Finite Group Databasebeta

← C67×#33033 #30712284

Subgroups — C67×#33033 #30712284

Order 17152.

854 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#475×C67C211yesmaximal, characteristic
2C2×#462×C67C211yesmaximal, characteristic
3C67×#2862C211yesmaximal, characteristic
4C42×Q8×C67C211yesmaximal, characteristic
5C67×#1988C211yesmaximal, characteristic
6C67×#1981C211yesmaximal, characteristic
7C67×#1927C214yesmaximal
8C67×#1878C214yesmaximal
9C67×#1831C211yesmaximal, characteristic
10C22×Q16×C67C2212yes—
11C2×C4⋊Q8×C67C2211yescharacteristic
12C2×C4⋊Q8×C67C2211yescharacteristic
13C2×C4×Q8×C67C2212yes—
14C2×C4×Q8×C67C2211yescharacteristic
15#475×C67—24no—
16#462×C67—24no—
17C2×#106×C67C2212yes—
18C2×C4⋊C8×C67C2211yescharacteristic
19C2×C4⋊C8×C67C2211yescharacteristic
20C2×#103×C67C2212yes—
21C2×#103×C67C2212yes—
22C2×#103×C67C2214yes—
23#391×C67—21no—
24#391×C67—21no—
25#391×C67C2214yes—
26#382×C67—22no—
27C4×C4⋊C4×C67—22no—
28C4×C4⋊C4×C67—21no—
29C4×C4⋊C4×C67C2211yescharacteristic
30C4×C4⋊C4×C67C2212yes—
31C43×C67—21no—
32C43×C67C2211yescharacteristic
33#340×C67C2214yes—
34#336×C67C2214yes—
35C22×Q8×C67—21no—
36C22×Q8×C67C2312yes—
37C22×Q8×C67C2311yescharacteristic
38C2×Q16×C67—24no—
39C2×Q16×C67—28no—
40C22×C8×C67C2312yes—
41C22×C8×C67C2312yes—
42C4⋊Q8×C67—22no—
43C4⋊Q8×C67—22no—
44C4⋊Q8×C67—22no—
45C4⋊Q8×C67—22no—
46C4×Q8×C67—24no—
47C4×Q8×C67—24no—
48C4×Q8×C67—22no—
49C4×Q8×C67D414yes—
50C2×C4⋊C4×C67—22no—
51C2×C4⋊C4×C67—21no—
52C2×C4⋊C4×C67—21no—
53C2×C4⋊C4×C67—21no—
54C2×C4⋊C4×C67—21no—
55C2×C4⋊C4×C67C2312yes—
56C2×C4⋊C4×C67C2311yescharacteristic
57C2×C4⋊C4×C67C2312yes—
58C2×C42×C67—22no—
59C2×C42×C67—21no—
60C2×C42×C67—21no—
61C2×C42×C67—22no—
62C2×C42×C67D411yescharacteristic
63C2×C42×C67D412yes—
64C2×C42×C67D411yescharacteristic
65C2×C42×C67C2312yes—
66C2×C42×C67C2311yescharacteristic
67#106×C67—24no—
68C4⋊C8×C67—22no—
69C4⋊C8×C67—22no—
70#103×C67—24no—
71#103×C67—24no—
72#103×C67—28no—
73#95×C67—22no—
74#95×C67—21no—
75#95×C67—24no—
76#95×C67—24no—
77C2×Q8×C67—21no—
78C2×Q8×C67—22no—
79C2×Q8×C67—22no—
80C2×Q8×C67—24no—
81C2×Q8×C67—22no—
82C2×Q8×C67—24no—
83C2×Q8×C67—42no—
84C2×Q8×C67—42no—
85C2×Q8×C67—42no—
86C2×Q8×C67C2×D412yes—
87C2×Q8×C67#4114yes—
88C22×C4×C67—21no—
89C22×C4×C67—21no—
90C22×C4×C67—22no—
91C22×C4×C67—22no—
92C22×C4×C67—22no—
93C22×C4×C67C2×D412yes—
94C22×C4×C67C2×D411yescharacteristic
95C22×C4×C67C2×D412yes—
96C22×C4×C67C2×D411yescharacteristic
97C22×C4×C67C2411yescharacteristic
98Q16×C67—48no—
99C2×C8×C67—22no—
100C2×C8×C67—22no—
101C2×C8×C67—24no—
102C2×C8×C67—24no—
103C4⋊C4×C67—28no—
104C4⋊C4×C67—24no—
105C4⋊C4×C67—21no—
106C4⋊C4×C67—22no—
107C4⋊C4×C67—22no—
108C4⋊C4×C67—22no—
109C4⋊C4×C67—42no—
110C4⋊C4×C67—42no—
111C4⋊C4×C67—42no—
112C4⋊C4×C67C2×D412yes—
113C4⋊C4×C67#4114yes—
114C42×C67—28no—
115C42×C67—24no—
116C42×C67—22no—
117C42×C67—22no—
118C42×C67—24no—
119C42×C67—22no—
120C42×C67—22no—
121C42×C67—42no—
122C42×C67—44no—
123C42×C67C2×D412yes—
124C42×C67#4112yes—
125C23×C67C22×D411yescharacteristic
126Q8×C67—24no—
127Q8×C67—22no—
128Q8×C67—42no—
129Q8×C67—44no—
130Q8×C67—42no—
131Q8×C67—84no—
132Q8×C67#12414yes—
133C2×C4×C67—22no—
134C2×C4×C67—22no—
135C2×C4×C67—24no—
136C2×C4×C67—22no—
137C2×C4×C67—22no—
138C2×C4×C67—22no—
139C2×C4×C67—22no—
140C2×C4×C67—24no—
141C2×C4×C67—22no—
142C2×C4×C67—22no—
143C2×C4×C67—42no—
144C2×C4×C67—42no—
145C2×C4×C67—42no—
146C2×C4×C67—42no—
147C2×C4×C67—42no—
148C2×C4×C67—44no—
149C2×C4×C67—42no—
150C2×C4×C67—42no—
151C2×C4×C67#12112yes—
152C2×C4×C67#12112yes—
153C2×C4×C67#13512yes—
154C2×C4×C67#13714yes—
155C2×C4×C67#13712yes—
156C2×C4×C67C22×D411yescharacteristic
157C2×C4×C67C2×#4112yes—
158C2×C4×C67C2×#4111yescharacteristic
159C2×C4×C67C2×#4112yes—
160C8×C67—42no—
161C8×C67—42no—
162C22×C67C2×#12111yescharacteristic
163C22×C67#52512yes—
164C22×C67C2×#13511yescharacteristic
165C22×C67C2×#13712yes—
166C22×C67C2×#13711yescharacteristic
167C4×C67—24no—
168C4×C67—24no—
169C4×C67—22no—
170C4×C67—42no—
171C4×C67—44no—
172C4×C67—42no—
173C4×C67—82no—
174C4×C67—82no—
175C4×C67#46414yes—
176C4×C67C2×#12411yescharacteristic
177C4×C67#52912yes—
178C4×C67#53211yescharacteristic
179#33033C6711yesmaximal, characteristic, direct product factor, Sylow
180C2×C67#285011yescharacteristic
181C2×C67C2×#46411yescharacteristic
182C2×C67#303911yescharacteristic
183C2×C67#310712yes—
184C2×C67#311312yes—
185C2×#475C2×C6711yescharacteristic
186C2×#462C2×C6711yescharacteristic
187#2862C2×C6711yescharacteristic
188C42×Q8C2×C6711yescharacteristic
189#1988C2×C6711yescharacteristic
190#1981C2×C6711yescharacteristic
191#1927C2×C6714yes—
192#1878C2×C6714yes—
193#1831C2×C6711yescharacteristic
194C67#3303311yescharacteristic, direct product factor, Sylow
195C22×Q16C22×C6712yes—
196C2×C4⋊Q8C22×C6711yescharacteristic
197C2×C4⋊Q8C22×C6711yescharacteristic
198C2×C4×Q8C22×C6712yes—
199C2×C4×Q8C22×C6711yescharacteristic
200#475—24no—
201#462—24no—
202C2×#106C22×C6712yes—
203C2×C4⋊C8C22×C6711yescharacteristic
204C2×C4⋊C8C22×C6711yescharacteristic
205C2×#103C22×C6712yes—
206C2×#103C22×C6714yes—
207C2×#103C22×C6712yes—
208#391—21no—
209#391—21no—
210#391C22×C6714yes—
211#382—22no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4C22×C6712yes—
215C4×C4⋊C4C22×C6711yescharacteristic
216C43—21no—
217C43C22×C6711yescharacteristic
218#340C22×C6714yes—
219#336C22×C6714yes—
220C22×Q8—21no—
221C22×Q8C23×C6712yes—
222C22×Q8C23×C6711yescharacteristic
223C2×Q16—24no—
224C2×Q16—28no—
225C22×C8C23×C6712yes—
226C22×C8C23×C6712yes—
227C4⋊Q8—22no—
228C4⋊Q8—22no—
229C4⋊Q8—22no—
230C4⋊Q8—22no—
231C4×Q8—24no—
232C4×Q8—24no—
233C4×Q8—22no—
234C4×Q8D4×C6714yes—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4C23×C6712yes—
241C2×C4⋊C4C23×C6711yescharacteristic
242C2×C4⋊C4C23×C6712yes—
243C2×C42—22no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—22no—
247C2×C42D4×C6711yescharacteristic
248C2×C42D4×C6712yes—
249C2×C42D4×C6711yescharacteristic
250C2×C42C23×C6712yes—
251C2×C42C23×C6711yescharacteristic
252#106—24no—
253C4⋊C8—22no—
254C4⋊C8—22no—
255#103—24no—
256#103—24no—
257#103—28no—
258#95—22no—
259#95—21no—
260#95—24no—
261#95—24no—
262C2×Q8—21no—
263C2×Q8—22no—
264C2×Q8—22no—
265C2×Q8—24no—
266C2×Q8—22no—
267C2×Q8—24no—
268C2×Q8—42no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8C2×D4×C6712yes—
272C2×Q8#41×C6714yes—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4C2×D4×C6712yes—
279C22×C4C2×D4×C6711yescharacteristic
280C22×C4C2×D4×C6712yes—
281C22×C4C2×D4×C6711yescharacteristic
282C22×C4C24×C6711yescharacteristic
283Q16—48no—
284C2×C8—22no—
285C2×C8—22no—
286C2×C8—24no—
287C2×C8—24no—
288C4⋊C4—28no—
289C4⋊C4—24no—
290C4⋊C4—21no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4C2×D4×C6712yes—
298C4⋊C4#41×C6714yes—
299C42—28no—
300C42—24no—
301C42—22no—
302C42—22no—
303C42—24no—
304C42—22no—
305C42—22no—
306C42—42no—
307C42—44no—
308C42C2×D4×C6712yes—
309C42#41×C6712yes—
310C23C22×D4×C6711yescharacteristic
311Q8—24no—
312Q8—22no—
313Q8—42no—
314Q8—44no—
315Q8—42no—
316Q8—84no—
317Q8#124×C6714yes—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—24no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—24no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—44no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4#121×C6712yes—
337C2×C4#121×C6712yes—
338C2×C4#135×C6712yes—
339C2×C4#137×C6714yes—
340C2×C4#137×C6712yes—
341C2×C4C22×D4×C6711yescharacteristic
342C2×C4C2×#41×C6712yes—
343C2×C4C2×#41×C6711yescharacteristic
344C2×C4C2×#41×C6712yes—
345C8—42no—
346C8—42no—
347C22C2×#121×C6711yescharacteristic
348C22#525×C6712yes—
349C22C2×#135×C6711yescharacteristic
350C22C2×#137×C6711yescharacteristic
351C22C2×#137×C6712yes—
352C4—22no—
353C4—24no—
354C4—24no—
355C4—42no—
356C4—42no—
357C4—44no—
358C4—82no—
359C4—82no—
360C4#464×C6714yes—
361C4C2×#124×C6711yescharacteristic
362C4#529×C6712yes—
363C4#532×C6711yescharacteristic
364C2C67×#285011yescharacteristic
365C2C2×#464×C6711yescharacteristic
366C2C67×#303911yescharacteristic
367C2C67×#310712yes—
368C2C67×#311312yes—
subgroupG/H
center, socle, prime radicalC23×C67C22×D4
commutator, Frattini, cosocleC22×C4C24×C67
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C67 ◁ C43×C67 ◁ G

Complements

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

NH ≅ G/Ntype
179. #33033C67direct (N × H)
194. C67#33033direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#475×C672
C2×#462×C672
C67×#28622
C42×Q8×C672
C67×#19882
C67×#19812
C67×#19272
C67×#18782
C67×#18312
#3303367