Finite Group Databasebeta

← C67×#33887 #30713138

Subgroups — C67×#33887 #30713138

Order 17152.

548 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3339C211yesmaximal, characteristic
2D4×Q16×C67C211yesmaximal, characteristic, complemented
3C67×#2245C211yesmaximal, characteristic, complemented
4C67×#2237C211yesmaximal, characteristic, complemented
5C67×#2236C211yesmaximal, characteristic, complemented
6C67×#2229C211yesmaximal, characteristic, complemented
7C67×#2220C211yesmaximal, characteristic, complemented
8C67×#2217C211yesmaximal, characteristic, complemented
9C67×#2208C211yesmaximal, characteristic, complemented
10C67×#2185C211yesmaximal, characteristic, complemented
11C67×#2185C211yesmaximal, characteristic, complemented
12C67×#2175C211yesmaximal, characteristic, complemented
13C67×#2168C211yesmaximal, characteristic, complemented
14C67×#2141C211yesmaximal, characteristic, complemented
15C2×#358×C67C211yesmaximal, characteristic, complemented
16C22×Q16×C67C2211yescharacteristic
17C22×Q16×C67C2211yescharacteristic
18#553×C67—21no—
19D4×Q8×C67—21no—
20C2×SD32×C67C2211yescharacteristic
21C2×SD32×C67C2211yescharacteristic
22C2×SD32×C67C2211yescharacteristic
23C2×M32×C67C2211yescharacteristic
24C22×C16×C67C2211yescharacteristic
25#499×C67C2211yescharacteristic
26#494×C67C2211yescharacteristic
27#482×C67—21no—
28#481×C67—21no—
29#478×C67—21no—
30#467×C67C2211yescharacteristic
31#467×C67C2211yescharacteristic
32#466×C67C2211yescharacteristic
33#466×C67C2211yescharacteristic
34#462×C67—21no—
35#451×C67—21no—
36#451×C67—21no—
37C4×Q16×C67C2211yescharacteristic
38C4×D8×C67C2211yescharacteristic
39C8×D4×C67C2211yescharacteristic
40#427×C67C2211yescharacteristic
41C2×#107×C67C2211yescharacteristic
42#367×C67C2211yescharacteristic
43#366×C67C2211yescharacteristic
44#366×C67C2211yescharacteristic
45C4⋊C16×C67C2211yescharacteristic
46#358×C67C2211yescharacteristic
47#358×C67C2211yescharacteristic
48#358×C67C2211yescharacteristic
49#358×C67C2211yescharacteristic
50#358×C67C2211yescharacteristic
51#358×C67C2211yescharacteristic
52#358×C67C2211yescharacteristic
53#358×C67C2211yescharacteristic
54#358×C67C2211yescharacteristic
55#357×C67C2211yescharacteristic
56C22⋊C16×C67C2211yescharacteristic
57C22⋊C16×C67C2211yescharacteristic
58#346×C67C2211yescharacteristic
59C22×Q8×C67—21no—
60C22×Q8×C67—21no—
61C2×Q16×C67—22no—
62C2×Q16×C67—22no—
63C2×Q16×C67—21no—
64C2×Q16×C67—21no—
65C2×Q16×C67—21no—
66C2×Q16×C67—41no—
67C2×Q16×C67—41no—
68C2×Q16×C67C2311yescharacteristic
69C2×Q16×C67C2311yescharacteristic
70C2×Q16×C67C2311yescharacteristic
71C2×D8×C67C2311yescharacteristic
72C22×C8×C67C2311yescharacteristic
73C22×C8×C67C2311yescharacteristic
74C4⋊Q8×C67—21no—
75C4⋊Q8×C67—41no—
76#126×C67—41no—
77#125×C67—21no—
78#123×C67—41no—
79C22⋊Q8×C67—21no—
80C22⋊Q8×C67—21no—
81C22⋊Q8×C67—41no—
82C22⋊Q8×C67—41no—
83#121×C67—21no—
84#121×C67—21no—
85#121×C67—41no—
86C4×Q8×C67—21no—
87C4×D4×C67—21no—
88C4×D4×C67—41no—
89C4×D4×C67—41no—
90C4×D4×C67D411yescharacteristic
91#117×C67—21no—
92C2×C4⋊C4×C67—21no—
93SD32×C67—22no—
94SD32×C67—41no—
95SD32×C67—41no—
96M32×C67—41no—
97C2×C16×C67—41no—
98C2×C16×C67C2311yescharacteristic
99C2×C16×C67C2311yescharacteristic
100C2×C16×C67C2311yescharacteristic
101C2×C16×C67C2311yescharacteristic
102#107×C67—21no—
103#107×C67C2311yescharacteristic
104#107×C67C2311yescharacteristic
105#107×C67C2311yescharacteristic
106#107×C67C2311yescharacteristic
107#106×C67—21no—
108#106×C67—21no—
109C4⋊C8×C67D411yescharacteristic
110#103×C67—21no—
111#103×C67—21no—
112#103×C67—21no—
113#102×C67—21no—
114#102×C67—21no—
115#102×C67—21no—
116C22⋊C8×C67D411yescharacteristic
117C22⋊C8×C67D411yescharacteristic
118C4×C8×C67C2311yescharacteristic
119C2×Q8×C67—21no—
120C2×Q8×C67—21no—
121C2×Q8×C67—21no—
122C2×Q8×C67—42no—
123C2×Q8×C67—42no—
124C2×Q8×C67—41no—
125C2×Q8×C67—41no—
126C2×Q8×C67—81no—
127C2×Q8×C67—81no—
128C2×D4×C67—21no—
129C2×D4×C67—41no—
130C2×D4×C67—41no—
131C2×D4×C67D811yescharacteristic
132C22×C4×C67—41no—
133C22×C4×C67—41no—
134C22×C4×C67—41no—
135C22×C4×C67—41no—
136C22×C4×C67C2×D411yescharacteristic
137C22×C4×C67C2×D411yescharacteristic
138Q16×C67—21no—
139Q16×C67—21no—
140Q16×C67—21no—
141Q16×C67—21no—
142Q16×C67—21no—
143Q16×C67—42no—
144Q16×C67—42no—
145Q16×C67—81no—
146Q16×C67#4112yes—
147D8×C67—21no—
148D8×C67—21no—
149C2×C8×C67—21no—
150C2×C8×C67—21no—
151C2×C8×C67—21no—
152C2×C8×C67—21no—
153C2×C8×C67C2×D411yescharacteristic
154C2×C8×C67C2×D411yescharacteristic
155C2×C8×C67C2×D411yescharacteristic
156C2×C8×C67C2411yescharacteristic
157C4⋊C4×C67—21no—
158C4⋊C4×C67—21no—
159C4⋊C4×C67—21no—
160C4⋊C4×C67—21no—
161C4⋊C4×C67—41no—
162C4⋊C4×C67—41no—
163C4⋊C4×C67—41no—
164C4⋊C4×C67—41no—
165C4⋊C4×C67—41no—
166C4⋊C4×C67—41no—
167C4⋊C4×C67—41no—
168C4⋊C4×C67—41no—
169C4⋊C4×C67D811yescharacteristic
170C22⋊C4×C67—41no—
171C22⋊C4×C67—41no—
172C22⋊C4×C67—41no—
173C22⋊C4×C67—41no—
174C22⋊C4×C67—41no—
175C22⋊C4×C67—41no—
176C22⋊C4×C67D811yescharacteristic
177C22⋊C4×C67D811yescharacteristic
178C42×C67—41no—
179C42×C67—41no—
180C42×C67C2×D411yescharacteristic
181C16×C67—21no—
182C16×C67—21no—
183C16×C67—21no—
184C16×C67—21no—
185C23×C67—41no—
186C23×C67C2×D811yescharacteristic
187C23×C67C2×D811yescharacteristic
188Q8×C67—22no—
189Q8×C67—41no—
190Q8×C67—41no—
191Q8×C67—41no—
192Q8×C67—41no—
193Q8×C67—41no—
194Q8×C67—82no—
195Q8×C67—82no—
196D4×C67—22no—
197D4×C67—41no—
198D4×C67—41no—
199D4×C67—82no—
200D4×C67—161no—
201C2×C4×C67—21no—
202C2×C4×C67—21no—
203C2×C4×C67—21no—
204C2×C4×C67—21no—
205C2×C4×C67—41no—
206C2×C4×C67—41no—
207C2×C4×C67—41no—
208C2×C4×C67—41no—
209C2×C4×C67—41no—
210C2×C4×C67—41no—
211C2×C4×C67—41no—
212C2×C4×C67—82no—
213C2×C4×C67—82no—
214C2×C4×C67—161no—
215C2×C4×C67—161no—
216C2×C4×C67C2×D811yescharacteristic
217C2×C4×C67C2×D811yescharacteristic
218C2×C4×C67C2×D811yescharacteristic
219C2×C4×C67C2×D811yescharacteristic
220C2×C4×C67C22×D411yescharacteristic
221C8×C67—21no—
222C8×C67—21no—
223C8×C67—21no—
224C8×C67C2×#4111yescharacteristic
225C8×C67H(5,𝔽2)11yescharacteristic
226C22×C67—21no—
227C22×C67—21no—
228C22×C67—21no—
229C22×C67—21no—
230C22×C67—21no—
231C22×C67—81no—
232C22×C67—81no—
233C22×C67—81no—
234C22×C67#51012yes—
235C22×C67C22×D811yescharacteristic
236C4×C67—21no—
237C4×C67—21no—
238C4×C67—21no—
239C4×C67—21no—
240C4×C67—42no—
241C4×C67—81no—
242C4×C67—81no—
243C4×C67—81no—
244C4×C67—81no—
245C4×C67—81no—
246C4×C67—81no—
247C4×C67#54611yescharacteristic
248C4×C67#57711yescharacteristic
249#33887C6711yesmaximal, characteristic, direct product factor, Sylow
250C2×C67—22no—
251C2×C67—41no—
252C2×C67—161no—
253C2×C67#329411yescharacteristic
254C2×C67C2×#51011yescharacteristic
255C2×C67#341611yescharacteristic
256#3339C2×C6711yescharacteristic
257D4×Q16C2×C6711yescharacteristic, complemented
258#2245C2×C6711yescharacteristic, complemented
259#2237C2×C6711yescharacteristic, complemented
260#2236C2×C6711yescharacteristic, complemented
261#2229C2×C6711yescharacteristic, complemented
262#2220C2×C6711yescharacteristic, complemented
263#2217C2×C6711yescharacteristic, complemented
264#2208C2×C6711yescharacteristic, complemented
265#2185C2×C6711yescharacteristic, complemented
266#2185C2×C6711yescharacteristic, complemented
267#2175C2×C6711yescharacteristic, complemented
268#2168C2×C6711yescharacteristic, complemented
269#2141C2×C6711yescharacteristic, complemented
270C2×#358C2×C6711yescharacteristic, complemented
271C67#3388711yescharacteristic, direct product factor, Sylow
272C22×Q16C22×C6711yescharacteristic
273C22×Q16C22×C6711yescharacteristic
274#553—21no—
275D4×Q8—21no—
276C2×SD32C22×C6711yescharacteristic
277C2×SD32C22×C6711yescharacteristic
278C2×SD32C22×C6711yescharacteristic
279C2×M32C22×C6711yescharacteristic
280C22×C16C22×C6711yescharacteristic
281#499C22×C6711yescharacteristic
282#494C22×C6711yescharacteristic
283#482—21no—
284#481—21no—
285#478—21no—
286#467C22×C6711yescharacteristic
287#467C22×C6711yescharacteristic
288#466C22×C6711yescharacteristic
289#466C22×C6711yescharacteristic
290#462—21no—
291#451—21no—
292#451—21no—
293C4×Q16C22×C6711yescharacteristic
294C4×D8C22×C6711yescharacteristic
295C8×D4C22×C6711yescharacteristic
296#427C22×C6711yescharacteristic
297C2×#107C22×C6711yescharacteristic
298#367C22×C6711yescharacteristic
299#366C22×C6711yescharacteristic
300#366C22×C6711yescharacteristic
301C4⋊C16C22×C6711yescharacteristic
302#358C22×C6711yescharacteristic
303#358C22×C6711yescharacteristic
304#358C22×C6711yescharacteristic
305#358C22×C6711yescharacteristic
306#358C22×C6711yescharacteristic
307#358C22×C6711yescharacteristic
308#358C22×C6711yescharacteristic
309#358C22×C6711yescharacteristic
310#358C22×C6711yescharacteristic
311#357C22×C6711yescharacteristic
312C22⋊C16C22×C6711yescharacteristic
313C22⋊C16C22×C6711yescharacteristic
314#346C22×C6711yescharacteristic
315C22×Q8—21no—
316C22×Q8—21no—
317C2×Q16—22no—
318C2×Q16—22no—
319C2×Q16—21no—
320C2×Q16—21no—
321C2×Q16—21no—
322C2×Q16—41no—
323C2×Q16—41no—
324C2×Q16C23×C6711yescharacteristic
325C2×Q16C23×C6711yescharacteristic
326C2×Q16C23×C6711yescharacteristic
327C2×D8C23×C6711yescharacteristic
328C22×C8C23×C6711yescharacteristic
329C22×C8C23×C6711yescharacteristic
330C4⋊Q8—21no—
331C4⋊Q8—41no—
332#126—41no—
333#125—21no—
334#123—41no—
335C22⋊Q8—21no—
336C22⋊Q8—21no—
337C22⋊Q8—41no—
338C22⋊Q8—41no—
339#121—21no—
340#121—21no—
341#121—41no—
342C4×Q8—21no—
343C4×D4—21no—
344C4×D4—41no—
345C4×D4—41no—
346C4×D4D4×C6711yescharacteristic
347#117—21no—
348C2×C4⋊C4—21no—
349SD32—22no—
350SD32—41no—
351SD32—41no—
352M32—41no—
353C2×C16—41no—
354C2×C16C23×C6711yescharacteristic
355C2×C16C23×C6711yescharacteristic
356C2×C16C23×C6711yescharacteristic
357C2×C16C23×C6711yescharacteristic
358#107—21no—
359#107C23×C6711yescharacteristic
360#107C23×C6711yescharacteristic
361#107C23×C6711yescharacteristic
362#107C23×C6711yescharacteristic
363#106—21no—
364#106—21no—
365C4⋊C8D4×C6711yescharacteristic
366#103—21no—
367#103—21no—
368#103—21no—
369#102—21no—
370#102—21no—
371#102—21no—
372C22⋊C8D4×C6711yescharacteristic
373C22⋊C8D4×C6711yescharacteristic
374C4×C8C23×C6711yescharacteristic
375C2×Q8—21no—
376C2×Q8—21no—
377C2×Q8—21no—
378C2×Q8—42no—
379C2×Q8—42no—
380C2×Q8—41no—
381C2×Q8—41no—
382C2×Q8—81no—
383C2×Q8—81no—
384C2×D4—21no—
385C2×D4—41no—
386C2×D4—41no—
387C2×D4D8×C6711yescharacteristic
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4C2×D4×C6711yescharacteristic
393C22×C4C2×D4×C6711yescharacteristic
394Q16—21no—
395Q16—21no—
396Q16—21no—
397Q16—21no—
398Q16—21no—
399Q16—42no—
400Q16—42no—
401Q16—81no—
402Q16#41×C6712yes—
403D8—21no—
404D8—21no—
405C2×C8—21no—
406C2×C8—21no—
407C2×C8—21no—
408C2×C8—21no—
409C2×C8C2×D4×C6711yescharacteristic
410C2×C8C2×D4×C6711yescharacteristic
411C2×C8C2×D4×C6711yescharacteristic
412C2×C8C24×C6711yescharacteristic
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—41no—
418C4⋊C4—41no—
419C4⋊C4—41no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—41no—
425C4⋊C4D8×C6711yescharacteristic
426C22⋊C4—41no—
427C22⋊C4—41no—
428C22⋊C4—41no—
429C22⋊C4—41no—
430C22⋊C4—41no—
431C22⋊C4—41no—
432C22⋊C4D8×C6711yescharacteristic
433C22⋊C4D8×C6711yescharacteristic
434C42—41no—
435C42—41no—
436C42C2×D4×C6711yescharacteristic
437C16—21no—
438C16—21no—
439C16—21no—
440C16—21no—
441C23—41no—
442C23C2×D8×C6711yescharacteristic
443C23C2×D8×C6711yescharacteristic
444Q8—22no—
445Q8—41no—
446Q8—41no—
447Q8—41no—
448Q8—41no—
449Q8—41no—
450Q8—82no—
451Q8—82no—
452D4—22no—
453D4—41no—
454D4—41no—
455D4—82no—
456D4—161no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—161no—
471C2×C4—161no—
472C2×C4C2×D8×C6711yescharacteristic
473C2×C4C2×D8×C6711yescharacteristic
474C2×C4C2×D8×C6711yescharacteristic
475C2×C4C2×D8×C6711yescharacteristic
476C2×C4C22×D4×C6711yescharacteristic
477C8—21no—
478C8—21no—
479C8—21no—
480C8C2×#41×C6711yescharacteristic
481C8H(5,𝔽2)×C6711yescharacteristic
482C22—21no—
483C22—21no—
484C22—21no—
485C22—21no—
486C22—21no—
487C22—81no—
488C22—81no—
489C22—81no—
490C22#510×C6712yes—
491C22C22×D8×C6711yescharacteristic
492C4—21no—
493C4—21no—
494C4—21no—
495C4—21no—
496C4—42no—
497C4—81no—
498C4—81no—
499C4—81no—
500C4—81no—
501C4—81no—
502C4—81no—
503C4#546×C6711yescharacteristic
504C4#577×C6711yescharacteristic
505C2—22no—
506C2—41no—
507C2—161no—
508C2C67×#329411yescharacteristic
509C2C2×#510×C6711yescharacteristic
510C2C67×#341611yescharacteristic
subgroupG/H
center, socleC22×C67C22×D8
commutator, Frattini, cosocleC2×C8C24×C67
prime radicalC67×#3339C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C67 ◁ C4×D4×C67 ◁ C8×D4×C67 ◁ G

Complements

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

NH ≅ G/Ntype
2. D4×Q16×C67C2semidirect
3. C67×#2245C2semidirect
4. C67×#2237C2semidirect
5. C67×#2236C2semidirect
6. C67×#2229C2semidirect
7. C67×#2220C2semidirect
8. C67×#2217C2semidirect
9. C67×#2208C2semidirect
10. C67×#2185C2semidirect
11. C67×#2185C2semidirect
12. C67×#2175C2semidirect
13. C67×#2168C2semidirect
14. C67×#2141C2semidirect
15. C2×#358×C67C2semidirect
249. #33887C67direct (N × H)
257. D4×Q16C2×C67semidirect
258. #2245C2×C67semidirect
259. #2237C2×C67semidirect
260. #2236C2×C67semidirect
261. #2229C2×C67semidirect
262. #2220C2×C67semidirect
263. #2217C2×C67semidirect
264. #2208C2×C67semidirect
265. #2185C2×C67semidirect
266. #2185C2×C67semidirect
267. #2175C2×C67semidirect
268. #2168C2×C67semidirect
269. #2141C2×C67semidirect
270. C2×#358C2×C67semidirect
271. C67#33887direct (N × H)

Maximal subgroups

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

subgroupindex
C67×#33392
D4×Q16×C672
C67×#22452
C67×#22372
C67×#22362
C67×#22292
C67×#22202
C67×#22172
C67×#22082
C67×#21852
C67×#21752
C67×#21682
C67×#21412
C2×#358×C672
#3388767