Finite Group Databasebeta

← #30766461

Subgroups — #30766461

Order 17152.

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

normal subgroups only

This group has 444 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
1C4×#14523960C211yesmaximal, characteristic
2#22558547C211yesmaximal, characteristic
3#22558236C214yesmaximal, complemented
4C67×#2487C211yesmaximal, characteristic
5C2×#14523960—21no—
6#16863360—21no—
7C4×#11761612—21no—
8#16863165C2212yes—
9#16863122C2212yes—
10#16863120—21no—
11C2×C4×D4×C67C2211yescharacteristic
12C22×C4⋊C4×C67C411yescharacteristic
13#400×C67C2212yes—
14#399×C67C412yes—
15#389×C67—21no—
16#386×C67—21no—
17#385×C67—21no—
18#380×C67—21no—
19#380×C67C411yescharacteristic
20C2×#11761612—22no—
21C2×#11761612—41no—
22#14523960—24no—
23C22⋊(C67⋊C8)—22no—
24C22⋊(C67⋊C8)—41no—
25#14523954—41no—
26C4×C67⋊C8—41no—
27C22×D4×C67D411yescharacteristic
28C23×C4×C67—21no—
29C23×C4×C67C2×C411yescharacteristic
30C23×C4×C67D411yescharacteristic
31C4×D4×C67—41no—
32C4×D4×C67—41no—
33C2×C4⋊C4×C67—21no—
34C2×C4⋊C4×C67—21no—
35C2×C4⋊C4×C67—22no—
36C2×C4⋊C4×C67—41no—
37C2×C4⋊C4×C67—41no—
38C2×C4⋊C4×C67C2×C412yes—
39C2×C4⋊C4×C67C2311yescharacteristic
40C2×C22⋊C4×C67—21no—
41C2×C22⋊C4×C67—22no—
42C2×C22⋊C4×C67C2×C411yescharacteristic
43C2×C22⋊C4×C67D411yescharacteristic
44C2×C42×C67D411yescharacteristic
45#95×C67—22no—
46#95×C67—22no—
47#95×C67C2×C412yes—
48#11761612—24no—
49#11761612—42no—
50#11761612—82no—
51C2×C67⋊C8—42no—
52C2×C67⋊C8—42no—
53C24×C67C22⋊C411yescharacteristic
54C24×C67C22⋊C411yescharacteristic
55C2×D4×C67—22no—
56C2×D4×C67—22no—
57C2×D4×C67—41no—
58C22×C4×C67—21no—
59C22×C4×C67—21no—
60C22×C4×C67—21no—
61C22×C4×C67—21no—
62C22×C4×C67—22no—
63C22×C4×C67—22no—
64C22×C4×C67—42no—
65C22×C4×C67—42no—
66C22×C4×C67—42no—
67C22×C4×C67—82no—
68C22×C4×C67C22⋊C411yescharacteristic
69C22×C4×C67C22⋊C411yescharacteristic
70C22×C4×C67C22×C411yescharacteristic
71C22×C4×C67C2×D411yescharacteristic
72C22×C4×C67C2×D411yescharacteristic
73C4⋊C4×C67—41no—
74C4⋊C4×C67—42no—
75C4⋊C4×C67—42no—
76C4⋊C4×C67—41no—
77C4⋊C4×C67—41no—
78C4⋊C4×C67—82no—
79C22⋊C4×C67—41no—
80C22⋊C4×C67—41no—
81C22⋊C4×C67—42no—
82C22⋊C4×C67—82no—
83C42×C67—21no—
84C42×C67—21no—
85C23×C67—21no—
86C23×C67—21no—
87C23×C67—21no—
88C23×C67—21no—
89C23×C67—21no—
90C23×C67—41no—
91C23×C67—41no—
92C23×C67—41no—
93C23×C67—41no—
94C23×C67#10412yes—
95C23×C67C2×C22⋊C411yescharacteristic
96D4×C67—44no—
97C2×C4×C67—21no—
98C2×C4×C67—21no—
99C2×C4×C67—22no—
100C2×C4×C67—21no—
101C2×C4×C67—21no—
102C2×C4×C67—21no—
103C2×C4×C67—42no—
104C2×C4×C67—42no—
105C2×C4×C67—42no—
106C2×C4×C67—41no—
107C2×C4×C67—41no—
108C2×C4×C67—41no—
109C2×C4×C67—42no—
110C2×C4×C67—42no—
111C2×C4×C67—42no—
112C2×C4×C67—42no—
113C2×C4×C67—42no—
114C2×C4×C67—42no—
115C2×C4×C67—81no—
116C2×C4×C67—84no—
117C2×C4×C67#9911yescharacteristic
118C2×C4×C67#9911yescharacteristic
119C2×C4×C67#13712yes—
120C67⋊C8—44no—
121C67⋊C8—82no—
122C22×C67—21no—
123C22×C67—21no—
124C22×C67—22no—
125C22×C67—42no—
126C22×C67—42no—
127C22×C67—42no—
128C22×C67—41no—
129C22×C67—41no—
130C22×C67—41no—
131C22×C67—41no—
132C22×C67—41no—
133C22×C67—81no—
134C22×C67C2×#9911yescharacteristic
135C22×C67#41911yescharacteristic
136C22×C67C2×#10411yescharacteristic
137C4×C67—21no—
138C4×C67—22no—
139C4×C67—41no—
140C4×C67—41no—
141C4×C67—41no—
142C4×C67—82no—
143C4×C67—82no—
144C4×C67#35212yes—
145#11790—671nomaximal, Sylow
146C2×C67—21no—
147C2×C67—21no—
148C2×C67—42no—
149C2×C67—81no—
150C2×C67#150811yescharacteristic
151C2×C67C2×#35211yescharacteristic
152C2×C67#213411yescharacteristic
153#2487D6711yescharacteristic
154C4×#100—671no—
155#1465—671no—
156#1353—674no—
157C67#1179011yescharacteristic, complemented, Sylow
158C2×C4×D4C2×D6711yescharacteristic
159C22×C4⋊C4Dic6711yescharacteristic
160C2×#100—1341no—
161#407—1341no—
162C4×M16—1341no—
163#400C2×D6712yes—
164#399Dic6712yes—
165#389—21no—
166#386—21no—
167#385—21no—
168#380—21no—
169#380Dic6711yescharacteristic
170#343—672no—
171#324—672no—
172#323—1341no—
173C22×D4C22⋊D6711yescharacteristic
174C23×C4—21no—
175C23×C4C2×Dic6711yescharacteristic
176C23×C4C22⋊D6711yescharacteristic
177C2×M16—2681no—
178C2×M16—1342no—
179C4×D4—41no—
180C4×D4—41no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4C2×Dic6712yes—
187C2×C4⋊C4C22×D6711yescharacteristic
188C2×C22⋊C4—21no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4C2×Dic6711yescharacteristic
191C2×C22⋊C4C22⋊D6711yescharacteristic
192C2×C42C22⋊D6711yescharacteristic
193#100—1344no—
194C22⋊C8—2681no—
195C22⋊C8—1342no—
196#97—2681no—
197C4×C8—2681no—
198#95—22no—
199#95—22no—
200#95C2×Dic6712yes—
201C24C22⋊Dic6711yescharacteristic
202C24C22⋊Dic6711yescharacteristic
203C2×D4—22no—
204C2×D4—22no—
205C2×D4—41no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—82no—
216C22×C4C22⋊Dic6711yescharacteristic
217C22×C4C22⋊Dic6711yescharacteristic
218C22×C4C22×Dic6711yescharacteristic
219C22×C4C2×C22⋊D6711yescharacteristic
220C22×C4C2×C22⋊D6711yescharacteristic
221M16—2682no—
222M16—5362no—
223M16—1344no—
224C2×C8—2682no—
225C2×C8—2682no—
226C4⋊C4—41no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—41no—
230C4⋊C4—41no—
231C4⋊C4—82no—
232C22⋊C4—41no—
233C22⋊C4—41no—
234C22⋊C4—42no—
235C22⋊C4—82no—
236C42—21no—
237C42—21no—
238C23—21no—
239C23—21no—
240C23—21no—
241C23—21no—
242C23—21no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23#1452397012yes—
248C23C2×C22⋊Dic6711yescharacteristic
249D4—44no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—81no—
269C2×C4—84no—
270C2×C4#1452395811yescharacteristic
271C2×C4#1452395811yescharacteristic
272C2×C4#1452407412yes—
273C8—2684no—
274C8—5362no—
275C22—21no—
276C22—21no—
277C22—22no—
278C22—42no—
279C22—42no—
280C22—42no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—41no—
286C22—81no—
287C22C2×#1452395811yescharacteristic
288C22#1686340511yescharacteristic
289C22C2×#1452397011yescharacteristic
290C4—22no—
291C4—21no—
292C4—41no—
293C4—41no—
294C4—41no—
295C4—82no—
296C4—82no—
297C4#1686318612yes—
298C2—21no—
299C2—21no—
300C2—42no—
301C2—81no—
302C2#2255872911yescharacteristic
303C2C2×#1686318611yescharacteristic
304C2#2256190111yescharacteristic
subgroupG/H
centerC22C2×#14523958
commutatorC22×C4×C67C22×C4
FrattiniC2×C4⋊C4C22×D67
FittingC67×#2487C2
socleC22×C67C2×#99
prime radicalC22×D4×C67D4
cosocleC2×C4⋊C4×C67C23
hypercenter#2487D67
hypocenterC67#11790
LCSG ▷ C22×C4×C67 ▷ C23×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2487

Complements

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

NH ≅ G/Ntype
3. #22558236C2semidirect
157. C67#11790semidirect

Maximal subgroups

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

subgroupindex
C4×#145239602
#225585472
#225582362
C67×#24872
#1179067