Finite Group Databasebeta

← C67×#1890 #22556321

Subgroups — C67×#1890 #22556321

Order 8576.

386 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 112 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×#123×C67C211yesmaximal, characteristic
2#518×C67C211yesmaximal, characteristic
3C2×#99×C67C211yesmaximal, characteristic
4#381×C67C212yesmaximal, complemented
5#342×C67C212yesmaximal, complemented
6C22×D4×C67C2211yescharacteristic
7C23×C4×C67C2211yescharacteristic
8#123×C67—22no—
9#123×C67C414yescomplemented
10C4×D4×C67—24no—
11#117×C67—21no—
12C2×C4⋊C4×C67—21no—
13C2×C22⋊C4×C67—21no—
14C2×C22⋊C4×C67—21no—
15C2×C22⋊C4×C67C2212yes—
16C2×C22⋊C4×C67C2212yes—
17C2×C22⋊C4×C67C2211yescharacteristic
18#99×C67—22no—
19#95×C67—24no—
20C24×C67D411yescharacteristic
21C24×C67C2311yescharacteristic
22C2×D4×C67—22no—
23C2×D4×C67—22no—
24C2×D4×C67C2×C412yes—
25C2×D4×C67D412yes—
26C22×C4×C67—21no—
27C22×C4×C67—21no—
28C22×C4×C67—21no—
29C22×C4×C67—22no—
30C22×C4×C67—22no—
31C22×C4×C67—22no—
32C22×C4×C67—22no—
33C22×C4×C67C2×C412yescomplemented
34C22×C4×C67D411yescharacteristic
35C22×C4×C67D414yes—
36C4⋊C4×C67—22no—
37C4⋊C4×C67—24no—
38C22⋊C4×C67—22no—
39C22⋊C4×C67—24no—
40C22⋊C4×C67—22no—
41C22⋊C4×C67—22no—
42C22⋊C4×C67—21no—
43C22⋊C4×C67—22no—
44C22⋊C4×C67—22no—
45C22⋊C4×C67C2×C412yescomplemented
46C42×C67—22no—
47C23×C67—21no—
48C23×C67—21no—
49C23×C67—42no—
50C23×C67—42no—
51C23×C67C22⋊C412yes—
52C23×C67C22×C411yescharacteristic
53C23×C67C2×D411yescharacteristic
54C23×C67C2×D412yes—
55C23×C67C2×D411yescharacteristic
56C23×C67#4112yes—
57D4×C67—24no—
58D4×C67—24no—
59C2×C4×C67—21no—
60C2×C4×C67—21no—
61C2×C4×C67—22no—
62C2×C4×C67—22no—
63C2×C4×C67—24no—
64C2×C4×C67—24no—
65C2×C4×C67—42no—
66C2×C4×C67—41no—
67C2×C4×C67—42no—
68C2×C4×C67—41no—
69C2×C4×C67—44no—
70C2×C4×C67—44no—
71C2×C4×C67—42no—
72C2×C4×C67—82no—
73C2×C4×C67C22⋊C412yes—
74C22×C67—24no—
75C22×C67—22no—
76C22×C67—22no—
77C22×C67—41no—
78C22×C67—41no—
79C22×C67—42no—
80C22×C67—42no—
81C22×C67—42no—
82C22×C67—81no—
83C22×C67C2×C22⋊C411yescharacteristic
84C22×C67C4×D412yes—
85C22×C67#12011yescharacteristic
86C22×C67#12112yes—
87C22×C67#12311yescharacteristic
88C4×C67—22no—
89C4×C67—42no—
90C4×C67—42no—
91C4×C67—44no—
92C4×C67—82no—
93C2×C67—21no—
94C2×C67—22no—
95C2×C67—21no—
96C2×C67—22no—
97C2×C67—42no—
98C2×C67#38611yescharacteristic
99C2×C67#45812yes—
100#1890C6711yesmaximal, characteristic, direct product factor, Sylow
101C67#189011yescharacteristic, direct product factor, Sylow
102C2×#123C2×C6711yescharacteristic
103#518C2×C6711yescharacteristic
104C2×#99C2×C6711yescharacteristic
105#381C2×C6712yescomplemented
106#342C2×C6712yescomplemented
107C22×D4C22×C6711yescharacteristic
108C23×C4C22×C6711yescharacteristic
109#123—22no—
110#123C4×C6714yescomplemented
111C4×D4—24no—
112#117—21no—
113C2×C4⋊C4—21no—
114C2×C22⋊C4—21no—
115C2×C22⋊C4—21no—
116C2×C22⋊C4C22×C6712yes—
117C2×C22⋊C4C22×C6712yes—
118C2×C22⋊C4C22×C6711yescharacteristic
119#99—22no—
120#95—24no—
121C24D4×C6711yescharacteristic
122C24C23×C6711yescharacteristic
123C2×D4—22no—
124C2×D4—22no—
125C2×D4C2×C4×C6712yes—
126C2×D4D4×C6712yes—
127C22×C4—22no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4C2×C4×C6712yescomplemented
135C22×C4D4×C6711yescharacteristic
136C22×C4D4×C6714yes—
137C4⋊C4—22no—
138C4⋊C4—24no—
139C22⋊C4—22no—
140C22⋊C4—24no—
141C22⋊C4—22no—
142C22⋊C4—22no—
143C22⋊C4—21no—
144C22⋊C4—22no—
145C22⋊C4—22no—
146C22⋊C4C2×C4×C6712yescomplemented
147C42—22no—
148C23—21no—
149C23—21no—
150C23—42no—
151C23—42no—
152C23C22⋊C4×C6712yes—
153C23C22×C4×C6711yescharacteristic
154C23C2×D4×C6711yescharacteristic
155C23C2×D4×C6712yes—
156C23C2×D4×C6711yescharacteristic
157C23#41×C6712yes—
158D4—24no—
159D4—24no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—24no—
165C2×C4—24no—
166C2×C4—42no—
167C2×C4—41no—
168C2×C4—42no—
169C2×C4—41no—
170C2×C4—44no—
171C2×C4—44no—
172C2×C4—42no—
173C2×C4—82no—
174C2×C4C22⋊C4×C6712yes—
175C22—24no—
176C22—22no—
177C22—22no—
178C22—41no—
179C22—41no—
180C22—42no—
181C22—42no—
182C22—42no—
183C22—81no—
184C22C2×C22⋊C4×C6711yescharacteristic
185C22C4×D4×C6712yes—
186C22#120×C6711yescharacteristic
187C22#121×C6712yes—
188C22#123×C6711yescharacteristic
189C4—22no—
190C4—42no—
191C4—44no—
192C4—42no—
193C4—82no—
194C2—21no—
195C2—22no—
196C2—21no—
197C2—22no—
198C2—42no—
199C2#386×C6711yescharacteristic
200C2#458×C6712yes—
subgroupG/H
center, socleC22×C67#120
commutatorC23C22×C4×C67
Frattini, cosocleC24C23×C67
prime radicalC22×D4×C67C22
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C67 ◁ C24×C67 ◁ G

Complements

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

NH ≅ G/Ntype
4. #381×C67C2semidirect
5. #342×C67C2semidirect
9. #123×C67C4semidirect
33. C22×C4×C67C2×C4semidirect
45. C22⋊C4×C67C2×C4semidirect
100. #1890C67direct (N × H)
101. C67#1890direct (N × H)
105. #381C2×C67semidirect
106. #342C2×C67semidirect
110. #123C4×C67semidirect
134. C22×C4C2×C4×C67semidirect
146. C22⋊C4C2×C4×C67semidirect

Maximal subgroups

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

subgroupindex
C2×#123×C672
#518×C672
C2×#99×C672
#381×C672
#342×C672
#189067