Finite Group Databasebeta

← #13054551

Subgroups — #13054551

Order 1344.

316 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 70 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×D7×Dic6C211yesmaximal, characteristic, complemented
2C2×#66633C211yesmaximal, characteristic, complemented
3C2×C7×#214C211yesmaximal, characteristic, complemented
4C3×#6501C211yesmaximal, characteristic
5#10620509C211yesmaximal, characteristic, complemented
6#10620507C211yesmaximal, characteristic, complemented
7(C2×D7)⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8#89942—31nomaximal, Hall
9C2×C3×C7×D4C2211yescharacteristic
10C2×Dic42C2211yescharacteristic
11C2×C7×Dic6C2211yescharacteristic
12C2×C3×C22⋊D7—21no—
13C2×C3×C4×D7C2211yescharacteristic
14C2×#4248—21no—
15C2×Dic3×D7—21no—
16D7×Dic6—22no—
17D7×Dic6—22no—
18#66633—22no—
19C2×C3⋊(C7⋊C8)C2211yescharacteristic
20C7×#214—22no—
21C2×C7×C3⋊C8C2211yescharacteristic
22C3×C22⋊Dic7—21no—
23C3×C4⋊Dic7C2211yescharacteristic
24C2×Q8×D7—31no—
25C2×C7×SD16—31no—
26#6501D311yescharacteristic
27C2×#1142—31no—
28#6409—31no—
29(C2×D7)⋊C8—31no—
30#6390—31no—
31#5040—71nomaximal, Hall
32C23×C3×C7—21no—
33C22×C3×D7D411yescharacteristic
34C3×C7×D4—21no—
35C3×C7×D4—21no—
36C2×C3×C4×C7C2311yescharacteristic
37C2×Dic21—21no—
38Dic42—21no—
39Dic42D412yescomplemented
40C2×C7×Dic3—21no—
41C7×Dic6—21no—
42C7×Dic6D412yescomplemented
43C3×C22⋊D7—42no—
44C2×C3×Dic7—21no—
45C2×C3×Dic7D411yescharacteristic
46C3×C4×D7—21no—
47C3×C4×D7—21no—
48#4248—24no—
49Dic3×D7—24no—
50C3⋊(C7⋊C8)—21no—
51C7×C3⋊C8—21no—
52C2×C7×Q8—31no—
53C2×C7×D4C2×D311yescharacteristic
54C2×C22⋊D7—21no—
55Q8×D7—62no—
56Q8×D7—62no—
57C2×C4×D7—61no—
58C2×C4×D7C2×D311yescharacteristic
59C2×Dic14—31no—
60C2×Dic14—61no—
61C7×SD16—62no—
62C2×C7×C8—31no—
63C22⋊Dic7—21no—
64#1142—62no—
65C4⋊Dic7C2×D311yescharacteristic
66C2×C7⋊C8—31no—
67C22×Dic6—71no—
68C3×#121—71no—
69C2×#214—71no—
70C2×#214D711yescharacteristic, complemented
71#830—71no—
72#810—71no—
73#810—71no—
74C22×C3×C7—41no—
75C22×C3×C7—41no—
76C22×C3×C7—41no—
77C22×C3×C7C2×D411yescharacteristic
78C2×C3×D7—21no—
79C2×C3×D7—21no—
80C2×C3×D7SD1612yescomplemented
81C3×C4×C7C2×D411yescharacteristic
82C3×C4×C7C2×D411yescharacteristic
83Dic21—22no—
84C3×Dic7—21no—
85C3×Dic7—41no—
86C7×Dic3—22no—
87#448—211noSylow
88C23×C7—21no—
89C22×D7#6711yescharacteristic
90C7×Q8—32no—
91C7×Q8—61no—
92C7×D4—21no—
93C7×D4—21no—
94C2×C4×C7—61no—
95C2×C4×C7C22×D311yescharacteristic
96C22⋊D7—42no—
97C2×Dic7—21no—
98C2×Dic7—61no—
99C2×Dic7#6711yescharacteristic
100C4×D7—21no—
101C4×D7—21no—
102C4×D7—64no—
103Dic14—32no—
104Dic14—64no—
105Dic14—61no—
106C7×C8—61no—
107C7⋊C8—61no—
108C2×C3×D4—141no—
109C2×C3×D4C2×D711yescharacteristic
110C22×C3×C4—71no—
111C22×Dic3—141no—
112C2×Dic6—71no—
113C2×Dic6—142no—
114C2×Dic6—141no—
115C2×Dic6—142no—
116C2×Dic6C2×D711yescharacteristic
117C3×C4⋊C4—71no—
118C3×C22⋊C4—141no—
119#214—22no—
120#214—142no—
121C2×C3⋊C8—71no—
122C2×C3⋊C8C2×D711yescharacteristic
123C2×C3×C7—81no—
124C2×C3×C7#12011yescharacteristic
125C2×C3×C7C2×SD1611yescharacteristic
126C2×C3×C7#13711yescharacteristic
127C3×D7—22no—
128C22×Q8—211no—
129C2×SD16—31no—
130C2×SD16—211no—
131#121—71no—
132#103—211no—
133#103—211no—
134C22⋊C8—211no—
135C22×C7—41no—
136C22×C7—41no—
137C22×C7—41no—
138C22×C7C2×#6711yescharacteristic
139C2×D7—21no—
140C2×D7—21no—
141C2×D7#21412yescomplemented
142C4×C7—62no—
143C4×C7D3×D411yescharacteristic
144C4×C7D3×D411yescharacteristic
145Dic7—21no—
146Dic7—41no—
147Dic7—62no—
148C23×C3—21no—
149C23×C3—71no—
150C3×D4—21no—
151C3×D4—21no—
152C3×D4—282no—
153C2×C3×C4—71no—
154C2×C3×C4—141no—
155C2×C3×C4—141no—
156C2×C3×C4—141no—
157C2×C3×C4C22×D711yescharacteristic
158C2×Dic3—21no—
159C2×Dic3—144no—
160C2×Dic3—141no—
161Dic6—21no—
162Dic6—72no—
163Dic6—144no—
164Dic6—141no—
165Dic6C22⋊D712yescomplemented
166C3⋊C8—21no—
167C3⋊C8—141no—
168C3×C7#44811yescharacteristic, complemented, Hall
169C2×Q8—31no—
170C2×Q8—211no—
171C2×Q8—421no—
172C2×Q8—422no—
173C2×Q8—422no—
174C2×D4—141no—
175C2×D4D3×D711yescharacteristic
176C22×C4—71no—
177C22×C4—421no—
178SD16—62no—
179SD16—422no—
180C2×C8—31no—
181C2×C8—211no—
182C4⋊C4—71no—
183C22⋊C4—141no—
184C2×C7—81no—
185C2×C7C2×#21411yescharacteristic
186C2×C7#93711yescharacteristic
187C2×C7#95111yescharacteristic
188D7—22no—
189C22×C3—41no—
190C22×C3—41no—
191C22×C3—41no—
192C22×C3—72no—
193C22×C3—141no—
194C22×C3—141no—
195C22×C3D4×D711yescharacteristic
196C3×C4—141no—
197C3×C4—281no—
198C3×C4D4×D711yescharacteristic
199C3×C4C2×C22⋊D711yescharacteristic
200Dic3—22no—
201Dic3—142no—
202C23—21no—
203C23—71no—
204Q8—32no—
205Q8—61no—
206Q8—212no—
207Q8—424no—
208Q8—421no—
209D4—21no—
210D4—21no—
211D4—282no—
212C2×C4—61no—
213C2×C4—71no—
214C2×C4—141no—
215C2×C4—141no—
216C2×C4—141no—
217C2×C4—424no—
218C2×C4—421no—
219C2×C4C2×D3×D711yescharacteristic
220C8—61no—
221C8—421no—
222C7#504011yescharacteristic, complemented, Sylow
223C2×C3—81no—
224C2×C3—142no—
225C2×C3D7×SD1611yescharacteristic
226C2×C3#647811yescharacteristic
227C2×C3#650011yescharacteristic
228C22—41no—
229C22—41no—
230C22—41no—
231C22—72no—
232C22—141no—
233C22—141no—
234C22D7×#6711yescharacteristic
235C4—62no—
236C4—141no—
237C4—281no—
238C4—422no—
239C4D3×C22⋊D711yescharacteristic
240C4#6669411yescharacteristic
241C3#8994211yescharacteristic, complemented, Sylow
242C2—81no—
243C2—142no—
244C2D7×#21411yescharacteristic
245C2#1062104611yescharacteristic
246C2#1062113911yescharacteristic
subgroupG/H
centerC22D7×#67
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×D4C22
socleC22×C3×C7C2×D4
prime radicalC3×#6501C2
hypercenterC2×D4D3×D7
hypocenterC3×C7#448
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×D7×Dic6C2semidirect
2. C2×#66633C2semidirect
3. C2×C7×#214C2semidirect
5. #10620509C2semidirect
6. #10620507C2semidirect
7. (C2×D7)⋊(C3⋊C8)C2semidirect
39. Dic42D4semidirect
42. C7×Dic6D4semidirect
70. C2×#214D7semidirect
80. C2×C3×D7SD16semidirect
141. C2×D7#214semidirect
165. Dic6C22⋊D7semidirect
168. C3×C7#448semidirect
222. C7#5040semidirect
241. C3#89942semidirect

Maximal subgroups

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

subgroupindex
C2×D7×Dic62
C2×#666332
C2×C7×#2142
C3×#65012
#106205092
#106205072
(C2×D7)⋊(C3⋊C8)2
#899423
#50407