Finite Group Databasebeta

← C67×#9454 #30688705

Subgroups — C67×#9454 #30688705

Order 17152.

448 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 126 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×#2983C211yesmaximal, characteristic
2C67×#2567C211yesmaximal, characteristic
3C67×#1556C212yesmaximal
4C67×#1467C211yesmaximal, characteristic
5C67×#1285C212yesmaximal
6C2×C4⋊Q8×C67C2211yescharacteristic
7C2×C4×Q8×C67C2211yescharacteristic
8#446×C67—24no—
9#432×C67—21no—
10#432×C67C2212yes—
11#432×C67C2212yes—
12#431×C67—21no—
13#395×C67—21no—
14#395×C67—21no—
15#391×C67C412yes—
16#389×C67—21no—
17#389×C67—21no—
18#384×C67C412yes—
19C4×C4⋊C4×C67C2211yescharacteristic
20#332×C67—24no—
21#324×C67—21no—
22#324×C67—21no—
23C22×Q8×C67—21no—
24C22×Q8×C67C2×C411yescharacteristic
25C4⋊Q8×C67—22no—
26C4⋊Q8×C67—22no—
27C4×Q8×C67—22no—
28C4×Q8×C67—22no—
29C2×C4⋊C4×C67—21no—
30C2×C4⋊C4×C67—21no—
31C2×C4⋊C4×C67—22no—
32C2×C4⋊C4×C67—22no—
33C2×C4⋊C4×C67C2×C412yes—
34C2×C4⋊C4×C67C2×C411yescharacteristic
35C2×C4⋊C4×C67D411yescharacteristic
36C2×C4⋊C4×C67D411yescharacteristic
37C2×C42×C67C2×C412yes—
38C2×C42×C67D411yescharacteristic
39C2×C42×C67D411yescharacteristic
40C2×C42×C67C2311yescharacteristic
41C4⋊C8×C67—22no—
42C4⋊C8×C67—22no—
43C4⋊C8×C67—22no—
44C4⋊C8×C67D414yes—
45C22⋊C8×C67—22no—
46C22⋊C8×C67—22no—
47#97×C67—22no—
48#97×C67—22no—
49#97×C67—22no—
50C4×C8×C67—22no—
51#95×C67—22no—
52#95×C67—22no—
53#95×C67—21no—
54C2×Q8×C67—22no—
55C2×Q8×C67—22no—
56C2×Q8×C67—41no—
57C2×Q8×C67—42no—
58C2×Q8×C67—82no—
59C22×C4×C67—22no—
60C22×C4×C67—22no—
61C22×C4×C67C22⋊C411yescharacteristic
62C22×C4×C67C22⋊C411yescharacteristic
63C22×C4×C67C22⋊C411yescharacteristic
64C22×C4×C67C22⋊C411yescharacteristic
65C22×C4×C67C22×C411yescharacteristic
66C22×C4×C67C2×D411yescharacteristic
67C22×C4×C67C2×D411yescharacteristic
68C2×C8×C67—24no—
69C2×C8×C67—42no—
70C4⋊C4×C67—24no—
71C4⋊C4×C67—22no—
72C4⋊C4×C67—22no—
73C4⋊C4×C67—22no—
74C4⋊C4×C67—24no—
75C4⋊C4×C67—41no—
76C4⋊C4×C67—41no—
77C4⋊C4×C67—82no—
78C4⋊C4×C67—82no—
79C42×C67—21no—
80C42×C67—21no—
81C42×C67—21no—
82C42×C67—21no—
83C42×C67—42no—
84C42×C67C2×D412yes—
85C42×C67#4112yes—
86C23×C67C2×C22⋊C411yescharacteristic
87Q8×C67—44no—
88Q8×C67—84no—
89C2×C4×C67—21no—
90C2×C4×C67—21no—
91C2×C4×C67—21no—
92C2×C4×C67—21no—
93C2×C4×C67—22no—
94C2×C4×C67—24no—
95C2×C4×C67—21no—
96C2×C4×C67—21no—
97C2×C4×C67—21no—
98C2×C4×C67—21no—
99C2×C4×C67—21no—
100C2×C4×C67—41no—
101C2×C4×C67—42no—
102C2×C4×C67—42no—
103C2×C4×C67—41no—
104C2×C4×C67—41no—
105C2×C4×C67—42no—
106C2×C4×C67—42no—
107C2×C4×C67—42no—
108C2×C4×C67#10112yes—
109C2×C4×C67C4×D412yes—
110C2×C4×C67#12011yescharacteristic
111C2×C4×C67#12112yes—
112C2×C4×C67#12311yescharacteristic
113C8×C67—44no—
114C8×C67—82no—
115C22×C67—21no—
116C22×C67—21no—
117C22×C67#38611yescharacteristic
118C22×C67C2×#10111yescharacteristic
119C22×C67#41311yescharacteristic
120C4×C67—21no—
121C4×C67—22no—
122C4×C67—22no—
123C4×C67—22no—
124C4×C67—44no—
125C4×C67—41no—
126C4×C67—41no—
127C4×C67—41no—
128C4×C67—82no—
129C4×C67#45612yes—
130#9454C6711yesmaximal, characteristic, direct product factor, Sylow
131C2×C67—21no—
132C2×C67—21no—
133C2×C67#188611yescharacteristic
134C2×C67#190111yescharacteristic
135C2×C67#191011yescharacteristic
136#2983C2×C6711yescharacteristic
137#2567C2×C6711yescharacteristic
138#1556C2×C6712yes—
139#1467C2×C6711yescharacteristic
140#1285C2×C6712yes—
141C67#945411yescharacteristic, direct product factor, Sylow
142C2×C4⋊Q8C22×C6711yescharacteristic
143C2×C4×Q8C22×C6711yescharacteristic
144#446—24no—
145#432—21no—
146#432C22×C6712yes—
147#432C22×C6712yes—
148#431—21no—
149#395—21no—
150#395—21no—
151#391C4×C6712yes—
152#389—21no—
153#389—21no—
154#384C4×C6712yes—
155C4×C4⋊C4C22×C6711yescharacteristic
156#332—24no—
157#324—21no—
158#324—21no—
159C22×Q8—21no—
160C22×Q8C2×C4×C6711yescharacteristic
161C4⋊Q8—22no—
162C4⋊Q8—22no—
163C4×Q8—22no—
164C4×Q8—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4C2×C4×C6712yes—
170C2×C4⋊C4C2×C4×C6711yescharacteristic
171C2×C4⋊C4D4×C6711yescharacteristic
172C2×C4⋊C4D4×C6711yescharacteristic
173C2×C42C2×C4×C6712yes—
174C2×C42D4×C6711yescharacteristic
175C2×C42D4×C6711yescharacteristic
176C2×C42C23×C6711yescharacteristic
177C4⋊C8—22no—
178C4⋊C8—22no—
179C4⋊C8—22no—
180C4⋊C8D4×C6714yes—
181C22⋊C8—22no—
182C22⋊C8—22no—
183#97—22no—
184#97—22no—
185#97—22no—
186C4×C8—22no—
187#95—22no—
188#95—22no—
189#95—21no—
190C2×Q8—22no—
191C2×Q8—22no—
192C2×Q8—41no—
193C2×Q8—42no—
194C2×Q8—82no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4C22⋊C4×C6711yescharacteristic
198C22×C4C22⋊C4×C6711yescharacteristic
199C22×C4C22⋊C4×C6711yescharacteristic
200C22×C4C22⋊C4×C6711yescharacteristic
201C22×C4C22×C4×C6711yescharacteristic
202C22×C4C2×D4×C6711yescharacteristic
203C22×C4C2×D4×C6711yescharacteristic
204C2×C8—24no—
205C2×C8—42no—
206C4⋊C4—24no—
207C4⋊C4—22no—
208C4⋊C4—22no—
209C4⋊C4—22no—
210C4⋊C4—24no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—82no—
214C4⋊C4—82no—
215C42—21no—
216C42—21no—
217C42—21no—
218C42—21no—
219C42—42no—
220C42C2×D4×C6712yes—
221C42#41×C6712yes—
222C23C2×C22⋊C4×C6711yescharacteristic
223Q8—44no—
224Q8—84no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—22no—
230C2×C4—24no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—41no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4#101×C6712yes—
245C2×C4C4×D4×C6712yes—
246C2×C4#120×C6711yescharacteristic
247C2×C4#121×C6712yes—
248C2×C4#123×C6711yescharacteristic
249C8—44no—
250C8—82no—
251C22—21no—
252C22—21no—
253C22#386×C6711yescharacteristic
254C22C2×#101×C6711yescharacteristic
255C22#413×C6711yescharacteristic
256C4—21no—
257C4—22no—
258C4—22no—
259C4—22no—
260C4—44no—
261C4—41no—
262C4—41no—
263C4—41no—
264C4—82no—
265C4#456×C6712yes—
266C2—21no—
267C2—21no—
268C2C67×#188611yescharacteristic
269C2C67×#190111yescharacteristic
270C2C67×#191011yescharacteristic
subgroupG/H
center, socleC22×C67#386
commutatorC22×C4C22×C4×C67
Frattini, cosocleC2×C42C23×C67
prime radicalC23×C67C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C67 ◁ C2×C42×C67 ◁ G

Complements

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

NH ≅ G/Ntype
130. #9454C67direct (N × H)
141. C67#9454direct (N × H)

Maximal subgroups

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

subgroupindex
C67×#29832
C67×#25672
C67×#15562
C67×#14672
C67×#12852
#945467