Finite Group Databasebeta

← #30823700

Subgroups — #30823700

Order 17152.

874 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22574019C211yesmaximal, characteristic
2#22569294C211yesmaximal, characteristic
3#22558772C212yesmaximal
4#22558759C214yesmaximal
5#22558720C214yesmaximal
6#22558707C212yesmaximal
7C67×#2356C211yesmaximal, characteristic
8Q8×Dic134C414yescomplemented
9#16864125C414yescomplemented
10#16864122—24no—
11#16864027—21no—
12#16864027—22no—
13#16863953C2214yes—
14#16863953C2212yes—
15#16863874—21no—
16#16863527—24no—
17#16863453C2214yes—
18#16863449C2212yes—
19#16863132C2214yes—
20#16863132C2218yes—
21#16863132C2214yes—
22C2×C4×Q8×C67C2211yescharacteristic
23#391×C67—21no—
24#387×C67—22no—
25#383×C67C2212yes—
26#382×C67C2211yescharacteristic
27C4×C4⋊C4×C67—21no—
28C4×C4⋊C4×C67C2211yescharacteristic
29#376×C67C2212yes—
30#14524043C2×C414yes—
31#14524043C2×C412yes—
32#14524043C2×C414yes—
33#14524043C2×C412yes—
34#14524042—28no—
35#14524042—24no—
36#14524035—22no—
37#14524035—24no—
38#14524034—28no—
39#14524034—24no—
40#14524022—24no—
41#14524022—22no—
42#14524022C2314yes—
43Q8×Dic67—24no—
44Q8×Dic67—24no—
45C4×Dic134—22no—
46C4×Dic134—24no—
47C4×Dic134C2×C418yes—
48C4×Dic134C2×C414yes—
49#14524002—24no—
50#14524002—22no—
51C4⋊(C67⋊C8)—22no—
52C4⋊(C67⋊C8)—24no—
53C22⋊(C67⋊C8)C2314yes—
54#14523954—24no—
55C4×C67⋊C8—22no—
56C22×Q8×C67D411yescharacteristic
57C4×Q8×C67C2×C414yes—
58C4×Q8×C67D414yes—
59C2×C4⋊C4×C67—21no—
60C2×C4⋊C4×C67—21no—
61C2×C4⋊C4×C67D411yescharacteristic
62C2×C4⋊C4×C67D412yes—
63C2×C42×C67—22no—
64C2×C42×C67—21no—
65C2×C42×C67C2312yes—
66C2×C42×C67C2311yescharacteristic
67#95×C67—22no—
68#95×C67—22no—
69#95×C67—21no—
70#95×C67—22no—
71#95×C67C2311yescharacteristic
72#95×C67C2312yes—
73#95×C67C2311yescharacteristic
74C2×Dic134—22no—
75C2×Dic134—24no—
76C2×Dic134C22×C414yes—
77C2×C67⋊C8—24no—
78C4⋊Dic67—24no—
79C4⋊Dic67—22no—
80C4⋊Dic67—24no—
81C4⋊Dic67—22no—
82C4⋊Dic67—22no—
83C4⋊Dic67—24no—
84C4⋊Dic67C22×C414yes—
85Dic67⋊C4—24no—
86Dic67⋊C4—28no—
87Dic67⋊C4—24no—
88C22⋊Dic67—24no—
89C4×Dic67—28no—
90C4×Dic67—24no—
91C2×Q8×C67—22no—
92C2×Q8×C67—22no—
93C2×Q8×C67C22⋊C414yes—
94C22×C4×C67—21no—
95C22×C4×C67—22no—
96C22×C4×C67—21no—
97C22×C4×C67C2×D412yes—
98C22×C4×C67C2×D412yes—
99C22×C4×C67C2×D411yescharacteristic
100C22×C4×C67C2×D411yescharacteristic
101C22×C4×C67C2411yescharacteristic
102C4⋊C4×C67—42no—
103C4⋊C4×C67—42no—
104C4⋊C4×C67C22⋊C418yes—
105C4⋊C4×C67C22⋊C414yes—
106C42×C67—44no—
107C42×C67—42no—
108C42×C67C22×C414yes—
109C42×C67C22×C412yes—
110C42×C67C2×D412yes—
111C42×C67C2×D414yes—
112C23×C67C22×D411yescharacteristic
113Q8×C67—28no—
114C2×C4×C67—22no—
115C2×C4×C67—22no—
116C2×C4×C67—22no—
117C2×C4×C67—42no—
118C2×C4×C67—41no—
119C2×C4×C67—41no—
120C2×C4×C67—41no—
121C2×C4×C67—41no—
122C2×C4×C67—42no—
123C2×C4×C67—41no—
124C2×C4×C67—41no—
125C2×C4×C67—42no—
126C2×C4×C67—41no—
127C2×C4×C67—42no—
128C2×C4×C67C2×C22⋊C412yes—
129C2×C4×C67C2×C22⋊C414yes—
130C2×C4×C67C2×C22⋊C414yes—
131C2×C4×C67C2×C22⋊C412yes—
132C2×C4×C67#13712yes—
133C2×C4×C67#13712yes—
134C2×C4×C67C23×C411yescharacteristic
135C2×C4×C67C22×D411yescharacteristic
136C2×Dic67—24no—
137C2×Dic67—24no—
138Dic134—44no—
139Dic134—84no—
140Dic134—82no—
141C67⋊C8—44no—
142C22×C67—21no—
143C22×C67—21no—
144C22×C67C22×C22⋊C411yescharacteristic
145C22×C67C2×#13711yescharacteristic
146C22×C67C2×#13711yescharacteristic
147C4×C67—21no—
148C4×C67—21no—
149C4×C67—24no—
150C4×C67—24no—
151C4×C67—22no—
152C4×C67—42no—
153C4×C67—81no—
154C4×C67—82no—
155C4×C67#41914yes—
156Dic67—44no—
157Dic67—44no—
158#17632—671nomaximal, Sylow
159C2×C67—21no—
160C2×C67—21no—
161C2×C67#288411yescharacteristic
162C2×C67C2×#41911yescharacteristic
163C2×C67#290111yescharacteristic
164#3502—671no—
165#2983—671no—
166#2356D6711yescharacteristic
167#1518—672no—
168#1515—674no—
169#1506—674no—
170#1503—672no—
171C67#1763211yescharacteristic, complemented, Sylow
172Q82—674no—
173#557—674no—
174#557—1344no—
175#544—1341no—
176#544—1342no—
177#533—672no—
178#533—674no—
179#519—1341no—
180C2×C4×Q8C2×D6711yescharacteristic
181#446—1344no—
182#432—674no—
183#431—672no—
184#391—21no—
185#387—22no—
186#383C2×D6712yes—
187#382C2×D6711yescharacteristic
188C4×C4⋊C4—21no—
189C4×C4⋊C4C2×D6711yescharacteristic
190#376C2×D6712yes—
191#328—674no—
192#328—674no—
193#328—678no—
194C22×Q8C22⋊D6711yescharacteristic
195C4⋊Q8—672no—
196C4⋊Q8—674no—
197C4⋊Q8—672no—
198C4⋊Q8—674no—
199C4⋊Q8—1348no—
200C4⋊Q8—1344no—
201#125—1348no—
202#125—1344no—
203#125—1342no—
204#125—1344no—
205C22⋊Q8—674no—
206C22⋊Q8—1344no—
207C22⋊Q8—1342no—
208C4×Q8—674no—
209C4×Q8—678no—
210C4×Q8—1344no—
211C4×Q8—1344no—
212C4×Q8—1342no—
213C4×Q8—1344no—
214C4×Q8C4×D6714yes—
215C4×Q8D26814yes—
216#117—1344no—
217#117—1342no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4C22⋊D6711yescharacteristic
221C2×C4⋊C4C22⋊D6712yes—
222C2×C42—22no—
223C2×C42—21no—
224C2×C42C22×D6712yes—
225C2×C42C22×D6711yescharacteristic
226C4⋊C8—1342no—
227C4⋊C8—1344no—
228C22⋊C8—674no—
229#97—1344no—
230C4×C8—1342no—
231#95—22no—
232#95—22no—
233#95—21no—
234#95—22no—
235#95C22×D6711yescharacteristic
236#95C22×D6712yes—
237#95C22×D6711yescharacteristic
238C2×Q8—22no—
239C2×Q8—22no—
240C2×Q8—674no—
241C2×Q8—1342no—
242C2×Q8—1344no—
243C2×Q8(C2×D67)⋊C414yes—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—21no—
247C22×C4C2×C22⋊D6712yes—
248C22×C4C2×C22⋊D6712yes—
249C22×C4C2×C22⋊D6711yescharacteristic
250C22×C4C2×C22⋊D6711yescharacteristic
251C22×C4C23×D6711yescharacteristic
252C2×C8—1344no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—674no—
256C4⋊C4—1344no—
257C4⋊C4—1342no—
258C4⋊C4—1344no—
259C4⋊C4—1342no—
260C4⋊C4—1344no—
261C4⋊C4—1348no—
262C4⋊C4—1342no—
263C4⋊C4—1344no—
264C4⋊C4—1344no—
265C4⋊C4(C2×D67)⋊C418yes—
266C4⋊C4(C2×D67)⋊C414yes—
267C22⋊C4—1344no—
268C42—44no—
269C42—42no—
270C42—1348no—
271C42—1344no—
272C42C2×C4×D6714yes—
273C42C2×C4×D6712yes—
274C42C2×D26812yes—
275C42C2×D26814yes—
276C23C22×C22⋊D6711yescharacteristic
277Q8—28no—
278Q8—2684no—
279Q8—5364no—
280Q8—5362no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—42no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—42no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—1344no—
296C2×C4—1344no—
297C2×C4C2×(C2×D67)⋊C414yes—
298C2×C4C2×(C2×D67)⋊C412yes—
299C2×C4C2×(C2×D67)⋊C414yes—
300C2×C4C2×(C2×D67)⋊C412yes—
301C2×C4#1452407312yes—
302C2×C4#1452407312yes—
303C2×C4C22×C4×D6711yescharacteristic
304C2×C4C22×D26811yescharacteristic
305C8—2684no—
306C22—21no—
307C22—21no—
308C22C22×(C2×D67)⋊C411yescharacteristic
309C22C2×#1452407311yescharacteristic
310C22C2×#1452407311yescharacteristic
311C4—24no—
312C4—24no—
313C4—22no—
314C4—21no—
315C4—21no—
316C4—42no—
317C4—81no—
318C4—82no—
319C4—2684no—
320C4—2684no—
321C4#1686340414yes—
322C2—21no—
323C2—21no—
324C2#2256867211yescharacteristic
325C2C2×#1686340411yescharacteristic
326C2#2256875711yescharacteristic
subgroupG/H
centerC22C22×(C2×D67)⋊C4
commutatorC2×C4×C67C23×C4
FrattiniC22×C4C23×D67
FittingC67×#2356C2
socleC22×C67C22×C22⋊C4
prime radicalC23×C67C22×D4
cosocleC22×C4×C67C24
hypercenter#2356D67
hypocenterC67#17632
LCSG ▷ C2×C4×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ #2356

Complements

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

NH ≅ G/Ntype
8. Q8×Dic134C4semidirect
9. #16864125C4semidirect
171. C67#17632semidirect

Maximal subgroups

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

subgroupindex
#225740192
#225692942
#225587722
#225587592
#225587202
#225587072
C67×#23562
#1763267