Finite Group Databasebeta

← #30766426

Subgroups — #30766426

Order 17152.

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

normal subgroups only

This group has 414 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×#14523962C211yesmaximal, characteristic
2#22558553C211yesmaximal, characteristic
3#22558215C214yesmaximal
4C67×#2546C211yesmaximal, characteristic
5#16863453—21no—
6C2×#14523962—21no—
7C4×#11761612—21no—
8#16863165C2212yes—
9#16863122—21no—
10#16863122C2212yes—
11C2×C4⋊Q8×C67—21no—
12C2×C4⋊Q8×C67C411yescharacteristic
13C2×C4×Q8×C67C2211yescharacteristic
14#398×C67C412yes—
15#398×C67C2212yes—
16#391×C67—21no—
17#389×C67—21no—
18#389×C67—21no—
19C4×C4⋊C4×C67C411yescharacteristic
20C2×#11761612—22no—
21C4⋊(C67⋊C8)—41no—
22#14523962—24no—
23C22⋊(C67⋊C8)—22no—
24#14523954—41no—
25#14523954—41no—
26C4×C67⋊C8—41no—
27C22×Q8×C67—21no—
28C22×Q8×C67D411yescharacteristic
29C4⋊Q8×C67—41no—
30C4⋊Q8×C67—41no—
31C4⋊Q8×C67—44no—
32C4×Q8×C67—41no—
33C4×Q8×C67—41no—
34C2×C4⋊C4×C67—21no—
35C2×C4⋊C4×C67—22no—
36C2×C4⋊C4×C67—21no—
37C2×C4⋊C4×C67—22no—
38C2×C4⋊C4×C67C2×C412yes—
39C2×C4⋊C4×C67C2×C411yescharacteristic
40C2×C4⋊C4×C67D411yescharacteristic
41C2×C4⋊C4×C67C2311yescharacteristic
42C2×C42×C67—21no—
43C2×C42×C67C2×C411yescharacteristic
44C2×C42×C67D411yescharacteristic
45C2×C42×C67D411yescharacteristic
46#95×C67—22no—
47#95×C67—22no—
48#95×C67C2×C412yes—
49#11761612—24no—
50#11761612—42no—
51C2×C67⋊C8—42no—
52C2×C67⋊C8—42no—
53C2×Q8×C67—22no—
54C2×Q8×C67—22no—
55C2×Q8×C67—41no—
56C2×Q8×C67—42no—
57C2×Q8×C67—42no—
58C2×Q8×C67—42no—
59C22×C4×C67—22no—
60C22×C4×C67—22no—
61C22×C4×C67C22⋊C411yescharacteristic
62C22×C4×C67C22⋊C411yescharacteristic
63C22×C4×C67C22⋊C411yescharacteristic
64C22×C4×C67C22⋊C411yescharacteristic
65C22×C4×C67C22×C411yescharacteristic
66C22×C4×C67C2×D411yescharacteristic
67C22×C4×C67C2×D411yescharacteristic
68C4⋊C4×C67—24no—
69C4⋊C4×C67—41no—
70C4⋊C4×C67—41no—
71C4⋊C4×C67—41no—
72C4⋊C4×C67—42no—
73C4⋊C4×C67—44no—
74C4⋊C4×C67—41no—
75C4⋊C4×C67—41no—
76C4⋊C4×C67—44no—
77C42×C67—21no—
78C42×C67—21no—
79C42×C67—21no—
80C42×C67—21no—
81C42×C67—21no—
82C42×C67—21no—
83C42×C67—42no—
84C23×C67C2×C22⋊C411yescharacteristic
85Q8×C67—44no—
86Q8×C67—84no—
87C2×C4×C67—21no—
88C2×C4×C67—21no—
89C2×C4×C67—21no—
90C2×C4×C67—21no—
91C2×C4×C67—22no—
92C2×C4×C67—24no—
93C2×C4×C67—21no—
94C2×C4×C67—21no—
95C2×C4×C67—21no—
96C2×C4×C67—21no—
97C2×C4×C67—21no—
98C2×C4×C67—21no—
99C2×C4×C67—42no—
100C2×C4×C67—42no—
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#9911yescharacteristic
108C2×C4×C67#9911yescharacteristic
109C2×C4×C67#10412yes—
110C2×C4×C67AGL(1,ℤ/8)12yes—
111C67⋊C8—44no—
112C67⋊C8—82no—
113C22×C67—21no—
114C22×C67—21no—
115C22×C67C2×#9911yescharacteristic
116C22×C67#41811yescharacteristic
117C22×C67C2×#10411yescharacteristic
118C4×C67—21no—
119C4×C67—22no—
120C4×C67—22no—
121C4×C67—44no—
122C4×C67—41no—
123C4×C67—41no—
124C4×C67—41no—
125C4×C67—41no—
126C4×C67—82no—
127C4×C67#35612yes—
128#11782—671nomaximal, Sylow
129C2×C67—21no—
130C2×C67—21no—
131C2×C67#150711yescharacteristic
132C2×C67#212311yescharacteristic
133C2×C67C2×#35611yescharacteristic
134#2546D6711yescharacteristic
135C4×#101—671no—
136#1467—671no—
137#1346—674no—
138C67#1178211yescharacteristic, complemented, Sylow
139C2×C4⋊Q8—21no—
140C2×C4⋊Q8Dic6711yescharacteristic
141C2×C4×Q8C2×D6711yescharacteristic
142#432—1341no—
143C2×#101—1341no—
144C4×M16—1341no—
145#398Dic6712yes—
146#398C2×D6712yes—
147#391—21no—
148#389—21no—
149#389—21no—
150C4×C4⋊C4Dic6711yescharacteristic
151#343—672no—
152#324—672no—
153#324—1341no—
154C22×Q8—21no—
155C22×Q8C22⋊D6711yescharacteristic
156C2×M16—1342no—
157C4⋊Q8—41no—
158C4⋊Q8—41no—
159C4⋊Q8—44no—
160C4×Q8—41no—
161C4×Q8—41no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4C2×Dic6712yes—
167C2×C4⋊C4C2×Dic6711yescharacteristic
168C2×C4⋊C4C22⋊D6711yescharacteristic
169C2×C4⋊C4C22×D6711yescharacteristic
170C2×C42—21no—
171C2×C42C2×Dic6711yescharacteristic
172C2×C42C22⋊D6711yescharacteristic
173C2×C42C22⋊D6711yescharacteristic
174C4⋊C8—2681no—
175#101—1344no—
176C22⋊C8—1342no—
177#97—2681no—
178#97—2681no—
179C4×C8—2681no—
180#95—22no—
181#95—22no—
182#95C2×Dic6712yes—
183C2×Q8—22no—
184C2×Q8—22no—
185C2×Q8—41no—
186C2×Q8—42no—
187C2×Q8—42no—
188C2×Q8—42no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4C22⋊Dic6711yescharacteristic
192C22×C4C22⋊Dic6711yescharacteristic
193C22×C4C22⋊Dic6711yescharacteristic
194C22×C4C22⋊Dic6711yescharacteristic
195C22×C4C22×Dic6711yescharacteristic
196C22×C4C2×C22⋊D6711yescharacteristic
197C22×C4C2×C22⋊D6711yescharacteristic
198M16—2682no—
199M16—1344no—
200C2×C8—2682no—
201C2×C8—2682no—
202C4⋊C4—24no—
203C4⋊C4—41no—
204C4⋊C4—41no—
205C4⋊C4—41no—
206C4⋊C4—42no—
207C4⋊C4—44no—
208C4⋊C4—41no—
209C4⋊C4—41no—
210C4⋊C4—44no—
211C42—21no—
212C42—21no—
213C42—21no—
214C42—21no—
215C42—21no—
216C42—21no—
217C42—42no—
218C23C2×C22⋊Dic6711yescharacteristic
219Q8—44no—
220Q8—84no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—24no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4#1452395811yescharacteristic
242C2×C4#1452395811yescharacteristic
243C2×C4#1452397012yes—
244C2×C4#1452406912yes—
245C8—2684no—
246C8—5362no—
247C22—21no—
248C22—21no—
249C22C2×#1452395811yescharacteristic
250C22#1686340111yescharacteristic
251C22C2×#1452397011yescharacteristic
252C4—22no—
253C4—22no—
254C4—21no—
255C4—44no—
256C4—41no—
257C4—41no—
258C4—41no—
259C4—41no—
260C4—82no—
261C4#1686319512yes—
262C2—21no—
263C2—21no—
264C2#2255872511yescharacteristic
265C2#2256186311yescharacteristic
266C2C2×#1686319511yescharacteristic
subgroupG/H
centerC22C2×#14523958
commutatorC22×C4×C67C22×C4
FrattiniC2×C4⋊C4C22×D67
FittingC67×#2546C2
socleC22×C67C2×#99
prime radicalC23×C67C2×C22⋊C4
cosocleC2×C4⋊C4×C67C23
hypercenter#2546D67
hypocenterC67#11782
LCSG ▷ C22×C4×C67 ▷ C23×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2546

Complements

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

NH ≅ G/Ntype
138. C67#11782semidirect

Maximal subgroups

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

subgroupindex
C4×#145239622
#225585532
#225582152
C67×#25462
#1178267