Finite Group Databasebeta

← #13062111

Subgroups — #13062111

Order 1344.

346 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 47 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
1C7×D4×A4C211yesmaximal, characteristic, complemented
2#10621506C212yesmaximal, complemented
3C2×C7⋊SL(2,ℤ/4)C212yesmaximal, complemented
4C4×C7⋊S4C211yesmaximal, characteristic, complemented
5#10621496C211yesmaximal, characteristic, complemented
6#90497—31nomaximal, Hall
7C22×C7×A4C2212yes—
8C2×C7⋊S4C2211yescharacteristic
9#66730—41nomaximal
10C4×C7×A4C2211yescharacteristic
11C7⋊SL(2,ℤ/4)C2212yes—
12C7⋊SL(2,ℤ/4)C2211yescharacteristic
13C22×C7×D4D311yescharacteristic, complemented
14C2×#1159—31no—
15#6516—32no—
16C2×C22⋊Dic7—32no—
17#6502—32no—
18#6501—31no—
19#6499—31no—
20#6498—32no—
21D4×Dic7—31no—
22C4×C22⋊D7—31no—
23#6487—31no—
24#5914—71nomaximal, Hall
25C2×C7×A4—22no—
26C2×C7×A4C2311yescharacteristic
27C7⋊S4—21no—
28C3×C7×D4—41no—
29#4268—42no—
30C2×Dic21—42no—
31C4×D21—41no—
32Dic42—41no—
33C24×C7C2×D312yes—
34C2×C7×D4—31no—
35C2×C7×D4—32no—
36C2×C7×D4—31no—
37C22×C4×C7C2×D311yescharacteristic
38C2×C22⋊D7—32no—
39C2×C22⋊D7—31no—
40C2×C22⋊D7—32no—
41C22×Dic7—32no—
42C22×Dic7—32no—
43#1159—62no—
44#1159—61no—
45#1159—61no—
46C2×C4×D7—31no—
47C2×Dic14—31no—
48C22⋊Dic7—32no—
49C22⋊Dic7—32no—
50C22⋊Dic7—32no—
51C22⋊Dic7—32no—
52C22⋊Dic7—32no—
53C22⋊Dic7—31no—
54(C2×D7)⋊C4—31no—
55C4⋊Dic7—31no—
56Dic7⋊C4—32no—
57Dic7⋊C4—31no—
58C4×Dic7—31no—
59D4×A4D711yescharacteristic, complemented
60GL(2,ℤ/4)—72no—
61C2×SL(2,ℤ/4)—72no—
62C4×S4—71no—
63A4⋊Q8—71no—
64C22×C3×C7—42no—
65C2×D21—41no—
66C7×A4#4111yescharacteristic, complemented
67C3×C4×C7—41no—
68Dic21—42no—
69Dic21—41no—
70#546—211noSylow
71C23×C7—22no—
72C23×C7—32no—
73C23×C7—32no—
74C23×C7—62no—
75C23×C7C22×D311yescharacteristic
76C22×D7—31no—
77C7×D4—32no—
78C7×D4—31no—
79C7×D4—61no—
80C7×D4S411yescharacteristic, complemented
81C2×C4×C7—31no—
82C2×C4×C7—31no—
83C22⋊D7—62no—
84C22⋊D7—61no—
85C22⋊D7—61no—
86C22⋊D7—62no—
87C22⋊D7—122no—
88C2×Dic7—31no—
89C2×Dic7—31no—
90C2×Dic7—32no—
91C2×Dic7—32no—
92C2×Dic7—31no—
93C2×Dic7—62no—
94C2×Dic7—62no—
95C2×Dic7—122no—
96C4×D7—61no—
97C4×D7—61no—
98Dic14—61no—
99Dic14—61no—
100C22×A4C2×D712yes—
101C2×S4—71no—
102#237—281no—
103C4×A4C2×D711yescharacteristic
104SL(2,ℤ/4)—72no—
105SL(2,ℤ/4)—71no—
106C2×C3×C7—41no—
107C2×C3×C7—82no—
108D21—81no—
109C2×#41—211no—
110C22×D4D2111yescharacteristic, complemented
111#124—211no—
112#123—212no—
113C22⋊Q8—211no—
114#121—211no—
115#121—212no—
116#120—212no—
117C4×D4—211no—
118C4×D4—211no—
119C2×C22⋊C4—212no—
120C22×C7—31no—
121C22×C7—31no—
122C22×C7—32no—
123C22×C7—62no—
124C22×C7—62no—
125C22×C7—62no—
126C22×C7—62no—
127C22×C7#23711yescharacteristic, complemented
128C22×C7C2×S412yes—
129C2×D7—61no—
130C2×D7—61no—
131C2×D7—61no—
132C4×C7—31no—
133C4×C7C2×S411yescharacteristic
134Dic7—61no—
135Dic7—62no—
136Dic7—61no—
137Dic7—62no—
138Dic7—61no—
139C2×A4—22no—
140C2×A4C22×D711yescharacteristic
141S4—141no—
142C3×D4—41no—
143#67—282no—
144C2×Dic3—282no—
145C4×D3—281no—
146Dic6—281no—
147C3×C7—41noHall
148C24C2×D2112yes—
149#41—422no—
150#41—421no—
151#41—421no—
152C2×Q8—211no—
153C2×D4—31no—
154C2×D4—32no—
155C2×D4—31no—
156C2×D4—212no—
157C2×D4—211no—
158C2×D4—212no—
159C22×C4—211no—
160C22×C4—212no—
161C22×C4—212no—
162C22×C4C2×D2111yescharacteristic
163C4⋊C4—211no—
164C4⋊C4—212no—
165C4⋊C4—211no—
166C22⋊C4—211no—
167C22⋊C4—212no—
168C22⋊C4—212no—
169C22⋊C4—212no—
170C22⋊C4—212no—
171C22⋊C4—211no—
172C22⋊C4—212no—
173C42—211no—
174C2×C7—22no—
175C2×C7—31no—
176C2×C7—31no—
177C2×C7—62no—
178C2×C7C22×S411yescharacteristic
179D7—121no—
180C22×C3—42no—
181C2×D3—281no—
182A4#115911yescharacteristic, complemented
183C3×C4—41no—
184Dic3—282no—
185Dic3—281no—
186C23—22no—
187C23—32no—
188C23—32no—
189C23—62no—
190C23—211no—
191C23C22×D2111yescharacteristic
192Q8—421no—
193Q8—421no—
194D4—32no—
195D4—31no—
196D4—61no—
197D4—421no—
198D4—422no—
199D4—421no—
200D4—422no—
201D4—842no—
202D4C7⋊S411yescharacteristic, complemented
203C2×C4—31no—
204C2×C4—31no—
205C2×C4—211no—
206C2×C4—211no—
207C2×C4—212no—
208C2×C4—211no—
209C2×C4—212no—
210C2×C4—421no—
211C2×C4—422no—
212C2×C4—422no—
213C2×C4—421no—
214C2×C4—842no—
215C7#591411yescharacteristic, complemented, Sylow
216C2×C3—41no—
217C2×C3—82no—
218D3—561no—
219C22—31no—
220C22—31no—
221C22—32no—
222C22—62no—
223C22—62no—
224C22—62no—
225C22—62no—
226C22—421no—
227C22—421no—
228C22—421no—
229C22#6673011yescharacteristic, complemented
230C22C2×C7⋊S412yes—
231C4—31no—
232C4—421no—
233C4—422no—
234C4—421no—
235C4—421no—
236C4—422no—
237C4C2×C7⋊S411yescharacteristic
238C3—41noSylow
239C2—22no—
240C2—31no—
241C2—31no—
242C2—62no—
243C2—841no—
244C2C22×C7⋊S411yescharacteristic
subgroupG/H
center, FrattiniC2C22×C7⋊S4
commutator, cosocleC2×C7×A4C23
FittingC22×C7×D4D3
socleC23×C7C22×D3
hypercenterD4C7⋊S4
hypocenterC7×A4#41
LCSG ▷ C2×C7×A4 ▷ C7×A4
UCSC1 ◁ C2 ◁ D4

Complements

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

NH ≅ G/Ntype
1. C7×D4×A4C2semidirect
2. #10621506C2semidirect
3. C2×C7⋊SL(2,ℤ/4)C2semidirect
4. C4×C7⋊S4C2semidirect
5. #10621496C2semidirect
13. C22×C7×D4D3semidirect
59. D4×A4D7semidirect
66. C7×A4#41semidirect
80. C7×D4S4semidirect
110. C22×D4D21semidirect
127. C22×C7#237semidirect
182. A4#1159semidirect
202. D4C7⋊S4semidirect
215. C7#5914semidirect
229. C22#66730semidirect

Maximal subgroups

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

subgroupindex
C7×D4×A42
#106215062
C2×C7⋊SL(2,ℤ/4)2
C4×C7⋊S42
#106214962
#904973
#667304
#59147