Finite Group Databasebeta

← C2×C67×#1320 #30690686

Subgroups — C2×C67×#1320 #30690686

Order 17152.

626 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#407×C67C211yesmaximal, characteristic
2C2×#349×C67C212yesmaximal, complemented
3C67×#1320C214yesmaximal, direct product factor
4C24×C4×C67C411yescharacteristic
5C22×M16×C67C411yescharacteristic
6C22×M16×C67C2211yescharacteristic
7C2×M32×C67—22no—
8#407×C67—22no—
9#407×C67C412yes—
10#407×C67C2212yes—
11C2×C22⋊C8×C67—22no—
12#349×C67—22no—
13#349×C67C2214yescomplemented
14C25×C67C811yescharacteristic
15C23×C4×C67—21no—
16C23×C4×C67—21no—
17C23×C4×C67—21no—
18C23×C4×C67—21no—
19C23×C4×C67—41no—
20C23×C4×C67—41no—
21C23×C4×C67—41no—
22C23×C4×C67—41no—
23C23×C4×C67C811yescharacteristic
24C23×C4×C67C812yes—
25C23×C4×C67C2×C411yescharacteristic
26C23×C4×C67C2×C412yes—
27C2×M16×C67—21no—
28C2×M16×C67—41no—
29C2×M16×C67—41no—
30C2×M16×C67—41no—
31C2×M16×C67—41no—
32C2×M16×C67C2×C412yes—
33C2×M16×C67C2×C411yescharacteristic
34C2×M16×C67D412yes—
35C2×M16×C67C2311yescharacteristic
36C22×C8×C67—21no—
37C22×C8×C67D412yes—
38M32×C67—24no—
39M32×C67—42no—
40C2×C16×C67—42no—
41C22⋊C8×C67—24no—
42C22⋊C8×C67—24no—
43C24×C67—21no—
44C24×C67—21no—
45C24×C67—41no—
46C24×C67—41no—
47C24×C67—44no—
48C24×C67C2×C811yescharacteristic
49C24×C67C2×C812yes—
50C22×C4×C67—21no—
51C22×C4×C67—21no—
52C22×C4×C67—21no—
53C22×C4×C67—21no—
54C22×C4×C67—21no—
55C22×C4×C67—21no—
56C22×C4×C67—22no—
57C22×C4×C67—22no—
58C22×C4×C67—22no—
59C22×C4×C67—22no—
60C22×C4×C67—42no—
61C22×C4×C67—42no—
62C22×C4×C67—42no—
63C22×C4×C67—42no—
64C22×C4×C67—42no—
65C22×C4×C67—42no—
66C22×C4×C67—42no—
67C22×C4×C67—42no—
68C22×C4×C67—42no—
69C22×C4×C67—42no—
70C22×C4×C67—41no—
71C22×C4×C67—41no—
72C22×C4×C67—41no—
73C22×C4×C67—41no—
74C22×C4×C67—41no—
75C22×C4×C67—41no—
76C22×C4×C67C22⋊C411yescharacteristic
77C22×C4×C67C22⋊C411yescharacteristic
78C22×C4×C67C2×C811yescharacteristic
79C22×C4×C67C2×C812yes—
80C22×C4×C67M1611yescharacteristic
81C22×C4×C67M1611yescharacteristic
82C22×C4×C67C22×C411yescharacteristic
83M16×C67—41no—
84M16×C67—42no—
85M16×C67—41no—
86M16×C67—42no—
87M16×C67—42no—
88C2×C8×C67—22no—
89C2×C8×C67—22no—
90C2×C8×C67—22no—
91C2×C8×C67—41no—
92C2×C8×C67—41no—
93C2×C8×C67C22⋊C412yes—
94C2×C8×C67C2×D412yes—
95C16×C67—44no—
96C23×C67—24no—
97C23×C67—21no—
98C23×C67—21no—
99C23×C67—22no—
100C23×C67—42no—
101C23×C67—42no—
102C23×C67—44no—
103C23×C67—42no—
104C23×C67—42no—
105C23×C67—44no—
106C23×C67—44no—
107C23×C67—48no—
108C23×C67—44no—
109C23×C67—41no—
110C23×C67—41no—
111C23×C67C22⋊C811yescharacteristic
112C23×C67C22⋊C811yescharacteristic
113C23×C67C22×C811yescharacteristic
114C2×C4×C67—21no—
115C2×C4×C67—21no—
116C2×C4×C67—21no—
117C2×C4×C67—21no—
118C2×C4×C67—21no—
119C2×C4×C67—21no—
120C2×C4×C67—21no—
121C2×C4×C67—21no—
122C2×C4×C67—21no—
123C2×C4×C67—21no—
124C2×C4×C67—22no—
125C2×C4×C67—44no—
126C2×C4×C67—44no—
127C2×C4×C67—42no—
128C2×C4×C67—41no—
129C2×C4×C67—41no—
130C2×C4×C67—42no—
131C2×C4×C67—42no—
132C2×C4×C67—42no—
133C2×C4×C67—42no—
134C2×C4×C67—42no—
135C2×C4×C67C22⋊C811yescharacteristic
136C2×C4×C67C22⋊C811yescharacteristic
137C2×C4×C67#9911yescharacteristic
138C2×C4×C67#9911yescharacteristic
139C2×C4×C67#10011yescharacteristic
140C2×C4×C67#10011yescharacteristic
141C2×C4×C67C2×C22⋊C411yescharacteristic
142C2×C4×C67C2×M1611yescharacteristic
143C8×C67—22no—
144C8×C67—22no—
145C8×C67—42no—
146C22×C67—22no—
147C22×C67—24no—
148C22×C67—21no—
149C22×C67—21no—
150C22×C67—41no—
151C22×C67—41no—
152C22×C67—42no—
153C22×C67—44no—
154C22×C67—42no—
155C22×C67—44no—
156C22×C67—44no—
157C22×C67—48no—
158C22×C67—44no—
159C22×C67—42no—
160C22×C67—42no—
161C22×C67#32311yescharacteristic
162C22×C67#32311yescharacteristic
163C22×C67C2×C22⋊C811yescharacteristic
164C4×C67—21no—
165C4×C67—21no—
166C4×C67—42no—
167C4×C67#32311yescharacteristic
168C4×C67#32311yescharacteristic
169C4×C67C2×#9911yescharacteristic
170C4×C67C2×#10011yescharacteristic
171C2×#1320C6711yesmaximal, characteristic, direct product factor, Sylow
172C2×C67—21no—
173C2×C67—21no—
174C2×C67—44no—
175C2×C67—41no—
176C2×C67—41no—
177C2×C67#132012yesdirect product factor
178C2×C67C2×#32311yescharacteristic
179C2×#407C2×C6711yescharacteristic
180C2×#349C2×C6712yescomplemented
181#1320C2×C6714yesdirect product factor
182C67C2×#132011yescharacteristic, direct product factor, Sylow
183C24×C4C4×C6711yescharacteristic
184C22×M16C4×C6711yescharacteristic
185C22×M16C22×C6711yescharacteristic
186C2×M32—22no—
187#407—22no—
188#407C4×C6712yes—
189#407C22×C6712yes—
190C2×C22⋊C8—22no—
191#349—22no—
192#349C22×C6714yescomplemented
193C25C8×C6711yescharacteristic
194C23×C4—21no—
195C23×C4—21no—
196C23×C4—21no—
197C23×C4—21no—
198C23×C4—41no—
199C23×C4—41no—
200C23×C4—41no—
201C23×C4—41no—
202C23×C4C8×C6711yescharacteristic
203C23×C4C8×C6712yes—
204C23×C4C2×C4×C6711yescharacteristic
205C23×C4C2×C4×C6712yes—
206C2×M16—21no—
207C2×M16—41no—
208C2×M16—41no—
209C2×M16—41no—
210C2×M16—41no—
211C2×M16C2×C4×C6712yes—
212C2×M16C2×C4×C6711yescharacteristic
213C2×M16D4×C6712yes—
214C2×M16C23×C6711yescharacteristic
215C22×C8—21no—
216C22×C8D4×C6712yes—
217M32—24no—
218M32—42no—
219C2×C16—42no—
220C22⋊C8—24no—
221C22⋊C8—24no—
222C24—21no—
223C24—21no—
224C24—41no—
225C24—41no—
226C24—44no—
227C24C2×C8×C6711yescharacteristic
228C24C2×C8×C6712yes—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4C22⋊C4×C6711yescharacteristic
256C22×C4C22⋊C4×C6711yescharacteristic
257C22×C4C2×C8×C6711yescharacteristic
258C22×C4C2×C8×C6712yes—
259C22×C4M16×C6711yescharacteristic
260C22×C4M16×C6711yescharacteristic
261C22×C4C22×C4×C6711yescharacteristic
262M16—41no—
263M16—42no—
264M16—41no—
265M16—42no—
266M16—42no—
267C2×C8—22no—
268C2×C8—22no—
269C2×C8—22no—
270C2×C8—41no—
271C2×C8—41no—
272C2×C8C22⋊C4×C6712yes—
273C2×C8C2×D4×C6712yes—
274C16—44no—
275C23—24no—
276C23—21no—
277C23—21no—
278C23—22no—
279C23—42no—
280C23—42no—
281C23—44no—
282C23—42no—
283C23—42no—
284C23—44no—
285C23—44no—
286C23—48no—
287C23—44no—
288C23—41no—
289C23—41no—
290C23C22⋊C8×C6711yescharacteristic
291C23C22⋊C8×C6711yescharacteristic
292C23C22×C8×C6711yescharacteristic
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—22no—
304C2×C4—44no—
305C2×C4—44no—
306C2×C4—42no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4C22⋊C8×C6711yescharacteristic
315C2×C4C22⋊C8×C6711yescharacteristic
316C2×C4#99×C6711yescharacteristic
317C2×C4#99×C6711yescharacteristic
318C2×C4#100×C6711yescharacteristic
319C2×C4#100×C6711yescharacteristic
320C2×C4C2×C22⋊C4×C6711yescharacteristic
321C2×C4C2×M16×C6711yescharacteristic
322C8—22no—
323C8—22no—
324C8—42no—
325C22—22no—
326C22—24no—
327C22—21no—
328C22—21no—
329C22—41no—
330C22—41no—
331C22—42no—
332C22—44no—
333C22—42no—
334C22—44no—
335C22—44no—
336C22—48no—
337C22—44no—
338C22—42no—
339C22—42no—
340C22#323×C6711yescharacteristic
341C22#323×C6711yescharacteristic
342C22C2×C22⋊C8×C6711yescharacteristic
343C4—21no—
344C4—21no—
345C4—42no—
346C4#323×C6711yescharacteristic
347C4#323×C6711yescharacteristic
348C4C2×#99×C6711yescharacteristic
349C4C2×#100×C6711yescharacteristic
350C2—21no—
351C2—21no—
352C2—44no—
353C2—41no—
354C2—41no—
355C2C67×#132012yesdirect product factor
356C2C2×#323×C6711yescharacteristic
subgroupG/H
centerC2×C4×C67#99
commutatorC23C22×C8×C67
Frattini, cosocleC2×M16C23×C67
socleC22×C67#323
prime radicalC25×C67C8
LCSG ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C67 ◁ C22×C4×C67 ◁ C22×M16×C67 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#349×C67C2semidirect
3. C67×#1320C2direct (N × H)
13. #349×C67C22semidirect
171. C2×#1320C67direct (N × H)
177. C2×C67#1320direct (N × H)
180. C2×#349C2×C67semidirect
181. #1320C2×C67direct (N × H)
182. C67C2×#1320direct (N × H)
192. #349C22×C67semidirect
355. C2C67×#1320direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#407×C672
C2×#349×C672
C67×#13202
C2×#132067