Finite Group Databasebeta

← C67×#25948 #30705199

Subgroups — C67×#25948 #30705199

Order 17152.

1082 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3501C211yesmaximal, characteristic
2C67×#2503C212yesmaximal, complemented
3C67×#2486C211yesmaximal, characteristic
4C67×#2030C214yesmaximal, complemented
5C67×#1996C212yesmaximal, complemented
6C67×#1890C214yesmaximal
7C4×#99×C67C211yesmaximal, characteristic
8#554×C67—22no—
9#554×C67—24no—
10D4×Q8×C67—22no—
11#544×C67—21no—
12#543×C67—21no—
13#543×C67C2212yes—
14#533×C67—21no—
15C2×#123×C67C2212yes—
16C2×C22⋊Q8×C67—21no—
17C2×C22⋊Q8×C67C2212yes—
18#518×C67C2212yes—
19C2×C4×D4×C67C2211yescharacteristic
20C22×C4⋊C4×C67C2211yescharacteristic
21C2×#99×C67C2212yes—
22#400×C67—24no—
23#398×C67—21no—
24#397×C67—21no—
25#391×C67—22no—
26#389×C67—21no—
27#382×C67—22no—
28#381×C67C2218yes—
29#381×C67C2212yes—
30#380×C67—21no—
31#380×C67C2211yescharacteristic
32C4×C22⋊C4×C67C2212yes—
33#342×C67C2218yes—
34#342×C67C2212yes—
35C22×Q8×C67—21no—
36C22×Q8×C67D412yes—
37C22×D4×C67C2311yescharacteristic
38C23×C4×C67D411yescharacteristic
39C23×C4×C67C2312yes—
40C23×C4×C67C2311yescharacteristic
41C4⋊Q8×C67—21no—
42C4⋊Q8×C67—21no—
43C4⋊Q8×C67—24no—
44#125×C67—22no—
45#125×C67—24no—
46#123×C67—28no—
47C22⋊Q8×C67—22no—
48C22⋊Q8×C67—22no—
49C22⋊Q8×C67—24no—
50C22⋊Q8×C67—24no—
51C22⋊Q8×C67—22no—
52C22⋊Q8×C67—22no—
53C22⋊Q8×C67—22no—
54C22⋊Q8×C67—22no—
55C22⋊Q8×C67—22no—
56C22⋊Q8×C67—22no—
57C22⋊Q8×C67—24no—
58C22⋊Q8×C67—24no—
59C4×Q8×C67—22no—
60C4×Q8×C67—22no—
61C4×D4×C67—22no—
62C4×D4×C67—22no—
63C4×D4×C67—24no—
64C4×D4×C67—24no—
65#117×C67—22no—
66C2×C4⋊C4×C67—21no—
67C2×C4⋊C4×C67—22no—
68C2×C4⋊C4×C67—21no—
69C2×C4⋊C4×C67—22no—
70C2×C4⋊C4×C67—21no—
71C2×C4⋊C4×C67—22no—
72C2×C4⋊C4×C67—22no—
73C2×C4⋊C4×C67—21no—
74C2×C4⋊C4×C67D411yescharacteristic
75C2×C4⋊C4×C67D412yes—
76C2×C4⋊C4×C67D414yes—
77C2×C22⋊C4×C67—22no—
78C2×C22⋊C4×C67—22no—
79C2×C22⋊C4×C67D411yescharacteristic
80C2×C22⋊C4×C67C2312yes—
81C2×C22⋊C4×C67C2314yes—
82C2×C22⋊C4×C67C2311yescharacteristic
83C2×C22⋊C4×C67C2314yes—
84C2×C42×C67—22no—
85C2×C42×C67D411yescharacteristic
86#99×C67—24no—
87#95×C67—22no—
88#95×C67—22no—
89#95×C67—21no—
90#95×C67—28no—
91#95×C67—22no—
92#95×C67—28no—
93C24×C67C2×D411yescharacteristic
94C24×C67C2411yescharacteristic
95C2×Q8×C67—22no—
96C2×Q8×C67—22no—
97C2×Q8×C67—22no—
98C2×Q8×C67—22no—
99C2×Q8×C67—44no—
100C2×Q8×C67—82no—
101C2×D4×C67—22no—
102C2×D4×C67—22no—
103C2×D4×C67#4114yes—
104C22×C4×C67—22no—
105C22×C4×C67—22no—
106C22×C4×C67—21no—
107C22×C4×C67—21no—
108C22×C4×C67—24no—
109C22×C4×C67—22no—
110C22×C4×C67—22no—
111C22×C4×C67—24no—
112C22×C4×C67—24no—
113C22×C4×C67—24no—
114C22×C4×C67—42no—
115C22×C4×C67—42no—
116C22×C4×C67C2×D412yes—
117C22×C4×C67C2×D411yescharacteristic
118C22×C4×C67C2×D411yescharacteristic
119C22×C4×C67C2×D411yescharacteristic
120C22×C4×C67C2×D414yes—
121C22×C4×C67C2×D414yes—
122C22×C4×C67C2×D411yescharacteristic
123C22×C4×C67C2×D411yescharacteristic
124C22×C4×C67C2×D412yes—
125C4⋊C4×C67—22no—
126C4⋊C4×C67—24no—
127C4⋊C4×C67—22no—
128C4⋊C4×C67—22no—
129C4⋊C4×C67—24no—
130C4⋊C4×C67—22no—
131C4⋊C4×C67—24no—
132C4⋊C4×C67—22no—
133C4⋊C4×C67—22no—
134C4⋊C4×C67—22no—
135C4⋊C4×C67—24no—
136C4⋊C4×C67—42no—
137C4⋊C4×C67—41no—
138C4⋊C4×C67—42no—
139C4⋊C4×C67—41no—
140C4⋊C4×C67—44no—
141C22⋊C4×C67—24no—
142C22⋊C4×C67—24no—
143C22⋊C4×C67—22no—
144C22⋊C4×C67—24no—
145C22⋊C4×C67—22no—
146C22⋊C4×C67—24no—
147C22⋊C4×C67—24no—
148C22⋊C4×C67—24no—
149C22⋊C4×C67—44no—
150C42×C67—24no—
151C42×C67—21no—
152C42×C67—21no—
153C42×C67—44no—
154C23×C67—21no—
155C23×C67—21no—
156C23×C67—41no—
157C23×C67—41no—
158C23×C67—41no—
159C23×C67—41no—
160C23×C67#12412yes—
161C23×C67C22×D411yescharacteristic
162C23×C67C22×D412yes—
163C23×C67C2×#4112yes—
164C23×C672−1+412yes—
165Q8×C67—44no—
166Q8×C67—82no—
167Q8×C67—82no—
168Q8×C67—82no—
169D4×C67—28no—
170C2×C4×C67—21no—
171C2×C4×C67—21no—
172C2×C4×C67—24no—
173C2×C4×C67—22no—
174C2×C4×C67—21no—
175C2×C4×C67—24no—
176C2×C4×C67—24no—
177C2×C4×C67—21no—
178C2×C4×C67—42no—
179C2×C4×C67—41no—
180C2×C4×C67—42no—
181C2×C4×C67—41no—
182C2×C4×C67—44no—
183C2×C4×C67—41no—
184C2×C4×C67—44no—
185C2×C4×C67—44no—
186C2×C4×C67—42no—
187C2×C4×C67—44no—
188C2×C4×C67—42no—
189C2×C4×C67—42no—
190C2×C4×C67—42no—
191C2×C4×C67—42no—
192C2×C4×C67—42no—
193C2×C4×C67—44no—
194C2×C4×C67—48no—
195C2×C4×C67—81no—
196C2×C4×C67—82no—
197C2×C4×C67—82no—
198C2×C4×C67#12011yescharacteristic
199C2×C4×C67#12012yes—
200C2×C4×C67#12011yescharacteristic
201C2×C4×C67#12412yes—
202C22×C67—22no—
203C22×C67—21no—
204C22×C67—22no—
205C22×C67—21no—
206C22×C67—22no—
207C22×C67—42no—
208C22×C67—42no—
209C22×C67—42no—
210C22×C67—41no—
211C22×C67—41no—
212C22×C67—81no—
213C22×C67C2×#12011yescharacteristic
214C22×C67C2×#12411yescharacteristic
215C22×C67#53611yescharacteristic
216C22×C67#54814yes—
217C4×C67—22no—
218C4×C67—21no—
219C4×C67—22no—
220C4×C67—41no—
221C4×C67—44no—
222C4×C67—44no—
223C4×C67—41no—
224C4×C67—82no—
225C4×C67—84no—
226C4×C67—82no—
227C4×C67#45812yes—
228#25948C6711yesmaximal, characteristic, direct product factor, Sylow
229C2×C67—21no—
230C2×C67—24no—
231C2×C67—21no—
232C2×C67—42no—
233C2×C67#243611yescharacteristic
234C2×C67C2×#45811yescharacteristic
235C2×C67#302811yescharacteristic
236#3501C2×C6711yescharacteristic
237#2503C2×C6712yescomplemented
238#2486C2×C6711yescharacteristic
239#2030C2×C6714yescomplemented
240#1996C2×C6712yescomplemented
241#1890C2×C6714yes—
242C4×#99C2×C6711yescharacteristic
243C67#2594811yescharacteristic, direct product factor, Sylow
244#554—22no—
245#554—24no—
246D4×Q8—22no—
247#544—21no—
248#543—21no—
249#543C22×C6712yes—
250#533—21no—
251C2×#123C22×C6712yes—
252C2×C22⋊Q8—21no—
253C2×C22⋊Q8C22×C6712yes—
254#518C22×C6712yes—
255C2×C4×D4C22×C6711yescharacteristic
256C22×C4⋊C4C22×C6711yescharacteristic
257C2×#99C22×C6712yes—
258#400—24no—
259#398—21no—
260#397—21no—
261#391—22no—
262#389—21no—
263#382—22no—
264#381C22×C6718yes—
265#381C22×C6712yes—
266#380—21no—
267#380C22×C6711yescharacteristic
268C4×C22⋊C4C22×C6712yes—
269#342C22×C6718yes—
270#342C22×C6712yes—
271C22×Q8—21no—
272C22×Q8D4×C6712yes—
273C22×D4C23×C6711yescharacteristic
274C23×C4D4×C6711yescharacteristic
275C23×C4C23×C6712yes—
276C23×C4C23×C6711yescharacteristic
277C4⋊Q8—21no—
278C4⋊Q8—21no—
279C4⋊Q8—24no—
280#125—22no—
281#125—24no—
282#123—28no—
283C22⋊Q8—22no—
284C22⋊Q8—22no—
285C22⋊Q8—24no—
286C22⋊Q8—24no—
287C22⋊Q8—22no—
288C22⋊Q8—22no—
289C22⋊Q8—22no—
290C22⋊Q8—22no—
291C22⋊Q8—22no—
292C22⋊Q8—22no—
293C22⋊Q8—24no—
294C22⋊Q8—24no—
295C4×Q8—22no—
296C4×Q8—22no—
297C4×D4—22no—
298C4×D4—22no—
299C4×D4—24no—
300C4×D4—24no—
301#117—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4D4×C6711yescharacteristic
311C2×C4⋊C4D4×C6712yes—
312C2×C4⋊C4D4×C6714yes—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4D4×C6711yescharacteristic
316C2×C22⋊C4C23×C6712yes—
317C2×C22⋊C4C23×C6714yes—
318C2×C22⋊C4C23×C6711yescharacteristic
319C2×C22⋊C4C23×C6714yes—
320C2×C42—22no—
321C2×C42D4×C6711yescharacteristic
322#99—24no—
323#95—22no—
324#95—22no—
325#95—21no—
326#95—28no—
327#95—22no—
328#95—28no—
329C24C2×D4×C6711yescharacteristic
330C24C24×C6711yescharacteristic
331C2×Q8—22no—
332C2×Q8—22no—
333C2×Q8—22no—
334C2×Q8—22no—
335C2×Q8—44no—
336C2×Q8—82no—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4#41×C6714yes—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4C2×D4×C6712yes—
353C22×C4C2×D4×C6711yescharacteristic
354C22×C4C2×D4×C6711yescharacteristic
355C22×C4C2×D4×C6711yescharacteristic
356C22×C4C2×D4×C6714yes—
357C22×C4C2×D4×C6714yes—
358C22×C4C2×D4×C6711yescharacteristic
359C22×C4C2×D4×C6711yescharacteristic
360C22×C4C2×D4×C6712yes—
361C4⋊C4—22no—
362C4⋊C4—24no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—24no—
366C4⋊C4—22no—
367C4⋊C4—24no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—24no—
372C4⋊C4—42no—
373C4⋊C4—41no—
374C4⋊C4—42no—
375C4⋊C4—41no—
376C4⋊C4—44no—
377C22⋊C4—24no—
378C22⋊C4—24no—
379C22⋊C4—22no—
380C22⋊C4—24no—
381C22⋊C4—22no—
382C22⋊C4—24no—
383C22⋊C4—24no—
384C22⋊C4—24no—
385C22⋊C4—44no—
386C42—24no—
387C42—21no—
388C42—21no—
389C42—44no—
390C23—21no—
391C23—21no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23—41no—
396C23#124×C6712yes—
397C23C22×D4×C6711yescharacteristic
398C23C22×D4×C6712yes—
399C23C2×#41×C6712yes—
400C232−1+4×C6712yes—
401Q8—44no—
402Q8—82no—
403Q8—82no—
404Q8—82no—
405D4—28no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—24no—
409C2×C4—22no—
410C2×C4—21no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—21no—
414C2×C4—42no—
415C2×C4—41no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—41no—
421C2×C4—44no—
422C2×C4—42no—
423C2×C4—44no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—44no—
430C2×C4—48no—
431C2×C4—81no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4#120×C6711yescharacteristic
435C2×C4#120×C6712yes—
436C2×C4#120×C6711yescharacteristic
437C2×C4#124×C6712yes—
438C22—22no—
439C22—21no—
440C22—22no—
441C22—21no—
442C22—22no—
443C22—42no—
444C22—42no—
445C22—42no—
446C22—41no—
447C22—41no—
448C22—81no—
449C22C2×#120×C6711yescharacteristic
450C22C2×#124×C6711yescharacteristic
451C22#536×C6711yescharacteristic
452C22#548×C6714yes—
453C4—21no—
454C4—22no—
455C4—22no—
456C4—41no—
457C4—44no—
458C4—44no—
459C4—41no—
460C4—82no—
461C4—84no—
462C4—82no—
463C4#458×C6712yes—
464C2—21no—
465C2—24no—
466C2—21no—
467C2—42no—
468C2C67×#243611yescharacteristic
469C2C2×#458×C6711yescharacteristic
470C2C67×#302811yescharacteristic
subgroupG/H
center, socleC22×C67C2×#120
commutator, Frattini, cosocleC24C24×C67
prime radicalC22×D4×C67C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C67 ◁ C23×C4×C67 ◁ G

Complements

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

NH ≅ G/Ntype
2. C67×#2503C2semidirect
4. C67×#2030C2semidirect
5. C67×#1996C2semidirect
228. #25948C67direct (N × H)
237. #2503C2×C67semidirect
239. #2030C2×C67semidirect
240. #1996C2×C67semidirect
243. C67#25948direct (N × H)

Maximal subgroups

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

subgroupindex
C67×#35012
C67×#25032
C67×#24862
C67×#20302
C67×#19962
C67×#18902
C4×#99×C672
#2594867