Finite Group Databasebeta

← C3×#89505 #13055166

Subgroups — C3×#89505 #13055166

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×#1157C211yesmaximal, characteristic
2C3×#6487C211yesmaximal, characteristic, complemented
3C2×C3×Dic28C211yesmaximal, characteristic, complemented
4C2×C3×#1132C211yesmaximal, characteristic, complemented
5C3×C7×C22⋊C8C211yesmaximal, characteristic, complemented
6C3×#6395C211yesmaximal, characteristic, complemented
7C3×#6390C211yesmaximal, characteristic, complemented
8#89505C311yesmaximal, characteristic, direct product factor, Hall
9C22×C3×C4×C7C2211yescharacteristic
10C2×C3×C22⋊D7—21no—
11C3×#1157—22no—
12C3×#1157—22no—
13C2×C3×D28C2211yescharacteristic
14C2×C3×C4×D7—21no—
15C2×C3×Dic14C2211yescharacteristic
16C2×C3×Dic14C2211yescharacteristic
17C2×C3×C7×C8C2211yescharacteristic
18C2×C3×C7×C8C2211yescharacteristic
19C3×C22⋊Dic7—21no—
20C3×C4⋊Dic7C2211yescharacteristic
21C3×Dic7⋊C4—21no—
22C3×Dic28—22no—
23C3×#1132—22no—
24C2×#1157C2×C311yescharacteristic
25#6487C2×C311yescharacteristic, complemented
26C2×Dic28C2×C311yescharacteristic, complemented
27C2×#1132C2×C311yescharacteristic, complemented
28C7×C22⋊C8C2×C311yescharacteristic, complemented
29#6395C2×C311yescharacteristic, complemented
30#6390C2×C311yescharacteristic, complemented
31C3×#452—71nomaximal, Hall
32C23×C3×C7D411yescharacteristic
33C22×C3×D7—21no—
34C2×C3×C4×C7—21no—
35C2×C3×C4×C7—21no—
36C2×C3×C4×C7D411yescharacteristic
37C2×C3×C4×C7C2311yescharacteristic
38C3×C22⋊D7—24no—
39C2×C3×Dic7—21no—
40C2×C3×Dic7—21no—
41C2×C3×Dic7—21no—
42C3×D28—21no—
43C3×D28D412yes—
44C3×C4×D7—24no—
45C3×Dic14—21no—
46C3×Dic14—21no—
47C3×Dic14—21no—
48C3×Dic14D412yes—
49C3×C7×C8—21no—
50C3×C7×C8—21no—
51C22×C4×C7C22×C311yescharacteristic
52C2×C22⋊D7—21no—
53#1157—22no—
54#1157—22no—
55C2×D28C22×C311yescharacteristic
56C2×C4×D7—21no—
57C2×Dic14C22×C311yescharacteristic
58C2×Dic14C22×C311yescharacteristic
59C2×C7×C8C22×C311yescharacteristic
60C2×C7×C8C22×C311yescharacteristic
61C22⋊Dic7—21no—
62C4⋊Dic7C22×C311yescharacteristic
63Dic7⋊C4—21no—
64Dic28—22no—
65#1132—22no—
66C2×C3×#41—71no—
67C2×C3×Q16—71no—
68C2×C3×SD16—71no—
69C3×C22⋊Q8—71no—
70C3×#103—71no—
71C3×#102—71no—
72C3×C22⋊C8D711yescharacteristic, complemented
73C22×C3×C7—21no—
74C22×C3×C7—21no—
75C22×C3×C7—21no—
76C22×C3×C7C2×D411yescharacteristic
77C2×C3×D7—22no—
78C2×C3×D7—41no—
79C2×C3×D7—41no—
80C3×C4×C7—21no—
81C3×C4×C7C2×D411yescharacteristic
82C3×C4×C7C2×D411yescharacteristic
83C3×Dic7—22no—
84C3×Dic7—41no—
85C3×Dic7—41no—
86#452—71noSylow
87C23×C7C3×D411yescharacteristic
88C22×D7—21no—
89C2×C4×C7—21no—
90C2×C4×C7—21no—
91C2×C4×C7C3×D411yescharacteristic
92C2×C4×C7C23×C311yescharacteristic
93C22⋊D7—24no—
94C2×Dic7—21no—
95C2×Dic7—21no—
96C2×Dic7—21no—
97D28—21no—
98D28C3×D412yes—
99C4×D7—24no—
100Dic14—21no—
101Dic14—21no—
102Dic14—21no—
103Dic14C3×D412yes—
104C7×C8—21no—
105C7×C8—21no—
106C3×#41—142no—
107C3×#41—142no—
108C2×C3×Q8—71no—
109C2×C3×Q8—71no—
110C2×C3×D4—71no—
111C2×C3×D4—141no—
112C22×C3×C4—141no—
113C22×C3×C4C2×D711yescharacteristic
114C3×Q16—142no—
115C3×SD16—142no—
116C2×C3×C8C2×D711yescharacteristic
117C2×C3×C8C2×D711yescharacteristic
118C3×C4⋊C4—71no—
119C3×C4⋊C4—141no—
120C3×C22⋊C4—141no—
121C2×C3×C7—41no—
122C2×C3×C7#12011yescharacteristic
123C2×C3×C7#13511yescharacteristic
124C2×C3×C7#13711yescharacteristic
125C3×D7—42no—
126C2×#41—71no—
127C2×Q16—71no—
128C2×SD16—71no—
129C22⋊Q8—71no—
130#103—71no—
131#102—71no—
132C22⋊C8C3×D711yescharacteristic, complemented
133C22×C7—21no—
134C22×C7—21no—
135C22×C7—21no—
136C22×C7C2×C3×D411yescharacteristic
137C2×D7—22no—
138C2×D7—41no—
139C2×D7—41no—
140C4×C7—21no—
141C4×C7C2×C3×D411yescharacteristic
142C4×C7C2×C3×D411yescharacteristic
143Dic7—22no—
144Dic7—41no—
145Dic7—41no—
146C23×C3—141no—
147C23×C3D2811yescharacteristic
148C3×Q8—72no—
149C3×Q8—141no—
150C3×Q8—141no—
151C3×Q8—141no—
152C3×D4—72no—
153C3×D4—144no—
154C3×D4—141no—
155C2×C3×C4—21no—
156C2×C3×C4—21no—
157C2×C3×C4—144no—
158C2×C3×C4—141no—
159C2×C3×C4—141no—
160C2×C3×C4—141no—
161C2×C3×C4D2811yescharacteristic
162C2×C3×C4C22×D711yescharacteristic
163C3×C8—21no—
164C3×C8—21no—
165C3×C7#45211yescharacteristic, complemented, Hall
166#41—142no—
167#41—142no—
168C2×Q8—71no—
169C2×Q8—71no—
170C2×D4—71no—
171C2×D4—141no—
172C22×C4—141no—
173C22×C4C2×C3×D711yescharacteristic
174Q16—142no—
175SD16—142no—
176C2×C8C2×C3×D711yescharacteristic
177C2×C8C2×C3×D711yescharacteristic
178C4⋊C4—71no—
179C4⋊C4—141no—
180C22⋊C4—141no—
181C2×C7—41no—
182C2×C7C3×#12011yescharacteristic
183C2×C7C3×#13511yescharacteristic
184C2×C7C3×#13711yescharacteristic
185D7—42no—
186C22×C3—21no—
187C22×C3—21no—
188C22×C3—21no—
189C22×C3—142no—
190C22×C3—281no—
191C22×C3—281no—
192C22×C3C2×D2811yescharacteristic
193C3×C4—21no—
194C3×C4—142no—
195C3×C4—281no—
196C3×C4—281no—
197C3×C4D4×D711yescharacteristic
198C3×C4D4×D711yescharacteristic
199C23—141no—
200C23C3×D2811yescharacteristic
201Q8—72no—
202Q8—141no—
203Q8—141no—
204Q8—141no—
205D4—72no—
206D4—144no—
207D4—141no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—144no—
211C2×C4—141no—
212C2×C4—141no—
213C2×C4—141no—
214C2×C4C3×D2811yescharacteristic
215C2×C4C22×C3×D711yescharacteristic
216C8—21no—
217C8—21no—
218C7C3×#45211yescharacteristic, complemented, Sylow
219C2×C3—41no—
220C2×C3—282no—
221C2×C3#644511yescharacteristic
222C2×C3#646711yescharacteristic
223C2×C3#647211yescharacteristic
224C22—21no—
225C22—21no—
226C22—21no—
227C22—142no—
228C22—281no—
229C22—281no—
230C22C2×C3×D2811yescharacteristic
231C4—21no—
232C4—142no—
233C4—281no—
234C4—281no—
235C4C3×D4×D711yescharacteristic
236C4C3×D4×D711yescharacteristic
237C3#8950511yescharacteristic, direct product factor, Sylow
238C2—41no—
239C2—282no—
240C2C3×#644511yescharacteristic
241C2C3×#646711yescharacteristic
242C2C3×#647211yescharacteristic
subgroupG/H
centerC22×C3C2×D28
commutator, cosocleC2×C4×C7C23×C3
FrattiniC2×C4C22×C3×D7
FittingC3×C7×C22⋊C8C2
socleC22×C3×C7C2×D4
prime radicalC2×C3×#1157C2
hypercenterC3×C22⋊C8D7
hypocenterC7C3×#452
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C3×#6487C2semidirect
3. C2×C3×Dic28C2semidirect
4. C2×C3×#1132C2semidirect
5. C3×C7×C22⋊C8C2semidirect
6. C3×#6395C2semidirect
7. C3×#6390C2semidirect
8. #89505C3direct (N × H)
25. #6487C2×C3semidirect
26. C2×Dic28C2×C3semidirect
27. C2×#1132C2×C3semidirect
28. C7×C22⋊C8C2×C3semidirect
29. #6395C2×C3semidirect
30. #6390C2×C3semidirect
72. C3×C22⋊C8D7semidirect
132. C22⋊C8C3×D7semidirect
165. C3×C7#452semidirect
218. C7C3×#452semidirect
237. C3#89505direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#11572
C3×#64872
C2×C3×Dic282
C2×C3×#11322
C3×C7×C22⋊C82
C3×#63952
C3×#63902
#895053
C3×#4527