Finite Group Databasebeta

← #30869384

Subgroups — #30869384

Order 17152.

750 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16863526C211yesmaximal, characteristic
2C2×C8×Dic134C211yesmaximal, characteristic
3C4×#14524043C211yesmaximal, characteristic
4#22561145C211yesmaximal, characteristic
5#22561145C211yesmaximal, characteristic
6#22560774C214yesmaximal
7#22560767C214yesmaximal
8#22560326C211yesmaximal, characteristic
9C67×#1744C211yesmaximal, characteristic
10C2×#14524043C411yescharacteristic
11C2×C4×Dic134C2211yescharacteristic
12C2×C4×Dic134C2211yescharacteristic
13#16863526—24no—
14C8×Dic134—24no—
15C2×C4⋊(C67⋊C8)C2211yescharacteristic
16C2×C4⋊(C67⋊C8)C2211yescharacteristic
17C2×Dic67⋊C8C2212yes—
18C2×Dic67⋊C8C2212yes—
19C2×#14523953C2212yes—
20C2×C8×Dic67C2212yes—
21#16863302C411yescharacteristic
22#16863302C411yescharacteristic
23#16863294C414yes—
24#16863266C411yescharacteristic
25C4×C4⋊Dic67C2214yes—
26#16863151C2214yes—
27#16863150C2214yes—
28#16863150C2214yes—
29C2×C4×C8×C67C2211yescharacteristic
30C2×C4×C8×C67C2211yescharacteristic
31C43×C67C2211yescharacteristic
32#336×C67C2212yes—
33#336×C67C2212yes—
34C22×Dic134C2×C411yescharacteristic
35C22×Dic134C2×C411yescharacteristic
36C22×C67⋊C8C2312yes—
37C22×C67⋊C8C2312yes—
38#14524043—24no—
39C4×Dic134—22no—
40C4×Dic134—22no—
41C4×Dic134—22no—
42C4×Dic134—22no—
43C2×C4⋊Dic67C2×C414yes—
44C2×C4⋊Dic67C2×C414yes—
45C2×C4⋊Dic67C2×C411yescharacteristic
46C2×C4⋊Dic67C2×C411yescharacteristic
47C2×Dic67⋊C4C2×C412yes—
48C2×Dic67⋊C4C2×C412yes—
49C2×C4×Dic67C2312yes—
50C2×C4×Dic67C2312yes—
51C4⋊(C67⋊C8)—22no—
52C4⋊(C67⋊C8)—22no—
53Dic67⋊C8—24no—
54Dic67⋊C8—24no—
55#14523953—24no—
56C8×Dic67—24no—
57C67⋊#95C2×C414yes—
58C67⋊#95C2×C414yes—
59C22×C8×C67C2312yes—
60C22×C8×C67C2311yescharacteristic
61C22×C8×C67C2311yescharacteristic
62C2×C42×C67C2×C411yescharacteristic
63C2×C42×C67C2×C412yes—
64C2×C42×C67C2×C411yescharacteristic
65C2×C42×C67C2312yes—
66C2×C42×C67C2311yescharacteristic
67C4×C8×C67—22no—
68C4×C8×C67D414yes—
69C2×Dic134—22no—
70C2×Dic134—22no—
71C2×Dic134—22no—
72C2×Dic134—22no—
73C2×Dic134—22no—
74C2×Dic134M1614yes—
75C22×Dic67C22×C412yes—
76C22×Dic67C22×C412yes—
77C22×Dic67C22×C412yes—
78C22×Dic67C22×C412yes—
79C2×C67⋊C8—22no—
80C2×C67⋊C8—22no—
81C2×C67⋊C8—24no—
82C2×C67⋊C8—24no—
83C4⋊Dic67—22no—
84C4⋊Dic67—22no—
85C4⋊Dic67—28no—
86C4⋊Dic67—28no—
87Dic67⋊C4—24no—
88Dic67⋊C4—24no—
89C4×Dic67—22no—
90C4×Dic67—22no—
91C4×Dic67—22no—
92C4×Dic67—22no—
93C22×C4×C67C22×C412yes—
94C22×C4×C67C22×C411yescharacteristic
95C22×C4×C67C22×C412yes—
96C22×C4×C67C22×C411yescharacteristic
97C22×C4×C67C2411yescharacteristic
98C2×C8×C67—22no—
99C2×C8×C67—22no—
100C2×C8×C67—22no—
101C2×C8×C67—22no—
102C2×C8×C67—22no—
103C2×C8×C67C2×D412yes—
104C2×C8×C67C2×D412yes—
105C42×C67—22no—
106C42×C67—24no—
107C42×C67—22no—
108C42×C67—22no—
109C42×C67—22no—
110C42×C67C2×D412yes—
111C42×C67#4112yes—
112C23×C67C23×C411yescharacteristic
113C2×C4×C67—22no—
114C2×C4×C67—22no—
115C2×C4×C67—22no—
116C2×C4×C67—22no—
117C2×C4×C67—22no—
118C2×C4×C67—22no—
119C2×C4×C67C4×D412yes—
120C2×C4×C67C4×D412yes—
121C2×C4×C67C2×M1612yes—
122C2×C4×C67#13114yes—
123C2×C4×C67#13112yes—
124C2×C4×C67C22×D411yescharacteristic
125C2×C4×C67C2×#4111yescharacteristic
126C2×C4×C672−1+412yes—
127C2×C4×C672−1+412yes—
128C2×Dic67—22no—
129C2×Dic67—22no—
130C2×Dic67—22no—
131C2×Dic67—22no—
132C2×Dic67—22no—
133C2×Dic67—22no—
134C2×Dic67—24no—
135C2×Dic67—24no—
136C2×Dic67—22no—
137C2×Dic67C2×M1614yes—
138Dic134—28no—
139Dic134—44no—
140C8×C67—22no—
141C8×C67—22no—
142C8×C67—42no—
143C67⋊C8—42no—
144C67⋊C8—42no—
145C22×C67C2×C4×D411yescharacteristic
146C22×C67#51912yes—
147C22×C67C22×M1611yescharacteristic
148C22×C67C2×#13112yes—
149C22×C67C2×#13111yescharacteristic
150C4×C67—22no—
151C4×C67—24no—
152C4×C67—42no—
153C4×C67#43514yes—
154C4×C67#53611yescharacteristic
155C4×C67#53611yescharacteristic
156C4×C67#54112yes—
157Dic67—24no—
158Dic67—42no—
159Dic67—42no—
160Dic67—42no—
161#22240—671nomaximal, Sylow
162C2×C67#232711yescharacteristic
163C2×C67C2×#43511yescharacteristic
164C2×C67#293111yescharacteristic
165C2×C67#298412yes—
166C2×C67#298612yes—
167C2×#446—671no—
168C2×C8×Q8—671no—
169C4×C4⋊Q8—671no—
170#1984—671no—
171#1984—671no—
172#1917—674no—
173#1916—674no—
174#1831—671no—
175#1744D6711yescharacteristic
176C67#2224011yescharacteristic, complemented, Sylow
177C2×C4⋊Q8—671no—
178C2×C4×Q8—671no—
179C2×C4×Q8—671no—
180#446—1344no—
181C8×Q8—1344no—
182C2×C4⋊C8—672no—
183C2×C4⋊C8—672no—
184C2×C4⋊C8—671no—
185C2×C4⋊C8—671no—
186C2×#97—672no—
187C2×C4×C8—672no—
188C2×C4×C8C2×D6711yescharacteristic
189C2×C4×C8C2×D6711yescharacteristic
190#391—671no—
191#391—671no—
192#389—674no—
193#384—671no—
194C4×C4⋊C4—674no—
195C43C2×D6711yescharacteristic
196#336—674no—
197#336—674no—
198#336—674no—
199#336C2×D6712yes—
200#336C2×D6712yes—
201C22×Q8—671no—
202C22×Q8—671no—
203C22×C8—672no—
204C22×C8—672no—
205C22×C8C22×D6712yes—
206C22×C8C22×D6711yescharacteristic
207C22×C8C22×D6711yescharacteristic
208C4⋊Q8—1344no—
209C4×Q8—1342no—
210C4×Q8—1342no—
211C4×Q8—1342no—
212C4×Q8—1342no—
213C2×C4⋊C4—672no—
214C2×C4⋊C4—672no—
215C2×C4⋊C4—671no—
216C2×C4⋊C4—671no—
217C2×C4⋊C4—674no—
218C2×C4⋊C4—674no—
219C2×C42—672no—
220C2×C42—672no—
221C2×C42C4×D6711yescharacteristic
222C2×C42C4×D6712yes—
223C2×C42C4×D6711yescharacteristic
224C2×C42C22×D6712yes—
225C2×C42C22×D6711yescharacteristic
226C4⋊C8—1344no—
227C4⋊C8—1344no—
228C4⋊C8—1342no—
229C4⋊C8—1342no—
230#97—1344no—
231C4×C8—22no—
232C4×C8—1344no—
233C4×C8D26814yes—
234#95—674no—
235#95—674no—
236C2×Q8—674no—
237C2×Q8—1342no—
238C2×Q8—1342no—
239C2×Q8—1342no—
240C2×Q8—1342no—
241C2×Q8—1342no—
242C22×C4—672no—
243C22×C4—672no—
244C22×C4—672no—
245C22×C4—672no—
246C22×C4C2×C4×D6712yes—
247C22×C4C2×C4×D6711yescharacteristic
248C22×C4C2×C4×D6712yes—
249C22×C4C2×C4×D6711yescharacteristic
250C22×C4C23×D6711yescharacteristic
251C2×C8—22no—
252C2×C8—22no—
253C2×C8—22no—
254C2×C8—22no—
255C2×C8—22no—
256C2×C8—1342no—
257C2×C8—1342no—
258C2×C8—1344no—
259C2×C8—1344no—
260C2×C8C2×D26812yes—
261C2×C8C2×D26812yes—
262C4⋊C4—1348no—
263C4⋊C4—1344no—
264C4⋊C4—1344no—
265C4⋊C4—1342no—
266C4⋊C4—1342no—
267C4⋊C4—1348no—
268C42—22no—
269C42—24no—
270C42—22no—
271C42—22no—
272C42—22no—
273C42—1342no—
274C42—1342no—
275C42—1342no—
276C42—1342no—
277C42C2×D26812yes—
278C42#1176162812yes—
279C23C22×C4×D6711yescharacteristic
280Q8—2684no—
281Q8—1348no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—674no—
289C2×C4—1342no—
290C2×C4—1342no—
291C2×C4—1342no—
292C2×C4—1342no—
293C2×C4—1344no—
294C2×C4—1344no—
295C2×C4—1342no—
296C2×C4—1342no—
297C2×C4—1342no—
298C2×C4C4×D26812yes—
299C2×C4C4×D26812yes—
300C2×C4M16×D6712yes—
301C2×C4#1452404914yes—
302C2×C4#1452405112yes—
303C2×C4C22×D26811yescharacteristic
304C2×C4C2×#1176162811yescharacteristic
305C2×C4#1452409012yes—
306C2×C4#1452409012yes—
307C8—22no—
308C8—22no—
309C8—42no—
310C8—2682no—
311C8—2682no—
312C22C2×C4×D26811yescharacteristic
313C22#1686387312yes—
314C22C2×M16×D6711yescharacteristic
315C22C2×#1452404912yes—
316C22C2×#1452405111yescharacteristic
317C4—24no—
318C4—22no—
319C4—42no—
320C4—2682no—
321C4—2682no—
322C4—2682no—
323C4—1344no—
324C4#1686346914yes—
325C4#1686397211yescharacteristic
326C4#1686397211yescharacteristic
327C4#1686400812yes—
328C2#2256292811yescharacteristic
329C2C2×#1686346911yescharacteristic
330C2#2256892211yescharacteristic
331C2#2256929612yes—
332C2#2256931012yes—
subgroupG/H
center, FrattiniC22×C4C23×D67
commutator, socle, prime radicalC23×C67C23×C4
FittingC67×#1744C2
cosocleC22×C4×C67C24
hypercenter#1744D67
hypocenterC67#22240
LCSG ▷ C23×C67 ▷ C67
UCSC1 ◁ C22×C4 ◁ #1744

Complements

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

NH ≅ G/Ntype
176. C67#22240semidirect

Maximal subgroups

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

subgroupindex
C2×#168635262
C2×C8×Dic1342
C4×#145240432
#225611452
#225607742
#225607672
#225603262
C67×#17442
#2224067