Finite Group Databasebeta

← C67×#10255 #30689506

Subgroups — C67×#10255 #30689506

Order 17152.

582 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 122 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×#3245C211yesmaximal, characteristic
2C67×#3037C212yesmaximal, complemented
3C67×#1984C211yesmaximal, characteristic, complemented
4C67×#1492C212yesmaximal, complemented
5C67×#1292C211yesmaximal, characteristic, complemented
6C2×#137×C67—22no—
7#532×C67C2212yes—
8C2×C4×Q8×C67C2211yescharacteristic
9#474×C67—22no—
10#466×C67—22no—
11#464×C67—24no—
12#462×C67—22no—
13#460×C67—22no—
14#459×C67—22no—
15#423×C67C2212yes—
16C2×C4⋊C8×C67C2211yescharacteristic
17#417×C67—22no—
18C2×C4×C8×C67C2211yescharacteristic
19#336×C67—22no—
20D4⋊C8×C67—24no—
21C2×#41×C67—22no—
22C22×Q8×C67D411yescharacteristic
23#137×C67—44no—
24#137×C67—44no—
25C2×Q16×C67—24no—
26C2×SD16×C67—24no—
27C2×D8×C67—22no—
28C2×M16×C67—22no—
29C22×C8×C67—21no—
30C22×C8×C67D412yes—
31C4⋊Q8×C67D412yes—
32#127×C67D412yes—
33#124×C67—22no—
34#121×C67—22no—
35#121×C67—22no—
36C4×Q8×C67—22no—
37C4×Q8×C67—22no—
38C4×D4×C67—22no—
39C4×D4×C67—22no—
40C2×C4⋊C4×C67—21no—
41C2×C4⋊C4×C67D411yescharacteristic
42C2×C42×C67—21no—
43C2×C42×C67C2311yescharacteristic
44#107×C67—22no—
45C4⋊C8×C67—22no—
46C4⋊C8×C67—22no—
47C4⋊C8×C67D414yes—
48#103×C67—24no—
49#102×C67—24no—
50#102×C67—24no—
51C4×C8×C67—22no—
52#41×C67—42no—
53#41×C67—42no—
54#41×C67—42no—
55#41×C67—42no—
56C2×Q8×C67—21no—
57C2×Q8×C67—22no—
58C2×Q8×C67—22no—
59C2×Q8×C67—22no—
60C2×Q8×C67D812yescomplemented
61C2×D4×C67—22no—
62C2×D4×C67—22no—
63C2×D4×C67—22no—
64C2×D4×C67—42no—
65C22×C4×C67—21no—
66C22×C4×C67—21no—
67C22×C4×C67—42no—
68C22×C4×C67C2×D411yescharacteristic
69C22×C4×C67C2×D411yescharacteristic
70C22×C4×C67#4111yescharacteristic
71Q16×C67—48no—
72SD16×C67—48no—
73D8×C67—82no—
74M16×C67—42no—
75M16×C67—42no—
76C2×C8×C67—22no—
77C2×C8×C67—22no—
78C2×C8×C67—24no—
79C2×C8×C67—41no—
80C2×C8×C67—41no—
81C2×C8×C67—42no—
82C4⋊C4×C67—22no—
83C4⋊C4×C67—22no—
84C4⋊C4×C67—22no—
85C4⋊C4×C67—42no—
86C22⋊C4×C67—42no—
87C22⋊C4×C67—42no—
88C42×C67—41no—
89C42×C67—41no—
90C42×C67C2×D412yes—
91C42×C67C2×D412yes—
92C23×C67—42no—
93C23×C67#12111yescharacteristic
94Q8×C67—24no—
95Q8×C67—42no—
96Q8×C67—42no—
97Q8×C67—42no—
98D4×C67—44no—
99D4×C67—42no—
100D4×C67—42no—
101D4×C67—42no—
102D4×C67—42no—
103D4×C67—82no—
104D4×C67—82no—
105C2×C4×C67—21no—
106C2×C4×C67—21no—
107C2×C4×C67—22no—
108C2×C4×C67—41no—
109C2×C4×C67—41no—
110C2×C4×C67—41no—
111C2×C4×C67—42no—
112C2×C4×C67—41no—
113C2×C4×C67—42no—
114C2×C4×C67—82no—
115C2×C4×C67—82no—
116C2×C4×C67#12012yes—
117C2×C4×C67#12011yescharacteristic
118C2×C4×C67#12112yes—
119C2×C4×C67#12711yescharacteristic
120C2×C4×C67C2×D811yescharacteristic
121C2×C4×C67#13511yescharacteristic
122C2×C4×C67AGL(1,ℤ/8)12yes—
123C8×C67—42no—
124C8×C67—44no—
125C8×C67—42no—
126C22×C67—21no—
127C22×C67—21no—
128C22×C67—82no—
129C22×C67—82no—
130C22×C67—82no—
131C22×C67#39211yescharacteristic
132C22×C67#46611yescharacteristic
133C22×C67#46911yescharacteristic
134C4×C67—21no—
135C4×C67—22no—
136C4×C67—42no—
137C4×C67—82no—
138C4×C67—81no—
139C4×C67#44712yes—
140C4×C67#44912yes—
141C4×C67#45511yescharacteristic
142C4×C67#45611yescharacteristic
143#10255C6711yesmaximal, characteristic, direct product factor, Sylow
144C2×C67—21no—
145C2×C67—21no—
146C2×C67—162no—
147C2×C67#199911yescharacteristic
148C2×C67#200711yescharacteristic
149C2×C67#201711yescharacteristic
150#3245C2×C6711yescharacteristic
151#3037C2×C6712yescomplemented
152#1984C2×C6711yescharacteristic, complemented
153#1492C2×C6712yescomplemented
154#1292C2×C6711yescharacteristic, complemented
155C67#1025511yescharacteristic, direct product factor, Sylow
156C2×#137—22no—
157#532C22×C6712yes—
158C2×C4×Q8C22×C6711yescharacteristic
159#474—22no—
160#466—22no—
161#464—24no—
162#462—22no—
163#460—22no—
164#459—22no—
165#423C22×C6712yes—
166C2×C4⋊C8C22×C6711yescharacteristic
167#417—22no—
168C2×C4×C8C22×C6711yescharacteristic
169#336—22no—
170D4⋊C8—24no—
171C2×#41—22no—
172C22×Q8D4×C6711yescharacteristic
173#137—44no—
174#137—44no—
175C2×Q16—24no—
176C2×SD16—24no—
177C2×D8—22no—
178C2×M16—22no—
179C22×C8—21no—
180C22×C8D4×C6712yes—
181C4⋊Q8D4×C6712yes—
182#127D4×C6712yes—
183#124—22no—
184#121—22no—
185#121—22no—
186C4×Q8—22no—
187C4×Q8—22no—
188C4×D4—22no—
189C4×D4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4D4×C6711yescharacteristic
192C2×C42—21no—
193C2×C42C23×C6711yescharacteristic
194#107—22no—
195C4⋊C8—22no—
196C4⋊C8—22no—
197C4⋊C8D4×C6714yes—
198#103—24no—
199#102—24no—
200#102—24no—
201C4×C8—22no—
202#41—42no—
203#41—42no—
204#41—42no—
205#41—42no—
206C2×Q8—21no—
207C2×Q8—22no—
208C2×Q8—22no—
209C2×Q8—22no—
210C2×Q8D8×C6712yescomplemented
211C2×D4—22no—
212C2×D4—22no—
213C2×D4—22no—
214C2×D4—42no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—42no—
218C22×C4C2×D4×C6711yescharacteristic
219C22×C4C2×D4×C6711yescharacteristic
220C22×C4#41×C6711yescharacteristic
221Q16—48no—
222SD16—48no—
223D8—82no—
224M16—42no—
225M16—42no—
226C2×C8—22no—
227C2×C8—22no—
228C2×C8—24no—
229C2×C8—41no—
230C2×C8—41no—
231C2×C8—42no—
232C4⋊C4—22no—
233C4⋊C4—22no—
234C4⋊C4—22no—
235C4⋊C4—42no—
236C22⋊C4—42no—
237C22⋊C4—42no—
238C42—41no—
239C42—41no—
240C42C2×D4×C6712yes—
241C42C2×D4×C6712yes—
242C23—42no—
243C23#121×C6711yescharacteristic
244Q8—24no—
245Q8—42no—
246Q8—42no—
247Q8—42no—
248D4—44no—
249D4—42no—
250D4—42no—
251D4—42no—
252D4—42no—
253D4—82no—
254D4—82no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—42no—
262C2×C4—41no—
263C2×C4—42no—
264C2×C4—82no—
265C2×C4—82no—
266C2×C4#120×C6712yes—
267C2×C4#120×C6711yescharacteristic
268C2×C4#121×C6712yes—
269C2×C4#127×C6711yescharacteristic
270C2×C4C2×D8×C6711yescharacteristic
271C2×C4#135×C6711yescharacteristic
272C2×C4AGL(1,ℤ/8)×C6712yes—
273C8—42no—
274C8—42no—
275C8—44no—
276C22—21no—
277C22—21no—
278C22—82no—
279C22—82no—
280C22—82no—
281C22#392×C6711yescharacteristic
282C22#466×C6711yescharacteristic
283C22#469×C6711yescharacteristic
284C4—21no—
285C4—22no—
286C4—42no—
287C4—82no—
288C4—81no—
289C4#447×C6712yes—
290C4#449×C6712yes—
291C4#455×C6711yescharacteristic
292C4#456×C6711yescharacteristic
293C2—21no—
294C2—21no—
295C2—162no—
296C2C67×#199911yescharacteristic
297C2C67×#200711yescharacteristic
298C2C67×#201711yescharacteristic
subgroupG/H
center, socleC22×C67#392
commutator, Frattini, cosocleC2×C42C23×C67
prime radicalC67×#3245C2
LCSG ▷ C2×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×#3037C2semidirect
3. C67×#1984C2semidirect
4. C67×#1492C2semidirect
5. C67×#1292C2semidirect
60. C2×Q8×C67D8semidirect
143. #10255C67direct (N × H)
151. #3037C2×C67semidirect
152. #1984C2×C67semidirect
153. #1492C2×C67semidirect
154. #1292C2×C67semidirect
155. C67#10255direct (N × H)
210. C2×Q8D8×C67semidirect

Maximal subgroups

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

subgroupindex
C67×#32452
C67×#30372
C67×#19842
C67×#14922
C67×#12922
#1025567