Finite Group Databasebeta

← #30746345

Subgroups — #30746345

Order 17152.

698 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22559971C212yesmaximal
2C2×#16863122C212yesmaximal
3#22558052C212yesmaximal
4C2×C4×C4⋊C4×C67C211yesmaximal, characteristic
5C2×C22⋊(C67⋊C8)C2214yes—
6C2×#14523954—22no—
7#16863151—22no—
8#16863122C418yescomplemented
9C22×C4⋊C4×C67C411yescharacteristic
10C22×C4⋊C4×C67C2211yescharacteristic
11C22×C42×C67C411yescharacteristic
12C22×C42×C67C411yescharacteristic
13C22×C42×C67C2211yescharacteristic
14C4×C4⋊C4×C67—22no—
15C4×C4⋊C4×C67—22no—
16C2×#95×C67C411yescharacteristic
17C2×#95×C67C2211yescharacteristic
18C22×C67⋊C8—24no—
19C22⋊(C67⋊C8)—24no—
20C22⋊(C67⋊C8)C2×C418yes—
21#14523954—44no—
22C23×C4×C67C2×C411yescharacteristic
23C23×C4×C67C2×C411yescharacteristic
24C23×C4×C67C2×C411yescharacteristic
25C23×C4×C67C2×C411yescharacteristic
26C23×C4×C67C2×C411yescharacteristic
27C23×C4×C67C2×C411yescharacteristic
28C23×C4×C67C2311yescharacteristic
29C2×C4⋊C4×C67—22no—
30C2×C4⋊C4×C67—22no—
31C2×C4⋊C4×C67—22no—
32C2×C4⋊C4×C67—22no—
33C2×C4⋊C4×C67—22no—
34C2×C4⋊C4×C67C2×C414yes—
35C2×C42×C67—21no—
36C2×C42×C67—21no—
37C2×C42×C67—22no—
38C2×C42×C67—22no—
39C2×C42×C67—21no—
40C2×C42×C67—21no—
41C2×C42×C67—42no—
42C2×C42×C67—42no—
43C2×C42×C67D412yes—
44C2×C42×C67D412yes—
45#95×C67—22no—
46#95×C67—22no—
47C2×C67⋊C8—28no—
48C2×C67⋊C8—44no—
49C2×C67⋊C8—44no—
50C24×C67C22×C411yescharacteristic
51C22×C4×C67—21no—
52C22×C4×C67—21no—
53C22×C4×C67—21no—
54C22×C4×C67—21no—
55C22×C4×C67—21no—
56C22×C4×C67—21no—
57C22×C4×C67—22no—
58C22×C4×C67—22no—
59C22×C4×C67—21no—
60C22×C4×C67—21no—
61C22×C4×C67—21no—
62C22×C4×C67—21no—
63C22×C4×C67—21no—
64C22×C4×C67—21no—
65C22×C4×C67—21no—
66C22×C4×C67—21no—
67C22×C4×C67—41no—
68C22×C4×C67—41no—
69C22×C4×C67—41no—
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×C67—41no—
77C22×C4×C67C4212yes—
78C22×C4×C67C4212yes—
79C22×C4×C67C22⋊C412yes—
80C22×C4×C67C22⋊C411yescharacteristic
81C22×C4×C67C22⋊C411yescharacteristic
82C22×C4×C67M1612yes—
83C22×C4×C67M1612yes—
84C22×C4×C67M1612yes—
85C22×C4×C67M1612yes—
86C22×C4×C67C22×C412yes—
87C22×C4×C67C2×D412yes—
88C22×C4×C67#4112yes—
89C4⋊C4×C67—28no—
90C4⋊C4×C67—28no—
91C42×C67—24no—
92C42×C67—24no—
93C42×C67—24no—
94C42×C67—42no—
95C42×C67—42no—
96C42×C67—42no—
97C23×C67—21no—
98C23×C67—21no—
99C23×C67—21no—
100C23×C67—21no—
101C23×C67C2×C4211yescharacteristic
102C23×C67C2×C22⋊C411yescharacteristic
103C23×C67#11711yescharacteristic
104C23×C67C2×M1611yescharacteristic
105C23×C67C2×M1611yescharacteristic
106C23×C67C2×M1611yescharacteristic
107C23×C67C2×M1611yescharacteristic
108C2×C4×C67—24no—
109C2×C4×C67—24no—
110C2×C4×C67—22no—
111C2×C4×C67—22no—
112C2×C4×C67—21no—
113C2×C4×C67—21no—
114C2×C4×C67—22no—
115C2×C4×C67—22no—
116C2×C4×C67—22no—
117C2×C4×C67—21no—
118C2×C4×C67—22no—
119C2×C4×C67—21no—
120C2×C4×C67—22no—
121C2×C4×C67—22no—
122C2×C4×C67—22no—
123C2×C4×C67—22no—
124C2×C4×C67—22no—
125C2×C4×C67—22no—
126C2×C4×C67—22no—
127C2×C4×C67—22no—
128C2×C4×C67—42no—
129C2×C4×C67—42no—
130C2×C4×C67—42no—
131C2×C4×C67—42no—
132C2×C4×C67—42no—
133C2×C4×C67—42no—
134C2×C4×C67—42no—
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—42no—
142C2×C4×C67—42no—
143C2×C4×C67—42no—
144C2×C4×C67#9714yescomplemented
145C2×C4×C67#9912yes—
146C2×C4×C67#10112yes—
147C2×C4×C67C4×D414yes—
148C67⋊C8—48no—
149C22×C67—21no—
150C22×C67—21no—
151C22×C67—21no—
152C22×C67—21no—
153C22×C67—21no—
154C22×C67—21no—
155C22×C67—21no—
156C22×C67—21no—
157C22×C67—21no—
158C22×C67—21no—
159C22×C67—21no—
160C22×C67—21no—
161C22×C67C4×C22⋊C411yescharacteristic
162C22×C67C2×#9711yescharacteristic
163C22×C67C4×M1611yescharacteristic
164C22×C67#40711yescharacteristic
165C22×C67#40711yescharacteristic
166C22×C67C2×#9911yescharacteristic
167C22×C67#41011yescharacteristic
168C22×C67C2×#10111yescharacteristic
169C22×C67#41311yescharacteristic
170C22×C67#43211yescharacteristic
171C22×C67#43211yescharacteristic
172C4×C67—22no—
173C4×C67—22no—
174C4×C67—24no—
175C4×C67—24no—
176C4×C67—42no—
177C4×C67—42no—
178C4×C67—42no—
179C4×C67—42no—
180#8153—671nomaximal, Sylow
181C2×C67—21no—
182C2×C67—21no—
183C2×C67—21no—
184C2×C67—21no—
185C2×C67#146311yescharacteristic
186C2×C67#146411yescharacteristic
187C2×C67#146511yescharacteristic
188C2×C67#146711yescharacteristic
189C2×C67#174911yescharacteristic
190C2×C67C4×#9911yescharacteristic
191C2×C67C4×#10111yescharacteristic
192C2×C4×C4⋊C4D6711yescharacteristic
193#1749—672no—
194C2×#324—672no—
195#1280—672no—
196C67#815311yescharacteristic, complemented, Sylow
197C22×C4⋊C4Dic6711yescharacteristic
198C22×C4⋊C4C2×D6711yescharacteristic
199C22×C42Dic6711yescharacteristic
200C22×C42Dic6711yescharacteristic
201C22×C42C2×D6711yescharacteristic
202C2×C22⋊C8—674no—
203C2×#97—1342no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—22no—
206C2×#95Dic6711yescharacteristic
207C2×#95C2×D6711yescharacteristic
208#336—1342no—
209#324—678no—
210C23×C4C2×Dic6711yescharacteristic
211C23×C4C2×Dic6711yescharacteristic
212C23×C4C2×Dic6711yescharacteristic
213C23×C4C2×Dic6711yescharacteristic
214C23×C4C2×Dic6711yescharacteristic
215C23×C4C2×Dic6711yescharacteristic
216C23×C4C22×D6711yescharacteristic
217C22×C8—1344no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4C4×D6714yes—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—42no—
231C2×C42—42no—
232C2×C42C22⋊D6712yes—
233C2×C42C22⋊D6712yes—
234C22⋊C8—678no—
235C22⋊C8—1344no—
236#97—2684no—
237#95—22no—
238#95—22no—
239C24C22×Dic6711yescharacteristic
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
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×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4C4×Dic6712yes—
267C22×C4C4×Dic6712yes—
268C22×C4C22⋊Dic6712yes—
269C22×C4C22⋊Dic6711yescharacteristic
270C22×C4C22⋊Dic6711yescharacteristic
271C22×C4#1176161212yes—
272C22×C4#1176161212yes—
273C22×C4#1176161212yes—
274C22×C4#1176161212yes—
275C22×C4C2×C4×D6712yes—
276C22×C4C2×C22⋊D6712yes—
277C22×C4#1176162812yes—
278C2×C8—2684no—
279C2×C8—2684no—
280C2×C8—1348no—
281C4⋊C4—28no—
282C4⋊C4—28no—
283C42—24no—
284C42—24no—
285C42—24no—
286C42—42no—
287C42—42no—
288C42—42no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23C2×C4×Dic6711yescharacteristic
294C23C2×C22⋊Dic6711yescharacteristic
295C23#1452400211yescharacteristic
296C23C2×#1176161211yescharacteristic
297C23C2×#1176161211yescharacteristic
298C23C2×#1176161211yescharacteristic
299C23C2×#1176161211yescharacteristic
300C2×C4—24no—
301C2×C4—24no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—21no—
310C2×C4—22no—
311C2×C4—21no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4#1452395414yescomplemented
337C2×C4#1452395812yes—
338C2×C4#1452396212yes—
339C2×C4C4×C22⋊D6714yes—
340C8—2688no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22C4×C22⋊Dic6711yescharacteristic
354C22C2×#1452395411yescharacteristic
355C22C4×#1176161211yescharacteristic
356C22#1686336011yescharacteristic
357C22#1686336011yescharacteristic
358C22C2×#1452395811yescharacteristic
359C22#1686337011yescharacteristic
360C22C2×#1452396211yescharacteristic
361C22#1686338011yescharacteristic
362C22#1686345311yescharacteristic
363C22#1686345311yescharacteristic
364C4—22no—
365C4—22no—
366C4—24no—
367C4—24no—
368C4—42no—
369C4—42no—
370C4—42no—
371C4—42no—
372C2—21no—
373C2—21no—
374C2—21no—
375C2—21no—
376C2#2255853911yescharacteristic
377C2#2255854311yescharacteristic
378C2#2255854711yescharacteristic
379C2#2255855311yescharacteristic
380C2#2255997111yescharacteristic
381C2C4×#1452395811yescharacteristic
382C2C4×#1452396211yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic67
commutatorC23×C67C2×C42
FrattiniC23×C4C22×D67
FittingC2×C4×C4⋊C4×C67C2
socleC23×C67C2×C22⋊C4
prime radicalC24×C67C22×C4
cosocleC23×C4×C67C23
hypercenterC2×C4×C4⋊C4D67
hypocenterC67#8153
LCSG ▷ C23×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
8. #16863122C4semidirect
144. C2×C4×C67#97semidirect
196. C67#8153semidirect
336. C2×C4#14523954semidirect

Maximal subgroups

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

subgroupindex
#225599712
C2×#168631222
#225580522
C2×C4×C4⋊C4×C672
#815367