Finite Group Databasebeta

← C67×#9010 #30688261

Subgroups — C67×#9010 #30688261

Order 17152.

782 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#2975C211yesmaximal, characteristic, complemented
2C67×#2388C211yesmaximal, characteristic
3C67×#1789C212yesmaximal, complemented
4C4×#99×C67C211yesmaximal, characteristic, complemented
5C67×#1297C212yesmaximal, complemented
6C22×#41×C67C2211yescharacteristic
7C2×C4×D4×C67—21no—
8C2×C4×D4×C67C2211yescharacteristic
9#436×C67—24no—
10#431×C67—21no—
11C2×#104×C67—24no—
12C2×#99×C67—21no—
13#407×C67—21no—
14#407×C67C2212yes—
15#395×C67C411yescharacteristic, complemented
16#394×C67—21no—
17#394×C67C411yescharacteristic, complemented
18#393×C67C411yescharacteristic, complemented
19#392×C67—21no—
20#392×C67C411yescharacteristic, complemented
21C4×C22⋊C4×C67—21no—
22C4×C22⋊C4×C67C2212yes—
23C4×C22⋊C4×C67C2211yescharacteristic
24#342×C67—21no—
25C2×#41×C67—22no—
26C2×#41×C67—22no—
27C2×#41×C67—44no—
28C22×Q8×C67C2×C411yescharacteristic
29C22×D4×C67—21no—
30C22×D4×C67—21no—
31C22×D4×C67C2×C411yescharacteristic
32C22×D4×C67D411yescharacteristic
33C23×C4×C67—21no—
34C23×C4×C67D411yescharacteristic
35C23×C4×C67C2311yescharacteristic
36C2×M16×C67—22no—
37C2×M16×C67D414yescomplemented
38C4×D4×C67—22no—
39C4×D4×C67—22no—
40C4×D4×C67—44no—
41C2×C4⋊C4×C67—21no—
42C2×C4⋊C4×C67C2×C411yescharacteristic
43C2×C22⋊C4×C67—22no—
44C2×C22⋊C4×C67—21no—
45C2×C22⋊C4×C67—21no—
46C2×C22⋊C4×C67—21no—
47C2×C22⋊C4×C67—21no—
48C2×C22⋊C4×C67—21no—
49C2×C22⋊C4×C67C2×C411yescharacteristic
50C2×C22⋊C4×C67D411yescharacteristic
51C2×C42×C67—21no—
52C2×C42×C67—22no—
53C2×C42×C67D411yescharacteristic
54C4⋊C8×C67—22no—
55#104×C67—48no—
56#99×C67—24no—
57C22⋊C8×C67—22no—
58C22⋊C8×C67—22no—
59C22⋊C8×C67—22no—
60C4×C8×C67—22no—
61#95×C67—21no—
62#95×C67—21no—
63#95×C67—22no—
64#95×C67C2×C411yescharacteristic
65#95×C67C2×C411yescharacteristic
66C24×C67—21no—
67C24×C67C22⋊C411yescharacteristic
68C24×C67C2×D411yescharacteristic
69#41×C67—28no—
70#41×C67—44no—
71#41×C67—44no—
72#41×C67—44no—
73C2×Q8×C67—22no—
74C2×Q8×C67—42no—
75C2×D4×C67—22no—
76C2×D4×C67—22no—
77C2×D4×C67—22no—
78C2×D4×C67—41no—
79C2×D4×C67—42no—
80C2×D4×C67—42no—
81C2×D4×C67—42no—
82C2×D4×C67—42no—
83C2×D4×C67—42no—
84C2×D4×C67—42no—
85C2×D4×C67—42no—
86C22×C4×C67—24no—
87C22×C4×C67—21no—
88C22×C4×C67—21no—
89C22×C4×C67—21no—
90C22×C4×C67—21no—
91C22×C4×C67—21no—
92C22×C4×C67—42no—
93C22×C4×C67—42no—
94C22×C4×C67—42no—
95C22×C4×C67—41no—
96C22×C4×C67—41no—
97C22×C4×C67—42no—
98C22×C4×C67C22⋊C411yescharacteristic
99C22×C4×C67C22⋊C411yescharacteristic
100C22×C4×C67C22⋊C411yescharacteristic
101C22×C4×C67C22×C411yescharacteristic
102C22×C4×C67C2×D412yes—
103C22×C4×C67C2×D411yescharacteristic
104C22×C4×C67#4112yes—
105M16×C67—44no—
106M16×C67—82no—
107C2×C8×C67—24no—
108C4⋊C4×C67—22no—
109C4⋊C4×C67—42no—
110C22⋊C4×C67—22no—
111C22⋊C4×C67—22no—
112C22⋊C4×C67—24no—
113C22⋊C4×C67—42no—
114C22⋊C4×C67—42no—
115C22⋊C4×C67—42no—
116C22⋊C4×C67—42no—
117C22⋊C4×C67—42no—
118C22⋊C4×C67—42no—
119C42×C67—21no—
120C42×C67—21no—
121C42×C67—42no—
122C42×C67—44no—
123C23×C67—21no—
124C23×C67—21no—
125C23×C67—22no—
126C23×C67—21no—
127C23×C67—21no—
128C23×C67—21no—
129C23×C67—41no—
130C23×C67—41no—
131C23×C67—41no—
132C23×C67—41no—
133C23×C67—41no—
134C23×C67—41no—
135C23×C67—41no—
136C23×C67—41no—
137C23×C67—41no—
138C23×C67—41no—
139C23×C67C2×C22⋊C411yescharacteristic
140C23×C67#12112yes—
141Q8×C67—24no—
142Q8×C67—42no—
143D4×C67—24no—
144D4×C67—44no—
145D4×C67—42no—
146D4×C67—48no—
147D4×C67—48no—
148C2×C4×C67—21no—
149C2×C4×C67—21no—
150C2×C4×C67—24no—
151C2×C4×C67—22no—
152C2×C4×C67—22no—
153C2×C4×C67—22no—
154C2×C4×C67—21no—
155C2×C4×C67—21no—
156C2×C4×C67—41no—
157C2×C4×C67—42no—
158C2×C4×C67—44no—
159C2×C4×C67—41no—
160C2×C4×C67—41no—
161C2×C4×C67—41no—
162C2×C4×C67—41no—
163C2×C4×C67—42no—
164C2×C4×C67—44no—
165C2×C4×C67—41no—
166C2×C4×C67—41no—
167C2×C4×C67—41no—
168C2×C4×C67—44no—
169C2×C4×C67—41no—
170C2×C4×C67—41no—
171C2×C4×C67—81no—
172C2×C4×C67—82no—
173C2×C4×C67—82no—
174C2×C4×C67#10412yescomplemented
175C2×C4×C67C4×D412yes—
176C2×C4×C67#12011yescharacteristic
177C2×C4×C67#12311yescharacteristic
178C8×C67—44no—
179C22×C67—21no—
180C22×C67—21no—
181C22×C67—22no—
182C22×C67—22no—
183C22×C67—22no—
184C22×C67—42no—
185C22×C67—42no—
186C22×C67—42no—
187C22×C67—41no—
188C22×C67—41no—
189C22×C67—42no—
190C22×C67—41no—
191C22×C67—81no—
192C22×C67—81no—
193C22×C67—81no—
194C22×C67—81no—
195C22×C67—81no—
196C22×C67—81no—
197C22×C67—81no—
198C22×C67#38611yescharacteristic
199C22×C67C2×#10411yescharacteristic
200C22×C67#42111yescharacteristic
201C4×C67—22no—
202C4×C67—21no—
203C4×C67—22no—
204C4×C67—42no—
205C4×C67—41no—
206C4×C67—42no—
207C4×C67—41no—
208C4×C67—81no—
209C4×C67—84no—
210C4×C67B(4,𝔽2)12yes—
211#9010C6711yesmaximal, characteristic, direct product factor, Sylow
212C2×C67—21no—
213C2×C67—21no—
214C2×C67—42no—
215C2×C67—81no—
216C2×C67—82no—
217C2×C67#187111yescharacteristic
218C2×C67#188411yescharacteristic
219C2×C67#188911yescharacteristic
220#2975C2×C6711yescharacteristic, complemented
221#2388C2×C6711yescharacteristic
222#1789C2×C6712yescomplemented
223C4×#99C2×C6711yescharacteristic, complemented
224#1297C2×C6712yescomplemented
225C67#901011yescharacteristic, direct product factor, Sylow
226C22×#41C22×C6711yescharacteristic
227C2×C4×D4—21no—
228C2×C4×D4C22×C6711yescharacteristic
229#436—24no—
230#431—21no—
231C2×#104—24no—
232C2×#99—21no—
233#407—21no—
234#407C22×C6712yes—
235#395C4×C6711yescharacteristic, complemented
236#394—21no—
237#394C4×C6711yescharacteristic, complemented
238#393C4×C6711yescharacteristic, complemented
239#392—21no—
240#392C4×C6711yescharacteristic, complemented
241C4×C22⋊C4—21no—
242C4×C22⋊C4C22×C6712yes—
243C4×C22⋊C4C22×C6711yescharacteristic
244#342—21no—
245C2×#41—22no—
246C2×#41—22no—
247C2×#41—44no—
248C22×Q8C2×C4×C6711yescharacteristic
249C22×D4—21no—
250C22×D4—21no—
251C22×D4C2×C4×C6711yescharacteristic
252C22×D4D4×C6711yescharacteristic
253C23×C4—21no—
254C23×C4D4×C6711yescharacteristic
255C23×C4C23×C6711yescharacteristic
256C2×M16—22no—
257C2×M16D4×C6714yescomplemented
258C4×D4—22no—
259C4×D4—22no—
260C4×D4—44no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4C2×C4×C6711yescharacteristic
263C2×C22⋊C4—22no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4C2×C4×C6711yescharacteristic
270C2×C22⋊C4D4×C6711yescharacteristic
271C2×C42—21no—
272C2×C42—22no—
273C2×C42D4×C6711yescharacteristic
274C4⋊C8—22no—
275#104—48no—
276#99—24no—
277C22⋊C8—22no—
278C22⋊C8—22no—
279C22⋊C8—22no—
280C4×C8—22no—
281#95—21no—
282#95—21no—
283#95—22no—
284#95C2×C4×C6711yescharacteristic
285#95C2×C4×C6711yescharacteristic
286C24—21no—
287C24C22⋊C4×C6711yescharacteristic
288C24C2×D4×C6711yescharacteristic
289#41—28no—
290#41—44no—
291#41—44no—
292#41—44no—
293C2×Q8—22no—
294C2×Q8—42no—
295C2×D4—22no—
296C2×D4—22no—
297C2×D4—22no—
298C2×D4—41no—
299C2×D4—42no—
300C2×D4—42no—
301C2×D4—42no—
302C2×D4—42no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—42no—
306C22×C4—24no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—42no—
318C22×C4C22⋊C4×C6711yescharacteristic
319C22×C4C22⋊C4×C6711yescharacteristic
320C22×C4C22⋊C4×C6711yescharacteristic
321C22×C4C22×C4×C6711yescharacteristic
322C22×C4C2×D4×C6712yes—
323C22×C4C2×D4×C6711yescharacteristic
324C22×C4#41×C6712yes—
325M16—44no—
326M16—82no—
327C2×C8—24no—
328C4⋊C4—22no—
329C4⋊C4—42no—
330C22⋊C4—22no—
331C22⋊C4—22no—
332C22⋊C4—24no—
333C22⋊C4—42no—
334C22⋊C4—42no—
335C22⋊C4—42no—
336C22⋊C4—42no—
337C22⋊C4—42no—
338C22⋊C4—42no—
339C42—21no—
340C42—21no—
341C42—42no—
342C42—44no—
343C23—21no—
344C23—21no—
345C23—22no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—41no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23C2×C22⋊C4×C6711yescharacteristic
360C23#121×C6712yes—
361Q8—24no—
362Q8—42no—
363D4—24no—
364D4—44no—
365D4—42no—
366D4—48no—
367D4—48no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—24no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—44no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—44no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—44no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—81no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4#104×C6712yescomplemented
395C2×C4C4×D4×C6712yes—
396C2×C4#120×C6711yescharacteristic
397C2×C4#123×C6711yescharacteristic
398C8—44no—
399C22—21no—
400C22—21no—
401C22—22no—
402C22—22no—
403C22—22no—
404C22—42no—
405C22—42no—
406C22—42no—
407C22—41no—
408C22—41no—
409C22—42no—
410C22—41no—
411C22—81no—
412C22—81no—
413C22—81no—
414C22—81no—
415C22—81no—
416C22—81no—
417C22—81no—
418C22#386×C6711yescharacteristic
419C22C2×#104×C6711yescharacteristic
420C22#421×C6711yescharacteristic
421C4—22no—
422C4—21no—
423C4—22no—
424C4—42no—
425C4—41no—
426C4—42no—
427C4—41no—
428C4—81no—
429C4—84no—
430C4B(4,𝔽2)×C6712yes—
431C2—21no—
432C2—21no—
433C2—42no—
434C2—81no—
435C2—82no—
436C2C67×#187111yescharacteristic
437C2C67×#188411yescharacteristic
438C2C67×#188911yescharacteristic
subgroupG/H
centerC2×C4×C67#120
commutatorC22×C4C22×C4×C67
Frattini, cosocleC23×C4C23×C67
socleC22×C67#386
prime radicalC67×#2388C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C67 ◁ C23×C4×C67 ◁ G

Complements

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

NH ≅ G/Ntype
1. C67×#2975C2semidirect
3. C67×#1789C2semidirect
4. C4×#99×C67C2semidirect
5. C67×#1297C2semidirect
15. #395×C67C4semidirect
17. #394×C67C4semidirect
18. #393×C67C4semidirect
20. #392×C67C4semidirect
37. C2×M16×C67D4semidirect
174. C2×C4×C67#104semidirect
211. #9010C67direct (N × H)
220. #2975C2×C67semidirect
222. #1789C2×C67semidirect
223. C4×#99C2×C67semidirect
224. #1297C2×C67semidirect
225. C67#9010direct (N × H)
235. #395C4×C67semidirect
237. #394C4×C67semidirect
238. #393C4×C67semidirect
240. #392C4×C67semidirect
257. C2×M16D4×C67semidirect
394. C2×C4#104×C67semidirect

Maximal subgroups

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

subgroupindex
C67×#29752
C67×#23882
C67×#17892
C4×#99×C672
C67×#12972
#901067