Finite Group Databasebeta

← #30883239

Subgroups — #30883239

Order 17152.

1506 conjugacy classes of subgroups; only the 275 normal subgroups are shown.

normal subgroups only

This group has 1506 conjugacy classes of subgroups — too many to draw. Only its 275 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#22562784C213yesmaximal
2#22561375C214yesmaximal, complemented
3#22561372C214yesmaximal, complemented
4C2×#16863163C213yesmaximal, complemented
5C2×#543×C67C211yesmaximal, characteristic, complemented
6C23×C22⋊D67C2211yescharacteristic
7C22×C22⋊Dic67C2213yes—
8#16863275—26no—
9#16863273—26no—
10#16863257C2216yes—
11#16863251C2216yes—
12#16863163C22112yes—
13#543×C67—26no—
14#543×C67C2214yes—
15C2×C22⋊Q8×C67—26no—
16C22×C22⋊C4×C67C2213yes—
17C24×D67C2311yescharacteristic
18C22×C22⋊D67—23no—
19C22×C22⋊D67—23no—
20C22×C22⋊D67—26no—
21C22×C22⋊D67Q814yescomplemented
22C23×Dic67—23no—
23C23×Dic67C2311yescharacteristic
24C2×C22⋊Dic67—23no—
25C2×C22⋊Dic67—212no—
26C2×C22⋊Dic67C2316yes—
27C2×(C2×D67)⋊C4—212no—
28C67⋊#95—212no—
29C25×C67C2311yescharacteristic
30C22×Q8×C67—22no—
31C23×C4×C67D416yes—
32C22⋊Q8×C67—224no—
33C22⋊Q8×C67—224no—
34C2×C4⋊C4×C67—26no—
35C2×C22⋊C4×C67—26no—
36C2×C22⋊C4×C67—26no—
37C2×C22⋊C4×C67D416yes—
38C2×C22⋊C4×C67C2316yes—
39C23×D67—23no—
40C23×D67—23no—
41C23×D67—82no—
42C23×D67C2×Q812yescomplemented
43C2×C22⋊D67—212no—
44C2×C22⋊D67—212no—
45C2×C22⋊D67—43no—
46C2×C22⋊D67—41no—
47C2×C22⋊D67—46no—
48C2×C22⋊D67—46no—
49C22×Dic67—23no—
50C22×Dic67—23no—
51C22×Dic67—26no—
52C22×Dic67—46no—
53C22×Dic67—86no—
54C22×Dic67C2×Q812yescomplemented
55C22⋊Dic67—212no—
56C22⋊Dic67—46no—
57(C2×D67)⋊C4—812no—
58C24×C67—82no—
59C24×C67C2×D413yes—
60C24×C67C2×D413yes—
61C24×C67C2×Q812yes—
62C24×C67#4116yes—
63C24×C67C2411yescharacteristic
64C2×Q8×C67—28no—
65C2×Q8×C67—84no—
66C22×C4×C67—26no—
67C22×C4×C67—26no—
68C22×C4×C67—86no—
69C22×C4×C67C2×D4112yes—
70C4⋊C4×C67—224no—
71C22⋊C4×C67—212no—
72C22⋊C4×C67—212no—
73C23×C67—46no—
74C23×C67—46no—
75C23×C67—42no—
76C23×C67—82no—
77C23×C67—86no—
78C23×C67#12013yes—
79C23×C67#12011yescharacteristic
80C23×C67C22⋊Q816yes—
81C23×C67C22⋊Q816yes—
82C23×C67#12416yes—
83C23×C67#12416yes—
84C23×C67C22×D413yes—
85C23×C67C22×Q811yescharacteristic
86C23×C67C2×#4113yes—
87C22×D67—26no—
88C22×D67—412no—
89C22×D67—41no—
90C22×D67—43no—
91C22×D67—82no—
92C22×D67—86no—
93Q8×C67—88no—
94C2×C4×C67—212no—
95C2×C4×C67—412no—
96C2×C4×C67—412no—
97C2×C4×C67—86no—
98C22⋊D67—412no—
99C22⋊D67—44no—
100C2×Dic67—26no—
101C2×Dic67—41no—
102C2×Dic67—43no—
103C2×Dic67—412no—
104C2×Dic67—46no—
105C2×Dic67—812no—
106C22×C67—26no—
107C22×C67—22no—
108C22×C67—26no—
109C22×C67—46no—
110C22×C67—41no—
111C22×C67—43no—
112C22×C67—46no—
113C22×C67—46no—
114C22×C67—46no—
115C22×C67#39316yes—
116C22×C67#39312yes—
117C22×C67C2×#12011yescharacteristic
118C22×C67C2×C22⋊Q813yes—
119C22×C67C2×#12413yes—
120C2×D67—412no—
121C2×D67—42no—
122C2×D67—86no—
123C2×D67—82no—
124C4×C67—412no—
125Dic67—42no—
126Dic67—86no—
127#23468—671nomaximal, Sylow
128C2×C67—23no—
129C2×C67—23no—
130C2×C67—22no—
131C2×C67—26no—
132C2×C67C2×#39311yescharacteristic
133C2×C67#302612yes—
134D67—82no—
135C2×#543D6711yescharacteristic, complemented
136#2308—673no—
137#2030—674no—
138#2029—674no—
139C2×#342—673no—
140C67#2346811yescharacteristic, complemented, Sylow
141C23×D4—671no—
142#543—26no—
143#543C2×D6714yes—
144C2×C22⋊Q8—26no—
145C22×C22⋊C4—673no—
146C22×C22⋊C4C2×D6713yes—
147#386—1346no—
148#386—1346no—
149#381—676no—
150#379—676no—
151#342—6712no—
152C25—671no—
153C25C22×D6711yescharacteristic
154C22×Q8—22no—
155C22×D4—674no—
156C22×D4—1343no—
157C22×D4—1343no—
158C22×D4—1346no—
159C23×C4—671no—
160C23×C4—1343no—
161C23×C4C22⋊D6716yes—
162C22⋊Q8—224no—
163C22⋊Q8—224no—
164C2×C4⋊C4—26no—
165C2×C22⋊C4—26no—
166C2×C22⋊C4—26no—
167C2×C22⋊C4—676no—
168C2×C22⋊C4—1343no—
169C2×C22⋊C4—13412no—
170C2×C22⋊C4—13412no—
171C2×C22⋊C4C22⋊D6716yes—
172C2×C22⋊C4C22×D6716yes—
173#95—13412no—
174C24—82no—
175C24—5362no—
176C24—672no—
177C24—1343no—
178C24—1343no—
179C24C2×C22⋊D6713yes—
180C24C2×C22⋊D6713yes—
181C24Q8×D6712yes—
182C24#1176162916yes—
183C24C23×D6711yescharacteristic
184C2×Q8—28no—
185C2×Q8—84no—
186C2×D4—2683no—
187C2×D4—2681no—
188C2×D4—2686no—
189C2×D4—2686no—
190C2×D4—13412no—
191C2×D4—13412no—
192C22×C4—26no—
193C22×C4—26no—
194C22×C4—86no—
195C22×C4—2686no—
196C22×C4—5366no—
197C22×C4—672no—
198C22×C4—1346no—
199C22×C4—1343no—
200C22×C4—1343no—
201C22×C4C2×C22⋊D67112yes—
202C4⋊C4—224no—
203C22⋊C4—212no—
204C22⋊C4—212no—
205C22⋊C4—2686no—
206C22⋊C4—53612no—
207C22⋊C4—13412no—
208C23—46no—
209C23—46no—
210C23—42no—
211C23—82no—
212C23—86no—
213C23—26812no—
214C23—2681no—
215C23—2683no—
216C23—5362no—
217C23—5366no—
218C23—1346no—
219C23#1452401213yes—
220C23#1452401211yescharacteristic
221C23#1452401916yes—
222C23#1452401916yes—
223C23#1452403216yes—
224C23#1452403216yes—
225C23C22×C22⋊D6713yes—
226C23C2×Q8×D6711yescharacteristic
227C23C2×#1176162913yes—
228Q8—88no—
229D4—26812no—
230D4—2684no—
231C2×C4—212no—
232C2×C4—412no—
233C2×C4—412no—
234C2×C4—86no—
235C2×C4—2681no—
236C2×C4—2683no—
237C2×C4—26812no—
238C2×C4—2686no—
239C2×C4—53612no—
240C2×C4—1346no—
241C22—22no—
242C22—26no—
243C22—26no—
244C22—46no—
245C22—41no—
246C22—43no—
247C22—46no—
248C22—46no—
249C22—46no—
250C22—26812no—
251C22—2682no—
252C22—5366no—
253C22—5362no—
254C22#1686330716yes—
255C22#1686330712yes—
256C22C2×#1452401211yescharacteristic
257C22C2×#1452401913yes—
258C22C2×#1452403213yes—
259C4—412no—
260C4—2682no—
261C4—5366no—
262C2—23no—
263C2—23no—
264C2—22no—
265C2—26no—
266C2—5362no—
267C2C2×#1686330711yescharacteristic
268C2#2256960612yes—
subgroupG/H
centerC22C2×#14524012
commutator, cosocleC24×C67C24
FrattiniC24C23×D67
FittingC2×#543×C67C2
socleC22×C67C2×#120
prime radicalC23×C22⋊D67C22
hypercenterC2×#543D67
hypocenterC67#23468
LCSG ▷ C24×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C25 ◁ C2×#543

Complements

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

NH ≅ G/Ntype
2. #22561375C2semidirect
3. #22561372C2semidirect
4. C2×#16863163C2semidirect
5. C2×#543×C67C2semidirect
21. C22×C22⋊D67Q8semidirect
42. C23×D67C2×Q8semidirect
54. C22×Dic67C2×Q8semidirect
135. C2×#543D67semidirect
140. C67#23468semidirect

Maximal subgroups

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

subgroupindex
#225627842
#225613752
#225613722
C2×#168631632
C2×#543×C672
#2346867