Finite Group Databasebeta

← C67×#9091 #30688342

Subgroups — C67×#9091 #30688342

Order 17152.

504 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#2975C211yesmaximal, characteristic
2C2×#440×C67C211yesmaximal, characteristic
3C67×#1479C211yesmaximal, characteristic, complemented
4C67×#1478C211yesmaximal, characteristic
5C67×#1461C211yesmaximal, characteristic, complemented
6C67×#1281C211yesmaximal, characteristic, complemented
7C67×#1275C211yesmaximal, characteristic, complemented
8C22×SD16×C67C411yescharacteristic
9C2×C4×Q8×C67C2211yescharacteristic
10C2×C4×D4×C67C2211yescharacteristic
11#440×C67—22no—
12#440×C67—22no—
13#436×C67—22no—
14#436×C67—22no—
15#431×C67C2211yescharacteristic
16#431×C67C2211yescharacteristic
17#431×C67C2211yescharacteristic
18#431×C67C2211yescharacteristic
19C2×#107×C67C411yescharacteristic
20C2×#103×C67C411yescharacteristic
21C2×#102×C67C411yescharacteristic
22#407×C67—22no—
23C2×#97×C67C2211yescharacteristic
24Q8⋊C8×C67—22no—
25D4⋊C8×C67—22no—
26#324×C67—22no—
27C22×Q8×C67C2×C411yescharacteristic
28C22×D4×C67C2×C411yescharacteristic
29C23×C4×C67—21no—
30C2×SD16×C67—22no—
31C2×SD16×C67—22no—
32C2×SD16×C67—22no—
33C2×M16×C67—24no—
34C22×C8×C67C2×C411yescharacteristic
35C22×C8×C67C2×C411yescharacteristic
36C4×Q8×C67—21no—
37C4×Q8×C67—21no—
38C4×Q8×C67—21no—
39C4×Q8×C67—21no—
40C4×D4×C67—21no—
41C4×D4×C67—21no—
42C4×D4×C67D412yes—
43C4×D4×C67D412yes—
44C2×C4⋊C4×C67—21no—
45C2×C4⋊C4×C67C2×C411yescharacteristic
46C2×C4⋊C4×C67C2×C411yescharacteristic
47C2×C22⋊C4×C67—21no—
48C2×C42×C67—21no—
49C2×C42×C67C2311yescharacteristic
50#107×C67—22no—
51C4⋊C8×C67—21no—
52C4⋊C8×C67—21no—
53C4⋊C8×C67—21no—
54C4⋊C8×C67—21no—
55#103×C67—22no—
56#102×C67—22no—
57C22⋊C8×C67—24no—
58C22⋊C8×C67—21no—
59C22⋊C8×C67—21no—
60C22⋊C8×C67—21no—
61C22⋊C8×C67D412yes—
62#97×C67—21no—
63#97×C67—21no—
64C4×C8×C67—21no—
65C4×C8×C67—21no—
66C4×C8×C67—21no—
67C4×C8×C67D412yes—
68C24×C67—21no—
69C2×Q8×C67—21no—
70C2×Q8×C67—21no—
71C2×Q8×C67—21no—
72C2×Q8×C67—21no—
73C2×Q8×C67—21no—
74C2×Q8×C67M1612yescomplemented
75C2×D4×C67—21no—
76C2×D4×C67—21no—
77C2×D4×C67—21no—
78C2×D4×C67—21no—
79C2×D4×C67—21no—
80C2×D4×C67C22⋊C412yes—
81C22×C4×C67—21no—
82C22×C4×C67—21no—
83C22×C4×C67—22no—
84C22×C4×C67—22no—
85C22×C4×C67—21no—
86C22×C4×C67—82no—
87C22×C4×C67C22×C411yescharacteristic
88C22×C4×C67C2×D411yescharacteristic
89C22×C4×C67#4111yescharacteristic
90SD16×C67—28no—
91M16×C67—84no—
92C2×C8×C67—21no—
93C2×C8×C67—21no—
94C2×C8×C67—21no—
95C2×C8×C67—22no—
96C2×C8×C67—22no—
97C2×C8×C67—21no—
98C2×C8×C67—21no—
99C2×C8×C67—41no—
100C2×C8×C67—41no—
101C2×C8×C67M1612yescomplemented
102C4⋊C4×C67—21no—
103C4⋊C4×C67—21no—
104C4⋊C4×C67—21no—
105C4⋊C4×C67—41no—
106C4⋊C4×C67—41no—
107C4⋊C4×C67C22⋊C412yes—
108C22⋊C4×C67—22no—
109C22⋊C4×C67—22no—
110C42×C67—41no—
111C42×C67—41no—
112C42×C67C2×D411yescharacteristic
113C42×C67C2×D411yescharacteristic
114C42×C67C2×D411yescharacteristic
115C42×C67#4111yescharacteristic
116C23×C67—22no—
117C23×C67—41no—
118C23×C67—41no—
119C23×C67—81no—
120C23×C67—81no—
121C23×C67C4×D411yescharacteristic
122Q8×C67—22no—
123Q8×C67—22no—
124Q8×C67—41no—
125Q8×C67—41no—
126D4×C67—22no—
127D4×C67—22no—
128D4×C67—41no—
129D4×C67—41no—
130C2×C4×C67—22no—
131C2×C4×C67—22no—
132C2×C4×C67—41no—
133C2×C4×C67—41no—
134C2×C4×C67—44no—
135C2×C4×C67—41no—
136C2×C4×C67—41no—
137C2×C4×C67—41no—
138C2×C4×C67—82no—
139C2×C4×C67—81no—
140C2×C4×C67—81no—
141C2×C4×C67C2×C22⋊C411yescharacteristic
142C2×C4×C67C4×D411yescharacteristic
143C2×C4×C67#12011yescharacteristic
144C2×C4×C67#12111yescharacteristic
145C2×C4×C67#12111yescharacteristic
146C2×C4×C67#12311yescharacteristic
147C2×C4×C67C2×M1611yescharacteristic
148C2×C4×C67C2×M1611yescharacteristic
149C2×C4×C67#13111yescharacteristic
150C2×C4×C67#13111yescharacteristic
151C8×C67—22no—
152C8×C67—42no—
153C8×C67—42no—
154C8×C67—41no—
155C8×C67—81no—
156C8×C67—81no—
157C22×C67—21no—
158C22×C67—21no—
159C22×C67—42no—
160C22×C67—42no—
161C22×C67—42no—
162C22×C67—82no—
163C22×C67—82no—
164C22×C67#38611yescharacteristic
165C22×C67#43511yescharacteristic
166C22×C67#43611yescharacteristic
167C4×C67—21no—
168C4×C67—21no—
169C4×C67—21no—
170C4×C67—21no—
171C4×C67—42no—
172C4×C67—42no—
173C4×C67—81no—
174C4×C67#40711yescharacteristic
175C4×C67#40811yescharacteristic
176C4×C67#43511yescharacteristic
177C4×C67#43611yescharacteristic
178#9091C6711yesmaximal, characteristic, direct product factor, Sylow
179C2×C67—21no—
180C2×C67—21no—
181C2×C67—82no—
182C2×C67#187011yescharacteristic
183C2×C67#190211yescharacteristic
184C2×C67#190411yescharacteristic
185#2975C2×C6711yescharacteristic
186C2×#440C2×C6711yescharacteristic
187#1479C2×C6711yescharacteristic, complemented
188#1478C2×C6711yescharacteristic
189#1461C2×C6711yescharacteristic, complemented
190#1281C2×C6711yescharacteristic, complemented
191#1275C2×C6711yescharacteristic, complemented
192C67#909111yescharacteristic, direct product factor, Sylow
193C22×SD16C4×C6711yescharacteristic
194C2×C4×Q8C22×C6711yescharacteristic
195C2×C4×D4C22×C6711yescharacteristic
196#440—22no—
197#440—22no—
198#436—22no—
199#436—22no—
200#431C22×C6711yescharacteristic
201#431C22×C6711yescharacteristic
202#431C22×C6711yescharacteristic
203#431C22×C6711yescharacteristic
204C2×#107C4×C6711yescharacteristic
205C2×#103C4×C6711yescharacteristic
206C2×#102C4×C6711yescharacteristic
207#407—22no—
208C2×#97C22×C6711yescharacteristic
209Q8⋊C8—22no—
210D4⋊C8—22no—
211#324—22no—
212C22×Q8C2×C4×C6711yescharacteristic
213C22×D4C2×C4×C6711yescharacteristic
214C23×C4—21no—
215C2×SD16—22no—
216C2×SD16—22no—
217C2×SD16—22no—
218C2×M16—24no—
219C22×C8C2×C4×C6711yescharacteristic
220C22×C8C2×C4×C6711yescharacteristic
221C4×Q8—21no—
222C4×Q8—21no—
223C4×Q8—21no—
224C4×Q8—21no—
225C4×D4—21no—
226C4×D4—21no—
227C4×D4D4×C6712yes—
228C4×D4D4×C6712yes—
229C2×C4⋊C4—21no—
230C2×C4⋊C4C2×C4×C6711yescharacteristic
231C2×C4⋊C4C2×C4×C6711yescharacteristic
232C2×C22⋊C4—21no—
233C2×C42—21no—
234C2×C42C23×C6711yescharacteristic
235#107—22no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240#103—22no—
241#102—22no—
242C22⋊C8—24no—
243C22⋊C8—21no—
244C22⋊C8—21no—
245C22⋊C8—21no—
246C22⋊C8D4×C6712yes—
247#97—21no—
248#97—21no—
249C4×C8—21no—
250C4×C8—21no—
251C4×C8—21no—
252C4×C8D4×C6712yes—
253C24—21no—
254C2×Q8—21no—
255C2×Q8—21no—
256C2×Q8—21no—
257C2×Q8—21no—
258C2×Q8—21no—
259C2×Q8M16×C6712yescomplemented
260C2×D4—21no—
261C2×D4—21no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—21no—
265C2×D4C22⋊C4×C6712yes—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—21no—
271C22×C4—82no—
272C22×C4C22×C4×C6711yescharacteristic
273C22×C4C2×D4×C6711yescharacteristic
274C22×C4#41×C6711yescharacteristic
275SD16—28no—
276M16—84no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—22no—
281C2×C8—22no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8M16×C6712yescomplemented
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4C22⋊C4×C6712yes—
293C22⋊C4—22no—
294C22⋊C4—22no—
295C42—41no—
296C42—41no—
297C42C2×D4×C6711yescharacteristic
298C42C2×D4×C6711yescharacteristic
299C42C2×D4×C6711yescharacteristic
300C42#41×C6711yescharacteristic
301C23—22no—
302C23—41no—
303C23—41no—
304C23—81no—
305C23—81no—
306C23C4×D4×C6711yescharacteristic
307Q8—22no—
308Q8—22no—
309Q8—41no—
310Q8—41no—
311D4—22no—
312D4—22no—
313D4—41no—
314D4—41no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—44no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—82no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4C2×C22⋊C4×C6711yescharacteristic
327C2×C4C4×D4×C6711yescharacteristic
328C2×C4#120×C6711yescharacteristic
329C2×C4#121×C6711yescharacteristic
330C2×C4#121×C6711yescharacteristic
331C2×C4#123×C6711yescharacteristic
332C2×C4C2×M16×C6711yescharacteristic
333C2×C4C2×M16×C6711yescharacteristic
334C2×C4#131×C6711yescharacteristic
335C2×C4#131×C6711yescharacteristic
336C8—22no—
337C8—42no—
338C8—42no—
339C8—41no—
340C8—81no—
341C8—81no—
342C22—21no—
343C22—21no—
344C22—42no—
345C22—42no—
346C22—42no—
347C22—82no—
348C22—82no—
349C22#386×C6711yescharacteristic
350C22#435×C6711yescharacteristic
351C22#436×C6711yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—42no—
357C4—42no—
358C4—81no—
359C4#407×C6711yescharacteristic
360C4#408×C6711yescharacteristic
361C4#435×C6711yescharacteristic
362C4#436×C6711yescharacteristic
363C2—21no—
364C2—21no—
365C2—82no—
366C2C67×#187011yescharacteristic
367C2C67×#190211yescharacteristic
368C2C67×#190411yescharacteristic
subgroupG/H
center, socleC22×C67#386
commutatorC22×C4C22×C4×C67
Frattini, cosocleC2×C42C23×C67
prime radicalC22×D4×C67C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C67 ◁ C2×C42×C67 ◁ G

Complements

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

NH ≅ G/Ntype
3. C67×#1479C2semidirect
5. C67×#1461C2semidirect
6. C67×#1281C2semidirect
7. C67×#1275C2semidirect
74. C2×Q8×C67M16semidirect
101. C2×C8×C67M16semidirect
178. #9091C67direct (N × H)
187. #1479C2×C67semidirect
189. #1461C2×C67semidirect
190. #1281C2×C67semidirect
191. #1275C2×C67semidirect
192. C67#9091direct (N × H)
259. C2×Q8M16×C67semidirect
286. C2×C8M16×C67semidirect

Maximal subgroups

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

subgroupindex
C67×#29752
C2×#440×C672
C67×#14792
C67×#14782
C67×#14612
C67×#12812
C67×#12752
#909167