Finite Group Databasebeta

← C67×#12184 #30691435

Subgroups — C67×#12184 #30691435

Order 17152.

372 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 74 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×#2844C211yesmaximal, characteristic, complemented
2C67×#2040C211yesmaximal, characteristic
3C67×#1604C212yesmaximal
4C67×#1354C212yesmaximal, complemented
5C67×#1324C211yesmaximal, characteristic, complemented
6#537×C67C2211yescharacteristic
7#448×C67—22no—
8C2×#104×C67—21no—
9#417×C67—21no—
10#400×C67—22no—
11#400×C67—21no—
12#395×C67—22no—
13#395×C67C2212yes—
14#382×C67—21no—
15#382×C67C2211yescharacteristic
16#343×C67C2211yescharacteristic
17#327×C67—22no—
18#324×C67C2212yes—
19C2×#41×C67—21no—
20C22×Q8×C67D412yes—
21C2×SD16×C67—42no—
22C2×M16×C67—21no—
23C22⋊Q8×C67—41no—
24#121×C67—22no—
25#121×C67—41no—
26C2×C4⋊C4×C67—21no—
27C2×C4⋊C4×C67—21no—
28C2×C4⋊C4×C67—21no—
29C2×C4⋊C4×C67—22no—
30C2×C4⋊C4×C67D411yescharacteristic
31C2×C4⋊C4×C67C2311yescharacteristic
32C2×C42×C67D411yescharacteristic
33#104×C67—42no—
34#103×C67—42no—
35#103×C67—41no—
36#102×C67—41no—
37C22⋊C8×C67—22no—
38#95×C67—22no—
39#95×C67—22no—
40#95×C67—21no—
41#95×C67—22no—
42#95×C67—22no—
43#95×C67D412yes—
44#41×C67—41no—
45#41×C67—41no—
46#41×C67—41no—
47#41×C67—41no—
48C2×Q8×C67—21no—
49C2×Q8×C67—42no—
50C2×Q8×C67—82no—
51C2×D4×C67—21no—
52C2×D4×C67—22no—
53C2×D4×C67—41no—
54C22×C4×C67—21no—
55C22×C4×C67—22no—
56C22×C4×C67—22no—
57C22×C4×C67—21no—
58C22×C4×C67—41no—
59C22×C4×C67C2×D411yescharacteristic
60C22×C4×C67C2×D412yes—
61SD16×C67—84no—
62M16×C67—41no—
63M16×C67—41no—
64C2×C8×C67—41no—
65C2×C8×C67—42no—
66C4⋊C4×C67—22no—
67C4⋊C4×C67—41no—
68C4⋊C4×C67—41no—
69C4⋊C4×C67—81no—
70C4⋊C4×C67—81no—
71C4⋊C4×C67—82no—
72C4⋊C4×C67—81no—
73C22⋊C4×C67—42no—
74C42×C67—21no—
75C42×C67SD1612yescomplemented
76C23×C67—41no—
77C23×C67#12011yescharacteristic
78Q8×C67—41no—
79Q8×C67—41no—
80Q8×C67—44no—
81Q8×C67—82no—
82D4×C67—41no—
83D4×C67—41no—
84D4×C67—82no—
85D4×C67—81no—
86D4×C67—81no—
87C2×C4×C67—21no—
88C2×C4×C67—21no—
89C2×C4×C67—22no—
90C2×C4×C67—22no—
91C2×C4×C67—41no—
92C2×C4×C67—41no—
93C2×C4×C67—42no—
94C2×C4×C67—42no—
95C2×C4×C67—42no—
96C2×C4×C67—42no—
97C2×C4×C67—42no—
98C2×C4×C67—42no—
99C2×C4×C67—42no—
100C2×C4×C67—41no—
101C2×C4×C67—41no—
102C2×C4×C67—42no—
103C2×C4×C67—81no—
104C2×C4×C67—81no—
105C2×C4×C67C2×SD1611yescharacteristic
106C2×C4×C67#13711yescharacteristic
107C8×C67—81no—
108C8×C67—82no—
109C22×C67—21no—
110C22×C67—21no—
111C22×C67—81no—
112C22×C67—81no—
113C22×C67—81no—
114C22×C67#44811yescharacteristic
115C22×C67#45411yescharacteristic
116C22×C67B(4,𝔽2)11yescharacteristic
117C4×C67—21no—
118C4×C67—21no—
119C4×C67—42no—
120C4×C67—42no—
121C4×C67—82no—
122C4×C67—81no—
123C4×C67—82no—
124C4×C67—81no—
125#12184C6711yesmaximal, characteristic, direct product factor, Sylow
126C2×C67—21no—
127C2×C67—21no—
128C2×C67—161no—
129C2×C67#159911yescharacteristic
130C2×C67#219911yescharacteristic
131C2×C67#220311yescharacteristic
132#2844C2×C6711yescharacteristic, complemented
133#2040C2×C6711yescharacteristic
134#1604C2×C6712yes—
135#1354C2×C6712yescomplemented
136#1324C2×C6711yescharacteristic, complemented
137C67#1218411yescharacteristic, direct product factor, Sylow
138#537C22×C6711yescharacteristic
139#448—22no—
140C2×#104—21no—
141#417—21no—
142#400—21no—
143#400—22no—
144#395—22no—
145#395C22×C6712yes—
146#382—21no—
147#382C22×C6711yescharacteristic
148#343C22×C6711yescharacteristic
149#327—22no—
150#324C22×C6712yes—
151C2×#41—21no—
152C22×Q8D4×C6712yes—
153C2×SD16—42no—
154C2×M16—21no—
155C22⋊Q8—41no—
156#121—22no—
157#121—41no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4D4×C6711yescharacteristic
163C2×C4⋊C4C23×C6711yescharacteristic
164C2×C42D4×C6711yescharacteristic
165#104—42no—
166#103—42no—
167#103—41no—
168#102—41no—
169C22⋊C8—22no—
170#95—22no—
171#95—22no—
172#95—21no—
173#95—22no—
174#95—22no—
175#95D4×C6712yes—
176#41—41no—
177#41—41no—
178#41—41no—
179#41—41no—
180C2×Q8—21no—
181C2×Q8—42no—
182C2×Q8—82no—
183C2×D4—21no—
184C2×D4—22no—
185C2×D4—41no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—41no—
191C22×C4C2×D4×C6711yescharacteristic
192C22×C4C2×D4×C6712yes—
193SD16—84no—
194M16—41no—
195M16—41no—
196C2×C8—42no—
197C2×C8—41no—
198C4⋊C4—22no—
199C4⋊C4—41no—
200C4⋊C4—41no—
201C4⋊C4—81no—
202C4⋊C4—81no—
203C4⋊C4—82no—
204C4⋊C4—81no—
205C22⋊C4—42no—
206C42—21no—
207C42SD16×C6712yescomplemented
208C23—41no—
209C23#120×C6711yescharacteristic
210Q8—41no—
211Q8—41no—
212Q8—44no—
213Q8—82no—
214D4—41no—
215D4—41no—
216D4—82no—
217D4—81no—
218D4—81no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—42no—
235C2×C4—81no—
236C2×C4—81no—
237C2×C4C2×SD16×C6711yescharacteristic
238C2×C4#137×C6711yescharacteristic
239C8—81no—
240C8—82no—
241C22—21no—
242C22—21no—
243C22—81no—
244C22—81no—
245C22—81no—
246C22#448×C6711yescharacteristic
247C22#454×C6711yescharacteristic
248C22B(4,𝔽2)×C6711yescharacteristic
249C4—21no—
250C4—21no—
251C4—42no—
252C4—42no—
253C4—82no—
254C4—81no—
255C4—82no—
256C4—81no—
257C2—21no—
258C2—21no—
259C2—161no—
260C2C67×#159911yescharacteristic
261C2C67×#219911yescharacteristic
262C2C67×#220311yescharacteristic
subgroupG/H
center, socleC22×C67B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C67
prime radical#537×C67C22
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C67 ◁ C23×C67 ◁ C2×C4⋊C4×C67 ◁ G

Complements

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

NH ≅ G/Ntype
1. C67×#2844C2semidirect
4. C67×#1354C2semidirect
5. C67×#1324C2semidirect
75. C42×C67SD16semidirect
125. #12184C67direct (N × H)
132. #2844C2×C67semidirect
135. #1354C2×C67semidirect
136. #1324C2×C67semidirect
137. C67#12184direct (N × H)
207. C42SD16×C67semidirect

Maximal subgroups

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

subgroupindex
C67×#28442
C67×#20402
C67×#16042
C67×#13542
C67×#13242
#1218467