Finite Group Databasebeta

← #30760583

Subgroups — #30760583

Order 17152.

350 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22561146C211yesmaximal, characteristic
2#22560336C211yesmaximal, characteristic
3#22560214C211yesmaximal, characteristic
4#22558093C211yesmaximal, characteristic
5#22558092C211yesmaximal, characteristic
6#22558043C211yesmaximal, characteristic
7C67×#1803C211yesmaximal, characteristic
8C2×C4⋊(C67⋊C8)C2211yescharacteristic
9C2×C4⋊(C67⋊C8)C2211yescharacteristic
10C2×#14523967—21no—
11C2×C4×C67⋊C8C2211yescharacteristic
12#16863266C2211yescharacteristic
13#16863158—21no—
14#16863158—21no—
15#16863158—21no—
16#16863151—21no—
17#16863151—21no—
18C2×C4×Q8×C67C2211yescharacteristic
19C2×C4⋊C8×C67C2211yescharacteristic
20C2×#103×C67—21no—
21#384×C67C2211yescharacteristic
22#340×C67—21no—
23C22×C67⋊C8—21no—
24C22×C67⋊C8—21no—
25C22×C67⋊C8D411yescharacteristic
26C22×C67⋊C8D411yescharacteristic
27C2×C4⋊Dic67—21no—
28C2×C4⋊Dic67—21no—
29C2×C4⋊Dic67—21no—
30C4⋊(C67⋊C8)—21no—
31C4⋊(C67⋊C8)—21no—
32C4⋊(C67⋊C8)—21no—
33C4⋊(C67⋊C8)—21no—
34#14523967—42no—
35C4×C67⋊C8—21no—
36C4×C67⋊C8—21no—
37C22×Q8×C67D411yescharacteristic
38C22×C8×C67—21no—
39C4×Q8×C67—21no—
40C4×Q8×C67—21no—
41C4×Q8×C67—21no—
42C4×Q8×C67—21no—
43C2×C4⋊C4×C67—21no—
44C2×C4⋊C4×C67—21no—
45C2×C4⋊C4×C67—21no—
46C2×C4⋊C4×C67—21no—
47C2×C4⋊C4×C67D411yescharacteristic
48C2×C42×C67—21no—
49C2×C42×C67C2311yescharacteristic
50C4⋊C8×C67—21no—
51C4⋊C8×C67—21no—
52#103×C67—42no—
53C22×Dic67—41no—
54C2×C67⋊C8—21no—
55C2×C67⋊C8—21no—
56C2×C67⋊C8—21no—
57C2×C67⋊C8—21no—
58C2×C67⋊C8—21no—
59C2×C67⋊C8—21no—
60C2×C67⋊C8—41no—
61C2×C67⋊C8—41no—
62C2×C67⋊C8—41no—
63C2×C67⋊C8—41no—
64C2×C67⋊C8—41no—
65C2×C67⋊C8—41no—
66C4⋊Dic67—41no—
67C4⋊Dic67—41no—
68C4⋊Dic67—41no—
69C4⋊Dic67—41no—
70C4⋊Dic67—81no—
71C2×Q8×C67—21no—
72C2×Q8×C67—21no—
73C2×Q8×C67—21no—
74C2×Q8×C67—21no—
75C2×Q8×C67—21no—
76C2×Q8×C67Q1612yes—
77C22×C4×C67—21no—
78C22×C4×C67—21no—
79C22×C4×C67—41no—
80C22×C4×C67C2×D411yescharacteristic
81C22×C4×C67C2×D411yescharacteristic
82C22×C4×C67#4111yescharacteristic
83C2×C8×C67—41no—
84C2×C8×C67—41no—
85C2×C8×C67—41no—
86C4⋊C4×C67—21no—
87C4⋊C4×C67—21no—
88C4⋊C4×C67—41no—
89C4⋊C4×C67—41no—
90C4⋊C4×C67—41no—
91C4⋊C4×C67—41no—
92C4⋊C4×C67—41no—
93C4⋊C4×C67—41no—
94C4⋊C4×C67—81no—
95C42×C67—41no—
96C42×C67—41no—
97C42×C67#4111yescharacteristic
98C42×C67#4111yescharacteristic
99C42×C67#4111yescharacteristic
100C42×C67#4111yescharacteristic
101C23×C67#12111yescharacteristic
102Q8×C67—22no—
103Q8×C67—22no—
104Q8×C67—41no—
105Q8×C67—41no—
106C2×C4×C67—21no—
107C2×C4×C67—21no—
108C2×C4×C67—22no—
109C2×C4×C67—41no—
110C2×C4×C67—41no—
111C2×C4×C67—41no—
112C2×C4×C67—41no—
113C2×C4×C67—41no—
114C2×C4×C67—81no—
115C2×C4×C67—81no—
116C2×C4×C67—81no—
117C2×C4×C67#12311yescharacteristic
118C2×C4×C67#12311yescharacteristic
119C2×C4×C67#12311yescharacteristic
120C2×C4×C67#12411yescharacteristic
121C2×C4×C67#12611yescharacteristic
122C2×C4×C67#12611yescharacteristic
123C2×C4×C67C2×Q1611yescharacteristic
124C2×C4×C67#13511yescharacteristic
125C2×C4×C67#13511yescharacteristic
126C2×C4×C67#13511yescharacteristic
127C2×C4×C67AGL(1,ℤ/8)11yescharacteristic
128C2×C4×C67AGL(1,ℤ/8)11yescharacteristic
129C2×C4×C67#13711yescharacteristic
130C2×C4×C67#13711yescharacteristic
131C2×Dic67—81no—
132C2×Dic67—81no—
133C2×Dic67—81no—
134C8×C67—81no—
135C67⋊C8—41no—
136C67⋊C8—41no—
137C67⋊C8—81no—
138C67⋊C8—81no—
139C22×C67#39711yescharacteristic
140C22×C67#46411yescharacteristic
141C22×C67#46411yescharacteristic
142C22×C67#46711yescharacteristic
143C22×C67#46911yescharacteristic
144C22×C67#47111yescharacteristic
145C22×C67#47111yescharacteristic
146C4×C67—21no—
147C4×C67—21no—
148C4×C67—42no—
149C4×C67—81no—
150C4×C67—161no—
151C4×C67#48211yescharacteristic
152C4×C67#48211yescharacteristic
153C4×C67#48411yescharacteristic
154C4×C67#48411yescharacteristic
155C4×C67#48511yescharacteristic
156C4×C67#48511yescharacteristic
157C4×C67#48811yescharacteristic
158C4×C67#49111yescharacteristic
159Dic67—161no—
160#10624—671nomaximal, Sylow
161C2×C67#168411yescharacteristic
162C2×C67#168411yescharacteristic
163C2×C67#170211yescharacteristic
164C2×C67#170211yescharacteristic
165C2×C67#207411yescharacteristic
166C2×C67#207611yescharacteristic
167C2×C67#207911yescharacteristic
168#1984—671no—
169#1834—671no—
170#1803—671no—
171#1803D6711yescharacteristic
172#1296—671no—
173#1296—671no—
174#1277—671no—
175C67#1062411yescharacteristic, complemented, Sylow
176C2×C4×Q8C2×D6711yescharacteristic
177C2×C4⋊C8—671no—
178C2×C4⋊C8—671no—
179C2×C4⋊C8C2×D6711yescharacteristic
180C2×#103—21no—
181C2×#103—1341no—
182C2×C4×C8—671no—
183#384—671no—
184#384C2×D6711yescharacteristic
185#340—21no—
186#340—1341no—
187#340—1341no—
188#340—1341no—
189#336—1341no—
190#336—1341no—
191C22×Q8C22⋊D6711yescharacteristic
192C22×C8—21no—
193C22×C8—671no—
194C22×C8—671no—
195C22×C8—1341no—
196C22×C8—1341no—
197C4×Q8—21no—
198C4×Q8—21no—
199C4×Q8—21no—
200C4×Q8—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—1341no—
206C2×C4⋊C4—1341no—
207C2×C4⋊C4—1341no—
208C2×C4⋊C4C22⋊D6711yescharacteristic
209C2×C42—21no—
210C2×C42C22×D6711yescharacteristic
211C4⋊C8—21no—
212C4⋊C8—21no—
213C4⋊C8—1341no—
214C4⋊C8—1341no—
215C4⋊C8—1341no—
216C4⋊C8—1341no—
217#103—42no—
218#103—2682no—
219C4×C8—1341no—
220C4×C8—1341no—
221C2×Q8—21no—
222C2×Q8—21no—
223C2×Q8—21no—
224C2×Q8—21no—
225C2×Q8—21no—
226C2×Q8#1176161812yes—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—41no—
230C22×C4—2681no—
231C22×C4D4×D6711yescharacteristic
232C22×C4C2×C22⋊D6711yescharacteristic
233C22×C4#1176162711yescharacteristic
234C2×C8—41no—
235C2×C8—41no—
236C2×C8—41no—
237C2×C8—2681no—
238C2×C8—2681no—
239C2×C8—2681no—
240C2×C8—2681no—
241C2×C8—2681no—
242C2×C8—2681no—
243C2×C8—1341no—
244C2×C8—1341no—
245C2×C8—1341no—
246C2×C8—1341no—
247C2×C8—1341no—
248C2×C8—1341no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—81no—
258C4⋊C4—2681no—
259C4⋊C4—2681no—
260C4⋊C4—2681no—
261C4⋊C4—2681no—
262C4⋊C4—5361no—
263C42—41no—
264C42—41no—
265C42#1176162711yescharacteristic
266C42#1176162711yescharacteristic
267C42#1176162811yescharacteristic
268C42#1176162811yescharacteristic
269C23#1452401511yescharacteristic
270Q8—22no—
271Q8—22no—
272Q8—41no—
273Q8—41no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—22no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—5361no—
286C2×C4—5361no—
287C2×C4—5361no—
288C2×C4#1452402411yescharacteristic
289C2×C4#1452402511yescharacteristic
290C2×C4#1452402611yescharacteristic
291C2×C4#1452402911yescharacteristic
292C2×C4#1452403611yescharacteristic
293C2×C4#1452403611yescharacteristic
294C2×C4C2×#1176161811yescharacteristic
295C2×C4#1452406311yescharacteristic
296C2×C4#1452406411yescharacteristic
297C2×C4#1452406611yescharacteristic
298C2×C4#1452406811yescharacteristic
299C2×C4#1452406911yescharacteristic
300C2×C4#1452407211yescharacteristic
301C2×C4#1452407511yescharacteristic
302C8—81no—
303C8—2681no—
304C8—2681no—
305C8—5361no—
306C8—5361no—
307C22#1686332411yescharacteristic
308C22#1686361611yescharacteristic
309C22#1686362011yescharacteristic
310C22#1686363511yescharacteristic
311C22#1686364711yescharacteristic
312C22#1686365711yescharacteristic
313C22#1686366111yescharacteristic
314C4—21no—
315C4—21no—
316C4—42no—
317C4—81no—
318C4—161no—
319C4—10721no—
320C4#1686371711yescharacteristic
321C4#1686372011yescharacteristic
322C4#1686372911yescharacteristic
323C4#1686373111yescharacteristic
324C4#1686373511yescharacteristic
325C4#1686373611yescharacteristic
326C4#1686375211yescharacteristic
327C4#1686376311yescharacteristic
328C2#2255966511yescharacteristic
329C2#2255966911yescharacteristic
330C2#2255977511yescharacteristic
331C2#2255977911yescharacteristic
332C2#2256163011yescharacteristic
333C2#2256164011yescharacteristic
334C2#2256165711yescharacteristic
subgroupG/H
centerC23#14524015
commutator, cosocleC2×C42×C67C23
FrattiniC2×C42C22×D67
FittingC67×#1803C2
socle, prime radicalC23×C67#121
hypercenter#1803D67
hypocenterC67#10624
LCSG ▷ C2×C42×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
175. C67#10624semidirect

Maximal subgroups

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

subgroupindex
#225611462
#225603362
#225602142
#225580932
#225580922
#225580432
C67×#18032
#1062467