Finite Group Databasebeta

← C67×#11558 #30690809

Subgroups — C67×#11558 #30690809

Order 17152.

446 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 106 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×#2486C211yesmaximal, characteristic
2C67×#1466C211yesmaximal, characteristic
3C67×#1459C211yesmaximal, characteristic
4C67×#1317C214yesmaximal, complemented
5C2×#123×C67C412yes—
6C2×C4×D4×C67C2211yescharacteristic
7C22×C4⋊C4×C67C411yescharacteristic
8#432×C67—21no—
9C2×C22⋊C8×C67—21no—
10#398×C67—21no—
11#397×C67—21no—
12#389×C67—21no—
13#381×C67C2212yes—
14#380×C67—21no—
15#380×C67C411yescharacteristic
16#324×C67—21no—
17#323×C67—21no—
18#323×C67C2212yes—
19#323×C67C2212yes—
20C22×D4×C67C2×C411yescharacteristic
21C23×C4×C67C2×C412yes—
22C23×C4×C67C2×C411yescharacteristic
23C23×C4×C67D411yescharacteristic
24C22×C8×C67—41no—
25#123×C67—44no—
26C4×D4×C67—41no—
27C4×D4×C67—41no—
28C2×C4⋊C4×C67—21no—
29C2×C4⋊C4×C67—22no—
30C2×C4⋊C4×C67—21no—
31C2×C4⋊C4×C67—21no—
32C2×C4⋊C4×C67—41no—
33C2×C4⋊C4×C67—41no—
34C2×C4⋊C4×C67D411yescharacteristic
35C2×C22⋊C4×C67—22no—
36C2×C22⋊C4×C67C2×C412yes—
37C2×C22⋊C4×C67D411yescharacteristic
38C2×C22⋊C4×C67C2311yescharacteristic
39C2×C42×C67D411yescharacteristic
40C4⋊C8×C67—41no—
41C22⋊C8×C67—22no—
42C22⋊C8×C67—22no—
43C22⋊C8×C67—41no—
44#97×C67—41no—
45#95×C67—21no—
46#95×C67—22no—
47#95×C67—22no—
48C24×C67C22⋊C411yescharacteristic
49C24×C67C22×C411yescharacteristic
50C2×D4×C67—22no—
51C2×D4×C67—22no—
52C2×D4×C67—41no—
53C22×C4×C67—21no—
54C22×C4×C67—21no—
55C22×C4×C67—21no—
56C22×C4×C67—21no—
57C22×C4×C67—22no—
58C22×C4×C67—22no—
59C22×C4×C67—24no—
60C22×C4×C67—42no—
61C22×C4×C67—42no—
62C22×C4×C67—42no—
63C22×C4×C67—42no—
64C22×C4×C67C22⋊C411yescharacteristic
65C22×C4×C67C22⋊C411yescharacteristic
66C22×C4×C67C22⋊C411yescharacteristic
67C22×C4×C67C2×D411yescharacteristic
68C22×C4×C67C2×D411yescharacteristic
69C2×C8×C67—42no—
70C2×C8×C67—42no—
71C2×C8×C67—82no—
72C4⋊C4×C67—41no—
73C4⋊C4×C67—42no—
74C4⋊C4×C67—42no—
75C4⋊C4×C67—44no—
76C4⋊C4×C67—41no—
77C4⋊C4×C67—41no—
78C22⋊C4×C67—41no—
79C22⋊C4×C67—41no—
80C22⋊C4×C67—44no—
81C22⋊C4×C67—42no—
82C42×C67—21no—
83C42×C67—21no—
84C23×C67—21no—
85C23×C67—21no—
86C23×C67—21no—
87C23×C67—21no—
88C23×C67—41no—
89C23×C67—41no—
90C23×C67—41no—
91C23×C67—41no—
92C23×C67#10012yes—
93C23×C67C2×C22⋊C411yescharacteristic
94C23×C67#13112yes—
95D4×C67—44no—
96C2×C4×C67—22no—
97C2×C4×C67—21no—
98C2×C4×C67—21no—
99C2×C4×C67—21no—
100C2×C4×C67—21no—
101C2×C4×C67—21no—
102C2×C4×C67—21no—
103C2×C4×C67—42no—
104C2×C4×C67—42no—
105C2×C4×C67—42no—
106C2×C4×C67—41no—
107C2×C4×C67—41no—
108C2×C4×C67—41no—
109C2×C4×C67—42no—
110C2×C4×C67—42no—
111C2×C4×C67—42no—
112C2×C4×C67—42no—
113C2×C4×C67—42no—
114C2×C4×C67—42no—
115C2×C4×C67—81no—
116C2×C4×C67—84no—
117C2×C4×C67#9911yescharacteristic
118C2×C4×C67#9911yescharacteristic
119C8×C67—82no—
120C8×C67—82no—
121C22×C67—21no—
122C22×C67—21no—
123C22×C67—22no—
124C22×C67—22no—
125C22×C67—42no—
126C22×C67—42no—
127C22×C67—42no—
128C22×C67—41no—
129C22×C67—41no—
130C22×C67—41no—
131C22×C67—41no—
132C22×C67—81no—
133C22×C67#40811yescharacteristic
134C22×C67C2×#9911yescharacteristic
135C22×C67C2×#10011yescharacteristic
136C4×C67—21no—
137C4×C67—41no—
138C4×C67—41no—
139C4×C67—41no—
140C4×C67—41no—
141C4×C67—82no—
142C4×C67—82no—
143C4×C67#35212yes—
144#11558C6711yesmaximal, characteristic, direct product factor, Sylow
145C2×C67—21no—
146C2×C67—21no—
147C2×C67—42no—
148C2×C67—42no—
149C2×C67#147111yescharacteristic
150C2×C67C2×#35211yescharacteristic
151C2×C67#212311yescharacteristic
152#2486C2×C6711yescharacteristic
153#1466C2×C6711yescharacteristic
154#1459C2×C6711yescharacteristic
155#1317C2×C6714yescomplemented
156C67#1155811yescharacteristic, direct product factor, Sylow
157C2×#123C4×C6712yes—
158C2×C4×D4C22×C6711yescharacteristic
159C22×C4⋊C4C4×C6711yescharacteristic
160#432—21no—
161C2×C22⋊C8—21no—
162#398—21no—
163#397—21no—
164#389—21no—
165#381C22×C6712yes—
166#380—21no—
167#380C4×C6711yescharacteristic
168#324—21no—
169#323—21no—
170#323C22×C6712yes—
171#323C22×C6712yes—
172C22×D4C2×C4×C6711yescharacteristic
173C23×C4C2×C4×C6712yes—
174C23×C4C2×C4×C6711yescharacteristic
175C23×C4D4×C6711yescharacteristic
176C22×C8—41no—
177#123—44no—
178C4×D4—41no—
179C4×D4—41no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4D4×C6711yescharacteristic
187C2×C22⋊C4—22no—
188C2×C22⋊C4C2×C4×C6712yes—
189C2×C22⋊C4D4×C6711yescharacteristic
190C2×C22⋊C4C23×C6711yescharacteristic
191C2×C42D4×C6711yescharacteristic
192C4⋊C8—41no—
193C22⋊C8—22no—
194C22⋊C8—22no—
195C22⋊C8—41no—
196#97—41no—
197#95—21no—
198#95—22no—
199#95—22no—
200C24C22⋊C4×C6711yescharacteristic
201C24C22×C4×C6711yescharacteristic
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—41no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—24no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4C22⋊C4×C6711yescharacteristic
217C22×C4C22⋊C4×C6711yescharacteristic
218C22×C4C22⋊C4×C6711yescharacteristic
219C22×C4C2×D4×C6711yescharacteristic
220C22×C4C2×D4×C6711yescharacteristic
221C2×C8—42no—
222C2×C8—42no—
223C2×C8—82no—
224C4⋊C4—41no—
225C4⋊C4—42no—
226C4⋊C4—44no—
227C4⋊C4—42no—
228C4⋊C4—41no—
229C4⋊C4—41no—
230C22⋊C4—41no—
231C22⋊C4—41no—
232C22⋊C4—44no—
233C22⋊C4—42no—
234C42—21no—
235C42—21no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23#100×C6712yes—
245C23C2×C22⋊C4×C6711yescharacteristic
246C23#131×C6712yes—
247D4—44no—
248C2×C4—22no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—81no—
268C2×C4—84no—
269C2×C4#99×C6711yescharacteristic
270C2×C4#99×C6711yescharacteristic
271C8—82no—
272C8—82no—
273C22—21no—
274C22—21no—
275C22—22no—
276C22—22no—
277C22—42no—
278C22—42no—
279C22—42no—
280C22—41no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—81no—
285C22#408×C6711yescharacteristic
286C22C2×#99×C6711yescharacteristic
287C22C2×#100×C6711yescharacteristic
288C4—21no—
289C4—41no—
290C4—41no—
291C4—41no—
292C4—41no—
293C4—82no—
294C4—82no—
295C4#352×C6712yes—
296C2—21no—
297C2—21no—
298C2—42no—
299C2—42no—
300C2C67×#147111yescharacteristic
301C2C2×#352×C6711yescharacteristic
302C2C67×#212311yescharacteristic
subgroupG/H
center, socleC22×C67C2×#99
commutatorC24C22×C4×C67
Frattini, cosocleC2×C22⋊C4C23×C67
prime radicalC22×D4×C67C2×C4
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C67 ◁ C22×C4×C67 ◁ C2×C4×D4×C67 ◁ G

Complements

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

NH ≅ G/Ntype
4. C67×#1317C2semidirect
144. #11558C67direct (N × H)
155. #1317C2×C67semidirect
156. C67#11558direct (N × H)

Maximal subgroups

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

subgroupindex
C67×#24862
C67×#14662
C67×#14592
C67×#13172
#1155867