Finite Group Databasebeta

← #22561620

Subgroups — #22561620

Order 8576.

238 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 83 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×#14523967C211yesmaximal, characteristic
2#16863303C211yesmaximal, characteristic
3#16863265C211yesmaximal, characteristic
4#16863159C211yesmaximal, characteristic
5#16863158C211yesmaximal, characteristic
6#16863150C211yesmaximal, characteristic
7C2×#103×C67C211yesmaximal, characteristic
8C22×C67⋊C8C2211yescharacteristic
9C2×C4⋊Dic67C2211yescharacteristic
10C2×C4⋊Dic67C2211yescharacteristic
11C2×Dic67⋊C4—21no—
12C2×Dic67⋊C4—21no—
13C2×C4×Dic67C2211yescharacteristic
14#14523967—22no—
15C67⋊#95—21no—
16C67⋊#95—21no—
17C22×Q8×C67C2211yescharacteristic
18C22×C8×C67C2211yescharacteristic
19C2×C4⋊C4×C67C2211yescharacteristic
20#103×C67—22no—
21C22×Dic67—21no—
22C22×Dic67—21no—
23C22×Dic67D411yescharacteristic
24C22×Dic67D411yescharacteristic
25C2×C67⋊C8—21no—
26C2×C67⋊C8—21no—
27C2×C67⋊C8—21no—
28C4⋊Dic67—21no—
29C4⋊Dic67—21no—
30C4⋊Dic67—21no—
31C4⋊Dic67—21no—
32Dic67⋊C4—42no—
33Dic67⋊C4—42no—
34C4×Dic67—21no—
35C4×Dic67—21no—
36C2×Q8×C67—21no—
37C2×Q8×C67—21no—
38C2×Q8×C67—21no—
39C2×Q8×C67—21no—
40C2×Q8×C67—21no—
41C2×Q8×C67D412yes—
42C22×C4×C67—21no—
43C22×C4×C67—21no—
44C22×C4×C67C2311yescharacteristic
45C2×C8×C67—21no—
46C2×C8×C67—21no—
47C2×C8×C67—21no—
48C4⋊C4×C67—21no—
49C4⋊C4×C67—21no—
50C23×C67C2×D411yescharacteristic
51Q8×C67—22no—
52Q8×C67—22no—
53Q8×C67—41no—
54Q8×C67—41no—
55C2×C4×C67—22no—
56C2×C4×C67—41no—
57C2×C4×C67—41no—
58C2×C4×C67—41no—
59C2×C4×C67—41no—
60C2×C4×C67—41no—
61C2×C4×C67C2×D411yescharacteristic
62C2×C4×C67#4111yescharacteristic
63C2×C4×C67#4111yescharacteristic
64C2×C4×C67#4111yescharacteristic
65C2×C4×C67#4111yescharacteristic
66C2×C4×C67#4111yescharacteristic
67C2×Dic67—21no—
68C2×Dic67—21no—
69C2×Dic67—21no—
70C2×Dic67—21no—
71C2×Dic67—41no—
72C2×Dic67—41no—
73C2×Dic67—41no—
74C2×Dic67—41no—
75C2×Dic67—41no—
76C2×Dic67—41no—
77C2×Dic67SD1612yes—
78C2×Dic67Q1612yes—
79C8×C67—41no—
80C67⋊C8—41no—
81C22×C67#12111yescharacteristic
82C22×C67#12311yescharacteristic
83C22×C67#12411yescharacteristic
84C22×C67C2×SD1611yescharacteristic
85C22×C67C2×Q1611yescharacteristic
86C22×C67AGL(1,ℤ/8)11yescharacteristic
87C22×C67#13711yescharacteristic
88C4×C67—42no—
89C4×C67—81no—
90C4×C67#12311yescharacteristic
91C4×C67#12311yescharacteristic
92C4×C67#12611yescharacteristic
93C4×C67#12611yescharacteristic
94Dic67—22no—
95Dic67—22no—
96Dic67—81no—
97Dic67—81no—
98C2×C67#39711yescharacteristic
99C2×C67#46011yescharacteristic
100C2×C67#46211yescharacteristic
101C2×C67#48311yescharacteristic
102C2×C67#48411yescharacteristic
103C2×C67#48711yescharacteristic
104C2×C67#48911yescharacteristic
105#2072—671nomaximal, Sylow
106C67#207211yescharacteristic, complemented, Sylow
107C2×#103—671no—
108C2×#103D6711yescharacteristic
109#391—671no—
110#384—671no—
111#340—671no—
112#340—671no—
113#336—671no—
114C22×Q8C2×D6711yescharacteristic
115C22×C8—671no—
116C22×C8C2×D6711yescharacteristic
117C2×C4⋊C4—671no—
118C2×C4⋊C4—671no—
119C2×C4⋊C4—1341no—
120C2×C4⋊C4—1341no—
121C2×C4⋊C4C2×D6711yescharacteristic
122C2×C42—671no—
123#103—22no—
124#103—1342no—
125#95—1341no—
126#95—1341no—
127C2×Q8—21no—
128C2×Q8—21no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8C22⋊D6712yes—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—671no—
136C22×C4—671no—
137C22×C4—1341no—
138C22×C4—1341no—
139C22×C4C22×D6711yescharacteristic
140C2×C8—21no—
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—1341no—
144C2×C8—1341no—
145C2×C8—1341no—
146C4⋊C4—21no—
147C4⋊C4—21no—
148C4⋊C4—2682no—
149C4⋊C4—2682no—
150C4⋊C4—1341no—
151C4⋊C4—1341no—
152C4⋊C4—1341no—
153C4⋊C4—1341no—
154C42—1341no—
155C42—1341no—
156C23D4×D6711yescharacteristic
157Q8—22no—
158Q8—22no—
159Q8—41no—
160Q8—41no—
161C2×C4—22no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—2681no—
168C2×C4—2681no—
169C2×C4—2681no—
170C2×C4—2681no—
171C2×C4—2681no—
172C2×C4—2681no—
173C2×C4—672no—
174C2×C4—672no—
175C2×C4—1341no—
176C2×C4—1341no—
177C2×C4—1341no—
178C2×C4—1341no—
179C2×C4C2×C22⋊D6711yescharacteristic
180C2×C4#1176162711yescharacteristic
181C2×C4#1176162711yescharacteristic
182C2×C4#1176162711yescharacteristic
183C2×C4#1176162811yescharacteristic
184C2×C4#1176162811yescharacteristic
185C8—41no—
186C8—2681no—
187C22#1452401511yescharacteristic
188C22#1452402511yescharacteristic
189C22#1452402911yescharacteristic
190C22SD16×D6711yescharacteristic
191C22Q16×D6711yescharacteristic
192C22#1452407011yescharacteristic
193C22#1452407511yescharacteristic
194C4—42no—
195C4—81no—
196C4—5361no—
197C4—5361no—
198C4—1342no—
199C4—1342no—
200C4#1452402411yescharacteristic
201C4#1452402611yescharacteristic
202C4#1452403611yescharacteristic
203C4#1452403611yescharacteristic
204C2#1686332411yescharacteristic
205C2#1686359411yescharacteristic
206C2#1686360411yescharacteristic
207C2#1686372511yescharacteristic
208C2#1686373011yescharacteristic
209C2#1686374511yescharacteristic
210C2#1686375711yescharacteristic
subgroupG/H
centerC23D4×D67
commutatorC22×C4×C67C23
FrattiniC22×C4C22×D67
FittingC2×#103×C67C2
socleC23×C67C2×D4
hypercenterC2×#103D67
hypocenterC67#2072
LCSG ▷ C22×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
106. C67#2072semidirect