Finite Group Databasebeta

← #30743137

Subgroups — #30743137

Order 17152.

342 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 71 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#22560507C211yesmaximal, characteristic
2#22558550C211yesmaximal, characteristic
3C67×#1799C211yesmaximal, characteristic
4C22×#11761612C411yescharacteristic, complemented
5#16863453—21no—
6C2×C22⋊(C67⋊C8)—21no—
7C2×C22⋊(C67⋊C8)C411yescharacteristic, complemented
8#16863151—21no—
9#16863120—21no—
10#16863120C412yescomplemented
11C2×C4×D4×C67C2211yescharacteristic
12C2×#99×C67C412yes—
13#380×C67—21no—
14#342×C67—21no—
15C2×#11761612—41no—
16C2×#11761612—42no—
17C22×C67⋊C8—21no—
18C22×C67⋊C8—21no—
19C4⋊(C67⋊C8)—41no—
20C22⋊(C67⋊C8)—22no—
21C22⋊(C67⋊C8)—41no—
22C22⋊(C67⋊C8)—42no—
23#14523954—41no—
24C22×D4×C67C2×C411yescharacteristic
25C23×C4×C67C2×C411yescharacteristic
26C23×C4×C67D411yescharacteristic
27C4×D4×C67—41no—
28C4×D4×C67—41no—
29C2×C4⋊C4×C67—41no—
30C2×C4⋊C4×C67Q811yescharacteristic
31C2×C22⋊C4×C67—22no—
32C2×C22⋊C4×C67C2×C411yescharacteristic
33C2×C22⋊C4×C67D411yescharacteristic
34C2×C42×C67D411yescharacteristic
35#99×C67—42no—
36#95×C67—41no—
37#11761612—42no—
38#11761612—42no—
39C2×C67⋊C8—22no—
40C2×C67⋊C8—41no—
41C2×C67⋊C8—41no—
42C2×C67⋊C8—42no—
43C2×C67⋊C8—41no—
44C2×C67⋊C8—41no—
45C2×C67⋊C8—41no—
46C24×C67C4211yescharacteristic
47C24×C67C22⋊C411yescharacteristic
48C2×D4×C67—21no—
49C2×D4×C67—21no—
50C2×D4×C67—41no—
51C2×D4×C67—41no—
52C22×C4×C67—21no—
53C22×C4×C67—21no—
54C22×C4×C67—21no—
55C22×C4×C67—21no—
56C22×C4×C67—41no—
57C22×C4×C67—41no—
58C22×C4×C67—41no—
59C22×C4×C67—41no—
60C22×C4×C67—42no—
61C22×C4×C67C22⋊C411yescharacteristic
62C22×C4×C67C22⋊C411yescharacteristic
63C22×C4×C67C4⋊C411yescharacteristic
64C22×C4×C67C4⋊C411yescharacteristic
65C22×C4×C67C4⋊C411yescharacteristic
66C4⋊C4×C67—41no—
67C4⋊C4×C67—81no—
68C4⋊C4×C67—81no—
69C22⋊C4×C67—41no—
70C22⋊C4×C67—41no—
71C22⋊C4×C67—42no—
72C22⋊C4×C67—42no—
73C42×C67—21no—
74C42×C67—21no—
75C23×C67—21no—
76C23×C67—21no—
77C23×C67—21no—
78C23×C67—21no—
79C23×C67—21no—
80C23×C67—41no—
81C23×C67—41no—
82C23×C67—41no—
83C23×C67—41no—
84C23×C67#9511yescharacteristic
85C23×C67#10011yescharacteristic
86C23×C67#10111yescharacteristic
87D4×C67—42no—
88D4×C67—42no—
89C2×C4×C67—21no—
90C2×C4×C67—21no—
91C2×C4×C67—21no—
92C2×C4×C67—21no—
93C2×C4×C67—21no—
94C2×C4×C67—21no—
95C2×C4×C67—21no—
96C2×C4×C67—21no—
97C2×C4×C67—42no—
98C2×C4×C67—41no—
99C2×C4×C67—41no—
100C2×C4×C67—41no—
101C2×C4×C67—41no—
102C2×C4×C67—41no—
103C2×C4×C67—41no—
104C2×C4×C67—41no—
105C2×C4×C67—81no—
106C2×C4×C67—82no—
107C2×C4×C67—82no—
108C2×C4×C67—82no—
109C2×C4×C67#9911yescharacteristic
110C2×C4×C67#9911yescharacteristic
111C67⋊C8—42no—
112C67⋊C8—81no—
113C67⋊C8—82no—
114C22×C67—21no—
115C22×C67—21no—
116C22×C67—21no—
117C22×C67—21no—
118C22×C67—41no—
119C22×C67—41no—
120C22×C67—41no—
121C22×C67—41no—
122C22×C67—41no—
123C22×C67—41no—
124C22×C67—41no—
125C22×C67—81no—
126C22×C67—81no—
127C22×C67—81no—
128C22×C67#34211yescharacteristic
129C22×C67#34311yescharacteristic
130C22×C67#34411yescharacteristic
131C4×C67—21no—
132C4×C67—41no—
133C4×C67—41no—
134C4×C67—41no—
135C4×C67—41no—
136C4×C67—162no—
137C4×C67AΣL(1,𝔽16)11yescharacteristic
138C4×C67#35211yescharacteristic
139#7320—671nomaximal, Sylow
140C2×C67—21no—
141C2×C67—21no—
142C2×C67—41no—
143C2×C67—41no—
144C2×C67—81no—
145C2×C67#130511yescharacteristic
146C2×C67#139011yescharacteristic
147C2×C67#139111yescharacteristic
148#1870—671no—
149#1799D6711yescharacteristic
150#1466—671no—
151C67#732011yescharacteristic, complemented, Sylow
152C22×M16—671no—
153C2×C4×D4C2×D6711yescharacteristic
154#432—1341no—
155C2×#99Dic6712yes—
156C2×C22⋊C8—671no—
157C2×C22⋊C8—1341no—
158#380—21no—
159#342—21no—
160#336—1341no—
161#323—672no—
162#323—1341no—
163C22×D4C2×Dic6711yescharacteristic
164C23×C4C4×D6711yescharacteristic
165C23×C4C22⋊D6711yescharacteristic
166C2×M16—2682no—
167C2×M16—2681no—
168C22×C8—1341no—
169C22×C8—1341no—
170C4×D4—41no—
171C4×D4—41no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4Dic13411yescharacteristic
174C2×C22⋊C4—22no—
175C2×C22⋊C4C4×D6711yescharacteristic
176C2×C22⋊C4C22⋊D6711yescharacteristic
177C2×C42D26811yescharacteristic
178C4⋊C8—2681no—
179#99—42no—
180C22⋊C8—2682no—
181C22⋊C8—2681no—
182C22⋊C8—1342no—
183#97—2681no—
184#95—41no—
185C24C4×Dic6711yescharacteristic
186C24C22⋊Dic6711yescharacteristic
187C2×D4—21no—
188C2×D4—21no—
189C2×D4—41no—
190C2×D4—41no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—42no—
200C22×C4(C2×D67)⋊C411yescharacteristic
201C22×C4(C2×D67)⋊C411yescharacteristic
202C22×C4Dic67⋊C411yescharacteristic
203C22×C4Dic67⋊C411yescharacteristic
204C22×C4C4⋊Dic6711yescharacteristic
205M16—2682no—
206M16—2682no—
207C2×C8—2681no—
208C2×C8—2681no—
209C2×C8—2682no—
210C2×C8—2681no—
211C2×C8—2681no—
212C2×C8—2681no—
213C2×C8—1342no—
214C4⋊C4—41no—
215C4⋊C4—81no—
216C4⋊C4—81no—
217C22⋊C4—41no—
218C22⋊C4—41no—
219C22⋊C4—42no—
220C22⋊C4—42no—
221C42—21no—
222C42—21no—
223C23—21no—
224C23—21no—
225C23—21no—
226C23—21no—
227C23—21no—
228C23—41no—
229C23—41no—
230C23—41no—
231C23—41no—
232C23C67⋊#9511yescharacteristic
233C23#1452396011yescharacteristic
234C23#1452396211yescharacteristic
235D4—42no—
236D4—42no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—42no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—81no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4—82no—
257C2×C4#1452395711yescharacteristic
258C2×C4#1452395711yescharacteristic
259C8—2682no—
260C8—5361no—
261C8—5362no—
262C22—21no—
263C22—21no—
264C22—21no—
265C22—21no—
266C22—41no—
267C22—41no—
268C22—41no—
269C22—41no—
270C22—41no—
271C22—41no—
272C22—41no—
273C22—81no—
274C22—81no—
275C22—81no—
276C22#1686316211yescharacteristic
277C22#1686316511yescharacteristic
278C22#1686316611yescharacteristic
279C4—21no—
280C4—41no—
281C4—41no—
282C4—41no—
283C4—41no—
284C4—162no—
285C4#1686318211yescharacteristic
286C4#1686318511yescharacteristic
287C2—21no—
288C2—21no—
289C2—41no—
290C2—41no—
291C2—81no—
292C2#2255811511yescharacteristic
293C2#2255832611yescharacteristic
294C2#2255832911yescharacteristic
subgroupG/H
centerC22#16863162
commutatorC24×C67C42
FrattiniC2×C4×D4C2×D67
FittingC67×#1799C2
socleC22×C67#342
prime radicalC22×D4×C67C2×C4
cosocleC2×C4×D4×C67C22
hypercenter#1799D67
hypocenterC67#7320
LCSG ▷ C24×C67 ▷ C23×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1799

Complements

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

NH ≅ G/Ntype
4. C22×#11761612C4semidirect
7. C2×C22⋊(C67⋊C8)C4semidirect
10. #16863120C4semidirect
151. C67#7320semidirect

Maximal subgroups

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

subgroupindex
#225605072
#225585502
C67×#17992
#732067