Finite Group Databasebeta

← C3×#89542 #13055202

Subgroups — C3×#89542 #13055202

Order 1344.

380 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 110 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×D4×D7C211yesmaximal, characteristic
2C2×C3×#1131C211yesmaximal, characteristic, complemented
3C3×#6455C211yesmaximal, characteristic, complemented
4C3×C7×#102C211yesmaximal, characteristic, complemented
5C3×#6406C211yesmaximal, characteristic, complemented
6C3×#6395C211yesmaximal, characteristic, complemented
7C3×#6383C211yesmaximal, characteristic, complemented
8#89542C311yesmaximal, characteristic, direct product factor, Hall
9C23×C3×D7—21no—
10C2×C3×C7×D4C2211yescharacteristic
11C2×C3×C22⋊D7—21no—
12C3×D4×D7—22no—
13C3×D4×D7C414yescomplemented
14C2×C3×D28C2211yescharacteristic
15C2×C3×C4×D7C2211yescharacteristic
16C2×C3×C7×C8C2211yescharacteristic, complemented
17C3×C7×C4⋊C4C2211yescharacteristic, complemented
18C3×(C2×D7)⋊C4—21no—
19C3×C4⋊Dic7C2211yescharacteristic, complemented
20C3×C4×Dic7—21no—
21C2×C3×C7⋊C8C2211yescharacteristic, complemented
22C3×#1131—22no—
23C2×D4×D7C2×C311yescharacteristic
24C2×#1131C2×C311yescharacteristic, complemented
25#6455C2×C311yescharacteristic, complemented
26C7×#102C2×C311yescharacteristic, complemented
27#6406C2×C311yescharacteristic, complemented
28#6395C2×C311yescharacteristic, complemented
29#6383C2×C311yescharacteristic, complemented
30C3×#418—71nomaximal, Hall
31C23×C3×C7—21no—
32C22×C3×D7—21no—
33C22×C3×D7—24no—
34C22×C3×D7—42no—
35C22×C3×D7—42no—
36C22×C3×D7D411yescharacteristic
37C3×C7×D4—21no—
38C3×C7×D4C2×C412yes—
39C2×C3×C4×C7—21no—
40C2×C3×C4×C7C2311yescharacteristic
41C3×C22⋊D7—24no—
42C2×C3×Dic7—21no—
43C2×C3×Dic7D411yescharacteristic
44C3×D28—21no—
45C3×D28C2×C412yes—
46C3×C4×D7C2×C412yes—
47C3×C4×D7D412yes—
48C3×C7×C8—21no—
49C3×C7⋊C8—21no—
50C23×D7—21no—
51C2×C7×D4C22×C311yescharacteristic
52C2×C22⋊D7—21no—
53D4×D7—22no—
54D4×D7C3×C414yescomplemented
55C2×D28C22×C311yescharacteristic
56C2×C4×D7C22×C311yescharacteristic
57C2×C7×C8C22×C311yescharacteristic, complemented
58C7×C4⋊C4C22×C311yescharacteristic, complemented
59(C2×D7)⋊C4—21no—
60C4⋊Dic7C22×C311yescharacteristic, complemented
61C4×Dic7—21no—
62C2×C7⋊C8C22×C311yescharacteristic, complemented
63#1131—22no—
64C22×C3×D4—71no—
65C2×C3×M16—71no—
66C3×#117—71no—
67C3×#102—71no—
68C3×#102—71no—
69C3×#102—71no—
70C3×#102D711yescharacteristic, complemented
71C22×C3×C7—22no—
72C22×C3×C7—41no—
73C22×C3×C7—41no—
74C22×C3×C7C2×D411yescharacteristic
75C2×C3×D7—21no—
76C2×C3×D7—21no—
77C2×C3×D7—22no—
78C2×C3×D7—48no—
79C2×C3×D7—41no—
80C2×C3×D7—41no—
81C2×C3×D7C22⋊C412yes—
82C3×C4×C7—41no—
83C3×C4×C7C22×C411yescharacteristic
84C3×C4×C7C2×D411yescharacteristic
85C3×Dic7—41no—
86C3×Dic7C22⋊C412yes—
87#418—71noSylow
88C23×C7—21no—
89C22×D7—21no—
90C22×D7—24no—
91C22×D7—42no—
92C22×D7—42no—
93C22×D7C3×D411yescharacteristic
94C7×D4—21no—
95C7×D4C2×C3×C412yes—
96C2×C4×C7—21no—
97C2×C4×C7C23×C311yescharacteristic
98C22⋊D7—24no—
99C2×Dic7—21no—
100C2×Dic7C3×D411yescharacteristic
101D28—21no—
102D28C2×C3×C412yes—
103C4×D7C2×C3×C412yes—
104C4×D7C3×D412yes—
105C7×C8—21no—
106C7⋊C8—21no—
107C24×C3—141no—
108C2×C3×D4—74no—
109C2×C3×D4—71no—
110C2×C3×D4—142no—
111C2×C3×D4—141no—
112C2×C3×D4C2×D711yescharacteristic
113C22×C3×C4—71no—
114C3×M16—142no—
115C2×C3×C8—71no—
116C2×C3×C8C2×D711yescharacteristic, complemented
117C3×C4⋊C4—71no—
118C3×C4⋊C4C2×D711yescharacteristic, complemented
119C3×C22⋊C4—141no—
120C3×C42—141no—
121C2×C3×C7—42no—
122C2×C3×C7C2×C22⋊C411yescharacteristic
123C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
124C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
125C3×D7—22no—
126C3×D7—42no—
127C22×D4—71no—
128C2×M16—71no—
129#117—71no—
130#102—71no—
131#102—71no—
132#102—71no—
133#102C3×D711yescharacteristic, complemented
134C22×C7—22no—
135C22×C7—41no—
136C22×C7—41no—
137C22×C7C2×C3×D411yescharacteristic
138C2×D7—21no—
139C2×D7—21no—
140C2×D7—22no—
141C2×D7—48no—
142C2×D7—41no—
143C2×D7—41no—
144C2×D7C3×C22⋊C412yes—
145C4×C7—41no—
146C4×C7C22×C3×C411yescharacteristic
147C4×C7C2×C3×D411yescharacteristic
148Dic7—41no—
149Dic7C3×C22⋊C412yes—
150C23×C3—21no—
151C23×C3—71no—
152C23×C3—144no—
153C23×C3—141no—
154C23×C3—282no—
155C23×C3—282no—
156C3×D4—21no—
157C3×D4—72no—
158C3×D4—144no—
159C3×D4—141no—
160C3×D4C4×D712yes—
161C2×C3×C4—21no—
162C2×C3×C4—71no—
163C2×C3×C4—72no—
164C2×C3×C4—72no—
165C2×C3×C4—141no—
166C2×C3×C4C22×D711yescharacteristic
167C3×C8—21no—
168C3×C8—141no—
169C3×C7#41811yescharacteristic, complemented, Hall
170C24—141no—
171C2×D4—74no—
172C2×D4—71no—
173C2×D4—142no—
174C2×D4—141no—
175C2×D4C2×C3×D711yescharacteristic
176C22×C4—71no—
177M16—142no—
178C2×C8—71no—
179C2×C8C2×C3×D711yescharacteristic, complemented
180C4⋊C4—71no—
181C4⋊C4C2×C3×D711yescharacteristic, complemented
182C22⋊C4—141no—
183C42—141no—
184C2×C7—42no—
185C2×C7C2×C3×C22⋊C411yescharacteristic
186C2×C7C3×AGL(1,ℤ/8)11yescharacteristic
187C2×C7C3×AGL(1,ℤ/8)11yescharacteristic
188D7—22no—
189D7—42no—
190C22×C3—22no—
191C22×C3—41no—
192C22×C3—41no—
193C22×C3—72no—
194C22×C3—141no—
195C22×C3—141no—
196C22×C3—142no—
197C22×C3—288no—
198C22×C3—281no—
199C22×C3—281no—
200C22×C3D4×D711yescharacteristic
201C3×C4—41no—
202C3×C4—72no—
203C3×C4—281no—
204C3×C4C2×C4×D711yescharacteristic
205C3×C4D4×D711yescharacteristic
206C23—21no—
207C23—71no—
208C23—144no—
209C23—141no—
210C23—282no—
211C23—282no—
212D4—21no—
213D4—72no—
214D4—144no—
215D4—141no—
216D4C3×C4×D712yes—
217C2×C4—21no—
218C2×C4—71no—
219C2×C4—72no—
220C2×C4—72no—
221C2×C4—141no—
222C2×C4C22×C3×D711yescharacteristic
223C8—21no—
224C8—141no—
225C7C3×#41811yescharacteristic, complemented, Sylow
226C2×C3—42no—
227C2×C3—142no—
228C2×C3—282no—
229C2×C3D7×C22⋊C411yescharacteristic
230C2×C3#647411yescharacteristic
231C2×C3#647711yescharacteristic
232C22—22no—
233C22—41no—
234C22—41no—
235C22—72no—
236C22—141no—
237C22—141no—
238C22—142no—
239C22—288no—
240C22—281no—
241C22—281no—
242C22C3×D4×D711yescharacteristic
243C4—41no—
244C4—72no—
245C4—281no—
246C4C2×C3×C4×D711yescharacteristic
247C4C3×D4×D711yescharacteristic
248C3#8954211yescharacteristic, direct product factor, Sylow
249C2—42no—
250C2—142no—
251C2—282no—
252C2C3×D7×C22⋊C411yescharacteristic
253C2C3×#647411yescharacteristic
254C2C3×#647711yescharacteristic
subgroupG/H
centerC22×C3D4×D7
commutatorC4×C7C22×C3×C4
FrattiniC2×C4C22×C3×D7
FittingC3×C7×#102C2
socleC22×C3×C7C2×D4
prime radicalC2×C3×D4×D7C2
cosocleC2×C4×C7C23×C3
hypercenterC3×#102D7
hypocenterC7C3×#418
LCSG ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×#102

Complements

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

NH ≅ G/Ntype
2. C2×C3×#1131C2semidirect
3. C3×#6455C2semidirect
4. C3×C7×#102C2semidirect
5. C3×#6406C2semidirect
6. C3×#6395C2semidirect
7. C3×#6383C2semidirect
8. #89542C3direct (N × H)
13. C3×D4×D7C4semidirect
16. C2×C3×C7×C8C22semidirect
17. C3×C7×C4⋊C4C22semidirect
19. C3×C4⋊Dic7C22semidirect
21. C2×C3×C7⋊C8C22semidirect
24. C2×#1131C2×C3semidirect
25. #6455C2×C3semidirect
26. C7×#102C2×C3semidirect
27. #6406C2×C3semidirect
28. #6395C2×C3semidirect
29. #6383C2×C3semidirect
54. D4×D7C3×C4semidirect
57. C2×C7×C8C22×C3semidirect
58. C7×C4⋊C4C22×C3semidirect
60. C4⋊Dic7C22×C3semidirect
62. C2×C7⋊C8C22×C3semidirect
70. C3×#102D7semidirect
116. C2×C3×C8C2×D7semidirect
118. C3×C4⋊C4C2×D7semidirect
133. #102C3×D7semidirect
169. C3×C7#418semidirect
179. C2×C8C2×C3×D7semidirect
181. C4⋊C4C2×C3×D7semidirect
225. C7C3×#418semidirect
248. C3#89542direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×D4×D72
C2×C3×#11312
C3×#64552
C3×C7×#1022
C3×#64062
C3×#63952
C3×#63832
#895423
C3×#4187