Finite Group Databasebeta

← C67×#61393 #30740644

Subgroups — C67×#61393 #30740644

Order 17152.

1958 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3480C212yesmaximal, complemented
2C67×#3396C213yesmaximal, complemented
3C67×#3395C211yesmaximal, characteristic, complemented
4C67×#3394C216yesmaximal, complemented
5C67×#3393C213yesmaximal, complemented
6C67×#3392C212yesmaximal, complemented
7C67×#3391C211yesmaximal, characteristic, complemented
8C67×#3147C213yesmaximal, complemented
9C67×#3035C216yesmaximal, complemented
10C2×C8×Q8×C67C211yesmaximal, characteristic, complemented
11C4×#135×C67C213yesmaximal, complemented
12C2×#135×C67C2213yes—
13C2×#135×C67C2216yes—
14#559×C67C2216yes—
15#557×C67C2212yes—
16#556×C67C2216yes—
17#553×C67—26no—
18#550×C67C2212yes—
19#548×C67—22no—
20#532×C67C2216yes—
21#529×C67—26no—
22C4×#41×C67—22no—
23C4×#41×C67C2216yes—
24C2×C4×Q8×C67C2211yescharacteristic
25#495×C67C2216yes—
26#495×C67C2216yes—
27#494×C67C2213yes—
28#493×C67C2213yes—
29#492×C67C2216yes—
30#464×C67C22112yes—
31#463×C67C22112yes—
32#462×C67C2216yes—
33#461×C67C2216yes—
34#460×C67C2216yes—
35#459×C67C2216yes—
36C8×Q8×C67C2211yescharacteristic
37C8×Q8×C67C2211yescharacteristic
38C8×Q8×C67C2213yes—
39C8×Q8×C67C2213yes—
40C4×Q16×C67C2213yes—
41C4×Q16×C67C2213yes—
42C4×SD16×C67C2216yes—
43C4×SD16×C67C2216yes—
44C4×D8×C67C2213yes—
45C4×D8×C67C2213yes—
46#427×C67C2211yescharacteristic
47C2×C4⋊C8×C67C2213yes—
48#416×C67C2216yes—
49C2×C4×C8×C67C2213yes—
50C2×#41×C67—22no—
51C2×#41×C67C2316yes—
52C22×Q8×C67D411yescharacteristic
53#135×C67—212no—
54#135×C67—412no—
55C2×Q16×C67C2313yes—
56C2×Q16×C67C2316yes—
57C2×SD16×C67C2316yes—
58C2×SD16×C67C23112yes—
59C2×D8×C67C2313yes—
60C2×D8×C67C2316yes—
61C22×C8×C67C2313yes—
62C22×C8×C67C2311yescharacteristic
63C4⋊Q8×C67C2316yes—
64#127×C67C2316yes—
65#126×C67—212no—
66#125×C67—26no—
67#124×C67—26no—
68#124×C67C23112yes—
69#123×C67—212no—
70C22⋊Q8×C67—26no—
71#121×C67—26no—
72#121×C67—26no—
73#121×C67—26no—
74C4×Q8×C67—22no—
75C4×Q8×C67D411yescharacteristic
76C4×Q8×C67D413yes—
77C4×Q8×C67C2316yes—
78C4×Q8×C67C2316yes—
79C4×Q8×C67C2311yescharacteristic
80C4×Q8×C67C2313yes—
81C4×D4×C67—26no—
82C4×D4×C67—26no—
83C4×D4×C67—26no—
84C4×D4×C67—26no—
85C4×D4×C67—26no—
86C4×D4×C67C2316yes—
87C4×D4×C67C2316yes—
88#117×C67—26no—
89#117×C67—26no—
90#117×C67C2312yes—
91C2×C4⋊C4×C67D413yes—
92C2×C42×C67—26no—
93C2×C42×C67C2313yes—
94#107×C67C2311yescharacteristic
95#107×C67C2311yescharacteristic
96#106×C67C2312yes—
97C4⋊C8×C67C2313yes—
98C4⋊C8×C67C2313yes—
99C4⋊C8×C67C2316yes—
100#103×C67C2316yes—
101#103×C67C2316yes—
102#102×C67C2316yes—
103#102×C67C2316yes—
104C4×C8×C67C2313yes—
105C4×C8×C67C2313yes—
106C4×C8×C67C2313yes—
107C4×C8×C67C2313yes—
108#41×C67—212no—
109#41×C67—212no—
110#41×C67—48no—
111C2×Q8×C67—24no—
112C2×Q8×C67C2×D413yes—
113C2×Q8×C67C2×D411yescharacteristic
114C2×Q8×C67C2416yes—
115C2×D4×C67—26no—
116C2×D4×C67—26no—
117C2×D4×C67—26no—
118C2×D4×C67C2416yes—
119C22×C4×C67—26no—
120C22×C4×C67—26no—
121C22×C4×C67—26no—
122C22×C4×C67C2×D413yes—
123C22×C4×C67C2×D413yes—
124C22×C4×C67C2411yescharacteristic
125Q16×C67—26no—
126Q16×C67—46no—
127SD16×C67—212no—
128SD16×C67—412no—
129D8×C67—26no—
130D8×C67—46no—
131C2×C8×C67—26no—
132C2×C8×C67#4114yescomplemented
133C2×C8×C67C2413yes—
134C2×C8×C67C2413yes—
135C2×C8×C67C2411yescharacteristic
136C2×C8×C67C2411yescharacteristic
137C4⋊C4×C67—26no—
138C4⋊C4×C67—26no—
139C4⋊C4×C67—26no—
140C4⋊C4×C67—26no—
141C4⋊C4×C67C2×D413yes—
142C4⋊C4×C67C2×D413yes—
143C4⋊C4×C67C2×D416yes—
144C4⋊C4×C67C2412yes—
145C4⋊C4×C67C2412yes—
146C22⋊C4×C67—22no—
147C22⋊C4×C67—212no—
148C22⋊C4×C67—26no—
149C22⋊C4×C67—26no—
150C22⋊C4×C67—212no—
151C42×C67—26no—
152C42×C67—26no—
153C42×C67—22no—
154C42×C67C2×D413yes—
155C42×C67C2×D413yes—
156C42×C67C2413yes—
157C42×C67C2413yes—
158C23×C67—26no—
159C23×C67C22×D411yescharacteristic
160Q8×C67—26no—
161Q8×C67—26no—
162Q8×C67—26no—
163Q8×C67#13514yescomplemented
164D4×C67—26no—
165D4×C67—26no—
166D4×C67—412no—
167D4×C67—412no—
168D4×C67—86no—
169C2×C4×C67—26no—
170C2×C4×C67—22no—
171C2×C4×C67—22no—
172C2×C4×C67—26no—
173C2×C4×C67—43no—
174C2×C4×C67—412no—
175C2×C4×C67—412no—
176C2×C4×C67—86no—
177C2×C4×C67C22×D413yes—
178C2×C4×C67C22×D413yes—
179C2×C4×C67C22×D413yes—
180C2×C4×C67C22×D413yes—
181C2×C4×C67C22×D411yescharacteristic
182C2×C4×C67C2×#4112yes—
183C2×C4×C67H(5,𝔽2)12yes—
184C2×C4×C67C2511yescharacteristic
185C8×C67—23no—
186C8×C67—23no—
187C8×C67C2×#4112yes—
188C8×C67C2×#4112yes—
189C22×C67—21no—
190C22×C67—21no—
191C22×C67—46no—
192C22×C67—46no—
193C22×C67—46no—
194C22×C67#55012yes—
195C22×C67C23×D411yescharacteristic
196C4×C67—23no—
197C4×C67—23no—
198C4×C67—23no—
199C4×C67—42no—
200C4×C67—46no—
201C4×C67—42no—
202C4×C67#55012yes—
203C4×C67C2×#13516yes—
204C4×C67C22×#4111yescharacteristic
205C4×C67C2×H(5,𝔽2)11yescharacteristic
206#61393C6711yesmaximal, characteristic, direct product factor, Sylow
207C2×C67—22no—
208C2×C67—86no—
209C2×C67C2×#55011yescharacteristic
210C2×C67C22×#13511yescharacteristic
211C2×C67#358411yescharacteristic
212#3480C2×C6712yescomplemented
213#3396C2×C6713yescomplemented
214#3395C2×C6711yescharacteristic, complemented
215#3394C2×C6716yescomplemented
216#3393C2×C6713yescomplemented
217#3392C2×C6712yescomplemented
218#3391C2×C6711yescharacteristic, complemented
219#3147C2×C6713yescomplemented
220#3035C2×C6716yescomplemented
221C2×C8×Q8C2×C6711yescharacteristic, complemented
222C4×#135C2×C6713yescomplemented
223C67#6139311yescharacteristic, direct product factor, Sylow
224C2×#135C22×C6713yes—
225C2×#135C22×C6716yes—
226#559C22×C6716yes—
227#557C22×C6712yes—
228#556C22×C6716yes—
229#553—26no—
230#550C22×C6712yes—
231#548—22no—
232#532C22×C6716yes—
233#529—26no—
234C4×#41—22no—
235C4×#41C22×C6716yes—
236C2×C4×Q8C22×C6711yescharacteristic
237#495C22×C6716yes—
238#495C22×C6716yes—
239#494C22×C6713yes—
240#493C22×C6713yes—
241#492C22×C6716yes—
242#464C22×C67112yes—
243#463C22×C67112yes—
244#462C22×C6716yes—
245#461C22×C6716yes—
246#460C22×C6716yes—
247#459C22×C6716yes—
248C8×Q8C22×C6711yescharacteristic
249C8×Q8C22×C6711yescharacteristic
250C8×Q8C22×C6713yes—
251C8×Q8C22×C6713yes—
252C4×Q16C22×C6713yes—
253C4×Q16C22×C6713yes—
254C4×SD16C22×C6716yes—
255C4×SD16C22×C6716yes—
256C4×D8C22×C6713yes—
257C4×D8C22×C6713yes—
258#427C22×C6711yescharacteristic
259C2×C4⋊C8C22×C6713yes—
260#416C22×C6716yes—
261C2×C4×C8C22×C6713yes—
262C2×#41—22no—
263C2×#41C23×C6716yes—
264C22×Q8D4×C6711yescharacteristic
265#135—212no—
266#135—412no—
267C2×Q16C23×C6713yes—
268C2×Q16C23×C6716yes—
269C2×SD16C23×C6716yes—
270C2×SD16C23×C67112yes—
271C2×D8C23×C6713yes—
272C2×D8C23×C6716yes—
273C22×C8C23×C6713yes—
274C22×C8C23×C6711yescharacteristic
275C4⋊Q8C23×C6716yes—
276#127C23×C6716yes—
277#126—212no—
278#125—26no—
279#124—26no—
280#124C23×C67112yes—
281#123—212no—
282C22⋊Q8—26no—
283#121—26no—
284#121—26no—
285#121—26no—
286C4×Q8—22no—
287C4×Q8D4×C6711yescharacteristic
288C4×Q8D4×C6713yes—
289C4×Q8C23×C6716yes—
290C4×Q8C23×C6716yes—
291C4×Q8C23×C6711yescharacteristic
292C4×Q8C23×C6713yes—
293C4×D4—26no—
294C4×D4—26no—
295C4×D4—26no—
296C4×D4—26no—
297C4×D4—26no—
298C4×D4C23×C6716yes—
299C4×D4C23×C6716yes—
300#117—26no—
301#117—26no—
302#117C23×C6712yes—
303C2×C4⋊C4D4×C6713yes—
304C2×C42—26no—
305C2×C42C23×C6713yes—
306#107C23×C6711yescharacteristic
307#107C23×C6711yescharacteristic
308#106C23×C6712yes—
309C4⋊C8C23×C6713yes—
310C4⋊C8C23×C6713yes—
311C4⋊C8C23×C6716yes—
312#103C23×C6716yes—
313#103C23×C6716yes—
314#102C23×C6716yes—
315#102C23×C6716yes—
316C4×C8C23×C6713yes—
317C4×C8C23×C6713yes—
318C4×C8C23×C6713yes—
319C4×C8C23×C6713yes—
320#41—212no—
321#41—212no—
322#41—48no—
323C2×Q8—24no—
324C2×Q8C2×D4×C6713yes—
325C2×Q8C2×D4×C6711yescharacteristic
326C2×Q8C24×C6716yes—
327C2×D4—26no—
328C2×D4—26no—
329C2×D4—26no—
330C2×D4C24×C6716yes—
331C22×C4—26no—
332C22×C4—26no—
333C22×C4—26no—
334C22×C4C2×D4×C6713yes—
335C22×C4C2×D4×C6713yes—
336C22×C4C24×C6711yescharacteristic
337Q16—26no—
338Q16—46no—
339SD16—212no—
340SD16—412no—
341D8—26no—
342D8—46no—
343C2×C8—26no—
344C2×C8#41×C6714yescomplemented
345C2×C8C24×C6713yes—
346C2×C8C24×C6713yes—
347C2×C8C24×C6711yescharacteristic
348C2×C8C24×C6711yescharacteristic
349C4⋊C4—26no—
350C4⋊C4—26no—
351C4⋊C4—26no—
352C4⋊C4—26no—
353C4⋊C4C2×D4×C6713yes—
354C4⋊C4C2×D4×C6713yes—
355C4⋊C4C2×D4×C6716yes—
356C4⋊C4C24×C6712yes—
357C4⋊C4C24×C6712yes—
358C22⋊C4—22no—
359C22⋊C4—212no—
360C22⋊C4—26no—
361C22⋊C4—26no—
362C22⋊C4—212no—
363C42—22no—
364C42—26no—
365C42—26no—
366C42C2×D4×C6713yes—
367C42C2×D4×C6713yes—
368C42C24×C6713yes—
369C42C24×C6713yes—
370C23—26no—
371C23C22×D4×C6711yescharacteristic
372Q8—26no—
373Q8—26no—
374Q8—26no—
375Q8#135×C6714yescomplemented
376D4—26no—
377D4—26no—
378D4—412no—
379D4—412no—
380D4—86no—
381C2×C4—26no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—26no—
385C2×C4—43no—
386C2×C4—412no—
387C2×C4—412no—
388C2×C4—86no—
389C2×C4C22×D4×C6713yes—
390C2×C4C22×D4×C6713yes—
391C2×C4C22×D4×C6713yes—
392C2×C4C22×D4×C6711yescharacteristic
393C2×C4C22×D4×C6713yes—
394C2×C4C2×#41×C6712yes—
395C2×C4H(5,𝔽2)×C6712yes—
396C2×C4C25×C6711yescharacteristic
397C8—23no—
398C8—23no—
399C8C2×#41×C6712yes—
400C8C2×#41×C6712yes—
401C22—21no—
402C22—21no—
403C22—46no—
404C22—46no—
405C22—46no—
406C22#550×C6712yes—
407C22C23×D4×C6711yescharacteristic
408C4—23no—
409C4—23no—
410C4—23no—
411C4—42no—
412C4—42no—
413C4—46no—
414C4#550×C6712yes—
415C4C2×#135×C6716yes—
416C4C22×#41×C6711yescharacteristic
417C4C2×H(5,𝔽2)×C6711yescharacteristic
418C2—22no—
419C2—86no—
420C2C2×#550×C6711yescharacteristic
421C2C22×#135×C6711yescharacteristic
422C2C67×#358411yescharacteristic
subgroupG/H
centerC2×C4×C67C22×D4
commutator, Frattini, cosocleC2×C4C25×C67
socleC22×C67C23×D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C67 ◁ C2×C4×Q8×C67 ◁ G

Complements

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

NH ≅ G/Ntype
1. C67×#3480C2semidirect
2. C67×#3396C2semidirect
3. C67×#3395C2semidirect
4. C67×#3394C2semidirect
5. C67×#3393C2semidirect
6. C67×#3392C2semidirect
7. C67×#3391C2semidirect
8. C67×#3147C2semidirect
9. C67×#3035C2semidirect
10. C2×C8×Q8×C67C2semidirect
11. C4×#135×C67C2semidirect
132. C2×C8×C67#41semidirect
163. Q8×C67#135semidirect
206. #61393C67direct (N × H)
212. #3480C2×C67semidirect
213. #3396C2×C67semidirect
214. #3395C2×C67semidirect
215. #3394C2×C67semidirect
216. #3393C2×C67semidirect
217. #3392C2×C67semidirect
218. #3391C2×C67semidirect
219. #3147C2×C67semidirect
220. #3035C2×C67semidirect
221. C2×C8×Q8C2×C67semidirect
222. C4×#135C2×C67semidirect
223. C67#61393direct (N × H)
344. C2×C8#41×C67semidirect
375. Q8#135×C67semidirect

Maximal subgroups

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

subgroupindex
C67×#34802
C67×#33962
C67×#33952
C67×#33942
C67×#33932
C67×#33922
C67×#33912
C67×#31472
C67×#30352
C2×C8×Q8×C672
C4×#135×C672
#6139367