Finite Group Databasebeta

← C67×#8761 #30688012

Subgroups — C67×#8761 #30688012

Order 17152.

530 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#2349C211yesmaximal, characteristic
2C67×#1799C211yesmaximal, characteristic
3C67×#1459C211yesmaximal, characteristic
4C67×#1294C212yesmaximal, complemented
5C67×#1289C212yesmaximal, complemented
6C2×C4×D4×C67—21no—
7C2×C4×D4×C67C2211yescharacteristic
8C2×#99×C67—21no—
9C2×C22⋊C8×C67C2212yes—
10#389×C67—21no—
11#385×C67—21no—
12#385×C67—21no—
13#382×C67C411yescharacteristic, complemented
14#380×C67C411yescharacteristic, complemented
15#380×C67C2212yes—
16#380×C67C2211yescharacteristic
17C4×C4⋊C4×C67C411yescharacteristic, complemented
18C2×#95×C67C411yescharacteristic, complemented
19C2×#95×C67C2211yescharacteristic
20#342×C67—21no—
21#324×C67—21no—
22#323×C67—21no—
23C22×D4×C67D411yescharacteristic
24C23×C4×C67—21no—
25C23×C4×C67—21no—
26C23×C4×C67C2×C411yescharacteristic
27C23×C4×C67C2311yescharacteristic
28C22×C8×C67—22no—
29C4×D4×C67—22no—
30C4×D4×C67—22no—
31C4×D4×C67—44no—
32C2×C4⋊C4×C67—21no—
33C2×C4⋊C4×C67—21no—
34C2×C4⋊C4×C67—21no—
35C2×C4⋊C4×C67—21no—
36C2×C4⋊C4×C67—22no—
37C2×C4⋊C4×C67D411yescharacteristic
38C2×C22⋊C4×C67—21no—
39C2×C22⋊C4×C67—21no—
40C2×C22⋊C4×C67—22no—
41C2×C22⋊C4×C67D411yescharacteristic
42C2×C22⋊C4×C67D411yescharacteristic
43C2×C42×C67—21no—
44C2×C42×C67C2×C411yescharacteristic
45#99×C67—42no—
46C22⋊C8×C67—22no—
47C22⋊C8×C67—22no—
48#95×C67—21no—
49#95×C67—21no—
50#95×C67—21no—
51#95×C67—22no—
52#95×C67C2×C411yescharacteristic
53#95×C67C2×C411yescharacteristic
54#95×C67C2×C411yescharacteristic
55#95×C67C2×C411yescharacteristic
56C24×C67C22⋊C411yescharacteristic
57C24×C67C2×D411yescharacteristic
58C2×D4×C67—22no—
59C2×D4×C67—22no—
60C2×D4×C67SD1612yes—
61C2×D4×C67Q1612yes—
62C22×C4×C67—21no—
63C22×C4×C67—21no—
64C22×C4×C67—21no—
65C22×C4×C67—21no—
66C22×C4×C67—21no—
67C22×C4×C67—21no—
68C22×C4×C67—42no—
69C22×C4×C67—42no—
70C22×C4×C67—42no—
71C22×C4×C67—42no—
72C22×C4×C67—42no—
73C22×C4×C67—81no—
74C22×C4×C67—81no—
75C22×C4×C67—81no—
76C22×C4×C67—81no—
77C22×C4×C67C22⋊C411yescharacteristic
78C22×C4×C67C22⋊C411yescharacteristic
79C22×C4×C67C22⋊C411yescharacteristic
80C22×C4×C67C22×C411yescharacteristic
81C22×C4×C67C2×D411yescharacteristic
82C22×C4×C67#4112yes—
83C22×C4×C67#4112yes—
84C2×C8×C67—42no—
85C2×C8×C67—82no—
86C4⋊C4×C67—22no—
87C4⋊C4×C67—42no—
88C4⋊C4×C67—42no—
89C4⋊C4×C67—81no—
90C4⋊C4×C67—81no—
91C4⋊C4×C67—82no—
92C22⋊C4×C67—22no—
93C22⋊C4×C67—22no—
94C22⋊C4×C67—42no—
95C22⋊C4×C67—42no—
96C22⋊C4×C67—42no—
97C22⋊C4×C67—42no—
98C42×C67—21no—
99C42×C67—21no—
100C42×C67—42no—
101C23×C67—21no—
102C23×C67—21no—
103C23×C67—41no—
104C23×C67—41no—
105C23×C67—41no—
106C23×C67—41no—
107C23×C67#10312yes—
108C23×C67C2×C22⋊C411yescharacteristic
109C23×C67#12312yes—
110C23×C67C2×SD1611yescharacteristic
111C23×C67C2×Q1611yescharacteristic
112C23×C67AGL(1,ℤ/8)12yes—
113D4×C67—24no—
114D4×C67—24no—
115C2×C4×C67—21no—
116C2×C4×C67—21no—
117C2×C4×C67—22no—
118C2×C4×C67—21no—
119C2×C4×C67—21no—
120C2×C4×C67—41no—
121C2×C4×C67—42no—
122C2×C4×C67—42no—
123C2×C4×C67—41no—
124C2×C4×C67—41no—
125C2×C4×C67—41no—
126C2×C4×C67—41no—
127C2×C4×C67—41no—
128C2×C4×C67—41no—
129C2×C4×C67—41no—
130C2×C4×C67—41no—
131C2×C4×C67—41no—
132C2×C4×C67—41no—
133C2×C4×C67—41no—
134C2×C4×C67—41no—
135C2×C4×C67—41no—
136C2×C4×C67—41no—
137C2×C4×C67—81no—
138C2×C4×C67—81no—
139C2×C4×C67—84no—
140C2×C4×C67—84no—
141C2×C4×C67—82no—
142C2×C4×C67—82no—
143C2×C4×C67—82no—
144C2×C4×C67#10312yes—
145C2×C4×C67#11711yescharacteristic
146C2×C4×C67#11711yescharacteristic
147C2×C4×C67#12312yes—
148C8×C67—82no—
149C22×C67—21no—
150C22×C67—22no—
151C22×C67—21no—
152C22×C67—22no—
153C22×C67—22no—
154C22×C67—42no—
155C22×C67—41no—
156C22×C67—41no—
157C22×C67—42no—
158C22×C67—42no—
159C22×C67—41no—
160C22×C67—41no—
161C22×C67#38111yescharacteristic
162C22×C67C2×#10311yescharacteristic
163C22×C67#41811yescharacteristic
164C22×C67#48112yes—
165C22×C67#48412yes—
166C4×C67—22no—
167C4×C67—22no—
168C4×C67—21no—
169C4×C67—21no—
170C4×C67—42no—
171C4×C67—41no—
172C4×C67—41no—
173C4×C67—81no—
174C4×C67—81no—
175C4×C67—81no—
176C4×C67—162no—
177#8761C6711yesmaximal, characteristic, direct product factor, Sylow
178C2×C67—21no—
179C2×C67—22no—
180C2×C67—21no—
181C2×C67—22no—
182C2×C67—42no—
183C2×C67#182411yescharacteristic
184C2×C67#182711yescharacteristic
185C2×C67#182811yescharacteristic
186#2349C2×C6711yescharacteristic
187#1799C2×C6711yescharacteristic
188#1459C2×C6711yescharacteristic
189#1294C2×C6712yescomplemented
190#1289C2×C6712yescomplemented
191C67#876111yescharacteristic, direct product factor, Sylow
192C2×C4×D4—21no—
193C2×C4×D4C22×C6711yescharacteristic
194C2×#99—21no—
195C2×C22⋊C8C22×C6712yes—
196#389—21no—
197#385—21no—
198#385—21no—
199#382C4×C6711yescharacteristic, complemented
200#380C4×C6711yescharacteristic, complemented
201#380C22×C6712yes—
202#380C22×C6711yescharacteristic
203C4×C4⋊C4C4×C6711yescharacteristic, complemented
204C2×#95C4×C6711yescharacteristic, complemented
205C2×#95C22×C6711yescharacteristic
206#342—21no—
207#324—21no—
208#323—21no—
209C22×D4D4×C6711yescharacteristic
210C23×C4—21no—
211C23×C4—21no—
212C23×C4C2×C4×C6711yescharacteristic
213C23×C4C23×C6711yescharacteristic
214C22×C8—22no—
215C4×D4—22no—
216C4×D4—22no—
217C4×D4—44no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4D4×C6711yescharacteristic
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4D4×C6711yescharacteristic
228C2×C22⋊C4D4×C6711yescharacteristic
229C2×C42—21no—
230C2×C42C2×C4×C6711yescharacteristic
231#99—42no—
232C22⋊C8—22no—
233C22⋊C8—22no—
234#95—21no—
235#95—21no—
236#95—21no—
237#95—22no—
238#95C2×C4×C6711yescharacteristic
239#95C2×C4×C6711yescharacteristic
240#95C2×C4×C6711yescharacteristic
241#95C2×C4×C6711yescharacteristic
242C24C22⋊C4×C6711yescharacteristic
243C24C2×D4×C6711yescharacteristic
244C2×D4—22no—
245C2×D4—22no—
246C2×D4SD16×C6712yes—
247C2×D4Q16×C6712yes—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—81no—
260C22×C4—81no—
261C22×C4—81no—
262C22×C4—81no—
263C22×C4C22⋊C4×C6711yescharacteristic
264C22×C4C22⋊C4×C6711yescharacteristic
265C22×C4C22⋊C4×C6711yescharacteristic
266C22×C4C22×C4×C6711yescharacteristic
267C22×C4C2×D4×C6711yescharacteristic
268C22×C4#41×C6712yes—
269C22×C4#41×C6712yes—
270C2×C8—42no—
271C2×C8—82no—
272C4⋊C4—22no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—81no—
276C4⋊C4—81no—
277C4⋊C4—82no—
278C22⋊C4—22no—
279C22⋊C4—22no—
280C22⋊C4—42no—
281C22⋊C4—42no—
282C22⋊C4—42no—
283C22⋊C4—42no—
284C42—21no—
285C42—21no—
286C42—42no—
287C23—21no—
288C23—21no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23#103×C6712yes—
294C23C2×C22⋊C4×C6711yescharacteristic
295C23#123×C6712yes—
296C23C2×SD16×C6711yescharacteristic
297C23C2×Q16×C6711yescharacteristic
298C23AGL(1,ℤ/8)×C6712yes—
299D4—24no—
300D4—24no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—41no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—84no—
326C2×C4—84no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4#103×C6712yes—
331C2×C4#117×C6711yescharacteristic
332C2×C4#117×C6711yescharacteristic
333C2×C4#123×C6712yes—
334C8—82no—
335C22—21no—
336C22—22no—
337C22—21no—
338C22—22no—
339C22—22no—
340C22—42no—
341C22—41no—
342C22—41no—
343C22—42no—
344C22—42no—
345C22—41no—
346C22—41no—
347C22#381×C6711yescharacteristic
348C22C2×#103×C6711yescharacteristic
349C22#418×C6711yescharacteristic
350C22#481×C6712yes—
351C22#484×C6712yes—
352C4—22no—
353C4—22no—
354C4—21no—
355C4—21no—
356C4—42no—
357C4—41no—
358C4—41no—
359C4—81no—
360C4—81no—
361C4—81no—
362C4—162no—
363C2—21no—
364C2—22no—
365C2—21no—
366C2—22no—
367C2—42no—
368C2C67×#182411yescharacteristic
369C2C67×#182711yescharacteristic
370C2C67×#182811yescharacteristic
subgroupG/H
center, socleC22×C67#381
commutatorC22×C4C22×C4×C67
Frattini, cosocleC23×C4C23×C67
prime radicalC22×D4×C67D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C67 ◁ C23×C4×C67 ◁ G

Complements

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

NH ≅ G/Ntype
4. C67×#1294C2semidirect
5. C67×#1289C2semidirect
13. #382×C67C4semidirect
14. #380×C67C4semidirect
17. C4×C4⋊C4×C67C4semidirect
18. C2×#95×C67C4semidirect
177. #8761C67direct (N × H)
189. #1294C2×C67semidirect
190. #1289C2×C67semidirect
191. C67#8761direct (N × H)
199. #382C4×C67semidirect
200. #380C4×C67semidirect
203. C4×C4⋊C4C4×C67semidirect
204. C2×#95C4×C67semidirect

Maximal subgroups

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

subgroupindex
C67×#23492
C67×#17992
C67×#14592
C67×#12942
C67×#12892
#876167