Finite Group Databasebeta

← C67×#10152 #30689403

Subgroups — C67×#10152 #30689403

Order 17152.

642 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
1C67×#3205C211yesmaximal, characteristic
2C2×#492×C67C211yesmaximal, characteristic, complemented
3C67×#3141C211yesmaximal, characteristic, complemented
4C67×#1965C211yesmaximal, characteristic, complemented
5C67×#1549C211yesmaximal, characteristic, complemented
6C67×#1492C211yesmaximal, characteristic, complemented
7C67×#1278C211yesmaximal, characteristic, complemented
8C22×SD16×C67—22no—
9#532×C67C2211yescharacteristic
10C2×C4⋊Q8×C67C2211yescharacteristic
11C2×#127×C67C2211yescharacteristic
12#493×C67—21no—
13#492×C67—22no—
14#492×C67—22no—
15#489×C67—22no—
16#487×C67—21no—
17#486×C67—21no—
18#486×C67—21no—
19#459×C67—22no—
20#447×C67—22no—
21#431×C67C2211yescharacteristic
22#423×C67C2211yescharacteristic
23#417×C67—22no—
24C2×#100×C67—22no—
25C4×M16×C67C2211yescharacteristic
26C2×C4×C8×C67C2211yescharacteristic
27#332×C67—22no—
28D4⋊C8×C67—22no—
29C2×#41×C67—21no—
30C22×Q8×C67—21no—
31C22×D4×C67—21no—
32C22×D4×C67—21no—
33C22×D4×C67D412yes—
34C2×SD16×C67—28no—
35C2×SD16×C67—48no—
36C2×D8×C67—24no—
37C2×M16×C67—21no—
38C2×M16×C67D412yes—
39C22×C8×C67—21no—
40C4⋊Q8×C67—21no—
41C4⋊Q8×C67—21no—
42C4⋊Q8×C67—21no—
43C4⋊Q8×C67D411yescharacteristic
44C4⋊Q8×C67D412yes—
45#127×C67—21no—
46#127×C67—21no—
47#127×C67—21no—
48#127×C67—21no—
49#127×C67D411yescharacteristic
50#124×C67—21no—
51#121×C67—21no—
52#121×C67—21no—
53C4×D4×C67—21no—
54C4×D4×C67—21no—
55C2×C4⋊C4×C67—21no—
56C2×C4⋊C4×C67—21no—
57C2×C42×C67C2311yescharacteristic
58C4⋊C8×C67—21no—
59C4⋊C8×C67—21no—
60C4⋊C8×C67D412yes—
61#103×C67—24no—
62#102×C67—24no—
63#102×C67—24no—
64#100×C67—44no—
65C22⋊C8×C67—21no—
66#97×C67—21no—
67C4×C8×C67—21no—
68C4×C8×C67—21no—
69C4×C8×C67—21no—
70C4×C8×C67D412yes—
71C24×C67—22no—
72#41×C67—42no—
73#41×C67—42no—
74C2×Q8×C67—21no—
75C2×Q8×C67—22no—
76C2×Q8×C67—44no—
77C2×Q8×C67—41no—
78C2×D4×C67—21no—
79C2×D4×C67—22no—
80C2×D4×C67—21no—
81C2×D4×C67—22no—
82C2×D4×C67—21no—
83C2×D4×C67—44no—
84C2×D4×C67—41no—
85C2×D4×C67—41no—
86C2×D4×C67—41no—
87C2×D4×C67—41no—
88C2×D4×C67—41no—
89C2×D4×C67—42no—
90C2×D4×C67—42no—
91C2×D4×C67—82no—
92C22×C4×C67—41no—
93C22×C4×C67—41no—
94C22×C4×C67C2×D411yescharacteristic
95C22×C4×C67C2×D411yescharacteristic
96C22×C4×C67#4111yescharacteristic
97SD16×C67—416no—
98D8×C67—84no—
99M16×C67—22no—
100M16×C67—22no—
101M16×C67—42no—
102C2×C8×C67—24no—
103C2×C8×C67—22no—
104C2×C8×C67—22no—
105C2×C8×C67—22no—
106C2×C8×C67—41no—
107C4⋊C4×C67—21no—
108C4⋊C4×C67—22no—
109C4⋊C4×C67—22no—
110C4⋊C4×C67—21no—
111C4⋊C4×C67—41no—
112C4⋊C4×C67—41no—
113C22⋊C4×C67—41no—
114C22⋊C4×C67—41no—
115C42×C67C2×D411yescharacteristic
116C42×C67C2×D411yescharacteristic
117C42×C67C2×D411yescharacteristic
118C42×C67C2×D411yescharacteristic
119C23×C67—41no—
120C23×C67—42no—
121C23×C67—44no—
122C23×C67—82no—
123C23×C67—82no—
124C23×C67#12111yescharacteristic
125Q8×C67—44no—
126Q8×C67—42no—
127Q8×C67—82no—
128D4×C67—44no—
129D4×C67—42no—
130D4×C67—42no—
131D4×C67—44no—
132D4×C67—42no—
133D4×C67—82no—
134D4×C67—82no—
135D4×C67—82no—
136D4×C67—82no—
137D4×C67—81no—
138D4×C67—81no—
139C2×C4×C67—22no—
140C2×C4×C67—22no—
141C2×C4×C67—41no—
142C2×C4×C67—42no—
143C2×C4×C67—82no—
144C2×C4×C67—81no—
145C2×C4×C67—81no—
146C2×C4×C67#12011yescharacteristic
147C2×C4×C67#12011yescharacteristic
148C2×C4×C67#12011yescharacteristic
149C2×C4×C67#12111yescharacteristic
150C2×C4×C67#12111yescharacteristic
151C2×C4×C67#12711yescharacteristic
152C2×C4×C67#13512yes—
153C2×C4×C67AGL(1,ℤ/8)12yes—
154C8×C67—24no—
155C8×C67—42no—
156C8×C67—42no—
157C8×C67—81no—
158C22×C67—21no—
159C22×C67—21no—
160C22×C67—82no—
161C22×C67—81no—
162C22×C67—81no—
163C22×C67—82no—
164C22×C67—81no—
165C22×C67—82no—
166C22×C67—164no—
167C22×C67#39211yescharacteristic
168C22×C67#46312yes—
169C4×C67—21no—
170C4×C67—21no—
171C4×C67—82no—
172C4×C67—81no—
173C4×C67#44912yes—
174C4×C67#45312yes—
175C4×C67#47112yes—
176C4×C67#49511yescharacteristic
177C4×C67#49611yescharacteristic
178#10152C6711yesmaximal, characteristic, direct product factor, Sylow
179C2×C67—22no—
180C2×C67—161no—
181C2×C67—162no—
182C2×C67#200211yescharacteristic
183C2×C67#201211yescharacteristic
184C2×C67#201911yescharacteristic
185#3205C2×C6711yescharacteristic
186C2×#492C2×C6711yescharacteristic, complemented
187#3141C2×C6711yescharacteristic, complemented
188#1965C2×C6711yescharacteristic, complemented
189#1549C2×C6711yescharacteristic, complemented
190#1492C2×C6711yescharacteristic, complemented
191#1278C2×C6711yescharacteristic, complemented
192C67#1015211yescharacteristic, direct product factor, Sylow
193C22×SD16—22no—
194#532C22×C6711yescharacteristic
195C2×C4⋊Q8C22×C6711yescharacteristic
196C2×#127C22×C6711yescharacteristic
197#493—21no—
198#492—22no—
199#492—22no—
200#489—22no—
201#487—21no—
202#486—21no—
203#486—21no—
204#459—22no—
205#447—22no—
206#431C22×C6711yescharacteristic
207#423C22×C6711yescharacteristic
208#417—22no—
209C2×#100—22no—
210C4×M16C22×C6711yescharacteristic
211C2×C4×C8C22×C6711yescharacteristic
212#332—22no—
213D4⋊C8—22no—
214C2×#41—21no—
215C22×Q8—21no—
216C22×D4—21no—
217C22×D4—21no—
218C22×D4D4×C6712yes—
219C2×SD16—28no—
220C2×SD16—48no—
221C2×D8—24no—
222C2×M16—21no—
223C2×M16D4×C6712yes—
224C22×C8—21no—
225C4⋊Q8—21no—
226C4⋊Q8—21no—
227C4⋊Q8—21no—
228C4⋊Q8D4×C6712yes—
229C4⋊Q8D4×C6711yescharacteristic
230#127—21no—
231#127—21no—
232#127—21no—
233#127—21no—
234#127D4×C6711yescharacteristic
235#124—21no—
236#121—21no—
237#121—21no—
238C4×D4—21no—
239C4×D4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C42C23×C6711yescharacteristic
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8D4×C6712yes—
246#103—24no—
247#102—24no—
248#102—24no—
249#100—44no—
250C22⋊C8—21no—
251#97—21no—
252C4×C8—21no—
253C4×C8—21no—
254C4×C8—21no—
255C4×C8D4×C6712yes—
256C24—22no—
257#41—42no—
258#41—42no—
259C2×Q8—21no—
260C2×Q8—22no—
261C2×Q8—44no—
262C2×Q8—41no—
263C2×D4—21no—
264C2×D4—22no—
265C2×D4—21no—
266C2×D4—22no—
267C2×D4—21no—
268C2×D4—44no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—82no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4C2×D4×C6711yescharacteristic
280C22×C4C2×D4×C6711yescharacteristic
281C22×C4#41×C6711yescharacteristic
282SD16—416no—
283D8—84no—
284M16—22no—
285M16—22no—
286M16—42no—
287C2×C8—24no—
288C2×C8—22no—
289C2×C8—22no—
290C2×C8—22no—
291C2×C8—41no—
292C4⋊C4—21no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—21no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C22⋊C4—41no—
299C22⋊C4—41no—
300C42C2×D4×C6711yescharacteristic
301C42C2×D4×C6711yescharacteristic
302C42C2×D4×C6711yescharacteristic
303C42C2×D4×C6711yescharacteristic
304C23—41no—
305C23—42no—
306C23—44no—
307C23—82no—
308C23—82no—
309C23#121×C6711yescharacteristic
310Q8—44no—
311Q8—42no—
312Q8—82no—
313D4—44no—
314D4—42no—
315D4—42no—
316D4—44no—
317D4—42no—
318D4—82no—
319D4—82no—
320D4—82no—
321D4—82no—
322D4—81no—
323D4—81no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—41no—
327C2×C4—42no—
328C2×C4—82no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4#120×C6711yescharacteristic
332C2×C4#120×C6711yescharacteristic
333C2×C4#120×C6711yescharacteristic
334C2×C4#121×C6711yescharacteristic
335C2×C4#121×C6711yescharacteristic
336C2×C4#127×C6711yescharacteristic
337C2×C4#135×C6712yes—
338C2×C4AGL(1,ℤ/8)×C6712yes—
339C8—24no—
340C8—42no—
341C8—42no—
342C8—81no—
343C22—21no—
344C22—21no—
345C22—82no—
346C22—81no—
347C22—81no—
348C22—82no—
349C22—81no—
350C22—82no—
351C22—164no—
352C22#392×C6711yescharacteristic
353C22#463×C6712yes—
354C4—21no—
355C4—21no—
356C4—82no—
357C4—81no—
358C4#449×C6712yes—
359C4#453×C6712yes—
360C4#471×C6712yes—
361C4#495×C6711yescharacteristic
362C4#496×C6711yescharacteristic
363C2—22no—
364C2—162no—
365C2—161no—
366C2C67×#200211yescharacteristic
367C2C67×#201211yescharacteristic
368C2C67×#201911yescharacteristic
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
2. C2×#492×C67C2semidirect
3. C67×#3141C2semidirect
4. C67×#1965C2semidirect
5. C67×#1549C2semidirect
6. C67×#1492C2semidirect
7. C67×#1278C2semidirect
178. #10152C67direct (N × H)
186. C2×#492C2×C67semidirect
187. #3141C2×C67semidirect
188. #1965C2×C67semidirect
189. #1549C2×C67semidirect
190. #1492C2×C67semidirect
191. #1278C2×C67semidirect
192. C67#10152direct (N × H)

Maximal subgroups

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

subgroupindex
C67×#32052
C2×#492×C672
C67×#31412
C67×#19652
C67×#15492
C67×#14922
C67×#12782
#1015267