Finite Group Databasebeta

← C2×C67×#1617 #30697369

Subgroups — C2×C67×#1617 #30697369

Order 17152.

984 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#544×C67C211yesmaximal, characteristic
2C2×#484×C67C212yesmaximal
3C2×#400×C67C211yesmaximal, characteristic
4C67×#1617C218yesmaximal, 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×C67C2211yescharacteristic
16C22×C4⋊C4×C67C2211yescharacteristic
17#484×C67—24no—
18#484×C67C2218yes—
19C2×#107×C67—22no—
20C2×#103×C67—22no—
21C2×C22⋊C8×C67C2212yes—
22#400×C67—22no—
23#400×C67C2214yes—
24C2×#95×C67—21no—
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—21no—
59C2×C4⋊C4×C67D412yes—
60C2×C4⋊C4×C67D412yes—
61C2×C4⋊C4×C67C2312yes—
62C2×C4⋊C4×C67C2312yes—
63C2×C22⋊C4×C67—22no—
64C2×C42×C67—22no—
65#107×C67—28no—
66#103×C67—28no—
67C22⋊C8×C67—22no—
68C22⋊C8×C67C2314yes—
69#95×C67—24no—
70#95×C67—24no—
71#95×C67—21no—
72#95×C67C2312yes—
73C24×C67C2×D411yescharacteristic
74C2×Q8×C67—24no—
75C2×Q8×C67—22no—
76C2×Q8×C67—44no—
77C2×Q8×C67—82no—
78C2×Q8×C67C2×D414yes—
79C22×C4×C67—21no—
80C22×C4×C67—21no—
81C22×C4×C67—22no—
82C22×C4×C67—22no—
83C22×C4×C67—22no—
84C22×C4×C67—21no—
85C22×C4×C67—21no—
86C22×C4×C67—22no—
87C22×C4×C67—22no—
88C22×C4×C67—42no—
89C22×C4×C67—42no—
90C22×C4×C67—42no—
91C22×C4×C67—42no—
92C22×C4×C67—44no—
93C22×C4×C67—41no—
94C22×C4×C67—41no—
95C22×C4×C67—42no—
96C22×C4×C67Q1614yes—
97C22×C4×C67C2×D412yes—
98C22×C4×C67C2×D412yes—
99C22×C4×C67C2×D412yes—
100C22×C4×C67C2×D411yescharacteristic
101C22×C4×C67C2×D412yes—
102C22×C4×C67C2411yescharacteristic
103C2×C8×C67—24no—
104C2×C8×C67—44no—
105C4⋊C4×C67—24no—
106C4⋊C4×C67—24no—
107C4⋊C4×C67—24no—
108C4⋊C4×C67—28no—
109C4⋊C4×C67—24no—
110C4⋊C4×C67—24no—
111C4⋊C4×C67—22no—
112C4⋊C4×C67—24no—
113C4⋊C4×C67—24no—
114C4⋊C4×C67—24no—
115C4⋊C4×C67—42no—
116C4⋊C4×C67C2×D414yes—
117C22⋊C4×C67—24no—
118C22⋊C4×C67—24no—
119C42×C67—28no—
120C23×C67—22no—
121C23×C67—22no—
122C23×C67#12011yescharacteristic
123C23×C67#12011yescharacteristic
124C23×C67C2×Q1612yes—
125C23×C67#13712yes—
126C23×C67C22×D411yescharacteristic
127Q8×C67—44no—
128Q8×C67—82no—
129Q8×C67—84no—
130C2×C4×C67—22no—
131C2×C4×C67—22no—
132C2×C4×C67—24no—
133C2×C4×C67—24no—
134C2×C4×C67—24no—
135C2×C4×C67—24no—
136C2×C4×C67—24no—
137C2×C4×C67—22no—
138C2×C4×C67—42no—
139C2×C4×C67—41no—
140C2×C4×C67—41no—
141C2×C4×C67—44no—
142C2×C4×C67—42no—
143C2×C4×C67—44no—
144C2×C4×C67—42no—
145C2×C4×C67—42no—
146C2×C4×C67—42no—
147C2×C4×C67—44no—
148C2×C4×C67—42no—
149C2×C4×C67—42no—
150C2×C4×C67—81no—
151C2×C4×C67—84no—
152C2×C4×C67—82no—
153C2×C4×C67—82no—
154C2×C4×C67#12012yes—
155C2×C4×C67C2×Q1614yes—
156C2×C4×C67C22×D412yes—
157C8×C67—44no—
158C22×C67—24no—
159C22×C67—24no—
160C22×C67—21no—
161C22×C67—21no—
162C22×C67—21no—
163C22×C67—21no—
164C22×C67#45112yes—
165C22×C67#45112yes—
166C22×C67#45412yes—
167C22×C67#45812yes—
168C22×C67C2×#12011yescharacteristic
169C22×C67C22×Q1611yescharacteristic
170C22×C67C2×#13711yescharacteristic
171C4×C67—24no—
172C4×C67—22no—
173C4×C67—22no—
174C4×C67—44no—
175C4×C67—42no—
176C4×C67—44no—
177C4×C67—82no—
178C4×C67—82no—
179C2×#1617C6711yesmaximal, characteristic, direct product factor, Sylow
180C2×C67—21no—
181C2×C67—21no—
182C2×C67—22no—
183C2×C67#161714yesdirect product factor
184C2×C67C2×#45111yescharacteristic
185C2×C67C2×#45411yescharacteristic
186C2×C67C2×#45811yescharacteristic
187C2×#544C2×C6711yescharacteristic
188C2×#484C2×C6712yes—
189C2×#400C2×C6711yescharacteristic
190#1617C2×C6718yesdirect product factor
191C2×#328C2×C6712yes—
192C2×#324C2×C6711yescharacteristic
193C67C2×#161711yescharacteristic, direct product factor, Sylow
194#544—24no—
195#544—22no—
196#544C22×C6714yes—
197C2×C4⋊Q8—22no—
198C2×#125—22no—
199C2×C22⋊Q8—21no—
200C2×C22⋊Q8C22×C6712yes—
201C2×#117—21no—
202C22×C4⋊C4C22×C6711yescharacteristic
203C22×C4⋊C4C22×C6711yescharacteristic
204#484—24no—
205#484C22×C6718yes—
206C2×#107—22no—
207C2×#103—22no—
208C2×C22⋊C8C22×C6712yes—
209#400—22no—
210#400C22×C6714yes—
211C2×#95—21no—
212C2×#95—21no—
213C2×#95C22×C6711yescharacteristic
214#328C22×C6718yes—
215#324C22×C6714yes—
216C22×Q8—21no—
217C22×Q8D4×C6712yes—
218C23×C4—21no—
219C23×C4—21no—
220C23×C4D4×C6711yescharacteristic
221C23×C4D4×C6711yescharacteristic
222C23×C4C23×C6711yescharacteristic
223C22×C8—22no—
224C4⋊Q8—28no—
225C4⋊Q8—28no—
226#125—28no—
227#125—28no—
228C22⋊Q8—24no—
229C22⋊Q8—24no—
230C22⋊Q8—22no—
231C22⋊Q8—22no—
232C22⋊Q8D4×C6714yes—
233C22⋊Q8C23×C6714yes—
234#117—24no—
235#117—24no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4D4×C6712yes—
247C2×C4⋊C4D4×C6712yes—
248C2×C4⋊C4C23×C6712yes—
249C2×C4⋊C4C23×C6712yes—
250C2×C22⋊C4—22no—
251C2×C42—22no—
252#107—28no—
253#103—28no—
254C22⋊C8—22no—
255C22⋊C8C23×C6714yes—
256#95—24no—
257#95—24no—
258#95—21no—
259#95C23×C6712yes—
260C24C2×D4×C6711yescharacteristic
261C2×Q8—24no—
262C2×Q8—22no—
263C2×Q8—44no—
264C2×Q8—82no—
265C2×Q8C2×D4×C6714yes—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—44no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—42no—
283C22×C4Q16×C6714yes—
284C22×C4C2×D4×C6712yes—
285C22×C4C2×D4×C6712yes—
286C22×C4C2×D4×C6712yes—
287C22×C4C2×D4×C6711yescharacteristic
288C22×C4C2×D4×C6712yes—
289C22×C4C24×C6711yescharacteristic
290C2×C8—24no—
291C2×C8—44no—
292C4⋊C4—24no—
293C4⋊C4—24no—
294C4⋊C4—24no—
295C4⋊C4—28no—
296C4⋊C4—24no—
297C4⋊C4—24no—
298C4⋊C4—22no—
299C4⋊C4—24no—
300C4⋊C4—24no—
301C4⋊C4—24no—
302C4⋊C4—42no—
303C4⋊C4C2×D4×C6714yes—
304C22⋊C4—24no—
305C22⋊C4—24no—
306C42—28no—
307C23—22no—
308C23—22no—
309C23#120×C6711yescharacteristic
310C23#120×C6711yescharacteristic
311C23C2×Q16×C6712yes—
312C23#137×C6712yes—
313C23C22×D4×C6711yescharacteristic
314Q8—44no—
315Q8—82no—
316Q8—84no—
317C2×C4—24no—
318C2×C4—24no—
319C2×C4—24no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—24no—
323C2×C4—24no—
324C2×C4—22no—
325C2×C4—42no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—44no—
329C2×C4—42no—
330C2×C4—44no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—44no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—81no—
338C2×C4—84no—
339C2×C4—82no—
340C2×C4—82no—
341C2×C4#120×C6712yes—
342C2×C4C2×Q16×C6714yes—
343C2×C4C22×D4×C6712yes—
344C8—44no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—24no—
350C22—24no—
351C22#451×C6712yes—
352C22#451×C6712yes—
353C22#454×C6712yes—
354C22#458×C6712yes—
355C22C2×#120×C6711yescharacteristic
356C22C22×Q16×C6711yescharacteristic
357C22C2×#137×C6711yescharacteristic
358C4—22no—
359C4—22no—
360C4—24no—
361C4—44no—
362C4—42no—
363C4—44no—
364C4—82no—
365C4—82no—
366C2—21no—
367C2—22no—
368C2—21no—
369C2C67×#161714yesdirect product factor
370C2C2×#451×C6711yescharacteristic
371C2C2×#454×C6711yescharacteristic
372C2C2×#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×#1617C2direct (N × H)
179. C2×#1617C67direct (N × H)
183. C2×C67#1617direct (N × H)
190. #1617C2×C67direct (N × H)
193. C67C2×#1617direct (N × H)
369. C2C67×#1617direct (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×#484×C672
C2×#400×C672
C67×#16172
C2×#328×C672
C2×#324×C672
C2×#161767