Finite Group Databasebeta

← #13054628

Subgroups — #13054628

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×#1161C211yesmaximal, characteristic, complemented
2C2×C7×#215C211yesmaximal, characteristic, complemented
3#10621011C211yesmaximal, characteristic, complemented
4C2×#66560C211yesmaximal, characteristic, complemented
5#10620674C211yesmaximal, characteristic, complemented
6#10620500C211yesmaximal, characteristic, complemented
7(C2×D7)⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8#89944—31nomaximal, Hall
9C2×C3×C7×Q8C2211yescharacteristic
10C2×C7×D12C2211yescharacteristic
11C3×#1161—22no—
12C3×#1161—22no—
13C2×C3×D28—21no—
14C2×C3×D28C2211yescharacteristic
15C2×C3×C4×D7—21no—
16C2×C3×C4×D7C2211yescharacteristic
17C2×#4245—21no—
18C4⋊Dic21C2211yescharacteristic
19C2×C3⋊(C7⋊C8)C2211yescharacteristic
20C7×#215—22no—
21C2×C7×C3⋊C8C2211yescharacteristic
22(C2×D3)⋊Dic7—21no—
23#66560—22no—
24C2×#1161D311yescharacteristic, complemented
25C2×C7×SD16—31no—
26#6501—31no—
27C2×#1141—31no—
28#6409—31no—
29#6395—31no—
30(C2×D7)⋊C8—31no—
31#5238—71nomaximal, Hall
32C22×D3×C7—21no—
33C22×C3×D7—21no—
34C22×C3×D7D411yescharacteristic
35C3×C7×Q8—21no—
36C3×C7×Q8D412yescomplemented
37C2×C3×C4×C7—21no—
38C2×C3×C4×C7C2311yescharacteristic
39C2×Dic21—21no—
40C7×D12—21no—
41C7×D12—21no—
42C2×C3×Dic7D411yescharacteristic
43C3×D28—24no—
44C3×D28—21no—
45C3×D28D412yescomplemented
46C3×C4×D7—24no—
47C3×C4×D7—21no—
48C3×C4×D7—21no—
49#4245—42no—
50C3⋊(C7⋊C8)—21no—
51C7×C3⋊C8—21no—
52C2×C7×Q8C2×D311yescharacteristic
53C2×C7×D4—31no—
54C2×C22⋊D7—61no—
55#1161—22no—
56#1161—22no—
57C2×D28—21no—
58C2×D28C2×D311yescharacteristic
59C2×C4×D7—21no—
60C2×C4×D7C2×D311yescharacteristic
61C7×SD16—62no—
62C2×C7×C8—31no—
63C22⋊Dic7—61no—
64#1141—62no—
65C4⋊Dic7—31no—
66C2×C7⋊C8—31no—
67C2×C3×#41—71no—
68C2×#215D711yescharacteristic, complemented
69C2×#213—71no—
70#930—71no—
71#835—71no—
72#832—71no—
73#830—71no—
74C22×C3×C7C2×D411yescharacteristic
75C2×D3×C7—41no—
76C2×D3×C7—41no—
77C2×D3×C7—41no—
78C2×C3×D7—21no—
79C2×C3×D7—22no—
80C2×C3×D7—21no—
81C2×C3×D7—21no—
82C2×C3×D7—41no—
83C2×C3×D7—41no—
84C3×C4×C7—22no—
85C3×C4×C7C2×D411yescharacteristic
86C3×C4×C7C2×D411yescharacteristic
87Dic21—41no—
88C3×Dic7—21no—
89#449—211noSylow
90C23×C7—61no—
91C22×D7—21no—
92C22×D7#6711yescharacteristic
93C7×Q8—21no—
94C7×Q8#6712yescomplemented
95C7×D4—61no—
96C7×D4—61no—
97C2×C4×C7—21no—
98C2×C4×C7C22×D311yescharacteristic
99C22⋊D7—122no—
100C2×Dic7—61no—
101C2×Dic7#6711yescharacteristic
102D28—24no—
103D28—21no—
104D28#6712yescomplemented
105C4×D7—24no—
106C4×D7—21no—
107C4×D7—21no—
108C7×C8—61no—
109C7⋊C8—61no—
110C3×#41—142no—
111C3×#41—142no—
112C2×C3×Q8C2×D711yescharacteristic
113C2×C3×D4—71no—
114C2×C3×D4—141no—
115C22×C3×C4—71no—
116C22×C3×C4—141no—
117C2×#67—141no—
118C2×D12C2×D711yescharacteristic
119#215—22no—
120#213—142no—
121(C2×D3)⋊C4—141no—
122C4⋊Dic3—71no—
123C2×C3⋊C8—71no—
124C2×C3⋊C8C2×D711yescharacteristic
125C2×C3×C7#12011yescharacteristic
126C2×C3×C7#13511yescharacteristic
127C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
128C3×D7—41no—
129C3×D7—42no—
130D3×C7—81no—
131C2×#41—71no—
132C2×SD16—31no—
133C2×D8—211no—
134#121—211no—
135#103—211no—
136#102—211no—
137C22⋊C8—211no—
138C22×C7—121no—
139C22×C7—121no—
140C22×C7—121no—
141C22×C7C2×#6711yescharacteristic
142C2×D7—21no—
143C2×D7—21no—
144C2×D7—21no—
145C2×D7—22no—
146C2×D7—41no—
147C2×D7—41no—
148C4×C7—22no—
149C4×C7C2×#6711yescharacteristic
150C4×C7C2×#6711yescharacteristic
151Dic7—21no—
152Dic7—121no—
153C23×C3—71no—
154C23×C3—141no—
155C22×D3—21no—
156C3×Q8—21no—
157C3×Q8C22⋊D712yescomplemented
158C3×D4—72no—
159C3×D4—144no—
160C3×D4—141no—
161C2×C3×C4—21no—
162C2×C3×C4—71no—
163C2×C3×C4—144no—
164C2×C3×C4—141no—
165C2×C3×C4—141no—
166C2×C3×C4C22×D711yescharacteristic
167#67—282no—
168C2×Dic3—141no—
169D12—21no—
170D12—21no—
171C3⋊C8—21no—
172C3⋊C8—141no—
173C3×C7#44911yescharacteristic, complemented, Hall
174#41—142no—
175#41—142no—
176C2×Q8D3×D711yescharacteristic
177C2×D4—31no—
178C2×D4—71no—
179C2×D4—141no—
180C2×D4—421no—
181C22×C4—71no—
182C22×C4—141no—
183SD16—62no—
184D8—422no—
185C2×C8—31no—
186C2×C8—211no—
187C4⋊C4—211no—
188C22⋊C4—421no—
189C2×C7—241no—
190C2×C7#94911yescharacteristic
191C2×C7#95011yescharacteristic
192C2×C7#95311yescharacteristic
193D7—41no—
194D7—42no—
195C22×C3—141no—
196C22×C3—141no—
197C22×C3—142no—
198C22×C3—141no—
199C22×C3—281no—
200C22×C3—281no—
201C22×C3D4×D711yescharacteristic
202C2×D3—41no—
203C2×D3—41no—
204C2×D3—41no—
205C3×C4—22no—
206C3×C4—141no—
207C3×C4D4×D711yescharacteristic
208C3×C4C2×C22⋊D711yescharacteristic
209Dic3—281no—
210C23—61no—
211C23—71no—
212C23—141no—
213Q8—21no—
214Q8#424512yescomplemented
215D4—61no—
216D4—61no—
217D4—72no—
218D4—144no—
219D4—141no—
220D4—842no—
221C2×C4—21no—
222C2×C4—71no—
223C2×C4—144no—
224C2×C4—141no—
225C2×C4—141no—
226C2×C4—421no—
227C2×C4C2×D3×D711yescharacteristic
228C8—61no—
229C8—421no—
230C7#523811yescharacteristic, complemented, Sylow
231C2×C3—281no—
232C2×C3—282no—
233C2×C3#647711yescharacteristic
234C2×C3#647911yescharacteristic
235C2×C3#650011yescharacteristic
236D3—81no—
237C22—121no—
238C22—121no—
239C22—121no—
240C22—141no—
241C22—141no—
242C22—142no—
243C22—141no—
244C22—281no—
245C22—281no—
246C22D7×#6711yescharacteristic
247C4—22no—
248C4—141no—
249C4—841no—
250C4C2×#424511yescharacteristic
251C4D7×#6711yescharacteristic
252C3#8994411yescharacteristic, complemented, Sylow
253C2—241no—
254C2—281no—
255C2—282no—
256C2#1062106411yescharacteristic
257C2#1062107611yescharacteristic
258C2#1062112511yescharacteristic
subgroupG/H
centerC22D7×#67
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×Q8C22
socleC22×C3×C7C2×D4
hypercenterC2×Q8D3×D7
hypocenterC3×C7#449
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×#1161C2semidirect
2. C2×C7×#215C2semidirect
3. #10621011C2semidirect
4. C2×#66560C2semidirect
5. #10620674C2semidirect
6. #10620500C2semidirect
7. (C2×D7)⋊(C3⋊C8)C2semidirect
24. C2×#1161D3semidirect
36. C3×C7×Q8D4semidirect
45. C3×D28D4semidirect
68. C2×#215D7semidirect
94. C7×Q8#67semidirect
104. D28#67semidirect
157. C3×Q8C22⋊D7semidirect
173. C3×C7#449semidirect
214. Q8#4245semidirect
230. C7#5238semidirect
252. C3#89944semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#11612
C2×C7×#2152
#106210112
C2×#665602
#106206742
#106205002
(C2×D7)⋊(C3⋊C8)2
#899443
#52387