Finite Group Databasebeta

← #30766517

Subgroups — #30766517

Order 17152.

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

normal subgroups only

This group has 478 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#22560289C211yesmaximal, characteristic
2#22558539C211yesmaximal, characteristic
3#22558215C214yesmaximal, complemented
4C2×#398×C67C211yesmaximal, characteristic
5#16863360—21no—
6#16863360—21no—
7#16863165—21no—
8#16863165C2212yes—
9#16863122—21no—
10#16863122C2212yes—
11C22×C4⋊C4×C67—21no—
12C22×C4⋊C4×C67C411yescharacteristic
13C22×C4⋊C4×C67C2211yescharacteristic
14#398×C67—21no—
15#398×C67—21no—
16#398×C67C412yes—
17#398×C67C2212yes—
18C2×#95×C67—21no—
19C2×#95×C67C411yescharacteristic
20C2×#11761612—22no—
21C2×#11761612—41no—
22C22⋊(C67⋊C8)—22no—
23C22⋊(C67⋊C8)—41no—
24C22⋊(C67⋊C8)—41no—
25C22⋊(C67⋊C8)—41no—
26C23×C4×C67—21no—
27C23×C4×C67—21no—
28C23×C4×C67C2×C411yescharacteristic
29C23×C4×C67D411yescharacteristic
30C23×C4×C67D411yescharacteristic
31C2×C4⋊C4×C67—22no—
32C2×C4⋊C4×C67—21no—
33C2×C4⋊C4×C67—22no—
34C2×C4⋊C4×C67—41no—
35C2×C4⋊C4×C67—41no—
36C2×C4⋊C4×C67—41no—
37C2×C4⋊C4×C67—41no—
38C2×C4⋊C4×C67—44no—
39C2×C4⋊C4×C67C2×C412yes—
40C2×C4⋊C4×C67C2×C411yescharacteristic
41C2×C4⋊C4×C67D411yescharacteristic
42C2×C4⋊C4×C67D411yescharacteristic
43C2×C4⋊C4×C67C2311yescharacteristic
44#95×C67—22no—
45#95×C67—21no—
46#95×C67—22no—
47#95×C67C2×C412yes—
48#11761612—42no—
49#11761612—42no—
50#11761612—82no—
51C2×C67⋊C8—42no—
52C2×C67⋊C8—42no—
53C24×C67C22⋊C411yescharacteristic
54C22×C4×C67—24no—
55C22×C4×C67—21no—
56C22×C4×C67—21no—
57C22×C4×C67—21no—
58C22×C4×C67—21no—
59C22×C4×C67—21no—
60C22×C4×C67—22no—
61C22×C4×C67—21no—
62C22×C4×C67—22no—
63C22×C4×C67—22no—
64C22×C4×C67—41no—
65C22×C4×C67—42no—
66C22×C4×C67—42no—
67C22×C4×C67—42no—
68C22×C4×C67—42no—
69C22×C4×C67—42no—
70C22×C4×C67—42no—
71C22×C4×C67—41no—
72C22×C4×C67—41no—
73C22×C4×C67C22⋊C411yescharacteristic
74C22×C4×C67C22⋊C411yescharacteristic
75C22×C4×C67C22⋊C411yescharacteristic
76C22×C4×C67C22×C411yescharacteristic
77C22×C4×C67C2×D411yescharacteristic
78C22×C4×C67C2×D411yescharacteristic
79C4⋊C4×C67—41no—
80C4⋊C4×C67—41no—
81C4⋊C4×C67—41no—
82C4⋊C4×C67—41no—
83C4⋊C4×C67—42no—
84C4⋊C4×C67—42no—
85C4⋊C4×C67—44no—
86C4⋊C4×C67—44no—
87C23×C67—22no—
88C23×C67—22no—
89C23×C67—21no—
90C23×C67—21no—
91C23×C67#9911yescharacteristic
92C23×C67#9911yescharacteristic
93C23×C67C2×C22⋊C411yescharacteristic
94C2×C4×C67—21no—
95C2×C4×C67—21no—
96C2×C4×C67—24no—
97C2×C4×C67—21no—
98C2×C4×C67—21no—
99C2×C4×C67—21no—
100C2×C4×C67—21no—
101C2×C4×C67—21no—
102C2×C4×C67—21no—
103C2×C4×C67—21no—
104C2×C4×C67—42no—
105C2×C4×C67—42no—
106C2×C4×C67—48no—
107C2×C4×C67—42no—
108C2×C4×C67—42no—
109C2×C4×C67—42no—
110C2×C4×C67—42no—
111C2×C4×C67—42no—
112C2×C4×C67—41no—
113C2×C4×C67—41no—
114C2×C4×C67—41no—
115C2×C4×C67—41no—
116C2×C4×C67—42no—
117C2×C4×C67—42no—
118C2×C4×C67—42no—
119C2×C4×C67—81no—
120C2×C4×C67—82no—
121C2×C4×C67—82no—
122C2×C4×C67AGL(1,ℤ/8)12yes—
123C67⋊C8—82no—
124C67⋊C8—82no—
125C22×C67—24no—
126C22×C67—24no—
127C22×C67—21no—
128C22×C67—21no—
129C22×C67—21no—
130C22×C67—21no—
131C22×C67—21no—
132C22×C67—21no—
133C22×C67#35212yes—
134C22×C67#35612yes—
135C22×C67C2×#9911yescharacteristic
136C22×C67#41811yescharacteristic
137C22×C67#42111yescharacteristic
138C4×C67—22no—
139C4×C67—44no—
140C4×C67—41no—
141C4×C67—41no—
142C4×C67—41no—
143C4×C67—41no—
144C4×C67—41no—
145C4×C67—82no—
146#11803—671nomaximal, Sylow
147C2×C67—21no—
148C2×C67—22no—
149C2×C67—22no—
150C2×C67—21no—
151C2×C67#151011yescharacteristic
152C2×C67C2×#35211yescharacteristic
153C2×C67C2×#35611yescharacteristic
154C2×#398D6711yescharacteristic
155#1821—671no—
156#1463—671no—
157#1346—674no—
158C67#1180311yescharacteristic, complemented, Sylow
159C22×C4⋊C4—21no—
160C22×C4⋊C4Dic6711yescharacteristic
161C22×C4⋊C4C2×D6711yescharacteristic
162#407—1341no—
163#407—1341no—
164#398—21no—
165#398—21no—
166#398Dic6712yes—
167#398C2×D6712yes—
168C2×#95—21no—
169C2×#95Dic6711yescharacteristic
170#343—672no—
171#343—1341no—
172#324—672no—
173#324—1341no—
174C23×C4—21no—
175C23×C4—21no—
176C23×C4C2×Dic6711yescharacteristic
177C23×C4C22⋊D6711yescharacteristic
178C23×C4C22⋊D6711yescharacteristic
179C2×M16—2681no—
180C2×M16—1342no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4C2×Dic6712yes—
190C2×C4⋊C4C2×Dic6711yescharacteristic
191C2×C4⋊C4C22⋊D6711yescharacteristic
192C2×C4⋊C4C22⋊D6711yescharacteristic
193C2×C4⋊C4C22×D6711yescharacteristic
194C22⋊C8—2681no—
195C22⋊C8—2681no—
196C22⋊C8—2681no—
197C22⋊C8—1342no—
198#95—22no—
199#95—21no—
200#95—22no—
201#95C2×Dic6712yes—
202C24C22⋊Dic6711yescharacteristic
203C22×C4—24no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—22no—
210C22×C4—21no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—41no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4C22⋊Dic6711yescharacteristic
223C22×C4C22⋊Dic6711yescharacteristic
224C22×C4C22⋊Dic6711yescharacteristic
225C22×C4C22×Dic6711yescharacteristic
226C22×C4C2×C22⋊D6711yescharacteristic
227C22×C4C2×C22⋊D6711yescharacteristic
228M16—2682no—
229M16—2682no—
230M16—5362no—
231C2×C8—2682no—
232C2×C8—2682no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—44no—
240C4⋊C4—44no—
241C23—21no—
242C23—21no—
243C23—22no—
244C23—22no—
245C23#1452395811yescharacteristic
246C23#1452395811yescharacteristic
247C23C2×C22⋊Dic6711yescharacteristic
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—24no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—48no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—81no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4#1452406912yes—
277C8—5362no—
278C8—5362no—
279C22—24no—
280C22—24no—
281C22—21no—
282C22—21no—
283C22—21no—
284C22—21no—
285C22—21no—
286C22—21no—
287C22#1686318612yes—
288C22#1686319512yes—
289C22C2×#1452395811yescharacteristic
290C22#1686340111yescharacteristic
291C22#1686341311yescharacteristic
292C4—22no—
293C4—44no—
294C4—41no—
295C4—41no—
296C4—41no—
297C4—41no—
298C4—41no—
299C4—82no—
300C2—21no—
301C2—21no—
302C2—22no—
303C2—22no—
304C2#2255873811yescharacteristic
305C2C2×#1686318611yescharacteristic
306C2C2×#1686319511yescharacteristic
subgroupG/H
centerC22C2×#14523958
commutatorC22×C4×C67C22×C4
FrattiniC2×C4⋊C4C22×D67
FittingC2×#398×C67C2
socleC22×C67C2×#99
prime radicalC24×C67C22⋊C4
cosocleC2×C4⋊C4×C67C23
hypercenterC2×#398D67
hypocenterC67#11803
LCSG ▷ C22×C4×C67 ▷ C23×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ C2×#398

Complements

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

NH ≅ G/Ntype
3. #22558215C2semidirect
158. C67#11803semidirect

Maximal subgroups

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

subgroupindex
#225602892
#225585392
#225582152
C2×#398×C672
#1180367