Finite Group Databasebeta

← #13062036

Subgroups — #13062036

Order 1344.

342 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 41 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
1C4×A4×D7C211yesmaximal, characteristic, complemented
2C4⋊(C7⋊S4)C211yesmaximal, characteristic, complemented
3C7×A4⋊Q8C211yesmaximal, characteristic, complemented
4#10621483C212yesmaximal, complemented
5(C7⋊S4)⋊C4C212yesmaximal, complemented
6#90318—31nomaximal, Hall
7C2×A4×D7C2211yescharacteristic
8C2×C7⋊S4C2212yes—
9C4×C7×A4C2211yescharacteristic
10#66677—41nomaximal
11A4×Dic7C2211yescharacteristic
12C7×SL(2,ℤ/4)C2212yes—
13C2×#1161—31no—
14C22×C4×D7D311yescharacteristic, complemented
15C7×C22⋊Q8—31no—
16#6509—31no—
17#6493—31no—
18#6458—31no—
19#6457—32no—
20#6456—32no—
21#6455—31no—
22#6445—32no—
23(C22⋊D7)⋊C4—32no—
24#5912—71nomaximal, Hall
25C2×C7×A4C2311yescharacteristic
26A4×D7—21no—
27C7⋊S4—22no—
28D84—41no—
29C7×Dic6—41no—
30C3×C4×D7—41no—
31#4246—42no—
32D21⋊C4—42no—
33C23×D7C2×D311yescharacteristic
34C2×C7×Q8—31no—
35C22×C4×C7C2×D311yescharacteristic
36C2×C22⋊D7—32no—
37C22×Dic7C2×D311yescharacteristic
38#1161—61no—
39#1161—61no—
40#1161—61no—
41#1161—61no—
42C2×D28—32no—
43C2×D28—32no—
44C2×D28—31no—
45C2×C4×D7—31no—
46C2×C4×D7—31no—
47C2×C4×D7—31no—
48C2×C4×D7—31no—
49C2×C4×D7—32no—
50C7×C4⋊C4—31no—
51C7×C4⋊C4—32no—
52C7×C22⋊C4—32no—
53(C2×D7)⋊C4—32no—
54(C2×D7)⋊C4—32no—
55(C2×D7)⋊C4—32no—
56(C2×D7)⋊C4—32no—
57C4⋊Dic7—31no—
58Dic7⋊C4—32no—
59C4×Dic7—32no—
60C2×C4×A4—71no—
61GL(2,ℤ/4)—72no—
62C4⋊S4—71no—
63C4×S4—72no—
64A4⋊Q8D711yescharacteristic, complemented
65C2×D21—42no—
66C2×C3×D7—41no—
67C7×A4#4111yescharacteristic, complemented
68C3×C4×C7—41no—
69C3×Dic7—41no—
70C7×Dic3—42no—
71#525—211noSylow
72C23×C7C22×D311yescharacteristic
73C22×D7—21no—
74C22×D7—31no—
75C22×D7—31no—
76C22×D7—32no—
77C22×D7—61no—
78C7×Q8—61no—
79C7×Q8—61no—
80C2×C4×C7—31no—
81C2×C4×C7—31no—
82C2×C4×C7—32no—
83C2×C4×C7—32no—
84C22⋊D7—62no—
85C22⋊D7—62no—
86C2×Dic7—31no—
87C2×Dic7—31no—
88C2×Dic7—32no—
89D28—61no—
90D28—64no—
91D28—62no—
92D28—62no—
93D28—61no—
94C4×D7—31no—
95C4×D7—31no—
96C4×D7—31no—
97C4×D7—61no—
98C4×D7—62no—
99C4×D7—62no—
100C4×D7S411yescharacteristic, complemented
101C22×A4—71no—
102C2×S4—72no—
103#235—281no—
104C4×A4—71no—
105C4×A4C2×D711yescharacteristic
106SL(2,ℤ/4)C2×D712yes—
107C2×C3×C7—41no—
108D21—82no—
109C3×D7—81no—
110C2×#41—211no—
111C23×C4—71no—
112#123—212no—
113C22⋊Q8—31no—
114C22⋊Q8—211no—
115#121—211no—
116#121—211no—
117#120—212no—
118C4×D4—212no—
119C4×D4—212no—
120#117—211no—
121C22×C7—31no—
122C22×C7—31no—
123C22×C7#23511yescharacteristic, complemented
124C2×D7—32no—
125C2×D7—31no—
126C2×D7—61no—
127C2×D7—61no—
128C2×D7—61no—
129C2×D7—62no—
130C2×D7—62no—
131C2×D7—62no—
132C2×D7C2×S411yescharacteristic
133C4×C7—31no—
134C4×C7—62no—
135C4×C7—62no—
136C4×C7C2×S411yescharacteristic
137Dic7—31no—
138Dic7—62no—
139Dic7C2×S411yescharacteristic
140C2×A4—141no—
141C2×A4C22×D711yescharacteristic
142S4—142no—
143C2×C3×C4—281no—
144#67—282no—
145D12—281no—
146C4×D3—282no—
147Dic6—41no—
148C3×C7—41noHall
149C24—71no—
150#41—421no—
151#41—421no—
152#41—421no—
153#41—421no—
154C2×Q8—31no—
155C2×D4—212no—
156C2×D4—212no—
157C2×D4—211no—
158C2×D4—212no—
159C22×C4—71no—
160C22×C4—211no—
161C22×C4—211no—
162C22×C4—211no—
163C22×C4—211no—
164C22×C4—212no—
165C22×C4D3×D711yescharacteristic
166C4⋊C4—31no—
167C4⋊C4—32no—
168C4⋊C4—212no—
169C4⋊C4—211no—
170C22⋊C4—32no—
171C22⋊C4—212no—
172C22⋊C4—212no—
173C22⋊C4—212no—
174C22⋊C4—212no—
175C42—212no—
176C2×C7—31no—
177C2×C7—31no—
178C2×C7C22×S411yescharacteristic
179D7—21no—
180D7—61no—
181D7—122no—
182C22×C3—281no—
183C2×D3—282no—
184A4#116111yescharacteristic, complemented
185C3×C4—41no—
186C3×C4—281no—
187Dic3—42no—
188C23—141no—
189C23—211no—
190C23—211no—
191C23—212no—
192C23—421no—
193C23C2×D3×D711yescharacteristic
194Q8—61no—
195Q8—61no—
196D4—421no—
197D4—422no—
198D4—424no—
199D4—422no—
200D4—422no—
201D4—422no—
202D4—421no—
203C2×C4—31no—
204C2×C4—31no—
205C2×C4—32no—
206C2×C4—32no—
207C2×C4—71no—
208C2×C4—211no—
209C2×C4—211no—
210C2×C4—211no—
211C2×C4—211no—
212C2×C4—211no—
213C2×C4—212no—
214C2×C4—421no—
215C2×C4—422no—
216C2×C4—422no—
217C7#591211yescharacteristic, complemented, Sylow
218C2×C3—41no—
219C2×C3—561no—
220D3—562no—
221C22—31no—
222C22—31no—
223C22—71no—
224C22—212no—
225C22—211no—
226C22—421no—
227C22—421no—
228C22—421no—
229C22—422no—
230C22—422no—
231C22—422no—
232C22#6667711yescharacteristic, complemented
233C4—31no—
234C4—62no—
235C4—62no—
236C4—71no—
237C4—211no—
238C4—422no—
239C4D7×S411yescharacteristic
240C3—41noSylow
241C2—31no—
242C2—31no—
243C2—141no—
244C2—421no—
245C2—842no—
246C2C2×D7×S411yescharacteristic
subgroupG/H
center, FrattiniC2C2×D7×S4
commutator, cosocleC2×C7×A4C23
FittingC22×C4×C7C2×D3
socleC23×C7C22×D3
hypercenterC4D7×S4
hypocenterC7×A4#41
LCSG ▷ C2×C7×A4 ▷ C7×A4
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
1. C4×A4×D7C2semidirect
2. C4⋊(C7⋊S4)C2semidirect
3. C7×A4⋊Q8C2semidirect
4. #10621483C2semidirect
5. (C7⋊S4)⋊C4C2semidirect
14. C22×C4×D7D3semidirect
64. A4⋊Q8D7semidirect
67. C7×A4#41semidirect
100. C4×D7S4semidirect
123. C22×C7#235semidirect
184. A4#1161semidirect
217. C7#5912semidirect
232. C22#66677semidirect

Maximal subgroups

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

subgroupindex
C4×A4×D72
C4⋊(C7⋊S4)2
C7×A4⋊Q82
#106214832
(C7⋊S4)⋊C42
#903183
#666774
#59127