Finite Group Databasebeta

← C67×#10148 #30689399

Subgroups — C67×#10148 #30689399

Order 17152.

614 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×#3227C211yesmaximal, characteristic, complemented
2C67×#3151C211yesmaximal, characteristic
3C2×#486×C67C211yesmaximal, characteristic, complemented
4C67×#1997C211yesmaximal, characteristic, complemented
5C67×#1545C211yesmaximal, characteristic, complemented
6C67×#1492C211yesmaximal, characteristic, complemented
7C67×#1278C211yesmaximal, characteristic, complemented
8C2×AGL(1,ℤ/8)×C67—22no—
9#532×C67C2211yescharacteristic
10C2×C4⋊Q8×C67C2211yescharacteristic
11C2×#127×C67C2211yescharacteristic
12#498×C67—21no—
13#496×C67—22no—
14#493×C67—21no—
15#492×C67—21no—
16#486×C67—22no—
17#486×C67—22no—
18#486×C67—21no—
19#481×C67—22no—
20#476×C67—22no—
21#431×C67C2211yescharacteristic
22#423×C67C2211yescharacteristic
23C2×#102×C67—22no—
24C4×M16×C67C2211yescharacteristic
25C2×C4×C8×C67C2211yescharacteristic
26#343×C67—22no—
27#331×C67—22no—
28D4⋊C8×C67—22no—
29C2×#41×C67—21no—
30C22×Q8×C67—21no—
31C22×D4×C67—21no—
32C22×D4×C67—21no—
33C22×D4×C67—21no—
34AGL(1,ℤ/8)×C67—44no—
35AGL(1,ℤ/8)×C67—44no—
36C2×SD16×C67—24no—
37C2×D8×C67—24no—
38C2×M16×C67—21no—
39C2×M16×C67D412yes—
40C22×C8×C67—21no—
41C4⋊Q8×C67—21no—
42C4⋊Q8×C67—21no—
43C4⋊Q8×C67—21no—
44C4⋊Q8×C67—21no—
45C4⋊Q8×C67D411yescharacteristic
46#127×C67—21no—
47#127×C67—21no—
48#127×C67—21no—
49#127×C67D411yescharacteristic
50#127×C67D412yes—
51#124×C67—21no—
52#121×C67—21no—
53#121×C67—21no—
54C4×D4×C67—21no—
55C4×D4×C67—21no—
56C2×C4⋊C4×C67—21no—
57C2×C4⋊C4×C67D412yes—
58C2×C42×C67C2311yescharacteristic
59#106×C67—24no—
60C4⋊C8×C67—21no—
61C4⋊C8×C67—21no—
62C4⋊C8×C67D412yes—
63#102×C67—24no—
64#102×C67—28no—
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—41no—
72#41×C67—42no—
73#41×C67—42no—
74C2×Q8×C67—21no—
75C2×Q8×C67—41no—
76C2×Q8×C67—41no—
77C2×Q8×C67—82no—
78C2×D4×C67—21no—
79C2×D4×C67—22no—
80C2×D4×C67—21no—
81C2×D4×C67—22no—
82C2×D4×C67—21no—
83C2×D4×C67—22no—
84C2×D4×C67—44no—
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—22no—
93C22×C4×C67—41no—
94C22×C4×C67C2×D411yescharacteristic
95C22×C4×C67C2×D411yescharacteristic
96C22×C4×C67#4111yescharacteristic
97SD16×C67—48no—
98D8×C67—48no—
99M16×C67—22no—
100M16×C67—42no—
101M16×C67SD1614yes—
102C2×C8×C67—22no—
103C2×C8×C67—22no—
104C2×C8×C67—22no—
105C2×C8×C67—42no—
106C2×C8×C67—41no—
107C4⋊C4×C67—22no—
108C4⋊C4×C67—22no—
109C4⋊C4×C67—21no—
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—82no—
122C23×C67—82no—
123C23×C67—82no—
124C23×C67#12111yescharacteristic
125Q8×C67—42no—
126Q8×C67—82no—
127Q8×C67—82no—
128D4×C67—44no—
129D4×C67—42no—
130D4×C67—44no—
131D4×C67—42no—
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—81no—
144C2×C4×C67—81no—
145C2×C4×C67—82no—
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×C67C2×SD1612yes—
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#46012yes—
169C4×C67—21no—
170C4×C67—21no—
171C4×C67—81no—
172C4×C67—82no—
173C4×C67#45012yes—
174C4×C67#45412yes—
175C4×C67#47212yes—
176C4×C67#49211yescharacteristic
177C4×C67#49611yescharacteristic
178#10148C6711yesmaximal, characteristic, direct product factor, Sylow
179C2×C67—22no—
180C2×C67—161no—
181C2×C67—162no—
182C2×C67#200311yescharacteristic
183C2×C67#201411yescharacteristic
184C2×C67#201811yescharacteristic
185#3227C2×C6711yescharacteristic, complemented
186#3151C2×C6711yescharacteristic
187C2×#486C2×C6711yescharacteristic, complemented
188#1997C2×C6711yescharacteristic, complemented
189#1545C2×C6711yescharacteristic, complemented
190#1492C2×C6711yescharacteristic, complemented
191#1278C2×C6711yescharacteristic, complemented
192C67#1014811yescharacteristic, direct product factor, Sylow
193C2×AGL(1,ℤ/8)—22no—
194#532C22×C6711yescharacteristic
195C2×C4⋊Q8C22×C6711yescharacteristic
196C2×#127C22×C6711yescharacteristic
197#498—21no—
198#496—22no—
199#493—21no—
200#492—21no—
201#486—22no—
202#486—21no—
203#486—22no—
204#481—22no—
205#476—22no—
206#431C22×C6711yescharacteristic
207#423C22×C6711yescharacteristic
208C2×#102—22no—
209C4×M16C22×C6711yescharacteristic
210C2×C4×C8C22×C6711yescharacteristic
211#343—22no—
212#331—22no—
213D4⋊C8—22no—
214C2×#41—21no—
215C22×Q8—21no—
216C22×D4—21no—
217C22×D4—21no—
218C22×D4—21no—
219AGL(1,ℤ/8)—44no—
220AGL(1,ℤ/8)—44no—
221C2×SD16—24no—
222C2×D8—24no—
223C2×M16—21no—
224C2×M16D4×C6712yes—
225C22×C8—21no—
226C4⋊Q8—21no—
227C4⋊Q8—21no—
228C4⋊Q8—21no—
229C4⋊Q8—21no—
230C4⋊Q8D4×C6711yescharacteristic
231#127—21no—
232#127—21no—
233#127—21no—
234#127D4×C6712yes—
235#127D4×C6711yescharacteristic
236#124—21no—
237#121—21no—
238#121—21no—
239C4×D4—21no—
240C4×D4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4D4×C6712yes—
243C2×C42C23×C6711yescharacteristic
244#106—24no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8D4×C6712yes—
248#102—28no—
249#102—24no—
250C22⋊C8—21no—
251#97—21no—
252C4×C8—21no—
253C4×C8—21no—
254C4×C8—21no—
255C4×C8D4×C6712yes—
256C24—41no—
257#41—42no—
258#41—42no—
259C2×Q8—21no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×Q8—82no—
263C2×D4—21no—
264C2×D4—22no—
265C2×D4—21no—
266C2×D4—22no—
267C2×D4—21no—
268C2×D4—22no—
269C2×D4—44no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—82no—
277C22×C4—22no—
278C22×C4—41no—
279C22×C4C2×D4×C6711yescharacteristic
280C22×C4C2×D4×C6711yescharacteristic
281C22×C4#41×C6711yescharacteristic
282SD16—48no—
283D8—48no—
284M16—22no—
285M16—42no—
286M16SD16×C6714yes—
287C2×C8—22no—
288C2×C8—22no—
289C2×C8—22no—
290C2×C8—42no—
291C2×C8—41no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—21no—
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—82no—
307C23—82no—
308C23—82no—
309C23#121×C6711yescharacteristic
310Q8—42no—
311Q8—82no—
312Q8—82no—
313D4—44no—
314D4—42no—
315D4—44no—
316D4—42no—
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—81no—
329C2×C4—81no—
330C2×C4—82no—
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×C4C2×SD16×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#460×C6712yes—
354C4—21no—
355C4—21no—
356C4—81no—
357C4—82no—
358C4#450×C6712yes—
359C4#454×C6712yes—
360C4#472×C6712yes—
361C4#492×C6711yescharacteristic
362C4#496×C6711yescharacteristic
363C2—22no—
364C2—162no—
365C2—161no—
366C2C67×#200311yescharacteristic
367C2C67×#201411yescharacteristic
368C2C67×#201811yescharacteristic
subgroupG/H
center, socleC22×C67#392
commutator, Frattini, cosocleC2×C42C23×C67
prime radicalC67×#3151C2
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×#3227C2semidirect
3. C2×#486×C67C2semidirect
4. C67×#1997C2semidirect
5. C67×#1545C2semidirect
6. C67×#1492C2semidirect
7. C67×#1278C2semidirect
178. #10148C67direct (N × H)
185. #3227C2×C67semidirect
187. C2×#486C2×C67semidirect
188. #1997C2×C67semidirect
189. #1545C2×C67semidirect
190. #1492C2×C67semidirect
191. #1278C2×C67semidirect
192. C67#10148direct (N × H)

Maximal subgroups

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

subgroupindex
C67×#32272
C67×#31512
C2×#486×C672
C67×#19972
C67×#15452
C67×#14922
C67×#12782
#1014867