Finite Group Databasebeta

← C67×#9442 #30688693

Subgroups — C67×#9442 #30688693

Order 17152.

456 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 118 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
1C67×#3142C211yesmaximal, characteristic
2C67×#2371C211yesmaximal, characteristic, complemented
3C67×#1534C211yesmaximal, characteristic, complemented
4C67×#1505C212yesmaximal
5C67×#1300C212yesmaximal, complemented
6#532×C67C2211yescharacteristic
7C2×#125×C67C2211yescharacteristic
8#491×C67C412yescomplemented
9#490×C67C412yescomplemented
10#488×C67—22no—
11#432×C67C2212yes—
12#417×C67—22no—
13C4×M16×C67C2211yescharacteristic
14#387×C67—22no—
15#387×C67—22no—
16C4×C4⋊C4×C67—22no—
17#376×C67C2212yes—
18#330×C67—22no—
19#327×C67—22no—
20#327×C67—22no—
21C2×#41×C67—21no—
22C2×M16×C67—21no—
23C4⋊Q8×C67C2×C411yescharacteristic
24#127×C67C2×C411yescharacteristic
25#125×C67—22no—
26#125×C67C2×C412yes—
27#125×C67D412yes—
28#124×C67—21no—
29#121×C67—22no—
30C4×D4×C67—22no—
31C2×C4⋊C4×C67—22no—
32C2×C4⋊C4×C67—22no—
33C2×C4⋊C4×C67D412yes—
34C2×C42×C67—22no—
35C2×C42×C67—22no—
36C2×C42×C67C2311yescharacteristic
37C4⋊C8×C67—22no—
38#103×C67—24no—
39#102×C67—24no—
40C22⋊C8×C67—22no—
41#97×C67—22no—
42#97×C67C2×C412yes—
43C4×C8×C67—21no—
44#95×C67—22no—
45#95×C67—22no—
46#95×C67—22no—
47#95×C67D412yes—
48#95×C67D412yes—
49#41×C67—44no—
50C2×Q8×C67—21no—
51C2×D4×C67—21no—
52C2×D4×C67—22no—
53C2×D4×C67—41no—
54C22×C4×C67—22no—
55C22×C4×C67—22no—
56C22×C4×C67—22no—
57C22×C4×C67—41no—
58C22×C4×C67C2×D411yescharacteristic
59C22×C4×C67C2×D412yes—
60M16×C67—42no—
61C2×C8×C67—22no—
62C2×C8×C67—42no—
63C4⋊C4×C67—24no—
64C4⋊C4×C67—24no—
65C4⋊C4×C67—22no—
66C4⋊C4×C67—22no—
67C4⋊C4×C67—44no—
68C4⋊C4×C67C22⋊C414yes—
69C22⋊C4×C67—42no—
70C42×C67—44no—
71C42×C67—44no—
72C42×C67C22×C411yescharacteristic
73C42×C67C2×D411yescharacteristic
74C42×C67#4112yes—
75C23×C67—41no—
76C23×C67#12011yescharacteristic
77Q8×C67—42no—
78D4×C67—42no—
79D4×C67—82no—
80D4×C67—82no—
81C2×C4×C67—22no—
82C2×C4×C67—24no—
83C2×C4×C67—24no—
84C2×C4×C67—42no—
85C2×C4×C67—42no—
86C2×C4×C67—42no—
87C2×C4×C67—42no—
88C2×C4×C67—42no—
89C2×C4×C67—42no—
90C2×C4×C67—42no—
91C2×C4×C67—41no—
92C2×C4×C67—42no—
93C2×C4×C67—82no—
94C2×C4×C67C2×C22⋊C411yescharacteristic
95C2×C4×C67C4×D412yes—
96C2×C4×C67#12112yes—
97C2×C4×C67#12311yescharacteristic
98C8×C67—42no—
99C8×C67—82no—
100C22×C67—22no—
101C22×C67—81no—
102C22×C67—82no—
103C22×C67#38611yescharacteristic
104C22×C67#45412yes—
105C4×C67—21no—
106C4×C67—21no—
107C4×C67—22no—
108C4×C67—22no—
109C4×C67—44no—
110C4×C67—44no—
111C4×C67—82no—
112C4×C67—81no—
113#9442C6711yesmaximal, characteristic, direct product factor, Sylow
114C2×C67—22no—
115C2×C67—161no—
116C2×C67#188811yescharacteristic
117C2×C67#190612yes—
118#3142C2×C6711yescharacteristic
119#2371C2×C6711yescharacteristic, complemented
120#1534C2×C6711yescharacteristic, complemented
121#1505C2×C6712yes—
122#1300C2×C6712yescomplemented
123C67#944211yescharacteristic, direct product factor, Sylow
124#532C22×C6711yescharacteristic
125C2×#125C22×C6711yescharacteristic
126#491C4×C6712yescomplemented
127#490C4×C6712yescomplemented
128#488—22no—
129#432C22×C6712yes—
130#417—22no—
131C4×M16C22×C6711yescharacteristic
132#387—22no—
133#387—22no—
134C4×C4⋊C4—22no—
135#376C22×C6712yes—
136#330—22no—
137#327—22no—
138#327—22no—
139C2×#41—21no—
140C2×M16—21no—
141C4⋊Q8C2×C4×C6711yescharacteristic
142#127C2×C4×C6711yescharacteristic
143#125—22no—
144#125C2×C4×C6712yes—
145#125D4×C6712yes—
146#124—21no—
147#121—22no—
148C4×D4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4D4×C6712yes—
152C2×C42—22no—
153C2×C42—22no—
154C2×C42C23×C6711yescharacteristic
155C4⋊C8—22no—
156#103—24no—
157#102—24no—
158C22⋊C8—22no—
159#97—22no—
160#97C2×C4×C6712yes—
161C4×C8—21no—
162#95—22no—
163#95—22no—
164#95—22no—
165#95D4×C6712yes—
166#95D4×C6712yes—
167#41—44no—
168C2×Q8—21no—
169C2×D4—21no—
170C2×D4—22no—
171C2×D4—41no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—41no—
176C22×C4C2×D4×C6711yescharacteristic
177C22×C4C2×D4×C6712yes—
178M16—42no—
179C2×C8—22no—
180C2×C8—42no—
181C4⋊C4—24no—
182C4⋊C4—24no—
183C4⋊C4—22no—
184C4⋊C4—22no—
185C4⋊C4—44no—
186C4⋊C4C22⋊C4×C6714yes—
187C22⋊C4—42no—
188C42—44no—
189C42—44no—
190C42C22×C4×C6711yescharacteristic
191C42C2×D4×C6711yescharacteristic
192C42#41×C6712yes—
193C23—41no—
194C23#120×C6711yescharacteristic
195Q8—42no—
196D4—42no—
197D4—82no—
198D4—82no—
199C2×C4—22no—
200C2×C4—24no—
201C2×C4—24no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—41no—
211C2×C4—82no—
212C2×C4C2×C22⋊C4×C6711yescharacteristic
213C2×C4C4×D4×C6712yes—
214C2×C4#121×C6712yes—
215C2×C4#123×C6711yescharacteristic
216C8—42no—
217C8—82no—
218C22—22no—
219C22—81no—
220C22—82no—
221C22#386×C6711yescharacteristic
222C22#454×C6712yes—
223C4—21no—
224C4—21no—
225C4—22no—
226C4—22no—
227C4—44no—
228C4—44no—
229C4—82no—
230C4—81no—
231C2—22no—
232C2—161no—
233C2C67×#188811yescharacteristic
234C2C67×#190612yes—
subgroupG/H
center, socleC22×C67#386
commutatorC42C22×C4×C67
Frattini, cosocleC2×C42C23×C67
prime radical#532×C67C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C67 ◁ C2×C42×C67 ◁ G

Complements

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

NH ≅ G/Ntype
2. C67×#2371C2semidirect
3. C67×#1534C2semidirect
5. C67×#1300C2semidirect
8. #491×C67C4semidirect
9. #490×C67C4semidirect
113. #9442C67direct (N × H)
119. #2371C2×C67semidirect
120. #1534C2×C67semidirect
122. #1300C2×C67semidirect
123. C67#9442direct (N × H)
126. #491C4×C67semidirect
127. #490C4×C67semidirect

Maximal subgroups

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

subgroupindex
C67×#31422
C67×#23712
C67×#15342
C67×#15052
C67×#13002
#944267