Finite Group Databasebeta

← C67×#36700 #30715951

Subgroups — C67×#36700 #30715951

Order 17152.

1356 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#2839C214yesmaximal, complemented
2C67×#2836C214yesmaximal
3C67×#2833C214yesmaximal, complemented
4C67×#2831C214yesmaximal
5C67×#2402C214yesmaximal, complemented
6C67×#2402C211yesmaximal, characteristic, complemented
7C67×#2401C214yesmaximal
8C67×#2399C211yesmaximal, characteristic, complemented
9C67×#2396C212yesmaximal
10C67×#2298C211yesmaximal, characteristic
11C67×#2295C212yesmaximal, complemented
12C2×C4×Q8×C67C2212yes—
13C2×#117×C67C2211yescharacteristic
14#401×C67C2214yes—
15#400×C67C2214yes—
16#400×C67C2214yes—
17#400×C67C2214yes—
18#400×C67C2214yes—
19#400×C67C2214yes—
20#400×C67C2214yes—
21#400×C67C2214yes—
22#400×C67C2212yes—
23#400×C67C2214yes—
24#399×C67C2212yes—
25#398×C67C2214yes—
26#398×C67C2214yes—
27#398×C67C2214yes—
28#398×C67C2214yes—
29#398×C67C2214yes—
30#398×C67C2214yes—
31#398×C67C2214yes—
32#397×C67C2214yes—
33#397×C67C2214yes—
34#397×C67C2214yes—
35#396×C67C2214yes—
36#396×C67C2214yes—
37#395×C67C2214yes—
38#395×C67C2212yes—
39#393×C67C2212yes—
40#391×C67C2214yes—
41#389×C67C2214yes—
42#388×C67C2214yes—
43#387×C67C2214yes—
44#387×C67C2214yes—
45#387×C67C2214yes—
46C4×C4⋊C4×C67C2212yes—
47C4×C4⋊C4×C67C2214yes—
48C4×C4⋊C4×C67C2212yes—
49C4×C4⋊C4×C67C2214yes—
50C4×C4⋊C4×C67C2214yes—
51C4×C4⋊C4×C67C2214yes—
52C4×C22⋊C4×C67C2212yes—
53C4×C22⋊C4×C67C2214yes—
54#376×C67C2214yes—
55#376×C67C2211yescharacteristic
56C43×C67C2211yescharacteristic
57C22×Q8×C67C2312yes—
58C23×C4×C67C2311yescharacteristic
59C4×Q8×C67—28no—
60#117×C67—24no—
61C2×C4⋊C4×C67C2312yes—
62C2×C4⋊C4×C67C2314yes—
63C2×C4⋊C4×C67C2312yes—
64C2×C4⋊C4×C67C2314yes—
65C2×C4⋊C4×C67C2314yes—
66C2×C4⋊C4×C67C2314yes—
67C2×C4⋊C4×C67C2314yes—
68C2×C4⋊C4×C67C2314yes—
69C2×C4⋊C4×C67C2314yes—
70C2×C4⋊C4×C67C2314yes—
71C2×C4⋊C4×C67C2314yes—
72C2×C4⋊C4×C67C2314yes—
73C2×C4⋊C4×C67C2314yes—
74C2×C22⋊C4×C67C2312yes—
75C2×C22⋊C4×C67C2314yes—
76C2×C22⋊C4×C67C2314yes—
77C2×C22⋊C4×C67C2314yes—
78C2×C42×C67C2311yescharacteristic
79C2×C42×C67C2314yes—
80C2×C42×C67C2312yes—
81C2×C42×C67C2314yes—
82C2×C42×C67C2312yes—
83C2×C42×C67C2314yes—
84C2×C42×C67C2311yescharacteristic
85#95×C67C2314yes—
86#95×C67C2314yes—
87#95×C67C2314yes—
88#95×C67C2314yes—
89#95×C67C2314yes—
90#95×C67C2314yes—
91#95×C67C2314yes—
92#95×C67C2314yes—
93#95×C67C2312yes—
94#95×C67C2312yes—
95#95×C67C2314yes—
96#95×C67C2314yes—
97#95×C67C2314yes—
98#95×C67C2314yes—
99#95×C67C2314yes—
100#95×C67C2314yes—
101#95×C67C2314yes—
102#95×C67C2314yes—
103#95×C67C2314yes—
104C24×C67C2411yescharacteristic
105C2×Q8×C67—24no—
106C2×Q8×C67—44no—
107C22×C4×C67—22no—
108C22×C4×C67—24no—
109C22×C4×C67C2414yes—
110C22×C4×C67C2412yes—
111C22×C4×C67C2414yes—
112C22×C4×C67C2414yes—
113C22×C4×C67C2414yes—
114C22×C4×C67C2414yes—
115C22×C4×C67C2414yes—
116C22×C4×C67C2411yescharacteristic
117C22×C4×C67C2412yes—
118C22×C4×C67C2411yescharacteristic
119C4⋊C4×C67—24no—
120C4⋊C4×C67—24no—
121C4⋊C4×C67—28no—
122C4⋊C4×C67—44no—
123C4⋊C4×C67—44no—
124C4⋊C4×C67—44no—
125C4⋊C4×C67—44no—
126C4⋊C4×C67—44no—
127C4⋊C4×C67—44no—
128C4⋊C4×C67—44no—
129C4⋊C4×C67—44no—
130C4⋊C4×C67—44no—
131C4⋊C4×C67—44no—
132C22⋊C4×C67—24no—
133C22⋊C4×C67—44no—
134C22⋊C4×C67—44no—
135C22⋊C4×C67—44no—
136C42×C67—22no—
137C42×C67—28no—
138C42×C67—24no—
139C42×C67—28no—
140C42×C67—24no—
141C42×C67—44no—
142C42×C67#4114yes—
143C23×C67—21no—
144C23×C67—22no—
145C23×C67—24no—
146C23×C67C2511yescharacteristic
147Q8×C67—48no—
148C2×C4×C67—24no—
149C2×C4×C67—22no—
150C2×C4×C67—24no—
151C2×C4×C67—24no—
152C2×C4×C67—21no—
153C2×C4×C67—24no—
154C2×C4×C67—24no—
155C2×C4×C67—24no—
156C2×C4×C67—24no—
157C2×C4×C67—22no—
158C2×C4×C67—24no—
159C2×C4×C67—24no—
160C2×C4×C67—24no—
161C2×C4×C67—24no—
162C2×C4×C67—24no—
163C2×C4×C67—24no—
164C2×C4×C67—24no—
165C2×C4×C67—24no—
166C2×C4×C67—24no—
167C2×C4×C67—24no—
168C2×C4×C67—24no—
169C2×C4×C67—24no—
170C2×C4×C67—24no—
171C2×C4×C67—44no—
172C2×C4×C67C2×#4114yes—
173C2×C4×C67C2×#4112yes—
174C2×C4×C672−1+414yes—
175C22×C67—41no—
176C22×C67—42no—
177C22×C67—44no—
178C22×C67C22×#4111yescharacteristic
179C22×C67C2×2−1+412yes—
180C22×C67#58514yes—
181C4×C67—24no—
182C4×C67—41no—
183C4×C67—42no—
184C4×C67—44no—
185C4×C67—44no—
186C4×C67—44no—
187C4×C67—44no—
188C4×C67—44no—
189C4×C67—44no—
190C4×C67#54114yes—
191#36700C6711yesmaximal, characteristic, direct product factor, Sylow
192C2×C67—81no—
193C2×C67C2×#54111yescharacteristic
194C2×C67#345512yes—
195C2×C67#353114yes—
196#2839C2×C6714yescomplemented
197#2836C2×C6714yes—
198#2833C2×C6714yescomplemented
199#2831C2×C6714yes—
200#2402C2×C6711yescharacteristic, complemented
201#2402C2×C6714yescomplemented
202#2401C2×C6714yes—
203#2399C2×C6711yescharacteristic, complemented
204#2396C2×C6712yes—
205#2298C2×C6711yescharacteristic
206#2295C2×C6712yescomplemented
207C67#3670011yescharacteristic, direct product factor, Sylow
208C2×C4×Q8C22×C6712yes—
209C2×#117C22×C6711yescharacteristic
210#401C22×C6714yes—
211#400C22×C6714yes—
212#400C22×C6714yes—
213#400C22×C6714yes—
214#400C22×C6714yes—
215#400C22×C6714yes—
216#400C22×C6714yes—
217#400C22×C6714yes—
218#400C22×C6712yes—
219#400C22×C6714yes—
220#399C22×C6712yes—
221#398C22×C6714yes—
222#398C22×C6714yes—
223#398C22×C6714yes—
224#398C22×C6714yes—
225#398C22×C6714yes—
226#398C22×C6714yes—
227#398C22×C6714yes—
228#397C22×C6714yes—
229#397C22×C6714yes—
230#397C22×C6714yes—
231#396C22×C6714yes—
232#396C22×C6714yes—
233#395C22×C6714yes—
234#395C22×C6712yes—
235#393C22×C6712yes—
236#391C22×C6714yes—
237#389C22×C6714yes—
238#388C22×C6714yes—
239#387C22×C6714yes—
240#387C22×C6714yes—
241#387C22×C6714yes—
242C4×C4⋊C4C22×C6714yes—
243C4×C4⋊C4C22×C6712yes—
244C4×C4⋊C4C22×C6714yes—
245C4×C4⋊C4C22×C6714yes—
246C4×C4⋊C4C22×C6714yes—
247C4×C4⋊C4C22×C6712yes—
248C4×C22⋊C4C22×C6714yes—
249C4×C22⋊C4C22×C6712yes—
250#376C22×C6711yescharacteristic
251#376C22×C6714yes—
252C43C22×C6711yescharacteristic
253C22×Q8C23×C6712yes—
254C23×C4C23×C6711yescharacteristic
255C4×Q8—28no—
256#117—24no—
257C2×C4⋊C4C23×C6712yes—
258C2×C4⋊C4C23×C6714yes—
259C2×C4⋊C4C23×C6712yes—
260C2×C4⋊C4C23×C6714yes—
261C2×C4⋊C4C23×C6714yes—
262C2×C4⋊C4C23×C6714yes—
263C2×C4⋊C4C23×C6714yes—
264C2×C4⋊C4C23×C6714yes—
265C2×C4⋊C4C23×C6714yes—
266C2×C4⋊C4C23×C6714yes—
267C2×C4⋊C4C23×C6714yes—
268C2×C4⋊C4C23×C6714yes—
269C2×C4⋊C4C23×C6714yes—
270C2×C22⋊C4C23×C6712yes—
271C2×C22⋊C4C23×C6714yes—
272C2×C22⋊C4C23×C6714yes—
273C2×C22⋊C4C23×C6714yes—
274C2×C42C23×C6711yescharacteristic
275C2×C42C23×C6714yes—
276C2×C42C23×C6712yes—
277C2×C42C23×C6712yes—
278C2×C42C23×C6714yes—
279C2×C42C23×C6714yes—
280C2×C42C23×C6711yescharacteristic
281#95C23×C6714yes—
282#95C23×C6714yes—
283#95C23×C6714yes—
284#95C23×C6714yes—
285#95C23×C6714yes—
286#95C23×C6714yes—
287#95C23×C6714yes—
288#95C23×C6714yes—
289#95C23×C6712yes—
290#95C23×C6712yes—
291#95C23×C6714yes—
292#95C23×C6714yes—
293#95C23×C6714yes—
294#95C23×C6714yes—
295#95C23×C6714yes—
296#95C23×C6714yes—
297#95C23×C6714yes—
298#95C23×C6714yes—
299#95C23×C6714yes—
300C24C24×C6711yescharacteristic
301C2×Q8—24no—
302C2×Q8—44no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4C24×C6711yescharacteristic
306C22×C4C24×C6714yes—
307C22×C4C24×C6714yes—
308C22×C4C24×C6714yes—
309C22×C4C24×C6714yes—
310C22×C4C24×C6714yes—
311C22×C4C24×C6712yes—
312C22×C4C24×C6714yes—
313C22×C4C24×C6712yes—
314C22×C4C24×C6711yescharacteristic
315C4⋊C4—24no—
316C4⋊C4—24no—
317C4⋊C4—28no—
318C4⋊C4—44no—
319C4⋊C4—44no—
320C4⋊C4—44no—
321C4⋊C4—44no—
322C4⋊C4—44no—
323C4⋊C4—44no—
324C4⋊C4—44no—
325C4⋊C4—44no—
326C4⋊C4—44no—
327C4⋊C4—44no—
328C22⋊C4—24no—
329C22⋊C4—44no—
330C22⋊C4—44no—
331C22⋊C4—44no—
332C42—22no—
333C42—28no—
334C42—24no—
335C42—24no—
336C42—28no—
337C42—44no—
338C42#41×C6714yes—
339C23—21no—
340C23—22no—
341C23—24no—
342C23C25×C6711yescharacteristic
343Q8—48no—
344C2×C4—24no—
345C2×C4—22no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—21no—
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—22no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—24no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—24no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—24no—
365C2×C4—24no—
366C2×C4—24no—
367C2×C4—44no—
368C2×C4C2×#41×C6714yes—
369C2×C4C2×#41×C6712yes—
370C2×C42−1+4×C6714yes—
371C22—44no—
372C22—41no—
373C22—42no—
374C22C22×#41×C6711yescharacteristic
375C22C2×2−1+4×C6712yes—
376C22#585×C6714yes—
377C4—24no—
378C4—44no—
379C4—44no—
380C4—44no—
381C4—44no—
382C4—41no—
383C4—44no—
384C4—42no—
385C4—44no—
386C4#541×C6714yes—
387C2—81no—
388C2C2×#541×C6711yescharacteristic
389C2C67×#345512yes—
390C2C67×#353114yes—
subgroupG/H
centerC22×C4×C67C24
commutator, Frattini, cosocleC23C25×C67
socleC23×C67C25
prime radicalC24×C67C24
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4×C67 ◁ G

Complements

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

NH ≅ G/Ntype
1. C67×#2839C2semidirect
3. C67×#2833C2semidirect
5. C67×#2402C2semidirect
6. C67×#2402C2semidirect
8. C67×#2399C2semidirect
11. C67×#2295C2semidirect
191. #36700C67direct (N × H)
196. #2839C2×C67semidirect
198. #2833C2×C67semidirect
200. #2402C2×C67semidirect
201. #2402C2×C67semidirect
203. #2399C2×C67semidirect
206. #2295C2×C67semidirect
207. C67#36700direct (N × H)

Maximal subgroups

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

subgroupindex
C67×#28392
C67×#28362
C67×#28332
C67×#28312
C67×#24022
C67×#24012
C67×#23992
C67×#23962
C67×#22982
C67×#22952
#3670067