Finite Group Databasebeta

← #30771842

Subgroups — #30771842

Order 17152.

432 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 119 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
1C2×#16863195C212yesmaximal
2#22560205C212yesmaximal
3#22558336C212yesmaximal
4C67×#2327C211yesmaximal, characteristic
5#16863420—22no—
6C2×#14523962C2214yes—
7#16863195—24no—
8#16863165—22no—
9C2×C4⋊Q8×C67C2211yescharacteristic
10C2×C4×Q8×C67C411yescharacteristic
11C2×C4×Q8×C67C2211yescharacteristic
12#391×C67C411yescharacteristic
13#391×C67C2211yescharacteristic
14#387×C67—22no—
15#387×C67—22no—
16#382×C67C411yescharacteristic
17#376×C67C411yescharacteristic
18C2×#11761612—24no—
19C4⋊(C67⋊C8)—44no—
20#14523962—24no—
21#14523962—24no—
22C22×Q8×C67C2×C411yescharacteristic
23C22×Q8×C67C2311yescharacteristic
24C4⋊Q8×C67—22no—
25C4⋊Q8×C67D412yes—
26C4⋊Q8×C67Q812yes—
27C4×Q8×C67—42no—
28C4×Q8×C67—42no—
29C2×C4⋊C4×C67—22no—
30C2×C4⋊C4×C67—21no—
31C2×C4⋊C4×C67C2×C411yescharacteristic
32C2×C4⋊C4×C67C2×C411yescharacteristic
33C2×C4⋊C4×C67D412yes—
34C2×C42×C67—21no—
35C2×C42×C67C2×C411yescharacteristic
36C2×C42×C67C2×C411yescharacteristic
37C2×C42×C67C2×C411yescharacteristic
38C2×C42×C67D412yes—
39#95×C67—21no—
40#95×C67—21no—
41#95×C67—22no—
42#95×C67—22no—
43#95×C67—22no—
44#11761612—44no—
45#11761612—44no—
46C2×C67⋊C8—44no—
47C2×Q8×C67—21no—
48C2×Q8×C67—41no—
49C2×Q8×C67—41no—
50C2×Q8×C67—41no—
51C2×Q8×C67—41no—
52C2×Q8×C67C4⋊C412yes—
53C2×Q8×C67C2×D411yescharacteristic
54C2×Q8×C67C2×Q811yescharacteristic
55C2×Q8×C67#4112yes—
56C22×C4×C67—21no—
57C22×C4×C67—21no—
58C22×C4×C67—21no—
59C22×C4×C67—21no—
60C22×C4×C67C22⋊C411yescharacteristic
61C22×C4×C67C22⋊C412yes—
62C22×C4×C67C22⋊C411yescharacteristic
63C22×C4×C67C22×C411yescharacteristic
64C22×C4×C67C2×D412yes—
65C4⋊C4×C67—24no—
66C4⋊C4×C67—24no—
67C4⋊C4×C67—41no—
68C4⋊C4×C67—42no—
69C4⋊C4×C67—41no—
70C4⋊C4×C67—42no—
71C42×C67—22no—
72C42×C67—24no—
73C42×C67—21no—
74C42×C67—41no—
75C42×C67—42no—
76C42×C67C4⋊C412yes—
77C23×C67C2×C22⋊C411yescharacteristic
78Q8×C67—41no—
79Q8×C67—41no—
80Q8×C67—42no—
81Q8×C67—42no—
82Q8×C67—42no—
83C2×C4×C67—21no—
84C2×C4×C67—22no—
85C2×C4×C67—21no—
86C2×C4×C67—22no—
87C2×C4×C67—22no—
88C2×C4×C67—22no—
89C2×C4×C67—21no—
90C2×C4×C67—22no—
91C2×C4×C67—41no—
92C2×C4×C67—41no—
93C2×C4×C67—41no—
94C2×C4×C67—41no—
95C2×C4×C67—41no—
96C2×C4×C67—41no—
97C2×C4×C67—42no—
98C2×C4×C67—41no—
99C2×C4×C67—41no—
100C2×C4×C67—42no—
101C2×C4×C67—41no—
102C2×C4×C67—41no—
103C2×C4×C67#9912yes—
104C2×C4×C67C2×C4⋊C411yescharacteristic
105C2×C4×C67#11711yescharacteristic
106C2×C4×C67#12112yes—
107C2×C4×C67C22⋊Q812yes—
108C67⋊C8—84no—
109C22×C67—21no—
110C22×C67—21no—
111C22×C67#38011yescharacteristic
112C22×C67C2×#9911yescharacteristic
113C22×C67#41011yescharacteristic
114C4×C67—22no—
115C4×C67—21no—
116C4×C67—21no—
117C4×C67—22no—
118C4×C67—22no—
119C4×C67—42no—
120C4×C67—42no—
121C4×C67—41no—
122C4×C67—42no—
123C4×C67—81no—
124C4×C67—81no—
125#12804—671nomaximal, Sylow
126C2×C67—21no—
127C2×C67—21no—
128C2×C67#179911yescharacteristic
129C2×C67#213311yescharacteristic
130C2×C67#213511yescharacteristic
131#2327D6711yescharacteristic
132C2×#356—672no—
133#1801—672no—
134#1394—672no—
135C67#1280411yescharacteristic, complemented, Sylow
136C2×C4⋊Q8C2×D6711yescharacteristic
137C2×C4×Q8Dic6711yescharacteristic
138C2×C4×Q8C2×D6711yescharacteristic
139#423—1342no—
140C2×#101—674no—
141#391Dic6711yescharacteristic
142#391C2×D6711yescharacteristic
143#387—22no—
144#387—22no—
145#382Dic6711yescharacteristic
146#376Dic6711yescharacteristic
147#356—1344no—
148#343—1342no—
149C22×Q8C2×Dic6711yescharacteristic
150C22×Q8C22×D6711yescharacteristic
151C2×M16—1344no—
152C4⋊Q8—22no—
153C4⋊Q8Dic13412yes—
154C4⋊Q8D26812yes—
155C4×Q8—42no—
156C4×Q8—42no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4C2×Dic6711yescharacteristic
160C2×C4⋊C4C2×Dic6711yescharacteristic
161C2×C4⋊C4C22⋊D6712yes—
162C2×C42—21no—
163C2×C42C2×Dic6711yescharacteristic
164C2×C42C2×Dic6711yescharacteristic
165C2×C42C2×Dic6711yescharacteristic
166C2×C42C22⋊D6712yes—
167C4⋊C8—2684no—
168#101—1344no—
169#101—1344no—
170#95—21no—
171#95—22no—
172#95—22no—
173#95—21no—
174#95—22no—
175C2×Q8—21no—
176C2×Q8—41no—
177C2×Q8—41no—
178C2×Q8—41no—
179C2×Q8—41no—
180C2×Q8C4⋊Dic6712yes—
181C2×Q8C2×D26811yescharacteristic
182C2×Q8C2×Dic13411yescharacteristic
183C2×Q8#1176162812yes—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4C22⋊Dic6712yes—
189C22×C4C22⋊Dic6711yescharacteristic
190C22×C4C22⋊Dic6711yescharacteristic
191C22×C4C22×Dic6711yescharacteristic
192C22×C4C2×C22⋊D6712yes—
193M16—2684no—
194M16—2684no—
195C2×C8—2684no—
196C4⋊C4—24no—
197C4⋊C4—24no—
198C4⋊C4—41no—
199C4⋊C4—42no—
200C4⋊C4—41no—
201C4⋊C4—42no—
202C42—24no—
203C42—22no—
204C42—21no—
205C42—41no—
206C42—42no—
207C42C4⋊Dic6712yes—
208C23C2×C22⋊Dic6711yescharacteristic
209Q8—41no—
210Q8—41no—
211Q8—42no—
212Q8—42no—
213Q8—42no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—42no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4#1452395812yes—
235C2×C4C2×C4⋊Dic6711yescharacteristic
236C2×C4#1452400211yescharacteristic
237C2×C4#1452401712yes—
238C2×C4#1452402212yes—
239C8—5364no—
240C22—21no—
241C22—21no—
242C22#1686325611yescharacteristic
243C22C2×#1452395811yescharacteristic
244C22#1686337011yescharacteristic
245C4—22no—
246C4—22no—
247C4—21no—
248C4—21no—
249C4—22no—
250C4—42no—
251C4—41no—
252C4—42no—
253C4—42no—
254C4—81no—
255C4—81no—
256C2—21no—
257C2—21no—
258C2#2256019711yescharacteristic
259C2#2256189811yescharacteristic
260C2#2256190411yescharacteristic
subgroupG/H
centerC22C2×#14523958
commutatorC22×C4×C67C22×C4
FrattiniC22×Q8C22×D67
FittingC67×#2327C2
socleC22×C67C2×#99
prime radicalC23×C67C2×C22⋊C4
cosocleC22×Q8×C67C23
hypercenter#2327D67
hypocenterC67#12804
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2327

Complements

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

NH ≅ G/Ntype
135. C67#12804semidirect

Maximal subgroups

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

subgroupindex
C2×#168631952
#225602052
#225583362
C67×#23272
#1280467