Finite Group Databasebeta

← #30745305

Subgroups — #30745305

Order 17152.

674 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16863447C211yesmaximal, characteristic
2C2×#16863418C211yesmaximal, characteristic
3C2×C4⋊C8×D67C211yesmaximal, characteristic
4#22558044C211yesmaximal, characteristic, complemented
5#22558043C211yesmaximal, characteristic, complemented
6#22558043C211yesmaximal, characteristic, complemented
7C67×#1277C211yesmaximal, characteristic, complemented
8C22×#11761611—21no—
9C22×C8×D67—21no—
10C2×C42×D67C2211yescharacteristic
11#16863447—22no—
12#16863447—22no—
13#16863418—22no—
14#16863418C414yes—
15C2×C4⋊(C67⋊C8)C2211yescharacteristic
16C2×C4⋊(C67⋊C8)C2211yescharacteristic
17C2×Dic67⋊C8—21no—
18C2×Dic67⋊C8—21no—
19C2×Dic67⋊C8C412yes—
20C4⋊C8×D67—22no—
21C4⋊C8×D67C414yes—
22C2×(C2×D67)⋊C8C412yes—
23C2×C4×C67⋊C8C2211yescharacteristic
24C2×C4⋊C8×C67C2211yescharacteristic
25C2×C4⋊C8×C67C2211yescharacteristic
26C2×C4×C8×C67C2211yescharacteristic
27C22×C4×D67C2×C411yescharacteristic
28C22×C4×D67D411yescharacteristic
29C22×C4×D67Q811yescharacteristic
30C2×#11761611—24no—
31C2×#11761611—42no—
32C2×#11761611—42no—
33C22×C67⋊C8—21no—
34C22×C67⋊C8—21no—
35C22×C67⋊C8C2×C412yes—
36C2×C8×D67—24no—
37C2×C8×D67—42no—
38C2×C8×D67—42no—
39C2×C4×Dic67C2×C411yescharacteristic
40C2×C4×Dic67D411yescharacteristic
41C2×C4×Dic67Q811yescharacteristic
42C42×D67C2×C412yes—
43C42×D67C2×C412yes—
44C42×D67D412yes—
45C42×D67D412yes—
46C4⋊(C67⋊C8)—21no—
47C4⋊(C67⋊C8)—21no—
48C4⋊(C67⋊C8)C2×C412yes—
49C4⋊(C67⋊C8)C2×C412yes—
50Dic67⋊C8—24no—
51Dic67⋊C8—24no—
52Dic67⋊C8—24no—
53(C2×D67)⋊C8—24no—
54C4×C67⋊C8—21no—
55C4×C67⋊C8—21no—
56C22×C8×C67—21no—
57C22×C8×C67—21no—
58C22×C8×C67C2×C412yes—
59C2×C42×C67C2311yescharacteristic
60C4⋊C8×C67—21no—
61C4⋊C8×C67—21no—
62C4⋊C8×C67C2×C412yes—
63C4⋊C8×C67C2×C412yes—
64C4×C8×C67—21no—
65C4×C8×C67—21no—
66C23×D67C4⋊C411yescharacteristic
67C22×Dic67C4⋊C411yescharacteristic
68C22×Dic67C4⋊C411yescharacteristic
69C22×Dic67C4⋊C411yescharacteristic
70C2×C4×D67—22no—
71C2×C4×D67—22no—
72C2×C4×D67—42no—
73C2×C4×D67C4212yes—
74C2×C4×D67C22⋊C412yes—
75C2×C4×D67C22⋊C412yes—
76C2×C4×D67C22⋊C412yes—
77C2×C4×D67C4⋊C412yes—
78C2×C4×D67C4⋊C412yes—
79C2×C4×D67D812yes—
80C2×C4×D67SD1612yes—
81C2×C4×D67SD1612yes—
82C2×C4×D67Q1612yes—
83#11761611—48no—
84C2×C67⋊C8—22no—
85C2×C67⋊C8—22no—
86C2×C67⋊C8—22no—
87C2×C67⋊C8—22no—
88C2×C67⋊C8—22no—
89C2×C67⋊C8—41no—
90C2×C67⋊C8—41no—
91C2×C67⋊C8—41no—
92C2×C67⋊C8—41no—
93C8×D67—48no—
94C4×Dic67C4212yes—
95C4×Dic67C22⋊C412yes—
96C4×Dic67C22⋊C412yes—
97C4×Dic67C22⋊C412yes—
98C4×Dic67C4⋊C412yes—
99C4×Dic67C4⋊C412yes—
100C22×C4×C67C22×C411yescharacteristic
101C22×C4×C67C2×D411yescharacteristic
102C22×C4×C67C2×Q811yescharacteristic
103C2×C8×C67—22no—
104C2×C8×C67—22no—
105C2×C8×C67—22no—
106C2×C8×C67—22no—
107C2×C8×C67—22no—
108C2×C8×C67—41no—
109C2×C8×C67—41no—
110C2×C8×C67—41no—
111C2×C8×C67—41no—
112C42×C67C22×C411yescharacteristic
113C42×C67C22×C411yescharacteristic
114C42×C67C2×D411yescharacteristic
115C42×C67C2×D411yescharacteristic
116C23×C67C2×C4⋊C411yescharacteristic
117C22×D67—22no—
118C22×D67—22no—
119C22×D67#9512yes—
120C22×D67#10612yes—
121C22×D67#10712yes—
122C2×C4×C67—22no—
123C2×C4×C67C2×C4211yescharacteristic
124C2×C4×C67C2×C22⋊C411yescharacteristic
125C2×C4×C67C2×C22⋊C411yescharacteristic
126C2×C4×C67C2×C22⋊C411yescharacteristic
127C2×C4×C67C2×C4⋊C411yescharacteristic
128C2×C4×C67C2×C4⋊C411yescharacteristic
129C2×C4×C67#13112yes—
130C2×C4×C67#13112yes—
131C2×C4×C67C2×D811yescharacteristic
132C2×C4×C67C2×SD1611yescharacteristic
133C2×C4×C67C2×SD1611yescharacteristic
134C2×C4×C67C2×Q1611yescharacteristic
135C2×Dic67—22no—
136C2×Dic67—22no—
137C2×Dic67#9512yes—
138C2×Dic67#9512yes—
139C2×Dic67#9512yes—
140C2×Dic67#10612yes—
141C2×Dic67#10712yes—
142C4×D67—22no—
143C4×D67—22no—
144C4×D67—24no—
145C4×D67—42no—
146C4×D67—42no—
147C4×D67#10214yes—
148C4×D67#10214yes—
149C4×D67#10314yes—
150C4×D67#10314yes—
151C8×C67—42no—
152C8×C67—42no—
153C8×C67—42no—
154C67⋊C8—42no—
155C67⋊C8—42no—
156C67⋊C8—42no—
157C22×C67C2×#9511yescharacteristic
158C22×C67#42412yes—
159C22×C67C2×#10611yescharacteristic
160C22×C67C2×#10711yescharacteristic
161C22×C67#43012yes—
162C2×D67—24no—
163C2×D67—24no—
164C2×D67—22no—
165C2×D67—22no—
166C2×D67#34014yes—
167C4×C67—21no—
168C4×C67—21no—
169C4×C67#40511yescharacteristic
170C4×C67#40511yescharacteristic
171C4×C67#40812yes—
172C4×C67C2×#10211yescharacteristic
173C4×C67C2×#10211yescharacteristic
174C4×C67C2×#10311yescharacteristic
175C4×C67C2×#10311yescharacteristic
176Dic67—22no—
177Dic67—22no—
178Dic67#34014yes—
179#7877—671nomaximal, Sylow
180C2×C67#157012yes—
181C2×C67#157112yes—
182C2×C67#172911yescharacteristic
183C2×C67C2×#34011yescharacteristic
184C2×C67#173811yescharacteristic
185D67—24no—
186C2×#431—671no—
187C2×#423—671no—
188C22×C4⋊C8—671no—
189#1277—671no—
190#1277—671no—
191#1277—671no—
192#1277D6711yescharacteristic, complemented
193C67#787711yescharacteristic, complemented, Sylow
194C22×M16—1341no—
195C23×C8—1341no—
196C22×C42—671no—
197#431—1342no—
198#431—1342no—
199#423—674no—
200#423—1342no—
201C2×C4⋊C8—671no—
202C2×C4⋊C8—671no—
203C2×C4⋊C8—674no—
204C2×C4⋊C8—672no—
205C2×C4⋊C8—1342no—
206C2×C4⋊C8—1341no—
207C2×C4⋊C8—1341no—
208C2×C4⋊C8C2×D6711yescharacteristic
209C2×C4⋊C8C2×D6711yescharacteristic
210C2×C22⋊C8—672no—
211C2×C4×C8—671no—
212C2×C4×C8C2×D6711yescharacteristic
213C23×C4—671no—
214C23×C4—671no—
215C23×C4—671no—
216C2×M16—2682no—
217C2×M16—2682no—
218C2×M16—1344no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—2682no—
222C22×C8—2682no—
223C22×C8—672no—
224C22×C8—1344no—
225C22×C8—1341no—
226C22×C8—1341no—
227C22×C8C4×D6712yes—
228C2×C42—671no—
229C2×C42—671no—
230C2×C42—671no—
231C2×C42—672no—
232C2×C42—672no—
233C2×C42—672no—
234C2×C42—672no—
235C2×C42C22×D6711yescharacteristic
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—672no—
239C4⋊C8—672no—
240C4⋊C8—1344no—
241C4⋊C8—1344no—
242C4⋊C8—1344no—
243C4⋊C8—1341no—
244C4⋊C8—1341no—
245C4⋊C8C4×D6712yes—
246C4⋊C8C4×D6712yes—
247C22⋊C8—1344no—
248C4×C8—21no—
249C4×C8—21no—
250C4×C8—1341no—
251C4×C8—1341no—
252C24—671no—
253C22×C4—2682no—
254C22×C4—672no—
255C22×C4—672no—
256C22×C4—671no—
257C22×C4—672no—
258C22×C4—672no—
259C22×C4—672no—
260C22×C4—672no—
261C22×C4—672no—
262C22×C4—672no—
263C22×C4—671no—
264C22×C4—671no—
265C22×C4—672no—
266C22×C4—672no—
267C22×C4—1342no—
268C22×C4—1342no—
269C22×C4C2×C4×D6711yescharacteristic
270C22×C4D4×D6711yescharacteristic
271C22×C4Q8×D6711yescharacteristic
272M16—2688no—
273C2×C8—22no—
274C2×C8—22no—
275C2×C8—22no—
276C2×C8—22no—
277C2×C8—22no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—2688no—
283C2×C8—2681no—
284C2×C8—2681no—
285C2×C8—2681no—
286C2×C8—2681no—
287C2×C8—1342no—
288C2×C8—1342no—
289C2×C8—1342no—
290C2×C8—1342no—
291C2×C8—1342no—
292C42—672no—
293C42—672no—
294C42—672no—
295C42—672no—
296C42—672no—
297C42—672no—
298C42C2×C4×D6711yescharacteristic
299C42C2×C4×D6711yescharacteristic
300C42D4×D6711yescharacteristic
301C42D4×D6711yescharacteristic
302C23—672no—
303C23—672no—
304C23—672no—
305C23—1342no—
306C23—1342no—
307C23C4⋊C4×D6711yescharacteristic
308C2×C4—22no—
309C2×C4—2682no—
310C2×C4—2682no—
311C2×C4—672no—
312C2×C4—672no—
313C2×C4—672no—
314C2×C4—674no—
315C2×C4—674no—
316C2×C4—674no—
317C2×C4—674no—
318C2×C4—672no—
319C2×C4—672no—
320C2×C4—1342no—
321C2×C4—1342no—
322C2×C4—1342no—
323C2×C4—1342no—
324C2×C4—1344no—
325C2×C4C42×D6711yescharacteristic
326C2×C4C22⋊C4×D6711yescharacteristic
327C2×C4C22⋊C4×D6711yescharacteristic
328C2×C4C22⋊C4×D6711yescharacteristic
329C2×C4C4⋊C4×D6711yescharacteristic
330C2×C4C4⋊C4×D6711yescharacteristic
331C2×C4#1452404912yes—
332C2×C4#1452405112yes—
333C2×C4D8×D6711yescharacteristic
334C2×C4SD16×D6711yescharacteristic
335C2×C4SD16×D6711yescharacteristic
336C2×C4Q16×D6711yescharacteristic
337C8—42no—
338C8—42no—
339C8—42no—
340C8—2682no—
341C8—2682no—
342C8—2682no—
343C22—674no—
344C22—1344no—
345C22—1344no—
346C22—1342no—
347C22—1342no—
348C22#95×D6711yescharacteristic
349C22#1686342112yes—
350C22#106×D6711yescharacteristic
351C22#107×D6711yescharacteristic
352C22#1686344312yes—
353C4—21no—
354C4—21no—
355C4—674no—
356C4—1342no—
357C4—1342no—
358C4#1686335011yescharacteristic
359C4#1686335311yescharacteristic
360C4#1686336112yes—
361C4#102×D6711yescharacteristic
362C4#102×D6711yescharacteristic
363C4#103×D6711yescharacteristic
364C4#103×D6711yescharacteristic
365C2—1344no—
366C2#2255897812yes—
367C2#2255898212yes—
368C2#2255990411yescharacteristic
369C2#340×D6711yescharacteristic
370C2#2255993411yescharacteristic
subgroupG/H
centerC22×C4D4×D67
commutatorC2×C4×C67C2×C42
FrattiniC2×C42C22×D67
FittingC67×#1277C2
socleC23×C67C2×C4⋊C4
prime radicalC23×D67C4⋊C4
cosocleC2×C42×C67C23
hypercenter#1277D67
hypocenterC67#7877
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ #1277

Complements

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

NH ≅ G/Ntype
4. #22558044C2semidirect
5. #22558043C2semidirect
6. #22558043C2semidirect
7. C67×#1277C2semidirect
192. #1277D67semidirect
193. C67#7877semidirect

Maximal subgroups

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

subgroupindex
C2×#168634472
C2×#168634182
C2×C4⋊C8×D672
#225580442
#225580432
C67×#12772
#787767