Finite Group Databasebeta

← #30765504

Subgroups — #30765504

Order 17152.

554 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 95 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22558550C211yesmaximal, characteristic
2#22558527C211yesmaximal, characteristic
3#22558157C214yesmaximal, complemented
4C67×#2409C211yesmaximal, characteristic
5#16863453—21no—
6C2×C22⋊(C67⋊C8)—21no—
7#16863122—21no—
8#16863120—21no—
9#16863120C2212yes—
10#16863120C2212yes—
11C23×Q8×C67C411yescharacteristic
12C2×C22⋊Q8×C67—21no—
13C2×C4×D4×C67C2211yescharacteristic
14#395×C67—21no—
15#393×C67—21no—
16#391×C67—21no—
17#386×C67C412yes—
18#385×C67C2212yes—
19#380×C67C411yescharacteristic
20C22×C67⋊C8—41no—
21C4⋊(C67⋊C8)—41no—
22C22⋊(C67⋊C8)—22no—
23C22⋊(C67⋊C8)—22no—
24C22⋊(C67⋊C8)—41no—
25#14523954—41no—
26C22×Q8×C67—21no—
27C22×Q8×C67—21no—
28C22×Q8×C67—41no—
29C22×Q8×C67—41no—
30C22×Q8×C67—44no—
31C22×D4×C67C2×C411yescharacteristic
32C23×C4×C67C2×C411yescharacteristic
33C23×C4×C67C2×C412yes—
34C23×C4×C67D411yescharacteristic
35C22⋊Q8×C67—44no—
36C4×D4×C67—41no—
37C4×D4×C67—41no—
38C2×C4⋊C4×C67—21no—
39C2×C4⋊C4×C67—22no—
40C2×C4⋊C4×C67D411yescharacteristic
41C2×C22⋊C4×C67—22no—
42C2×C22⋊C4×C67C2×C412yes—
43C2×C22⋊C4×C67D411yescharacteristic
44C2×C22⋊C4×C67C2311yescharacteristic
45C2×C42×C67D411yescharacteristic
46#95×C67—21no—
47#95×C67—22no—
48#95×C67—22no—
49C2×C67⋊C8—42no—
50C2×C67⋊C8—42no—
51C2×C67⋊C8—82no—
52C24×C67C22⋊C411yescharacteristic
53C24×C67C22×C411yescharacteristic
54C2×Q8×C67—24no—
55C2×Q8×C67—42no—
56C2×Q8×C67—44no—
57C2×Q8×C67—42no—
58C2×Q8×C67—44no—
59C2×Q8×C67—48no—
60C2×Q8×C67—44no—
61C2×Q8×C67—41no—
62C2×Q8×C67—41no—
63C2×D4×C67—22no—
64C2×D4×C67—22no—
65C2×D4×C67—41no—
66C22×C4×C67—22no—
67C22×C4×C67—21no—
68C22×C4×C67—21no—
69C22×C4×C67—21no—
70C22×C4×C67—21no—
71C22×C4×C67—22no—
72C22×C4×C67—42no—
73C22×C4×C67—44no—
74C22×C4×C67—42no—
75C22×C4×C67—42no—
76C22×C4×C67C22⋊C411yescharacteristic
77C22×C4×C67C22⋊C411yescharacteristic
78C22×C4×C67C22⋊C411yescharacteristic
79C22×C4×C67C2×D411yescharacteristic
80C22×C4×C67C2×D411yescharacteristic
81C4⋊C4×C67—41no—
82C4⋊C4×C67—42no—
83C4⋊C4×C67—44no—
84C22⋊C4×C67—24no—
85C22⋊C4×C67—41no—
86C22⋊C4×C67—41no—
87C22⋊C4×C67—44no—
88C42×C67—21no—
89C42×C67—21no—
90C23×C67—21no—
91C23×C67—21no—
92C23×C67—21no—
93C23×C67—21no—
94C23×C67—41no—
95C23×C67—41no—
96C23×C67—41no—
97C23×C67—41no—
98C23×C67#10112yes—
99C23×C67C2×C22⋊C411yescharacteristic
100C23×C67#13112yes—
101Q8×C67—44no—
102Q8×C67—48no—
103Q8×C67—44no—
104D4×C67—44no—
105C2×C4×C67—22no—
106C2×C4×C67—24no—
107C2×C4×C67—21no—
108C2×C4×C67—21no—
109C2×C4×C67—21no—
110C2×C4×C67—21no—
111C2×C4×C67—21no—
112C2×C4×C67—21no—
113C2×C4×C67—42no—
114C2×C4×C67—42no—
115C2×C4×C67—44no—
116C2×C4×C67—42no—
117C2×C4×C67—42no—
118C2×C4×C67—42no—
119C2×C4×C67—41no—
120C2×C4×C67—41no—
121C2×C4×C67—41no—
122C2×C4×C67—44no—
123C2×C4×C67—42no—
124C2×C4×C67—42no—
125C2×C4×C67—42no—
126C2×C4×C67—81no—
127C2×C4×C67#9911yescharacteristic
128C2×C4×C67#9911yescharacteristic
129C67⋊C8—82no—
130C67⋊C8—82no—
131C22×C67—21no—
132C22×C67—21no—
133C22×C67—22no—
134C22×C67—22no—
135C22×C67—42no—
136C22×C67—42no—
137C22×C67—42no—
138C22×C67—41no—
139C22×C67—41no—
140C22×C67—41no—
141C22×C67—41no—
142C22×C67—81no—
143C22×C67#40811yescharacteristic
144C22×C67C2×#9911yescharacteristic
145C22×C67C2×#10111yescharacteristic
146C4×C67—21no—
147C4×C67—41no—
148C4×C67—44no—
149C4×C67—41no—
150C4×C67—41no—
151C4×C67—41no—
152C4×C67—82no—
153C4×C67AΣL(1,𝔽16)12yes—
154#11575—671nomaximal, Sylow
155C2×C67—21no—
156C2×C67—21no—
157C2×C67—42no—
158C2×C67—42no—
159C2×C67#147211yescharacteristic
160C2×C67C2×AΣL(1,𝔽16)11yescharacteristic
161C2×C67#212311yescharacteristic
162#2409D6711yescharacteristic
163#1466—671no—
164#1459—671no—
165#1326—674no—
166C67#1157511yescharacteristic, complemented, Sylow
167C23×Q8Dic6711yescharacteristic
168C2×C22⋊Q8—21no—
169C2×C4×D4C2×D6711yescharacteristic
170#432—1341no—
171C2×C22⋊C8—1341no—
172#395—21no—
173#393—21no—
174#391—21no—
175#386Dic6712yes—
176#385C2×D6712yes—
177#380Dic6711yescharacteristic
178#324—1341no—
179#323—672no—
180#323—672no—
181#323—1341no—
182C22×Q8—21no—
183C22×Q8—21no—
184C22×Q8—41no—
185C22×Q8—41no—
186C22×Q8—44no—
187C22×D4C2×Dic6711yescharacteristic
188C23×C4C2×Dic6711yescharacteristic
189C23×C4C2×Dic6712yes—
190C23×C4C22⋊D6711yescharacteristic
191C22×C8—2681no—
192C22⋊Q8—44no—
193C4×D4—41no—
194C4×D4—41no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4C22⋊D6711yescharacteristic
198C2×C22⋊C4—22no—
199C2×C22⋊C4C2×Dic6712yes—
200C2×C22⋊C4C22⋊D6711yescharacteristic
201C2×C22⋊C4C22×D6711yescharacteristic
202C2×C42C22⋊D6711yescharacteristic
203C4⋊C8—2681no—
204C22⋊C8—2681no—
205C22⋊C8—1342no—
206C22⋊C8—1342no—
207#97—2681no—
208#95—21no—
209#95—22no—
210#95—22no—
211C24C22⋊Dic6711yescharacteristic
212C24C22×Dic6711yescharacteristic
213C2×Q8—24no—
214C2×Q8—42no—
215C2×Q8—44no—
216C2×Q8—42no—
217C2×Q8—44no—
218C2×Q8—48no—
219C2×Q8—44no—
220C2×Q8—41no—
221C2×Q8—41no—
222C2×D4—22no—
223C2×D4—22no—
224C2×D4—41no—
225C22×C4—22no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—22no—
231C22×C4—42no—
232C22×C4—44no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4C22⋊Dic6711yescharacteristic
236C22×C4C22⋊Dic6711yescharacteristic
237C22×C4C22⋊Dic6711yescharacteristic
238C22×C4C2×C22⋊D6711yescharacteristic
239C22×C4C2×C22⋊D6711yescharacteristic
240C2×C8—2682no—
241C2×C8—2682no—
242C2×C8—5362no—
243C4⋊C4—41no—
244C4⋊C4—42no—
245C4⋊C4—44no—
246C22⋊C4—24no—
247C22⋊C4—41no—
248C22⋊C4—41no—
249C22⋊C4—44no—
250C42—21no—
251C42—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23#1452396212yes—
261C23C2×C22⋊Dic6711yescharacteristic
262C23#1452405012yes—
263Q8—44no—
264Q8—48no—
265Q8—44no—
266D4—44no—
267C2×C4—22no—
268C2×C4—24no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—44no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—44no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—81no—
289C2×C4#1452395811yescharacteristic
290C2×C4#1452395811yescharacteristic
291C8—5362no—
292C8—5362no—
293C22—21no—
294C22—21no—
295C22—22no—
296C22—22no—
297C22—42no—
298C22—42no—
299C22—42no—
300C22—41no—
301C22—41no—
302C22—41no—
303C22—41no—
304C22—81no—
305C22#1686336311yescharacteristic
306C22C2×#1452395811yescharacteristic
307C22C2×#1452396211yescharacteristic
308C4—21no—
309C4—41no—
310C4—44no—
311C4—41no—
312C4—41no—
313C4—41no—
314C4—82no—
315C4#1686318312yes—
316C2—21no—
317C2—21no—
318C2—42no—
319C2—42no—
320C2#2255857311yescharacteristic
321C2C2×#1686318311yescharacteristic
322C2#2256186311yescharacteristic
subgroupG/H
centerC22C2×#14523958
commutatorC24×C67C22×C4
FrattiniC2×C22⋊C4C22×D67
FittingC67×#2409C2
socleC22×C67C2×#99
prime radicalC22×D4×C67C2×C4
cosocleC2×C22⋊C4×C67C23
hypercenter#2409D67
hypocenterC67#11575
LCSG ▷ C24×C67 ▷ C22×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2409

Complements

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

NH ≅ G/Ntype
3. #22558157C2semidirect
166. C67#11575semidirect

Maximal subgroups

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

subgroupindex
#225585502
#225585272
#225581572
C67×#24092
#1157567