Finite Group Databasebeta

← C67×#10254 #30689505

Subgroups — C67×#10254 #30689505

Order 17152.

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

normal subgroups only

This group has 634 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, complemented
2C67×#3036C212yesmaximal, complemented
3C67×#1954C211yesmaximal, characteristic, complemented
4C67×#1492C212yesmaximal, complemented
5C67×#1292C211yesmaximal, characteristic, complemented
6C2×AGL(1,ℤ/8)×C67—22no—
7#532×C67C2212yes—
8C2×C4×D4×C67C2211yescharacteristic
9#474×C67—22no—
10#466×C67—22no—
11#463×C67—24no—
12#461×C67—22no—
13#459×C67—22no—
14#459×C67—22no—
15#423×C67C2212yes—
16C2×C4⋊C8×C67C2211yescharacteristic
17#417×C67—22no—
18C2×C22⋊C8×C67—22no—
19C2×C4×C8×C67C2211yescharacteristic
20D4⋊C8×C67—24no—
21C2×#41×C67—22no—
22C22×D4×C67D411yescharacteristic
23C23×C4×C67—21no—
24AGL(1,ℤ/8)×C67—44no—
25AGL(1,ℤ/8)×C67—44no—
26C2×SD16×C67—24no—
27C2×D8×C67—24no—
28C2×D8×C67—22no—
29C2×M16×C67—22no—
30C22×C8×C67—21no—
31C22×C8×C67D412yes—
32C4⋊Q8×C67D412yes—
33#127×C67D412yes—
34#124×C67—22no—
35#121×C67—22no—
36#121×C67—22no—
37C4×D4×C67—22no—
38C4×D4×C67—22no—
39C4×D4×C67—22no—
40C4×D4×C67—22no—
41C2×C4⋊C4×C67D411yescharacteristic
42C2×C22⋊C4×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—
51C22⋊C8×C67—44no—
52C4×C8×C67—22no—
53C24×C67—21no—
54#41×C67—42no—
55#41×C67—42no—
56#41×C67—42no—
57#41×C67—42no—
58C2×Q8×C67—22no—
59C2×D4×C67—22no—
60C2×D4×C67—22no—
61C2×D4×C67—22no—
62C2×D4×C67—22no—
63C2×D4×C67—21no—
64C2×D4×C67—22no—
65C2×D4×C67—42no—
66C2×D4×C67D812yescomplemented
67C22×C4×C67—21no—
68C22×C4×C67—42no—
69C22×C4×C67—42no—
70C22×C4×C67—41no—
71C22×C4×C67—42no—
72C22×C4×C67—41no—
73C22×C4×C67C2×D411yescharacteristic
74C22×C4×C67C2×D411yescharacteristic
75C22×C4×C67#4111yescharacteristic
76SD16×C67—48no—
77D8×C67—48no—
78D8×C67—82no—
79M16×C67—42no—
80M16×C67—42no—
81C2×C8×C67—22no—
82C2×C8×C67—22no—
83C2×C8×C67—24no—
84C2×C8×C67—41no—
85C2×C8×C67—41no—
86C2×C8×C67—42no—
87C4⋊C4×C67—22no—
88C4⋊C4×C67—22no—
89C4⋊C4×C67—22no—
90C22⋊C4×C67—42no—
91C22⋊C4×C67—42no—
92C22⋊C4×C67—42no—
93C42×C67C2×D412yes—
94C42×C67C2×D412yes—
95C23×C67—22no—
96C23×C67—42no—
97C23×C67—42no—
98C23×C67—41no—
99C23×C67—41no—
100C23×C67—42no—
101C23×C67#12111yescharacteristic
102Q8×C67—42no—
103Q8×C67—42no—
104D4×C67—24no—
105D4×C67—44no—
106D4×C67—42no—
107D4×C67—42no—
108D4×C67—42no—
109D4×C67—42no—
110D4×C67—42no—
111D4×C67—82no—
112D4×C67—82no—
113C2×C4×C67—21no—
114C2×C4×C67—21no—
115C2×C4×C67—41no—
116C2×C4×C67—42no—
117C2×C4×C67—41no—
118C2×C4×C67—42no—
119C2×C4×C67—82no—
120C2×C4×C67—82no—
121C2×C4×C67—82no—
122C2×C4×C67—81no—
123C2×C4×C67—81no—
124C2×C4×C67#12012yes—
125C2×C4×C67#12011yescharacteristic
126C2×C4×C67#12112yes—
127C2×C4×C67#12711yescharacteristic
128C2×C4×C67C2×D811yescharacteristic
129C2×C4×C67#13511yescharacteristic
130C2×C4×C67AGL(1,ℤ/8)12yes—
131C8×C67—42no—
132C8×C67—44no—
133C8×C67—42no—
134C22×C67—21no—
135C22×C67—21no—
136C22×C67—44no—
137C22×C67—42no—
138C22×C67—82no—
139C22×C67—81no—
140C22×C67—81no—
141C22×C67—82no—
142C22×C67—82no—
143C22×C67—82no—
144C22×C67#39211yescharacteristic
145C22×C67#46611yescharacteristic
146C22×C67#46911yescharacteristic
147C4×C67—21no—
148C4×C67—22no—
149C4×C67—82no—
150C4×C67—81no—
151C4×C67#44712yes—
152C4×C67#44912yes—
153C4×C67#45311yescharacteristic
154C4×C67#45411yescharacteristic
155#10254C6711yesmaximal, characteristic, direct product factor, Sylow
156C2×C67—21no—
157C2×C67—21no—
158C2×C67—82no—
159C2×C67—162no—
160C2×C67#199911yescharacteristic
161C2×C67#200611yescharacteristic
162C2×C67#201711yescharacteristic
163#3245C2×C6711yescharacteristic, complemented
164#3036C2×C6712yescomplemented
165#1954C2×C6711yescharacteristic, complemented
166#1492C2×C6712yescomplemented
167#1292C2×C6711yescharacteristic, complemented
168C67#1025411yescharacteristic, direct product factor, Sylow
169C2×AGL(1,ℤ/8)—22no—
170#532C22×C6712yes—
171C2×C4×D4C22×C6711yescharacteristic
172#474—22no—
173#466—22no—
174#463—24no—
175#461—22no—
176#459—22no—
177#459—22no—
178#423C22×C6712yes—
179C2×C4⋊C8C22×C6711yescharacteristic
180#417—22no—
181C2×C22⋊C8—22no—
182C2×C4×C8C22×C6711yescharacteristic
183D4⋊C8—24no—
184C2×#41—22no—
185C22×D4D4×C6711yescharacteristic
186C23×C4—21no—
187AGL(1,ℤ/8)—44no—
188AGL(1,ℤ/8)—44no—
189C2×SD16—24no—
190C2×D8—24no—
191C2×D8—22no—
192C2×M16—22no—
193C22×C8—21no—
194C22×C8D4×C6712yes—
195C4⋊Q8D4×C6712yes—
196#127D4×C6712yes—
197#124—22no—
198#121—22no—
199#121—22no—
200C4×D4—22no—
201C4×D4—22no—
202C4×D4—22no—
203C4×D4—22no—
204C2×C4⋊C4D4×C6711yescharacteristic
205C2×C22⋊C4—21no—
206C2×C42C23×C6711yescharacteristic
207#107—22no—
208C4⋊C8—22no—
209C4⋊C8—22no—
210C4⋊C8D4×C6714yes—
211#103—24no—
212#102—24no—
213#102—24no—
214C22⋊C8—44no—
215C4×C8—22no—
216C24—21no—
217#41—42no—
218#41—42no—
219#41—42no—
220#41—42no—
221C2×Q8—22no—
222C2×D4—22no—
223C2×D4—22no—
224C2×D4—22no—
225C2×D4—22no—
226C2×D4—21no—
227C2×D4—22no—
228C2×D4—42no—
229C2×D4D8×C6712yescomplemented
230C22×C4—21no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—41no—
234C22×C4—42no—
235C22×C4—41no—
236C22×C4C2×D4×C6711yescharacteristic
237C22×C4C2×D4×C6711yescharacteristic
238C22×C4#41×C6711yescharacteristic
239SD16—48no—
240D8—48no—
241D8—82no—
242M16—42no—
243M16—42no—
244C2×C8—22no—
245C2×C8—22no—
246C2×C8—24no—
247C2×C8—41no—
248C2×C8—41no—
249C2×C8—42no—
250C4⋊C4—22no—
251C4⋊C4—22no—
252C4⋊C4—22no—
253C22⋊C4—42no—
254C22⋊C4—42no—
255C22⋊C4—42no—
256C42C2×D4×C6712yes—
257C42C2×D4×C6712yes—
258C23—22no—
259C23—42no—
260C23—42no—
261C23—41no—
262C23—41no—
263C23—42no—
264C23#121×C6711yescharacteristic
265Q8—42no—
266Q8—42no—
267D4—24no—
268D4—44no—
269D4—42no—
270D4—42no—
271D4—42no—
272D4—42no—
273D4—42no—
274D4—82no—
275D4—82no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—41no—
281C2×C4—42no—
282C2×C4—82no—
283C2×C4—82no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4—82no—
287C2×C4#120×C6712yes—
288C2×C4#120×C6711yescharacteristic
289C2×C4#121×C6712yes—
290C2×C4#127×C6711yescharacteristic
291C2×C4C2×D8×C6711yescharacteristic
292C2×C4#135×C6711yescharacteristic
293C2×C4AGL(1,ℤ/8)×C6712yes—
294C8—42no—
295C8—42no—
296C8—44no—
297C22—21no—
298C22—21no—
299C22—44no—
300C22—42no—
301C22—82no—
302C22—81no—
303C22—81no—
304C22—82no—
305C22—82no—
306C22—82no—
307C22#392×C6711yescharacteristic
308C22#466×C6711yescharacteristic
309C22#469×C6711yescharacteristic
310C4—21no—
311C4—22no—
312C4—82no—
313C4—81no—
314C4#447×C6712yes—
315C4#449×C6712yes—
316C4#453×C6711yescharacteristic
317C4#454×C6711yescharacteristic
318C2—21no—
319C2—21no—
320C2—82no—
321C2—162no—
322C2C67×#199911yescharacteristic
323C2C67×#200611yescharacteristic
324C2C67×#201711yescharacteristic
subgroupG/H
center, socleC22×C67#392
commutator, Frattini, cosocleC2×C42C23×C67
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
1. C67×#3245C2semidirect
2. C67×#3036C2semidirect
3. C67×#1954C2semidirect
4. C67×#1492C2semidirect
5. C67×#1292C2semidirect
66. C2×D4×C67D8semidirect
155. #10254C67direct (N × H)
163. #3245C2×C67semidirect
164. #3036C2×C67semidirect
165. #1954C2×C67semidirect
166. #1492C2×C67semidirect
167. #1292C2×C67semidirect
168. C67#10254direct (N × H)
229. C2×D4D8×C67semidirect

Maximal subgroups

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

subgroupindex
C67×#32452
C67×#30362
C67×#19542
C67×#14922
C67×#12922
#1025467