Finite Group Databasebeta

← C67×#18997 #30698248

Subgroups — C67×#18997 #30698248

Order 17152.

1066 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 1066 conjugacy classes of subgroups — too many to draw. Only its 222 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×H(3,𝔽4)×C67C211yesmaximal, characteristic
2C67×#3215C212yesmaximal, complemented
3C67×#3140C211yesmaximal, characteristic, complemented
4C67×#1659C218yesmaximal, complemented
5C67×#1533C211yesmaximal, characteristic, complemented
6C67×#1527C212yesmaximal, complemented
7H(3,𝔽4)×C67—24no—
8H(3,𝔽4)×C67—22no—
9H(3,𝔽4)×C67C2214yes—
10#533×C67C2211yescharacteristic, complemented
11C2×#124×C67—21no—
12C2×#124×C67—22no—
13C2×#124×C67C2212yes—
14C2×#124×C67C2211yescharacteristic
15C2×#120×C67—21no—
16C2×#120×C67—22no—
17#497×C67C2214yescomplemented
18#489×C67C2214yescomplemented
19#489×C67C2214yescomplemented
20#488×C67—22no—
21#464×C67—24no—
22#451×C67—22no—
23#450×C67—22no—
24#432×C67C2212yescomplemented
25#419×C67—21no—
26#418×C67—21no—
27C4×M16×C67C2211yescharacteristic, complemented
28#329×C67C2218yescomplemented
29#329×C67C2214yescomplemented
30C25×C67—21no—
31C22×Q8×C67D411yescharacteristic
32C22×Q8×C67D412yes—
33C22×D4×C67—22no—
34C22×D4×C67—21no—
35C22×D4×C67D411yescharacteristic
36C22×D4×C67D412yes—
37C2×Q16×C67—24no—
38C2×SD16×C67—24no—
39C2×M16×C67—21no—
40C4⋊Q8×C67C2312yescomplemented
41#125×C67—21no—
42#124×C67—24no—
43#124×C67—28no—
44#124×C67—24no—
45#124×C67—28no—
46#124×C67—22no—
47#124×C67—22no—
48#124×C67—22no—
49#124×C67D414yes—
50#124×C67D412yes—
51#124×C67C2312yes—
52#124×C67C2314yes—
53C22⋊Q8×C67—22no—
54#120×C67—24no—
55#120×C67—28no—
56#120×C67—24no—
57#120×C67—28no—
58C4×Q8×C67—22no—
59#117×C67—21no—
60C2×C22⋊C4×C67—21no—
61C2×C22⋊C4×C67—22no—
62C2×C22⋊C4×C67—22no—
63C2×C22⋊C4×C67—22no—
64C2×C22⋊C4×C67—22no—
65C2×C42×C67—21no—
66C2×C42×C67C2311yescharacteristic
67C4⋊C8×C67—22no—
68#103×C67—24no—
69#103×C67—24no—
70#102×C67—24no—
71#102×C67—24no—
72C22⋊C8×C67—22no—
73#97×C67C2312yescomplemented
74#97×C67C2314yescomplemented
75C4×C8×C67—21no—
76C24×C67—24no—
77C24×C67—21no—
78C24×C67—22no—
79C24×C67—84no—
80C24×C67—82no—
81C2×Q8×C67—21no—
82C2×Q8×C67—22no—
83C2×Q8×C67—22no—
84C2×Q8×C67—42no—
85C2×Q8×C67—82no—
86C2×Q8×C67C2×D412yes—
87C2×Q8×C67C2×D414yes—
88C2×D4×C67—24no—
89C2×D4×C67—28no—
90C2×D4×C67—21no—
91C2×D4×C67—22no—
92C2×D4×C67—42no—
93C2×D4×C67—82no—
94C2×D4×C67—84no—
95C2×D4×C67—82no—
96C2×D4×C67C2×D412yes—
97C2×D4×C67C2×D414yes—
98C22×C4×C67—22no—
99C22×C4×C67—21no—
100C22×C4×C67C2×D411yescharacteristic
101C22×C4×C67C2×D412yes—
102Q16×C67—84no—
103SD16×C67—84no—
104M16×C67—42no—
105C2×C8×C67—22no—
106C2×C8×C67—24no—
107C4⋊C4×C67—22no—
108C4⋊C4×C67—22no—
109C4⋊C4×C67—42no—
110C22⋊C4×C67—28no—
111C22⋊C4×C67—24no—
112C22⋊C4×C67—24no—
113C22⋊C4×C67—24no—
114C22⋊C4×C67—24no—
115C22⋊C4×C67—24no—
116C22⋊C4×C67—24no—
117C22⋊C4×C67—24no—
118C22⋊C4×C67—41no—
119C42×C67—24no—
120C42×C67—41no—
121C42×C67C2×D412yes—
122C42×C67C2×D411yescharacteristic
123C42×C67C2411yescharacteristic
124C23×C67—22no—
125C23×C67—24no—
126C23×C67—44no—
127C23×C67—44no—
128C23×C67—44no—
129C23×C67—42no—
130C23×C67—44no—
131C23×C67—84no—
132C23×C67—88no—
133C23×C67—816no—
134C23×C67#12011yescharacteristic
135Q8×C67—44no—
136Q8×C67—42no—
137Q8×C67—44no—
138Q8×C67—81no—
139Q8×C67—82no—
140D4×C67—44no—
141D4×C67—42no—
142D4×C67—81no—
143D4×C67—82no—
144D4×C67—84no—
145D4×C67—88no—
146C2×C4×C67—22no—
147C2×C4×C67—22no—
148C2×C4×C67—24no—
149C2×C4×C67—42no—
150C2×C4×C67—44no—
151C2×C4×C67—42no—
152C2×C4×C67—42no—
153C2×C4×C67#12011yescharacteristic
154C2×C4×C67#12012yes—
155C2×C4×C67C22×D411yescharacteristic
156C2×C4×C67C22×D412yes—
157C8×C67—42no—
158C8×C67—44no—
159C22×C67—22no—
160C22×C67—42no—
161C22×C67—44no—
162C22×C67—44no—
163C22×C67—44no—
164C22×C67—44no—
165C22×C67—84no—
166C22×C67—88no—
167C22×C67—816no—
168C22×C67#45512yes—
169C22×C67C2×#12011yescharacteristic
170C4×C67—21no—
171C4×C67—21no—
172C4×C67—22no—
173C4×C67—22no—
174C4×C67—42no—
175C4×C67—44no—
176C4×C67—82no—
177#18997C6711yesmaximal, characteristic, direct product factor, Sylow
178C2×C67—22no—
179C2×C67—82no—
180C2×C67—84no—
181C2×C67C2×#45511yescharacteristic
182C2×C67#301912yes—
183C2×H(3,𝔽4)C2×C6711yescharacteristic
184#3215C2×C6712yescomplemented
185#3140C2×C6711yescharacteristic, complemented
186#1659C2×C6718yescomplemented
187#1533C2×C6711yescharacteristic, complemented
188#1527C2×C6712yescomplemented
189C67#1899711yescharacteristic, direct product factor, Sylow
190H(3,𝔽4)—24no—
191H(3,𝔽4)—22no—
192H(3,𝔽4)C22×C6714yes—
193#533C22×C6711yescharacteristic, complemented
194C2×#124—21no—
195C2×#124—22no—
196C2×#124C22×C6711yescharacteristic
197C2×#124C22×C6712yes—
198C2×#120—21no—
199C2×#120—22no—
200#497C22×C6714yescomplemented
201#489C22×C6714yescomplemented
202#489C22×C6714yescomplemented
203#488—22no—
204#464—24no—
205#451—22no—
206#450—22no—
207#432C22×C6712yescomplemented
208#419—21no—
209#418—21no—
210C4×M16C22×C6711yescharacteristic, complemented
211#329C22×C6718yescomplemented
212#329C22×C6714yescomplemented
213C25—21no—
214C22×Q8D4×C6711yescharacteristic
215C22×Q8D4×C6712yes—
216C22×D4—22no—
217C22×D4—21no—
218C22×D4D4×C6711yescharacteristic
219C22×D4D4×C6712yes—
220C2×Q16—24no—
221C2×SD16—24no—
222C2×M16—21no—
223C4⋊Q8C23×C6712yescomplemented
224#125—21no—
225#124—24no—
226#124—28no—
227#124—24no—
228#124—28no—
229#124—22no—
230#124—22no—
231#124—22no—
232#124D4×C6714yes—
233#124D4×C6712yes—
234#124C23×C6712yes—
235#124C23×C6714yes—
236C22⋊Q8—22no—
237#120—24no—
238#120—28no—
239#120—24no—
240#120—28no—
241C4×Q8—22no—
242#117—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C42—21no—
249C2×C42C23×C6711yescharacteristic
250C4⋊C8—22no—
251#103—24no—
252#103—24no—
253#102—24no—
254#102—24no—
255C22⋊C8—22no—
256#97C23×C6712yescomplemented
257#97C23×C6714yescomplemented
258C4×C8—21no—
259C24—24no—
260C24—21no—
261C24—22no—
262C24—84no—
263C24—82no—
264C2×Q8—21no—
265C2×Q8—22no—
266C2×Q8—22no—
267C2×Q8—42no—
268C2×Q8—82no—
269C2×Q8C2×D4×C6712yes—
270C2×Q8C2×D4×C6714yes—
271C2×D4—24no—
272C2×D4—28no—
273C2×D4—21no—
274C2×D4—22no—
275C2×D4—42no—
276C2×D4—82no—
277C2×D4—84no—
278C2×D4—82no—
279C2×D4C2×D4×C6712yes—
280C2×D4C2×D4×C6714yes—
281C22×C4—22no—
282C22×C4—21no—
283C22×C4C2×D4×C6711yescharacteristic
284C22×C4C2×D4×C6712yes—
285Q16—84no—
286SD16—84no—
287M16—42no—
288C2×C8—22no—
289C2×C8—24no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—42no—
293C22⋊C4—28no—
294C22⋊C4—24no—
295C22⋊C4—24no—
296C22⋊C4—24no—
297C22⋊C4—24no—
298C22⋊C4—24no—
299C22⋊C4—24no—
300C22⋊C4—24no—
301C22⋊C4—41no—
302C42—24no—
303C42—41no—
304C42C2×D4×C6711yescharacteristic
305C42C2×D4×C6712yes—
306C42C24×C6711yescharacteristic
307C23—22no—
308C23—24no—
309C23—44no—
310C23—44no—
311C23—44no—
312C23—42no—
313C23—44no—
314C23—84no—
315C23—88no—
316C23—816no—
317C23#120×C6711yescharacteristic
318Q8—44no—
319Q8—42no—
320Q8—44no—
321Q8—81no—
322Q8—82no—
323D4—44no—
324D4—42no—
325D4—81no—
326D4—82no—
327D4—84no—
328D4—88no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—24no—
332C2×C4—42no—
333C2×C4—44no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4#120×C6711yescharacteristic
337C2×C4#120×C6712yes—
338C2×C4C22×D4×C6711yescharacteristic
339C2×C4C22×D4×C6712yes—
340C8—42no—
341C8—44no—
342C22—22no—
343C22—42no—
344C22—44no—
345C22—44no—
346C22—44no—
347C22—44no—
348C22—84no—
349C22—88no—
350C22—816no—
351C22#455×C6712yes—
352C22C2×#120×C6711yescharacteristic
353C4—21no—
354C4—21no—
355C4—22no—
356C4—22no—
357C4—42no—
358C4—44no—
359C4—82no—
360C2—22no—
361C2—82no—
362C2—84no—
363C2C2×#455×C6711yescharacteristic
364C2C67×#301912yes—
subgroupG/H
center, socleC22×C67C2×#120
commutator, Frattini, cosocleC42C24×C67
prime radicalC2×H(3,𝔽4)×C67C2
LCSG ▷ 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. C67×#3215C2semidirect
3. C67×#3140C2semidirect
4. C67×#1659C2semidirect
5. C67×#1533C2semidirect
6. C67×#1527C2semidirect
10. #533×C67C22semidirect
17. #497×C67C22semidirect
18. #489×C67C22semidirect
19. #489×C67C22semidirect
24. #432×C67C22semidirect
27. C4×M16×C67C22semidirect
28. #329×C67C22semidirect
29. #329×C67C22semidirect
40. C4⋊Q8×C67C23semidirect
73. #97×C67C23semidirect
74. #97×C67C23semidirect
177. #18997C67direct (N × H)
184. #3215C2×C67semidirect
185. #3140C2×C67semidirect
186. #1659C2×C67semidirect
187. #1533C2×C67semidirect
188. #1527C2×C67semidirect
189. C67#18997direct (N × H)
193. #533C22×C67semidirect
200. #497C22×C67semidirect
201. #489C22×C67semidirect
202. #489C22×C67semidirect
207. #432C22×C67semidirect
210. C4×M16C22×C67semidirect
211. #329C22×C67semidirect
212. #329C22×C67semidirect
223. C4⋊Q8C23×C67semidirect
256. #97C23×C67semidirect
257. #97C23×C67semidirect

Maximal subgroups

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

subgroupindex
C2×H(3,𝔽4)×C672
C67×#32152
C67×#31402
C67×#16592
C67×#15332
C67×#15272
#1899767