Finite Group Databasebeta

← #13052890

Subgroups — #13052890

Order 1344.

304 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 66 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×#66676C211yesmaximal, characteristic
2C3×#6460C211yesmaximal, characteristic, complemented
3C2×#66570C211yesmaximal, characteristic, complemented
4C2×#66564C211yesmaximal, characteristic, complemented
5#10620649C211yesmaximal, characteristic, complemented
6C7×#809C211yesmaximal, characteristic, complemented
7(C2×D7)⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8#89555—31nomaximal, Hall
9C2×Dic42C2211yescharacteristic
10C2×C7×D12C2211yescharacteristic
11C2×C3×C4×D7C2211yescharacteristic
12C2×C3×Dic14C2211yescharacteristic
13C2×#4245—21no—
14C2×D3×Dic7—21no—
15#66676—22no—
16#66676—22no—
17C3×C7×C4⋊C4C2211yescharacteristic
18C2×C3⋊(C7⋊C8)C2211yescharacteristic
19C2×C7×C3⋊C8C2211yescharacteristic
20C3×(C2×D7)⋊C4—21no—
21C3×C4⋊Dic7—21no—
22#66570—22no—
23#66564—22no—
24C2×#1159—31no—
25C2×#1142—31no—
26C2×Dic28—31no—
27#6460D311yescharacteristic, complemented
28C7×#102—31no—
29(C2×D7)⋊C8—31no—
30#6384—31no—
31#5047—71nomaximal, Hall
32C22×D3×C7—21no—
33C22×C3×D7D411yescharacteristic
34C2×C3×C4×C7—21no—
35C2×C3×C4×C7C2311yescharacteristic
36C2×Dic21—21no—
37Dic42—21no—
38Dic42D412yes—
39C7×D12—21no—
40C7×D12D412yes—
41C2×C3×Dic7—21no—
42C2×C3×Dic7D411yescharacteristic
43C3×C4×D7—21no—
44C3×C4×D7—21no—
45C3×Dic14—21no—
46C3×Dic14—21no—
47#4245—24no—
48D3×Dic7—24no—
49C3⋊(C7⋊C8)—21no—
50C7×C3⋊C8—21no—
51C2×C7×D4—31no—
52C2×C22⋊D7—61no—
53C22×Dic7—61no—
54#1159—62no—
55#1159—62no—
56C2×C4×D7C2×D311yescharacteristic
57C2×Dic14—31no—
58C2×Dic14C2×D311yescharacteristic
59C2×C7×C8—31no—
60C7×C4⋊C4C2×D311yescharacteristic
61#1142—62no—
62(C2×D7)⋊C4—21no—
63C4⋊Dic7—21no—
64C2×C7⋊C8—31no—
65Dic28—62no—
66C2×#235—71no—
67C3×C22⋊Q8—71no—
68C2×#216—71no—
69C2×#215—71no—
70#830—71no—
71#810—71no—
72#809D711yescharacteristic, complemented
73C22×C3×C7C2×D411yescharacteristic
74C2×D3×C7—22no—
75C2×D3×C7—41no—
76C2×D3×C7—41no—
77C2×C3×D7—21no—
78C2×C3×D7—21no—
79C2×C3×D7—21no—
80C3×C4×C7—41no—
81C3×C4×C7C2×D411yescharacteristic
82C3×C4×C7C2×D411yescharacteristic
83Dic21—22no—
84C3×Dic7—21no—
85C3×Dic7—41no—
86#452—211noSylow
87C23×C7—61no—
88C22×D7#6711yescharacteristic
89C7×D4—32no—
90C7×D4—61no—
91C2×C4×C7—21no—
92C2×C4×C7C22×D311yescharacteristic
93C22⋊D7—64no—
94C2×Dic7—21no—
95C2×Dic7—64no—
96C2×Dic7—61no—
97C2×Dic7#6711yescharacteristic
98C4×D7—21no—
99C4×D7—21no—
100Dic14—21no—
101Dic14—21no—
102Dic14—32no—
103Dic14—61no—
104C7×C8—61no—
105C7⋊C8—61no—
106C2×C3×Q8—71no—
107C22×C3×C4—71no—
108C2×#67—141no—
109#235—142no—
110#235—142no—
111C2×D12C2×D711yescharacteristic
112C2×C4×D3—141no—
113C2×Dic6—71no—
114C3×C4⋊C4—141no—
115C3×C4⋊C4C2×D711yescharacteristic
116C3×C22⋊C4—141no—
117#216—142no—
118#215—142no—
119C2×C3⋊C8—71no—
120C2×C3⋊C8C2×D711yescharacteristic
121C2×C3×C7#12011yescharacteristic
122C2×C3×C7#13511yescharacteristic
123C2×C3×C7#13711yescharacteristic
124C3×D7—41no—
125D3×C7—42no—
126C2×#41—211no—
127C2×Q16—211no—
128C2×SD16—211no—
129C22⋊Q8—71no—
130#103—211no—
131#102—31no—
132C22⋊C8—211no—
133C22×C7—62no—
134C22×C7—121no—
135C22×C7—121no—
136C22×C7C2×#6711yescharacteristic
137C2×D7—21no—
138C2×D7—21no—
139C2×D7—21no—
140C4×C7—41no—
141C4×C7D3×D411yescharacteristic
142C4×C7D3×D411yescharacteristic
143Dic7—21no—
144Dic7—41no—
145Dic7—62no—
146C23×C3—71no—
147C22×D3—21no—
148C3×Q8—141no—
149C3×Q8—141no—
150C2×C3×C4—21no—
151C2×C3×C4—71no—
152C2×C3×C4—141no—
153C2×C3×C4—141no—
154C2×C3×C4—141no—
155C2×C3×C4C22×D711yescharacteristic
156#67—144no—
157C2×Dic3—141no—
158D12—21no—
159D12D2812yes—
160C4×D3—144no—
161Dic6—72no—
162Dic6—141no—
163C3⋊C8—21no—
164C3⋊C8—141no—
165C3×C7#45211yescharacteristic, complemented, Hall
166#41—422no—
167#41—422no—
168C2×Q8—71no—
169C2×Q8—211no—
170C2×D4—31no—
171C2×D4—421no—
172C22×C4—71no—
173C22×C4—421no—
174Q16—422no—
175SD16—422no—
176C2×C8—31no—
177C2×C8—211no—
178C4⋊C4—141no—
179C4⋊C4D3×D711yescharacteristic
180C22⋊C4—141no—
181C2×C7—122no—
182C2×C7#93711yescharacteristic
183C2×C7#94211yescharacteristic
184C2×C7#95011yescharacteristic
185D7—41no—
186C22×C3—141no—
187C22×C3—141no—
188C22×C3—141no—
189C22×C3D4×D711yescharacteristic
190C2×D3—22no—
191C2×D3—41no—
192C2×D3—41no—
193C3×C4—41no—
194C3×C4—141no—
195C3×C4—281no—
196C3×C4C2×D2811yescharacteristic
197C3×C4D4×D711yescharacteristic
198Dic3—142no—
199C23—61no—
200C23—71no—
201Q8—141no—
202Q8—141no—
203Q8—212no—
204Q8—421no—
205D4—32no—
206D4—61no—
207D4—424no—
208C2×C4—21no—
209C2×C4—71no—
210C2×C4—141no—
211C2×C4—141no—
212C2×C4—141no—
213C2×C4—424no—
214C2×C4—421no—
215C2×C4C2×D3×D711yescharacteristic
216C8—61no—
217C8—421no—
218C7#504711yescharacteristic, complemented, Sylow
219C2×C3—281no—
220C2×C3#644511yescharacteristic
221C2×C3#647511yescharacteristic
222C2×C3#647811yescharacteristic
223D3—42no—
224C22—62no—
225C22—121no—
226C22—121no—
227C22—141no—
228C22—141no—
229C22—141no—
230C22D7×#6711yescharacteristic
231C4—41no—
232C4—141no—
233C4—281no—
234C4—422no—
235C4D3×D2811yescharacteristic
236C4#6668111yescharacteristic
237C3#8955511yescharacteristic, complemented, Sylow
238C2—122no—
239C2—281no—
240C2#1062103611yescharacteristic
241C2#1062105211yescharacteristic
242C2#1062107511yescharacteristic
subgroupG/H
centerC22D7×#67
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC3×C7×C4⋊C4C22
socleC22×C3×C7C2×D4
prime radicalC2×#66676C2
hypercenterC4⋊C4D3×D7
hypocenterC3×C7#452
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C3×#6460C2semidirect
3. C2×#66570C2semidirect
4. C2×#66564C2semidirect
5. #10620649C2semidirect
6. C7×#809C2semidirect
7. (C2×D7)⋊(C3⋊C8)C2semidirect
27. #6460D3semidirect
72. #809D7semidirect
165. C3×C7#452semidirect
218. C7#5047semidirect
237. C3#89555semidirect

Maximal subgroups

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

subgroupindex
C2×#666762
C3×#64602
C2×#665702
C2×#665642
#106206492
C7×#8092
(C2×D7)⋊(C3⋊C8)2
#895553
#50477