Finite Group Databasebeta

← #30746379

Subgroups — #30746379

Order 17152.

582 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×C67⋊C8C211yesmaximal, characteristic
2#22559987C212yesmaximal, complemented
3#22558527C211yesmaximal, characteristic
4#22558057C212yesmaximal, complemented
5C4×#99×C67C211yesmaximal, characteristic
6C23×C67⋊C8C411yescharacteristic, complemented
7C23×C67⋊C8C2211yescharacteristic
8D4×C67⋊C8—22no—
9D4×C67⋊C8—22no—
10C2×C4⋊(C67⋊C8)C411yescharacteristic, complemented
11C2×C22⋊(C67⋊C8)C411yescharacteristic, complemented
12C2×C22⋊(C67⋊C8)C2211yescharacteristic
13C2×C22⋊(C67⋊C8)C2212yes—
14C2×#14523954—22no—
15C2×C4×C67⋊C8C411yescharacteristic, complemented
16#16863151—22no—
17#16863122C412yescomplemented
18#16863120C412yescomplemented
19C2×C4×D4×C67C2211yescharacteristic
20C2×#99×C67C412yes—
21C4×C22⋊C4×C67C2212yes—
22#342×C67C412yes—
23C22×C67⋊C8—21no—
24C22×C67⋊C8—21no—
25C22×C67⋊C8—22no—
26C22×C67⋊C8—42no—
27C22×C67⋊C8—42no—
28C22×C67⋊C8C2×C411yescharacteristic
29C22×C67⋊C8C2×C411yescharacteristic
30C22×C67⋊C8C2×C411yescharacteristic
31C22×C67⋊C8C2×C411yescharacteristic
32C22×C67⋊C8D412yes—
33C22×C67⋊C8D412yes—
34C4⋊(C67⋊C8)—22no—
35C22⋊(C67⋊C8)—22no—
36C22⋊(C67⋊C8)—22no—
37C22⋊(C67⋊C8)—22no—
38C22⋊(C67⋊C8)C2×C414yes—
39#14523954—44no—
40C4×C67⋊C8—21no—
41C4×C67⋊C8—21no—
42C22×D4×C67C2×C411yescharacteristic
43C23×C4×C67C2×C411yescharacteristic
44C23×C4×C67C2311yescharacteristic
45C4×D4×C67—22no—
46C4×D4×C67—22no—
47C2×C4⋊C4×C67C2×C411yescharacteristic
48C2×C22⋊C4×C67C2×C411yescharacteristic
49C2×C22⋊C4×C67C2×C414yes—
50C2×C22⋊C4×C67C2×C411yescharacteristic
51C2×C42×C67—22no—
52C2×C42×C67C2×C411yescharacteristic
53#99×C67—24no—
54#95×C67—22no—
55C2×C67⋊C8—24no—
56C2×C67⋊C8—22no—
57C2×C67⋊C8—21no—
58C2×C67⋊C8—21no—
59C2×C67⋊C8—24no—
60C2×C67⋊C8—41no—
61C2×C67⋊C8—42no—
62C2×C67⋊C8—42no—
63C2×C67⋊C8—42no—
64C2×C67⋊C8—42no—
65C2×C67⋊C8—42no—
66C2×C67⋊C8—41no—
67C2×C67⋊C8—41no—
68C2×C67⋊C8C22⋊C411yescharacteristic
69C2×C67⋊C8C22⋊C412yes—
70C2×C67⋊C8C22⋊C411yescharacteristic
71C24×C67C4211yescharacteristic
72C24×C67C22×C411yescharacteristic
73C2×D4×C67—22no—
74C2×D4×C67—22no—
75C2×D4×C67M1614yes—
76C22×C4×C67—21no—
77C22×C4×C67—21no—
78C22×C4×C67—24no—
79C22×C4×C67—41no—
80C22×C4×C67—41no—
81C22×C4×C67—41no—
82C22×C4×C67—41no—
83C22×C4×C67C4211yescharacteristic
84C22×C4×C67C4211yescharacteristic
85C22×C4×C67C4211yescharacteristic
86C22×C4×C67C22×C411yescharacteristic
87C22×C4×C67C22×C411yescharacteristic
88C22×C4×C67C2×D412yes—
89C22×C4×C67#4112yes—
90C4⋊C4×C67—22no—
91C22⋊C4×C67—22no—
92C22⋊C4×C67—22no—
93C22⋊C4×C67—24no—
94C22⋊C4×C67—24no—
95C42×C67—21no—
96C42×C67—21no—
97C42×C67—42no—
98C42×C67—42no—
99C23×C67—21no—
100C23×C67—21no—
101C23×C67—41no—
102C23×C67—41no—
103C23×C67—41no—
104C23×C67—41no—
105C23×C67#9712yescomplemented
106C23×C67C2×C4211yescharacteristic
107C23×C67C4×D412yes—
108C23×C67C2×M1612yes—
109C23×C67#13112yes—
110D4×C67—28no—
111C2×C4×C67—21no—
112C2×C4×C67—21no—
113C2×C4×C67—22no—
114C2×C4×C67—21no—
115C2×C4×C67—22no—
116C2×C4×C67—21no—
117C2×C4×C67—21no—
118C2×C4×C67—21no—
119C2×C4×C67—41no—
120C2×C4×C67—42no—
121C2×C4×C67—42no—
122C2×C4×C67—41no—
123C2×C4×C67—41no—
124C2×C4×C67—41no—
125C2×C4×C67—41no—
126C2×C4×C67—44no—
127C2×C4×C67—44no—
128C2×C4×C67—44no—
129C2×C4×C67#9712yescomplemented
130C2×C4×C67C2×C22⋊C411yescharacteristic
131C2×C4×C67#11711yescharacteristic
132C2×C4×C67C4×D412yes—
133C67⋊C8—21no—
134C67⋊C8—22no—
135C67⋊C8—44no—
136C67⋊C8—41no—
137C67⋊C8—41no—
138C67⋊C8#9912yescomplemented
139C22×C67—22no—
140C22×C67—21no—
141C22×C67—22no—
142C22×C67—21no—
143C22×C67—22no—
144C22×C67—42no—
145C22×C67—42no—
146C22×C67—42no—
147C22×C67—42no—
148C22×C67—41no—
149C22×C67—41no—
150C22×C67C4×C22⋊C411yescharacteristic
151C22×C67C2×#9711yescharacteristic
152C22×C67#40511yescharacteristic
153C22×C67#43514yes—
154C4×C67—22no—
155C4×C67—22no—
156C4×C67—21no—
157C4×C67—41no—
158C4×C67—41no—
159C4×C67—84no—
160C4×C67C2×#9911yescharacteristic
161C4×C67#41011yescharacteristic
162#8160—671nomaximal, Sylow
163C2×C67—24no—
164C2×C67—21no—
165C2×C67—21no—
166C2×C67—42no—
167C2×C67#175211yescharacteristic
168C2×C67C4×#9911yescharacteristic
169C2×C67#175711yescharacteristic
170C2×C8×D4—671no—
171C4×#99D6711yescharacteristic
172#1752—672no—
173#1459—671no—
174#1282—672no—
175C67#816011yescharacteristic, complemented, Sylow
176C23×C8—671no—
177C23×C8—671no—
178C2×C4×D4C2×D6711yescharacteristic
179C8×D4—1342no—
180C8×D4—1342no—
181C2×C4⋊C8—671no—
182C2×#99Dic6712yes—
183C2×C22⋊C8—672no—
184C2×C22⋊C8—671no—
185C2×C22⋊C8—671no—
186C2×#97—1342no—
187C2×C4×C8—671no—
188C4×C22⋊C4C2×D6712yes—
189#342Dic6712yes—
190#336—1342no—
191#324—672no—
192#323—672no—
193C22×D4C2×Dic6711yescharacteristic
194C23×C4C4×D6711yescharacteristic
195C23×C4C22×D6711yescharacteristic
196C22×C8—2682no—
197C22×C8—2682no—
198C22×C8—671no—
199C22×C8—672no—
200C22×C8—671no—
201C22×C8—671no—
202C22×C8—671no—
203C22×C8—672no—
204C22×C8—1341no—
205C22×C8—1341no—
206C22×C8—1342no—
207C4×D4—22no—
208C4×D4—22no—
209C2×C4⋊C4C4×D6711yescharacteristic
210C2×C22⋊C4C4×D6711yescharacteristic
211C2×C22⋊C4C2×Dic6714yes—
212C2×C22⋊C4C2×Dic6711yescharacteristic
213C2×C42—22no—
214C2×C42C4×D6711yescharacteristic
215C4⋊C8—1342no—
216#99—24no—
217C22⋊C8—674no—
218C22⋊C8—1342no—
219C22⋊C8—1342no—
220C22⋊C8—1342no—
221#97—2684no—
222C4×C8—1341no—
223C4×C8—1341no—
224#95—22no—
225C24C4×Dic6711yescharacteristic
226C24C22×Dic6711yescharacteristic
227C2×D4—22no—
228C2×D4—22no—
229C2×D4#1176161214yes—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—24no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4C4×Dic6711yescharacteristic
238C22×C4C4×Dic6711yescharacteristic
239C22×C4C4×Dic6711yescharacteristic
240C22×C4C2×C4×D6711yescharacteristic
241C22×C4C2×C4×D6711yescharacteristic
242C22×C4D4×D6712yes—
243C22×C4#1176162712yes—
244C2×C8—2682no—
245C2×C8—2682no—
246C2×C8—2682no—
247C2×C8—2682no—
248C2×C8—2681no—
249C2×C8—2681no—
250C2×C8—2682no—
251C2×C8—2681no—
252C2×C8—671no—
253C2×C8—671no—
254C2×C8—672no—
255C2×C8—1341no—
256C2×C8—1341no—
257C2×C8—1344no—
258C2×C8—1344no—
259C2×C8—1342no—
260C4⋊C4—22no—
261C22⋊C4—22no—
262C22⋊C4—22no—
263C22⋊C4—24no—
264C22⋊C4—24no—
265C42—21no—
266C42—21no—
267C42—42no—
268C42—42no—
269C23—21no—
270C23—21no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23—41no—
275C23#1452395412yescomplemented
276C23C2×C4×Dic6711yescharacteristic
277C23D4×Dic6712yes—
278C23C2×#1176161212yes—
279C23#1452405012yes—
280D4—28no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—22no—
284C2×C4—21no—
285C2×C4—22no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—41no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—44no—
297C2×C4—44no—
298C2×C4—44no—
299C2×C4#1452395412yescomplemented
300C2×C4C22⋊C4×D6711yescharacteristic
301C2×C4#1452399911yescharacteristic
302C2×C4(C22⋊D67)⋊C412yes—
303C8—2684no—
304C8—2681no—
305C8—2681no—
306C8—672no—
307C8—1341no—
308C8—1342no—
309C22—22no—
310C22—21no—
311C22—22no—
312C22—21no—
313C22—22no—
314C22—42no—
315C22—42no—
316C22—42no—
317C22—42no—
318C22—41no—
319C22—41no—
320C22C22⋊C4×Dic6711yescharacteristic
321C22C2×#1452395411yescharacteristic
322C22#1686335211yescharacteristic
323C22#1686346814yes—
324C4—22no—
325C4—22no—
326C4—21no—
327C4—41no—
328C4—41no—
329C4—84no—
330C4#99×D6711yescharacteristic
331C4#1686336811yescharacteristic
332C2—24no—
333C2—21no—
334C2—21no—
335C2—42no—
336C2#2255998711yescharacteristic
337C2#99×Dic6711yescharacteristic
338C2#2256001111yescharacteristic
subgroupG/H
centerC2×C4C22⋊C4×D67
commutatorC23×C67C2×C42
FrattiniC23×C4C22×D67
FittingC4×#99×C67C2
socleC22×C67C4×C22⋊C4
prime radicalC22×D4×C67C2×C4
cosocleC23×C4×C67C23
hypercenterC4×#99D67
hypocenterC67#8160
LCSG ▷ C23×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C4×#99

Complements

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

NH ≅ G/Ntype
2. #22559987C2semidirect
4. #22558057C2semidirect
6. C23×C67⋊C8C4semidirect
10. C2×C4⋊(C67⋊C8)C4semidirect
11. C2×C22⋊(C67⋊C8)C4semidirect
15. C2×C4×C67⋊C8C4semidirect
17. #16863122C4semidirect
18. #16863120C4semidirect
105. C23×C67#97semidirect
129. C2×C4×C67#97semidirect
138. C67⋊C8#99semidirect
175. C67#8160semidirect
275. C23#14523954semidirect
299. C2×C4#14523954semidirect

Maximal subgroups

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

subgroupindex
C2×D4×C67⋊C82
#225599872
#225585272
#225580572
C4×#99×C672
#816067