Finite Group Databasebeta

← C67×#10284 #30689535

Subgroups — C67×#10284 #30689535

Order 17152.

678 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3226C211yesmaximal, characteristic, complemented
2C67×#3205C211yesmaximal, characteristic
3C67×#2858C211yesmaximal, characteristic, complemented
4C67×#1962C211yesmaximal, characteristic, complemented
5C67×#1546C211yesmaximal, characteristic, complemented
6C67×#1545C211yesmaximal, characteristic, complemented
7C67×#1276C211yesmaximal, characteristic, complemented
8#532×C67C2211yescharacteristic
9C2×C4⋊Q8×C67C2211yescharacteristic
10C2×#127×C67C2211yescharacteristic
11C2×#124×C67—22no—
12#500×C67—21no—
13#493×C67—21no—
14#486×C67—21no—
15#486×C67—21no—
16#480×C67—22no—
17#474×C67—22no—
18#459×C67—22no—
19#447×C67—22no—
20#431×C67C2211yescharacteristic
21#431×C67C2211yescharacteristic
22#423×C67C2211yescharacteristic
23C2×#104×C67—22no—
24#390×C67—22no—
25#390×C67—22no—
26C43×C67C2211yescharacteristic
27#332×C67—22no—
28#331×C67—22no—
29C2×#41×C67—21no—
30C22×Q8×C67—21no—
31C22×D4×C67—21no—
32C22×D4×C67—21no—
33C22×D4×C67D412yes—
34C2×D8×C67—24no—
35C2×M16×C67—21no—
36C4⋊Q8×C67—21no—
37C4⋊Q8×C67—21no—
38C4⋊Q8×C67—21no—
39C4⋊Q8×C67—21no—
40C4⋊Q8×C67D411yescharacteristic
41#127×C67—21no—
42#127×C67—21no—
43#127×C67—21no—
44#127×C67—21no—
45#127×C67D411yescharacteristic
46#124×C67—21no—
47#124×C67—44no—
48#124×C67—44no—
49#121×C67—21no—
50#121×C67—21no—
51C4×D4×C67—21no—
52C4×D4×C67—21no—
53C2×C4⋊C4×C67—21no—
54C2×C4⋊C4×C67D412yes—
55C2×C22⋊C4×C67—28no—
56C2×C42×C67—21no—
57C2×C42×C67—21no—
58C2×C42×C67D412yes—
59C2×C42×C67C2311yescharacteristic
60#107×C67—24no—
61C4⋊C8×C67—21no—
62C4⋊C8×C67—21no—
63C4⋊C8×C67D412yes—
64C4⋊C8×C67D412yes—
65#104×C67—44no—
66#102×C67—24no—
67#102×C67—24no—
68C22⋊C8×C67—21no—
69C22⋊C8×C67—21no—
70C4×C8×C67—21no—
71C4×C8×C67—21no—
72C24×C67—22no—
73#41×C67—42no—
74#41×C67—42no—
75C2×Q8×C67—21no—
76C2×Q8×C67—41no—
77C2×Q8×C67—41no—
78C2×Q8×C67—42no—
79C2×Q8×C67—42no—
80C2×D4×C67—21no—
81C2×D4×C67—22no—
82C2×D4×C67—21no—
83C2×D4×C67—21no—
84C2×D4×C67—22no—
85C2×D4×C67—44no—
86C2×D4×C67—44no—
87C2×D4×C67—41no—
88C2×D4×C67—41no—
89C2×D4×C67—41no—
90C2×D4×C67—41no—
91C2×D4×C67—41no—
92C2×D4×C67—42no—
93C2×D4×C67—42no—
94C22×C4×C67—22no—
95C22×C4×C67—22no—
96C22×C4×C67—41no—
97C22×C4×C67C2×D411yescharacteristic
98C22×C4×C67C2×D411yescharacteristic
99C22×C4×C67C2×D411yescharacteristic
100D8×C67—84no—
101M16×C67—42no—
102C2×C8×C67—22no—
103C2×C8×C67—22no—
104C2×C8×C67—41no—
105C4⋊C4×C67—21no—
106C4⋊C4×C67—22no—
107C4⋊C4×C67—21no—
108C4⋊C4×C67—22no—
109C4⋊C4×C67—41no—
110C4⋊C4×C67—41no—
111C22⋊C4×C67—416no—
112C22⋊C4×C67—41no—
113C22⋊C4×C67—41no—
114C42×C67—22no—
115C42×C67—42no—
116C42×C67—42no—
117C42×C67D814yescomplemented
118C42×C67C2×D411yescharacteristic
119C42×C67C2×D411yescharacteristic
120C42×C67C2×D411yescharacteristic
121C42×C67#4111yescharacteristic
122C23×C67—44no—
123C23×C67—44no—
124C23×C67—41no—
125C23×C67—42no—
126C23×C67—44no—
127C23×C67#12011yescharacteristic
128Q8×C67—42no—
129Q8×C67—82no—
130Q8×C67—81no—
131Q8×C67—81no—
132D4×C67—42no—
133D4×C67—42no—
134D4×C67—44no—
135D4×C67—42no—
136D4×C67—82no—
137D4×C67—82no—
138D4×C67—82no—
139D4×C67—82no—
140D4×C67—82no—
141D4×C67—81no—
142D4×C67—81no—
143C2×C4×C67—22no—
144C2×C4×C67—22no—
145C2×C4×C67—24no—
146C2×C4×C67—42no—
147C2×C4×C67—42no—
148C2×C4×C67—41no—
149C2×C4×C67—42no—
150C2×C4×C67—44no—
151C2×C4×C67—81no—
152C2×C4×C67—81no—
153C2×C4×C67#12011yescharacteristic
154C2×C4×C67#12011yescharacteristic
155C2×C4×C67#12111yescharacteristic
156C2×C4×C67#12111yescharacteristic
157C2×C4×C67#12111yescharacteristic
158C2×C4×C67#12711yescharacteristic
159C2×C4×C67C2×D812yes—
160C2×C4×C67AGL(1,ℤ/8)12yes—
161C8×C67—42no—
162C8×C67—42no—
163C8×C67—81no—
164C22×C67—21no—
165C22×C67—21no—
166C22×C67—88no—
167C22×C67—81no—
168C22×C67—82no—
169C22×C67—82no—
170C22×C67—81no—
171C22×C67—81no—
172C22×C67—82no—
173C22×C67#39211yescharacteristic
174C22×C67#44712yes—
175C4×C67—21no—
176C4×C67—21no—
177C4×C67—44no—
178C4×C67—82no—
179C4×C67—81no—
180C4×C67#45412yes—
181C4×C67#45512yes—
182C4×C67#45912yes—
183C4×C67#49311yescharacteristic
184C4×C67#49611yescharacteristic
185#10284C6711yesmaximal, characteristic, direct product factor, Sylow
186C2×C67—22no—
187C2×C67—161no—
188C2×C67—162no—
189C2×C67#200311yescharacteristic
190C2×C67#200411yescharacteristic
191C2×C67#201111yescharacteristic
192#3226C2×C6711yescharacteristic, complemented
193#3205C2×C6711yescharacteristic
194#2858C2×C6711yescharacteristic, complemented
195#1962C2×C6711yescharacteristic, complemented
196#1546C2×C6711yescharacteristic, complemented
197#1545C2×C6711yescharacteristic, complemented
198#1276C2×C6711yescharacteristic, complemented
199C67#1028411yescharacteristic, direct product factor, Sylow
200#532C22×C6711yescharacteristic
201C2×C4⋊Q8C22×C6711yescharacteristic
202C2×#127C22×C6711yescharacteristic
203C2×#124—22no—
204#500—21no—
205#493—21no—
206#486—21no—
207#486—21no—
208#480—22no—
209#474—22no—
210#459—22no—
211#447—22no—
212#431C22×C6711yescharacteristic
213#431C22×C6711yescharacteristic
214#423C22×C6711yescharacteristic
215C2×#104—22no—
216#390—22no—
217#390—22no—
218C43C22×C6711yescharacteristic
219#332—22no—
220#331—22no—
221C2×#41—21no—
222C22×Q8—21no—
223C22×D4—21no—
224C22×D4—21no—
225C22×D4D4×C6712yes—
226C2×D8—24no—
227C2×M16—21no—
228C4⋊Q8—21no—
229C4⋊Q8—21no—
230C4⋊Q8—21no—
231C4⋊Q8—21no—
232C4⋊Q8D4×C6711yescharacteristic
233#127—21no—
234#127—21no—
235#127—21no—
236#127—21no—
237#127D4×C6711yescharacteristic
238#124—21no—
239#124—44no—
240#124—44no—
241#121—21no—
242#121—21no—
243C4×D4—21no—
244C4×D4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4D4×C6712yes—
247C2×C22⋊C4—28no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42D4×C6712yes—
251C2×C42C23×C6711yescharacteristic
252#107—24no—
253C4⋊C8—21no—
254C4⋊C8—21no—
255C4⋊C8D4×C6712yes—
256C4⋊C8D4×C6712yes—
257#104—44no—
258#102—24no—
259#102—24no—
260C22⋊C8—21no—
261C22⋊C8—21no—
262C4×C8—21no—
263C4×C8—21no—
264C24—22no—
265#41—42no—
266#41—42no—
267C2×Q8—21no—
268C2×Q8—41no—
269C2×Q8—41no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×D4—21no—
273C2×D4—22no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4—22no—
277C2×D4—44no—
278C2×D4—44no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—42no—
285C2×D4—42no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—41no—
289C22×C4C2×D4×C6711yescharacteristic
290C22×C4C2×D4×C6711yescharacteristic
291C22×C4C2×D4×C6711yescharacteristic
292D8—84no—
293M16—42no—
294C2×C8—22no—
295C2×C8—22no—
296C2×C8—41no—
297C4⋊C4—21no—
298C4⋊C4—22no—
299C4⋊C4—21no—
300C4⋊C4—22no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C22⋊C4—416no—
304C22⋊C4—41no—
305C22⋊C4—41no—
306C42—22no—
307C42—42no—
308C42—42no—
309C42D8×C6714yescomplemented
310C42C2×D4×C6711yescharacteristic
311C42C2×D4×C6711yescharacteristic
312C42C2×D4×C6711yescharacteristic
313C42#41×C6711yescharacteristic
314C23—44no—
315C23—44no—
316C23—41no—
317C23—42no—
318C23—44no—
319C23#120×C6711yescharacteristic
320Q8—42no—
321Q8—82no—
322Q8—81no—
323Q8—81no—
324D4—42no—
325D4—42no—
326D4—44no—
327D4—42no—
328D4—82no—
329D4—82no—
330D4—82no—
331D4—82no—
332D4—82no—
333D4—81no—
334D4—81no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—24no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—44no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4#120×C6711yescharacteristic
346C2×C4#120×C6711yescharacteristic
347C2×C4#121×C6711yescharacteristic
348C2×C4#121×C6711yescharacteristic
349C2×C4#121×C6711yescharacteristic
350C2×C4#127×C6711yescharacteristic
351C2×C4C2×D8×C6712yes—
352C2×C4AGL(1,ℤ/8)×C6712yes—
353C8—42no—
354C8—42no—
355C8—81no—
356C22—21no—
357C22—21no—
358C22—88no—
359C22—81no—
360C22—82no—
361C22—82no—
362C22—81no—
363C22—81no—
364C22—82no—
365C22#392×C6711yescharacteristic
366C22#447×C6712yes—
367C4—21no—
368C4—21no—
369C4—44no—
370C4—82no—
371C4—81no—
372C4#454×C6712yes—
373C4#455×C6712yes—
374C4#459×C6712yes—
375C4#493×C6711yescharacteristic
376C4#496×C6711yescharacteristic
377C2—22no—
378C2—162no—
379C2—161no—
380C2C67×#200311yescharacteristic
381C2C67×#200411yescharacteristic
382C2C67×#201111yescharacteristic
subgroupG/H
center, socleC22×C67#392
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
1. C67×#3226C2semidirect
3. C67×#2858C2semidirect
4. C67×#1962C2semidirect
5. C67×#1546C2semidirect
6. C67×#1545C2semidirect
7. C67×#1276C2semidirect
117. C42×C67D8semidirect
185. #10284C67direct (N × H)
192. #3226C2×C67semidirect
194. #2858C2×C67semidirect
195. #1962C2×C67semidirect
196. #1546C2×C67semidirect
197. #1545C2×C67semidirect
198. #1276C2×C67semidirect
199. C67#10284direct (N × H)
309. C42D8×C67semidirect

Maximal subgroups

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

subgroupindex
C67×#32262
C67×#32052
C67×#28582
C67×#19622
C67×#15462
C67×#15452
C67×#12762
#1028467