Finite Group Databasebeta

← C67×#10290 #30689541

Subgroups — C67×#10290 #30689541

Order 17152.

912 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 912 conjugacy classes of subgroups — too many to draw. Only its 130 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×#3208C212yesmaximal, complemented
2C67×#2867C211yesmaximal, characteristic
3C67×#1994C211yesmaximal, characteristic, complemented
4C67×#1545C212yesmaximal, complemented
5C67×#1276C211yesmaximal, characteristic, complemented
6#533×C67C2211yescharacteristic
7C2×#127×C67—24no—
8C2×#127×C67—22no—
9C2×#127×C67C2212yes—
10#492×C67—22no—
11#486×C67—22no—
12#460×C67—24no—
13#450×C67—24no—
14#431×C67C2212yes—
15#423×C67C2211yescharacteristic
16C43×C67C2211yescharacteristic
17#339×C67—22no—
18#331×C67—24no—
19C22×D4×C67—28no—
20C22×D4×C67—22no—
21C22×D4×C67—22no—
22C22×D4×C67D414yes—
23C2×SD16×C67—28no—
24C2×M16×C67—21no—
25C4⋊Q8×C67D412yescomplemented
26#127×C67—22no—
27#127×C67—22no—
28#127×C67—22no—
29#127×C67—22no—
30#127×C67—416no—
31#127×C67—44no—
32#127×C67—44no—
33#125×C67—21no—
34C22⋊Q8×C67—22no—
35C4×Q8×C67—22no—
36#117×C67—21no—
37C2×C42×C67—22no—
38C2×C42×C67D412yes—
39C2×C42×C67C2311yescharacteristic
40C4⋊C8×C67—22no—
41C4⋊C8×C67D414yescomplemented
42#102×C67—28no—
43C22⋊C8×C67—22no—
44C4×C8×C67—22no—
45C24×C67—24no—
46C2×Q8×C67—22no—
47C2×D4×C67—24no—
48C2×D4×C67—24no—
49C2×D4×C67—48no—
50C2×D4×C67—432no—
51C2×D4×C67—416no—
52C2×D4×C67—48no—
53C2×D4×C67—42no—
54C2×D4×C67—42no—
55C2×D4×C67—42no—
56C2×D4×C67—42no—
57C2×D4×C67—44no—
58C2×D4×C67—44no—
59C22×C4×C67—22no—
60C22×C4×C67C2×D412yes—
61C22×C4×C67C2×D411yescharacteristic
62SD16×C67—88no—
63M16×C67—42no—
64C2×C8×C67—24no—
65C2×C8×C67—41no—
66C4⋊C4×C67—22no—
67C4⋊C4×C67—22no—
68C4⋊C4×C67—42no—
69C22⋊C4×C67—41no—
70C42×C67—28no—
71C42×C67—22no—
72C42×C67—41no—
73C42×C67SD1614yescomplemented
74C42×C67C2×D411yescharacteristic
75C42×C67C2×D412yes—
76C42×C67#4111yescharacteristic
77C23×C67—48no—
78C23×C67—48no—
79C23×C67—44no—
80C23×C67—48no—
81C23×C67#12011yescharacteristic
82Q8×C67—44no—
83D4×C67—48no—
84D4×C67—84no—
85D4×C67—832no—
86D4×C67—84no—
87D4×C67—84no—
88D4×C67—84no—
89D4×C67—82no—
90D4×C67—82no—
91C2×C4×C67—28no—
92C2×C4×C67—24no—
93C2×C4×C67—24no—
94C2×C4×C67—42no—
95C2×C4×C67#12012yes—
96C2×C4×C67#12112yes—
97C2×C4×C67#12111yescharacteristic
98C2×C4×C67#12711yescharacteristic
99C2×C4×C67C2×SD1612yes—
100C2×C4×C67AGL(1,ℤ/8)12yes—
101C8×C67—44no—
102C8×C67—81no—
103C22×C67—22no—
104C22×C67—816no—
105C22×C67—84no—
106C22×C67—84no—
107C22×C67—84no—
108C22×C67#39211yescharacteristic
109C22×C67#45012yes—
110C4×C67—28no—
111C4×C67—22no—
112C4×C67—82no—
113C4×C67#45314yes—
114C4×C67#46012yes—
115C4×C67#49211yescharacteristic
116C4×C67#49611yescharacteristic
117#10290C6711yesmaximal, characteristic, direct product factor, Sylow
118C2×C67—22no—
119C2×C67—164no—
120C2×C67#200212yes—
121C2×C67#201411yescharacteristic
122#3208C2×C6712yescomplemented
123#2867C2×C6711yescharacteristic
124#1994C2×C6711yescharacteristic, complemented
125#1545C2×C6712yescomplemented
126#1276C2×C6711yescharacteristic, complemented
127C67#1029011yescharacteristic, direct product factor, Sylow
128#533C22×C6711yescharacteristic
129C2×#127—24no—
130C2×#127—22no—
131C2×#127C22×C6712yes—
132#492—22no—
133#486—22no—
134#460—24no—
135#450—24no—
136#431C22×C6712yes—
137#423C22×C6711yescharacteristic
138C43C22×C6711yescharacteristic
139#339—22no—
140#331—24no—
141C22×D4—28no—
142C22×D4—22no—
143C22×D4—22no—
144C22×D4D4×C6714yes—
145C2×SD16—28no—
146C2×M16—21no—
147C4⋊Q8D4×C6712yescomplemented
148#127—22no—
149#127—22no—
150#127—22no—
151#127—22no—
152#127—416no—
153#127—44no—
154#127—44no—
155#125—21no—
156C22⋊Q8—22no—
157C4×Q8—22no—
158#117—21no—
159C2×C42—22no—
160C2×C42D4×C6712yes—
161C2×C42C23×C6711yescharacteristic
162C4⋊C8—22no—
163C4⋊C8D4×C6714yescomplemented
164#102—28no—
165C22⋊C8—22no—
166C4×C8—22no—
167C24—24no—
168C2×Q8—22no—
169C2×D4—24no—
170C2×D4—24no—
171C2×D4—48no—
172C2×D4—432no—
173C2×D4—416no—
174C2×D4—48no—
175C2×D4—42no—
176C2×D4—42no—
177C2×D4—42no—
178C2×D4—42no—
179C2×D4—44no—
180C2×D4—44no—
181C22×C4—22no—
182C22×C4C2×D4×C6712yes—
183C22×C4C2×D4×C6711yescharacteristic
184SD16—88no—
185M16—42no—
186C2×C8—24no—
187C2×C8—41no—
188C4⋊C4—22no—
189C4⋊C4—22no—
190C4⋊C4—42no—
191C22⋊C4—41no—
192C42—28no—
193C42—22no—
194C42—41no—
195C42SD16×C6714yescomplemented
196C42C2×D4×C6711yescharacteristic
197C42C2×D4×C6712yes—
198C42#41×C6711yescharacteristic
199C23—48no—
200C23—48no—
201C23—48no—
202C23—44no—
203C23#120×C6711yescharacteristic
204Q8—44no—
205D4—48no—
206D4—84no—
207D4—832no—
208D4—84no—
209D4—84no—
210D4—84no—
211D4—82no—
212D4—82no—
213C2×C4—28no—
214C2×C4—24no—
215C2×C4—24no—
216C2×C4—42no—
217C2×C4#120×C6712yes—
218C2×C4#121×C6712yes—
219C2×C4#121×C6711yescharacteristic
220C2×C4#127×C6711yescharacteristic
221C2×C4C2×SD16×C6712yes—
222C2×C4AGL(1,ℤ/8)×C6712yes—
223C8—44no—
224C8—81no—
225C22—22no—
226C22—816no—
227C22—84no—
228C22—84no—
229C22—84no—
230C22#392×C6711yescharacteristic
231C22#450×C6712yes—
232C4—28no—
233C4—22no—
234C4—82no—
235C4#453×C6714yes—
236C4#460×C6712yes—
237C4#492×C6711yescharacteristic
238C4#496×C6711yescharacteristic
239C2—22no—
240C2—164no—
241C2C67×#200212yes—
242C2C67×#201411yescharacteristic
subgroupG/H
center, socleC22×C67#392
commutator, Frattini, cosocleC2×C42C23×C67
prime radicalC67×#2867C2
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C67 ◁ C2×C42×C67 ◁ G

Complements

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

NH ≅ G/Ntype
1. C67×#3208C2semidirect
3. C67×#1994C2semidirect
4. C67×#1545C2semidirect
5. C67×#1276C2semidirect
25. C4⋊Q8×C67D4semidirect
41. C4⋊C8×C67D4semidirect
73. C42×C67SD16semidirect
117. #10290C67direct (N × H)
122. #3208C2×C67semidirect
124. #1994C2×C67semidirect
125. #1545C2×C67semidirect
126. #1276C2×C67semidirect
127. C67#10290direct (N × H)
147. C4⋊Q8D4×C67semidirect
163. C4⋊C8D4×C67semidirect
195. C42SD16×C67semidirect

Maximal subgroups

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

subgroupindex
C67×#32082
C67×#28672
C67×#19942
C67×#15452
C67×#12762
#1029067