Finite Group Databasebeta

← #22568270

Subgroups — #22568270

Order 8576.

430 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 430 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
1C2×#14524043C211yesmaximal, characteristic, complemented
2C2×#14524037C211yesmaximal, characteristic
3#16863336C212yesmaximal, complemented
4#16863333C212yesmaximal
5#16863328C212yesmaximal, complemented
6#16863323C212yesmaximal
7#16863317C212yesmaximal, complemented
8#16863309C212yesmaximal
9#383×C67C211yesmaximal, characteristic, complemented
10C22×Dic134C2212yes—
11#14524043—24no—
12#14524037—24no—
13C2×C4⋊Dic67C2212yes—
14C2×C4⋊Dic67C2212yes—
15C2×C4⋊Dic67C2214yes—
16C2×Dic67⋊C4C2211yescharacteristic
17C2×Dic67⋊C4C2211yescharacteristic
18C2×Dic67⋊C4C2211yescharacteristic
19C2×(C2×D67)⋊C4C2211yescharacteristic
20C2×(C2×D67)⋊C4C2211yescharacteristic
21C2×(C2×D67)⋊C4C2211yescharacteristic
22C2×(C2×D67)⋊C4C2214yes—
23C67⋊#95C2214yes—
24C67⋊#95C2214yes—
25C2×C42×C67C2211yescharacteristic
26#95×C67C2212yes—
27#95×C67C2212yes—
28#95×C67C2212yes—
29C23×D67C2311yescharacteristic
30C2×Dic134—24no—
31C2×Dic134—44no—
32C22×Dic67C2311yescharacteristic
33C22×Dic67C2311yescharacteristic
34C22×Dic67C2314yes—
35C22×Dic67C2311yescharacteristic
36C4⋊Dic67—24no—
37C4⋊Dic67—24no—
38C4⋊Dic67—44no—
39Dic67⋊C4—22no—
40Dic67⋊C4—22no—
41Dic67⋊C4—22no—
42(C2×D67)⋊C4—22no—
43(C2×D67)⋊C4—22no—
44(C2×D67)⋊C4—22no—
45(C2×D67)⋊C4—44no—
46C22×C4×C67C2314yes—
47C22×C4×C67C2311yescharacteristic
48C22×C4×C67C2311yescharacteristic
49C22×C4×C67C2311yescharacteristic
50C42×C67D414yes—
51C23×C67C2411yescharacteristic
52C22×D67—21no—
53C22×D67—22no—
54C22×D67—22no—
55C22×D67—22no—
56C2×C4×C67—22no—
57C2×C4×C67—22no—
58C2×C4×C67—22no—
59C2×C4×C67—24no—
60C2×C4×C67—24no—
61C2×C4×C67—24no—
62C2×C4×C67C2×D412yes—
63C2×C4×C67C2×D412yes—
64C2×C4×C67C2×D412yes—
65C2×Dic67—21no—
66C2×Dic67—21no—
67C2×Dic67—21no—
68C2×Dic67—24no—
69C2×Dic67—24no—
70C2×Dic67—22no—
71C2×Dic67—24no—
72C2×Dic67—22no—
73C2×Dic67—22no—
74Dic134—48no—
75C22×C67C22×D411yescharacteristic
76C22×C67H(5,𝔽2)12yes—
77C22×C672−1+412yes—
78C22×C672−1+412yes—
79C2×D67—41no—
80C2×D67—41no—
81C2×D67—41no—
82C2×D67—44no—
83C4×C67—22no—
84C4×C67—22no—
85C4×C67—22no—
86C4×C67—44no—
87Dic67—44no—
88Dic67—41no—
89Dic67—41no—
90Dic67—41no—
91C2×C67#53511yescharacteristic
92C2×C67#53611yescharacteristic
93C2×C67#53611yescharacteristic
94C2×C67#56314yes—
95D67—81no—
96#2821—671nomaximal, Sylow
97C67#282111yescharacteristic, complemented, Sylow
98C2×C4⋊Q8—671no—
99C2×#126—671no—
100#400—672no—
101#399—672no—
102#398—672no—
103#397—672no—
104#395—672no—
105#393—672no—
106#383D6711yescharacteristic, complemented
107C22×Q8—672no—
108C4⋊Q8—1344no—
109#126—1344no—
110C2×C4⋊C4—674no—
111C2×C4⋊C4—671no—
112C2×C4⋊C4—671no—
113C2×C4⋊C4—671no—
114C2×C4⋊C4—672no—
115C2×C4⋊C4—672no—
116C2×C22⋊C4—674no—
117C2×C22⋊C4—671no—
118C2×C22⋊C4—671no—
119C2×C22⋊C4—671no—
120C2×C42C2×D6711yescharacteristic
121#95—674no—
122#95—674no—
123#95C2×D6712yes—
124#95C2×D6712yes—
125#95C2×D6712yes—
126C24—671no—
127C2×Q8—2684no—
128C2×Q8—1344no—
129C22×C4—671no—
130C22×C4—671no—
131C22×C4—674no—
132C22×C4—671no—
133C22×C4C22×D6714yes—
134C22×C4C22×D6711yescharacteristic
135C22×C4C22×D6711yescharacteristic
136C22×C4C22×D6711yescharacteristic
137C4⋊C4—2684no—
138C4⋊C4—1342no—
139C4⋊C4—1342no—
140C4⋊C4—1342no—
141C4⋊C4—1344no—
142C4⋊C4—1344no—
143C22⋊C4—2684no—
144C22⋊C4—1342no—
145C22⋊C4—1342no—
146C22⋊C4—1342no—
147C42D26814yes—
148C23—1341no—
149C23—1342no—
150C23—1342no—
151C23—1342no—
152C23C23×D6711yescharacteristic
153Q8—2688no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—24no—
158C2×C4—24no—
159C2×C4—24no—
160C2×C4—1341no—
161C2×C4—1341no—
162C2×C4—1341no—
163C2×C4—1344no—
164C2×C4—1344no—
165C2×C4—1342no—
166C2×C4—1344no—
167C2×C4—1342no—
168C2×C4—1342no—
169C2×C4C2×D26812yes—
170C2×C4C2×D26812yes—
171C2×C4C2×D26812yes—
172C22—2684no—
173C22—2681no—
174C22—2681no—
175C22—2681no—
176C22C22×D26811yescharacteristic
177C22#1452408912yes—
178C22#1452409012yes—
179C22#1452409012yes—
180C4—22no—
181C4—22no—
182C4—22no—
183C4—44no—
184C4—2684no—
185C4—2681no—
186C4—2681no—
187C4—2681no—
188C2—5361no—
189C2#1686396511yescharacteristic
190C2#1686397211yescharacteristic
191C2#1686397211yescharacteristic
192C2#1686414814yes—
subgroupG/H
center, FrattiniC23C23×D67
commutator, socleC23×C67C24
Fitting#383×C67C2
hypercenter#383D67
hypocenterC67#2821
LCSG ▷ C23×C67 ▷ C67
UCSC1 ◁ C23 ◁ #383

Complements

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

NH ≅ G/Ntype
1. C2×#14524043C2semidirect
3. #16863336C2semidirect
5. #16863328C2semidirect
7. #16863317C2semidirect
9. #383×C67C2semidirect
97. C67#2821semidirect
106. #383D67semidirect