Finite Group Databasebeta

← #30766994

Subgroups — #30766994

Order 17152.

372 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 107 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#22560205C211yesmaximal, characteristic
2#22558533C211yesmaximal, characteristic
3#22558236C214yesmaximal
4C67×#2568C211yesmaximal, characteristic
5#16863449—21no—
6#16863420—21no—
7C2×#14523962—21no—
8#16863165C2212yes—
9#16863122—21no—
10#16863122C2212yes—
11C2×C4⋊Q8×C67C411yescharacteristic
12C2×C4×Q8×C67C2211yescharacteristic
13#400×C67C412yes—
14#400×C67C2212yes—
15#391×C67—21no—
16#391×C67—21no—
17#391×C67—21no—
18#382×C67—21no—
19C4×C4⋊C4×C67C411yescharacteristic
20C2×#11761612—22no—
21C4⋊(C67⋊C8)—41no—
22C4⋊(C67⋊C8)—41no—
23C4⋊(C67⋊C8)—41no—
24#14523962—42no—
25C22⋊(C67⋊C8)—22no—
26C4×C67⋊C8—41no—
27C22×Q8×C67—21no—
28C22×Q8×C67D411yescharacteristic
29C4⋊Q8×C67—41no—
30C4⋊Q8×C67—41no—
31C4×Q8×C67—41no—
32C4×Q8×C67—41no—
33C2×C4⋊C4×C67—21no—
34C2×C4⋊C4×C67—21no—
35C2×C4⋊C4×C67C2×C412yes—
36C2×C4⋊C4×C67C2×C411yescharacteristic
37C2×C4⋊C4×C67D411yescharacteristic
38C2×C4⋊C4×C67C2311yescharacteristic
39C2×C42×C67—21no—
40C2×C42×C67C2×C411yescharacteristic
41C2×C42×C67D411yescharacteristic
42C2×C42×C67D411yescharacteristic
43#95×C67—22no—
44#95×C67—22no—
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—
52C2×Q8×C67—41no—
53C2×Q8×C67—41no—
54C2×Q8×C67—44no—
55C2×Q8×C67—41no—
56C2×Q8×C67—41no—
57C2×Q8×C67SD1612yes—
58C2×Q8×C67Q1612yes—
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—41no—
69C4⋊C4×C67—41no—
70C4⋊C4×C67—41no—
71C4⋊C4×C67—42no—
72C4⋊C4×C67—42no—
73C4⋊C4×C67—42no—
74C42×C67—21no—
75C42×C67—21no—
76C42×C67—21no—
77C42×C67—21no—
78C42×C67—21no—
79C42×C67—21no—
80C42×C67—42no—
81C23×C67C2×C22⋊C411yescharacteristic
82Q8×C67—42no—
83Q8×C67—42no—
84Q8×C67—82no—
85Q8×C67—82no—
86C2×C4×C67—21no—
87C2×C4×C67—21no—
88C2×C4×C67—21no—
89C2×C4×C67—21no—
90C2×C4×C67—21no—
91C2×C4×C67—21no—
92C2×C4×C67—22no—
93C2×C4×C67—21no—
94C2×C4×C67—21no—
95C2×C4×C67—21no—
96C2×C4×C67—41no—
97C2×C4×C67—41no—
98C2×C4×C67—41no—
99C2×C4×C67—42no—
100C2×C4×C67—42no—
101C2×C4×C67—44no—
102C2×C4×C67—42no—
103C2×C4×C67—42no—
104C2×C4×C67#9911yescharacteristic
105C2×C4×C67#9911yescharacteristic
106C2×C4×C67#10312yes—
107C2×C4×C67#10312yes—
108C2×C4×C67C2×SD1611yescharacteristic
109C2×C4×C67C2×Q1611yescharacteristic
110C67⋊C8—82no—
111C67⋊C8—82no—
112C22×C67—21no—
113C22×C67—21no—
114C22×C67C2×#9911yescharacteristic
115C22×C67C2×#10311yescharacteristic
116C22×C67#42111yescharacteristic
117C4×C67—21no—
118C4×C67—21no—
119C4×C67—22no—
120C4×C67—22no—
121C4×C67—21no—
122C4×C67—41no—
123C4×C67—41no—
124C4×C67—41no—
125C4×C67—82no—
126C4×C67—82no—
127C4×C67#35512yes—
128#11907—671nomaximal, Sylow
129C2×C67—21no—
130C2×C67—21no—
131C2×C67#152011yescharacteristic
132C2×C67#212311yescharacteristic
133C2×C67C2×#35511yescharacteristic
134#2568D6711yescharacteristic
135#1801—671no—
136#1461—671no—
137#1353—674no—
138C67#1190711yescharacteristic, complemented, Sylow
139C2×C4⋊Q8Dic6711yescharacteristic
140C2×C4×Q8C2×D6711yescharacteristic
141#431—1341no—
142#423—1341no—
143C2×#101—1341no—
144#400Dic6712yes—
145#400C2×D6712yes—
146#391—21no—
147#391—21no—
148#391—21no—
149#382—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—41no—
160C4×Q8—41no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4C2×Dic6712yes—
164C2×C4⋊C4C2×Dic6711yescharacteristic
165C2×C4⋊C4C22⋊D6711yescharacteristic
166C2×C4⋊C4C22×D6711yescharacteristic
167C2×C42—21no—
168C2×C42C2×Dic6711yescharacteristic
169C2×C42C22⋊D6711yescharacteristic
170C2×C42C22⋊D6711yescharacteristic
171C4⋊C8—2681no—
172C4⋊C8—2681no—
173C4⋊C8—2681no—
174#101—2682no—
175C22⋊C8—1342no—
176C4×C8—2681no—
177#95—22no—
178#95—22no—
179#95—22no—
180#95—22no—
181#95C2×Dic6712yes—
182C2×Q8—41no—
183C2×Q8—41no—
184C2×Q8—44no—
185C2×Q8—41no—
186C2×Q8—41no—
187C2×Q8#1176161512yes—
188C2×Q8#1176161812yes—
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—2682no—
200C2×C8—2682no—
201C2×C8—2682no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4—41no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C42—21no—
209C42—21no—
210C42—21no—
211C42—21no—
212C42—21no—
213C42—21no—
214C42—42no—
215C23C2×C22⋊Dic6711yescharacteristic
216Q8—42no—
217Q8—42no—
218Q8—82no—
219Q8—82no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—42no—
234C2×C4—44no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4#1452395811yescharacteristic
239C2×C4#1452395811yescharacteristic
240C2×C4#1452396712yes—
241C2×C4#1452396712yes—
242C2×C4C2×#1176161511yescharacteristic
243C2×C4C2×#1176161811yescharacteristic
244C8—5362no—
245C8—5362no—
246C22—21no—
247C22—21no—
248C22C2×#1452395811yescharacteristic
249C22C2×#1452396711yescharacteristic
250C22#1686341311yescharacteristic
251C4—21no—
252C4—22no—
253C4—22no—
254C4—21no—
255C4—21no—
256C4—41no—
257C4—41no—
258C4—41no—
259C4—82no—
260C4—82no—
261C4#1686319312yes—
262C2—21no—
263C2—21no—
264C2#2255878111yescharacteristic
265C2#2256186311yescharacteristic
266C2C2×#1686319311yescharacteristic
subgroupG/H
centerC22C2×#14523958
commutatorC22×C4×C67C22×C4
FrattiniC2×C4⋊C4C22×D67
FittingC67×#2568C2
socleC22×C67C2×#99
prime radicalC23×C67C2×C22⋊C4
cosocleC2×C4⋊C4×C67C23
hypercenter#2568D67
hypocenterC67#11907
LCSG ▷ C22×C4×C67 ▷ C23×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2568

Complements

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

NH ≅ G/Ntype
138. C67#11907semidirect

Maximal subgroups

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

subgroupindex
#225602052
#225585332
#225582362
C67×#25682
#1190767