Finite Group Databasebeta

← #30747294

Subgroups — #30747294

Order 17152.

876 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22560277C211yesmaximal, characteristic
2C2×#16863133C214yesmaximal
3#22558436C211yesmaximal, characteristic
4C2×#533×C67C211yesmaximal, characteristic
5C2×C22⋊(C67⋊C8)C2212yes—
6C2×C67⋊#95—21no—
7C2×C67⋊#95—21no—
8C2×C67⋊#95C2212yes—
9#16863151—21no—
10#16863151—21no—
11#16863133—28no—
12#533×C67—28no—
13C2×C4⋊Q8×C67C411yescharacteristic, complemented
14C2×C4⋊Q8×C67C411yescharacteristic, complemented
15C2×#125×C67—21no—
16C2×C22⋊Q8×C67—21no—
17C2×C22⋊Q8×C67C2212yes—
18C2×C4×Q8×C67—21no—
19C2×C4×Q8×C67C412yescomplemented
20C2×#117×C67—21no—
21C22×C42×C67C2211yescharacteristic
22C23×Dic67—22no—
23C22×C67⋊C8—22no—
24C22⋊(C67⋊C8)—24no—
25C67⋊#95—24no—
26C67⋊#95—24no—
27C67⋊#95—42no—
28C22×Q8×C67—21no—
29C22×Q8×C67C2×C412yes—
30C23×C4×C67D411yescharacteristic
31C23×C4×C67D411yescharacteristic
32C23×C4×C67C2311yescharacteristic
33C4⋊Q8×C67—24no—
34C4⋊Q8×C67—24no—
35#125×C67—28no—
36C22⋊Q8×C67—28no—
37C22⋊Q8×C67—24no—
38C22⋊Q8×C67D418yes—
39C4×Q8×C67—28no—
40C4×Q8×C67—24no—
41C4×Q8×C67—24no—
42#117×C67—28no—
43C2×C4⋊C4×C67—21no—
44C2×C4⋊C4×C67—21no—
45C2×C4⋊C4×C67—21no—
46C2×C4⋊C4×C67—22no—
47C2×C4⋊C4×C67—22no—
48C2×C4⋊C4×C67C2×C412yes—
49C2×C22⋊C4×C67—22no—
50C2×C42×C67—22no—
51C2×C42×C67—24no—
52C2×C42×C67C2×C411yescharacteristic
53C2×C42×C67C2×C411yescharacteristic
54C2×C42×C67D411yescharacteristic
55C2×C42×C67D411yescharacteristic
56C22×Dic67—24no—
57C22×Dic67—42no—
58C22×Dic67—42no—
59C22×Dic67—42no—
60C22×Dic67—42no—
61C22×Dic67—42no—
62C22×Dic67—42no—
63C2×C67⋊C8—42no—
64C2×C67⋊C8—42no—
65C2×C67⋊C8—42no—
66C24×C67C2×D411yescharacteristic
67C2×Q8×C67—24no—
68C2×Q8×C67—22no—
69C2×Q8×C67—42no—
70C2×Q8×C67—42no—
71C2×Q8×C67—42no—
72C2×Q8×C67—42no—
73C2×Q8×C67C22⋊C414yes—
74C22×C4×C67—22no—
75C22×C4×C67—22no—
76C22×C4×C67—22no—
77C22×C4×C67—22no—
78C22×C4×C67—22no—
79C22×C4×C67—22no—
80C22×C4×C67—41no—
81C22×C4×C67—41no—
82C22×C4×C67—41no—
83C22×C4×C67C22⋊C411yescharacteristic
84C22×C4×C67C22⋊C411yescharacteristic
85C22×C4×C67C22⋊C411yescharacteristic
86C22×C4×C67C22⋊C411yescharacteristic
87C22×C4×C67SD1612yes—
88C22×C4×C67Q1612yes—
89C22×C4×C67C22×C411yescharacteristic
90C22×C4×C67C2×D411yescharacteristic
91C22×C4×C67C2×D414yes—
92C4⋊C4×C67—24no—
93C4⋊C4×C67—24no—
94C4⋊C4×C67—28no—
95C4⋊C4×C67—24no—
96C4⋊C4×C67—22no—
97C4⋊C4×C67—28no—
98C4⋊C4×C67C22⋊C414yes—
99C22⋊C4×C67—28no—
100C42×C67—28no—
101C42×C67—22no—
102C42×C67—22no—
103C42×C67—22no—
104C42×C67—22no—
105C23×C67—21no—
106C23×C67—21no—
107C23×C67—21no—
108C23×C67—21no—
109C23×C67C2×C22⋊C411yescharacteristic
110C23×C67#12012yes—
111C23×C67C2×SD1611yescharacteristic
112C23×C67C2×Q1611yescharacteristic
113C23×C67#13712yes—
114Q8×C67—44no—
115Q8×C67—44no—
116Q8×C67—48no—
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—24no—
128C2×C4×C67—24no—
129C2×C4×C67—21no—
130C2×C4×C67—24no—
131C2×C4×C67—24no—
132C2×C4×C67—21no—
133C2×C4×C67—21no—
134C2×C4×C67—44no—
135C2×C4×C67—44no—
136C2×C4×C67—42no—
137C2×C4×C67—42no—
138C2×C4×C67—42no—
139C2×C4×C67—42no—
140C2×C4×C67#10312yes—
141C2×C4×C67#10312yes—
142C2×C4×C67#10412yes—
143C2×C4×C67C2×C22⋊C412yes—
144C2×C4×C67#12012yes—
145C2×Dic67—44no—
146C2×Dic67—44no—
147C2×Dic67—44no—
148C2×Dic67—82no—
149C2×Dic67—82no—
150C2×Dic67—82no—
151C2×Dic67—82no—
152C67⋊C8—82no—
153C22×C67—21no—
154C22×C67—21no—
155C22×C67—22no—
156C22×C67—22no—
157C22×C67—21no—
158C22×C67—21no—
159C22×C67—22no—
160C22×C67—22no—
161C22×C67#37911yescharacteristic
162C22×C67#41011yescharacteristic
163C22×C67#41011yescharacteristic
164C22×C67C2×#10311yescharacteristic
165C22×C67#41911yescharacteristic
166C22×C67C2×#10411yescharacteristic
167C22×C67#42111yescharacteristic
168C22×C67#44812yes—
169C22×C67#45112yes—
170C4×C67—22no—
171C4×C67—22no—
172C4×C67—22no—
173C4×C67—22no—
174C4×C67—22no—
175C4×C67—44no—
176C4×C67—41no—
177C4×C67—44no—
178Dic67—84no—
179#8333—671nomaximal, Sylow
180C2×C67—22no—
181C2×C67—22no—
182C2×C67#150311yescharacteristic
183C2×C67#150611yescharacteristic
184C2×C67#151511yescharacteristic
185C2×C67#151811yescharacteristic
186C2×C67#178211yescharacteristic
187C2×C67#178711yescharacteristic
188C2×C67#178911yescharacteristic
189C2×#533D6711yescharacteristic
190#1817—671no—
191C2×#328—674no—
192#1434—671no—
193C67#833311yescharacteristic, complemented, Sylow
194#533—28no—
195C2×C4⋊Q8Dic6711yescharacteristic, complemented
196C2×C4⋊Q8Dic6711yescharacteristic, complemented
197C2×#125—21no—
198C2×C22⋊Q8—21no—
199C2×C22⋊Q8C2×D6712yes—
200C2×C4×Q8—21no—
201C2×C4×Q8Dic6712yescomplemented
202C2×#117—21no—
203C22×C42C2×D6711yescharacteristic
204C2×C22⋊C8—672no—
205C2×#95—672no—
206C2×#95—1341no—
207C2×#95—1341no—
208#336—1341no—
209#336—1341no—
210#328—1348no—
211C22×Q8—21no—
212C22×Q8C2×Dic6712yes—
213C23×C4—1342no—
214C23×C4C22⋊D6711yescharacteristic
215C23×C4C22⋊D6711yescharacteristic
216C23×C4C22×D6711yescharacteristic
217C22×C8—1342no—
218C4⋊Q8—24no—
219C4⋊Q8—24no—
220#125—28no—
221C22⋊Q8—28no—
222C22⋊Q8—24no—
223C22⋊Q8C22⋊D6718yes—
224C4×Q8—28no—
225C4×Q8—24no—
226C4×Q8—24no—
227#117—28no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4C2×Dic6712yes—
234C2×C22⋊C4—22no—
235C2×C42—22no—
236C2×C42—24no—
237C2×C42C2×Dic6711yescharacteristic
238C2×C42C2×Dic6711yescharacteristic
239C2×C42C22⋊D6711yescharacteristic
240C2×C42C22⋊D6711yescharacteristic
241C22⋊C8—1344no—
242#95—2682no—
243#95—1344no—
244#95—1344no—
245C24C2×C22⋊D6711yescharacteristic
246C2×Q8—24no—
247C2×Q8—22no—
248C2×Q8—42no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×Q8—42no—
252C2×Q8C22⋊Dic6714yes—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—2682no—
263C22×C4—2682no—
264C22×C4—2682no—
265C22×C4—2682no—
266C22×C4—2682no—
267C22×C4—2682no—
268C22×C4—1344no—
269C22×C4C22⋊Dic6711yescharacteristic
270C22×C4C22⋊Dic6711yescharacteristic
271C22×C4C22⋊Dic6711yescharacteristic
272C22×C4C22⋊Dic6711yescharacteristic
273C22×C4#1176161512yes—
274C22×C4#1176161812yes—
275C22×C4C22×Dic6711yescharacteristic
276C22×C4C2×C22⋊D6711yescharacteristic
277C22×C4C2×C22⋊D6714yes—
278C2×C8—2682no—
279C2×C8—2682no—
280C2×C8—2682no—
281C4⋊C4—24no—
282C4⋊C4—24no—
283C4⋊C4—28no—
284C4⋊C4—24no—
285C4⋊C4—22no—
286C4⋊C4—28no—
287C4⋊C4C22⋊Dic6714yes—
288C22⋊C4—28no—
289C42—28no—
290C42—22no—
291C42—22no—
292C42—22no—
293C42—22no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23C2×C22⋊Dic6711yescharacteristic
299C23#1452401212yes—
300C23C2×#1176161511yescharacteristic
301C23C2×#1176161811yescharacteristic
302C23#1452407412yes—
303Q8—44no—
304Q8—44no—
305Q8—48no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—24no—
310C2×C4—24no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—24no—
320C2×C4—24no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—44no—
324C2×C4—44no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—2684no—
330C2×C4—2684no—
331C2×C4—2684no—
332C2×C4—5362no—
333C2×C4—5362no—
334C2×C4—5362no—
335C2×C4—5362no—
336C2×C4#1452396712yes—
337C2×C4#1452396712yes—
338C2×C4#1452397012yes—
339C2×C4C2×C22⋊Dic6712yes—
340C2×C4#1452401212yes—
341C8—5362no—
342C22—22no—
343C22—22no—
344C22—21no—
345C22—21no—
346C22—22no—
347C22—22no—
348C22—21no—
349C22—21no—
350C22#1686325211yescharacteristic
351C22#1686337011yescharacteristic
352C22#1686337011yescharacteristic
353C22C2×#1452396711yescharacteristic
354C22#1686340511yescharacteristic
355C22C2×#1452397011yescharacteristic
356C22#1686341311yescharacteristic
357C22#1686353512yes—
358C22#1686355212yes—
359C4—22no—
360C4—22no—
361C4—22no—
362C4—22no—
363C4—22no—
364C4—44no—
365C4—41no—
366C4—44no—
367C4—5364no—
368C2—22no—
369C2—22no—
370C2#2255870811yescharacteristic
371C2#2255872111yescharacteristic
372C2#2255876011yescharacteristic
373C2#2255877311yescharacteristic
374C2#2256012711yescharacteristic
375C2#2256014411yescharacteristic
376C2#2256015311yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic67
commutatorC22×C4×C67C22×C4
FrattiniC23×C4C22×D67
FittingC2×#533×C67C2
socleC23×C67C2×C22⋊C4
prime radicalC24×C67C2×D4
cosocleC23×C4×C67C23
hypercenterC2×#533D67
hypocenterC67#8333
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#533

Complements

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

NH ≅ G/Ntype
13. C2×C4⋊Q8×C67C4semidirect
14. C2×C4⋊Q8×C67C4semidirect
19. C2×C4×Q8×C67C4semidirect
193. C67#8333semidirect
195. C2×C4⋊Q8Dic67semidirect
196. C2×C4⋊Q8Dic67semidirect
201. C2×C4×Q8Dic67semidirect

Maximal subgroups

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

subgroupindex
#225602772
C2×#168631332
#225584362
C2×#533×C672
#833367