Finite Group Databasebeta

← C67×#33703 #30712954

Subgroups — C67×#33703 #30712954

Order 17152.

570 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3164C211yesmaximal, characteristic, complemented
2C67×#3156C211yesmaximal, characteristic
3C67×#2245C212yesmaximal, complemented
4C67×#2243C212yesmaximal, complemented
5C67×#2242C212yesmaximal, complemented
6C67×#2241C212yesmaximal, complemented
7C67×#2141C211yesmaximal, characteristic, complemented
8C67×#2139C211yesmaximal, characteristic
9C67×#2138C211yesmaximal, characteristic
10C2×#358×C67C211yesmaximal, characteristic, complemented
11C67×#2108C211yesmaximal, characteristic, complemented
12C2×#137×C67—21no—
13C2×#135×C67C2211yescharacteristic
14C22×Q16×C67C2211yescharacteristic
15#544×C67—21no—
16#536×C67—21no—
17C2×M32×C67C2211yescharacteristic
18C22×C16×C67C2211yescharacteristic
19#501×C67—22no—
20#500×C67C2212yes—
21#499×C67C2212yes—
22#497×C67—22no—
23#495×C67C2212yes—
24#494×C67C2212yes—
25#428×C67—21no—
26#427×C67C2211yescharacteristic
27C2×#107×C67C2211yescharacteristic
28#405×C67C2211yescharacteristic
29#358×C67C2211yescharacteristic
30#358×C67C2212yes—
31#358×C67C2212yes—
32#358×C67C2212yes—
33#358×C67C2212yes—
34#358×C67C2212yes—
35#358×C67C2211yescharacteristic
36#357×C67C2211yescharacteristic
37#357×C67C2212yes—
38#357×C67C2211yescharacteristic
39#346×C67C2212yes—
40C4×C16×C67C2212yes—
41C2×#41×C67—21no—
42C22×Q8×C67—21no—
43#137×C67—44no—
44#135×C67—22no—
45#135×C67—22no—
46C2×Q16×C67—22no—
47C2×Q16×C67—21no—
48C2×Q16×C67—22no—
49C2×Q16×C67—22no—
50C2×Q16×C67C2311yescharacteristic
51C2×Q16×C67C2312yes—
52C2×SD16×C67—22no—
53C2×SD16×C67—21no—
54C2×D8×C67C2311yescharacteristic
55C2×M16×C67D411yescharacteristic
56C22×C8×C67C2311yescharacteristic
57C4⋊Q8×C67—22no—
58C4⋊Q8×C67—22no—
59C4⋊Q8×C67—41no—
60#125×C67—22no—
61#125×C67—41no—
62#124×C67—22no—
63#123×C67—42no—
64C22⋊Q8×C67—41no—
65C22⋊Q8×C67—41no—
66C22⋊Q8×C67—42no—
67#117×C67—21no—
68#117×C67D411yescharacteristic
69C2×C4⋊C4×C67—21no—
70M32×C67—22no—
71C2×C16×C67—22no—
72C2×C16×C67C2311yescharacteristic
73C2×C16×C67C2311yescharacteristic
74C2×C16×C67C2312yes—
75#107×C67—22no—
76#107×C67—21no—
77#107×C67C2311yescharacteristic
78#107×C67C2312yes—
79#107×C67C2311yescharacteristic
80#106×C67—22no—
81#106×C67—21no—
82#97×C67D412yes—
83C4×C8×C67C2312yes—
84#41×C67—42no—
85#41×C67—42no—
86C2×Q8×C67—22no—
87C2×Q8×C67—21no—
88C2×Q8×C67—41no—
89C2×Q8×C67—42no—
90C2×Q8×C67—42no—
91C2×Q8×C67—42no—
92C2×D4×C67—21no—
93C2×D4×C67—41no—
94C22×C4×C67—41no—
95C22×C4×C67—41no—
96C22×C4×C67—41no—
97C22×C4×C67C2×D411yescharacteristic
98Q16×C67—22no—
99Q16×C67—22no—
100Q16×C67—21no—
101Q16×C67—21no—
102Q16×C67—42no—
103Q16×C67—44no—
104SD16×C67—42no—
105SD16×C67—44no—
106D8×C67—21no—
107D8×C67—21no—
108M16×C67SD1614yes—
109C2×C8×C67C2×D411yescharacteristic
110C2×C8×C67C2×D412yes—
111C2×C8×C67#4112yes—
112C2×C8×C67#4112yes—
113C2×C8×C67C2411yescharacteristic
114C4⋊C4×C67—21no—
115C4⋊C4×C67—22no—
116C4⋊C4×C67—21no—
117C4⋊C4×C67—41no—
118C4⋊C4×C67—42no—
119C4⋊C4×C67—42no—
120C4⋊C4×C67—42no—
121C4⋊C4×C67—42no—
122C4⋊C4×C67—42no—
123C4⋊C4×C67D811yescharacteristic
124C4⋊C4×C67D811yescharacteristic
125C22⋊C4×C67—41no—
126C22⋊C4×C67—42no—
127C22⋊C4×C67—42no—
128C22⋊C4×C67D812yes—
129C42×C67—41no—
130C42×C67C2×D412yes—
131C16×C67—21no—
132C16×C67—21no—
133C16×C67—22no—
134C23×C67—41no—
135C23×C67C2×D811yescharacteristic
136Q8×C67—41no—
137Q8×C67—42no—
138Q8×C67—42no—
139Q8×C67—41no—
140Q8×C67—82no—
141Q8×C67—162no—
142D4×C67—41no—
143D4×C67—41no—
144D4×C67—82no—
145C2×C4×C67—41no—
146C2×C4×C67—41no—
147C2×C4×C67—42no—
148C2×C4×C67—42no—
149C2×C4×C67—41no—
150C2×C4×C67—82no—
151C2×C4×C67—82no—
152C2×C4×C67—161no—
153C2×C4×C67#12412yes—
154C2×C4×C67C2×D812yes—
155C2×C4×C67C2×D812yes—
156C2×C4×C67C2×D811yescharacteristic
157C2×C4×C67C2×SD1612yes—
158C2×C4×C67C22×D411yescharacteristic
159C8×C67#12412yes—
160C8×C67C2×SD1614yes—
161C8×C67C2×#4111yescharacteristic
162C8×C67C2×#4111yescharacteristic
163C22×C67—21no—
164C22×C67—21no—
165C22×C67—81no—
166C22×C67—81no—
167C22×C67—81no—
168C22×C67#48612yes—
169C22×C67C22×D811yescharacteristic
170C4×C67—22no—
171C4×C67—22no—
172C4×C67—81no—
173C4×C67—82no—
174C4×C67—82no—
175C4×C67—81no—
176C4×C67—81no—
177C4×C67#48612yes—
178C4×C67C2×#12411yescharacteristic
179C4×C67C22×SD1611yescharacteristic
180#33703C6711yesmaximal, characteristic, direct product factor, Sylow
181C2×C67—22no—
182C2×C67—161no—
183C2×C67C2×#48611yescharacteristic
184C2×C67#341611yescharacteristic
185C2×C67#341711yescharacteristic
186#3164C2×C6711yescharacteristic, complemented
187#3156C2×C6711yescharacteristic
188#2245C2×C6712yescomplemented
189#2243C2×C6712yescomplemented
190#2242C2×C6712yescomplemented
191#2241C2×C6712yescomplemented
192#2141C2×C6711yescharacteristic, complemented
193#2139C2×C6711yescharacteristic
194#2138C2×C6711yescharacteristic
195C2×#358C2×C6711yescharacteristic, complemented
196#2108C2×C6711yescharacteristic, complemented
197C67#3370311yescharacteristic, direct product factor, Sylow
198C2×#137—21no—
199C2×#135C22×C6711yescharacteristic
200C22×Q16C22×C6711yescharacteristic
201#544—21no—
202#536—21no—
203C2×M32C22×C6711yescharacteristic
204C22×C16C22×C6711yescharacteristic
205#501—22no—
206#500C22×C6712yes—
207#499C22×C6712yes—
208#497—22no—
209#495C22×C6712yes—
210#494C22×C6712yes—
211#428—21no—
212#427C22×C6711yescharacteristic
213C2×#107C22×C6711yescharacteristic
214#405C22×C6711yescharacteristic
215#358C22×C6712yes—
216#358C22×C6712yes—
217#358C22×C6711yescharacteristic
218#358C22×C6711yescharacteristic
219#358C22×C6712yes—
220#358C22×C6712yes—
221#358C22×C6712yes—
222#357C22×C6711yescharacteristic
223#357C22×C6711yescharacteristic
224#357C22×C6712yes—
225#346C22×C6712yes—
226C4×C16C22×C6712yes—
227C2×#41—21no—
228C22×Q8—21no—
229#137—44no—
230#135—22no—
231#135—22no—
232C2×Q16—21no—
233C2×Q16—22no—
234C2×Q16—22no—
235C2×Q16—22no—
236C2×Q16C23×C6712yes—
237C2×Q16C23×C6711yescharacteristic
238C2×SD16—21no—
239C2×SD16—22no—
240C2×D8C23×C6711yescharacteristic
241C2×M16D4×C6711yescharacteristic
242C22×C8C23×C6711yescharacteristic
243C4⋊Q8—22no—
244C4⋊Q8—22no—
245C4⋊Q8—41no—
246#125—22no—
247#125—41no—
248#124—22no—
249#123—42no—
250C22⋊Q8—41no—
251C22⋊Q8—41no—
252C22⋊Q8—42no—
253#117—21no—
254#117D4×C6711yescharacteristic
255C2×C4⋊C4—21no—
256M32—22no—
257C2×C16—22no—
258C2×C16C23×C6711yescharacteristic
259C2×C16C23×C6712yes—
260C2×C16C23×C6711yescharacteristic
261#107—21no—
262#107—22no—
263#107C23×C6712yes—
264#107C23×C6711yescharacteristic
265#107C23×C6711yescharacteristic
266#106—21no—
267#106—22no—
268#97D4×C6712yes—
269C4×C8C23×C6712yes—
270#41—42no—
271#41—42no—
272C2×Q8—22no—
273C2×Q8—21no—
274C2×Q8—41no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×D4—21no—
279C2×D4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4C2×D4×C6711yescharacteristic
284Q16—22no—
285Q16—22no—
286Q16—21no—
287Q16—21no—
288Q16—42no—
289Q16—44no—
290SD16—42no—
291SD16—44no—
292D8—21no—
293D8—21no—
294M16SD16×C6714yes—
295C2×C8C2×D4×C6711yescharacteristic
296C2×C8C2×D4×C6712yes—
297C2×C8#41×C6712yes—
298C2×C8#41×C6712yes—
299C2×C8C24×C6711yescharacteristic
300C4⋊C4—21no—
301C4⋊C4—22no—
302C4⋊C4—21no—
303C4⋊C4—41no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4D8×C6711yescharacteristic
310C4⋊C4D8×C6711yescharacteristic
311C22⋊C4—41no—
312C22⋊C4—42no—
313C22⋊C4—42no—
314C22⋊C4D8×C6712yes—
315C42—41no—
316C42C2×D4×C6712yes—
317C16—21no—
318C16—21no—
319C16—22no—
320C23—41no—
321C23C2×D8×C6711yescharacteristic
322Q8—41no—
323Q8—42no—
324Q8—42no—
325Q8—41no—
326Q8—82no—
327Q8—162no—
328D4—41no—
329D4—41no—
330D4—82no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—41no—
336C2×C4—82no—
337C2×C4—82no—
338C2×C4—161no—
339C2×C4#124×C6712yes—
340C2×C4C2×D8×C6712yes—
341C2×C4C2×D8×C6712yes—
342C2×C4C2×D8×C6711yescharacteristic
343C2×C4C2×SD16×C6712yes—
344C2×C4C22×D4×C6711yescharacteristic
345C8#124×C6712yes—
346C8C2×SD16×C6714yes—
347C8C2×#41×C6711yescharacteristic
348C8C2×#41×C6711yescharacteristic
349C22—21no—
350C22—21no—
351C22—81no—
352C22—81no—
353C22—81no—
354C22#486×C6712yes—
355C22C22×D8×C6711yescharacteristic
356C4—22no—
357C4—22no—
358C4—81no—
359C4—82no—
360C4—82no—
361C4—81no—
362C4—81no—
363C4#486×C6712yes—
364C4C2×#124×C6711yescharacteristic
365C4C22×SD16×C6711yescharacteristic
366C2—22no—
367C2—161no—
368C2C2×#486×C6711yescharacteristic
369C2C67×#341611yescharacteristic
370C2C67×#341711yescharacteristic
subgroupG/H
center, socleC22×C67C22×D8
commutator, Frattini, cosocleC2×C8C24×C67
prime radicalC2×#135×C67C22
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C67 ◁ #117×C67 ◁ #405×C67 ◁ G

Complements

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

NH ≅ G/Ntype
1. C67×#3164C2semidirect
3. C67×#2245C2semidirect
4. C67×#2243C2semidirect
5. C67×#2242C2semidirect
6. C67×#2241C2semidirect
7. C67×#2141C2semidirect
10. C2×#358×C67C2semidirect
11. C67×#2108C2semidirect
180. #33703C67direct (N × H)
186. #3164C2×C67semidirect
188. #2245C2×C67semidirect
189. #2243C2×C67semidirect
190. #2242C2×C67semidirect
191. #2241C2×C67semidirect
192. #2141C2×C67semidirect
195. C2×#358C2×C67semidirect
196. #2108C2×C67semidirect
197. C67#33703direct (N × H)

Maximal subgroups

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

subgroupindex
C67×#31642
C67×#31562
C67×#22452
C67×#22432
C67×#22422
C67×#22412
C67×#21412
C67×#21392
C67×#21382
C2×#358×C672
C67×#21082
#3370367