Finite Group Databasebeta

← C67×#33587 #30712838

Subgroups — C67×#33587 #30712838

Order 17152.

712 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#510×C67C211yesmaximal, characteristic, complemented
2C22×SD32×C67C211yesmaximal, characteristic, complemented
3C67×#3155C211yesmaximal, characteristic
4C67×#2950C211yesmaximal, characteristic, complemented
5C67×#2211C214yesmaximal, complemented
6C67×#2207C214yesmaximal, complemented
7C67×#2151C211yesmaximal, characteristic, complemented
8C67×#2140C211yesmaximal, characteristic, complemented
9C67×#2138C211yesmaximal, characteristic, complemented
10C2×AGL(1,ℤ/8)×C67—21no—
11C2×#135×C67C2211yescharacteristic
12C22×Q16×C67C2211yescharacteristic
13C22×D8×C67C2211yescharacteristic
14#535×C67—21no—
15#519×C67—21no—
16#510×C67—24no—
17C2×Q32×C67C2212yescomplemented
18C2×SD32×C67—24no—
19C2×SD32×C67C2214yes—
20C2×SD32×C67C2212yescomplemented
21C2×M32×C67C2211yescharacteristic
22C22×C16×C67C2211yescharacteristic
23#496×C67—22no—
24#495×C67C2212yes—
25#493×C67C2212yes—
26#441×C67—22no—
27C4×Q16×C67C2214yescomplemented
28#427×C67C2211yescharacteristic
29#419×C67—21no—
30#405×C67C2211yescharacteristic
31C4⋊C16×C67C2214yescomplemented
32#358×C67C2212yescomplemented
33#357×C67C2214yes—
34#357×C67C2212yescomplemented
35C2×#41×C67—21no—
36C22×Q8×C67—21no—
37C22×D4×C67—21no—
38AGL(1,ℤ/8)×C67—44no—
39#135×C67—22no—
40#135×C67—22no—
41C2×Q16×C67—21no—
42C2×Q16×C67—22no—
43C2×Q16×C67D414yescomplemented
44C2×Q16×C67C2311yescharacteristic
45C2×Q16×C67C2312yes—
46C2×SD16×C67—22no—
47C2×SD16×C67—21no—
48C2×D8×C67—22no—
49C2×D8×C67—22no—
50C2×D8×C67—21no—
51C2×D8×C67—22no—
52C2×D8×C67C2311yescharacteristic
53C2×D8×C67C2312yes—
54C2×M16×C67D411yescharacteristic
55C22×C8×C67C2311yescharacteristic
56#127×C67—22no—
57#124×C67—22no—
58#121×C67—42no—
59#120×C67—42no—
60C4×Q8×C67—24no—
61C4×Q8×C67—42no—
62#117×C67—21no—
63#117×C67—42no—
64#117×C67D411yescharacteristic
65Q32×C67—24no—
66SD32×C67—28no—
67SD32×C67—24no—
68M32×C67—22no—
69C2×C16×C67—22no—
70C2×C16×C67C2312yes—
71C2×C16×C67C2312yes—
72#107×C67C2312yes—
73#106×C67—21no—
74#103×C67—24no—
75#97×C67D412yes—
76C4×C8×C67C2312yes—
77C24×C67—41no—
78#41×C67—42no—
79#41×C67—42no—
80C2×Q8×C67—24no—
81C2×Q8×C67—22no—
82C2×Q8×C67—21no—
83C2×Q8×C67—41no—
84C2×Q8×C67—42no—
85C2×D4×C67—22no—
86C2×D4×C67—21no—
87C2×D4×C67—41no—
88C2×D4×C67—42no—
89C2×D4×C67—41no—
90C2×D4×C67—44no—
91C2×D4×C67—42no—
92C22×C4×C67—41no—
93C22×C4×C67—41no—
94C22×C4×C67C2×D411yescharacteristic
95Q16×C67—24no—
96Q16×C67—21no—
97Q16×C67—22no—
98Q16×C67—21no—
99Q16×C67C2×D414yes—
100SD16×C67—44no—
101SD16×C67—42no—
102D8×C67—22no—
103D8×C67—21no—
104D8×C67—22no—
105D8×C67—21no—
106D8×C67—44no—
107D8×C67—42no—
108M16×C67—22no—
109C2×C8×C67C2×D412yes—
110C2×C8×C67C2×D411yescharacteristic
111C2×C8×C67C2×D412yes—
112C2×C8×C67#4112yes—
113C2×C8×C67C2411yescharacteristic
114C4⋊C4×C67—22no—
115C4⋊C4×C67—44no—
116C4⋊C4×C67D812yes—
117C22⋊C4×C67—41no—
118C22⋊C4×C67—44no—
119C22⋊C4×C67D812yes—
120C42×C67—41no—
121C42×C67—44no—
122C42×C67C2×D412yes—
123C16×C67—22no—
124C16×C67—22no—
125C23×C67—42no—
126C23×C67—41no—
127C23×C67—82no—
128C23×C67—162no—
129C23×C67C2×D811yescharacteristic
130Q8×C67—24no—
131Q8×C67—42no—
132Q8×C67—44no—
133Q8×C67—41no—
134Q8×C67—41no—
135D4×C67—42no—
136D4×C67—41no—
137D4×C67—41no—
138D4×C67—42no—
139D4×C67—82no—
140D4×C67—82no—
141D4×C67—164no—
142C2×C4×C67—42no—
143C2×C4×C67—44no—
144C2×C4×C67—42no—
145C2×C4×C67—41no—
146C2×C4×C67—82no—
147C2×C4×C67#12112yes—
148C2×C4×C67C2×D814yes—
149C2×C4×C67C2×D811yescharacteristic
150C2×C4×C67AGL(1,ℤ/8)12yes—
151C2×C4×C67C22×D411yescharacteristic
152C8×C67—22no—
153C8×C67#12112yes—
154C8×C67C22×D411yescharacteristic
155C8×C67C2×#4111yescharacteristic
156C22×C67—22no—
157C22×C67—82no—
158C22×C67—84no—
159C22×C67—81no—
160C22×C67—82no—
161C22×C67—164no—
162C22×C67#45912yes—
163C22×C67C22×D811yescharacteristic
164C4×C67—24no—
165C4×C67—44no—
166C4×C67—82no—
167C4×C67—81no—
168C4×C67#45912yes—
169C4×C67C2×#12111yescharacteristic
170C4×C67C2×AGL(1,ℤ/8)11yescharacteristic
171#33587C6711yesmaximal, characteristic, direct product factor, Sylow
172C2×C67—22no—
173C2×C67—161no—
174C2×C67—162no—
175C2×C67C2×#45911yescharacteristic
176C2×C67#341612yes—
177C2×#510C2×C6711yescharacteristic, complemented
178C22×SD32C2×C6711yescharacteristic, complemented
179#3155C2×C6711yescharacteristic
180#2950C2×C6711yescharacteristic, complemented
181#2211C2×C6714yescomplemented
182#2207C2×C6714yescomplemented
183#2151C2×C6711yescharacteristic, complemented
184#2140C2×C6711yescharacteristic, complemented
185#2138C2×C6711yescharacteristic, complemented
186C67#3358711yescharacteristic, direct product factor, Sylow
187C2×AGL(1,ℤ/8)—21no—
188C2×#135C22×C6711yescharacteristic
189C22×Q16C22×C6711yescharacteristic
190C22×D8C22×C6711yescharacteristic
191#535—21no—
192#519—21no—
193#510—24no—
194C2×Q32C22×C6712yescomplemented
195C2×SD32—24no—
196C2×SD32C22×C6714yes—
197C2×SD32C22×C6712yescomplemented
198C2×M32C22×C6711yescharacteristic
199C22×C16C22×C6711yescharacteristic
200#496—22no—
201#495C22×C6712yes—
202#493C22×C6712yes—
203#441—22no—
204C4×Q16C22×C6714yescomplemented
205#427C22×C6711yescharacteristic
206#419—21no—
207#405C22×C6711yescharacteristic
208C4⋊C16C22×C6714yescomplemented
209#358C22×C6712yescomplemented
210#357C22×C6714yes—
211#357C22×C6712yescomplemented
212C2×#41—21no—
213C22×Q8—21no—
214C22×D4—21no—
215AGL(1,ℤ/8)—44no—
216#135—22no—
217#135—22no—
218C2×Q16—21no—
219C2×Q16—22no—
220C2×Q16D4×C6714yescomplemented
221C2×Q16C23×C6711yescharacteristic
222C2×Q16C23×C6712yes—
223C2×SD16—21no—
224C2×SD16—22no—
225C2×D8—22no—
226C2×D8—21no—
227C2×D8—22no—
228C2×D8—22no—
229C2×D8C23×C6712yes—
230C2×D8C23×C6711yescharacteristic
231C2×M16D4×C6711yescharacteristic
232C22×C8C23×C6711yescharacteristic
233#127—22no—
234#124—22no—
235#121—42no—
236#120—42no—
237C4×Q8—24no—
238C4×Q8—42no—
239#117—21no—
240#117—42no—
241#117D4×C6711yescharacteristic
242Q32—24no—
243SD32—28no—
244SD32—24no—
245M32—22no—
246C2×C16—22no—
247C2×C16C23×C6712yes—
248C2×C16C23×C6712yes—
249#107C23×C6712yes—
250#106—21no—
251#103—24no—
252#97D4×C6712yes—
253C4×C8C23×C6712yes—
254C24—41no—
255#41—42no—
256#41—42no—
257C2×Q8—24no—
258C2×Q8—22no—
259C2×Q8—21no—
260C2×Q8—41no—
261C2×Q8—42no—
262C2×D4—22no—
263C2×D4—21no—
264C2×D4—41no—
265C2×D4—42no—
266C2×D4—41no—
267C2×D4—44no—
268C2×D4—42no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4C2×D4×C6711yescharacteristic
272Q16—24no—
273Q16—21no—
274Q16—22no—
275Q16—21no—
276Q16C2×D4×C6714yes—
277SD16—44no—
278SD16—42no—
279D8—22no—
280D8—21no—
281D8—22no—
282D8—21no—
283D8—44no—
284D8—42no—
285M16—22no—
286C2×C8C2×D4×C6712yes—
287C2×C8C2×D4×C6711yescharacteristic
288C2×C8C2×D4×C6712yes—
289C2×C8#41×C6712yes—
290C2×C8C24×C6711yescharacteristic
291C4⋊C4—22no—
292C4⋊C4—44no—
293C4⋊C4D8×C6712yes—
294C22⋊C4—41no—
295C22⋊C4—44no—
296C22⋊C4D8×C6712yes—
297C42—41no—
298C42—44no—
299C42C2×D4×C6712yes—
300C16—22no—
301C16—22no—
302C23—42no—
303C23—41no—
304C23—82no—
305C23—162no—
306C23C2×D8×C6711yescharacteristic
307Q8—24no—
308Q8—44no—
309Q8—42no—
310Q8—41no—
311Q8—41no—
312D4—42no—
313D4—41no—
314D4—41no—
315D4—42no—
316D4—82no—
317D4—82no—
318D4—164no—
319C2×C4—44no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—82no—
324C2×C4#121×C6712yes—
325C2×C4C2×D8×C6714yes—
326C2×C4C2×D8×C6711yescharacteristic
327C2×C4AGL(1,ℤ/8)×C6712yes—
328C2×C4C22×D4×C6711yescharacteristic
329C8—22no—
330C8#121×C6712yes—
331C8C22×D4×C6711yescharacteristic
332C8C2×#41×C6711yescharacteristic
333C22—22no—
334C22—81no—
335C22—82no—
336C22—82no—
337C22—84no—
338C22—164no—
339C22#459×C6712yes—
340C22C22×D8×C6711yescharacteristic
341C4—24no—
342C4—44no—
343C4—81no—
344C4—82no—
345C4#459×C6712yes—
346C4C2×#121×C6711yescharacteristic
347C4C2×AGL(1,ℤ/8)×C6711yescharacteristic
348C2—22no—
349C2—162no—
350C2—161no—
351C2C2×#459×C6711yescharacteristic
352C2C67×#341612yes—
subgroupG/H
center, socleC22×C67C22×D8
commutator, Frattini, cosocleC2×C8C24×C67
prime radicalC67×#3155C2
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. C2×#510×C67C2semidirect
2. C22×SD32×C67C2semidirect
4. C67×#2950C2semidirect
5. C67×#2211C2semidirect
6. C67×#2207C2semidirect
7. C67×#2151C2semidirect
8. C67×#2140C2semidirect
9. C67×#2138C2semidirect
17. C2×Q32×C67C22semidirect
20. C2×SD32×C67C22semidirect
27. C4×Q16×C67C22semidirect
31. C4⋊C16×C67C22semidirect
32. #358×C67C22semidirect
34. #357×C67C22semidirect
43. C2×Q16×C67D4semidirect
171. #33587C67direct (N × H)
177. C2×#510C2×C67semidirect
178. C22×SD32C2×C67semidirect
180. #2950C2×C67semidirect
181. #2211C2×C67semidirect
182. #2207C2×C67semidirect
183. #2151C2×C67semidirect
184. #2140C2×C67semidirect
185. #2138C2×C67semidirect
186. C67#33587direct (N × H)
194. C2×Q32C22×C67semidirect
197. C2×SD32C22×C67semidirect
204. C4×Q16C22×C67semidirect
208. C4⋊C16C22×C67semidirect
209. #358C22×C67semidirect
211. #357C22×C67semidirect
220. C2×Q16D4×C67semidirect

Maximal subgroups

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

subgroupindex
C2×#510×C672
C22×SD32×C672
C67×#31552
C67×#29502
C67×#22112
C67×#22072
C67×#21512
C67×#21402
C67×#21382
#3358767