Finite Group Databasebeta

← #30754192

Subgroups — #30754192

Order 17152.

382 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22561169C211yesmaximal, characteristic
2#22561157C211yesmaximal, characteristic
3C4⋊C4×C67⋊C8C211yesmaximal, characteristic
4C2×#16863144C211yesmaximal, characteristic
5#22558092C211yesmaximal, characteristic
6#22558043C211yesmaximal, characteristic
7C67×#1947C211yesmaximal, characteristic
8C2×#14524043C2211yescharacteristic
9C2×C4⋊(C67⋊C8)C2211yescharacteristic
10C2×C4⋊(C67⋊C8)C2211yescharacteristic
11C2×C4×C67⋊C8—21no—
12C2×C4×C67⋊C8C2211yescharacteristic
13#16863158—21no—
14#16863158—21no—
15#16863158—21no—
16#16863158—21no—
17#16863151—21no—
18#16863144C414yescomplemented
19C2×#107×C67—21no—
20C2×C4⋊C8×C67C2211yescharacteristic
21#384×C67C2211yescharacteristic
22C4×C4⋊C4×C67C2211yescharacteristic
23#340×C67—21no—
24C22×Dic134—21no—
25C22×C67⋊C8—21no—
26C22×C67⋊C8—21no—
27C22×C67⋊C8D411yescharacteristic
28C22×C67⋊C8D411yescharacteristic
29#14524043—21no—
30#14524043—21no—
31#14524043—21no—
32#14524043C2×C412yes—
33C2×C4⋊Dic67—21no—
34C2×C4⋊Dic67—21no—
35C4⋊(C67⋊C8)—21no—
36C4⋊(C67⋊C8)—21no—
37C4⋊(C67⋊C8)—21no—
38C4⋊(C67⋊C8)C2×C412yes—
39C4×C67⋊C8—21no—
40C4×C67⋊C8—21no—
41C4×C67⋊C8—42no—
42C22×C8×C67—21no—
43C2×C4⋊C4×C67—21no—
44C2×C4⋊C4×C67—21no—
45C2×C4⋊C4×C67—21no—
46C2×C4⋊C4×C67D411yescharacteristic
47C2×C4⋊C4×C67D411yescharacteristic
48C2×C42×C67—21no—
49C2×C42×C67C2311yescharacteristic
50#107×C67—42no—
51C4⋊C8×C67—21no—
52C4⋊C8×C67C2×C412yes—
53#95×C67—21no—
54C2×Dic134—22no—
55C2×Dic134—41no—
56C2×Dic134—81no—
57C2×Dic134—81no—
58C22×Dic67—41no—
59C2×C67⋊C8—21no—
60C2×C67⋊C8—21no—
61C2×C67⋊C8—22no—
62C2×C67⋊C8—21no—
63C2×C67⋊C8—21no—
64C2×C67⋊C8—21no—
65C2×C67⋊C8—41no—
66C2×C67⋊C8—41no—
67C2×C67⋊C8—41no—
68C2×C67⋊C8—41no—
69C2×C67⋊C8—41no—
70C2×C67⋊C8Q1612yes—
71C4⋊Dic67—22no—
72C4⋊Dic67—22no—
73C4⋊Dic67—41no—
74C4⋊Dic67—41no—
75C22×C4×C67—21no—
76C22×C4×C67—21no—
77C22×C4×C67—41no—
78C22×C4×C67C2×D411yescharacteristic
79C22×C4×C67C2×D411yescharacteristic
80C22×C4×C67#4111yescharacteristic
81C2×C8×C67—22no—
82C2×C8×C67—41no—
83C2×C8×C67—41no—
84C4⋊C4×C67—21no—
85C4⋊C4×C67—21no—
86C4⋊C4×C67—41no—
87C4⋊C4×C67—41no—
88C4⋊C4×C67—41no—
89C4⋊C4×C67—41no—
90C4⋊C4×C67—81no—
91C4⋊C4×C67SD1612yes—
92C4⋊C4×C67Q1612yes—
93C42×C67—41no—
94C42×C67—41no—
95C42×C67C22×C411yescharacteristic
96C42×C67#4111yescharacteristic
97C42×C67#4111yescharacteristic
98C42×C67#4111yescharacteristic
99C23×C67#12111yescharacteristic
100C2×C4×C67—21no—
101C2×C4×C67—21no—
102C2×C4×C67—22no—
103C2×C4×C67—22no—
104C2×C4×C67—41no—
105C2×C4×C67—41no—
106C2×C4×C67—41no—
107C2×C4×C67—41no—
108C2×C4×C67—81no—
109C2×C4×C67—81no—
110C2×C4×C67—81no—
111C2×C4×C67#11711yescharacteristic
112C2×C4×C67C4×D411yescharacteristic
113C2×C4×C67C4×D411yescharacteristic
114C2×C4×C67#12311yescharacteristic
115C2×C4×C67#12411yescharacteristic
116C2×C4×C67#12611yescharacteristic
117C2×C4×C67C2×SD1611yescharacteristic
118C2×C4×C67C2×Q1611yescharacteristic
119C2×C4×C67C2×Q1611yescharacteristic
120C2×C4×C67#13511yescharacteristic
121C2×C4×C67#13511yescharacteristic
122C2×C4×C67#13511yescharacteristic
123C2×C4×C67AGL(1,ℤ/8)11yescharacteristic
124C2×C4×C67#13711yescharacteristic
125C2×Dic67—42no—
126C2×Dic67—81no—
127C2×Dic67—81no—
128Dic134—82no—
129Dic134—82no—
130C8×C67—42no—
131C67⋊C8—22no—
132C67⋊C8—42no—
133C67⋊C8—41no—
134C67⋊C8—81no—
135C22×C67#38811yescharacteristic
136C22×C67#46211yescharacteristic
137C22×C67#46311yescharacteristic
138C22×C67#46511yescharacteristic
139C22×C67#46711yescharacteristic
140C22×C67#47111yescharacteristic
141C22×C67#47211yescharacteristic
142C4×C67—21no—
143C4×C67—21no—
144C4×C67—42no—
145C4×C67—42no—
146C4×C67—161no—
147C4×C67C4×SD1611yescharacteristic
148C4×C67C4×Q1611yescharacteristic
149C4×C67C4×Q1611yescharacteristic
150C4×C67#44011yescharacteristic
151C4×C67#48411yescharacteristic
152C4×C67#48511yescharacteristic
153C4×C67#48711yescharacteristic
154C4×C67#48811yescharacteristic
155Dic67—82no—
156#9568—671nomaximal, Sylow
157C2×C67#167211yescharacteristic
158C2×C67#167611yescharacteristic
159C2×C67#167711yescharacteristic
160C2×C67#168311yescharacteristic
161C2×C67#192811yescharacteristic
162C2×C67#193111yescharacteristic
163C2×C67#193311yescharacteristic
164#1990—671no—
165#1987—671no—
166#1947D6711yescharacteristic
167C8×C4⋊C4—671no—
168C2×#333—671no—
169#1296—671no—
170#1277—671no—
171C67#956811yescharacteristic, complemented, Sylow
172C2×C4⋊Q8—671no—
173C2×#107—21no—
174C2×C4⋊C8—671no—
175C2×C4⋊C8—671no—
176C2×C4⋊C8C2×D6711yescharacteristic
177C2×C4×C8—671no—
178C2×C4×C8—1341no—
179#384C2×D6711yescharacteristic
180C4×C4⋊C4C2×D6711yescharacteristic
181#340—21no—
182#340—1341no—
183#340—1341no—
184#340—1341no—
185#340—1341no—
186#336—1341no—
187#333—674no—
188C22×Q8—1341no—
189C22×C8—21no—
190C22×C8—671no—
191C22×C8—671no—
192C22×C8—1341no—
193C22×C8—1341no—
194C4⋊Q8—672no—
195C4⋊Q8—1341no—
196C4⋊Q8—1341no—
197C4⋊Q8—1341no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—1341no—
202C2×C4⋊C4—1341no—
203C2×C4⋊C4C22⋊D6711yescharacteristic
204C2×C4⋊C4C22⋊D6711yescharacteristic
205C2×C42—21no—
206C2×C42C22×D6711yescharacteristic
207#107—42no—
208C4⋊C8—21no—
209C4⋊C8—672no—
210C4⋊C8—1341no—
211C4⋊C8—1341no—
212C4⋊C8—1341no—
213C4⋊C8C4×D6712yes—
214C4×C8—2682no—
215C4×C8—1341no—
216C4×C8—1341no—
217#95—21no—
218C2×Q8—2681no—
219C2×Q8—5361no—
220C2×Q8—5361no—
221C2×Q8—1342no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—41no—
225C22×C4—2681no—
226C22×C4D4×D6711yescharacteristic
227C22×C4C2×C22⋊D6711yescharacteristic
228C22×C4#1176162711yescharacteristic
229C2×C8—22no—
230C2×C8—41no—
231C2×C8—41no—
232C2×C8—2681no—
233C2×C8—2681no—
234C2×C8—2681no—
235C2×C8—2681no—
236C2×C8—2681no—
237C2×C8—672no—
238C2×C8—1342no—
239C2×C8—1341no—
240C2×C8—1341no—
241C2×C8—1341no—
242C2×C8—1341no—
243C2×C8—1341no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—81no—
251C4⋊C4—2681no—
252C4⋊C4—2681no—
253C4⋊C4—1342no—
254C4⋊C4—1342no—
255C4⋊C4#1176161612yes—
256C4⋊C4#1176161812yes—
257C42—41no—
258C42—41no—
259C42C2×C4×D6711yescharacteristic
260C42#1176162811yescharacteristic
261C42#1176162911yescharacteristic
262C42#1176162911yescharacteristic
263C23#1452401511yescharacteristic
264Q8—5362no—
265Q8—5362no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—81no—
275C2×C4—81no—
276C2×C4—81no—
277C2×C4—2682no—
278C2×C4—5361no—
279C2×C4—5361no—
280C2×C4#1452400111yescharacteristic
281C2×C4C4×C22⋊D6711yescharacteristic
282C2×C4#1452400511yescharacteristic
283C2×C4#1452402811yescharacteristic
284C2×C4#1452403211yescharacteristic
285C2×C4#1452403811yescharacteristic
286C2×C4C2×#1176161611yescharacteristic
287C2×C4Q16×D6711yescharacteristic
288C2×C4C2×#1176161811yescharacteristic
289C2×C4#1452406211yescharacteristic
290C2×C4#1452406311yescharacteristic
291C2×C4#1452406311yescharacteristic
292C2×C4#1452406711yescharacteristic
293C2×C4#1452407511yescharacteristic
294C8—42no—
295C8—2682no—
296C8—2681no—
297C8—5361no—
298C8—1342no—
299C22#1686328611yescharacteristic
300C22#1686360411yescharacteristic
301C22#1686360911yescharacteristic
302C22#1686362311yescharacteristic
303C22#1686363511yescharacteristic
304C22#1686366111yescharacteristic
305C22#1686366411yescharacteristic
306C4—21no—
307C4—21no—
308C4—42no—
309C4—42no—
310C4—161no—
311C4—5362no—
312C4C4×#1176161611yescharacteristic
313C4C4×#1176161811yescharacteristic
314C4#1686349011yescharacteristic
315C4#1686349511yescharacteristic
316C4#1686373311yescharacteristic
317C4#1686373911yescharacteristic
318C4#1686374811yescharacteristic
319C4#1686375311yescharacteristic
320C2#2255958911yescharacteristic
321C2#2255961111yescharacteristic
322C2#2255961611yescharacteristic
323C2#2255965811yescharacteristic
324C2#2256084911yescharacteristic
325C2#2256087011yescharacteristic
326C2#2256088411yescharacteristic
subgroupG/H
centerC23#14524015
commutatorC42×C67C22×C4
FrattiniC2×C42C22×D67
FittingC67×#1947C2
socle, prime radicalC23×C67#121
cosocleC2×C42×C67C23
hypercenter#1947D67
hypocenterC67#9568
LCSG ▷ C42×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1947

Complements

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

NH ≅ G/Ntype
18. #16863144C4semidirect
171. C67#9568semidirect

Maximal subgroups

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

subgroupindex
#225611692
#225611572
C4⋊C4×C67⋊C82
C2×#168631442
#225580922
#225580432
C67×#19472
#956867