Finite Group Databasebeta

← #30859142

Subgroups — #30859142

Order 17152.

1002 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
1C2×#16863422C211yesmaximal, characteristic
2C2×#16863419C211yesmaximal, characteristic
3C2×#16863352C211yesmaximal, characteristic
4#22562693C211yesmaximal, characteristic
5#22560930C214yesmaximal, complemented
6#22560925C214yesmaximal, complemented
7#22559907C211yesmaximal, characteristic
8C2×#16863150C211yesmaximal, characteristic
9C2×#424×C67C211yesmaximal, characteristic
10C22×#11761612C2211yescharacteristic
11C23×C67⋊C8C2211yescharacteristic
12C2×#14524002C2211yescharacteristic
13C22×C4×Dic67C2211yescharacteristic
14#16863422—24no—
15#16863419—24no—
16C2×Dic67⋊C8C2214yes—
17C2×Dic67⋊C8C2214yes—
18#16863352—24no—
19C2×#14523954C2212yes—
20C2×C4×C67⋊C8C2212yes—
21#16863295C412yes—
22#16863295C412yes—
23#16863271C414yes—
24C4×C4⋊Dic67C2214yes—
25#16863150C2212yes—
26#16863150C2212yes—
27#16863150C2212yes—
28#16863150C2212yes—
29C22×M16×C67C2211yescharacteristic
30C23×C8×C67C2211yescharacteristic
31C2×#117×C67C2211yescharacteristic
32#424×C67—24no—
33C2×C4⋊C8×C67C2214yes—
34C23×Dic67C2×C412yes—
35C2×#11761612—22no—
36C2×#11761612—22no—
37C2×#11761612D414yes—
38C22×C67⋊C8—22no—
39C22×C67⋊C8—22no—
40C22×C67⋊C8Q814yes—
41C22×C67⋊C8C2312yes—
42C22×C67⋊C8C2311yescharacteristic
43C22×C67⋊C8C2311yescharacteristic
44#14524002—22no—
45#14524002—22no—
46C2×C4⋊Dic67C2×C414yes—
47C2×C4⋊Dic67C2×C412yes—
48C2×C4⋊Dic67C2×C414yes—
49C2×C22⋊Dic67C2×C412yes—
50C2×C4×Dic67—22no—
51C2×C4×Dic67—22no—
52C2×C4×Dic67C2×C411yescharacteristic
53C2×C4×Dic67C2×C411yescharacteristic
54C2×C4×Dic67C2312yes—
55C2×C4×Dic67C2312yes—
56Dic67⋊C8—28no—
57Dic67⋊C8—28no—
58#14523954—24no—
59C4×C67⋊C8—24no—
60C67⋊#95C2×C414yes—
61C67⋊#95C2×C414yes—
62C23×C4×C67C2311yescharacteristic
63C2×M16×C67—22no—
64C2×M16×C67—22no—
65C2×M16×C67—22no—
66C22×C8×C67—22no—
67C22×C8×C67—22no—
68C22×C8×C67—22no—
69C22×C8×C67C2312yes—
70C22×C8×C67C2312yes—
71#117×C67D414yes—
72#117×C67Q814yes—
73C2×C4⋊C4×C67C2×C412yes—
74C2×C22⋊C4×C67C2×C412yes—
75C2×C42×C67C2312yes—
76C4⋊C8×C67—28no—
77C22×Dic67—24no—
78C22×Dic67—24no—
79C22×Dic67—42no—
80C22×Dic67C22×C412yes—
81C22×Dic67C22×C414yes—
82C22×Dic67C22×C412yes—
83#11761612—28no—
84C2×C67⋊C8—28no—
85C2×C67⋊C8—21no—
86C2×C67⋊C8—22no—
87C2×C67⋊C8—21no—
88C2×C67⋊C8—21no—
89C2×C67⋊C8—21no—
90C2×C67⋊C8—22no—
91C2×C67⋊C8C2×D414yes—
92C2×C67⋊C8C2×Q812yes—
93C2×C67⋊C8C2×Q812yes—
94C4⋊Dic67—24no—
95C4⋊Dic67—28no—
96C4⋊Dic67—44no—
97C22⋊Dic67—24no—
98C4×Dic67—22no—
99C4×Dic67—22no—
100C4×Dic67—22no—
101C4×Dic67—22no—
102C4×Dic67—22no—
103C4×Dic67—22no—
104C24×C67C22×C411yescharacteristic
105C22×C4×C67C22×C414yes—
106C22×C4×C67C22×C411yescharacteristic
107C22×C4×C67C2×D412yes—
108C22×C4×C67C2×D412yes—
109C22×C4×C67C2×Q812yes—
110C22×C4×C67C2×Q812yes—
111C22×C4×C67#4112yes—
112C22×C4×C67#4112yes—
113C22×C4×C67C2411yescharacteristic
114M16×C67—44no—
115C2×C8×C67—22no—
116C2×C8×C67—22no—
117C2×C8×C67—22no—
118C2×C8×C67—22no—
119C2×C8×C67—22no—
120C2×C8×C67—22no—
121C2×C8×C67—44no—
122C4⋊C4×C67C4⋊C418yes—
123C22⋊C4×C67C4⋊C418yes—
124C42×C67C2×D414yes—
125C42×C67C2×Q814yes—
126C23×C67—22no—
127C23×C67—22no—
128C23×C67C2×C4⋊C412yes—
129C23×C67C4×D412yes—
130C23×C67C4×Q812yes—
131C23×C67C23×C411yescharacteristic
132C2×C4×C67—24no—
133C2×C4×C67—24no—
134C2×C4×C67C2×C4⋊C418yes—
135C2×C4×C67C2×C4⋊C412yes—
136C2×C4×C67C4×D412yes—
137C2×C4×C67C4×Q812yes—
138C2×C4×C67#12114yes—
139C2×C4×C67C22⋊Q814yes—
140C2×C4×C67#12514yes—
141C2×C4×C67C4⋊Q814yes—
142C2×C4×C67C22×D411yescharacteristic
143C2×C4×C67C22×D411yescharacteristic
144C2×C4×C67C22×Q811yescharacteristic
145C2×C4×C67C22×Q811yescharacteristic
146C2×C4×C67C2×#4111yescharacteristic
147C2×C4×C67C2×#4111yescharacteristic
148C2×Dic67—22no—
149C2×Dic67—22no—
150C2×Dic67—22no—
151C2×Dic67—24no—
152C2×Dic67—24no—
153C2×Dic67—22no—
154C2×Dic67—22no—
155C2×Dic67—24no—
156C2×Dic67—48no—
157C2×Dic67#13114yes—
158C8×C67—42no—
159C8×C67—42no—
160C67⋊C8—24no—
161C67⋊C8—22no—
162C67⋊C8—22no—
163C22×C67—24no—
164C22×C67—24no—
165C22×C67—22no—
166C22×C67—22no—
167C22×C67#38914yes—
168C22×C67C22×C4⋊C411yescharacteristic
169C22×C67C2×C4×D411yescharacteristic
170C22×C67C2×C4×Q811yescharacteristic
171C22×C67C2×#13112yes—
172C22×C67#56812yes—
173C4×C67—28no—
174C4×C67#38914yes—
175C4×C67C2×#12111yescharacteristic
176C4×C67C2×C22⋊Q811yescharacteristic
177C4×C67C2×#12511yescharacteristic
178C4×C67C2×C4⋊Q811yescharacteristic
179Dic67—24no—
180Dic67—44no—
181Dic67—42no—
182#21148—671nomaximal, Sylow
183C2×C67—24no—
184C2×C67C2×#38911yescharacteristic
185C2×C67#290612yes—
186C2×C67#293312yes—
187C2×C67#296011yescharacteristic
188C2×C67#296111yescharacteristic
189C2×#424—671no—
190C2×#424D6711yescharacteristic
191C2×#423—671no—
192C2×#405—671no—
193#2296—671no—
194#1940—674no—
195#1939—674no—
196#1729—671no—
197C2×#336—671no—
198C67#2114811yescharacteristic, complemented, Sylow
199C22×M16—671no—
200C22×M16C2×D6711yescharacteristic
201C23×C8—671no—
202C23×C8C2×D6711yescharacteristic
203C2×#117—671no—
204C2×#117C2×D6711yescharacteristic
205C22×C42—671no—
206#424—24no—
207#424—1344no—
208#423—1344no—
209C2×C4⋊C8—674no—
210C2×C4⋊C8—674no—
211C2×C4⋊C8C2×D6714yes—
212#405—1344no—
213C2×#97—672no—
214C2×C4×C8—672no—
215#389—672no—
216#389—672no—
217#385—674no—
218C4×C4⋊C4—674no—
219#336—672no—
220#336—672no—
221#336—672no—
222#336—672no—
223C23×C4—672no—
224C23×C4C22×D6711yescharacteristic
225C2×M16—22no—
226C2×M16—22no—
227C2×M16—22no—
228C2×M16—674no—
229C2×M16—1342no—
230C2×M16—1342no—
231C22×C8—22no—
232C22×C8—22no—
233C22×C8—22no—
234C22×C8—674no—
235C22×C8—671no—
236C22×C8—671no—
237C22×C8—672no—
238C22×C8—1342no—
239C22×C8—1342no—
240C22×C8C22×D6712yes—
241C22×C8C22×D6712yes—
242#117—1342no—
243#117—1342no—
244#117Dic13414yes—
245#117C22⋊D6714yes—
246C2×C4⋊C4—674no—
247C2×C4⋊C4—672no—
248C2×C4⋊C4—674no—
249C2×C4⋊C4C4×D6712yes—
250C2×C22⋊C4—672no—
251C2×C22⋊C4C4×D6712yes—
252C2×C42—671no—
253C2×C42—671no—
254C2×C42—672no—
255C2×C42—672no—
256C2×C42—1342no—
257C2×C42—1342no—
258C2×C42C22×D6712yes—
259C4⋊C8—28no—
260C4⋊C8—1348no—
261C4⋊C8—1348no—
262#97—1344no—
263C4×C8—1344no—
264#95—674no—
265#95—674no—
266C24C2×C4×D6711yescharacteristic
267C22×C4—2682no—
268C22×C4—672no—
269C22×C4—672no—
270C22×C4—674no—
271C22×C4—1344no—
272C22×C4—1344no—
273C22×C4C2×C4×D6714yes—
274C22×C4C2×C4×D6711yescharacteristic
275C22×C4D4×D6712yes—
276C22×C4C2×C22⋊D6712yes—
277C22×C4Q8×D6712yes—
278C22×C4C2×Dic13412yes—
279C22×C4#1176162712yes—
280C22×C4#1176162912yes—
281C22×C4C23×D6711yescharacteristic
282M16—44no—
283M16—1348no—
284C2×C8—22no—
285C2×C8—22no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—22no—
289C2×C8—22no—
290C2×C8—44no—
291C2×C8—674no—
292C2×C8—672no—
293C2×C8—672no—
294C2×C8—1348no—
295C2×C8—1341no—
296C2×C8—1342no—
297C2×C8—1341no—
298C2×C8—1342no—
299C2×C8—1341no—
300C2×C8—1341no—
301C4⋊C4—2684no—
302C4⋊C4—1348no—
303C4⋊C4—1344no—
304C4⋊C4Dic67⋊C418yes—
305C22⋊C4—1344no—
306C22⋊C4Dic67⋊C418yes—
307C42—1342no—
308C42—1342no—
309C42—1342no—
310C42—1342no—
311C42—1342no—
312C42—1342no—
313C42C2×C22⋊D6714yes—
314C42C2×Dic13414yes—
315C23—22no—
316C23—22no—
317C23C2×Dic67⋊C412yes—
318C23#1452400512yes—
319C23#1452400812yes—
320C23C22×C4×D6711yescharacteristic
321C2×C4—24no—
322C2×C4—24no—
323C2×C4—2688no—
324C2×C4—674no—
325C2×C4—1342no—
326C2×C4—1344no—
327C2×C4—1344no—
328C2×C4—1342no—
329C2×C4—1342no—
330C2×C4—1344no—
331C2×C4—1342no—
332C2×C4—1342no—
333C2×C4C2×Dic67⋊C418yes—
334C2×C4C2×Dic67⋊C412yes—
335C2×C4#1452400512yes—
336C2×C4#1452400812yes—
337C2×C4#1452401414yes—
338C2×C4#1452401914yes—
339C2×C4#1452403414yes—
340C2×C4#1452404214yes—
341C2×C4C2×D4×D6711yescharacteristic
342C2×C4C22×C22⋊D6711yescharacteristic
343C2×C4C2×Q8×D6711yescharacteristic
344C2×C4C22×Dic13411yescharacteristic
345C2×C4C2×#1176162711yescharacteristic
346C2×C4C2×#1176162911yescharacteristic
347C8—42no—
348C8—42no—
349C8—1344no—
350C8—1342no—
351C8—1342no—
352C22—24no—
353C22—22no—
354C22—22no—
355C22—24no—
356C22#1686329214yes—
357C22C22×Dic67⋊C411yescharacteristic
358C22C2×#1452400511yescharacteristic
359C22C2×#1452400811yescharacteristic
360C22#131×D6712yes—
361C22#1686416112yes—
362C4—28no—
363C4—2682no—
364C4—2684no—
365C4—1344no—
366C4#1686329214yes—
367C4C2×#1452401411yescharacteristic
368C4C2×#1452401911yescharacteristic
369C4C2×#1452403411yescharacteristic
370C4C2×#1452404211yescharacteristic
371C2—24no—
372C2C2×#1686329211yescharacteristic
373C2#2256877512yes—
374C2#2256893312yes—
375C2#2256914211yescharacteristic
376C2#2256914811yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D67
commutator, socleC23×C67C23×C4
FittingC2×#424×C67C2
prime radicalC24×C67C22×C4
cosocleC22×C4×C67C24
hypercenterC2×#424D67
hypocenterC67#21148
LCSG ▷ C23×C67 ▷ C67
UCSC1 ◁ C22×C4 ◁ C2×#424

Complements

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

NH ≅ G/Ntype
5. #22560930C2semidirect
6. #22560925C2semidirect
198. C67#21148semidirect

Maximal subgroups

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

subgroupindex
C2×#168634222
C2×#168634192
C2×#168633522
#225626932
#225609302
#225609252
#225599072
C2×#168631502
C2×#424×C672
#2114867