Finite Group Databasebeta

← C67×#19065 #30698316

Subgroups — C67×#19065 #30698316

Order 17152.

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

normal subgroups only

This group has 920 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
1C67×#3548C211yesmaximal, characteristic
2C67×#3216C212yesmaximal, complemented
3C67×#3162C211yesmaximal, characteristic, complemented
4C67×#1661C218yesmaximal, complemented
5C67×#1537C211yesmaximal, characteristic, complemented
6C67×#1530C212yesmaximal, complemented
7#562×C67C2214yes—
8#554×C67—22no—
9#550×C67—22no—
10#547×C67—22no—
11#540×C67—22no—
12#537×C67—24no—
13#533×C67C2211yescharacteristic
14#533×C67C2211yescharacteristic
15#532×C67C2212yes—
16#501×C67—22no—
17#501×C67C2214yes—
18#497×C67C2214yes—
19#489×C67C2214yes—
20#462×C67—22no—
21#460×C67—22no—
22#452×C67—24no—
23#432×C67C2212yes—
24#427×C67—22no—
25C2×#97×C67C2211yescharacteristic
26#330×C67C2214yes—
27#329×C67C2218yes—
28C2×#41×C67—24no—
29C2×#41×C67D414yes—
30C2×Q16×C67—24no—
31C2×SD16×C67—24no—
32C22×C8×C67—21no—
33C4⋊Q8×C67D412yes—
34C4⋊Q8×C67D412yes—
35C4⋊Q8×C67C2312yes—
36#127×C67—22no—
37#127×C67D412yes—
38#125×C67—22no—
39#125×C67—21no—
40#125×C67C2312yes—
41#124×C67C2314yes—
42#123×C67—28no—
43C22⋊Q8×C67—24no—
44C22⋊Q8×C67—28no—
45C22⋊Q8×C67—22no—
46C22⋊Q8×C67—22no—
47#121×C67—28no—
48#121×C67—24no—
49#121×C67—28no—
50#121×C67—22no—
51#121×C67—22no—
52C4×Q8×C67—22no—
53C4×Q8×C67—22no—
54C4×D4×C67—24no—
55C4×D4×C67—22no—
56C4×D4×C67—22no—
57#117×C67—21no—
58#117×C67D412yes—
59C2×C4⋊C4×C67—24no—
60C2×C42×C67C2311yescharacteristic
61#107×C67—24no—
62#106×C67—24no—
63C4⋊C8×C67—22no—
64#103×C67—24no—
65#102×C67—24no—
66C22⋊C8×C67—22no—
67#97×C67—21no—
68#97×C67C2314yes—
69#97×C67C2312yes—
70#41×C67—84no—
71#41×C67—88no—
72C2×Q8×C67—22no—
73C2×Q8×C67—22no—
74C2×Q8×C67C2×D414yes—
75C2×D4×C67—28no—
76C2×D4×C67—24no—
77C2×D4×C67—22no—
78C2×D4×C67—22no—
79C2×D4×C67C2×D414yes—
80C22×C4×C67—28no—
81C22×C4×C67—24no—
82C22×C4×C67C2×D411yescharacteristic
83C22×C4×C67C2×D412yes—
84Q16×C67—84no—
85SD16×C67—84no—
86C2×C8×C67—24no—
87C2×C8×C67—22no—
88C2×C8×C67—24no—
89C4⋊C4×C67—28no—
90C4⋊C4×C67—22no—
91C4⋊C4×C67—22no—
92C4⋊C4×C67—22no—
93C4⋊C4×C67—24no—
94C4⋊C4×C67—22no—
95C4⋊C4×C67—22no—
96C4⋊C4×C67—42no—
97C4⋊C4×C67C2×D414yes—
98C22⋊C4×C67—28no—
99C22⋊C4×C67—22no—
100C22⋊C4×C67—24no—
101C22⋊C4×C67—24no—
102C22⋊C4×C67—41no—
103C42×C67—22no—
104C42×C67—41no—
105C42×C67C2×D412yes—
106C42×C67C2×D411yescharacteristic
107C42×C67C2411yescharacteristic
108C23×C67—24no—
109C23×C67#12011yescharacteristic
110Q8×C67—44no—
111Q8×C67—44no—
112Q8×C67—42no—
113Q8×C67—42no—
114D4×C67—44no—
115D4×C67—44no—
116D4×C67—44no—
117D4×C67—84no—
118D4×C67—88no—
119C2×C4×C67—22no—
120C2×C4×C67—24no—
121C2×C4×C67—24no—
122C2×C4×C67—42no—
123C2×C4×C67—48no—
124C2×C4×C67—42no—
125C2×C4×C67—88no—
126C2×C4×C67#12012yes—
127C2×C4×C67#12011yescharacteristic
128C2×C4×C67C22×D411yescharacteristic
129C2×C4×C67C22×D412yes—
130C8×C67—24no—
131C8×C67—44no—
132C22×C67—21no—
133C22×C67—22no—
134C22×C67—44no—
135C22×C67—44no—
136C22×C67—44no—
137C22×C67C2×#12011yescharacteristic
138C4×C67—21no—
139C4×C67—22no—
140C4×C67—22no—
141C4×C67—44no—
142C4×C67—44no—
143C4×C67—82no—
144C4×C67#45612yes—
145#19065C6711yesmaximal, characteristic, direct product factor, Sylow
146C2×C67—41no—
147C2×C67—84no—
148C2×C67C2×#45611yescharacteristic
149C2×C67#301912yes—
150#3548C2×C6711yescharacteristic
151#3216C2×C6712yescomplemented
152#3162C2×C6711yescharacteristic, complemented
153#1661C2×C6718yescomplemented
154#1537C2×C6711yescharacteristic, complemented
155#1530C2×C6712yescomplemented
156C67#1906511yescharacteristic, direct product factor, Sylow
157#562C22×C6714yes—
158#554—22no—
159#550—22no—
160#547—22no—
161#540—22no—
162#537—24no—
163#533C22×C6711yescharacteristic
164#533C22×C6711yescharacteristic
165#532C22×C6712yes—
166#501—22no—
167#501C22×C6714yes—
168#497C22×C6714yes—
169#489C22×C6714yes—
170#462—22no—
171#460—22no—
172#452—24no—
173#432C22×C6712yes—
174#427—22no—
175C2×#97C22×C6711yescharacteristic
176#330C22×C6714yes—
177#329C22×C6718yes—
178C2×#41—24no—
179C2×#41D4×C6714yes—
180C2×Q16—24no—
181C2×SD16—24no—
182C22×C8—21no—
183C4⋊Q8D4×C6712yes—
184C4⋊Q8D4×C6712yes—
185C4⋊Q8C23×C6712yes—
186#127—22no—
187#127D4×C6712yes—
188#125—22no—
189#125—21no—
190#125C23×C6712yes—
191#124C23×C6714yes—
192#123—28no—
193C22⋊Q8—24no—
194C22⋊Q8—28no—
195C22⋊Q8—22no—
196C22⋊Q8—22no—
197#121—28no—
198#121—24no—
199#121—28no—
200#121—22no—
201#121—22no—
202C4×Q8—22no—
203C4×Q8—22no—
204C4×D4—24no—
205C4×D4—22no—
206C4×D4—22no—
207#117—21no—
208#117D4×C6712yes—
209C2×C4⋊C4—24no—
210C2×C42C23×C6711yescharacteristic
211#107—24no—
212#106—24no—
213C4⋊C8—22no—
214#103—24no—
215#102—24no—
216C22⋊C8—22no—
217#97—21no—
218#97C23×C6712yes—
219#97C23×C6714yes—
220#41—84no—
221#41—88no—
222C2×Q8—22no—
223C2×Q8—22no—
224C2×Q8C2×D4×C6714yes—
225C2×D4—28no—
226C2×D4—24no—
227C2×D4—22no—
228C2×D4—22no—
229C2×D4C2×D4×C6714yes—
230C22×C4—28no—
231C22×C4—24no—
232C22×C4C2×D4×C6711yescharacteristic
233C22×C4C2×D4×C6712yes—
234Q16—84no—
235SD16—84no—
236C2×C8—24no—
237C2×C8—22no—
238C2×C8—24no—
239C4⋊C4—28no—
240C4⋊C4—22no—
241C4⋊C4—22no—
242C4⋊C4—22no—
243C4⋊C4—24no—
244C4⋊C4—22no—
245C4⋊C4—22no—
246C4⋊C4—42no—
247C4⋊C4C2×D4×C6714yes—
248C22⋊C4—28no—
249C22⋊C4—22no—
250C22⋊C4—24no—
251C22⋊C4—24no—
252C22⋊C4—41no—
253C42—22no—
254C42—41no—
255C42C2×D4×C6711yescharacteristic
256C42C2×D4×C6712yes—
257C42C24×C6711yescharacteristic
258C23—24no—
259C23#120×C6711yescharacteristic
260Q8—44no—
261Q8—44no—
262Q8—42no—
263Q8—42no—
264D4—44no—
265D4—44no—
266D4—44no—
267D4—84no—
268D4—88no—
269C2×C4—22no—
270C2×C4—24no—
271C2×C4—24no—
272C2×C4—42no—
273C2×C4—48no—
274C2×C4—42no—
275C2×C4—88no—
276C2×C4#120×C6712yes—
277C2×C4#120×C6711yescharacteristic
278C2×C4C22×D4×C6711yescharacteristic
279C2×C4C22×D4×C6712yes—
280C8—24no—
281C8—44no—
282C22—21no—
283C22—22no—
284C22—44no—
285C22—44no—
286C22—44no—
287C22C2×#120×C6711yescharacteristic
288C4—21no—
289C4—22no—
290C4—22no—
291C4—44no—
292C4—44no—
293C4—82no—
294C4#456×C6712yes—
295C2—41no—
296C2—84no—
297C2C2×#456×C6711yescharacteristic
298C2C67×#301912yes—
subgroupG/H
center, socleC22×C67C2×#120
commutator, Frattini, cosocleC42C24×C67
prime radicalC67×#3548C2
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×#3216C2semidirect
3. C67×#3162C2semidirect
4. C67×#1661C2semidirect
5. C67×#1537C2semidirect
6. C67×#1530C2semidirect
145. #19065C67direct (N × H)
151. #3216C2×C67semidirect
152. #3162C2×C67semidirect
153. #1661C2×C67semidirect
154. #1537C2×C67semidirect
155. #1530C2×C67semidirect
156. C67#19065direct (N × H)

Maximal subgroups

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

subgroupindex
C67×#35482
C67×#32162
C67×#31622
C67×#16612
C67×#15372
C67×#15302
#1906567