Finite Group Databasebeta

← #31008874

Subgroups — #31008874

Order 17152.

658 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16863814C211yesmaximal, characteristic
2#22568806C211yesmaximal, characteristic
3#22568805C211yesmaximal, characteristic
4#22561997C211yesmaximal, characteristic, complemented
5#22561996C212yesmaximal, complemented
6#22561994C212yesmaximal, complemented
7#22561992C212yesmaximal, complemented
8#22561991C211yesmaximal, characteristic, complemented
9C2×#16863221C211yesmaximal, characteristic, complemented
10#366×D67C212yesmaximal, complemented
11C67×#2159C211yesmaximal, characteristic, complemented
12C2×#14524049C2211yescharacteristic
13#16863877—21no—
14#16863876—21no—
15#16863814C418yescomplemented
16C2×#14523982C2212yes—
17C22×C67⋊C16C2211yescharacteristic
18C2×C16×D67C2212yes—
19#16863439—21no—
20#16863437—22no—
21#16863436—21no—
22#16863435—22no—
23#16863434C2211yescharacteristic
24#16863433C2212yes—
25#16863433C2212yes—
26C2×#14523977C2211yescharacteristic
27#107×D67C2212yes—
28#107×D67C2212yes—
29#16863224C2212yes—
30#16863223C2211yescharacteristic
31#16863223C2211yescharacteristic
32#16863222C2212yes—
33#16863221C2211yescharacteristic
34#16863221C2212yes—
35#16863221C2212yes—
36#16863221C2211yescharacteristic
37C2×M32×C67C2211yescharacteristic
38#427×C67C2211yescharacteristic
39C2×#107×C67C2211yescharacteristic
40#367×C67C2212yes—
41#366×C67C2212yes—
42C2×#11761628D411yescharacteristic
43#14524049C2×C414yes—
44#14524049Q814yes—
45C2×#11761612D411yescharacteristic
46C2×#11761611D412yes—
47C2×C8×D67C2312yes—
48C4×Dic134—41no—
49#14524008—41no—
50C4×D268—41no—
51#14524005—41no—
52C4×C22⋊D67—42no—
53(C22⋊D67)⋊C4—42no—
54#14524002—21no—
55#14524001—22no—
56#14524001—22no—
57C2×C4⋊Dic67—21no—
58C4⋊C4×D67—22no—
59C4⋊C4×D67—22no—
60#14523982C2×C418yescomplemented
61C2×C67⋊C16C2×C414yescomplemented
62C2×C67⋊C16C2311yescharacteristic
63C2×C67⋊C16C2311yescharacteristic
64C16×D67C2×C418yescomplemented
65#14523977—21no—
66#14523977C2311yescharacteristic
67#14523977C2312yes—
68#14523977C2311yescharacteristic
69#14523976—22no—
70#14523976—22no—
71#14523975—21no—
72#14523974—22no—
73#14523974—22no—
74C22×C8×C67C2311yescharacteristic
75#117×C67—21no—
76C2×C4⋊C4×C67—21no—
77M32×C67C2×C414yescomplemented
78C2×C16×C67C2312yes—
79#107×C67—21no—
80#107×C67C2311yescharacteristic
81#107×C67C2312yes—
82#107×C67C2311yescharacteristic
83#106×C67—21no—
84#11761628C4⋊C414yes—
85#11761628Q1614yes—
86C2×Dic134D811yescharacteristic
87C2×D268D811yescharacteristic
88C2×C22⋊D67D812yes—
89C22×Dic67—41no—
90C2×C4×D67—42no—
91C2×C4×D67—42no—
92C2×C4×D67C2×D412yes—
93#11761612C4⋊C414yes—
94#11761611C4⋊C418yes—
95C2×C67⋊C8C2×D412yes—
96C8×D67C22×C414yes—
97C8×D67C2×Q814yes—
98C4⋊Dic67—21no—
99C4⋊Dic67—22no—
100C4⋊Dic67—21no—
101C4⋊Dic67—41no—
102Dic67⋊C4—42no—
103Dic67⋊C4—42no—
104C22⋊Dic67—41no—
105(C2×D67)⋊C4—42no—
106(C2×D67)⋊C4—42no—
107C4×Dic67—41no—
108C4×Dic67—42no—
109C4×Dic67—42no—
110C67⋊C16C22×C412yes—
111C67⋊C16C22×C412yes—
112C22×C4×C67—41no—
113C22×C4×C67C2×D411yescharacteristic
114C2×C8×C67C22×C412yes—
115C2×C8×C67C2×D411yescharacteristic
116C2×C8×C67C2×Q812yes—
117C2×C8×C67C2411yescharacteristic
118C4⋊C4×C67—21no—
119C4⋊C4×C67—21no—
120C4⋊C4×C67—22no—
121C4⋊C4×C67—41no—
122C22⋊C4×C67—41no—
123C42×C67—41no—
124C16×C67C22×C414yes—
125C23×C67C2×D811yescharacteristic
126C22×D67C2×D812yes—
127C2×C4×C67—41no—
128C2×C4×C67—41no—
129C2×C4×C67—42no—
130C2×C4×C67—161no—
131C2×C4×C67C2×C4⋊C412yes—
132C2×C4×C67C2×D811yescharacteristic
133C2×C4×C67C2×Q1612yes—
134C2×C4×C67C22×D411yescharacteristic
135C22⋊D67#10718yes—
136C2×Dic67—41no—
137C2×Dic67—42no—
138C2×Dic67—41no—
139C2×Dic67—161no—
140C2×Dic67C2×D812yes—
141D268#10714yes—
142C4×D67—162no—
143C4×D67—162no—
144C4×D67C2×C4⋊C414yes—
145C4×D67C2×Q1614yes—
146Dic134#10714yes—
147C8×C67C2×C4⋊C412yes—
148C8×C67C23×C411yescharacteristic
149C8×C67C22×Q811yescharacteristic
150C67⋊C8C2×C4⋊C414yes—
151C22×C67—21no—
152C22×C67—21no—
153C22×C67C2×#10712yes—
154C22×C67C22×D811yescharacteristic
155C2×D67—22no—
156C2×D67—22no—
157C2×D67C2×#10714yes—
158C4×C67—81no—
159C4×C67—82no—
160C4×C67—81no—
161C4×C67C2×#10712yes—
162C4×C67C22×C4⋊C411yescharacteristic
163C4×C67C22×Q1611yescharacteristic
164Dic67—81no—
165Dic67—82no—
166Dic67—81no—
167Dic67C2×#10714yes—
168#33410—671nomaximal, Sylow
169C2×C67—22no—
170C2×C67C22×#10711yescharacteristic
171C2×C67#341511yescharacteristic
172C2×C67#341711yescharacteristic
173D67—24no—
174C2×#504—671no—
175#2913—671no—
176#2913—671no—
177#2159—671no—
178#2159—672no—
179#2159—672no—
180#2159D6711yescharacteristic, complemented
181#2158—671no—
182#2158—672no—
183C2×#366—671no—
184C2×#366—672no—
185C67#3341011yescharacteristic, complemented, Sylow
186C2×#131—671no—
187#520—1341no—
188#520—1341no—
189#504—678no—
190C2×M32—672no—
191C2×M32C2×D6711yescharacteristic
192C22×C16—671no—
193C22×C16—672no—
194#428—1342no—
195#428—1342no—
196#428—1341no—
197#428—1341no—
198#427—671no—
199#427—672no—
200#427—672no—
201#427C2×D6711yescharacteristic
202C2×#107—671no—
203C2×#107—672no—
204C2×#107—672no—
205C2×#107C2×D6711yescharacteristic
206#367—672no—
207#367—671no—
208#367—671no—
209#367C2×D6712yes—
210#366—672no—
211#366—672no—
212#366—671no—
213#366—672no—
214#366—671no—
215#366C2×D6712yes—
216C2×#41—671no—
217#131—674no—
218#131—674no—
219C2×M16—671no—
220C2×M16—672no—
221C22×C8—672no—
222C22×C8C22×D6711yescharacteristic
223C4×Q8—2681no—
224C4×Q8—2681no—
225C4×D4—2681no—
226C4×D4—2682no—
227C4×D4—2681no—
228C4×D4—2682no—
229#117—21no—
230#117—1342no—
231#117—1342no—
232#117—1341no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—1342no—
235C2×C4⋊C4—1342no—
236C2×C4⋊C4—1341no—
237M32—678no—
238M32C4×D6714yescomplemented
239C2×C16—674no—
240C2×C16—671no—
241C2×C16—671no—
242C2×C16—678no—
243C2×C16C22×D6712yes—
244#107—21no—
245#107—671no—
246#107—672no—
247#107—671no—
248#107—1342no—
249#107—1342no—
250#107—1341no—
251#107C22×D6712yes—
252#107C22×D6711yescharacteristic
253#107C22×D6711yescharacteristic
254#106—21no—
255#106—1342no—
256#106—1342no—
257#106—1341no—
258#41—674no—
259#41—674no—
260C2×Q8—671no—
261C2×D4—672no—
262C2×D4—671no—
263C22×C4—41no—
264C22×C4—2682no—
265C22×C4—2682no—
266C22×C4—2681no—
267C22×C4—672no—
268C22×C4D4×D6711yescharacteristic
269M16—674no—
270M16—678no—
271C2×C8—672no—
272C2×C8—674no—
273C2×C8—674no—
274C2×C8C2×C4×D6712yes—
275C2×C8D4×D6711yescharacteristic
276C2×C8Q8×D6712yes—
277C2×C8C23×D6711yescharacteristic
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—22no—
281C4⋊C4—41no—
282C4⋊C4—2682no—
283C4⋊C4—2682no—
284C4⋊C4—2681no—
285C4⋊C4—1341no—
286C4⋊C4—1342no—
287C4⋊C4—1341no—
288C22⋊C4—41no—
289C22⋊C4—2682no—
290C22⋊C4—2682no—
291C22⋊C4—2681no—
292C42—41no—
293C42—2682no—
294C42—2682no—
295C42—2681no—
296C16—672no—
297C16—672no—
298C16C2×C4×D6714yes—
299C23—672no—
300C23D8×D6711yescharacteristic
301Q8—674no—
302D4—678no—
303D4—674no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—42no—
307C2×C4—2681no—
308C2×C4—2682no—
309C2×C4—2681no—
310C2×C4—161no—
311C2×C4—10722no—
312C2×C4—10722no—
313C2×C4—10721no—
314C2×C4—672no—
315C2×C4—674no—
316C2×C4—674no—
317C2×C4C4⋊C4×D6712yes—
318C2×C4D8×D6711yescharacteristic
319C2×C4Q16×D6712yes—
320C2×C4C2×D4×D6711yescharacteristic
321C8—674no—
322C8C4⋊C4×D6712yes—
323C8C22×C4×D6711yescharacteristic
324C8C2×Q8×D6711yescharacteristic
325C22—21no—
326C22—21no—
327C22—674no—
328C22—1342no—
329C22—1342no—
330C22#107×D6712yes—
331C22C2×D8×D6711yescharacteristic
332C4—82no—
333C4—81no—
334C4—81no—
335C4—5361no—
336C4—5362no—
337C4—5361no—
338C4—674no—
339C4#107×D6712yes—
340C4C2×C4⋊C4×D6711yescharacteristic
341C4C2×Q16×D6711yescharacteristic
342C2—22no—
343C2—1344no—
344C2C2×#107×D6711yescharacteristic
345C2#2257334911yescharacteristic
346C2#2257336111yescharacteristic
subgroupG/H
centerC22C2×D8×D67
commutatorC8×C67C23×C4
FrattiniC2×C8C23×D67
FittingC67×#2159C2
socleC22×C67C22×D8
prime radicalC2×#11761628D4
cosocleC2×C8×C67C24
hypercenter#2159D67
hypocenterC67#33410
LCSG ▷ C8×C67 ▷ C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2159

Complements

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

NH ≅ G/Ntype
4. #22561997C2semidirect
5. #22561996C2semidirect
6. #22561994C2semidirect
7. #22561992C2semidirect
8. #22561991C2semidirect
9. C2×#16863221C2semidirect
10. #366×D67C2semidirect
11. C67×#2159C2semidirect
15. #16863814C4semidirect
60. #14523982C2×C4semidirect
61. C2×C67⋊C16C2×C4semidirect
64. C16×D67C2×C4semidirect
77. M32×C67C2×C4semidirect
180. #2159D67semidirect
185. C67#33410semidirect
238. M32C4×D67semidirect

Maximal subgroups

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

subgroupindex
C2×#168638142
#225688062
#225688052
#225619972
#225619962
#225619942
#225619922
#225619912
C2×#168632212
#366×D672
C67×#21592
#3341067