Finite Group Databasebeta

← #13054484

Subgroups — #13054484

Order 1344.

380 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 96 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
2#10620920C211yesmaximal, characteristic, complemented
3C2×#66557C211yesmaximal, characteristic, complemented
4#10620671C211yesmaximal, characteristic, complemented
5C7×#832C211yesmaximal, characteristic, complemented
6#10620500C211yesmaximal, characteristic, complemented
7#10620499C211yesmaximal, characteristic, complemented
8#89542—31nomaximal, 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
16C4⋊Dic21C2211yescharacteristic, complemented
17C2×C3⋊(C7⋊C8)C2211yescharacteristic, complemented
18C7×C4⋊Dic3C2211yescharacteristic, complemented
19C2×C7×C3⋊C8C2211yescharacteristic, complemented
20(C2×D7)⋊Dic3—21no—
21Dic3×Dic7—21no—
22#66557—22no—
23C2×D4×D7D311yescharacteristic
24C2×#1131—31no—
25#6455—31no—
26C7×#102—31no—
27#6406—31no—
28#6395—31no—
29#6383—31no—
30#5215—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×C7C2311yescharacteristic
40C2×Dic21—21no—
41C2×C7×Dic3—21no—
42C3×C22⋊D7—24no—
43C2×C3×Dic7D411yescharacteristic
44C3×D28—21no—
45C3×D28C2×C412yes—
46C3×C4×D7C2×C412yes—
47C3×C4×D7D412yes—
48C3⋊(C7⋊C8)—21no—
49C7×C3⋊C8—21no—
50C23×D7—21no—
51C2×C7×D4C2×D311yescharacteristic
52C2×C22⋊D7—21no—
53D4×D7—22no—
54D4×D7Dic314yescomplemented
55C2×D28C2×D311yescharacteristic
56C2×C4×D7C2×D311yescharacteristic
57C2×C7×C8—31no—
58C7×C4⋊C4—31no—
59(C2×D7)⋊C4—61no—
60C4⋊Dic7—31no—
61C4×Dic7—61no—
62C2×C7⋊C8—31no—
63#1131—62no—
64C22×C3×D4—71no—
65#926—71no—
66C2×#208—71no—
67#832—71no—
68#832—71no—
69#832—71no—
70#832D711yescharacteristic, 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×C7C22×C411yescharacteristic
83C3×C4×C7C2×D411yescharacteristic
84Dic21—41no—
85C3×Dic7C22⋊C412yes—
86C7×Dic3—41no—
87#418—211noSylow
88C23×C7—21no—
89C22×D7—21no—
90C22×D7—24no—
91C22×D7—42no—
92C22×D7—42no—
93C22×D7#6711yescharacteristic
94C7×D4—21no—
95C7×D4C2×Dic312yes—
96C2×C4×C7—61no—
97C2×C4×C7C22×D311yescharacteristic
98C22⋊D7—24no—
99C2×Dic7—61no—
100C2×Dic7#6711yescharacteristic
101D28—21no—
102D28C2×Dic312yes—
103C4×D7C2×Dic312yes—
104C4×D7#6712yes—
105C7×C8—61no—
106C7⋊C8—61no—
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—
114#217—141no—
115C4⋊Dic3—71no—
116C4⋊Dic3C2×D711yescharacteristic, complemented
117C4×Dic3—141no—
118#208—142no—
119C2×C3⋊C8—71no—
120C2×C3⋊C8C2×D711yescharacteristic, complemented
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—211no—
129#117—211no—
130#102—31no—
131#102—211no—
132#102—211no—
133#102—211no—
134C22×C7—22no—
135C22×C7—41no—
136C22×C7—41no—
137C22×C7C2×#6711yescharacteristic
138C2×D7—21no—
139C2×D7—21no—
140C2×D7—22no—
141C2×D7—48no—
142C2×D7—41no—
143C2×D7—41no—
144C2×D7#21712yes—
145C4×C7—121no—
146C4×C7C22×Dic311yescharacteristic
147C4×C7C2×#6711yescharacteristic
148Dic7—121no—
149Dic7#21712yes—
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—71no—
162C2×C3×C4—72no—
163C2×C3×C4—72no—
164C2×C3×C4C22×D711yescharacteristic
165C2×Dic3—21no—
166C2×Dic3—141no—
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×D4D3×D711yescharacteristic
176C22×C4—71no—
177M16—422no—
178C2×C8—31no—
179C2×C8—211no—
180C4⋊C4—31no—
181C4⋊C4—211no—
182C22⋊C4—421no—
183C42—421no—
184C2×C7—42no—
185C2×C7#93211yescharacteristic
186C2×C7#93211yescharacteristic
187C2×C7C2×#21711yescharacteristic
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—72no—
202C3×C4C2×C4×D711yescharacteristic
203C3×C4D4×D711yescharacteristic
204Dic3—41no—
205Dic3—281no—
206C23—21no—
207C23—71no—
208C23—144no—
209C23—141no—
210C23—282no—
211C23—282no—
212D4—21no—
213D4—72no—
214D4—144no—
215D4—141no—
216D4Dic3×D712yes—
217C2×C4—61no—
218C2×C4—71no—
219C2×C4—72no—
220C2×C4—72no—
221C2×C4—421no—
222C2×C4C2×D3×D711yescharacteristic
223C8—61no—
224C8—421no—
225C7#521511yescharacteristic, 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—
242C22D7×#6711yescharacteristic
243C4—72no—
244C4—121no—
245C4—841no—
246C4C2×Dic3×D711yescharacteristic
247C4D7×#6711yescharacteristic
248C3#8954211yescharacteristic, complemented, Sylow
249C2—42no—
250C2—142no—
251C2—282no—
252C2#1062105111yescharacteristic
253C2#1062105311yescharacteristic
254C2D7×#21711yescharacteristic
subgroupG/H
centerC22D7×#67
commutatorC3×C4×C7C22×C4
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×D4C22
socleC22×C3×C7C2×D4
prime radicalC2×C3×D4×D7C2
cosocleC2×C3×C4×C7C23
hypercenterC2×D4D3×D7
hypocenterC3×C7#418
LCSG ▷ C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. #10620920C2semidirect
3. C2×#66557C2semidirect
4. #10620671C2semidirect
5. C7×#832C2semidirect
6. #10620500C2semidirect
7. #10620499C2semidirect
13. C3×D4×D7C4semidirect
16. C4⋊Dic21C22semidirect
17. C2×C3⋊(C7⋊C8)C22semidirect
18. C7×C4⋊Dic3C22semidirect
19. C2×C7×C3⋊C8C22semidirect
54. D4×D7Dic3semidirect
70. #832D7semidirect
116. C4⋊Dic3C2×D7semidirect
120. C2×C3⋊C8C2×D7semidirect
169. C3×C7#418semidirect
225. C7#5215semidirect
248. C3#89542semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D4×D72
#106209202
C2×#665572
#106206712
C7×#8322
#106205002
#106204992
#895423
#52157