Finite Group Databasebeta

← #30874328

Subgroups — #30874328

Order 17152.

862 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
1C2×#16863514C211yesmaximal, characteristic
2#22569083C212yesmaximal
3#22568805C212yesmaximal, complemented
4#22560992C212yesmaximal, complemented
5#22560434C214yesmaximal, complemented
6#22560274C212yesmaximal, complemented
7#22560088C211yesmaximal, characteristic, complemented
8C67×#1845C211yesmaximal, characteristic, complemented
9C2×#14524065C2211yescharacteristic
10C2×#14524049C2212yes—
11#16863877C2214yes—
12#16863514C418yescomplemented
13#16863495—24no—
14#16863490—22no—
15#16863478—22no—
16C2×#14523979C2211yescharacteristic
17C2×#14523979C2211yescharacteristic
18#16863437—24no—
19#16863436C2214yes—
20#16863434C2211yescharacteristic
21#16863433—22no—
22#107×D67—22no—
23C2×#14523971C2212yes—
24#16863395C2214yes—
25#16863351C2212yes—
26#16863165C2214yes—
27#16863156C2212yes—
28C2×#107×C67C2212yes—
29C2×#107×C67C2212yes—
30C2×C4×C8×C67C2211yescharacteristic
31#384×C67C2212yes—
32C2×#11761628C2312yes—
33#14524065C2×C414yes—
34#14524065Q814yes—
35C2×Dic268D411yescharacteristic
36C2×#11761617D412yes—
37C2×D536D411yescharacteristic
38#14524049—24no—
39#14524049C2×C418yescomplemented
40C2×#11761612C2312yes—
41C2×#11761612C2312yes—
42C2×#11761611—22no—
43C2×C8×D67—22no—
44#14524008—24no—
45#14524005—24no—
46C4×C22⋊D67—44no—
47#14524002C2312yes—
48#14524001—24no—
49C4⋊C4×D67—24no—
50#14523979—22no—
51#14523979C2×C414yescomplemented
52#14523977D411yescharacteristic
53#14523977D411yescharacteristic
54#14523976—24no—
55#14523975D412yes—
56#14523974—24no—
57#14523971C2×C418yescomplemented
58#14523966—24no—
59#14523963—24no—
60#14523953—22no—
61C8×Dic67—22no—
62C22×C8×C67C2311yescharacteristic
63C22×C8×C67C2311yescharacteristic
64C2×C4⋊C4×C67—42no—
65C2×C4⋊C4×C67C2314yes—
66C2×C42×C67C2311yescharacteristic
67#107×C67—22no—
68#107×C67—22no—
69#107×C67—22no—
70#107×C67—22no—
71C4×C8×C67C2×C414yescomplemented
72#11761628—22no—
73#11761628—22no—
74#11761628C22×C414yes—
75#11761628C2×Q814yes—
76C2×Dic134C2×D412yes—
77C2×D268C2×D412yes—
78C2×C22⋊D67—22no—
79C2×C4×D67—22no—
80C2×C4×D67—44no—
81Dic268C4⋊C414yescomplemented
82#11761617C4⋊C418yescomplemented
83D536C4⋊C414yescomplemented
84#11761612—22no—
85#11761612—22no—
86#11761612—22no—
87#11761612C22×C414yes—
88#11761611—28no—
89C2×C67⋊C8—22no—
90C2×C67⋊C8—22no—
91C8×D67—28no—
92C4⋊Dic67C2×D412yes—
93C4⋊Dic67C2×D412yes—
94Dic67⋊C4—44no—
95C22⋊Dic67—22no—
96(C2×D67)⋊C4—44no—
97C4×Dic67—22no—
98C4×Dic67—44no—
99C22×C4×C67—21no—
100C22×C4×C67—24no—
101C22×C4×C67C2411yescharacteristic
102C2×C8×C67C22×C412yes—
103C2×C8×C67C22×C412yes—
104C2×C8×C67C2×D411yescharacteristic
105C2×C8×C67C2×D411yescharacteristic
106C2×C8×C67C2×D411yescharacteristic
107C2×C8×C67C2×D411yescharacteristic
108C2×C8×C67C2×Q812yes—
109C2×C8×C67#4112yes—
110C4⋊C4×C67—24no—
111C4⋊C4×C67—24no—
112C4⋊C4×C67—84no—
113C42×C67—21no—
114C42×C67C22×C412yes—
115C23×C67H(5,𝔽2)11yescharacteristic
116C22×D67—22no—
117C2×C4×C67—22no—
118C2×C4×C67—22no—
119C2×C4×C67—22no—
120C2×C4×C67—44no—
121C2×C4×C67—84no—
122C2×C4×C67C23×C411yescharacteristic
123C2×C4×C67C22×D411yescharacteristic
124C2×C4×C67C22×D411yescharacteristic
125C2×C4×C67C22×Q811yescharacteristic
126C2×C4×C67C2×#4111yescharacteristic
127C2×C4×C672−1+411yescharacteristic
128C22⋊D67—24no—
129C22⋊D67—24no—
130C2×Dic67—22no—
131C2×Dic67—22no—
132C2×Dic67—22no—
133D268—22no—
134D268C2×C4⋊C414yes—
135C4×D67—24no—
136C4×D67—24no—
137C4×D67—164no—
138Dic134—22no—
139Dic134C2×C4⋊C414yes—
140C8×C67C2×C4⋊C412yes—
141C8×C67C2×C4⋊C412yes—
142C8×C67C4×D412yes—
143C8×C67C4×D412yes—
144C67⋊C8—24no—
145C67⋊C8—42no—
146C22×C67—21no—
147C22×C67—21no—
148C22×C67#52011yescharacteristic
149C22×C67D4211yescharacteristic
150C22×C67#55411yescharacteristic
151C2×D67—24no—
152C2×D67—42no—
153C2×D67—42no—
154C4×C67—24no—
155C4×C67—84no—
156C4×C67C22×C4⋊C411yescharacteristic
157C4×C67C2×C4×D411yescharacteristic
158C4×C67#54711yescharacteristic
159C4×C67D4×Q811yescharacteristic
160Dic67—24no—
161Dic67—42no—
162Dic67—42no—
163#22692—671nomaximal, Sylow
164C2×C67—21no—
165C2×C67—21no—
166C2×C67D4×C4⋊C411yescharacteristic
167C2×C67#329211yescharacteristic
168C2×C67#329311yescharacteristic
169D67—44no—
170C2×#443—671no—
171#2951—672no—
172#2913—672no—
173#1952—672no—
174#1859—674no—
175#1845D6711yescharacteristic, complemented
176#1816—672no—
177#1772—671no—
178C67#2269211yescharacteristic, complemented, Sylow
179C2×#135—671no—
180C2×#131—672no—
181#520—674no—
182#443—678no—
183#440—1344no—
184C4×Q16—1342no—
185C4×D8—1342no—
186C2×#108—671no—
187C2×#108—671no—
188#428—674no—
189#428—1344no—
190#427—671no—
191#427—1342no—
192C2×#107—1342no—
193C2×#107C2×D6712yes—
194C2×#107C2×D6712yes—
195C2×#104—672no—
196#417—674no—
197#405—672no—
198C2×C4×C8C2×D6711yescharacteristic
199#384C2×D6712yes—
200#343—674no—
201#339—672no—
202C2×#41—672no—
203#135—674no—
204#135—674no—
205C2×Q16—671no—
206C2×SD16—672no—
207C2×D8—671no—
208#131—678no—
209#131—1344no—
210C2×M16—672no—
211C2×M16—672no—
212C2×M16—1342no—
213C22×C8—1342no—
214C22×C8C22×D6711yescharacteristic
215C22×C8C22×D6711yescharacteristic
216C4×Q8—1344no—
217C4×D4—2684no—
218C4×D4—1344no—
219#117—672no—
220#117—1344no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—1344no—
223C2×C4⋊C4C22×D6714yes—
224C2×C42C22×D6711yescharacteristic
225#108—674no—
226#108—1342no—
227#107—22no—
228#107—22no—
229#107—22no—
230#107—22no—
231#107—671no—
232#107—671no—
233#107—1344no—
234#106—672no—
235#106—1344no—
236#104—678no—
237#103—1344no—
238#102—1344no—
239#97—1342no—
240C4×C8—1342no—
241C4×C8C4×D6714yescomplemented
242#41—674no—
243#41—674no—
244#41—1342no—
245#41—1342no—
246C2×Q8—672no—
247C2×D4—672no—
248C2×D4—1342no—
249C22×C4—21no—
250C22×C4—24no—
251C22×C4—2684no—
252C22×C4—1342no—
253C22×C4C23×D6711yescharacteristic
254Q16—674no—
255SD16—678no—
256D8—674no—
257M16—674no—
258M16—1342no—
259M16—1342no—
260M16—1342no—
261M16—1348no—
262C2×C8—1342no—
263C2×C8—1342no—
264C2×C8—1348no—
265C2×C8C2×C4×D6712yes—
266C2×C8C2×C4×D6712yes—
267C2×C8D4×D6711yescharacteristic
268C2×C8D4×D6711yescharacteristic
269C2×C8D4×D6711yescharacteristic
270C2×C8D4×D6711yescharacteristic
271C2×C8Q8×D6712yes—
272C2×C8#1176162912yes—
273C4⋊C4—24no—
274C4⋊C4—24no—
275C4⋊C4—84no—
276C4⋊C4—2684no—
277C4⋊C4—672no—
278C4⋊C4—672no—
279C22⋊C4—2684no—
280C22⋊C4—1342no—
281C42—21no—
282C42—2684no—
283C42—1342no—
284C42C2×C4×D6712yes—
285C23—1342no—
286C23#1452408811yescharacteristic
287Q8—674no—
288Q8—1342no—
289D4—674no—
290D4—1344no—
291D4—1344no—
292D4—1342no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—44no—
297C2×C4—84no—
298C2×C4—10724no—
299C2×C4—1342no—
300C2×C4—1342no—
301C2×C4—1342no—
302C2×C4—1344no—
303C2×C4—1344no—
304C2×C4C22×C4×D6711yescharacteristic
305C2×C4C2×D4×D6711yescharacteristic
306C2×C4C2×D4×D6711yescharacteristic
307C2×C4C2×Q8×D6711yescharacteristic
308C2×C4C2×#1176162911yescharacteristic
309C2×C4#1452409111yescharacteristic
310C8—2682no—
311C8—1344no—
312C8C4⋊C4×D6712yes—
313C8C4⋊C4×D6712yes—
314C8#1452400512yes—
315C8#1452400512yes—
316C22—21no—
317C22—21no—
318C22—2682no—
319C22—2682no—
320C22—1344no—
321C22#1686387711yescharacteristic
322C22#1686403111yescharacteristic
323C22#1686410711yescharacteristic
324C4—24no—
325C4—84no—
326C4—2682no—
327C4—2682no—
328C4—1344no—
329C4C2×C4⋊C4×D6711yescharacteristic
330C4C2×#1452400511yescharacteristic
331C4#1686404911yescharacteristic
332C4#1686406111yescharacteristic
333C2—21no—
334C2—21no—
335C2—2684no—
336C2#2256310911yescharacteristic
337C2#2257200811yescharacteristic
338C2#2257201311yescharacteristic
subgroupG/H
centerC22#16864031
commutatorC2×C4×C67C23×C4
FrattiniC22×C4C23×D67
FittingC67×#1845C2
socleC22×C67D42
prime radicalC2×#14524065C22
cosocleC22×C4×C67C24
hypercenter#1845D67
hypocenterC67#22692
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1845

Complements

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

NH ≅ G/Ntype
3. #22568805C2semidirect
4. #22560992C2semidirect
5. #22560434C2semidirect
6. #22560274C2semidirect
7. #22560088C2semidirect
8. C67×#1845C2semidirect
12. #16863514C4semidirect
39. #14524049C2×C4semidirect
51. #14523979C2×C4semidirect
57. #14523971C2×C4semidirect
71. C4×C8×C67C2×C4semidirect
81. Dic268C4⋊C4semidirect
82. #11761617C4⋊C4semidirect
83. D536C4⋊C4semidirect
175. #1845D67semidirect
178. C67#22692semidirect
241. C4×C8C4×D67semidirect

Maximal subgroups

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

subgroupindex
C2×#168635142
#225690832
#225688052
#225609922
#225604342
#225602742
#225600882
C67×#18452
#2269267