Finite Group Databasebeta

← #13054365

Subgroups — #13054365

Order 1344.

264 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 82 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
1D4×Dic21C211yesmaximal, characteristic, complemented
2C2×C7×#213C211yesmaximal, characteristic
3#10620997C211yesmaximal, characteristic, complemented
4C3×#6406C211yesmaximal, characteristic, complemented
5#10620522C211yesmaximal, characteristic, complemented
6#10620502C211yesmaximal, characteristic, complemented
7#10620489C211yesmaximal, characteristic, complemented
8#89924—31nomaximal, Hall
9C2×C3×C7×D4C2211yescharacteristic
10C22×Dic21—21no—
11C2×C7×D12C2211yescharacteristic
12C2×D3×Dic7—21no—
13#66636—21no—
14C4⋊Dic21C2211yescharacteristic
15C4×Dic21C2211yescharacteristic
16C7×#213C414yescomplemented
17C2×C7×C3⋊C8C2211yescharacteristic
18C3×C4⋊Dic7C2211yescharacteristic
19C2×C3×C7⋊C8C2211yescharacteristic
20(C2×D3)⋊Dic7—21no—
21C2×C7×D8—31no—
22D4×Dic7—31no—
23D4×Dic7—31no—
24#6406—31no—
25#6406D311yescharacteristic, complemented
26#6391—31no—
27#6389—31no—
28#4785—71nomaximal, Hall
29C23×C3×C7—21no—
30C22×D3×C7—21no—
31C3×C7×D4—21no—
32C3×C7×D4C2×C412yescomplemented
33C2×C3×C4×C7C2311yescharacteristic
34C2×Dic21—21no—
35C2×Dic21—42no—
36C2×Dic21D411yescharacteristic
37C2×Dic21D411yescharacteristic
38C7×D12—21no—
39C7×D12C2×C412yescomplemented
40C2×C3×Dic7—21no—
41D3×Dic7—42no—
42C3×C7⋊C8—21no—
43C7×C3⋊C8C2×C412yescomplemented
44C2×C7×D4—31no—
45C2×C7×D4C2×D311yescharacteristic
46C22×Dic7—61no—
47C22×Dic7—61no—
48C7×D8—34no—
49C2×C7×C8—31no—
50C22⋊Dic7—61no—
51C22⋊Dic7—61no—
52C4⋊Dic7—31no—
53C4⋊Dic7C2×D311yescharacteristic
54C4×Dic7—31no—
55C2×C7⋊C8C2×D311yescharacteristic
56D4×Dic3—71no—
57C2×#213D711yescharacteristic
58#893—71no—
59C3×#102—71no—
60#821—71no—
61#815—71no—
62#808—71no—
63C22×C3×C7—22no—
64C22×C3×C7—41no—
65C22×C3×C7—41no—
66C22×C3×C7C2×D411yescharacteristic
67C2×D3×C7—22no—
68C2×D3×C7—41no—
69C2×D3×C7—41no—
70C3×C4×C7C22×C411yescharacteristic
71C3×C4×C7#4111yescharacteristic
72Dic21—21no—
73Dic21—21no—
74Dic21—41no—
75C3×Dic7—41no—
76#442—211noSylow
77C23×C7—21no—
78C23×C7—61no—
79C7×D4—21no—
80C7×D4—32no—
81C7×D4—61no—
82C7×D4C4×D312yescomplemented
83C2×C4×C7C22×D311yescharacteristic
84C2×Dic7—21no—
85C2×Dic7—31no—
86C2×Dic7—31no—
87C2×Dic7—61no—
88C2×Dic7—122no—
89C2×Dic7—122no—
90C7×C8—32no—
91C7⋊C8—21no—
92C2×C3×D4C2×D711yescharacteristic
93C22×Dic3—141no—
94C2×D12C2×D711yescharacteristic
95C2×C4×D3—141no—
96C2×C3×C8—71no—
97C3×C4⋊C4—71no—
98#217—141no—
99#213Dic714yescomplemented
100(C2×D3)⋊C4—141no—
101C4⋊Dic3—71no—
102C4×Dic3—71no—
103C2×C3⋊C8C2×D711yescharacteristic
104C2×C3×C7—42no—
105C2×C3×C7C4×D411yescharacteristic
106C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
107C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
108D3×C7—42no—
109C2×D8—31no—
110C4×D4—211no—
111C4×D4—211no—
112#106—211no—
113#102—71no—
114#102—211no—
115#97—211no—
116C22×C7—22no—
117C22×C7—41no—
118C22×C7—41no—
119C22×C7—62no—
120C22×C7—121no—
121C22×C7—121no—
122C22×C7D3×D411yescharacteristic
123C4×C7C2×C4×D311yescharacteristic
124C4×C7#23711yescharacteristic
125Dic7—41no—
126Dic7—61no—
127Dic7—61no—
128Dic7—121no—
129C23×C3—21no—
130C22×D3—21no—
131C3×D4—21no—
132C3×D4C2×Dic712yescomplemented
133C2×C3×C4—141no—
134C2×C3×C4C22×D711yescharacteristic
135C2×Dic3—71no—
136C2×Dic3—71no—
137C2×Dic3—141no—
138C2×Dic3—282no—
139D12—21no—
140D12C2×Dic712yescomplemented
141C4×D3—282no—
142C3×C8—141no—
143C3⋊C8C2×Dic712yescomplemented
144C3×C7#44211yescharacteristic, complemented, Hall
145C2×D4—31no—
146C2×D4D3×D711yescharacteristic
147C22×C4—421no—
148C22×C4—421no—
149D8—34no—
150C2×C8—31no—
151C2×C8—71no—
152C4⋊C4—71no—
153C4⋊C4—211no—
154C22⋊C4—421no—
155C22⋊C4—421no—
156C42—211no—
157C2×C7—42no—
158C2×C7—122no—
159C2×C7#67⋊C411yescharacteristic
160C2×C7#91111yescharacteristic
161C2×C7#91411yescharacteristic
162C22×C3—22no—
163C22×C3—41no—
164C22×C3—41no—
165C22×C3D4×D711yescharacteristic
166C2×D3—22no—
167C2×D3—41no—
168C2×D3—41no—
169C3×C4—281no—
170C3×C4#115911yescharacteristic
171C3×C4C22×Dic711yescharacteristic
172Dic3—141no—
173Dic3—141no—
174Dic3—281no—
175C23—21no—
176C23—61no—
177D4—21no—
178D4—32no—
179D4—61no—
180D4D3×Dic712yescomplemented
181C2×C4—141no—
182C2×C4—211no—
183C2×C4—211no—
184C2×C4—421no—
185C2×C4—842no—
186C2×C4—842no—
187C2×C4C2×D3×D711yescharacteristic
188C8—32no—
189C8—141no—
190C7#478511yescharacteristic, complemented, Sylow
191C2×C3—42no—
192C2×C3#647411yescharacteristic
193C2×C3#647411yescharacteristic
194C2×C3D4×Dic711yescharacteristic
195D3—42no—
196C22—22no—
197C22—41no—
198C22—41no—
199C22—62no—
200C22—121no—
201C22—121no—
202C22#6669411yescharacteristic
203C4—281no—
204C4—421no—
205C4—421no—
206C4—841no—
207C4#6668411yescharacteristic
208C4C2×D3×Dic711yescharacteristic
209C3#8992411yescharacteristic, complemented, Sylow
210C2—42no—
211C2—122no—
212C2#1062104411yescharacteristic
213C2#1062106211yescharacteristic
214C2#67⋊Dic711yescharacteristic
subgroupG/H
centerC22#66694
commutatorC3×C4×C7C22×C4
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×D4C22
socleC22×C3×C7C2×D4
prime radicalC2×C7×#213C2
cosocleC2×C3×C4×C7C23
hypercenterC2×D4D3×D7
hypocenterC3×C7#442
LCSG ▷ 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. D4×Dic21C2semidirect
3. #10620997C2semidirect
4. C3×#6406C2semidirect
5. #10620522C2semidirect
6. #10620502C2semidirect
7. #10620489C2semidirect
16. C7×#213C4semidirect
25. #6406D3semidirect
32. C3×C7×D4C2×C4semidirect
39. C7×D12C2×C4semidirect
43. C7×C3⋊C8C2×C4semidirect
82. C7×D4C4×D3semidirect
99. #213Dic7semidirect
132. C3×D4C2×Dic7semidirect
140. D12C2×Dic7semidirect
143. C3⋊C8C2×Dic7semidirect
144. C3×C7#442semidirect
180. D4D3×Dic7semidirect
190. C7#4785semidirect
209. C3#89924semidirect

Maximal subgroups

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

subgroupindex
D4×Dic212
C2×C7×#2132
#106209972
C3×#64062
#106205222
#106205022
#106204892
#899243
#47857