Finite Group Databasebeta

← #30762315

Subgroups — #30762315

Order 17152.

978 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14523961C212yesmaximal
2C2×#16863302C211yesmaximal, characteristic
3C2×#16863164C212yesmaximal
4#22559903C211yesmaximal, characteristic
5C2×C4×M16×C67C211yesmaximal, characteristic
6C23×Dic134C2211yescharacteristic
7C22×#11761612C2212yes—
8C22×C4⋊Dic67C2211yescharacteristic
9C2×#14523961—24no—
10C2×#14523961—24no—
11C2×#14523961C418yescomplemented
12#16863302—22no—
13#16863302—21no—
14#16863302—21no—
15C2×C67⋊#95C414yes—
16#16863164—24no—
17#16863151—22no—
18C22×M16×C67C2212yes—
19C22×C42×C67C2211yescharacteristic
20C4×M16×C67—24no—
21C2×#97×C67—21no—
22C2×C4×C8×C67—21no—
23C22×Dic134—21no—
24C22×Dic134—21no—
25C22×Dic134—22no—
26C22×Dic134—22no—
27C22×Dic134—22no—
28C22×Dic134—22no—
29C22×Dic134—22no—
30C22×Dic134C2×C414yes—
31C23×Dic67C2×C412yes—
32C23×Dic67C2×C412yes—
33C2×#11761612—22no—
34C2×#11761612—22no—
35C2×#11761612—22no—
36C2×#11761612—22no—
37C2×#11761612—22no—
38C2×#11761612C2×C414yes—
39C22×C67⋊C8—22no—
40C2×C4⋊Dic67—21no—
41C2×C4⋊Dic67—21no—
42C2×C4⋊Dic67—22no—
43C2×C4⋊Dic67—22no—
44#14523961—216no—
45C67⋊#95—24no—
46C67⋊#95—24no—
47C23×C4×C67C2×C412yes—
48C23×C4×C67C2311yescharacteristic
49C2×M16×C67—22no—
50C2×M16×C67—22no—
51C2×M16×C67—22no—
52C2×M16×C67C2×C414yes—
53C2×M16×C67D414yes—
54C2×M16×C67D414yes—
55C22×C8×C67—22no—
56C2×C42×C67—22no—
57C2×C42×C67—22no—
58C2×C42×C67D412yes—
59C2×C42×C67D411yescharacteristic
60C2×C42×C67D411yescharacteristic
61#97×C67—24no—
62C4×C8×C67—24no—
63C2×Dic134—24no—
64C2×Dic134—24no—
65C2×Dic134—24no—
66C2×Dic134—24no—
67C2×Dic134—24no—
68C2×Dic134—24no—
69C2×Dic134—41no—
70C2×Dic134—42no—
71C2×Dic134—42no—
72C2×Dic134—42no—
73C2×Dic134—42no—
74C2×Dic134—41no—
75C2×Dic134—41no—
76C2×Dic134—42no—
77C2×Dic134—42no—
78C2×Dic134—41no—
79C22×Dic67—22no—
80C22×Dic67—22no—
81C22×Dic67—24no—
82C22×Dic67—24no—
83C22×Dic67—22no—
84C22×Dic67—24no—
85C22×Dic67—44no—
86C22×Dic67C4214yes—
87#11761612—24no—
88#11761612—24no—
89#11761612—42no—
90#11761612—42no—
91C2×C67⋊C8—24no—
92C2×C67⋊C8—42no—
93C2×C67⋊C8—42no—
94C4⋊Dic67—42no—
95C4⋊Dic67—41no—
96C4⋊Dic67—41no—
97C24×C67C22×C411yescharacteristic
98C22×C4×C67—24no—
99C22×C4×C67—24no—
100C22×C4×C67—42no—
101C22×C4×C67C22⋊C412yes—
102C22×C4×C67C22⋊C412yes—
103C22×C4×C67C22×C412yes—
104C22×C4×C67C2×D412yes—
105C22×C4×C67C2×D411yescharacteristic
106C22×C4×C67C2×D412yes—
107C22×C4×C67C2×D411yescharacteristic
108C22×C4×C67#4112yes—
109C22×C4×C67#4112yes—
110C22×C4×C67#4112yes—
111M16×C67—24no—
112M16×C67—24no—
113M16×C67—24no—
114M16×C67C22⋊C418yes—
115C2×C8×C67—24no—
116C2×C8×C67—24no—
117C2×C8×C67—24no—
118C42×C67—21no—
119C42×C67—21no—
120C42×C67—24no—
121C42×C67—21no—
122C42×C67—21no—
123C23×C67—22no—
124C23×C67—22no—
125C23×C67C2×C4211yescharacteristic
126C23×C67C2×C22⋊C411yescharacteristic
127C23×C67#11711yescharacteristic
128C23×C67C4×D412yes—
129C23×C67C4×D412yes—
130C2×C4×C67—22no—
131C2×C4×C67—22no—
132C2×C4×C67—22no—
133C2×C4×C67—22no—
134C2×C4×C67—22no—
135C2×C4×C67—48no—
136C2×C4×C67#10114yescomplemented
137C2×C4×C67C2×C22⋊C412yes—
138C2×C4×C67#11712yes—
139C2×C4×C67C4×D412yes—
140C2×C4×C67C4×D412yes—
141C2×C4×C67C4×D412yes—
142C2×C4×C67C4×D412yes—
143C2×C4×C67#12011yescharacteristic
144C2×C4×C67#12112yes—
145C2×C4×C67#12112yes—
146C2×C4×C67#12112yes—
147C2×C4×C67#12312yes—
148C2×C4×C67#12311yescharacteristic
149C2×C4×C67#12411yescharacteristic
150C2×C4×C67#12412yes—
151C2×C4×C67#12612yes—
152C2×C4×C67#12711yescharacteristic
153C2×Dic67—28no—
154C2×Dic67—24no—
155C2×Dic67—42no—
156C2×Dic67—44no—
157C2×Dic67—42no—
158C2×Dic67—42no—
159C2×Dic67—44no—
160C2×Dic67—84no—
161Dic134—44no—
162Dic134—44no—
163Dic134—44no—
164Dic134—44no—
165C8×C67—28no—
166C67⋊C8—44no—
167C22×C67—24no—
168C22×C67—22no—
169C22×C67—22no—
170C22×C67—22no—
171C22×C67—22no—
172C22×C67C4×C22⋊C412yes—
173C22×C67C4×C22⋊C411yescharacteristic
174C22×C67#38611yescharacteristic
175C22×C67#38812yes—
176C22×C67#39011yescharacteristic
177C22×C67C2×#10112yes—
178C22×C67#41312yes—
179C4×C67—24no—
180C4×C67—42no—
181C4×C67#38612yes—
182C4×C67#38812yes—
183C4×C67#38812yes—
184C4×C67#39012yes—
185Dic67—44no—
186Dic67—82no—
187#10963—671nomaximal, Sylow
188C2×C67—22no—
189C2×C67—22no—
190C2×C67#143711yescharacteristic
191C2×C67C4×#10112yes—
192C2×C67#190112yes—
193C2×C67#196611yescharacteristic
194C2×C67#196711yescharacteristic
195C22×#101—672no—
196C2×C4×M16D6711yescharacteristic
197C2×#391—671no—
198C2×#343—672no—
199#1728—671no—
200C67#1096311yescharacteristic, complemented, Sylow
201C23×Q8—671no—
202C22×M16—672no—
203C22×M16C2×D6712yes—
204C22×C4⋊C4—671no—
205C22×C42C2×D6711yescharacteristic
206C2×#101—678no—
207C2×#101—1344no—
208C2×#101—1344no—
209C4×M16—24no—
210C2×#97—21no—
211C2×C4×C8—21no—
212#391—1342no—
213#391—1341no—
214#391—1341no—
215C2×#95—674no—
216#343—1344no—
217#336—1342no—
218C22×Q8—674no—
219C22×Q8—1341no—
220C22×Q8—1342no—
221C22×Q8—1341no—
222C22×Q8—1342no—
223C22×Q8—1342no—
224C22×Q8—1342no—
225C22×Q8—1342no—
226C23×C4—672no—
227C23×C4—672no—
228C23×C4C4×D6712yes—
229C23×C4C22×D6711yescharacteristic
230C2×M16—22no—
231C2×M16—22no—
232C2×M16—22no—
233C2×M16—674no—
234C2×M16—1342no—
235C2×M16—1342no—
236C2×M16—1342no—
237C2×M16—1342no—
238C2×M16—1342no—
239C2×M16C4×D6714yes—
240C2×M16D26814yes—
241C2×M16C22⋊D6714yes—
242C22×C8—22no—
243C22×C8—1342no—
244C2×C4⋊C4—1341no—
245C2×C4⋊C4—1342no—
246C2×C4⋊C4—1341no—
247C2×C4⋊C4—1342no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42D26812yes—
251C2×C42C22⋊D6711yescharacteristic
252C2×C42C22⋊D6711yescharacteristic
253#101—13416no—
254#97—24no—
255C4×C8—24no—
256#95—1344no—
257#95—1344no—
258C24C2×C4×D6711yescharacteristic
259C2×Q8—2681no—
260C2×Q8—2682no—
261C2×Q8—2682no—
262C2×Q8—2682no—
263C2×Q8—2681no—
264C2×Q8—2682no—
265C2×Q8—2682no—
266C2×Q8—2681no—
267C2×Q8—2682no—
268C2×Q8—2681no—
269C2×Q8—1344no—
270C2×Q8—1344no—
271C2×Q8—1344no—
272C2×Q8—1344no—
273C2×Q8—1344no—
274C2×Q8—1344no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—42no—
278C22×C4—2684no—
279C22×C4—674no—
280C22×C4—1342no—
281C22×C4—1342no—
282C22×C4—1342no—
283C22×C4—1344no—
284C22×C4—1344no—
285C22×C4—1344no—
286C22×C4(C2×D67)⋊C412yes—
287C22×C4(C2×D67)⋊C412yes—
288C22×C4C2×C4×D6712yes—
289C22×C4C2×C22⋊D6712yes—
290C22×C4C2×C22⋊D6711yescharacteristic
291C22×C4C2×D26812yes—
292C22×C4C2×D26811yescharacteristic
293C22×C4#1176162812yes—
294C22×C4#1176162812yes—
295C22×C4#1176162812yes—
296M16—24no—
297M16—24no—
298M16—24no—
299M16—2682no—
300M16—2682no—
301M16—1344no—
302M16—1344no—
303M16(C2×D67)⋊C418yes—
304C2×C8—24no—
305C2×C8—24no—
306C2×C8—24no—
307C2×C8—2682no—
308C2×C8—2682no—
309C2×C8—1344no—
310C4⋊C4—2682no—
311C4⋊C4—2681no—
312C4⋊C4—2681no—
313C42—21no—
314C42—21no—
315C42—24no—
316C42—21no—
317C42—21no—
318C23—22no—
319C23—22no—
320C23C42×D6711yescharacteristic
321C23C2×(C2×D67)⋊C411yescharacteristic
322C23#1452400011yescharacteristic
323C23C4×C22⋊D6712yes—
324C23C4×D26812yes—
325Q8—2684no—
326Q8—2684no—
327Q8—2684no—
328Q8—2684no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—48no—
335C2×C4—2682no—
336C2×C4—2684no—
337C2×C4—2682no—
338C2×C4—2682no—
339C2×C4—2684no—
340C2×C4—5364no—
341C2×C4—1348no—
342C2×C4—1344no—
343C2×C4#1452396114yescomplemented
344C2×C4C2×(C2×D67)⋊C412yes—
345C2×C4#1452400012yes—
346C2×C4C4×C22⋊D6712yes—
347C2×C4C4×C22⋊D6712yes—
348C2×C4C4×D26812yes—
349C2×C4C4×D26812yes—
350C2×C4#1452401211yescharacteristic
351C2×C4#1452401712yes—
352C2×C4#1452401712yes—
353C2×C4#1452401712yes—
354C2×C4#1452402612yes—
355C2×C4#1452402611yescharacteristic
356C2×C4#1452403111yescharacteristic
357C2×C4#1452403112yes—
358C2×C4#1452403712yes—
359C2×C4#1452404011yescharacteristic
360C8—28no—
361C8—2684no—
362C22—22no—
363C22—22no—
364C22—22no—
365C22—22no—
366C22—24no—
367C22C4×(C2×D67)⋊C412yes—
368C22C4×(C2×D67)⋊C411yescharacteristic
369C22#1686327511yescharacteristic
370C22#1686328712yes—
371C22#1686329811yescharacteristic
372C22C2×#1452396112yes—
373C22#1686337812yes—
374C4—24no—
375C4—42no—
376C4—2684no—
377C4—5362no—
378C4#1686327512yes—
379C4#1686328712yes—
380C4#1686328712yes—
381C4#1686329812yes—
382C2—22no—
383C2—22no—
384C2#2255844711yescharacteristic
385C2C4×#1452396112yes—
386C2#2256067712yes—
387C2#2256105311yescharacteristic
388C2#2256105811yescharacteristic
subgroupG/H
centerC23C2×(C2×D67)⋊C4
commutatorC23×C67C2×C42
FrattiniC23×C4C22×D67
FittingC2×C4×M16×C67C2
socleC23×C67C2×C22⋊C4
prime radicalC24×C67C22×C4
cosocleC23×C4×C67C23
hypercenterC2×C4×M16D67
hypocenterC67#10963
LCSG ▷ C23×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×C4×M16

Complements

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

NH ≅ G/Ntype
11. C2×#14523961C4semidirect
136. C2×C4×C67#101semidirect
200. C67#10963semidirect
343. C2×C4#14523961semidirect

Maximal subgroups

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

subgroupindex
C22×#145239612
C2×#168633022
C2×#168631642
#225599032
C2×C4×M16×C672
#1096367