Finite Group Databasebeta

← #30751462

Subgroups — #30751462

Order 17152.

530 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16863453C211yesmaximal, characteristic
2#22559971C211yesmaximal, characteristic
3C2×#16863138C212yesmaximal, complemented
4#22558090C212yesmaximal, complemented
5C67×#2851C211yesmaximal, characteristic
6#16863453—24no—
7C2×C4⋊(C67⋊C8)—21no—
8C2×C22⋊(C67⋊C8)—21no—
9C2×C22⋊(C67⋊C8)—21no—
10C2×#14523954C2212yes—
11C2×#14523954C2212yes—
12#16863151—21no—
13#16863138—24no—
14C2×#125×C67C2211yescharacteristic
15C22×C42×C67C2211yescharacteristic
16#389×C67C411yescharacteristic
17#389×C67C411yescharacteristic
18#387×C67—21no—
19#387×C67—21no—
20#385×C67—21no—
21#385×C67—21no—
22#382×C67C2211yescharacteristic
23#380×C67C411yescharacteristic
24#380×C67C411yescharacteristic
25C22×C67⋊C8—22no—
26C22×C67⋊C8—22no—
27C4⋊(C67⋊C8)—24no—
28C22⋊(C67⋊C8)—24no—
29C22⋊(C67⋊C8)—24no—
30#14523954—22no—
31#14523954—22no—
32#14523954—22no—
33#14523954D414yes—
34C23×C4×C67C2×C411yescharacteristic
35C23×C4×C67D411yescharacteristic
36C23×C4×C67D411yescharacteristic
37#125×C67—22no—
38#125×C67—22no—
39C2×C4⋊C4×C67—21no—
40C2×C4⋊C4×C67—21no—
41C2×C4⋊C4×C67—21no—
42C2×C4⋊C4×C67—21no—
43C2×C4⋊C4×C67C2×C411yescharacteristic
44C2×C4⋊C4×C67C2×C411yescharacteristic
45C2×C4⋊C4×C67C2×C411yescharacteristic
46C2×C4⋊C4×C67C2×C411yescharacteristic
47C2×C22⋊C4×C67—21no—
48C2×C22⋊C4×C67—21no—
49C2×C42×C67—22no—
50C2×C42×C67—22no—
51C2×C42×C67C2×C411yescharacteristic
52C2×C42×C67D411yescharacteristic
53C2×C42×C67D411yescharacteristic
54C2×C42×C67C2311yescharacteristic
55#95×C67—21no—
56#95×C67—21no—
57#95×C67—21no—
58#95×C67—21no—
59#95×C67—21no—
60#95×C67—21no—
61C2×C67⋊C8—24no—
62C2×C67⋊C8—24no—
63C2×C67⋊C8—42no—
64C2×C67⋊C8—42no—
65C2×C67⋊C8—42no—
66C2×C67⋊C8—42no—
67C24×C67C22⋊C411yescharacteristic
68C22×C4×C67—22no—
69C22×C4×C67—22no—
70C22×C4×C67—22no—
71C22×C4×C67—21no—
72C22×C4×C67—22no—
73C22×C4×C67—21no—
74C22×C4×C67—21no—
75C22×C4×C67—21no—
76C22×C4×C67—21no—
77C22×C4×C67—21no—
78C22×C4×C67—41no—
79C22×C4×C67—41no—
80C22×C4×C67—41no—
81C22×C4×C67—41no—
82C22×C4×C67C22⋊C411yescharacteristic
83C22×C4×C67C22⋊C411yescharacteristic
84C22×C4×C67C22⋊C411yescharacteristic
85C22×C4×C67C22×C411yescharacteristic
86C22×C4×C67C2×D411yescharacteristic
87C22×C4×C67C2×D411yescharacteristic
88C4⋊C4×C67—22no—
89C4⋊C4×C67—22no—
90C4⋊C4×C67—41no—
91C4⋊C4×C67—41no—
92C4⋊C4×C67—42no—
93C4⋊C4×C67—42no—
94C4⋊C4×C67—41no—
95C4⋊C4×C67—42no—
96C4⋊C4×C67—41no—
97C22⋊C4×C67—42no—
98C22⋊C4×C67—42no—
99C42×C67—22no—
100C42×C67—21no—
101C42×C67—22no—
102C42×C67—21no—
103C42×C67C2×D412yes—
104C42×C67#4112yes—
105C23×C67—21no—
106C23×C67—21no—
107C23×C67—21no—
108C23×C67—21no—
109C23×C67—21no—
110C23×C67—21no—
111C23×C67#10112yes—
112C23×C67C2×C22⋊C411yescharacteristic
113C2×C4×C67—21no—
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—21no—
125C2×C4×C67—21no—
126C2×C4×C67—21no—
127C2×C4×C67—44no—
128C2×C4×C67—44no—
129C2×C4×C67—42no—
130C2×C4×C67—42no—
131C2×C4×C67—41no—
132C2×C4×C67—41no—
133C2×C4×C67—41no—
134C2×C4×C67—41no—
135C2×C4×C67—41no—
136C2×C4×C67—41no—
137C2×C4×C67—41no—
138C2×C4×C67—41no—
139C2×C4×C67—41no—
140C2×C4×C67—41no—
141C2×C4×C67—41no—
142C2×C4×C67—41no—
143C2×C4×C67#10412yes—
144C2×C4×C67C4×D412yes—
145C2×C4×C67#12011yescharacteristic
146C2×C4×C67#12112yes—
147C2×C4×C67#12311yescharacteristic
148C67⋊C8—44no—
149C67⋊C8—44no—
150C22×C67—22no—
151C22×C67—22no—
152C22×C67—22no—
153C22×C67—41no—
154C22×C67—41no—
155C22×C67—41no—
156C22×C67—41no—
157C22×C67#38611yescharacteristic
158C22×C67C2×#10111yescharacteristic
159C22×C67#41311yescharacteristic
160C22×C67C2×#10411yescharacteristic
161C22×C67#42111yescharacteristic
162C22×C67#42111yescharacteristic
163C22×C67#42111yescharacteristic
164C4×C67—22no—
165C4×C67—22no—
166C4×C67—21no—
167C4×C67—21no—
168C4×C67—22no—
169C4×C67—41no—
170C4×C67—41no—
171C4×C67—81no—
172C4×C67—81no—
173C4×C67—81no—
174C4×C67—81no—
175#9126—671nomaximal, Sylow
176C2×C67—42no—
177C2×C67#152911yescharacteristic
178C2×C67#153111yescharacteristic
179C2×C67#153411yescharacteristic
180C2×C67#153511yescharacteristic
181C2×C67#187111yescharacteristic
182C2×C67#189711yescharacteristic
183C2×C67#190111yescharacteristic
184C2×#432—671no—
185#2851D6711yescharacteristic
186#1749—671no—
187C2×#330—672no—
188#1295—672no—
189C67#912611yescharacteristic, complemented, Sylow
190C2×#125C2×D6711yescharacteristic
191C22×C42C2×D6711yescharacteristic
192#432—1344no—
193C2×C4⋊C8—1341no—
194C2×C22⋊C8—1341no—
195C2×C22⋊C8—1341no—
196C2×#97—672no—
197C2×#97—672no—
198#389Dic6711yescharacteristic
199#389Dic6711yescharacteristic
200#387—21no—
201#387—21no—
202#385—21no—
203#385—21no—
204#382C2×D6711yescharacteristic
205#380Dic6711yescharacteristic
206#380Dic6711yescharacteristic
207#336—1341no—
208#330—1344no—
209C23×C4C2×Dic6711yescharacteristic
210C23×C4C22⋊D6711yescharacteristic
211C23×C4C22⋊D6711yescharacteristic
212C22×C8—1342no—
213C22×C8—1342no—
214#125—22no—
215#125—22no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4C2×Dic6711yescharacteristic
221C2×C4⋊C4C2×Dic6711yescharacteristic
222C2×C4⋊C4C2×Dic6711yescharacteristic
223C2×C4⋊C4C2×Dic6711yescharacteristic
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42C2×Dic6711yescharacteristic
229C2×C42C22⋊D6711yescharacteristic
230C2×C42C22⋊D6711yescharacteristic
231C2×C42C22×D6711yescharacteristic
232C4⋊C8—1344no—
233C22⋊C8—1344no—
234C22⋊C8—1344no—
235#97—674no—
236#97—1342no—
237#97—1342no—
238#97—1342no—
239#95—21no—
240#95—21no—
241#95—21no—
242#95—21no—
243#95—21no—
244#95—21no—
245C24C22⋊Dic6711yescharacteristic
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—21no—
250C22×C4—22no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4C22⋊Dic6711yescharacteristic
261C22×C4C22⋊Dic6711yescharacteristic
262C22×C4C22⋊Dic6711yescharacteristic
263C22×C4C22×Dic6711yescharacteristic
264C22×C4C2×C22⋊D6711yescharacteristic
265C22×C4C2×C22⋊D6711yescharacteristic
266C2×C8—2682no—
267C2×C8—2682no—
268C2×C8—2682no—
269C2×C8—2682no—
270C2×C8—1344no—
271C2×C8—1344no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—41no—
279C4⋊C4—42no—
280C4⋊C4—41no—
281C22⋊C4—42no—
282C22⋊C4—42no—
283C42—22no—
284C42—21no—
285C42—22no—
286C42—21no—
287C42D4×D6712yes—
288C42#1176162712yes—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23#1452396212yes—
296C23C2×C22⋊Dic6711yescharacteristic
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—44no—
312C2×C4—44no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4#1452397012yes—
328C2×C4D4×Dic6712yes—
329C2×C4#1452401111yescharacteristic
330C2×C4#1452401512yes—
331C2×C4#1452402411yescharacteristic
332C8—2684no—
333C8—2684no—
334C22—22no—
335C22—22no—
336C22—22no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22#1686327611yescharacteristic
342C22C2×#1452396211yescharacteristic
343C22#1686338011yescharacteristic
344C22C2×#1452397011yescharacteristic
345C22#1686341311yescharacteristic
346C22#1686341311yescharacteristic
347C22#1686341311yescharacteristic
348C4—22no—
349C4—22no—
350C4—21no—
351C4—21no—
352C4—22no—
353C4—41no—
354C4—41no—
355C4—81no—
356C4—81no—
357C4—81no—
358C4—81no—
359C2—42no—
360C2#2255881611yescharacteristic
361C2#2255882411yescharacteristic
362C2#2255883611yescharacteristic
363C2#2255884011yescharacteristic
364C2#2256051211yescharacteristic
365C2#2256065811yescharacteristic
366C2#2256067811yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic67
commutatorC22×C4×C67C22×C4
FrattiniC2×C42C22×D67
FittingC67×#2851C2
socleC23×C67C2×C22⋊C4
prime radicalC24×C67C22⋊C4
cosocleC2×C42×C67C23
hypercenter#2851D67
hypocenterC67#9126
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2851

Complements

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

NH ≅ G/Ntype
3. C2×#16863138C2semidirect
4. #22558090C2semidirect
189. C67#9126semidirect

Maximal subgroups

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

subgroupindex
C2×#168634532
#225599712
C2×#168631382
#225580902
C67×#28512
#912667