Finite Group Databasebeta

← #13054553

Subgroups — #13054553

Order 1344.

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

normal subgroups only

This group has 316 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×D21⋊Q8C211yesmaximal, characteristic, complemented
2#10621421C211yesmaximal, characteristic
3C2×C7×#214C211yesmaximal, characteristic, complemented
4C2×C3×#1142C211yesmaximal, characteristic, complemented
5#10620508C211yesmaximal, characteristic, complemented
6#10620506C211yesmaximal, characteristic, complemented
7#10620492C211yesmaximal, characteristic, complemented
8#89942—31nomaximal, Hall
9C2×C3×C7×D4C2211yescharacteristic
10C2×#4268—21no—
11C2×C4×D21C2211yescharacteristic
12C2×C7×Dic6C2211yescharacteristic
13C2×C3×Dic14C2211yescharacteristic
14C2×#4248—21no—
15C2×D21⋊C4—21no—
16D21⋊Q8—22no—
17D21⋊Q8—22no—
18#66636—21no—
19C4⋊Dic21C2211yescharacteristic
20C7×#214—22no—
21C2×C7×C3⋊C8C2211yescharacteristic
22C3×#1142—22no—
23C2×C3×C7⋊C8C2211yescharacteristic
24C2×Q8×D7—31no—
25C2×C7×SD16—31no—
26#6501—31no—
27C2×#1142D311yescharacteristic, complemented
28#6409—31no—
29(C2×D7)⋊C8—31no—
30#6390—31no—
31#5170—71nomaximal, Hall
32C23×C3×C7—21no—
33C22×D21D411yescharacteristic
34C3×C7×D4—21no—
35C3×C7×D4—21no—
36C2×C3×C4×C7C2311yescharacteristic
37#4268—42no—
38C2×Dic21—21no—
39C2×Dic21D411yescharacteristic
40C4×D21—21no—
41C4×D21—21no—
42C2×C7×Dic3—21no—
43C7×Dic6—21no—
44C7×Dic6D412yescomplemented
45C2×C3×Dic7—21no—
46C3×Dic14—21no—
47C3×Dic14D412yescomplemented
48#4248—24no—
49D21⋊C4—24no—
50C3×C7⋊C8—21no—
51C7×C3⋊C8—21no—
52C2×C7×Q8—31no—
53C2×C7×D4C2×D311yescharacteristic
54C2×C22⋊D7—61no—
55Q8×D7—62no—
56Q8×D7—62no—
57C2×C4×D7—31no—
58C2×C4×D7—61no—
59C2×Dic14—61no—
60C2×Dic14C2×D311yescharacteristic
61C7×SD16—62no—
62C2×C7×C8—31no—
63C22⋊Dic7—61no—
64#1142—22no—
65C4⋊Dic7—31no—
66C2×C7⋊C8C2×D311yescharacteristic
67C2×D3×Q8—71no—
68C2×C3×SD16—71no—
69#938—71no—
70C2×#214D711yescharacteristic, complemented
71#835—71no—
72(C2×D3)⋊C8—71no—
73#816—71no—
74C22×C3×C7—41no—
75C22×C3×C7—41no—
76C22×C3×C7—41no—
77C22×C3×C7C2×D411yescharacteristic
78C2×D21—21no—
79C2×D21—21no—
80C2×D21SD1612yescomplemented
81C3×C4×C7C2×D411yescharacteristic
82C3×C4×C7C2×D411yescharacteristic
83Dic21—21no—
84Dic21—41no—
85C3×Dic7—22no—
86C7×Dic3—22no—
87#448—211noSylow
88C23×C7—21no—
89C22×D7—31no—
90C7×Q8—32no—
91C7×Q8—61no—
92C7×D4—21no—
93C7×D4—21no—
94C2×C4×C7—61no—
95C2×C4×C7C22×D311yescharacteristic
96C22⋊D7—122no—
97C2×Dic7—21no—
98C2×Dic7—31no—
99C2×Dic7—61no—
100C4×D7—64no—
101C4×D7—61no—
102C4×D7—61no—
103Dic14—21no—
104Dic14—64no—
105Dic14#6712yescomplemented
106C7×C8—61no—
107C7⋊C8—21no—
108C2×C3×Q8—71no—
109C2×C3×D4C2×D711yescharacteristic
110C2×#67—141no—
111D3×Q8—142no—
112D3×Q8—142no—
113C2×C4×D3—71no—
114C2×C4×D3—141no—
115C2×Dic6—141no—
116C2×Dic6C2×D711yescharacteristic
117C3×SD16—142no—
118C2×C3×C8—71no—
119#217—141no—
120#214—22no—
121C4⋊Dic3—71no—
122C2×C3⋊C8C2×D711yescharacteristic
123C2×C3×C7—81no—
124C2×C3×C7#12011yescharacteristic
125C2×C3×C7C2×SD1611yescharacteristic
126C2×C3×C7#13711yescharacteristic
127D21—22no—
128C22×Q8—211no—
129C2×SD16—31no—
130C2×SD16—71no—
131#121—211no—
132#103—211no—
133#103—211no—
134C22⋊C8—211no—
135C22×C7—41no—
136C22×C7—41no—
137C22×C7—41no—
138C22×C7D3×D411yescharacteristic
139C2×D7—32no—
140C2×D7—61no—
141C2×D7—61no—
142C4×C7—62no—
143C4×C7D3×D411yescharacteristic
144C4×C7C2×#6711yescharacteristic
145Dic7—22no—
146Dic7—61no—
147Dic7—121no—
148C23×C3—21no—
149C22×D3—71no—
150C3×Q8—72no—
151C3×Q8—141no—
152C3×D4—21no—
153C3×D4—21no—
154C2×C3×C4—141no—
155C2×C3×C4C22×D711yescharacteristic
156#67—282no—
157C2×Dic3—21no—
158C2×Dic3—71no—
159C2×Dic3—141no—
160C4×D3—144no—
161C4×D3—141no—
162C4×D3—141no—
163Dic6—21no—
164Dic6—144no—
165Dic6C22⋊D712yescomplemented
166C3×C8—141no—
167C3⋊C8—21no—
168C3×C7#44811yescharacteristic, complemented, Hall
169C2×Q8—31no—
170C2×Q8—71no—
171C2×Q8—421no—
172C2×Q8—422no—
173C2×Q8—422no—
174C2×D4—421no—
175C2×D4D3×D711yescharacteristic
176C22×C4—211no—
177C22×C4—421no—
178SD16—62no—
179SD16—142no—
180C2×C8—31no—
181C2×C8—71no—
182C4⋊C4—211no—
183C22⋊C4—421no—
184C2×C7—81no—
185C2×C7D3×SD1611yescharacteristic
186C2×C7#91511yescharacteristic
187C2×C7#93711yescharacteristic
188D7—62no—
189C22×C3—41no—
190C22×C3—41no—
191C22×C3—41no—
192C22×C3D4×D711yescharacteristic
193C2×D3—72no—
194C2×D3—141no—
195C2×D3—141no—
196C3×C4—142no—
197C3×C4D4×D711yescharacteristic
198C3×C4C2×C22⋊D711yescharacteristic
199Dic3—22no—
200Dic3—141no—
201Dic3—281no—
202C23—21no—
203C23—211no—
204Q8—32no—
205Q8—61no—
206Q8—72no—
207Q8—141no—
208Q8—424no—
209D4—21no—
210D4—21no—
211D4—842no—
212C2×C4—61no—
213C2×C4—141no—
214C2×C4—211no—
215C2×C4—424no—
216C2×C4—421no—
217C2×C4—421no—
218C2×C4—421no—
219C2×C4C2×D3×D711yescharacteristic
220C8—61no—
221C8—141no—
222C7#517011yescharacteristic, complemented, Sylow
223C2×C3—81no—
224C2×C3D7×SD1611yescharacteristic
225C2×C3#647811yescharacteristic
226C2×C3#650011yescharacteristic
227D3—142no—
228C22—41no—
229C22—41no—
230C22—41no—
231C22—212no—
232C22—421no—
233C22—421no—
234C22#6669411yescharacteristic
235C4—62no—
236C4—142no—
237C4—421no—
238C4—841no—
239C4D7×#6711yescharacteristic
240C4D3×C22⋊D711yescharacteristic
241C3#8994211yescharacteristic, complemented, Sylow
242C2—81no—
243C2—422no—
244C2#1062104911yescharacteristic
245C2#1062105011yescharacteristic
246C2#1062113911yescharacteristic
subgroupG/H
centerC22#66694
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×D4C22
socleC22×C3×C7C2×D4
prime radical#10621421C2
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×D21⋊Q8C2semidirect
3. C2×C7×#214C2semidirect
4. C2×C3×#1142C2semidirect
5. #10620508C2semidirect
6. #10620506C2semidirect
7. #10620492C2semidirect
27. C2×#1142D3semidirect
44. C7×Dic6D4semidirect
47. C3×Dic14D4semidirect
70. C2×#214D7semidirect
80. C2×D21SD16semidirect
105. Dic14#67semidirect
165. Dic6C22⋊D7semidirect
168. C3×C7#448semidirect
222. C7#5170semidirect
241. C3#89942semidirect

Maximal subgroups

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

subgroupindex
C2×D21⋊Q82
#106214212
C2×C7×#2142
C2×C3×#11422
#106205082
#106205062
#106204922
#899423
#51707