Finite Group Databasebeta

← C67×#10500 #30689751

Subgroups — C67×#10500 #30689751

Order 17152.

508 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3205C211yesmaximal, characteristic
2C67×#1982C211yesmaximal, characteristic, complemented
3C67×#1980C211yesmaximal, characteristic, complemented
4C67×#1965C211yesmaximal, characteristic, complemented
5C67×#1565C211yesmaximal, characteristic, complemented
6C67×#1547C211yesmaximal, characteristic, complemented
7C67×#1303C211yesmaximal, characteristic, complemented
8#532×C67C2211yescharacteristic
9C2×#127×C67C2211yescharacteristic
10#493×C67—21no—
11#486×C67—21no—
12#459×C67—21no—
13#459×C67—21no—
14#447×C67—22no—
15#431×C67C2211yescharacteristic
16#423×C67C2211yescharacteristic
17C2×C4⋊C8×C67C2211yescharacteristic
18C2×#104×C67—21no—
19C2×#104×C67—21no—
20C2×#102×C67—22no—
21C2×#100×C67—21no—
22C2×#100×C67—21no—
23C4×M16×C67C2211yescharacteristic
24C4×C4⋊C4×C67C2211yescharacteristic
25#335×C67—21no—
26#334×C67—21no—
27#332×C67—21no—
28#331×C67—21no—
29C2×#41×C67—21no—
30C22×D4×C67—21no—
31C22×D4×C67—21no—
32C22×D4×C67D412yes—
33C2×D8×C67—22no—
34C2×D8×C67—22no—
35C2×M16×C67—21no—
36C2×M16×C67D411yescharacteristic
37C2×M16×C67D411yescharacteristic
38C22×C8×C67—21no—
39C4⋊Q8×C67D411yescharacteristic
40#127×C67—21no—
41#127×C67—21no—
42#127×C67—21no—
43#127×C67—21no—
44#127×C67D411yescharacteristic
45#124×C67—21no—
46#121×C67—21no—
47#121×C67—21no—
48C4×D4×C67—21no—
49C4×D4×C67—21no—
50C2×C4⋊C4×C67—21no—
51C2×C42×C67—21no—
52C2×C42×C67C2311yescharacteristic
53C4⋊C8×C67—21no—
54C4⋊C8×C67—21no—
55C4⋊C8×C67—21no—
56C4⋊C8×C67—21no—
57C4⋊C8×C67D411yescharacteristic
58C4⋊C8×C67D411yescharacteristic
59#104×C67—42no—
60#104×C67—42no—
61#102×C67—22no—
62#102×C67—22no—
63#102×C67—44no—
64#100×C67—42no—
65#100×C67—42no—
66C22⋊C8×C67—21no—
67#97×C67—21no—
68C4×C8×C67—21no—
69C4×C8×C67—21no—
70#95×C67—21no—
71C24×C67—22no—
72#41×C67—41no—
73#41×C67—41no—
74#41×C67—41no—
75#41×C67—41no—
76C2×Q8×C67—21no—
77C2×D4×C67—22no—
78C2×D4×C67—21no—
79C2×D4×C67—21no—
80C2×D4×C67—22no—
81C2×D4×C67—21no—
82C2×D4×C67—42no—
83C2×D4×C67—42no—
84C2×D4×C67—41no—
85C2×D4×C67—41no—
86C2×D4×C67—41no—
87C2×D4×C67—41no—
88C2×D4×C67—41no—
89C2×D4×C67—41no—
90C2×D4×C67—41no—
91C2×D4×C67—41no—
92C2×D4×C67—41no—
93C2×D4×C67—82no—
94C22×C4×C67—41no—
95C22×C4×C67—41no—
96C22×C4×C67C2×D411yescharacteristic
97C22×C4×C67C2×D411yescharacteristic
98C22×C4×C67#4111yescharacteristic
99D8×C67—82no—
100D8×C67—82no—
101M16×C67—21no—
102M16×C67—21no—
103M16×C67—21no—
104M16×C67—41no—
105M16×C67—41no—
106M16×C67D812yescomplemented
107C2×C8×C67—21no—
108C2×C8×C67—21no—
109C2×C8×C67—21no—
110C2×C8×C67—21no—
111C2×C8×C67—42no—
112C2×C8×C67—41no—
113C2×C8×C67—41no—
114C4⋊C4×C67—22no—
115C4⋊C4×C67—21no—
116C4⋊C4×C67—21no—
117C4⋊C4×C67—41no—
118C22⋊C4×C67—41no—
119C22⋊C4×C67—41no—
120C42×C67—41no—
121C42×C67—41no—
122C42×C67C2×D411yescharacteristic
123C42×C67C2×D411yescharacteristic
124C42×C67#4111yescharacteristic
125C42×C67#4111yescharacteristic
126C23×C67—41no—
127C23×C67—42no—
128C23×C67—42no—
129C23×C67—42no—
130C23×C67—82no—
131C23×C67—82no—
132C23×C67#12111yescharacteristic
133Q8×C67—41no—
134Q8×C67—41no—
135D4×C67—44no—
136D4×C67—41no—
137D4×C67—42no—
138D4×C67—41no—
139D4×C67—42no—
140D4×C67—41no—
141D4×C67—41no—
142D4×C67—81no—
143D4×C67—82no—
144D4×C67—82no—
145D4×C67—81no—
146D4×C67—81no—
147D4×C67—81no—
148D4×C67—81no—
149D4×C67—81no—
150D4×C67—81no—
151C2×C4×C67—21no—
152C2×C4×C67—21no—
153C2×C4×C67—41no—
154C2×C4×C67—42no—
155C2×C4×C67—41no—
156C2×C4×C67—81no—
157C2×C4×C67—81no—
158C2×C4×C67—81no—
159C2×C4×C67—81no—
160C2×C4×C67#12011yescharacteristic
161C2×C4×C67#12111yescharacteristic
162C2×C4×C67#12111yescharacteristic
163C2×C4×C67#12311yescharacteristic
164C2×C4×C67#12311yescharacteristic
165C2×C4×C67#12411yescharacteristic
166C2×C4×C67C2×D811yescharacteristic
167C2×C4×C67#13511yescharacteristic
168C2×C4×C67AGL(1,ℤ/8)11yescharacteristic
169C2×C4×C67AGL(1,ℤ/8)11yescharacteristic
170C8×C67—22no—
171C8×C67—41no—
172C8×C67—41no—
173C8×C67—41no—
174C8×C67—81no—
175C8×C67—81no—
176C22×C67—21no—
177C22×C67—21no—
178C22×C67—82no—
179C22×C67—81no—
180C22×C67—81no—
181C22×C67—82no—
182C22×C67—82no—
183C22×C67—81no—
184C22×C67—162no—
185C22×C67—162no—
186C22×C67#39411yescharacteristic
187C22×C67#45911yescharacteristic
188C22×C67#46311yescharacteristic
189C4×C67—21no—
190C4×C67—21no—
191C4×C67—21no—
192C4×C67—82no—
193C4×C67—81no—
194C4×C67#45311yescharacteristic
195C4×C67#45511yescharacteristic
196C4×C67#45911yescharacteristic
197C4×C67#46311yescharacteristic
198C4×C67#46611yescharacteristic
199C4×C67#46911yescharacteristic
200#10500C6711yesmaximal, characteristic, direct product factor, Sylow
201C2×C67—21no—
202C2×C67—21no—
203C2×C67—161no—
204C2×C67—162no—
205C2×C67#203711yescharacteristic
206C2×C67#204811yescharacteristic
207C2×C67#205411yescharacteristic
208#3205C2×C6711yescharacteristic
209#1982C2×C6711yescharacteristic, complemented
210#1980C2×C6711yescharacteristic, complemented
211#1965C2×C6711yescharacteristic, complemented
212#1565C2×C6711yescharacteristic, complemented
213#1547C2×C6711yescharacteristic, complemented
214#1303C2×C6711yescharacteristic, complemented
215C67#1050011yescharacteristic, direct product factor, Sylow
216#532C22×C6711yescharacteristic
217C2×#127C22×C6711yescharacteristic
218#493—21no—
219#486—21no—
220#459—21no—
221#459—21no—
222#447—22no—
223#431C22×C6711yescharacteristic
224#423C22×C6711yescharacteristic
225C2×C4⋊C8C22×C6711yescharacteristic
226C2×#104—21no—
227C2×#104—21no—
228C2×#102—22no—
229C2×#100—21no—
230C2×#100—21no—
231C4×M16C22×C6711yescharacteristic
232C4×C4⋊C4C22×C6711yescharacteristic
233#335—21no—
234#334—21no—
235#332—21no—
236#331—21no—
237C2×#41—21no—
238C22×D4—21no—
239C22×D4—21no—
240C22×D4D4×C6712yes—
241C2×D8—22no—
242C2×D8—22no—
243C2×M16—21no—
244C2×M16D4×C6711yescharacteristic
245C2×M16D4×C6711yescharacteristic
246C22×C8—21no—
247C4⋊Q8D4×C6711yescharacteristic
248#127—21no—
249#127—21no—
250#127—21no—
251#127—21no—
252#127D4×C6711yescharacteristic
253#124—21no—
254#121—21no—
255#121—21no—
256C4×D4—21no—
257C4×D4—21no—
258C2×C4⋊C4—21no—
259C2×C42—21no—
260C2×C42C23×C6711yescharacteristic
261C4⋊C8—21no—
262C4⋊C8—21no—
263C4⋊C8—21no—
264C4⋊C8—21no—
265C4⋊C8D4×C6711yescharacteristic
266C4⋊C8D4×C6711yescharacteristic
267#104—42no—
268#104—42no—
269#102—22no—
270#102—22no—
271#102—44no—
272#100—42no—
273#100—42no—
274C22⋊C8—21no—
275#97—21no—
276C4×C8—21no—
277C4×C8—21no—
278#95—21no—
279C24—22no—
280#41—41no—
281#41—41no—
282#41—41no—
283#41—41no—
284C2×Q8—21no—
285C2×D4—22no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—22no—
289C2×D4—21no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—82no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4C2×D4×C6711yescharacteristic
305C22×C4C2×D4×C6711yescharacteristic
306C22×C4#41×C6711yescharacteristic
307D8—82no—
308D8—82no—
309M16—21no—
310M16—21no—
311M16—21no—
312M16—41no—
313M16—41no—
314M16D8×C6712yescomplemented
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—42no—
320C2×C8—41no—
321C2×C8—41no—
322C4⋊C4—22no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—41no—
326C22⋊C4—41no—
327C22⋊C4—41no—
328C42—41no—
329C42—41no—
330C42C2×D4×C6711yescharacteristic
331C42C2×D4×C6711yescharacteristic
332C42#41×C6711yescharacteristic
333C42#41×C6711yescharacteristic
334C23—41no—
335C23—42no—
336C23—42no—
337C23—42no—
338C23—82no—
339C23—82no—
340C23#121×C6711yescharacteristic
341Q8—41no—
342Q8—41no—
343D4—44no—
344D4—41no—
345D4—42no—
346D4—41no—
347D4—42no—
348D4—41no—
349D4—41no—
350D4—81no—
351D4—82no—
352D4—82no—
353D4—81no—
354D4—81no—
355D4—81no—
356D4—81no—
357D4—81no—
358D4—81no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—41no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4#120×C6711yescharacteristic
369C2×C4#121×C6711yescharacteristic
370C2×C4#121×C6711yescharacteristic
371C2×C4#123×C6711yescharacteristic
372C2×C4#123×C6711yescharacteristic
373C2×C4#124×C6711yescharacteristic
374C2×C4C2×D8×C6711yescharacteristic
375C2×C4#135×C6711yescharacteristic
376C2×C4AGL(1,ℤ/8)×C6711yescharacteristic
377C2×C4AGL(1,ℤ/8)×C6711yescharacteristic
378C8—22no—
379C8—41no—
380C8—41no—
381C8—41no—
382C8—81no—
383C8—81no—
384C22—21no—
385C22—21no—
386C22—82no—
387C22—81no—
388C22—82no—
389C22—81no—
390C22—81no—
391C22—82no—
392C22—162no—
393C22—162no—
394C22#394×C6711yescharacteristic
395C22#459×C6711yescharacteristic
396C22#463×C6711yescharacteristic
397C4—21no—
398C4—21no—
399C4—21no—
400C4—82no—
401C4—81no—
402C4#453×C6711yescharacteristic
403C4#455×C6711yescharacteristic
404C4#459×C6711yescharacteristic
405C4#463×C6711yescharacteristic
406C4#466×C6711yescharacteristic
407C4#469×C6711yescharacteristic
408C2—21no—
409C2—21no—
410C2—162no—
411C2—161no—
412C2C67×#203711yescharacteristic
413C2C67×#204811yescharacteristic
414C2C67×#205411yescharacteristic
subgroupG/H
center, socleC22×C67#394
commutator, Frattini, cosocleC2×C42C23×C67
prime radicalC67×#3205C2
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C67 ◁ C2×C42×C67 ◁ G

Complements

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

NH ≅ G/Ntype
2. C67×#1982C2semidirect
3. C67×#1980C2semidirect
4. C67×#1965C2semidirect
5. C67×#1565C2semidirect
6. C67×#1547C2semidirect
7. C67×#1303C2semidirect
106. M16×C67D8semidirect
200. #10500C67direct (N × H)
209. #1982C2×C67semidirect
210. #1980C2×C67semidirect
211. #1965C2×C67semidirect
212. #1565C2×C67semidirect
213. #1547C2×C67semidirect
214. #1303C2×C67semidirect
215. C67#10500direct (N × H)
314. M16D8×C67semidirect

Maximal subgroups

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

subgroupindex
C67×#32052
C67×#19822
C67×#19802
C67×#19652
C67×#15652
C67×#15472
C67×#13032
#1050067