4C2211yescharacteristic
21C2×(C2×D67)⋊C4C2211yescharacteristic
22C2×(C2×D67)⋊C4C2214yes—
23C67⋊#95C2214yes—
24C67⋊#95C2214yes—
25C23×C4×C67C2211yescharacteristic
26C2×C22⋊C4×C67C2212yes—
27#95×C67C2212yes—
28#95×C67C2212yes—
29C23×D67C2311yescharacteristic
30C2×C22⋊D67—22no—
31C2×C22⋊D67—24no—
32C22×Dic67—24no—
33C22×Dic67—44no—
34C22×Dic67C2311yescharacteristic
35C22×Dic67C2311yescharacteristic
36C22×Dic67C2314yes—
37C22×Dic67C2311yescharacteristic
38C4⋊Dic67—24no—
39C4⋊Dic67—24no—
40C4⋊Dic67—44no—
41Dic67⋊C4—22no—
42Dic67⋊C4—22no—
43C22⋊Dic67—22no—
44(C2×D67)⋊C4—22no—
45(C2×D67)⋊C4—22no—
46(C2×D67)⋊C4—44no—
47C24×C67C2311yescharacteristic
48C22×C4×C67—24no—
49C22×C4×C67D414yes—
50C22×C4×C67C2314yes—
51C22×C4×C67C2311yescharacteristic
52C22×C4×C67C2311yescharacteristic
53C22⋊C4×C67—24no—
54C23×C67—22no—
55C23×C67—22no—
56C23×C67C2×D412yes—
57C23×C67#4114yes—
58C23×C67C2411yescharacteristic
59C22×D67—21no—
60C22×D67—22no—
61C22×D67—22no—
62C22×D67—22no—
63C2×C4×C67—28no—
64C2×C4×C67—22no—
65C2×C4×C67—22no—
66C2×C4×C67—24no—
67C2×C4×C67—24no—
68C2×C4×C67—24no—
69C2×C4×C67C2×D412yes—
70C2×C4×C67C2×D412yes—
71C22⋊D67—44no—
72C2×Dic67—21no—
73C2×Dic67—21no—
74C2×Dic67—21no—
75C2×Dic67—24no—
76C2×Dic67—22no—
77C2×Dic67—24no—
78C2×Dic67—22no—
79C2×Dic67—24no—
80C2×Dic67—22no—
81C2×Dic67—48no—
82C22×C67—22no—
83C22×C67—22no—
84C22×C67—28no—
85C22×C67#12314yes—
86C22×C67C22×D411yescharacteristic
87C22×C67C2×#4112yes—
88C22×C67H(5,𝔽2)12yes—
89C22×C672−1+412yes—
90C2×D67—41no—
91C2×D67—41no—
92C2×D67—41no—
93C2×D67—44no—
94C4×C67—22no—
95C4×C67—22no—
96C4×C67—44no—
97Dic67—41no—
98Dic67—41no—
99Dic67—44no—
100Dic67—41no—
101C2×C67—24no—
102C2×C67C2×#12311yescharacteristic
103C2×C67#53511yescharacteristic
104C2×C67#53611yescharacteristic
105C2×C67#53914yes—
106D67—81no—
107#2623—671nomaximal, Sylow
108C67#262311yescharacteristic, complemented, Sylow
109C2×#123—671no—
110C22×C4⋊C4—671no—
111#400—672no—
112#399—672no—
113#398—672no—
114#397—672no—
115#386—672no—
116#385—672no—
117#381D6711yescharacteristic, complemented
118C22×D4—671no—
119C23×C4—672no—
120C23×C4C2×D6711yescharacteristic
121#123—1344no—
122C2×C4⋊C4—674no—
123C2×C4⋊C4—672no—
124C2×C4⋊C4—672no—
125C2×C4⋊C4—671no—
126C2×C4⋊C4—671no—
127C2×C4⋊C4—1344no—
128C2×C22⋊C4—674no—
129C2×C22⋊C4—671no—
130C2×C22⋊C4—671no—
131C2×C22⋊C4—671no—
132C2×C22⋊C4C2×D6712yes—
133#95—674no—
134#95—674no—
135#95C2×D6712yes—
136#95C2×D6712yes—
137C24—671no—
138C24C22×D6711yescharacteristic
139C2×D4—1342no—
140C2×D4—1344no—
141C22×C4—24no—
142C22×C4—2684no—
143C22×C4—671no—
144C22×C4—671no—
145C22×C4—674no—
146C22×C4—671no—
147C22×C4—1344no—
148C22×C4D26814yes—
149C22×C4C22×D6714yes—
150C22×C4C22×D6711yescharacteristic
151C22×C4C22×D6711yescharacteristic
152C4⋊C4—2684no—
153C4⋊C4—1344no—
154C4⋊C4—1344no—
155C4⋊C4—1342no—
156C4⋊C4—1342no—
157C22⋊C4—24no—
158C22⋊C4—2684no—
159C22⋊C4—1342no—
160C22⋊C4—1342no—
161C22⋊C4—1342no—
162C23—22no—
163C23—22no—
164C23—1341no—
165C23—1342no—
166C23—1342no—
167C23—1342no—
168C23C2×D26812yes—
169C23#1176162714yes—
170C23C23×D6711yescharacteristic
171D4—2684no—
172C2×C4—28no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—24no—
176C2×C4—24no—
177C2×C4—24no—
178C2×C4—2688no—
179C2×C4—1341no—
180C2×C4—1341no—
181C2×C4—1344no—
182C2×C4—1341no—
183C2×C4—1342no—
184C2×C4—1344no—
185C2×C4—1342no—
186C2×C4—1344no—
187C2×C4—1342no—
188C2×C4C2×D26812yes—
189C2×C4C2×D26812yes—
190C22—22no—
191C22—22no—
192C22—28no—
193C22—2681no—
194C22—2681no—
195C22—2681no—
196C22—2684no—
197C22#1452402714yes—
198C22C22×D26811yescharacteristic
199C22C2×#1176162712yes—
200C22#1452408912yes—
201C22#1452409012yes—
202C4—22no—
203C4—22no—
204C4—44no—
205C4—2681no—
206C4—2684no—
207C4—2681no—
208C4—2681no—
209C2—24no—
210C2—5361no—
211C2C2×#1452402711yescharacteristic
212C2#1686396511yescharacteristic
213C2#1686397211yescharacteristic
214C2#1686399014yes—
subgroupG/H
center, FrattiniC23C23×D67
commutator, socleC23×C67C24
Fitting#381×C67C2
hypercenter#381D67
hypocenterC67#2623
LCSG ▷ C23×C67 ▷ C67
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
2. C22×C4⋊Dic67C2semidirect
3. #16863336C2semidirect
4. #16863333C2semidirect
5. #16863328C2semidirect
6. #16863323C2semidirect
8. #16863271C2semidirect
9. #381×C67C2semidirect
108. C67#2623semidirect
117. #381D67semidirect