Finite Group Databasebeta

← C67×#9390 #30688641

Subgroups — C67×#9390 #30688641

Order 17152.

446 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 118 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×#2921C211yesmaximal, characteristic
2C67×#2611C211yesmaximal, characteristic
3C67×#1528C212yesmaximal, complemented
4C67×#1464C211yesmaximal, characteristic
5C67×#1300C212yesmaximal, complemented
6C2×#125×C67C2211yescharacteristic
7C2×#117×C67C2211yescharacteristic
8#433×C67—24no—
9#432×C67C2212yes—
10#423×C67—21no—
11#407×C67—21no—
12C4×M16×C67C2212yes—
13#400×C67C411yescharacteristic
14#399×C67C411yescharacteristic
15#398×C67C411yescharacteristic
16#397×C67C411yescharacteristic
17#389×C67—21no—
18#388×C67—21no—
19#387×C67—21no—
20#385×C67—21no—
21#376×C67C2211yescharacteristic
22#330×C67—24no—
23#324×C67—21no—
24#323×C67—21no—
25C23×C4×C67D411yescharacteristic
26C2×M16×C67—22no—
27#125×C67—22no—
28#125×C67—22no—
29#117×C67—22no—
30#117×C67—22no—
31C2×C4⋊C4×C67—21no—
32C2×C4⋊C4×C67—21no—
33C2×C4⋊C4×C67—21no—
34C2×C4⋊C4×C67—21no—
35C2×C4⋊C4×C67C2×C411yescharacteristic
36C2×C4⋊C4×C67C2×C411yescharacteristic
37C2×C4⋊C4×C67C2×C411yescharacteristic
38C2×C4⋊C4×C67D411yescharacteristic
39C2×C22⋊C4×C67—21no—
40C2×C22⋊C4×C67—21no—
41C2×C22⋊C4×C67C2×C411yescharacteristic
42C2×C22⋊C4×C67D411yescharacteristic
43C2×C42×C67—21no—
44C2×C42×C67D411yescharacteristic
45C2×C42×C67C2311yescharacteristic
46C4⋊C8×C67—24no—
47C4⋊C8×C67—22no—
48C22⋊C8×C67—24no—
49C22⋊C8×C67—22no—
50#97×C67—22no—
51#97×C67D414yes—
52C4×C8×C67—22no—
53#95×C67—21no—
54#95×C67—21no—
55#95×C67—21no—
56#95×C67—21no—
57#95×C67—21no—
58#95×C67C2×C411yescharacteristic
59#95×C67C2×C411yescharacteristic
60C24×C67C22⋊C411yescharacteristic
61C22×C4×C67—21no—
62C22×C4×C67—21no—
63C22×C4×C67—21no—
64C22×C4×C67—21no—
65C22×C4×C67—21no—
66C22×C4×C67—21no—
67C22×C4×C67—41no—
68C22×C4×C67—41no—
69C22×C4×C67C22⋊C411yescharacteristic
70C22×C4×C67C22⋊C411yescharacteristic
71C22×C4×C67C22⋊C411yescharacteristic
72C22×C4×C67C22×C411yescharacteristic
73C22×C4×C67C2×D411yescharacteristic
74C22×C4×C67C2×D411yescharacteristic
75M16×C67—42no—
76M16×C67—42no—
77C2×C8×C67—24no—
78C2×C8×C67—42no—
79C4⋊C4×C67—22no—
80C4⋊C4×C67—22no—
81C4⋊C4×C67—22no—
82C4⋊C4×C67—22no—
83C4⋊C4×C67—42no—
84C4⋊C4×C67—42no—
85C4⋊C4×C67—41no—
86C4⋊C4×C67—42no—
87C4⋊C4×C67—41no—
88C22⋊C4×C67—22no—
89C22⋊C4×C67—22no—
90C22⋊C4×C67—42no—
91C22⋊C4×C67—42no—
92C42×C67—21no—
93C42×C67—21no—
94C42×C67—42no—
95C42×C67C2×D412yes—
96C42×C67#4112yes—
97C23×C67—21no—
98C23×C67—21no—
99C23×C67—21no—
100C23×C67—41no—
101C23×C67—41no—
102C23×C67C2×C22⋊C411yescharacteristic
103C2×C4×C67—21no—
104C2×C4×C67—21no—
105C2×C4×C67—22no—
106C2×C4×C67—21no—
107C2×C4×C67—21no—
108C2×C4×C67—21no—
109C2×C4×C67—21no—
110C2×C4×C67—21no—
111C2×C4×C67—41no—
112C2×C4×C67—41no—
113C2×C4×C67—42no—
114C2×C4×C67—42no—
115C2×C4×C67—41no—
116C2×C4×C67—41no—
117C2×C4×C67—41no—
118C2×C4×C67—41no—
119C2×C4×C67—41no—
120C2×C4×C67—41no—
121C2×C4×C67—41no—
122C2×C4×C67—41no—
123C2×C4×C67—41no—
124C2×C4×C67—41no—
125C2×C4×C67—41no—
126C2×C4×C67—41no—
127C2×C4×C67—41no—
128C2×C4×C67C4×D412yes—
129C2×C4×C67#12011yescharacteristic
130C2×C4×C67#12112yes—
131C2×C4×C67#12311yescharacteristic
132C8×C67—44no—
133C8×C67—82no—
134C22×C67—21no—
135C22×C67—21no—
136C22×C67—42no—
137C22×C67—41no—
138C22×C67—41no—
139C22×C67—41no—
140C22×C67—81no—
141C22×C67#38611yescharacteristic
142C22×C67#42111yescharacteristic
143C22×C67#42111yescharacteristic
144C4×C67—22no—
145C4×C67—21no—
146C4×C67—21no—
147C4×C67—22no—
148C4×C67—42no—
149C4×C67—41no—
150C4×C67—41no—
151C4×C67—41no—
152C4×C67—81no—
153C4×C67—81no—
154C4×C67—81no—
155#9390C6711yesmaximal, characteristic, direct product factor, Sylow
156C2×C67—21no—
157C2×C67—21no—
158C2×C67—81no—
159C2×C67#188411yescharacteristic
160C2×C67#189711yescharacteristic
161C2×C67#191511yescharacteristic
162#2921C2×C6711yescharacteristic
163#2611C2×C6711yescharacteristic
164#1528C2×C6712yescomplemented
165#1464C2×C6711yescharacteristic
166#1300C2×C6712yescomplemented
167C67#939011yescharacteristic, direct product factor, Sylow
168C2×#125C22×C6711yescharacteristic
169C2×#117C22×C6711yescharacteristic
170#433—24no—
171#432C22×C6712yes—
172#423—21no—
173#407—21no—
174C4×M16C22×C6712yes—
175#400C4×C6711yescharacteristic
176#399C4×C6711yescharacteristic
177#398C4×C6711yescharacteristic
178#397C4×C6711yescharacteristic
179#389—21no—
180#388—21no—
181#387—21no—
182#385—21no—
183#376C22×C6711yescharacteristic
184#330—24no—
185#324—21no—
186#323—21no—
187C23×C4D4×C6711yescharacteristic
188C2×M16—22no—
189#125—22no—
190#125—22no—
191#117—22no—
192#117—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4C2×C4×C6711yescharacteristic
198C2×C4⋊C4C2×C4×C6711yescharacteristic
199C2×C4⋊C4C2×C4×C6711yescharacteristic
200C2×C4⋊C4D4×C6711yescharacteristic
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4C2×C4×C6711yescharacteristic
204C2×C22⋊C4D4×C6711yescharacteristic
205C2×C42—21no—
206C2×C42D4×C6711yescharacteristic
207C2×C42C23×C6711yescharacteristic
208C4⋊C8—24no—
209C4⋊C8—22no—
210C22⋊C8—24no—
211C22⋊C8—22no—
212#97—22no—
213#97D4×C6714yes—
214C4×C8—22no—
215#95—21no—
216#95—21no—
217#95—21no—
218#95—21no—
219#95—21no—
220#95C2×C4×C6711yescharacteristic
221#95C2×C4×C6711yescharacteristic
222C24C22⋊C4×C6711yescharacteristic
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4C22⋊C4×C6711yescharacteristic
232C22×C4C22⋊C4×C6711yescharacteristic
233C22×C4C22⋊C4×C6711yescharacteristic
234C22×C4C22×C4×C6711yescharacteristic
235C22×C4C2×D4×C6711yescharacteristic
236C22×C4C2×D4×C6711yescharacteristic
237M16—42no—
238M16—42no—
239C2×C8—24no—
240C2×C8—42no—
241C4⋊C4—22no—
242C4⋊C4—22no—
243C4⋊C4—22no—
244C4⋊C4—22no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—41no—
248C4⋊C4—42no—
249C4⋊C4—41no—
250C22⋊C4—22no—
251C22⋊C4—22no—
252C22⋊C4—42no—
253C22⋊C4—42no—
254C42—21no—
255C42—21no—
256C42—42no—
257C42C2×D4×C6712yes—
258C42#41×C6712yes—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—41no—
263C23—41no—
264C23C2×C22⋊C4×C6711yescharacteristic
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4C4×D4×C6712yes—
291C2×C4#120×C6711yescharacteristic
292C2×C4#121×C6712yes—
293C2×C4#123×C6711yescharacteristic
294C8—44no—
295C8—82no—
296C22—21no—
297C22—21no—
298C22—42no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—81no—
303C22#386×C6711yescharacteristic
304C22#421×C6711yescharacteristic
305C22#421×C6711yescharacteristic
306C4—22no—
307C4—21no—
308C4—21no—
309C4—22no—
310C4—42no—
311C4—41no—
312C4—41no—
313C4—41no—
314C4—81no—
315C4—81no—
316C4—81no—
317C2—21no—
318C2—21no—
319C2—81no—
320C2C67×#188411yescharacteristic
321C2C67×#189711yescharacteristic
322C2C67×#191511yescharacteristic
subgroupG/H
center, socleC22×C67#386
commutatorC22×C4C22×C4×C67
Frattini, cosocleC2×C42C23×C67
prime radicalC24×C67C22⋊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
3. C67×#1528C2semidirect
5. C67×#1300C2semidirect
155. #9390C67direct (N × H)
164. #1528C2×C67semidirect
166. #1300C2×C67semidirect
167. C67#9390direct (N × H)

Maximal subgroups

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

subgroupindex
C67×#29212
C67×#26112
C67×#15282
C67×#14642
C67×#13002
#939067