Finite Group Databasebeta

← #30749237

Subgroups — #30749237

Order 17152.

482 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
1#22559971C211yesmaximal, characteristic
2#22559971C211yesmaximal, characteristic
3#22558090C212yesmaximal, complemented
4#22558090C212yesmaximal, complemented
5C67×#2283C211yesmaximal, characteristic
6C2×C22⋊(C67⋊C8)—21no—
7C2×C22⋊(C67⋊C8)—21no—
8C2×#14523954C2212yes—
9C2×#14523954C2212yes—
10#16863151—21no—
11#16863151—21no—
12C22×C42×C67C2211yescharacteristic
13#383×C67—21no—
14#382×C67C2211yescharacteristic
15#382×C67C2211yescharacteristic
16#381×C67—21no—
17#380×C67C411yescharacteristic
18#380×C67C411yescharacteristic
19C4×C4⋊C4×C67C411yescharacteristic
20C4×C4⋊C4×C67C411yescharacteristic
21C4×C22⋊C4×C67—21no—
22#376×C67—21no—
23C22×C67⋊C8—22no—
24C22×C67⋊C8—22no—
25C22⋊(C67⋊C8)—24no—
26C22⋊(C67⋊C8)—24no—
27#14523954—24no—
28#14523954—24no—
29C23×C4×C67C2×C411yescharacteristic
30C23×C4×C67D411yescharacteristic
31C23×C4×C67D411yescharacteristic
32C2×C4⋊C4×C67C2×C411yescharacteristic
33C2×C4⋊C4×C67C2×C411yescharacteristic
34C2×C4⋊C4×C67C2×C411yescharacteristic
35C2×C4⋊C4×C67C2×C411yescharacteristic
36C2×C22⋊C4×C67—21no—
37C2×C22⋊C4×C67—21no—
38C2×C42×C67—24no—
39C2×C42×C67—21no—
40C2×C42×C67—21no—
41C2×C42×C67C2×C411yescharacteristic
42C2×C42×C67D411yescharacteristic
43C2×C42×C67D411yescharacteristic
44C2×C42×C67C2311yescharacteristic
45#95×C67—21no—
46#95×C67—21no—
47#95×C67—21no—
48#95×C67—21no—
49#95×C67—21no—
50#95×C67—21no—
51#95×C67—21no—
52#95×C67—21no—
53C2×C67⋊C8—24no—
54C2×C67⋊C8—24no—
55C2×C67⋊C8—42no—
56C2×C67⋊C8—42no—
57C2×C67⋊C8—42no—
58C2×C67⋊C8—42no—
59C24×C67C22⋊C411yescharacteristic
60C22×C4×C67—22no—
61C22×C4×C67—22no—
62C22×C4×C67—22no—
63C22×C4×C67—22no—
64C22×C4×C67—22no—
65C22×C4×C67—21no—
66C22×C4×C67—21no—
67C22×C4×C67—21no—
68C22×C4×C67—21no—
69C22×C4×C67—41no—
70C22×C4×C67—41no—
71C22×C4×C67—41no—
72C22×C4×C67—41no—
73C22×C4×C67C22⋊C411yescharacteristic
74C22×C4×C67C22⋊C411yescharacteristic
75C22×C4×C67C22⋊C411yescharacteristic
76C22×C4×C67C22×C411yescharacteristic
77C22×C4×C67C2×D411yescharacteristic
78C22×C4×C67C2×D411yescharacteristic
79C4⋊C4×C67—41no—
80C4⋊C4×C67—41no—
81C4⋊C4×C67—41no—
82C4⋊C4×C67—41no—
83C22⋊C4×C67—42no—
84C22⋊C4×C67—42no—
85C42×C67—22no—
86C42×C67—22no—
87C42×C67—22no—
88C42×C67—42no—
89C42×C67—42no—
90C42×C67#4114yes—
91C23×C67—21no—
92C23×C67—21no—
93C23×C67—21no—
94C23×C67—21no—
95C23×C67—21no—
96C23×C67—21no—
97C23×C67#10412yes—
98C23×C67C2×C22⋊C411yescharacteristic
99C2×C4×C67—21no—
100C2×C4×C67—21no—
101C2×C4×C67—21no—
102C2×C4×C67—21no—
103C2×C4×C67—21no—
104C2×C4×C67—21no—
105C2×C4×C67—21no—
106C2×C4×C67—21no—
107C2×C4×C67—21no—
108C2×C4×C67—21no—
109C2×C4×C67—21no—
110C2×C4×C67—21no—
111C2×C4×C67—21no—
112C2×C4×C67—21no—
113C2×C4×C67—44no—
114C2×C4×C67—44no—
115C2×C4×C67—44no—
116C2×C4×C67—41no—
117C2×C4×C67—41no—
118C2×C4×C67—41no—
119C2×C4×C67—41no—
120C2×C4×C67—41no—
121C2×C4×C67—41no—
122C2×C4×C67—41no—
123C2×C4×C67—41no—
124C2×C4×C67—41no—
125C2×C4×C67—41no—
126C2×C4×C67—41no—
127C2×C4×C67—41no—
128C2×C4×C67#10412yes—
129C2×C4×C67#11712yes—
130C2×C4×C67#12312yes—
131C2×C4×C67#12312yes—
132C67⋊C8—44no—
133C67⋊C8—44no—
134C22×C67—22no—
135C22×C67—22no—
136C22×C67—22no—
137C22×C67—41no—
138C22×C67—41no—
139C22×C67—41no—
140C22×C67—41no—
141C22×C67#38111yescharacteristic
142C22×C67#41311yescharacteristic
143C22×C67#41311yescharacteristic
144C22×C67C2×#10411yescharacteristic
145C22×C67C2×#10411yescharacteristic
146C22×C67#42111yescharacteristic
147C22×C67#42111yescharacteristic
148C4×C67—22no—
149C4×C67—22no—
150C4×C67—22no—
151C4×C67—22no—
152C4×C67—41no—
153C4×C67—41no—
154C4×C67—81no—
155C4×C67—81no—
156C4×C67—81no—
157C4×C67—81no—
158#8720—671nomaximal, Sylow
159C2×C67—42no—
160C2×C67#152411yescharacteristic
161C2×C67#152911yescharacteristic
162C2×C67#153211yescharacteristic
163C2×C67#153511yescharacteristic
164C2×C67#182211yescharacteristic
165C2×C67#182611yescharacteristic
166C2×C67#182611yescharacteristic
167#2283D6711yescharacteristic
168#1749—671no—
169#1749—671no—
170#1295—672no—
171#1295—672no—
172C67#872011yescharacteristic, complemented, Sylow
173C22×C42C2×D6711yescharacteristic
174C2×C22⋊C8—1341no—
175C2×C22⋊C8—1341no—
176C2×#97—672no—
177C2×#97—672no—
178#383—21no—
179#382C2×D6711yescharacteristic
180#382C2×D6711yescharacteristic
181#381—21no—
182#380Dic6711yescharacteristic
183#380Dic6711yescharacteristic
184C4×C4⋊C4Dic6711yescharacteristic
185C4×C4⋊C4Dic6711yescharacteristic
186C4×C22⋊C4—21no—
187#376—21no—
188#336—1341no—
189#336—1341no—
190C23×C4C2×Dic6711yescharacteristic
191C23×C4C22⋊D6711yescharacteristic
192C23×C4C22⋊D6711yescharacteristic
193C22×C8—1342no—
194C22×C8—1342no—
195C2×C4⋊C4C2×Dic6711yescharacteristic
196C2×C4⋊C4C2×Dic6711yescharacteristic
197C2×C4⋊C4C2×Dic6711yescharacteristic
198C2×C4⋊C4C2×Dic6711yescharacteristic
199C2×C22⋊C4—21no—
200C2×C22⋊C4—21no—
201C2×C42—24no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42C2×Dic6711yescharacteristic
205C2×C42C22⋊D6711yescharacteristic
206C2×C42C22⋊D6711yescharacteristic
207C2×C42C22×D6711yescharacteristic
208C22⋊C8—1344no—
209C22⋊C8—1344no—
210#97—1344no—
211#97—1344no—
212#95—21no—
213#95—21no—
214#95—21no—
215#95—21no—
216#95—21no—
217#95—21no—
218#95—21no—
219#95—21no—
220C24C22⋊Dic6711yescharacteristic
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4C22⋊Dic6711yescharacteristic
235C22×C4C22⋊Dic6711yescharacteristic
236C22×C4C22⋊Dic6711yescharacteristic
237C22×C4C22×Dic6711yescharacteristic
238C22×C4C2×C22⋊D6711yescharacteristic
239C22×C4C2×C22⋊D6711yescharacteristic
240C2×C8—2682no—
241C2×C8—2682no—
242C2×C8—2682no—
243C2×C8—2682no—
244C2×C8—1344no—
245C2×C8—1344no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C22⋊C4—42no—
251C22⋊C4—42no—
252C42—22no—
253C42—22no—
254C42—22no—
255C42—42no—
256C42—42no—
257C42#1176162814yes—
258C23—21no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23#1452397012yes—
265C23C2×C22⋊Dic6711yescharacteristic
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—44no—
281C2×C4—44no—
282C2×C4—44no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4#1452397012yes—
296C2×C4#1452400212yes—
297C2×C4#1452402612yes—
298C2×C4#1452402612yes—
299C8—2684no—
300C8—2684no—
301C22—22no—
302C22—22no—
303C22—22no—
304C22—41no—
305C22—41no—
306C22—41no—
307C22—41no—
308C22#1686325911yescharacteristic
309C22#1686338011yescharacteristic
310C22#1686338011yescharacteristic
311C22C2×#1452397011yescharacteristic
312C22C2×#1452397011yescharacteristic
313C22#1686341311yescharacteristic
314C22#1686341311yescharacteristic
315C4—22no—
316C4—22no—
317C4—22no—
318C4—22no—
319C4—41no—
320C4—41no—
321C4—81no—
322C4—81no—
323C4—81no—
324C4—81no—
325C2—42no—
326C2#2255879711yescharacteristic
327C2#2255881611yescharacteristic
328C2#2255882911yescharacteristic
329C2#2255884011yescharacteristic
330C2#2256029211yescharacteristic
331C2#2256030811yescharacteristic
332C2#2256030811yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic67
commutatorC22×C4×C67C22×C4
FrattiniC2×C42C22×D67
FittingC67×#2283C2
socleC23×C67C2×C22⋊C4
prime radicalC24×C67C22⋊C4
cosocleC2×C42×C67C23
hypercenter#2283D67
hypocenterC67#8720
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2283

Complements

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

NH ≅ G/Ntype
3. #22558090C2semidirect
4. #22558090C2semidirect
172. C67#8720semidirect

Maximal subgroups

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

subgroupindex
#225599712
#225580902
C67×#22832
#872067