Finite Group Databasebeta

← C67×#7307 #30686558

Subgroups — C67×#7307 #30686558

Order 17152.

794 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 794 conjugacy classes of subgroups — too many to draw. Only its 86 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×#1781C212yesmaximal
2C67×#1438C211yesmaximal, characteristic
3C23×D4×C67C2211yescharacteristic
4C22×C22⋊C4×C67—22no—
5C2×#99×C67—22no—
6C2×#99×C67C414yescomplemented
7#379×C67—22no—
8C2×#95×C67C412yescomplemented
9C25×C67D412yes—
10C22×D4×C67—42no—
11C22×D4×C67—44no—
12C22×D4×C67C2×C412yes—
13C22×D4×C67D411yescharacteristic
14C22×D4×C67Q811yescharacteristic
15C23×C4×C67—22no—
16C23×C4×C67C2×C411yescharacteristic
17C2×C22⋊C4×C67—24no—
18C2×C22⋊C4×C67—48no—
19C2×C22⋊C4×C67—44no—
20C2×C22⋊C4×C67—44no—
21#99×C67—44no—
22#99×C67—44no—
23#95×C67—42no—
24C24×C67—24no—
25C24×C67—24no—
26C24×C67—42no—
27C24×C67—44no—
28C24×C67—44no—
29C24×C67C22⋊C412yes—
30C24×C67C22⋊C411yescharacteristic
31C24×C67C4⋊C412yes—
32C2×D4×C67—24no—
33C2×D4×C67—24no—
34C2×D4×C67—42no—
35C2×D4×C67—42no—
36C2×D4×C67—41no—
37C2×D4×C67—42no—
38C2×D4×C67—41no—
39C2×D4×C67—416no—
40C22×C4×C67—22no—
41C22×C4×C67—24no—
42C22×C4×C67—42no—
43C22×C4×C67—44no—
44C22×C4×C67—44no—
45C22×C4×C67—42no—
46C22×C4×C67—42no—
47C22×C4×C67—44no—
48C22×C4×C67C4211yescharacteristic
49C22×C4×C67C4⋊C411yescharacteristic
50C22⋊C4×C67—416no—
51C22⋊C4×C67—44no—
52C22⋊C4×C67—44no—
53C22⋊C4×C67—84no—
54C22⋊C4×C67—84no—
55C23×C67—22no—
56C23×C67—22no—
57C23×C67—22no—
58C23×C67—22no—
59C23×C67—22no—
60C23×C67—22no—
61C23×C67—22no—
62C23×C67—22no—
63C23×C67—48no—
64C23×C67—44no—
65C23×C67—44no—
66C23×C67—44no—
67C23×C67—48no—
68C23×C67—44no—
69C23×C67—44no—
70C23×C67—44no—
71C23×C67—84no—
72C23×C67—82no—
73C23×C67—84no—
74C23×C67—82no—
75C23×C67#9511yescharacteristic
76C23×C67#9912yes—
77C23×C67#9914yes—
78D4×C67—44no—
79D4×C67—48no—
80D4×C67—44no—
81C2×C4×C67—22no—
82C2×C4×C67—22no—
83C2×C4×C67—21no—
84C2×C4×C67—21no—
85C2×C4×C67—44no—
86C2×C4×C67—48no—
87C2×C4×C67—44no—
88C2×C4×C67—84no—
89C2×C4×C67—82no—
90C2×C4×C67—82no—
91C2×C4×C67—84no—
92C2×C4×C67—88no—
93C22×C67—22no—
94C22×C67—24no—
95C22×C67—22no—
96C22×C67—22no—
97C22×C67—22no—
98C22×C67—22no—
99C22×C67—24no—
100C22×C67—44no—
101C22×C67—44no—
102C22×C67—44no—
103C22×C67—42no—
104C22×C67—44no—
105C22×C67—82no—
106C22×C67—88no—
107C22×C67—88no—
108C22×C67—82no—
109C22×C67#34212yes—
110C22×C67#34211yescharacteristic
111C22×C67#35312yes—
112C22×C67#35412yes—
113C4×C67—41no—
114C4×C67—41no—
115C4×C67—84no—
116C4×C67—164no—
117#7307C6711yesmaximal, characteristic, direct product factor, Sylow
118C2×C67—22no—
119C2×C67—22no—
120C2×C67—22no—
121C2×C67—44no—
122C2×C67—82no—
123C2×C67#130411yescharacteristic
124C2×C67#139312yes—
125#1781C2×C6712yes—
126#1438C2×C6711yescharacteristic
127C67#730711yescharacteristic, direct product factor, Sylow
128C23×D4C22×C6711yescharacteristic
129C22×C22⋊C4—22no—
130C2×#99—22no—
131C2×#99C4×C6714yescomplemented
132#379—22no—
133C2×#95C4×C6712yescomplemented
134C25D4×C6712yes—
135C22×D4—42no—
136C22×D4—44no—
137C22×D4C2×C4×C6712yes—
138C22×D4D4×C6711yescharacteristic
139C22×D4Q8×C6711yescharacteristic
140C23×C4—22no—
141C23×C4C2×C4×C6711yescharacteristic
142C2×C22⋊C4—24no—
143C2×C22⋊C4—48no—
144C2×C22⋊C4—44no—
145C2×C22⋊C4—44no—
146#99—44no—
147#99—44no—
148#95—42no—
149C24—24no—
150C24—24no—
151C24—42no—
152C24—44no—
153C24—44no—
154C24C22⋊C4×C6711yescharacteristic
155C24C22⋊C4×C6712yes—
156C24C4⋊C4×C6712yes—
157C2×D4—24no—
158C2×D4—24no—
159C2×D4—42no—
160C2×D4—42no—
161C2×D4—41no—
162C2×D4—41no—
163C2×D4—42no—
164C2×D4—416no—
165C22×C4—22no—
166C22×C4—24no—
167C22×C4—42no—
168C22×C4—44no—
169C22×C4—44no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—44no—
173C22×C4C42×C6711yescharacteristic
174C22×C4C4⋊C4×C6711yescharacteristic
175C22⋊C4—416no—
176C22⋊C4—44no—
177C22⋊C4—44no—
178C22⋊C4—84no—
179C22⋊C4—84no—
180C23—22no—
181C23—22no—
182C23—22no—
183C23—22no—
184C23—22no—
185C23—22no—
186C23—22no—
187C23—22no—
188C23—48no—
189C23—44no—
190C23—44no—
191C23—44no—
192C23—48no—
193C23—44no—
194C23—44no—
195C23—44no—
196C23—84no—
197C23—82no—
198C23—84no—
199C23—82no—
200C23#95×C6711yescharacteristic
201C23#99×C6712yes—
202C23#99×C6714yes—
203D4—44no—
204D4—48no—
205D4—44no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—44no—
211C2×C4—48no—
212C2×C4—44no—
213C2×C4—84no—
214C2×C4—82no—
215C2×C4—82no—
216C2×C4—84no—
217C2×C4—88no—
218C22—22no—
219C22—24no—
220C22—22no—
221C22—22no—
222C22—22no—
223C22—22no—
224C22—24no—
225C22—44no—
226C22—44no—
227C22—44no—
228C22—42no—
229C22—44no—
230C22—82no—
231C22—88no—
232C22—88no—
233C22—82no—
234C22#342×C6712yes—
235C22#342×C6711yescharacteristic
236C22#353×C6712yes—
237C22#354×C6712yes—
238C4—41no—
239C4—41no—
240C4—84no—
241C4—164no—
242C2—22no—
243C2—22no—
244C2—22no—
245C2—44no—
246C2—82no—
247C2C67×#130411yescharacteristic
248C2C67×#139312yes—
subgroupG/H
center, socleC22×C67#342
commutatorC22×C4C42×C67
Frattini, cosocleC23×D4C22×C67
prime radicalC23×D4×C67C22
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C67 ◁ C24×C67 ◁ C23×D4×C67 ◁ G

Complements

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

NH ≅ G/Ntype
6. C2×#99×C67C4semidirect
8. C2×#95×C67C4semidirect
117. #7307C67direct (N × H)
127. C67#7307direct (N × H)
131. C2×#99C4×C67semidirect
133. C2×#95C4×C67semidirect

Maximal subgroups

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

subgroupindex
C67×#17812
C67×#14382
#730767