Finite Group Databasebeta

← C2×C67×#1611 #30697368

Subgroups — C2×C67×#1611 #30697368

Order 17152.

1028 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#544×C67C211yesmaximal, characteristic
2C2×#483×C67C212yesmaximal
3C2×#398×C67C211yesmaximal, characteristic
4C67×#1611C218yesmaximal, direct product factor
5C2×#328×C67C212yesmaximal
6C2×#324×C67C211yesmaximal, characteristic
7#544×C67—24no—
8#544×C67—22no—
9#544×C67C2214yes—
10C2×C4⋊Q8×C67—22no—
11C2×#125×C67—22no—
12C2×C22⋊Q8×C67—21no—
13C2×C22⋊Q8×C67C2212yes—
14C2×#117×C67—21no—
15C22×C4⋊C4×C67—21no—
16C22×C4⋊C4×C67C2211yescharacteristic
17C22×C4⋊C4×C67C2211yescharacteristic
18#483×C67—24no—
19#483×C67C2218yes—
20C2×#106×C67—22no—
21C2×#103×C67—22no—
22C2×C22⋊C8×C67C2212yes—
23#398×C67—22no—
24#398×C67C2214yes—
25C2×#95×C67—21no—
26C2×#95×C67C2211yescharacteristic
27#328×C67C2218yes—
28#324×C67C2214yes—
29C22×Q8×C67—21no—
30C22×Q8×C67D412yes—
31C23×C4×C67—21no—
32C23×C4×C67—21no—
33C23×C4×C67D411yescharacteristic
34C23×C4×C67D411yescharacteristic
35C23×C4×C67C2311yescharacteristic
36C22×C8×C67—22no—
37C4⋊Q8×C67—28no—
38C4⋊Q8×C67—28no—
39#125×C67—28no—
40#125×C67—28no—
41C22⋊Q8×C67—24no—
42C22⋊Q8×C67—24no—
43C22⋊Q8×C67—22no—
44C22⋊Q8×C67—22no—
45C22⋊Q8×C67D414yes—
46C22⋊Q8×C67C2314yes—
47#117×C67—24no—
48#117×C67—24no—
49C2×C4⋊C4×C67—21no—
50C2×C4⋊C4×C67—21no—
51C2×C4⋊C4×C67—21no—
52C2×C4⋊C4×C67—22no—
53C2×C4⋊C4×C67—22no—
54C2×C4⋊C4×C67—22no—
55C2×C4⋊C4×C67—22no—
56C2×C4⋊C4×C67—22no—
57C2×C4⋊C4×C67—22no—
58C2×C4⋊C4×C67—24no—
59C2×C4⋊C4×C67—21no—
60C2×C4⋊C4×C67—44no—
61C2×C4⋊C4×C67D412yes—
62C2×C4⋊C4×C67D412yes—
63C2×C4⋊C4×C67C2312yes—
64C2×C4⋊C4×C67C2312yes—
65C2×C22⋊C4×C67—22no—
66C2×C42×C67—22no—
67#106×C67—28no—
68#103×C67—28no—
69C22⋊C8×C67—22no—
70C22⋊C8×C67C2314yes—
71#95×C67—24no—
72#95×C67—21no—
73#95×C67C2312yes—
74C24×C67C2×D411yescharacteristic
75C2×Q8×C67—24no—
76C2×Q8×C67—22no—
77C2×Q8×C67—44no—
78C2×Q8×C67—82no—
79C2×Q8×C67C2×D414yes—
80C22×C4×C67—21no—
81C22×C4×C67—21no—
82C22×C4×C67—22no—
83C22×C4×C67—24no—
84C22×C4×C67—22no—
85C22×C4×C67—22no—
86C22×C4×C67—21no—
87C22×C4×C67—22no—
88C22×C4×C67—21no—
89C22×C4×C67—22no—
90C22×C4×C67—42no—
91C22×C4×C67—42no—
92C22×C4×C67—42no—
93C22×C4×C67—42no—
94C22×C4×C67—41no—
95C22×C4×C67—41no—
96C22×C4×C67—42no—
97C22×C4×C67—42no—
98C22×C4×C67SD1614yes—
99C22×C4×C67C2×D412yes—
100C22×C4×C67C2×D412yes—
101C22×C4×C67C2×D412yes—
102C22×C4×C67C2×D411yescharacteristic
103C22×C4×C67C2×D412yes—
104C22×C4×C67C2411yescharacteristic
105C2×C8×C67—24no—
106C2×C8×C67—44no—
107C4⋊C4×C67—24no—
108C4⋊C4×C67—24no—
109C4⋊C4×C67—24no—
110C4⋊C4×C67—28no—
111C4⋊C4×C67—24no—
112C4⋊C4×C67—24no—
113C4⋊C4×C67—22no—
114C4⋊C4×C67—24no—
115C4⋊C4×C67—24no—
116C4⋊C4×C67—24no—
117C4⋊C4×C67—42no—
118C4⋊C4×C67—48no—
119C4⋊C4×C67C2×D414yes—
120C22⋊C4×C67—24no—
121C22⋊C4×C67—24no—
122C42×C67—28no—
123C23×C67—22no—
124C23×C67—22no—
125C23×C67#12011yescharacteristic
126C23×C67#12011yescharacteristic
127C23×C67C2×SD1612yes—
128C23×C67#13712yes—
129C23×C67C22×D411yescharacteristic
130Q8×C67—44no—
131Q8×C67—82no—
132Q8×C67—84no—
133C2×C4×C67—22no—
134C2×C4×C67—22no—
135C2×C4×C67—24no—
136C2×C4×C67—24no—
137C2×C4×C67—24no—
138C2×C4×C67—24no—
139C2×C4×C67—24no—
140C2×C4×C67—24no—
141C2×C4×C67—22no—
142C2×C4×C67—42no—
143C2×C4×C67—41no—
144C2×C4×C67—41no—
145C2×C4×C67—48no—
146C2×C4×C67—42no—
147C2×C4×C67—42no—
148C2×C4×C67—44no—
149C2×C4×C67—42no—
150C2×C4×C67—44no—
151C2×C4×C67—42no—
152C2×C4×C67—42no—
153C2×C4×C67—42no—
154C2×C4×C67—42no—
155C2×C4×C67—81no—
156C2×C4×C67—82no—
157C2×C4×C67—82no—
158C2×C4×C67#12012yes—
159C2×C4×C67C2×SD1614yes—
160C2×C4×C67C22×D412yes—
161C8×C67—44no—
162C22×C67—24no—
163C22×C67—24no—
164C22×C67—21no—
165C22×C67—21no—
166C22×C67—21no—
167C22×C67—21no—
168C22×C67#44812yes—
169C22×C67#44812yes—
170C22×C67#45612yes—
171C22×C67#45812yes—
172C22×C67C2×#12011yescharacteristic
173C22×C67C22×SD1611yescharacteristic
174C22×C67C2×#13711yescharacteristic
175C4×C67—24no—
176C4×C67—22no—
177C4×C67—22no—
178C4×C67—44no—
179C4×C67—44no—
180C4×C67—42no—
181C4×C67—44no—
182C4×C67—82no—
183C2×#1611C6711yesmaximal, characteristic, direct product factor, Sylow
184C2×C67—21no—
185C2×C67—21no—
186C2×C67—22no—
187C2×C67#161114yesdirect product factor
188C2×C67C2×#44811yescharacteristic
189C2×C67C2×#45611yescharacteristic
190C2×C67C2×#45811yescharacteristic
191C2×#544C2×C6711yescharacteristic
192C2×#483C2×C6712yes—
193C2×#398C2×C6711yescharacteristic
194#1611C2×C6718yesdirect product factor
195C2×#328C2×C6712yes—
196C2×#324C2×C6711yescharacteristic
197C67C2×#161111yescharacteristic, direct product factor, Sylow
198#544—24no—
199#544—22no—
200#544C22×C6714yes—
201C2×C4⋊Q8—22no—
202C2×#125—22no—
203C2×C22⋊Q8—21no—
204C2×C22⋊Q8C22×C6712yes—
205C2×#117—21no—
206C22×C4⋊C4—21no—
207C22×C4⋊C4C22×C6711yescharacteristic
208C22×C4⋊C4C22×C6711yescharacteristic
209#483—24no—
210#483C22×C6718yes—
211C2×#106—22no—
212C2×#103—22no—
213C2×C22⋊C8C22×C6712yes—
214#398—22no—
215#398C22×C6714yes—
216C2×#95—21no—
217C2×#95C22×C6711yescharacteristic
218#328C22×C6718yes—
219#324C22×C6714yes—
220C22×Q8—21no—
221C22×Q8D4×C6712yes—
222C23×C4—21no—
223C23×C4—21no—
224C23×C4D4×C6711yescharacteristic
225C23×C4D4×C6711yescharacteristic
226C23×C4C23×C6711yescharacteristic
227C22×C8—22no—
228C4⋊Q8—28no—
229C4⋊Q8—28no—
230#125—28no—
231#125—28no—
232C22⋊Q8—24no—
233C22⋊Q8—24no—
234C22⋊Q8—22no—
235C22⋊Q8—22no—
236C22⋊Q8D4×C6714yes—
237C22⋊Q8C23×C6714yes—
238#117—24no—
239#117—24no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4D4×C6712yes—
253C2×C4⋊C4D4×C6712yes—
254C2×C4⋊C4C23×C6712yes—
255C2×C4⋊C4C23×C6712yes—
256C2×C22⋊C4—22no—
257C2×C42—22no—
258#106—28no—
259#103—28no—
260C22⋊C8—22no—
261C22⋊C8C23×C6714yes—
262#95—24no—
263#95—21no—
264#95C23×C6712yes—
265C24C2×D4×C6711yescharacteristic
266C2×Q8—24no—
267C2×Q8—22no—
268C2×Q8—44no—
269C2×Q8—82no—
270C2×Q8C2×D4×C6714yes—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—24no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—21no—
280C22×C4—22no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4SD16×C6714yes—
290C22×C4C2×D4×C6712yes—
291C22×C4C2×D4×C6712yes—
292C22×C4C2×D4×C6712yes—
293C22×C4C2×D4×C6711yescharacteristic
294C22×C4C2×D4×C6712yes—
295C22×C4C24×C6711yescharacteristic
296C2×C8—24no—
297C2×C8—44no—
298C4⋊C4—24no—
299C4⋊C4—24no—
300C4⋊C4—24no—
301C4⋊C4—28no—
302C4⋊C4—24no—
303C4⋊C4—24no—
304C4⋊C4—22no—
305C4⋊C4—24no—
306C4⋊C4—24no—
307C4⋊C4—24no—
308C4⋊C4—42no—
309C4⋊C4—48no—
310C4⋊C4C2×D4×C6714yes—
311C22⋊C4—24no—
312C22⋊C4—24no—
313C42—28no—
314C23—22no—
315C23—22no—
316C23#120×C6711yescharacteristic
317C23#120×C6711yescharacteristic
318C23C2×SD16×C6712yes—
319C23#137×C6712yes—
320C23C22×D4×C6711yescharacteristic
321Q8—44no—
322Q8—82no—
323Q8—84no—
324C2×C4—24no—
325C2×C4—24no—
326C2×C4—24no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—24no—
332C2×C4—22no—
333C2×C4—42no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—42no—
337C2×C4—48no—
338C2×C4—44no—
339C2×C4—42no—
340C2×C4—44no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—81no—
347C2×C4—82no—
348C2×C4—82no—
349C2×C4#120×C6712yes—
350C2×C4C2×SD16×C6714yes—
351C2×C4C22×D4×C6712yes—
352C8—44no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—24no—
358C22—24no—
359C22#448×C6712yes—
360C22#448×C6712yes—
361C22#456×C6712yes—
362C22#458×C6712yes—
363C22C2×#120×C6711yescharacteristic
364C22C22×SD16×C6711yescharacteristic
365C22C2×#137×C6711yescharacteristic
366C4—22no—
367C4—22no—
368C4—24no—
369C4—44no—
370C4—44no—
371C4—42no—
372C4—44no—
373C4—82no—
374C2—21no—
375C2—22no—
376C2—21no—
377C2C67×#161114yesdirect product factor
378C2C2×#448×C6711yescharacteristic
379C2C2×#456×C6711yescharacteristic
380C2C2×#458×C6711yescharacteristic
subgroupG/H
center, socleC23×C67#120
commutator, Frattini, cosocleC22×C4C24×C67
prime radicalC24×C67C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C67 ◁ C23×C4×C67 ◁ G

Complements

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

NH ≅ G/Ntype
4. C67×#1611C2direct (N × H)
183. C2×#1611C67direct (N × H)
187. C2×C67#1611direct (N × H)
194. #1611C2×C67direct (N × H)
197. C67C2×#1611direct (N × H)
377. C2C67×#1611direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#544×C672
C2×#483×C672
C2×#398×C672
C67×#16112
C2×#328×C672
C2×#324×C672
C2×#161167