Finite Group Databasebeta

← #13052885

Subgroups — #13052885

Order 1344.

316 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 78 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×#66670C211yesmaximal, characteristic
2#10621375C211yesmaximal, characteristic, complemented
3C2×D21⋊C8C211yesmaximal, characteristic, complemented
4C7×#810C211yesmaximal, characteristic, complemented
5C3×#6383C211yesmaximal, characteristic, complemented
6#10620508C211yesmaximal, characteristic, complemented
7#10620500C211yesmaximal, characteristic, complemented
8#89576—31nomaximal, Hall
9C2×C4×D21C2211yescharacteristic
10C2×C7×Dic6C2211yescharacteristic
11C2×C3×D28C2211yescharacteristic
12C2×#4246—21no—
13C2×Dic3×D7—21no—
14#66670—22no—
15#66670C414yescomplemented
16C3×C7×C4⋊C4C2211yescharacteristic
17#66631—21no—
18C4⋊Dic21C2211yescharacteristic
19C4×Dic21—21no—
20C2×C7×C3⋊C8C2211yescharacteristic
21C2×C3×C7⋊C8C2211yescharacteristic
22D21⋊C8—22no—
23C2×#1161—31no—
24C2×C8×D7—31no—
25#6455—31no—
26C7×#103—31no—
27#6409—31no—
28#6395—31no—
29#6383D311yescharacteristic, complemented
30#4772—71nomaximal, Hall
31C22×D21D411yescharacteristic
32C22×C3×D7—21no—
33C2×C3×C4×C7—21no—
34C2×C3×C4×C7C2311yescharacteristic
35C2×Dic21—21no—
36C2×Dic21D411yescharacteristic
37C4×D21C2×C412yes—
38C4×D21D412yes—
39C2×C7×Dic3—21no—
40C7×Dic6—21no—
41C7×Dic6C2×C412yes—
42C3×D28—21no—
43C3×D28C2×C412yes—
44#4246—24no—
45Dic3×D7—24no—
46C3×C7⋊C8—21no—
47C7×C3⋊C8—21no—
48C2×C7×Q8—31no—
49#1161—34no—
50#1161—62no—
51C2×D28—61no—
52C2×D28C2×D311yescharacteristic
53C2×C4×D7—31no—
54C2×C4×D7—61no—
55C2×C7×C8—31no—
56C7×C4⋊C4C2×D311yescharacteristic
57(C2×D7)⋊C4—61no—
58C4⋊Dic7—31no—
59C4×Dic7—61no—
60C2×C7⋊C8C2×D311yescharacteristic
61C8×D7—62no—
62C2×#237—71no—
63C2×D3×C8—71no—
64#892—71no—
65C3×#102—71no—
66#832—71no—
67#816—71no—
68#810D711yescharacteristic, complemented
69C22×C3×C7C2×D411yescharacteristic
70C2×D21—21no—
71C2×D21—21no—
72C2×D21C22⋊C412yes—
73C2×C3×D7—22no—
74C2×C3×D7—41no—
75C2×C3×D7—41no—
76C3×C4×C7—41no—
77C3×C4×C7C22×C411yescharacteristic
78C3×C4×C7C2×D411yescharacteristic
79Dic21—41no—
80Dic21C22⋊C412yes—
81C7×Dic3—22no—
82#416—211noSylow
83C22×D7—21no—
84C22×D7—31no—
85C7×Q8—32no—
86C7×Q8—61no—
87C2×C4×C7—21no—
88C2×C4×C7—61no—
89C2×C4×C7C22×D311yescharacteristic
90C2×Dic7—31no—
91C2×Dic7—61no—
92D28—21no—
93D28—64no—
94D28C4×D312yes—
95C4×D7—32no—
96C4×D7—32no—
97C4×D7—64no—
98C7×C8—61no—
99C7⋊C8—21no—
100C2×C3×D4—71no—
101C2×#67—141no—
102C22×Dic3—141no—
103#237—74no—
104#237—142no—
105C2×C4×D3—71no—
106C2×Dic6C2×D711yescharacteristic
107C2×C3×C8—71no—
108C3×C4⋊C4C2×D711yescharacteristic
109(C2×D3)⋊C4—141no—
110C4⋊Dic3—71no—
111C4×Dic3—141no—
112C2×C3⋊C8C2×D711yescharacteristic
113D3×C8—142no—
114C2×C3×C7C2×C22⋊C411yescharacteristic
115C2×C3×C7#13511yescharacteristic
116C2×C3×C7#13511yescharacteristic
117D21—22no—
118C3×D7—42no—
119C2×#41—211no—
120C22×C8—211no—
121#117—211no—
122#103—31no—
123#103—211no—
124#102—71no—
125#102—211no—
126C22×C7D3×D411yescharacteristic
127C2×D7—22no—
128C2×D7—32no—
129C2×D7—41no—
130C2×D7—41no—
131C2×D7—61no—
132C2×D7—61no—
133C4×C7—41no—
134C4×C7—62no—
135C4×C7C2×C4×D311yescharacteristic
136C4×C7D3×D411yescharacteristic
137Dic7—32no—
138Dic7—121no—
139C23×C3—141no—
140C22×D3—71no—
141C3×D4—72no—
142C3×D4—141no—
143C2×C3×C4—21no—
144C2×C3×C4C22×D711yescharacteristic
145#67—144no—
146C2×Dic3—21no—
147C2×Dic3—71no—
148C2×Dic3—144no—
149C2×Dic3—141no—
150C4×D3—72no—
151C4×D3—72no—
152Dic6—21no—
153Dic6C4×D712yes—
154C3×C8—141no—
155C3⋊C8—21no—
156C3×C7#41611yescharacteristic, complemented, Hall
157#41—214no—
158#41—422no—
159C2×Q8—31no—
160C2×D4—71no—
161C2×D4—421no—
162C22×C4—211no—
163C22×C4—421no—
164C2×C8—31no—
165C2×C8—71no—
166C2×C8—422no—
167C4⋊C4—211no—
168C4⋊C4D3×D711yescharacteristic
169C22⋊C4—421no—
170C42—421no—
171C2×C7D3×C22⋊C411yescharacteristic
172C2×C7#91211yescharacteristic
173C2×C7#91611yescharacteristic
174D7—42no—
175D7—62no—
176C22×C3—142no—
177C22×C3—281no—
178C22×C3—281no—
179C22×C3D4×D711yescharacteristic
180C2×D3—72no—
181C2×D3—141no—
182C2×D3—141no—
183C3×C4—41no—
184C3×C4C2×C4×D711yescharacteristic
185C3×C4D4×D711yescharacteristic
186Dic3—22no—
187Dic3—72no—
188Dic3—281no—
189C23—141no—
190C23—211no—
191Q8—32no—
192Q8—61no—
193D4—72no—
194D4—141no—
195D4—424no—
196C2×C4—21no—
197C2×C4—61no—
198C2×C4—211no—
199C2×C4—212no—
200C2×C4—212no—
201C2×C4—424no—
202C2×C4—421no—
203C2×C4C2×D3×D711yescharacteristic
204C8—61no—
205C8—141no—
206C7#477211yescharacteristic, complemented, Sylow
207C2×C3—282no—
208C2×C3D7×C22⋊C411yescharacteristic
209C2×C3#647911yescharacteristic
210C2×C3#648211yescharacteristic
211D3—142no—
212C22—142no—
213C22—212no—
214C22—281no—
215C22—281no—
216C22—421no—
217C22—421no—
218C22#6669411yescharacteristic
219C4—41no—
220C4—62no—
221C4—212no—
222C4—841no—
223C4C4×D3×D711yescharacteristic
224C4#6668111yescharacteristic
225C3#8957611yescharacteristic, complemented, Sylow
226C2—282no—
227C2—422no—
228C2#1062103511yescharacteristic
229C2#1062105911yescharacteristic
230C2#1062108311yescharacteristic
subgroupG/H
centerC22#66694
commutatorC3×C4×C7C22×C4
FrattiniC2×C4C2×D3×D7
FittingC3×C7×C4⋊C4C22
socleC22×C3×C7C2×D4
prime radicalC2×#66670C2
cosocleC2×C3×C4×C7C23
hypercenterC4⋊C4D3×D7
hypocenterC3×C7#416
LCSG ▷ C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #10621375C2semidirect
3. C2×D21⋊C8C2semidirect
4. C7×#810C2semidirect
5. C3×#6383C2semidirect
6. #10620508C2semidirect
7. #10620500C2semidirect
15. #66670C4semidirect
29. #6383D3semidirect
68. #810D7semidirect
156. C3×C7#416semidirect
206. C7#4772semidirect
225. C3#89576semidirect

Maximal subgroups

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

subgroupindex
C2×#666702
#106213752
C2×D21⋊C82
C7×#8102
C3×#63832
#106205082
#106205002
#895763
#47727