Finite Group Databasebeta

← #30775437

Subgroups — #30775437

Order 17152.

410 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
1C22×#14523989C211yesmaximal, characteristic
2C2×C4⋊(C67⋊C16)C211yesmaximal, characteristic
3C2×#16863200C211yesmaximal, characteristic
4C2×#16863199C211yesmaximal, characteristic
5#22561070C211yesmaximal, characteristic
6#22558300C211yesmaximal, characteristic
7C67×#1969C211yesmaximal, characteristic
8C22×Dic268C2211yescharacteristic
9C2×#14523989—22no—
10C2×#14523989—22no—
11C2×#14523989—22no—
12C22×C67⋊C16C2211yescharacteristic
13C22×C67⋊C16C2211yescharacteristic
14C2×#14523977C2211yescharacteristic
15C2×#14523968—21no—
16#16863302—21no—
17C4⋊(C67⋊C16)—22no—
18#16863200—22no—
19#16863199—22no—
20C22×Q16×C67C2211yescharacteristic
21C2×#107×C67C2211yescharacteristic
22C2×#103×C67—21no—
23C2×C4×C8×C67C2211yescharacteristic
24#391×C67—21no—
25C22×Dic134—21no—
26C2×Dic268—21no—
27C2×Dic268—21no—
28C2×Dic268—21no—
29C2×Dic268—21no—
30C2×Dic268—21no—
31C2×Dic268D412yes—
32C2×C4⋊Dic67—21no—
33#14523989—28no—
34C2×C67⋊C16—21no—
35C2×C67⋊C16—21no—
36C2×C67⋊C16—21no—
37C2×C67⋊C16—21no—
38C2×C67⋊C16—21no—
39C2×C67⋊C16D412yes—
40#14523977—21no—
41#14523977—21no—
42#14523968—42no—
43C67⋊#95—41no—
44C67⋊#95—41no—
45C22×Q8×C67—21no—
46C2×Q16×C67—21no—
47C2×Q16×C67—21no—
48C2×Q16×C67—21no—
49C2×Q16×C67—21no—
50C2×Q16×C67—21no—
51C2×Q16×C67D412yes—
52C22×C8×C67D411yescharacteristic
53C22×C8×C67C2311yescharacteristic
54C2×C4⋊C4×C67—21no—
55C2×C42×C67D411yescharacteristic
56#107×C67—21no—
57#107×C67—21no—
58#103×C67—42no—
59C4×C8×C67—21no—
60C4×C8×C67—21no—
61#95×C67—41no—
62#95×C67—41no—
63C2×Dic134—22no—
64C2×Dic134—41no—
65C2×Dic134—41no—
66C2×Dic134—41no—
67C2×Dic134—41no—
68C2×Dic134—41no—
69C22×Dic67—41no—
70C22×Dic67—41no—
71Dic268—22no—
72Dic268—22no—
73Dic268—41no—
74Dic268—41no—
75C4⋊Dic67—41no—
76C4⋊Dic67—41no—
77C67⋊C16—22no—
78C67⋊C16—41no—
79C2×Q8×C67—22no—
80C2×Q8×C67—41no—
81C2×Q8×C67—41no—
82C2×Q8×C67—41no—
83C2×Q8×C67—41no—
84C2×Q8×C67—41no—
85C22×C4×C67—41no—
86C22×C4×C67—41no—
87C22×C4×C67D811yescharacteristic
88C22×C4×C67D811yescharacteristic
89C22×C4×C67C2×D411yescharacteristic
90Q16×C67—22no—
91Q16×C67—22no—
92Q16×C67—41no—
93Q16×C67—41no—
94C2×C8×C67—21no—
95C2×C8×C67—21no—
96C2×C8×C67—21no—
97C2×C8×C67C2×D411yescharacteristic
98C2×C8×C67C2×D411yescharacteristic
99C2×C8×C67C2×D411yescharacteristic
100C2×C8×C67#4111yescharacteristic
101C2×C8×C67#4111yescharacteristic
102C2×C8×C67#4111yescharacteristic
103C4⋊C4×C67—41no—
104C4⋊C4×C67—41no—
105C42×C67—21no—
106C42×C67—21no—
107C23×C67C2×D811yescharacteristic
108Q8×C67—42no—
109Q8×C67—42no—
110Q8×C67—81no—
111Q8×C67—81no—
112C2×C4×C67—21no—
113C2×C4×C67—21no—
114C2×C4×C67—21no—
115C2×C4×C67—21no—
116C2×C4×C67—21no—
117C2×C4×C67—42no—
118C2×C4×C67—81no—
119C2×C4×C67—81no—
120C2×C4×C67—81no—
121C2×C4×C67—81no—
122C2×C4×C67—81no—
123C2×C4×C67Q3212yescomplemented
124C2×C4×C67#12111yescharacteristic
125C2×C4×C67#12111yescharacteristic
126C2×C4×C67#12111yescharacteristic
127C2×C4×C67#13511yescharacteristic
128C2×C4×C67AGL(1,ℤ/8)11yescharacteristic
129C2×C4×C67AGL(1,ℤ/8)11yescharacteristic
130C2×Dic67—42no—
131C2×Dic67—81no—
132C2×Dic67—81no—
133C2×Dic67—81no—
134C2×Dic67—81no—
135C2×Dic67—81no—
136Dic134—42no—
137Dic134—42no—
138Dic134—81no—
139Dic134—81no—
140C8×C67—41no—
141C8×C67#12011yescharacteristic
142C8×C67#12311yescharacteristic
143C8×C67#12311yescharacteristic
144C8×C67#12411yescharacteristic
145C22×C67#45911yescharacteristic
146C22×C67#45911yescharacteristic
147C22×C67#46611yescharacteristic
148C22×C67C2×Q3211yescharacteristic
149C22×C67#50811yescharacteristic
150C22×C67#51011yescharacteristic
151C22×C67#51011yescharacteristic
152C4×C67—22no—
153C4×C67—41no—
154C4×C67—82no—
155C4×C67—161no—
156C4×C67#39411yescharacteristic
157C4×C67#46311yescharacteristic
158C4×C67#46311yescharacteristic
159C4×C67#46911yescharacteristic
160Dic67—82no—
161Dic67—161no—
162#13450—671nomaximal, Sylow
163C2×C67#205411yescharacteristic
164C2×C67#220911yescharacteristic
165C2×C67#220911yescharacteristic
166C2×C67#221111yescharacteristic
167C2×C67#221111yescharacteristic
168C2×C67#221611yescharacteristic
169C2×C67#221911yescharacteristic
170C22×Q32—671no—
171C2×C4⋊C16—671no—
172C2×#358—671no—
173C2×#358—671no—
174#1969—671no—
175#1969D6711yescharacteristic
176#1380—671no—
177C67#1345011yescharacteristic, complemented, Sylow
178C22×Q16—671no—
179C22×Q16C2×D6711yescharacteristic
180C2×Q32—1342no—
181C2×Q32—1342no—
182C2×Q32—1342no—
183C22×C16—671no—
184C22×C16—671no—
185C2×#107—671no—
186C2×#107C2×D6711yescharacteristic
187C2×#103—21no—
188C2×#103—1341no—
189C2×C4×C8C2×D6711yescharacteristic
190#391—21no—
191#391—1341no—
192C4⋊C16—1342no—
193#358—1342no—
194#358—1342no—
195C22×Q8—21no—
196C22×Q8—1341no—
197C2×Q16—21no—
198C2×Q16—21no—
199C2×Q16—21no—
200C2×Q16—21no—
201C2×Q16—21no—
202C2×Q16—672no—
203C2×Q16—1341no—
204C2×Q16—1341no—
205C2×Q16—1341no—
206C2×Q16—1341no—
207C2×Q16—1341no—
208C2×Q16D26812yes—
209C22×C8C22⋊D6711yescharacteristic
210C22×C8C22×D6711yescharacteristic
211C2×C4⋊C4—21no—
212C2×C4⋊C4—1341no—
213C2×C42C22⋊D6711yescharacteristic
214Q32—1348no—
215C2×C16—672no—
216C2×C16—1341no—
217C2×C16—1341no—
218C2×C16—1341no—
219C2×C16—1341no—
220C2×C16—1341no—
221#107—21no—
222#107—21no—
223#107—1341no—
224#107—1341no—
225#103—42no—
226#103—2682no—
227C4×C8—21no—
228C4×C8—21no—
229#95—41no—
230#95—41no—
231#95—2681no—
232#95—2681no—
233C2×Q8—22no—
234C2×Q8—41no—
235C2×Q8—41no—
236C2×Q8—41no—
237C2×Q8—41no—
238C2×Q8—41no—
239C2×Q8—2681no—
240C2×Q8—2681no—
241C2×Q8—2681no—
242C2×Q8—2681no—
243C2×Q8—2681no—
244C2×Q8—1342no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—2681no—
248C22×C4—2681no—
249C22×C4#1176161311yescharacteristic
250C22×C4#1176161311yescharacteristic
251C22×C4C2×C22⋊D6711yescharacteristic
252Q16—22no—
253Q16—22no—
254Q16—41no—
255Q16—41no—
256Q16—2681no—
257Q16—2681no—
258Q16—1342no—
259Q16—1342no—
260C2×C8—21no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8D4×D6711yescharacteristic
264C2×C8D4×D6711yescharacteristic
265C2×C8C2×D26811yescharacteristic
266C2×C8#1176162711yescharacteristic
267C2×C8#1176162711yescharacteristic
268C2×C8#1176162811yescharacteristic
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—2681no—
272C4⋊C4—2681no—
273C42—21no—
274C42—21no—
275C16—2681no—
276C16—1342no—
277C23C2×#1176161311yescharacteristic
278Q8—42no—
279Q8—42no—
280Q8—81no—
281Q8—81no—
282Q8—2682no—
283Q8—2682no—
284Q8—5361no—
285Q8—5361no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—42no—
292C2×C4—81no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—2682no—
298C2×C4—5361no—
299C2×C4—5361no—
300C2×C4—5361no—
301C2×C4—5361no—
302C2×C4—5361no—
303C2×C4#1452398912yescomplemented
304C2×C4#1452401411yescharacteristic
305C2×C4#1452401511yescharacteristic
306C2×C4#1452401711yescharacteristic
307C2×C4#1452406311yescharacteristic
308C2×C4#1452406811yescharacteristic
309C2×C4#1452406911yescharacteristic
310C8—41no—
311C8#1452401311yescharacteristic
312C8#1452402511yescharacteristic
313C8#1452402711yescharacteristic
314C8#1452402911yescharacteristic
315C22#1686358811yescharacteristic
316C22#1686359111yescharacteristic
317C22#1686363011yescharacteristic
318C22C2×#1452398911yescharacteristic
319C22#1686382611yescharacteristic
320C22#1686383611yescharacteristic
321C22#1686383711yescharacteristic
322C4—22no—
323C4—41no—
324C4—82no—
325C4—161no—
326C4—5362no—
327C4—10721no—
328C4#1686331211yescharacteristic
329C4#1686360811yescharacteristic
330C4#1686361111yescharacteristic
331C4#1686364711yescharacteristic
332C2#2256151911yescharacteristic
333C2#2256225611yescharacteristic
334C2#2256225911yescharacteristic
335C2#2256226811yescharacteristic
336C2#2256227111yescharacteristic
337C2#2256229811yescharacteristic
338C2#2256231511yescharacteristic
subgroupG/H
centerC23C2×#11761613
commutator, cosocleC22×C8×C67C23
FrattiniC22×C8C22×D67
FittingC67×#1969C2
socle, prime radicalC23×C67C2×D8
hypercenter#1969D67
hypocenterC67#13450
LCSG ▷ C22×C8×C67 ▷ C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ #1969

Complements

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

NH ≅ G/Ntype
123. C2×C4×C67Q32semidirect
177. C67#13450semidirect
303. C2×C4#14523989semidirect

Maximal subgroups

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

subgroupindex
C22×#145239892
C2×C4⋊(C67⋊C16)2
C2×#168632002
C2×#168631992
#225610702
#225583002
C67×#19692
#1345067