Finite Group Databasebeta

← #13054184

Subgroups — #13054184

Order 1344.

324 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 72 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, complemented
2C7×#930C211yesmaximal, characteristic, complemented
3C2×#66564C211yesmaximal, characteristic, complemented
4C2×#66560C211yesmaximal, characteristic, complemented
5#10620669C211yesmaximal, characteristic, complemented
6#10620505C211yesmaximal, characteristic, complemented
7#10620499C211yesmaximal, characteristic, complemented
8#89809—31nomaximal, Hall
9C22×C3×C4×C7C2211yescharacteristic
10C2×C7×#67—21no—
11C2×C7×D12C2211yescharacteristic
12C2×C3×C22⋊D7—21no—
13C3×#1157—22no—
14C3×#1157—22no—
15C2×C3×D28C2211yescharacteristic
16C2×C3×C4×D7—21no—
17C2×C3×Dic14C2211yescharacteristic
18C2×C3⋊(C7⋊C8)C2211yescharacteristic
19C2×C3⋊(C7⋊C8)C2211yescharacteristic
20C7×(C2×D3)⋊C4—21no—
21C7×C4⋊Dic3C2211yescharacteristic
22#66564—22no—
23#66560—22no—
24C2×#1157D311yescharacteristic, complemented
25C7×#121—31no—
26C2×#1142—31no—
27C2×#1141—31no—
28C22⋊(C7⋊C8)—31no—
29#6384—31no—
30#6383—31no—
31#5238—71nomaximal, Hall
32C23×C3×C7D411yescharacteristic
33C22×D3×C7—21no—
34C22×C3×D7—21no—
35C2×C3×C4×C7—21no—
36C2×C3×C4×C7—21no—
37C2×C3×C4×C7D411yescharacteristic
38C2×C3×C4×C7C2311yescharacteristic
39C7×#67—42no—
40C2×C7×Dic3—21no—
41C7×D12—21no—
42C7×D12—21no—
43C3×C22⋊D7—24no—
44C2×C3×Dic7—21no—
45C3×D28—21no—
46C3×D28D412yescomplemented
47C3×C4×D7—24no—
48C3×Dic14—21no—
49C3×Dic14D412yescomplemented
50C3⋊(C7⋊C8)—21no—
51C3⋊(C7⋊C8)—21no—
52C2×C7×D4—31no—
53C2×C7×D4—61no—
54C22×C4×C7C2×D311yescharacteristic
55C2×C22⋊D7—21no—
56#1157—22no—
57#1157—22no—
58C2×D28C2×D311yescharacteristic
59C2×C4×D7—21no—
60C2×Dic14C2×D311yescharacteristic
61C7×C4⋊C4—31no—
62C7×C22⋊C4—61no—
63#1142—62no—
64#1141—62no—
65C2×C7⋊C8—31no—
66C2×C7⋊C8—31no—
67C2×C3×#41—71no—
68C2×#215—71no—
69C2×#213—71no—
70#930D711yescharacteristic, complemented
71#835—71no—
72#832—71no—
73#830—71no—
74C22×C3×C7—21no—
75C22×C3×C7—21no—
76C22×C3×C7—21no—
77C22×C3×C7C2×D411yescharacteristic
78C2×D3×C7—41no—
79C2×D3×C7—41no—
80C2×D3×C7—41no—
81C2×C3×D7—22no—
82C2×C3×D7—41no—
83C2×C3×D7—41no—
84C3×C4×C7—21no—
85C3×C4×C7C2×D411yescharacteristic
86C3×C4×C7C2×D411yescharacteristic
87C3×Dic7—22no—
88C7×Dic3—41no—
89#449—211noSylow
90C23×C7—61no—
91C23×C7#6711yescharacteristic
92C22×D7—21no—
93C7×D4—61no—
94C7×D4—61no—
95C7×D4—122no—
96C2×C4×C7—21no—
97C2×C4×C7—21no—
98C2×C4×C7—61no—
99C2×C4×C7#6711yescharacteristic
100C2×C4×C7C22×D311yescharacteristic
101C22⋊D7—24no—
102C2×Dic7—21no—
103D28—21no—
104D28#6712yescomplemented
105C4×D7—24no—
106Dic14—21no—
107Dic14#6712yescomplemented
108C7⋊C8—61no—
109C7⋊C8—61no—
110C3×#41—142no—
111C3×#41—142no—
112C2×C3×Q8—71no—
113C2×C3×D4—71no—
114C2×C3×D4—141no—
115C22×C3×C4—141no—
116C22×C3×C4C2×D711yescharacteristic
117C2×#67—21no—
118C2×D12C2×D711yescharacteristic
119#215—142no—
120#213—142no—
121(C2×D3)⋊C4—21no—
122C4⋊Dic3C2×D711yescharacteristic
123C2×C3⋊C8—71no—
124C2×C3⋊C8—71no—
125C2×C3×C7—41no—
126C2×C3×C7#12011yescharacteristic
127C2×C3×C7#13511yescharacteristic
128C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
129C3×D7—42no—
130D3×C7—81no—
131C2×#41—71no—
132C2×SD16—211no—
133C2×D8—211no—
134#121—31no—
135#103—211no—
136#102—211no—
137C22⋊C8—211no—
138C22×C7—21no—
139C22×C7—21no—
140C22×C7—21no—
141C22×C7—121no—
142C22×C7—121no—
143C22×C7—121no—
144C22×C7C2×#6711yescharacteristic
145C2×D7—22no—
146C2×D7—41no—
147C2×D7—41no—
148C4×C7—21no—
149C4×C7—121no—
150C4×C7C2×#6711yescharacteristic
151C4×C7C2×#6711yescharacteristic
152Dic7—22no—
153C23×C3—141no—
154C23×C3C22⋊D711yescharacteristic
155C22×D3—21no—
156C3×Q8—72no—
157C3×Q8—141no—
158C3×D4—72no—
159C3×D4—144no—
160C3×D4—141no—
161C2×C3×C4—21no—
162C2×C3×C4—21no—
163C2×C3×C4—144no—
164C2×C3×C4—141no—
165C2×C3×C4C22⋊D711yescharacteristic
166C2×C3×C4C22×D711yescharacteristic
167#67—42no—
168C2×Dic3—21no—
169D12—21no—
170D12—21no—
171C3⋊C8—141no—
172C3⋊C8—141no—
173C3×C7#44911yescharacteristic, complemented, Hall
174#41—142no—
175#41—142no—
176C2×Q8—71no—
177C2×D4—31no—
178C2×D4—61no—
179C2×D4—71no—
180C2×D4—141no—
181C22×C4—141no—
182C22×C4D3×D711yescharacteristic
183SD16—422no—
184D8—422no—
185C2×C8—211no—
186C2×C8—211no—
187C4⋊C4—31no—
188C22⋊C4—61no—
189C2×C7—41no—
190C2×C7—241no—
191C2×C7#94911yescharacteristic
192C2×C7#95011yescharacteristic
193C2×C7#95311yescharacteristic
194D7—42no—
195C22×C3—21no—
196C22×C3—21no—
197C22×C3—21no—
198C22×C3—142no—
199C22×C3—281no—
200C22×C3—281no—
201C22×C3C2×C22⋊D711yescharacteristic
202C2×D3—41no—
203C2×D3—41no—
204C2×D3—41no—
205C3×C4—21no—
206C3×C4—142no—
207C3×C4D4×D711yescharacteristic
208C3×C4D4×D711yescharacteristic
209Dic3—41no—
210C23—61no—
211C23—141no—
212C23#424511yescharacteristic
213Q8—72no—
214Q8—141no—
215D4—61no—
216D4—61no—
217D4—72no—
218D4—122no—
219D4—144no—
220D4—141no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—61no—
224C2×C4—144no—
225C2×C4—141no—
226C2×C4#424511yescharacteristic
227C2×C4C2×D3×D711yescharacteristic
228C8—421no—
229C8—421no—
230C7#523811yescharacteristic, complemented, Sylow
231C2×C3—41no—
232C2×C3—282no—
233C2×C3#649511yescharacteristic
234C2×C3#650011yescharacteristic
235C2×C3#651311yescharacteristic
236D3—81no—
237C22—21no—
238C22—21no—
239C22—21no—
240C22—121no—
241C22—121no—
242C22—121no—
243C22—142no—
244C22—281no—
245C22—281no—
246C22C2×#424511yescharacteristic
247C4—21no—
248C4—121no—
249C4—142no—
250C4D7×#6711yescharacteristic
251C4D7×#6711yescharacteristic
252C3#8980911yescharacteristic, complemented, Sylow
253C2—41no—
254C2—241no—
255C2—282no—
256C2#1062085911yescharacteristic
257C2#1062086111yescharacteristic
258C2#1062112511yescharacteristic
subgroupG/H
centerC22C2×#4245
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC22×C3×C4×C7C22
socleC22×C3×C7C2×D4
hypercenterC22×C4D3×D7
hypocenterC3×C7#449
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#1157C2semidirect
2. C7×#930C2semidirect
3. C2×#66564C2semidirect
4. C2×#66560C2semidirect
5. #10620669C2semidirect
6. #10620505C2semidirect
7. #10620499C2semidirect
24. C2×#1157D3semidirect
46. C3×D28D4semidirect
49. C3×Dic14D4semidirect
70. #930D7semidirect
104. D28#67semidirect
107. Dic14#67semidirect
173. C3×C7#449semidirect
230. C7#5238semidirect
252. C3#89809semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#11572
C7×#9302
C2×#665642
C2×#665602
#106206692
#106205052
#106204992
#898093
#52387