Finite Group Databasebeta

← #30766569

Subgroups — #30766569

Order 17152.

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

normal subgroups only

This group has 586 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#22558239C214yesmaximal, complemented
4C2×#395×C67C211yesmaximal, characteristic
5#16863360—21no—
6#16863360—21no—
7#16863165—21no—
8#16863165C2212yes—
9#16863122—21no—
10#16863122C2212yes—
11C23×Q8×C67C411yescharacteristic
12C22×C4⋊C4×C67—21no—
13C22×C4⋊C4×C67C2211yescharacteristic
14#395×C67—21no—
15#395×C67—21no—
16#395×C67C412yes—
17#395×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—
26C22×Q8×C67—21no—
27C22×Q8×C67—44no—
28C22×Q8×C67—41no—
29C22×Q8×C67—41no—
30C22×Q8×C67C2×C412yes—
31C23×C4×C67—21no—
32C23×C4×C67—21no—
33C23×C4×C67C2×C411yescharacteristic
34C23×C4×C67D411yescharacteristic
35C23×C4×C67D411yescharacteristic
36C2×C4⋊C4×C67—22no—
37C2×C4⋊C4×C67—22no—
38C2×C4⋊C4×C67—41no—
39C2×C4⋊C4×C67—41no—
40C2×C4⋊C4×C67—44no—
41C2×C4⋊C4×C67C2×C411yescharacteristic
42C2×C4⋊C4×C67D411yescharacteristic
43C2×C4⋊C4×C67D411yescharacteristic
44C2×C4⋊C4×C67C2311yescharacteristic
45#95×C67—22no—
46#95×C67—21no—
47#95×C67—22no—
48#95×C67C2×C412yes—
49#11761612—42no—
50#11761612—42no—
51#11761612—82no—
52C2×C67⋊C8—42no—
53C2×C67⋊C8—42no—
54C24×C67C22⋊C411yescharacteristic
55C2×Q8×C67—44no—
56C2×Q8×C67—44no—
57C2×Q8×C67—42no—
58C2×Q8×C67—42no—
59C2×Q8×C67—48no—
60C2×Q8×C67—44no—
61C2×Q8×C67—44no—
62C22×C4×C67—24no—
63C22×C4×C67—24no—
64C22×C4×C67—21no—
65C22×C4×C67—21no—
66C22×C4×C67—21no—
67C22×C4×C67—21no—
68C22×C4×C67—21no—
69C22×C4×C67—22no—
70C22×C4×C67—22no—
71C22×C4×C67—21no—
72C22×C4×C67—22no—
73C22×C4×C67—41no—
74C22×C4×C67—42no—
75C22×C4×C67—42no—
76C22×C4×C67—42no—
77C22×C4×C67—41no—
78C22×C4×C67—41no—
79C22×C4×C67—42no—
80C22×C4×C67—42no—
81C22×C4×C67C22⋊C411yescharacteristic
82C22×C4×C67C22⋊C411yescharacteristic
83C22×C4×C67C22⋊C411yescharacteristic
84C22×C4×C67C22×C411yescharacteristic
85C22×C4×C67C2×D411yescharacteristic
86C22×C4×C67C2×D411yescharacteristic
87C4⋊C4×C67—41no—
88C4⋊C4×C67—41no—
89C4⋊C4×C67—41no—
90C4⋊C4×C67—41no—
91C4⋊C4×C67—44no—
92C4⋊C4×C67—44no—
93C23×C67—22no—
94C23×C67—22no—
95C23×C67—21no—
96C23×C67—21no—
97C23×C67#9911yescharacteristic
98C23×C67#9911yescharacteristic
99C23×C67C2×C22⋊C411yescharacteristic
100Q8×C67—44no—
101Q8×C67—44no—
102Q8×C67—48no—
103C2×C4×C67—21no—
104C2×C4×C67—21no—
105C2×C4×C67—24no—
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—21no—
114C2×C4×C67—42no—
115C2×C4×C67—42no—
116C2×C4×C67—42no—
117C2×C4×C67—48no—
118C2×C4×C67—42no—
119C2×C4×C67—44no—
120C2×C4×C67—42no—
121C2×C4×C67—44no—
122C2×C4×C67—42no—
123C2×C4×C67—42no—
124C2×C4×C67—42no—
125C2×C4×C67—42no—
126C2×C4×C67—41no—
127C2×C4×C67—41no—
128C2×C4×C67—41no—
129C2×C4×C67—41no—
130C2×C4×C67—81no—
131C2×C4×C67#13712yes—
132C67⋊C8—82no—
133C67⋊C8—82no—
134C22×C67—24no—
135C22×C67—24no—
136C22×C67—21no—
137C22×C67—21no—
138C22×C67—21no—
139C22×C67—21no—
140C22×C67—21no—
141C22×C67—21no—
142C22×C67AΣL(1,𝔽16)12yes—
143C22×C67#35612yes—
144C22×C67C2×#9911yescharacteristic
145C22×C67#41911yescharacteristic
146C22×C67#42111yescharacteristic
147C4×C67—22no—
148C4×C67—44no—
149C4×C67—41no—
150C4×C67—44no—
151C4×C67—41no—
152C4×C67—41no—
153C4×C67—41no—
154C4×C67—41no—
155#11815—671nomaximal, Sylow
156C2×C67—21no—
157C2×C67—22no—
158C2×C67—22no—
159C2×C67—21no—
160C2×C67#151111yescharacteristic
161C2×C67C2×AΣL(1,𝔽16)11yescharacteristic
162C2×C67C2×#35611yescharacteristic
163C2×#395D6711yescharacteristic
164#1821—671no—
165#1463—671no—
166#1354—674no—
167C67#1181511yescharacteristic, complemented, Sylow
168C23×Q8Dic6711yescharacteristic
169C22×C4⋊C4—21no—
170C22×C4⋊C4C2×D6711yescharacteristic
171#407—1341no—
172#407—1341no—
173#395—21no—
174#395—21no—
175#395Dic6712yes—
176#395C2×D6712yes—
177C2×#95—21no—
178C2×#95Dic6711yescharacteristic
179#343—672no—
180#343—1341no—
181#324—672no—
182#324—1341no—
183C22×Q8—21no—
184C22×Q8—44no—
185C22×Q8—41no—
186C22×Q8—41no—
187C22×Q8C2×Dic6712yes—
188C23×C4—21no—
189C23×C4—21no—
190C23×C4C2×Dic6711yescharacteristic
191C23×C4C22⋊D6711yescharacteristic
192C23×C4C22⋊D6711yescharacteristic
193C2×M16—2681no—
194C2×M16—1342no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4C2×Dic6711yescharacteristic
201C2×C4⋊C4C22⋊D6711yescharacteristic
202C2×C4⋊C4C22⋊D6711yescharacteristic
203C2×C4⋊C4C22×D6711yescharacteristic
204C22⋊C8—2681no—
205C22⋊C8—2681no—
206C22⋊C8—2681no—
207C22⋊C8—1342no—
208#95—22no—
209#95—21no—
210#95—22no—
211#95C2×Dic6712yes—
212C24C22⋊Dic6711yescharacteristic
213C2×Q8—44no—
214C2×Q8—44no—
215C2×Q8—42no—
216C2×Q8—42no—
217C2×Q8—48no—
218C2×Q8—44no—
219C2×Q8—44no—
220C22×C4—24no—
221C22×C4—24no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—21no—
230C22×C4—22no—
231C22×C4—41no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4C22⋊Dic6711yescharacteristic
240C22×C4C22⋊Dic6711yescharacteristic
241C22×C4C22⋊Dic6711yescharacteristic
242C22×C4C22×Dic6711yescharacteristic
243C22×C4C2×C22⋊D6711yescharacteristic
244C22×C4C2×C22⋊D6711yescharacteristic
245M16—2682no—
246M16—2682no—
247M16—5362no—
248C2×C8—2682no—
249C2×C8—2682no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—44no—
255C4⋊C4—44no—
256C23—21no—
257C23—21no—
258C23—22no—
259C23—22no—
260C23#1452395811yescharacteristic
261C23#1452395811yescharacteristic
262C23C2×C22⋊Dic6711yescharacteristic
263Q8—44no—
264Q8—44no—
265Q8—48no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—24no—
269C2×C4—24no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—44no—
282C2×C4—42no—
283C2×C4—48no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—44no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—81no—
294C2×C4#1452407412yes—
295C8—5362no—
296C8—5362no—
297C22—24no—
298C22—24no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22#1686318312yes—
306C22#1686319512yes—
307C22C2×#1452395811yescharacteristic
308C22#1686340511yescharacteristic
309C22#1686341311yescharacteristic
310C4—22no—
311C4—44no—
312C4—41no—
313C4—41no—
314C4—44no—
315C4—41no—
316C4—41no—
317C4—41no—
318C2—21no—
319C2—21no—
320C2—22no—
321C2—22no—
322C2#2255874211yescharacteristic
323C2C2×#1686318311yescharacteristic
324C2C2×#1686319511yescharacteristic
subgroupG/H
centerC22C2×#14523958
commutatorC22×C4×C67C22×C4
FrattiniC2×C4⋊C4C22×D67
FittingC2×#395×C67C2
socleC22×C67C2×#99
prime radicalC24×C67C22⋊C4
cosocleC2×C4⋊C4×C67C23
hypercenterC2×#395D67
hypocenterC67#11815
LCSG ▷ C22×C4×C67 ▷ C23×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ C2×#395

Complements

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

NH ≅ G/Ntype
3. #22558239C2semidirect
167. C67#11815semidirect

Maximal subgroups

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

subgroupindex
#225602892
#225585392
#225582392
C2×#395×C672
#1181567