Finite Group Databasebeta

← C67×#1508 #22555939

Subgroups — C67×#1508 #22555939

Order 8576.

294 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 96 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×C22⋊Q8×C67C211yesmaximal, characteristic
2#407×C67C211yesmaximal, characteristic
3C4×C22⋊C4×C67C211yesmaximal, characteristic
4#328×C67C214yesmaximal, complemented
5C22×Q8×C67C411yescharacteristic, complemented
6C23×C4×C67C2211yescharacteristic
7C2×M16×C67—21no—
8C22⋊Q8×C67—21no—
9C22⋊Q8×C67—21no—
10C22⋊Q8×C67C412yescomplemented
11C22⋊Q8×C67C2212yes—
12C2×C4⋊C4×C67—21no—
13C2×C4⋊C4×C67C411yescharacteristic, complemented
14C2×C22⋊C4×C67—21no—
15C2×C22⋊C4×C67—21no—
16C2×C42×C67—21no—
17C22⋊C8×C67—21no—
18C22⋊C8×C67C2212yes—
19#95×C67C2212yes—
20C24×C67D411yescharacteristic
21C2×Q8×C67—21no—
22C2×Q8×C67—42no—
23C2×Q8×C67C2×C412yes—
24C22×C4×C67—21no—
25C22×C4×C67—21no—
26C22×C4×C67—22no—
27C22×C4×C67—42no—
28C22×C4×C67C2×C411yescharacteristic
29C22×C4×C67C2×C411yescharacteristic
30C22×C4×C67D411yescharacteristic
31C22×C4×C67D411yescharacteristic
32C22×C4×C67D411yescharacteristic
33C22×C4×C67C2311yescharacteristic
34M16×C67—42no—
35C2×C8×C67—22no—
36C4⋊C4×C67—22no—
37C4⋊C4×C67—21no—
38C4⋊C4×C67—22no—
39C4⋊C4×C67C2×C412yes—
40C22⋊C4×C67—24no—
41C22⋊C4×C67—22no—
42C22⋊C4×C67—22no—
43C42×C67—42no—
44C23×C67—22no—
45C23×C67—22no—
46C23×C67—41no—
47C23×C67C22⋊C411yescharacteristic
48C23×C67C22⋊C411yescharacteristic
49C23×C67C2×D411yescharacteristic
50Q8×C67—42no—
51Q8×C67—42no—
52C2×C4×C67—22no—
53C2×C4×C67—22no—
54C2×C4×C67—22no—
55C2×C4×C67—24no—
56C2×C4×C67—22no—
57C2×C4×C67—22no—
58C2×C4×C67—41no—
59C2×C4×C67—41no—
60C2×C4×C67—41no—
61C2×C4×C67—42no—
62C2×C4×C67—42no—
63C2×C4×C67—42no—
64C2×C4×C67—42no—
65C2×C4×C67C22⋊C411yescharacteristic
66C2×C4×C67C22⋊C411yescharacteristic
67C2×C4×C67C22×C411yescharacteristic
68C2×C4×C67C2×D411yescharacteristic
69C8×C67—42no—
70C22×C67—21no—
71C22×C67—21no—
72C22×C67—21no—
73C22×C67—21no—
74C22×C67—21no—
75C22×C67—21no—
76C22×C67—21no—
77C22×C67—44no—
78C22×C67#10412yes—
79C22×C67C2×C22⋊C411yescharacteristic
80C22×C67#13712yes—
81C4×C67—21no—
82C4×C67—21no—
83C4×C67—21no—
84C4×C67—44no—
85C4×C67—42no—
86C4×C67—42no—
87C4×C67#9912yes—
88C2×C67—22no—
89C2×C67—22no—
90C2×C67—41no—
91C2×C67C2×#9911yescharacteristic
92C2×C67#41911yescharacteristic
93C2×C67C2×#10411yescharacteristic
94#1508C6711yesmaximal, characteristic, direct product factor, Sylow
95C67#150811yescharacteristic, direct product factor, Sylow
96C2×C22⋊Q8C2×C6711yescharacteristic
97#407C2×C6711yescharacteristic
98C4×C22⋊C4C2×C6711yescharacteristic
99#328C2×C6714yescomplemented
100C22×Q8C4×C6711yescharacteristic, complemented
101C23×C4C22×C6711yescharacteristic
102C2×M16—21no—
103C22⋊Q8—21no—
104C22⋊Q8—21no—
105C22⋊Q8C4×C6712yescomplemented
106C22⋊Q8C22×C6712yes—
107C2×C4⋊C4—21no—
108C2×C4⋊C4C4×C6711yescharacteristic, complemented
109C2×C22⋊C4—21no—
110C2×C22⋊C4—21no—
111C2×C42—21no—
112C22⋊C8—21no—
113C22⋊C8C22×C6712yes—
114#95C22×C6712yes—
115C24D4×C6711yescharacteristic
116C2×Q8—21no—
117C2×Q8—42no—
118C2×Q8C2×C4×C6712yes—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—22no—
122C22×C4—42no—
123C22×C4C2×C4×C6711yescharacteristic
124C22×C4C2×C4×C6711yescharacteristic
125C22×C4D4×C6711yescharacteristic
126C22×C4D4×C6711yescharacteristic
127C22×C4D4×C6711yescharacteristic
128C22×C4C23×C6711yescharacteristic
129M16—42no—
130C2×C8—22no—
131C4⋊C4—22no—
132C4⋊C4—21no—
133C4⋊C4—22no—
134C4⋊C4C2×C4×C6712yes—
135C22⋊C4—24no—
136C22⋊C4—22no—
137C22⋊C4—22no—
138C42—42no—
139C23—22no—
140C23—22no—
141C23—41no—
142C23C22⋊C4×C6711yescharacteristic
143C23C22⋊C4×C6711yescharacteristic
144C23C2×D4×C6711yescharacteristic
145Q8—42no—
146Q8—42no—
147C2×C4—22no—
148C2×C4—22no—
149C2×C4—22no—
150C2×C4—24no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—42no—
157C2×C4—42no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4C22⋊C4×C6711yescharacteristic
161C2×C4C22⋊C4×C6711yescharacteristic
162C2×C4C22×C4×C6711yescharacteristic
163C2×C4C2×D4×C6711yescharacteristic
164C8—42no—
165C22—21no—
166C22—21no—
167C22—21no—
168C22—21no—
169C22—21no—
170C22—21no—
171C22—21no—
172C22—44no—
173C22#104×C6712yes—
174C22C2×C22⋊C4×C6711yescharacteristic
175C22#137×C6712yes—
176C4—21no—
177C4—21no—
178C4—21no—
179C4—44no—
180C4—42no—
181C4—42no—
182C4#99×C6712yes—
183C2—22no—
184C2—22no—
185C2—41no—
186C2C2×#99×C6711yescharacteristic
187C2#419×C6711yescharacteristic
188C2C2×#104×C6711yescharacteristic
subgroupG/H
center, socleC22×C67C2×C22⋊C4
commutatorC2×C4C22×C4×C67
Frattini, cosocleC22×C4C23×C67
prime radicalC24×C67D4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C67 ◁ C23×C4×C67 ◁ G

Complements

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

NH ≅ G/Ntype
4. #328×C67C2semidirect
5. C22×Q8×C67C4semidirect
10. C22⋊Q8×C67C4semidirect
13. C2×C4⋊C4×C67C4semidirect
94. #1508C67direct (N × H)
95. C67#1508direct (N × H)
99. #328C2×C67semidirect
100. C22×Q8C4×C67semidirect
105. C22⋊Q8C4×C67semidirect
108. C2×C4⋊C4C4×C67semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×C22⋊Q8×C672
#407×C672
C4×C22⋊C4×C672
#328×C672
#150867