Finite Group Databasebeta

← #13054655

Subgroups — #13054655

Order 1344.

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

normal subgroups only

This group has 296 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×Q8×D21C211yesmaximal, characteristic
2C2×C7×#216C211yesmaximal, characteristic, complemented
3#10620945C211yesmaximal, characteristic
4C2×#66571C211yesmaximal, characteristic, complemented
5C3×#6409C211yesmaximal, characteristic, complemented
6#10620509C211yesmaximal, characteristic, complemented
7#10620492C211yesmaximal, characteristic
8#89958—31nomaximal, Hall
9C2×C3×C7×Q8C2211yescharacteristic
10Q8×D21—22no—
11Q8×D21—22no—
12C2×C4×D21—21no—
13C2×C4×D21C2211yescharacteristic
14C2×Dic42—21no—
15C2×Dic42C2211yescharacteristic
16C2×C7×Dic6C2211yescharacteristic
17C7×#216—22no—
18C2×C7×C3⋊C8C2211yescharacteristic
19C3×C4⋊Dic7C2211yescharacteristic
20C2×C3×C7⋊C8C2211yescharacteristic
21Dic7⋊Dic3—21no—
22#66575—21no—
23#66571—22no—
24C2×Q8×D7—31no—
25C2×C7×Q16—31no—
26#6509—31no—
27C2×#1144—31no—
28#6409D311yescharacteristic, complemented
29(C2×D7)⋊C8—31no—
30#6390—31no—
31#4809—71nomaximal, Hall
32C22×D21D411yescharacteristic
33C3×C7×Q8—21no—
34C3×C7×Q8D412yes—
35C2×C3×C4×C7—21no—
36C2×C3×C4×C7C2311yescharacteristic
37C2×Dic21—21no—
38C2×Dic21D411yescharacteristic
39C4×D21—24no—
40C4×D21—21no—
41C4×D21—21no—
42Dic42—24no—
43Dic42—21no—
44Dic42D412yes—
45C2×C7×Dic3—21no—
46C7×Dic6—21no—
47C7×Dic6—21no—
48C2×C3×Dic7—21no—
49C3×C7⋊C8—21no—
50C7×C3⋊C8—21no—
51C2×C7×Q8—31no—
52C2×C7×Q8C2×D311yescharacteristic
53Q8×D7—62no—
54Q8×D7—62no—
55C2×C4×D7—31no—
56C2×C4×D7—61no—
57C2×Dic14—31no—
58C2×Dic14—61no—
59C7×Q16—62no—
60C2×C7×C8—31no—
61#1144—62no—
62(C2×D7)⋊C4—61no—
63C4⋊Dic7C2×D311yescharacteristic
64Dic7⋊C4—61no—
65C2×C7⋊C8C2×D311yescharacteristic
66C2×D3×Q8—71no—
67C2×#216D711yescharacteristic, complemented
68C2×Dic12—71no—
69#897—71no—
70C3×#103—71no—
71(C2×D3)⋊C8—71no—
72#810—71no—
73C22×C3×C7C2×D411yescharacteristic
74C2×D21—21no—
75C2×D21—21no—
76C2×D21Q1612yescomplemented
77C3×C4×C7—22no—
78C3×C4×C7C2×D411yescharacteristic
79C3×C4×C7C2×D411yescharacteristic
80Dic21—21no—
81Dic21—22no—
82C3×Dic7—41no—
83C7×Dic3—41no—
84#451—211noSylow
85C22×D7—31no—
86C7×Q8—21no—
87C7×Q8—61no—
88C7×Q8—61no—
89C7×Q8D1212yes—
90C2×C4×C7—21no—
91C2×C4×C7—61no—
92C2×C4×C7C22×D311yescharacteristic
93C2×Dic7—21no—
94C2×Dic7—31no—
95C2×Dic7—61no—
96C4×D7—64no—
97C4×D7—61no—
98C4×D7—61no—
99Dic14—32no—
100Dic14—64no—
101Dic14—61no—
102C7×C8—61no—
103C7⋊C8—21no—
104C2×C3×Q8C2×D711yescharacteristic
105D3×Q8—142no—
106D3×Q8—142no—
107C2×C4×D3—71no—
108C2×C4×D3—141no—
109C2×Dic6—71no—
110C2×Dic6—141no—
111C2×Dic6C2×D711yescharacteristic
112C2×C3×C8—71no—
113C3×C4⋊C4—71no—
114#216—22no—
115(C2×D3)⋊C4—141no—
116C4⋊Dic3—141no—
117C2×C3⋊C8C2×D711yescharacteristic
118Dic12—142no—
119C2×C3×C7#12011yescharacteristic
120C2×C3×C7C2×Q1611yescharacteristic
121C2×C3×C7#13711yescharacteristic
122D21—22no—
123C22×Q8—211no—
124C2×Q16—31no—
125C2×Q16—211no—
126C22⋊Q8—211no—
127#103—71no—
128#103—211no—
129C22⋊C8—211no—
130C22×C7D3×D411yescharacteristic
131C2×D7—32no—
132C2×D7—61no—
133C2×D7—61no—
134C4×C7—22no—
135C4×C7—121no—
136C4×C7C2×D1211yescharacteristic
137C4×C7D3×D411yescharacteristic
138Dic7—41no—
139Dic7—62no—
140Dic7—61no—
141C22×D3—71no—
142C3×Q8—21no—
143C3×Q8C22⋊D712yes—
144C2×C3×C4—21no—
145C2×C3×C4—141no—
146C2×C3×C4C22×D711yescharacteristic
147C2×Dic3—21no—
148C2×Dic3—71no—
149C2×Dic3—141no—
150C4×D3—144no—
151C4×D3—141no—
152C4×D3—141no—
153Dic6—21no—
154Dic6—21no—
155Dic6—72no—
156Dic6—144no—
157Dic6—141no—
158C3×C8—141no—
159C3⋊C8—21no—
160C3×C7#45111yescharacteristic, complemented, Hall
161C2×Q8—31no—
162C2×Q8—211no—
163C2×Q8—421no—
164C2×Q8—422no—
165C2×Q8—422no—
166C2×Q8D3×D711yescharacteristic
167C22×C4—211no—
168C22×C4—421no—
169Q16—62no—
170Q16—422no—
171C2×C8—31no—
172C2×C8—71no—
173C4⋊C4—71no—
174C4⋊C4—421no—
175C22⋊C4—421no—
176C2×C7#88211yescharacteristic
177C2×C7#91511yescharacteristic
178C2×C7D3×Q1611yescharacteristic
179D7—62no—
180C22×C3D4×D711yescharacteristic
181C2×D3—72no—
182C2×D3—141no—
183C2×D3—141no—
184C3×C4—22no—
185C3×C4—281no—
186C3×C4D4×D711yescharacteristic
187C3×C4C2×C22⋊D711yescharacteristic
188Dic3—41no—
189Dic3—142no—
190Dic3—141no—
191C23—211no—
192Q8—21no—
193Q8—61no—
194Q8—61no—
195Q8—212no—
196Q8—424no—
197Q8—421no—
198Q8#424712yes—
199C2×C4—21no—
200C2×C4—61no—
201C2×C4—141no—
202C2×C4—211no—
203C2×C4—424no—
204C2×C4—421no—
205C2×C4—421no—
206C2×C4—421no—
207C2×C4C2×D3×D711yescharacteristic
208C8—61no—
209C8—141no—
210C7#480911yescharacteristic, complemented, Sylow
211C2×C3D7×Q1611yescharacteristic
212C2×C3#648111yescharacteristic
213C2×C3#650011yescharacteristic
214D3—142no—
215C22—212no—
216C22—421no—
217C22—421no—
218C22#6669411yescharacteristic
219C4—22no—
220C4—121no—
221C4—281no—
222C4—421no—
223C4—422no—
224C4C2×#424711yescharacteristic
225C4#6669411yescharacteristic
226C3#8995811yescharacteristic, complemented, Sylow
227C2—422no—
228C2#1062107311yescharacteristic
229C2#1062108411yescharacteristic
230C2#1062113511yescharacteristic
subgroupG/H
centerC22#66694
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×Q8C22
socleC22×C3×C7C2×D4
prime radicalC22×D21D4
hypercenterC2×Q8D3×D7
hypocenterC3×C7#451
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
2. C2×C7×#216C2semidirect
4. C2×#66571C2semidirect
5. C3×#6409C2semidirect
6. #10620509C2semidirect
28. #6409D3semidirect
67. C2×#216D7semidirect
76. C2×D21Q16semidirect
160. C3×C7#451semidirect
210. C7#4809semidirect
226. C3#89958semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×D212
C2×C7×#2162
#106209452
C2×#665712
C3×#64092
#106205092
#106204922
#899583
#48097