Finite Group Databasebeta

← #30744245

Subgroups — #30744245

Order 17152.

438 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
1#22560352C211yesmaximal, characteristic, complemented
2C4×C4⋊(C67⋊C8)C211yesmaximal, characteristic, complemented
3#22558032C212yesmaximal, complemented
4#22558030C212yesmaximal
5C67×#2860C211yesmaximal, characteristic
6C2×C4⋊(C67⋊C8)—21no—
7C2×C4×C67⋊C8—21no—
8#16863151—21no—
9#16863151—21no—
10#16863151C2212yes—
11#16863151C2212yes—
12C2×#126×C67—21no—
13#390×C67C411yescharacteristic
14#390×C67C411yescharacteristic
15#390×C67C2211yescharacteristic
16#389×C67C411yescharacteristic
17#389×C67C411yescharacteristic
18#389×C67C2211yescharacteristic
19#388×C67—21no—
20#388×C67—21no—
21#388×C67—21no—
22C43×C67C2211yescharacteristic
23C22×C67⋊C8—22no—
24C22×C67⋊C8—22no—
25C4⋊(C67⋊C8)—24no—
26C4×C67⋊C8—42no—
27C22×D4×C67C2×C411yescharacteristic
28#126×C67—42no—
29#126×C67—42no—
30C2×C4⋊C4×C67—21no—
31C2×C4⋊C4×C67—21no—
32C2×C4⋊C4×C67—21no—
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×C22⋊C4×C67—21no—
39C2×C22⋊C4×C67—21no—
40C2×C22⋊C4×C67—21no—
41C2×C22⋊C4×C67—21no—
42C2×C42×C67C2×C411yescharacteristic
43C2×C42×C67C2×C411yescharacteristic
44C2×C42×C67D411yescharacteristic
45C2×C42×C67D411yescharacteristic
46C2×C42×C67D411yescharacteristic
47C2×C42×C67D411yescharacteristic
48C2×C42×C67C2311yescharacteristic
49#95×C67—21no—
50#95×C67—21no—
51#95×C67—21no—
52C2×C67⋊C8—24no—
53C2×C67⋊C8—42no—
54C2×C67⋊C8—42no—
55C2×C67⋊C8—42no—
56C2×C67⋊C8—42no—
57C2×C67⋊C8—42no—
58C24×C67—21no—
59C2×D4×C67—22no—
60C2×D4×C67—41no—
61C2×D4×C67—41no—
62C22×C4×C67—21no—
63C22×C4×C67—21no—
64C22×C4×C67—21no—
65C22×C4×C67C22⋊C411yescharacteristic
66C22×C4×C67C22⋊C411yescharacteristic
67C22×C4×C67C22⋊C411yescharacteristic
68C22×C4×C67C22⋊C411yescharacteristic
69C22×C4×C67C22×C411yescharacteristic
70C22×C4×C67C2×D411yescharacteristic
71C22×C4×C67C2×D411yescharacteristic
72C4⋊C4×C67—22no—
73C4⋊C4×C67—41no—
74C4⋊C4×C67—41no—
75C4⋊C4×C67—42no—
76C4⋊C4×C67—42no—
77C4⋊C4×C67—41no—
78C4⋊C4×C67—41no—
79C22⋊C4×C67—42no—
80C22⋊C4×C67—42no—
81C22⋊C4×C67—42no—
82C22⋊C4×C67—42no—
83C22⋊C4×C67—41no—
84C22⋊C4×C67—41no—
85C22⋊C4×C67—42no—
86C42×C67—22no—
87C42×C67—22no—
88C42×C67—22no—
89C42×C67—41no—
90C42×C67—41no—
91C42×C67—41no—
92C42×C67D812yes—
93C42×C67SD1612yes—
94C23×C67—41no—
95C23×C67—41no—
96C23×C67—41no—
97C23×C67—41no—
98C23×C67—41no—
99C23×C67—41no—
100C23×C67—41no—
101C23×C67C2×C22⋊C411yescharacteristic
102D4×C67—82no—
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—21no—
114C2×C4×C67—21no—
115C2×C4×C67—21no—
116C2×C4×C67—21no—
117C2×C4×C67—21no—
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#10212yes—
128C2×C4×C67#10212yes—
129C2×C4×C67#10412yes—
130C2×C4×C67#13112yes—
131C2×C4×C67C2×D811yescharacteristic
132C2×C4×C67C2×SD1611yescharacteristic
133C2×C4×C67#13512yes—
134C67⋊C8—44no—
135C67⋊C8—82no—
136C22×C67—81no—
137C22×C67—81no—
138C22×C67—81no—
139C22×C67—81no—
140C22×C67—81no—
141C22×C67—81no—
142C22×C67—81no—
143C22×C67#40811yescharacteristic
144C22×C67#41011yescharacteristic
145C22×C67#41311yescharacteristic
146C22×C67C2×#10211yescharacteristic
147C22×C67#41611yescharacteristic
148C22×C67C2×#10411yescharacteristic
149C22×C67#42111yescharacteristic
150C4×C67—22no—
151C4×C67—22no—
152C4×C67—22no—
153C4×C67—22no—
154C4×C67—21no—
155C4×C67—21no—
156C4×C67—22no—
157C4×C67—41no—
158C4×C67—81no—
159C4×C67—81no—
160C4×C67—81no—
161#7672—671nomaximal, Sylow
162C2×C67—161no—
163C2×C67#147311yescharacteristic
164C2×C67#148011yescharacteristic
165C2×C67#149411yescharacteristic
166C2×C67#150211yescharacteristic
167C2×C67#152111yescharacteristic
168C2×C67#152911yescharacteristic
169C2×C67#154011yescharacteristic
170#2860D6711yescharacteristic
171#1839—671no—
172C4×C4⋊C8—671no—
173#1271—672no—
174#1270—672no—
175C67#767211yescharacteristic, complemented, Sylow
176C2×#126—21no—
177C2×C4⋊C8—1341no—
178C2×C4×C8—1341no—
179#390Dic6711yescharacteristic
180#390Dic6711yescharacteristic
181#390C2×D6711yescharacteristic
182#389Dic6711yescharacteristic
183#389Dic6711yescharacteristic
184#389C2×D6711yescharacteristic
185#388—21no—
186#388—21no—
187#388—21no—
188C43C2×D6711yescharacteristic
189#336—672no—
190#336—672no—
191#336—1341no—
192#336—1341no—
193C22×D4C2×Dic6711yescharacteristic
194C22×C8—1342no—
195C22×C8—1342no—
196#126—42no—
197#126—42no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4C2×Dic6711yescharacteristic
202C2×C4⋊C4C2×Dic6711yescharacteristic
203C2×C4⋊C4C2×Dic6711yescharacteristic
204C2×C22⋊C4—21no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C42C2×Dic6711yescharacteristic
211C2×C42C2×Dic6711yescharacteristic
212C2×C42C22⋊D6711yescharacteristic
213C2×C42C22⋊D6711yescharacteristic
214C2×C42C22⋊D6711yescharacteristic
215C2×C42C22⋊D6711yescharacteristic
216C2×C42C22×D6711yescharacteristic
217C4⋊C8—1344no—
218C4×C8—2682no—
219#95—21no—
220#95—21no—
221#95—21no—
222C24—21no—
223C2×D4—22no—
224C2×D4—41no—
225C2×D4—41no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4C22⋊Dic6711yescharacteristic
230C22×C4C22⋊Dic6711yescharacteristic
231C22×C4C22⋊Dic6711yescharacteristic
232C22×C4C22⋊Dic6711yescharacteristic
233C22×C4C22×Dic6711yescharacteristic
234C22×C4C2×C22⋊D6711yescharacteristic
235C22×C4C2×C22⋊D6711yescharacteristic
236C2×C8—2682no—
237C2×C8—2682no—
238C2×C8—2682no—
239C2×C8—2682no—
240C2×C8—2682no—
241C2×C8—1344no—
242C4⋊C4—22no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C22⋊C4—42no—
250C22⋊C4—42no—
251C22⋊C4—42no—
252C22⋊C4—42no—
253C22⋊C4—41no—
254C22⋊C4—41no—
255C22⋊C4—42no—
256C42—22no—
257C42—22no—
258C42—22no—
259C42—41no—
260C42—41no—
261C42—41no—
262C42#1176161312yes—
263C42#1176161612yes—
264C23—41no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23C2×C22⋊Dic6711yescharacteristic
272D4—82no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4#1452396412yes—
298C2×C4#1452396412yes—
299C2×C4#1452397012yes—
300C2×C4#1452405012yes—
301C2×C4C2×#1176161311yescharacteristic
302C2×C4C2×#1176161611yescharacteristic
303C2×C4#1452406312yes—
304C8—2684no—
305C8—5362no—
306C22—81no—
307C22—81no—
308C22—81no—
309C22—81no—
310C22—81no—
311C22—81no—
312C22—81no—
313C22#1686336311yescharacteristic
314C22#1686337011yescharacteristic
315C22#1686338011yescharacteristic
316C22C2×#1452396411yescharacteristic
317C22#1686339211yescharacteristic
318C22C2×#1452397011yescharacteristic
319C22#1686341311yescharacteristic
320C4—22no—
321C4—22no—
322C4—22no—
323C4—22no—
324C4—22no—
325C4—21no—
326C4—21no—
327C4—41no—
328C4—81no—
329C4—81no—
330C4—81no—
331C2—161no—
332C2#2255857811yescharacteristic
333C2#2255860811yescharacteristic
334C2#2255866911yescharacteristic
335C2#2255870411yescharacteristic
336C2#2255878611yescharacteristic
337C2#2255881611yescharacteristic
338C2#2255885811yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic67
commutatorC22×C4×C67C22×C4
FrattiniC2×C42C22×D67
FittingC67×#2860C2
socleC23×C67C2×C22⋊C4
prime radicalC22×D4×C67C2×C4
cosocleC2×C42×C67C23
hypercenter#2860D67
hypocenterC67#7672
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C23 ◁ C43 ◁ #2860

Complements

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

NH ≅ G/Ntype
1. #22560352C2semidirect
2. C4×C4⋊(C67⋊C8)C2semidirect
3. #22558032C2semidirect
175. C67#7672semidirect

Maximal subgroups

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

subgroupindex
#225603522
C4×C4⋊(C67⋊C8)2
#225580322
#225580302
C67×#28602
#767267