Finite Group Databasebeta

← #30766535

Subgroups — #30766535

Order 17152.

388 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 87 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#22560292C211yesmaximal, characteristic
2#22558543C211yesmaximal, characteristic
3#22558215C214yesmaximal, complemented
4C67×#2621C211yesmaximal, characteristic
5#16863453—21no—
6#16863360—21no—
7#16863165—21no—
8#16863165C2212yes—
9#16863122C2212yes—
10#16863120—21no—
11C2×C4⋊Q8×C67C411yescharacteristic
12C2×C22⋊Q8×C67—21no—
13C2×#117×C67C2211yescharacteristic
14#398×C67—21no—
15#398×C67C2212yes—
16#397×C67—21no—
17#397×C67C412yes—
18#391×C67—21no—
19#382×C67C411yescharacteristic
20C2×#11761612—22no—
21C4⋊(C67⋊C8)—41no—
22C22⋊(C67⋊C8)—22no—
23C22⋊(C67⋊C8)—41no—
24C22⋊(C67⋊C8)—41no—
25#14523954—41no—
26C22×Q8×C67—21no—
27C23×C4×C67D411yescharacteristic
28C4⋊Q8×C67—41no—
29C4⋊Q8×C67—41no—
30C22⋊Q8×C67—44no—
31#117×C67—41no—
32#117×C67—41no—
33C2×C4⋊C4×C67—21no—
34C2×C4⋊C4×C67—21no—
35C2×C4⋊C4×C67—22no—
36C2×C4⋊C4×C67C2×C412yes—
37C2×C4⋊C4×C67D411yescharacteristic
38C2×C4⋊C4×C67C2311yescharacteristic
39C2×C22⋊C4×C67—22no—
40C2×C22⋊C4×C67C2×C411yescharacteristic
41C2×C22⋊C4×C67D411yescharacteristic
42C2×C42×C67C2×C411yescharacteristic
43C2×C42×C67D411yescharacteristic
44#95×C67—21no—
45#95×C67—22no—
46#95×C67—22no—
47#95×C67C2×C412yes—
48#11761612—42no—
49#11761612—42no—
50C2×C67⋊C8—42no—
51C2×C67⋊C8—42no—
52C24×C67C22⋊C411yescharacteristic
53C2×Q8×C67—42no—
54C2×Q8×C67—42no—
55C2×Q8×C67—41no—
56C2×Q8×C67—41no—
57C22×C4×C67—21no—
58C22×C4×C67—21no—
59C22×C4×C67—22no—
60C22×C4×C67—22no—
61C22×C4×C67—22no—
62C22×C4×C67—41no—
63C22×C4×C67C22⋊C411yescharacteristic
64C22×C4×C67C22⋊C411yescharacteristic
65C22×C4×C67C22⋊C411yescharacteristic
66C22×C4×C67C22×C411yescharacteristic
67C22×C4×C67C2×D411yescharacteristic
68C22×C4×C67C2×D411yescharacteristic
69C4⋊C4×C67—41no—
70C4⋊C4×C67—41no—
71C4⋊C4×C67—42no—
72C4⋊C4×C67—42no—
73C4⋊C4×C67—42no—
74C4⋊C4×C67—44no—
75C22⋊C4×C67—41no—
76C22⋊C4×C67—41no—
77C22⋊C4×C67—44no—
78C42×C67—21no—
79C42×C67—21no—
80C42×C67—21no—
81C42×C67—21no—
82C23×C67—21no—
83C23×C67—21no—
84C23×C67—21no—
85C23×C67—41no—
86C23×C67—41no—
87C23×C67C2×C22⋊C411yescharacteristic
88Q8×C67—82no—
89Q8×C67—82no—
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—21no—
98C2×C4×C67—42no—
99C2×C4×C67—42no—
100C2×C4×C67—41no—
101C2×C4×C67—41no—
102C2×C4×C67—41no—
103C2×C4×C67—41no—
104C2×C4×C67—42no—
105C2×C4×C67—42no—
106C2×C4×C67—42no—
107C2×C4×C67—42no—
108C2×C4×C67—42no—
109C2×C4×C67—42no—
110C2×C4×C67#9911yescharacteristic
111C2×C4×C67#9911yescharacteristic
112C2×C4×C67AGL(1,ℤ/8)12yes—
113C67⋊C8—82no—
114C67⋊C8—82no—
115C22×C67—21no—
116C22×C67—21no—
117C22×C67—42no—
118C22×C67—41no—
119C22×C67—41no—
120C22×C67—41no—
121C22×C67—81no—
122C22×C67C2×#9911yescharacteristic
123C22×C67#41811yescharacteristic
124C22×C67#42111yescharacteristic
125C4×C67—22no—
126C4×C67—21no—
127C4×C67—21no—
128C4×C67—41no—
129C4×C67—41no—
130C4×C67—41no—
131C4×C67—41no—
132C4×C67—82no—
133C4×C67—82no—
134#11807—671nomaximal, Sylow
135C2×C67—21no—
136C2×C67—21no—
137C2×C67—81no—
138C2×C67#151011yescharacteristic
139C2×C67#212311yescharacteristic
140C2×C67#213511yescharacteristic
141#2621D6711yescharacteristic
142#1822—671no—
143#1464—671no—
144#1346—674no—
145C67#1180711yescharacteristic, complemented, Sylow
146C2×C4⋊Q8Dic6711yescharacteristic
147C2×C22⋊Q8—21no—
148C2×#117C2×D6711yescharacteristic
149#432—1341no—
150#407—1341no—
151#398—21no—
152#398C2×D6712yes—
153#397—21no—
154#397Dic6712yes—
155#391—21no—
156#382Dic6711yescharacteristic
157#343—672no—
158#343—1341no—
159#324—672no—
160#323—1341no—
161C22×Q8—21no—
162C23×C4C22⋊D6711yescharacteristic
163C2×M16—1342no—
164C4⋊Q8—41no—
165C4⋊Q8—41no—
166C22⋊Q8—44no—
167#117—41no—
168#117—41no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4C2×Dic6712yes—
173C2×C4⋊C4C22⋊D6711yescharacteristic
174C2×C4⋊C4C22×D6711yescharacteristic
175C2×C22⋊C4—22no—
176C2×C22⋊C4C2×Dic6711yescharacteristic
177C2×C22⋊C4C22⋊D6711yescharacteristic
178C2×C42C2×Dic6711yescharacteristic
179C2×C42C22⋊D6711yescharacteristic
180C4⋊C8—2681no—
181C22⋊C8—2681no—
182C22⋊C8—2681no—
183C22⋊C8—1342no—
184#97—2681no—
185#95—21no—
186#95—22no—
187#95—22no—
188#95C2×Dic6712yes—
189C24C22⋊Dic6711yescharacteristic
190C2×Q8—42no—
191C2×Q8—42no—
192C2×Q8—41no—
193C2×Q8—41no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—41no—
200C22×C4C22⋊Dic6711yescharacteristic
201C22×C4C22⋊Dic6711yescharacteristic
202C22×C4C22⋊Dic6711yescharacteristic
203C22×C4C22×Dic6711yescharacteristic
204C22×C4C2×C22⋊D6711yescharacteristic
205C22×C4C2×C22⋊D6711yescharacteristic
206M16—2682no—
207M16—2682no—
208C2×C8—2682no—
209C2×C8—2682no—
210C4⋊C4—41no—
211C4⋊C4—41no—
212C4⋊C4—42no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—44no—
216C22⋊C4—41no—
217C22⋊C4—41no—
218C22⋊C4—44no—
219C42—21no—
220C42—21no—
221C42—21no—
222C42—21no—
223C23—21no—
224C23—21no—
225C23—21no—
226C23—41no—
227C23—41no—
228C23C2×C22⋊Dic6711yescharacteristic
229Q8—82no—
230Q8—82no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4#1452395811yescharacteristic
252C2×C4#1452395811yescharacteristic
253C2×C4#1452406912yes—
254C8—5362no—
255C8—5362no—
256C22—21no—
257C22—21no—
258C22—42no—
259C22—41no—
260C22—41no—
261C22—41no—
262C22—81no—
263C22C2×#1452395811yescharacteristic
264C22#1686340111yescharacteristic
265C22#1686341311yescharacteristic
266C4—22no—
267C4—21no—
268C4—21no—
269C4—41no—
270C4—41no—
271C4—41no—
272C4—41no—
273C4—82no—
274C4—82no—
275C2—21no—
276C2—21no—
277C2—81no—
278C2#2255873811yescharacteristic
279C2#2256186311yescharacteristic
280C2#2256190411yescharacteristic
subgroupG/H
centerC22C2×#14523958
commutatorC22×C4×C67C22×C4
FrattiniC2×C4⋊C4C22×D67
FittingC67×#2621C2
socleC22×C67C2×#99
prime radicalC24×C67C22⋊C4
cosocleC2×C4⋊C4×C67C23
hypercenter#2621D67
hypocenterC67#11807
LCSG ▷ C22×C4×C67 ▷ C23×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2621

Complements

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

NH ≅ G/Ntype
3. #22558215C2semidirect
145. C67#11807semidirect

Maximal subgroups

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

subgroupindex
#225602922
#225585432
#225582152
C67×#26212
#1180767