Finite Group Databasebeta

← #22561900

Subgroups — #22561900

Order 8576.

262 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 65 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#16863960C211yesmaximal, characteristic
2#16863380C211yesmaximal, characteristic, complemented
3#16863192C214yesmaximal, complemented
4#413×C67C211yesmaximal, characteristic, complemented
5C22×D268C411yescharacteristic
6C2×#11761612—21no—
7#14524039C412yes—
8#14524032C2212yes—
9#14524017—21no—
10#14524017—21no—
11#14524011—21no—
12#14524011—21no—
13#14524002C411yescharacteristic
14#14523962C2212yescomplemented
15#14523960—21no—
16C2×#41×C67C2211yescharacteristic
17C2×M16×C67—21no—
18#101×C67C2212yescomplemented
19#100×C67—21no—
20C23×D67—21no—
21C2×D268—21no—
22C2×D268—41no—
23C2×D268—41no—
24C2×D268C2×C412yes—
25C2×C22⋊D67—22no—
26C2×C22⋊D67—22no—
27#11761612—22no—
28#11761612—41no—
29C2×C67⋊C8—41no—
30C4⋊Dic67—21no—
31C22⋊Dic67—21no—
32(C2×D67)⋊C4—22no—
33(C2×D67)⋊C4—22no—
34C4×Dic67C2×C412yes—
35#41×C67—41no—
36#41×C67—41no—
37C2×Q8×C67C2311yescharacteristic
38C2×D4×C67C2×C411yescharacteristic
39C2×D4×C67D411yescharacteristic
40C2×D4×C67D411yescharacteristic
41C22×C4×C67C2×C411yescharacteristic
42C22×C4×C67D411yescharacteristic
43C22×C4×C67D411yescharacteristic
44M16×C67—22no—
45M16×C67—41no—
46C2×C8×C67—41no—
47C23×C67C22⋊C411yescharacteristic
48C23×C67C22⋊C411yescharacteristic
49C23×C67C22⋊C411yescharacteristic
50C22×D67—22no—
51C22×D67—42no—
52C22×D67—82no—
53Q8×C67—41no—
54D4×C67—41no—
55D4×C67—41no—
56D4×C67—41no—
57C2×C4×C67—21no—
58C2×C4×C67—21no—
59C2×C4×C67—21no—
60C2×C4×C67—21no—
61C2×C4×C67—21no—
62C2×C4×C67—21no—
63C2×C4×C67C22⋊C411yescharacteristic
64C2×C4×C67C22×C411yescharacteristic
65C2×C4×C67C2×D411yescharacteristic
66C2×C4×C67C2×D411yescharacteristic
67C22⋊D67—82no—
68C22⋊D67—82no—
69C2×Dic67—22no—
70D268—42no—
71D268—41no—
72D268—41no—
73C8×C67—42no—
74C67⋊C8—42no—
75C22×C67—21no—
76C22×C67—21no—
77C22×C67—21no—
78C22×C67—41no—
79C22×C67—41no—
80C22×C67—41no—
81C22×C67C2×C22⋊C411yescharacteristic
82C2×D67—42no—
83C2×D67—42no—
84C2×D67—42no—
85C2×D67—84no—
86C4×C67—21no—
87C4×C67—21no—
88C4×C67—21no—
89C4×C67#9911yescharacteristic
90C4×C67#9911yescharacteristic
91Dic67—42no—
92C2×C67—21no—
93C2×C67—41no—
94C2×C67—41no—
95C2×C67—41no—
96C2×C67C2×#9911yescharacteristic
97D67—82no—
98#2134—671nomaximal, Sylow
99C67#213411yescharacteristic, complemented, Sylow
100#535—671no—
101#413—671no—
102#413D6711yescharacteristic, complemented
103#355—674no—
104C2×#41C2×D6711yescharacteristic
105C22×D4—671no—
106C2×M16—21no—
107C2×M16—1341no—
108#127—672no—
109#124—672no—
110#121—1341no—
111#121—1341no—
112#120—1341no—
113#120—1341no—
114#117—671no—
115#101—672no—
116#101C2×D6712yescomplemented
117#100—21no—
118#100—1341no—
119C24—1341no—
120#41—41no—
121#41—41no—
122C2×Q8C22×D6711yescharacteristic
123C2×D4—2681no—
124C2×D4—2681no—
125C2×D4—672no—
126C2×D4—1342no—
127C2×D4—1342no—
128C2×D4—1341no—
129C2×D4C4×D6711yescharacteristic
130C2×D4D26811yescharacteristic
131C2×D4C22⋊D6711yescharacteristic
132C22×C4C4×D6711yescharacteristic
133C22×C4D26811yescharacteristic
134C22×C4C22⋊D6711yescharacteristic
135M16—22no—
136M16—41no—
137M16—2681no—
138M16—1342no—
139C2×C8—41no—
140C2×C8—2681no—
141C4⋊C4—1341no—
142C22⋊C4—1341no—
143C22⋊C4—1342no—
144C22⋊C4—1342no—
145C42—672no—
146C23—2682no—
147C23—5362no—
148C23—1342no—
149C23(C2×D67)⋊C411yescharacteristic
150C23(C2×D67)⋊C411yescharacteristic
151C23(C2×D67)⋊C411yescharacteristic
152Q8—41no—
153D4—41no—
154D4—41no—
155D4—41no—
156D4—2682no—
157D4—2681no—
158D4—2681no—
159D4—5362no—
160D4—5362no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—1342no—
168C2×C4(C2×D67)⋊C411yescharacteristic
169C2×C4C2×C4×D6711yescharacteristic
170C2×C4C2×C22⋊D6711yescharacteristic
171C2×C4C2×D26811yescharacteristic
172C8—42no—
173C8—2682no—
174C22—21no—
175C22—21no—
176C22—21no—
177C22—41no—
178C22—41no—
179C22—41no—
180C22—2682no—
181C22—2682no—
182C22—2682no—
183C22—5364no—
184C22C2×(C2×D67)⋊C411yescharacteristic
185C4—21no—
186C4—21no—
187C4—21no—
188C4—2682no—
189C4#1452395711yescharacteristic
190C4#1452395711yescharacteristic
191C2—21no—
192C2—41no—
193C2—41no—
194C2—41no—
195C2—5362no—
196C2C2×#1452395711yescharacteristic
subgroupG/H
centerC2C2×#14523957
commutatorC2×C4×C67C22×C4
FrattiniC2×Q8C22×D67
Fitting#413×C67C2
socleC2×C67C2×#99
hypercenter#413D67
hypocenterC67#2134
LCSG ▷ C2×C4×C67 ▷ C22×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #413

Complements

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

NH ≅ G/Ntype
2. #16863380C2semidirect
3. #16863192C2semidirect
4. #413×C67C2semidirect
14. #14523962C22semidirect
18. #101×C67C22semidirect
99. C67#2134semidirect
102. #413D67semidirect
116. #101C2×D67semidirect