Finite Group Databasebeta

← #13054391

Subgroups — #13054391

Order 1344.

268 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 68 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×C7×#213C211yesmaximal, characteristic, complemented
2C3×#6501C211yesmaximal, characteristic, complemented
3#10621011C211yesmaximal, characteristic, complemented
4C2×D7×C3⋊C8C211yesmaximal, characteristic, complemented
5#10620671C211yesmaximal, characteristic, complemented
6#10620525C211yesmaximal, characteristic, complemented
7#10620501C211yesmaximal, characteristic, complemented
8#89929—31nomaximal, Hall
9C2×C3×C7×D4C2211yescharacteristic
10C2×C7×D12C2211yescharacteristic
11C2×C3×C22⋊D7—21no—
12C2×C3×C4×D7C2211yescharacteristic
13C2×#4245—21no—
14C4⋊Dic21C2211yescharacteristic
15C2×C3⋊(C7⋊C8)C2211yescharacteristic
16C7×#213—22no—
17C2×C7×C3⋊C8C2211yescharacteristic
18C3×C22⋊Dic7—21no—
19C3×C4⋊Dic7C2211yescharacteristic
20(C2×D3)⋊Dic7—21no—
21D7×C3⋊C8—22no—
22C2×C7×D8—31no—
23#6501—31no—
24#6501D311yescharacteristic, complemented
25C2×C8×D7—31no—
26#6406—31no—
27#6406—31no—
28#6392—31no—
29#5038—71nomaximal, Hall
30C23×C3×C7—21no—
31C22×D3×C7—21no—
32C22×C3×D7D411yescharacteristic
33C3×C7×D4—21no—
34C3×C7×D4—21no—
35C2×C3×C4×C7C2311yescharacteristic
36C2×Dic21—21no—
37C7×D12—21no—
38C7×D12—21no—
39C3×C22⋊D7—42no—
40C2×C3×Dic7—21no—
41C2×C3×Dic7D411yescharacteristic
42C3×C4×D7—21no—
43C3×C4×D7—21no—
44#4245—42no—
45C3⋊(C7⋊C8)—21no—
46C7×C3⋊C8D412yescomplemented
47C2×C7×D4—31no—
48C2×C7×D4C2×D311yescharacteristic
49C2×C22⋊D7—21no—
50C2×C22⋊D7—61no—
51C2×C4×D7C2×D311yescharacteristic
52C7×D8—62no—
53C2×C7×C8—31no—
54C22⋊Dic7—21no—
55C22⋊Dic7—61no—
56C4⋊Dic7—31no—
57C4⋊Dic7C2×D311yescharacteristic
58C2×C7⋊C8—31no—
59C8×D7—62no—
60C3×#121—71no—
61C2×#213D711yescharacteristic, complemented
62#930—71no—
63C22×C3⋊C8—71no—
64#832—71no—
65#809—71no—
66#807—71no—
67C22×C3×C7—41no—
68C22×C3×C7—41no—
69C22×C3×C7—41no—
70C22×C3×C7C2×D411yescharacteristic
71C2×D3×C7—41no—
72C2×D3×C7—41no—
73C2×D3×C7—41no—
74C2×C3×D7—21no—
75C2×C3×D7—21no—
76C2×C3×D7D812yescomplemented
77C3×C4×C7C2×D411yescharacteristic
78C3×C4×C7#4111yescharacteristic
79Dic21—41no—
80C3×Dic7—21no—
81C3×Dic7—41no—
82#466—211noSylow
83C23×C7—21no—
84C23×C7—61no—
85C22×D7#6711yescharacteristic
86C7×D4—21no—
87C7×D4—21no—
88C7×D4—61no—
89C7×D4—61no—
90C2×C4×C7C22×D311yescharacteristic
91C22⋊D7—42no—
92C22⋊D7—122no—
93C2×Dic7—21no—
94C2×Dic7—61no—
95C2×Dic7#6711yescharacteristic
96C4×D7—21no—
97C4×D7—21no—
98C7×C8—32no—
99C7⋊C8—61no—
100C2×C3×D4—141no—
101C2×C3×D4C2×D711yescharacteristic
102C22×C3×C4—71no—
103C2×#67—141no—
104C2×D12C2×D711yescharacteristic
105C3×C4⋊C4—71no—
106C3×C22⋊C4—141no—
107#213—22no—
108(C2×D3)⋊C4—141no—
109C4⋊Dic3—71no—
110C2×C3⋊C8—71no—
111C2×C3⋊C8—142no—
112C2×C3⋊C8C2×D711yescharacteristic
113C2×C3×C7—81no—
114C2×C3×C7#12111yescharacteristic
115C2×C3×C7C2×D811yescharacteristic
116C2×C3×C7#13511yescharacteristic
117C3×D7—22no—
118D3×C7—81no—
119C2×D8—31no—
120C22×C8—211no—
121#121—71no—
122#121—211no—
123#107—211no—
124#102—211no—
125#102—211no—
126C22×C7—41no—
127C22×C7—41no—
128C22×C7—41no—
129C22×C7—121no—
130C22×C7—121no—
131C22×C7—121no—
132C22×C7C2×#6711yescharacteristic
133C2×D7—21no—
134C2×D7—21no—
135C2×D7#21312yescomplemented
136C4×C7D3×D411yescharacteristic
137C4×C7#23711yescharacteristic
138Dic7—21no—
139Dic7—41no—
140Dic7—121no—
141C23×C3—21no—
142C23×C3—71no—
143C22×D3—21no—
144C3×D4—21no—
145C3×D4—21no—
146C3×D4—282no—
147C2×C3×C4—71no—
148C2×C3×C4—141no—
149C2×C3×C4—141no—
150C2×C3×C4—141no—
151C2×C3×C4C22×D711yescharacteristic
152#67—282no—
153C2×Dic3—141no—
154D12—21no—
155D12—21no—
156C3⋊C8—141no—
157C3⋊C8C22⋊D712yescomplemented
158C3×C7#46611yescharacteristic, complemented, Hall
159C2×D4—31no—
160C2×D4—141no—
161C2×D4—421no—
162C2×D4D3×D711yescharacteristic
163C22×C4—71no—
164D8—62no—
165C2×C8—31no—
166C2×C8—211no—
167C2×C8—422no—
168C4⋊C4—71no—
169C4⋊C4—211no—
170C22⋊C4—141no—
171C22⋊C4—421no—
172C2×C7—81no—
173C2×C7—241no—
174C2×C7C2×#21311yescharacteristic
175C2×C7#93911yescharacteristic
176C2×C7#95011yescharacteristic
177D7—22no—
178C22×C3—41no—
179C22×C3—41no—
180C22×C3—41no—
181C22×C3—72no—
182C22×C3—141no—
183C22×C3—141no—
184C22×C3D4×D711yescharacteristic
185C2×D3—41no—
186C2×D3—41no—
187C2×D3—41no—
188C3×C4—141no—
189C3×C4—281no—
190C3×C4#115911yescharacteristic
191C3×C4C2×C22⋊D711yescharacteristic
192Dic3—281no—
193C23—21no—
194C23—61no—
195C23—71no—
196D4—21no—
197D4—21no—
198D4—61no—
199D4—61no—
200D4—282no—
201D4—842no—
202C2×C4—71no—
203C2×C4—141no—
204C2×C4—141no—
205C2×C4—141no—
206C2×C4—421no—
207C2×C4C2×D3×D711yescharacteristic
208C8—32no—
209C8—421no—
210C7#503811yescharacteristic, complemented, Sylow
211C2×C3—81no—
212C2×C3—142no—
213C2×C3D7×D811yescharacteristic
214C2×C3#647511yescharacteristic
215C2×C3#650111yescharacteristic
216D3—81no—
217C22—41no—
218C22—41no—
219C22—41no—
220C22—72no—
221C22—121no—
222C22—121no—
223C22—121no—
224C22—141no—
225C22—141no—
226C22D7×#6711yescharacteristic
227C4—141no—
228C4—281no—
229C4—841no—
230C4#6668411yescharacteristic
231C4D3×C22⋊D711yescharacteristic
232C3#8992911yescharacteristic, complemented, Sylow
233C2—81no—
234C2—142no—
235C2—241no—
236C2D7×#21311yescharacteristic
237C2#1062105211yescharacteristic
238C2#1062111111yescharacteristic
subgroupG/H
centerC22D7×#67
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×D4C22
socleC22×C3×C7C2×D4
hypercenterC2×D4D3×D7
hypocenterC3×C7#466
LCSG ▷ C2×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
1. C2×C7×#213C2semidirect
2. C3×#6501C2semidirect
3. #10621011C2semidirect
4. C2×D7×C3⋊C8C2semidirect
5. #10620671C2semidirect
6. #10620525C2semidirect
7. #10620501C2semidirect
24. #6501D3semidirect
46. C7×C3⋊C8D4semidirect
61. C2×#213D7semidirect
76. C2×C3×D7D8semidirect
135. C2×D7#213semidirect
157. C3⋊C8C22⋊D7semidirect
158. C3×C7#466semidirect
210. C7#5038semidirect
232. C3#89929semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#2132
C3×#65012
#106210112
C2×D7×C3⋊C82
#106206712
#106205252
#106205012
#899293
#50387