Finite Group Databasebeta

← C67×#33023 #30712274

Subgroups — C67×#33023 #30712274

Order 17152.

1314 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3428C211yesmaximal, characteristic
2C67×#2484C212yesmaximal, complemented
3C67×#2406C211yesmaximal, characteristic
4C67×#1890C214yesmaximal, complemented
5C67×#1867C214yesmaximal, complemented
6C67×#1820C212yesmaximal, complemented
7C67×#1782C211yesmaximal, characteristic
8C22×#41×C67C2211yescharacteristic
9C2×#123×C67C2212yes—
10#520×C67—22no—
11#520×C67—22no—
12#519×C67—21no—
13#518×C67—21no—
14#518×C67C2214yes—
15C4×#41×C67—24no—
16C2×#117×C67—22no—
17C2×#117×C67C2212yes—
18C22×C4⋊C4×C67C2211yescharacteristic
19C2×#99×C67C2214yes—
20#400×C67—22no—
21#398×C67—22no—
22#395×C67—21no—
23#394×C67—21no—
24#387×C67—24no—
25#386×C67C2212yes—
26#385×C67—22no—
27#385×C67C2214yes—
28#381×C67C2214yes—
29#381×C67C2212yes—
30#381×C67C2211yescharacteristic
31#342×C67C2218yes—
32C2×#41×C67—28no—
33C2×#41×C67—22no—
34C2×#41×C67—22no—
35C22×Q8×C67D411yescharacteristic
36C22×D4×C67D411yescharacteristic
37C22×D4×C67C2312yes—
38C23×C4×C67D412yes—
39C23×C4×C67C2311yescharacteristic
40C23×C4×C67C2311yescharacteristic
41C23×C4×C67C2311yescharacteristic
42#123×C67—28no—
43C4×Q8×C67—24no—
44C4×Q8×C67—24no—
45C4×D4×C67—24no—
46C4×D4×C67—24no—
47C4×D4×C67—28no—
48C4×D4×C67—28no—
49#117×C67—24no—
50#117×C67—24no—
51#117×C67—22no—
52#117×C67—24no—
53#117×C67—22no—
54#117×C67—22no—
55#117×C67—22no—
56C2×C4⋊C4×C67—22no—
57C2×C4⋊C4×C67—22no—
58C2×C4⋊C4×C67—21no—
59C2×C4⋊C4×C67—22no—
60C2×C4⋊C4×C67—21no—
61C2×C4⋊C4×C67—22no—
62C2×C4⋊C4×C67—22no—
63C2×C4⋊C4×C67—24no—
64C2×C4⋊C4×C67D412yes—
65C2×C4⋊C4×C67D412yes—
66C2×C22⋊C4×C67—24no—
67C2×C22⋊C4×C67—22no—
68C2×C22⋊C4×C67—22no—
69C2×C22⋊C4×C67C2312yes—
70C2×C22⋊C4×C67C2314yes—
71C2×C22⋊C4×C67C2314yes—
72C2×C42×C67—24no—
73C2×C42×C67D414yes—
74#99×C67—28no—
75#95×C67—21no—
76#95×C67—22no—
77#95×C67—24no—
78#95×C67—24no—
79#95×C67—21no—
80#95×C67—22no—
81#95×C67—22no—
82#95×C67—24no—
83C24×C67C2×D412yes—
84C24×C67C2411yescharacteristic
85#41×C67—216no—
86#41×C67—216no—
87C2×Q8×C67—22no—
88C2×Q8×C67—24no—
89C2×D4×C67—24no—
90C2×D4×C67—24no—
91C2×D4×C67—24no—
92C2×D4×C67—22no—
93C2×D4×C67#4118yes—
94C22×C4×C67—24no—
95C22×C4×C67—24no—
96C22×C4×C67—22no—
97C22×C4×C67—22no—
98C22×C4×C67—24no—
99C22×C4×C67—21no—
100C22×C4×C67—22no—
101C22×C4×C67—24no—
102C22×C4×C67—24no—
103C22×C4×C67—22no—
104C22×C4×C67—21no—
105C22×C4×C67—24no—
106C22×C4×C67—22no—
107C22×C4×C67—24no—
108C22×C4×C67—42no—
109C22×C4×C67C2×D412yes—
110C22×C4×C67C2×D412yes—
111C22×C4×C67C2×D411yescharacteristic
112C22×C4×C67C2×D411yescharacteristic
113C22×C4×C67C2×D412yes—
114C22×C4×C67C2×D414yes—
115C22×C4×C67C2×D414yes—
116C4⋊C4×C67—24no—
117C4⋊C4×C67—24no—
118C4⋊C4×C67—22no—
119C4⋊C4×C67—22no—
120C4⋊C4×C67—22no—
121C4⋊C4×C67—22no—
122C4⋊C4×C67—48no—
123C4⋊C4×C67—41no—
124C4⋊C4×C67—42no—
125C4⋊C4×C67—41no—
126C4⋊C4×C67—44no—
127C22⋊C4×C67—28no—
128C22⋊C4×C67—24no—
129C22⋊C4×C67—24no—
130C22⋊C4×C67—24no—
131C22⋊C4×C67—28no—
132C22⋊C4×C67—44no—
133C22⋊C4×C67—44no—
134C42×C67—28no—
135C42×C67—24no—
136C42×C67—24no—
137C23×C67—22no—
138C23×C67—22no—
139C23×C67—41no—
140C23×C67—41no—
141C23×C67—42no—
142C23×C67—42no—
143C23×C67#12314yes—
144C23×C67C22×D411yescharacteristic
145C23×C67C22×D412yes—
146C23×C67C2×#4114yes—
147Q8×C67—28no—
148D4×C67—28no—
149D4×C67—216no—
150C2×C4×C67—28no—
151C2×C4×C67—28no—
152C2×C4×C67—22no—
153C2×C4×C67—22no—
154C2×C4×C67—22no—
155C2×C4×C67—22no—
156C2×C4×C67—24no—
157C2×C4×C67—24no—
158C2×C4×C67—22no—
159C2×C4×C67—24no—
160C2×C4×C67—24no—
161C2×C4×C67—24no—
162C2×C4×C67—42no—
163C2×C4×C67—41no—
164C2×C4×C67—41no—
165C2×C4×C67—44no—
166C2×C4×C67—44no—
167C2×C4×C67—42no—
168C2×C4×C67—44no—
169C2×C4×C67—44no—
170C2×C4×C67—42no—
171C2×C4×C67—41no—
172C2×C4×C67—42no—
173C2×C4×C67—42no—
174C2×C4×C67—44no—
175C2×C4×C67—44no—
176C2×C4×C67—84no—
177C2×C4×C67—84no—
178C2×C4×C67#12011yescharacteristic
179C2×C4×C67#12012yes—
180C2×C4×C67#12011yescharacteristic
181C2×C4×C67#12314yes—
182C22×C67—22no—
183C22×C67—21no—
184C22×C67—22no—
185C22×C67—21no—
186C22×C67—24no—
187C22×C67—44no—
188C22×C67—44no—
189C22×C67—42no—
190C22×C67—42no—
191C22×C67—82no—
192C22×C67C2×#12011yescharacteristic
193C22×C67C2×#12312yes—
194C22×C67#52514yes—
195C4×C67—21no—
196C4×C67—21no—
197C4×C67—22no—
198C4×C67—24no—
199C4×C67—42no—
200C4×C67—44no—
201C4×C67—44no—
202C4×C67—82no—
203C4×C67—84no—
204C4×C67—81no—
205#33023C6711yesmaximal, characteristic, direct product factor, Sylow
206C2×C67—22no—
207C2×C67—24no—
208C2×C67—44no—
209C2×C67#284911yescharacteristic
210C2×C67#302511yescharacteristic
211C2×C67#302811yescharacteristic
212#3428C2×C6711yescharacteristic
213#2484C2×C6712yescomplemented
214#2406C2×C6711yescharacteristic
215#1890C2×C6714yescomplemented
216#1867C2×C6714yescomplemented
217#1820C2×C6712yescomplemented
218#1782C2×C6711yescharacteristic
219C67#3302311yescharacteristic, direct product factor, Sylow
220C22×#41C22×C6711yescharacteristic
221C2×#123C22×C6712yes—
222#520—22no—
223#520—22no—
224#519—21no—
225#518—21no—
226#518C22×C6714yes—
227C4×#41—24no—
228C2×#117—22no—
229C2×#117C22×C6712yes—
230C22×C4⋊C4C22×C6711yescharacteristic
231C2×#99C22×C6714yes—
232#400—22no—
233#398—22no—
234#395—21no—
235#394—21no—
236#387—24no—
237#386C22×C6712yes—
238#385—22no—
239#385C22×C6714yes—
240#381C22×C6711yescharacteristic
241#381C22×C6712yes—
242#381C22×C6714yes—
243#342C22×C6718yes—
244C2×#41—28no—
245C2×#41—22no—
246C2×#41—22no—
247C22×Q8D4×C6711yescharacteristic
248C22×D4D4×C6711yescharacteristic
249C22×D4C23×C6712yes—
250C23×C4D4×C6712yes—
251C23×C4C23×C6711yescharacteristic
252C23×C4C23×C6711yescharacteristic
253C23×C4C23×C6711yescharacteristic
254#123—28no—
255C4×Q8—24no—
256C4×Q8—24no—
257C4×D4—24no—
258C4×D4—24no—
259C4×D4—28no—
260C4×D4—28no—
261#117—24no—
262#117—24no—
263#117—22no—
264#117—24no—
265#117—22no—
266#117—22no—
267#117—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4D4×C6712yes—
277C2×C4⋊C4D4×C6712yes—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4C23×C6712yes—
282C2×C22⋊C4C23×C6714yes—
283C2×C22⋊C4C23×C6714yes—
284C2×C42—24no—
285C2×C42D4×C6714yes—
286#99—28no—
287#95—21no—
288#95—22no—
289#95—24no—
290#95—24no—
291#95—21no—
292#95—22no—
293#95—22no—
294#95—24no—
295C24C2×D4×C6712yes—
296C24C24×C6711yescharacteristic
297#41—216no—
298#41—216no—
299C2×Q8—22no—
300C2×Q8—24no—
301C2×D4—24no—
302C2×D4—24no—
303C2×D4—24no—
304C2×D4—22no—
305C2×D4#41×C6718yes—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—21no—
319C22×C4—24no—
320C22×C4—42no—
321C22×C4C2×D4×C6712yes—
322C22×C4C2×D4×C6712yes—
323C22×C4C2×D4×C6711yescharacteristic
324C22×C4C2×D4×C6711yescharacteristic
325C22×C4C2×D4×C6712yes—
326C22×C4C2×D4×C6714yes—
327C22×C4C2×D4×C6714yes—
328C4⋊C4—24no—
329C4⋊C4—24no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—48no—
335C4⋊C4—41no—
336C4⋊C4—42no—
337C4⋊C4—41no—
338C4⋊C4—44no—
339C22⋊C4—28no—
340C22⋊C4—24no—
341C22⋊C4—24no—
342C22⋊C4—28no—
343C22⋊C4—24no—
344C22⋊C4—44no—
345C22⋊C4—44no—
346C42—28no—
347C42—24no—
348C42—24no—
349C23—22no—
350C23—22no—
351C23—41no—
352C23—41no—
353C23—42no—
354C23—42no—
355C23#123×C6714yes—
356C23C22×D4×C6711yescharacteristic
357C23C22×D4×C6712yes—
358C23C2×#41×C6714yes—
359Q8—28no—
360D4—28no—
361D4—216no—
362C2×C4—28no—
363C2×C4—28no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—22no—
371C2×C4—24no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—42no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—42no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—42no—
383C2×C4—41no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—84no—
389C2×C4—84no—
390C2×C4#120×C6711yescharacteristic
391C2×C4#120×C6712yes—
392C2×C4#120×C6711yescharacteristic
393C2×C4#123×C6714yes—
394C22—22no—
395C22—21no—
396C22—22no—
397C22—21no—
398C22—24no—
399C22—44no—
400C22—44no—
401C22—42no—
402C22—42no—
403C22—82no—
404C22C2×#120×C6711yescharacteristic
405C22C2×#123×C6712yes—
406C22#525×C6714yes—
407C4—21no—
408C4—21no—
409C4—22no—
410C4—24no—
411C4—42no—
412C4—44no—
413C4—44no—
414C4—82no—
415C4—84no—
416C4—81no—
417C2—22no—
418C2—24no—
419C2—44no—
420C2C67×#284911yescharacteristic
421C2C67×#302511yescharacteristic
422C2C67×#302811yescharacteristic
subgroupG/H
center, socleC22×C67C2×#120
commutator, Frattini, cosocleC24C24×C67
prime radicalC22×#41×C67C22
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C67 ◁ C23×C4×C67 ◁ G

Complements

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

NH ≅ G/Ntype
2. C67×#2484C2semidirect
4. C67×#1890C2semidirect
5. C67×#1867C2semidirect
6. C67×#1820C2semidirect
205. #33023C67direct (N × H)
213. #2484C2×C67semidirect
215. #1890C2×C67semidirect
216. #1867C2×C67semidirect
217. #1820C2×C67semidirect
219. C67#33023direct (N × H)

Maximal subgroups

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

subgroupindex
C67×#34282
C67×#24842
C67×#24062
C67×#18902
C67×#18672
C67×#18202
C67×#17822
#3302367