Finite Group Databasebeta

← C67×#60253 #30739504

Subgroups — C67×#60253 #30739504

Order 17152.

1702 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#41×C67C211yesmaximal, characteristic
2C67×#2986C216yesmaximal, complemented
3C67×#2985C216yesmaximal, complemented
4C67×#2984C212yesmaximal, complemented
5C67×#2982C212yesmaximal, complemented
6Q8×M16×C67C211yesmaximal, characteristic, complemented
7C67×#2962C213yesmaximal, complemented
8C67×#2960C213yesmaximal, complemented
9C2×C8×Q8×C67C211yesmaximal, characteristic, complemented
10C67×#2925C213yesmaximal
11C67×#2922C213yesmaximal
12Q82×C67C411yescharacteristic
13#557×C67C413yes—
14#554×C67C416yes—
15#554×C67C413yes—
16D4×Q8×C67C411yescharacteristic
17D4×Q8×C67C412yes—
18#533×C67C2213yes—
19#533×C67C2216yes—
20C4×#41×C67C2213yes—
21C2×C4×Q8×C67C2211yescharacteristic
22C2×C4×Q8×C67C2212yes—
23#446×C67C2216yes—
24#446×C67C2216yes—
25#446×C67C2216yes—
26#446×C67C22112yes—
27#446×C67C2216yes—
28C8×Q8×C67C2211yescharacteristic
29C8×Q8×C67C2211yescharacteristic
30C8×Q8×C67C2213yes—
31C8×Q8×C67C2213yes—
32C8×Q8×C67C2216yes—
33C8×Q8×C67C2216yes—
34C8×Q8×C67C2216yes—
35C8×Q8×C67C2212yes—
36#433×C67C2216yes—
37#433×C67C2216yes—
38#433×C67C22112yes—
39#432×C67C2216yes—
40#431×C67C2216yes—
41#431×C67C2216yes—
42#431×C67C2216yes—
43#424×C67C2213yes—
44#424×C67C2213yes—
45#423×C67C2213yes—
46C2×C4⋊C8×C67C2213yes—
47#408×C67C2211yescharacteristic
48#408×C67C2213yes—
49#405×C67C2216yes—
50C4×M16×C67C2213yes—
51C2×C4×C8×C67C2213yes—
52C2×#41×C67C2311yescharacteristic
53C22×Q8×C67C2×C411yescharacteristic
54C22×Q8×C67C2×C412yes—
55C2×M16×C67C2311yescharacteristic
56C2×M16×C67C2313yes—
57C22×C8×C67C2311yescharacteristic
58C22×C8×C67C2313yes—
59C4⋊Q8×C67C2×C413yes—
60C4⋊Q8×C67C2×C416yes—
61C4⋊Q8×C67C2×C413yes—
62C4⋊Q8×C67C2×C416yes—
63#125×C67C2×C416yes—
64#125×C67C2×C4112yes—
65C22⋊Q8×C67C2×C416yes—
66C22⋊Q8×C67C2×C416yes—
67C22⋊Q8×C67C2×C416yes—
68C22⋊Q8×C67C2×C416yes—
69C22⋊Q8×C67C2×C4112yes—
70C4×Q8×C67C2×C411yescharacteristic
71C4×Q8×C67C2×C412yes—
72C4×Q8×C67C2×C413yes—
73C4×Q8×C67C2×C416yes—
74C4×Q8×C67C2×C413yes—
75C4×Q8×C67C2×C413yes—
76C4×Q8×C67C2311yescharacteristic
77C4×Q8×C67C2313yes—
78C4×Q8×C67C2316yes—
79C4×Q8×C67C2316yes—
80C4×D4×C67C2×C416yes—
81C4×D4×C67C2×C413yes—
82C4×D4×C67C2×C416yes—
83C4×D4×C67C2×C413yes—
84#117×C67C2313yes—
85#117×C67C2316yes—
86C2×C4⋊C4×C67C2×C413yes—
87C2×C4⋊C4×C67C2×C416yes—
88C2×C42×C67C2313yes—
89C2×C42×C67C2316yes—
90C4⋊C8×C67C2313yes—
91C4⋊C8×C67C2313yes—
92C4⋊C8×C67C2316yes—
93C4⋊C8×C67C2316yes—
94C4⋊C8×C67C2316yes—
95C4⋊C8×C67C2316yes—
96C4⋊C8×C67C2316yes—
97C4⋊C8×C67C2316yes—
98C4⋊C8×C67C2316yes—
99C22⋊C8×C67C2312yes—
100C22⋊C8×C67C2312yes—
101C22⋊C8×C67C2316yes—
102C22⋊C8×C67C2316yes—
103#97×C67C2316yes—
104#97×C67C23112yes—
105C4×C8×C67C2313yes—
106C4×C8×C67C2313yes—
107C4×C8×C67C2313yes—
108C4×C8×C67C2313yes—
109C4×C8×C67C2316yes—
110C4×C8×C67C23112yes—
111#41×C67—22no—
112#41×C67—22no—
113C2×Q8×C67—24no—
114C2×Q8×C67—28no—
115C2×Q8×C67C22×C411yescharacteristic
116C2×Q8×C67C22×C413yes—
117C2×Q8×C67C22×C416yes—
118C2×Q8×C67C22×C416yes—
119C2×Q8×C67C22×C411yescharacteristic
120C2×D4×C67C22×C412yes—
121C2×D4×C67C22×C411yescharacteristic
122C22×C4×C67C22×C416yes—
123C22×C4×C67C22×C413yes—
124C22×C4×C67C22×C416yes—
125C22×C4×C67C22×C413yes—
126C22×C4×C67C2412yes—
127C22×C4×C67C2411yescharacteristic
128M16×C67—22no—
129M16×C67—26no—
130C2×C8×C67—26no—
131C2×C8×C67—22no—
132C2×C8×C67C2413yes—
133C2×C8×C67C2411yescharacteristic
134C2×C8×C67C2411yescharacteristic
135C2×C8×C67C2413yes—
136C2×C8×C67C2416yes—
137C2×C8×C67C2412yes—
138C4⋊C4×C67C22×C413yes—
139C4⋊C4×C67C22×C416yes—
140C4⋊C4×C67C22×C4112yes—
141C4⋊C4×C67C22×C413yes—
142C4⋊C4×C67C22×C416yes—
143C4⋊C4×C67C22×C416yes—
144C4⋊C4×C67C22×C416yes—
145C4⋊C4×C67C22×C413yes—
146C4⋊C4×C67C22×C413yes—
147C4⋊C4×C67C22×C4112yes—
148C4⋊C4×C67C22×C416yes—
149C22⋊C4×C67C22×C416yes—
150C22⋊C4×C67C22×C416yes—
151C22⋊C4×C67C22×C416yes—
152C42×C67C22×C416yes—
153C42×C67C22×C413yes—
154C42×C67C22×C416yes—
155C42×C67C22×C413yes—
156C42×C67C2413yes—
157C42×C67C2413yes—
158C42×C67C2416yes—
159C23×C67C23×C412yes—
160C23×C67C23×C411yescharacteristic
161Q8×C67—22no—
162Q8×C67—26no—
163Q8×C67—212no—
164Q8×C67—46no—
165Q8×C67#13114yes—
166D4×C67—24no—
167D4×C67—22no—
168C2×C4×C67—26no—
169C2×C4×C67—212no—
170C2×C4×C67—22no—
171C2×C4×C67—22no—
172C2×C4×C67—43no—
173C2×C4×C67—46no—
174C2×C4×C67C23×C412yes—
175C2×C4×C67C23×C413yes—
176C2×C4×C67C23×C416yes—
177C2×C4×C67C23×C413yes—
178C2×C4×C67C23×C416yes—
179C2×C4×C67C23×C413yes—
180C2×C4×C67C23×C411yescharacteristic
181C2×C4×C67C23×C413yes—
182C2×C4×C672−1+412yes—
183C2×C4×C672−1+412yes—
184C2×C4×C67C2511yescharacteristic
185C8×C67—23no—
186C8×C67—21no—
187C8×C67—21no—
188C8×C67—23no—
189C8×C67—22no—
190C8×C67—26no—
191C22×C67—21no—
192C22×C67—21no—
193C22×C67—22no—
194C22×C67—22no—
195C22×C67—22no—
196C22×C67#51912yes—
197C22×C67C24×C411yescharacteristic
198C4×C67—22no—
199C4×C67—23no—
200C4×C67—26no—
201C4×C67—23no—
202C4×C67—23no—
203C4×C67—26no—
204C4×C67#51912yes—
205C4×C67C2×#13116yes—
206C4×C67C2×2−1+411yescharacteristic
207C4×C67C2×2−1+411yescharacteristic
208#60253C6711yesmaximal, characteristic, direct product factor, Sylow
209C2×C67—22no—
210C2×C67—42no—
211C2×C67C2×#51911yescharacteristic
212C2×C67C22×#13111yescharacteristic
213C2×C67#357311yescharacteristic
214Q8×#41C2×C6711yescharacteristic
215#2986C2×C6716yescomplemented
216#2985C2×C6716yescomplemented
217#2984C2×C6712yescomplemented
218#2982C2×C6712yescomplemented
219Q8×M16C2×C6711yescharacteristic, complemented
220#2962C2×C6713yescomplemented
221#2960C2×C6713yescomplemented
222C2×C8×Q8C2×C6711yescharacteristic, complemented
223#2925C2×C6713yes—
224#2922C2×C6713yes—
225C67#6025311yescharacteristic, direct product factor, Sylow
226Q82C4×C6711yescharacteristic
227#557C4×C6713yes—
228#554C4×C6716yes—
229#554C4×C6713yes—
230D4×Q8C4×C6711yescharacteristic
231D4×Q8C4×C6712yes—
232#533C22×C6713yes—
233#533C22×C6716yes—
234C4×#41C22×C6713yes—
235C2×C4×Q8C22×C6711yescharacteristic
236C2×C4×Q8C22×C6712yes—
237#446C22×C6716yes—
238#446C22×C6716yes—
239#446C22×C6716yes—
240#446C22×C67112yes—
241#446C22×C6716yes—
242C8×Q8C22×C6711yescharacteristic
243C8×Q8C22×C6711yescharacteristic
244C8×Q8C22×C6713yes—
245C8×Q8C22×C6713yes—
246C8×Q8C22×C6716yes—
247C8×Q8C22×C6716yes—
248C8×Q8C22×C6716yes—
249C8×Q8C22×C6712yes—
250#433C22×C6716yes—
251#433C22×C6716yes—
252#433C22×C67112yes—
253#432C22×C6716yes—
254#431C22×C6716yes—
255#431C22×C6716yes—
256#431C22×C6716yes—
257#424C22×C6713yes—
258#424C22×C6713yes—
259#423C22×C6713yes—
260C2×C4⋊C8C22×C6713yes—
261#408C22×C6711yescharacteristic
262#408C22×C6713yes—
263#405C22×C6716yes—
264C4×M16C22×C6713yes—
265C2×C4×C8C22×C6713yes—
266C2×#41C23×C6711yescharacteristic
267C22×Q8C2×C4×C6711yescharacteristic
268C22×Q8C2×C4×C6712yes—
269C2×M16C23×C6711yescharacteristic
270C2×M16C23×C6713yes—
271C22×C8C23×C6711yescharacteristic
272C22×C8C23×C6713yes—
273C4⋊Q8C2×C4×C6713yes—
274C4⋊Q8C2×C4×C6716yes—
275C4⋊Q8C2×C4×C6713yes—
276C4⋊Q8C2×C4×C6716yes—
277#125C2×C4×C6716yes—
278#125C2×C4×C67112yes—
279C22⋊Q8C2×C4×C6716yes—
280C22⋊Q8C2×C4×C6716yes—
281C22⋊Q8C2×C4×C6716yes—
282C22⋊Q8C2×C4×C6716yes—
283C22⋊Q8C2×C4×C67112yes—
284C4×Q8C2×C4×C6711yescharacteristic
285C4×Q8C2×C4×C6712yes—
286C4×Q8C2×C4×C6713yes—
287C4×Q8C2×C4×C6716yes—
288C4×Q8C2×C4×C6713yes—
289C4×Q8C2×C4×C6713yes—
290C4×Q8C23×C6711yescharacteristic
291C4×Q8C23×C6713yes—
292C4×Q8C23×C6716yes—
293C4×Q8C23×C6716yes—
294C4×D4C2×C4×C6716yes—
295C4×D4C2×C4×C6713yes—
296C4×D4C2×C4×C6716yes—
297C4×D4C2×C4×C6713yes—
298#117C23×C6713yes—
299#117C23×C6716yes—
300C2×C4⋊C4C2×C4×C6713yes—
301C2×C4⋊C4C2×C4×C6716yes—
302C2×C42C23×C6713yes—
303C2×C42C23×C6716yes—
304C4⋊C8C23×C6713yes—
305C4⋊C8C23×C6713yes—
306C4⋊C8C23×C6716yes—
307C4⋊C8C23×C6716yes—
308C4⋊C8C23×C6716yes—
309C4⋊C8C23×C6716yes—
310C4⋊C8C23×C6716yes—
311C4⋊C8C23×C6716yes—
312C4⋊C8C23×C6716yes—
313C22⋊C8C23×C6712yes—
314C22⋊C8C23×C6712yes—
315C22⋊C8C23×C6716yes—
316C22⋊C8C23×C6716yes—
317#97C23×C6716yes—
318#97C23×C67112yes—
319C4×C8C23×C6713yes—
320C4×C8C23×C6713yes—
321C4×C8C23×C6713yes—
322C4×C8C23×C6713yes—
323C4×C8C23×C6716yes—
324C4×C8C23×C67112yes—
325#41—22no—
326#41—22no—
327C2×Q8—24no—
328C2×Q8—28no—
329C2×Q8C22×C4×C6711yescharacteristic
330C2×Q8C22×C4×C6713yes—
331C2×Q8C22×C4×C6716yes—
332C2×Q8C22×C4×C6716yes—
333C2×Q8C22×C4×C6711yescharacteristic
334C2×D4C22×C4×C6712yes—
335C2×D4C22×C4×C6711yescharacteristic
336C22×C4C22×C4×C6716yes—
337C22×C4C22×C4×C6713yes—
338C22×C4C22×C4×C6716yes—
339C22×C4C22×C4×C6713yes—
340C22×C4C24×C6712yes—
341C22×C4C24×C6711yescharacteristic
342M16—22no—
343M16—26no—
344C2×C8—26no—
345C2×C8—22no—
346C2×C8C24×C6711yescharacteristic
347C2×C8C24×C6713yes—
348C2×C8C24×C6711yescharacteristic
349C2×C8C24×C6712yes—
350C2×C8C24×C6716yes—
351C2×C8C24×C6713yes—
352C4⋊C4C22×C4×C6713yes—
353C4⋊C4C22×C4×C6716yes—
354C4⋊C4C22×C4×C67112yes—
355C4⋊C4C22×C4×C6713yes—
356C4⋊C4C22×C4×C6716yes—
357C4⋊C4C22×C4×C6716yes—
358C4⋊C4C22×C4×C6716yes—
359C4⋊C4C22×C4×C6713yes—
360C4⋊C4C22×C4×C6713yes—
361C4⋊C4C22×C4×C67112yes—
362C4⋊C4C22×C4×C6716yes—
363C22⋊C4C22×C4×C6716yes—
364C22⋊C4C22×C4×C6716yes—
365C22⋊C4C22×C4×C6716yes—
366C42C22×C4×C6716yes—
367C42C22×C4×C6713yes—
368C42C22×C4×C6716yes—
369C42C22×C4×C6713yes—
370C42C24×C6713yes—
371C42C24×C6713yes—
372C42C24×C6716yes—
373C23C23×C4×C6712yes—
374C23C23×C4×C6711yescharacteristic
375Q8—22no—
376Q8—26no—
377Q8—212no—
378Q8—46no—
379Q8#131×C6714yes—
380D4—24no—
381D4—22no—
382C2×C4—26no—
383C2×C4—212no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—43no—
387C2×C4—46no—
388C2×C4C23×C4×C6712yes—
389C2×C4C23×C4×C6713yes—
390C2×C4C23×C4×C6716yes—
391C2×C4C23×C4×C6713yes—
392C2×C4C23×C4×C6716yes—
393C2×C4C23×C4×C6713yes—
394C2×C4C23×C4×C6711yescharacteristic
395C2×C4C23×C4×C6713yes—
396C2×C42−1+4×C6712yes—
397C2×C42−1+4×C6712yes—
398C2×C4C25×C6711yescharacteristic
399C8—23no—
400C8—21no—
401C8—21no—
402C8—23no—
403C8—22no—
404C8—26no—
405C22—21no—
406C22—21no—
407C22—22no—
408C22—22no—
409C22—22no—
410C22#519×C6712yes—
411C22C24×C4×C6711yescharacteristic
412C4—23no—
413C4—26no—
414C4—23no—
415C4—23no—
416C4—26no—
417C4—22no—
418C4#519×C6712yes—
419C4C2×#131×C6716yes—
420C4C2×2−1+4×C6711yescharacteristic
421C4C2×2−1+4×C6711yescharacteristic
422C2—22no—
423C2—42no—
424C2C2×#519×C6711yescharacteristic
425C2C22×#131×C6711yescharacteristic
426C2C67×#357311yescharacteristic
subgroupG/H
centerC2×C4×C67C25
commutatorC22C24×C4×C67
Frattini, cosocleC2×C4C25×C67
socleC22×C67C24×C4
prime radicalC2×#41×C67C23
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C67 ◁ G

Complements

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

NH ≅ G/Ntype
2. C67×#2986C2semidirect
3. C67×#2985C2semidirect
4. C67×#2984C2semidirect
5. C67×#2982C2semidirect
6. Q8×M16×C67C2semidirect
7. C67×#2962C2semidirect
8. C67×#2960C2semidirect
9. C2×C8×Q8×C67C2semidirect
208. #60253C67direct (N × H)
215. #2986C2×C67semidirect
216. #2985C2×C67semidirect
217. #2984C2×C67semidirect
218. #2982C2×C67semidirect
219. Q8×M16C2×C67semidirect
220. #2962C2×C67semidirect
221. #2960C2×C67semidirect
222. C2×C8×Q8C2×C67semidirect
225. C67#60253direct (N × H)

Maximal subgroups

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

subgroupindex
Q8×#41×C672
C67×#29862
C67×#29852
C67×#29842
C67×#29822
Q8×M16×C672
C67×#29622
C67×#29602
C2×C8×Q8×C672
C67×#29252
C67×#29222
#6025367