Finite Group Databasebeta

← C3×#89582 #13055249

Subgroups — C3×#89582 #13055249

Order 1344.

304 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 86 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×C3×#1161C211yesmaximal, characteristic
2C2×C3×#1144C211yesmaximal, characteristic, complemented
3C2×C3×#1132C211yesmaximal, characteristic, complemented
4C3×#6460C211yesmaximal, characteristic, complemented
5C3×C7×#103C211yesmaximal, characteristic, complemented
6C3×(C2×D7)⋊C8C211yesmaximal, characteristic, complemented
7C3×#6383C211yesmaximal, characteristic, complemented
8#89582C311yesmaximal, characteristic, direct product factor, Hall
9C2×C3×C7×Q8C2211yescharacteristic
10C3×#1161—22no—
11C3×#1161—22no—
12C2×C3×D28—21no—
13C2×C3×D28C2211yescharacteristic
14C2×C3×C4×D7—21no—
15C2×C3×C4×D7C2211yescharacteristic
16C2×C3×Dic14C2211yescharacteristic
17C2×C3×C7×C8C2211yescharacteristic
18C3×C7×C4⋊C4C2211yescharacteristic
19C3×#1144—22no—
20C3×(C2×D7)⋊C4—21no—
21C3×C4⋊Dic7—21no—
22C2×C3×C7⋊C8C2211yescharacteristic
23C3×#1132—22no—
24C2×#1161C2×C311yescharacteristic
25C2×#1144C2×C311yescharacteristic, complemented
26C2×#1132C2×C311yescharacteristic, complemented
27#6460C2×C311yescharacteristic, complemented
28C7×#103C2×C311yescharacteristic, complemented
29(C2×D7)⋊C8C2×C311yescharacteristic, complemented
30#6383C2×C311yescharacteristic, complemented
31C3×#452—71nomaximal, Hall
32C22×C3×D7—21no—
33C22×C3×D7D411yescharacteristic
34C3×C7×Q8—21no—
35C3×C7×Q8D412yes—
36C2×C3×C4×C7—21no—
37C2×C3×C4×C7—21no—
38C2×C3×C4×C7C2311yescharacteristic
39C2×C3×Dic7—21no—
40C2×C3×Dic7D411yescharacteristic
41C3×D28—24no—
42C3×D28—21no—
43C3×D28D412yes—
44C3×C4×D7—24no—
45C3×C4×D7—21no—
46C3×C4×D7—21no—
47C3×Dic14—21no—
48C3×Dic14—21no—
49C3×C7×C8—21no—
50C3×C7⋊C8—21no—
51C2×C7×Q8C22×C311yescharacteristic
52#1161—22no—
53#1161—22no—
54C2×D28—21no—
55C2×D28C22×C311yescharacteristic
56C2×C4×D7—21no—
57C2×C4×D7C22×C311yescharacteristic
58C2×Dic14C22×C311yescharacteristic
59C2×C7×C8C22×C311yescharacteristic
60C7×C4⋊C4C22×C311yescharacteristic
61#1144—22no—
62(C2×D7)⋊C4—21no—
63C4⋊Dic7—21no—
64C2×C7⋊C8C22×C311yescharacteristic
65#1132—22no—
66C2×C3×#41—71no—
67C2×C3×Q16—71no—
68C2×C3×SD16—71no—
69C3×C22⋊Q8—71no—
70C3×#103D711yescharacteristic, complemented
71C3×#102—71no—
72C3×C22⋊C8—71no—
73C22×C3×C7C2×D411yescharacteristic
74C2×C3×D7—21no—
75C2×C3×D7—22no—
76C2×C3×D7—21no—
77C2×C3×D7—21no—
78C2×C3×D7—41no—
79C2×C3×D7—41no—
80C3×C4×C7—22no—
81C3×C4×C7—41no—
82C3×C4×C7C2×D411yescharacteristic
83C3×C4×C7C2×D411yescharacteristic
84C3×Dic7—21no—
85C3×Dic7—41no—
86#452—71noSylow
87C22×D7—21no—
88C22×D7C3×D411yescharacteristic
89C7×Q8—21no—
90C7×Q8C3×D412yes—
91C2×C4×C7—21no—
92C2×C4×C7—21no—
93C2×C4×C7C23×C311yescharacteristic
94C2×Dic7—21no—
95C2×Dic7C3×D411yescharacteristic
96D28—24no—
97D28—21no—
98D28C3×D412yes—
99C4×D7—24no—
100C4×D7—21no—
101C4×D7—21no—
102Dic14—21no—
103Dic14—21no—
104C7×C8—21no—
105C7⋊C8—21no—
106C3×#41—142no—
107C3×#41—142no—
108C2×C3×Q8—71no—
109C2×C3×Q8C2×D711yescharacteristic
110C2×C3×D4—71no—
111C2×C3×D4—141no—
112C22×C3×C4—71no—
113C22×C3×C4—141no—
114C3×Q16—142no—
115C3×SD16—142no—
116C2×C3×C8—71no—
117C2×C3×C8C2×D711yescharacteristic
118C3×C4⋊C4—141no—
119C3×C4⋊C4C2×D711yescharacteristic
120C3×C22⋊C4—141no—
121C2×C3×C7#12011yescharacteristic
122C2×C3×C7#13511yescharacteristic
123C2×C3×C7#13711yescharacteristic
124C3×D7—41no—
125C3×D7—42no—
126C2×#41—71no—
127C2×Q16—71no—
128C2×SD16—71no—
129C22⋊Q8—71no—
130#103C3×D711yescharacteristic, complemented
131#102—71no—
132C22⋊C8—71no—
133C22×C7C2×C3×D411yescharacteristic
134C2×D7—21no—
135C2×D7—22no—
136C2×D7—21no—
137C2×D7—21no—
138C2×D7—41no—
139C2×D7—41no—
140C4×C7—22no—
141C4×C7—41no—
142C4×C7C2×C3×D411yescharacteristic
143C4×C7C2×C3×D411yescharacteristic
144Dic7—21no—
145Dic7—41no—
146C23×C3—71no—
147C23×C3—141no—
148C3×Q8—21no—
149C3×Q8—141no—
150C3×Q8—141no—
151C3×Q8D2812yes—
152C3×D4—72no—
153C3×D4—144no—
154C3×D4—141no—
155C2×C3×C4—21no—
156C2×C3×C4—21no—
157C2×C3×C4—71no—
158C2×C3×C4—144no—
159C2×C3×C4—141no—
160C2×C3×C4—141no—
161C2×C3×C4—141no—
162C2×C3×C4C22×D711yescharacteristic
163C3×C8—21no—
164C3×C8—141no—
165C3×C7#45211yescharacteristic, complemented, Hall
166#41—142no—
167#41—142no—
168C2×Q8—71no—
169C2×Q8C2×C3×D711yescharacteristic
170C2×D4—71no—
171C2×D4—141no—
172C22×C4—71no—
173C22×C4—141no—
174Q16—142no—
175SD16—142no—
176C2×C8—71no—
177C2×C8C2×C3×D711yescharacteristic
178C4⋊C4—141no—
179C4⋊C4C2×C3×D711yescharacteristic
180C22⋊C4—141no—
181C2×C7C3×#12011yescharacteristic
182C2×C7C3×#13511yescharacteristic
183C2×C7C3×#13711yescharacteristic
184D7—41no—
185D7—42no—
186C22×C3—142no—
187C22×C3—141no—
188C22×C3—141no—
189C22×C3—141no—
190C22×C3—281no—
191C22×C3—281no—
192C22×C3D4×D711yescharacteristic
193C3×C4—22no—
194C3×C4—41no—
195C3×C4—141no—
196C3×C4—281no—
197C3×C4C2×D2811yescharacteristic
198C3×C4D4×D711yescharacteristic
199C23—71no—
200C23—141no—
201Q8—21no—
202Q8—141no—
203Q8—141no—
204Q8C3×D2812yes—
205D4—72no—
206D4—144no—
207D4—141no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—71no—
211C2×C4—144no—
212C2×C4—141no—
213C2×C4—141no—
214C2×C4—141no—
215C2×C4C22×C3×D711yescharacteristic
216C8—21no—
217C8—141no—
218C7C3×#45211yescharacteristic, complemented, Sylow
219C2×C3—281no—
220C2×C3—282no—
221C2×C3#644511yescharacteristic
222C2×C3#647911yescharacteristic
223C2×C3#648111yescharacteristic
224C22—142no—
225C22—141no—
226C22—141no—
227C22—141no—
228C22—281no—
229C22—281no—
230C22C3×D4×D711yescharacteristic
231C4—22no—
232C4—41no—
233C4—141no—
234C4—281no—
235C4C2×C3×D2811yescharacteristic
236C4C3×D4×D711yescharacteristic
237C3#8958211yescharacteristic, direct product factor, Sylow
238C2—281no—
239C2—282no—
240C2C3×#644511yescharacteristic
241C2C3×#647911yescharacteristic
242C2C3×#648111yescharacteristic
subgroupG/H
centerC22×C3D4×D7
commutator, cosocleC2×C4×C7C23×C3
FrattiniC2×C4C22×C3×D7
FittingC3×C7×#103C2
socleC22×C3×C7C2×D4
prime radicalC2×C3×#1161C2
hypercenterC3×#103D7
hypocenterC7C3×#452
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×#103

Complements

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

NH ≅ G/Ntype
2. C2×C3×#1144C2semidirect
3. C2×C3×#1132C2semidirect
4. C3×#6460C2semidirect
5. C3×C7×#103C2semidirect
6. C3×(C2×D7)⋊C8C2semidirect
7. C3×#6383C2semidirect
8. #89582C3direct (N × H)
25. C2×#1144C2×C3semidirect
26. C2×#1132C2×C3semidirect
27. #6460C2×C3semidirect
28. C7×#103C2×C3semidirect
29. (C2×D7)⋊C8C2×C3semidirect
30. #6383C2×C3semidirect
70. C3×#103D7semidirect
130. #103C3×D7semidirect
165. C3×C7#452semidirect
218. C7C3×#452semidirect
237. C3#89582direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#11612
C2×C3×#11442
C2×C3×#11322
C3×#64602
C3×C7×#1032
C3×(C2×D7)⋊C82
C3×#63832
#895823
C3×#4527