Finite Group Databasebeta

← #13062109

Subgroups — #13062109

Order 1344.

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

normal subgroups only

This group has 342 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
1C2×C4×C7×A4C211yesmaximal, characteristic, complemented
2#10621506C212yesmaximal, complemented
3C4⋊(C7⋊S4)C211yesmaximal, characteristic, complemented
4C4×C7⋊S4C212yesmaximal, complemented
5#10621496C211yesmaximal, characteristic, complemented
6#90482—31nomaximal, Hall
7C22×C7×A4C2211yescharacteristic
8C2×C7⋊S4C2212yes—
9#66728—41nomaximal
10C4×C7×A4C2211yescharacteristic
11C4×C7×A4C2211yescharacteristic
12C7⋊SL(2,ℤ/4)C2212yes—
13C23×C4×C7D311yescharacteristic, complemented
14C2×#1157—31no—
15#6516—32no—
16#6493—31no—
17#6493—31no—
18#6492—32no—
19C4×C22⋊D7—32no—
20C4×C22⋊D7—32no—
21#6489—31no—
22#6487—31no—
23#6487—31no—
24#5912—71nomaximal, Hall
25C2×C7×A4—21no—
26C2×C7×A4C2311yescharacteristic
27C7⋊S4—22no—
28C2×C3×C4×C7—41no—
29#4268—42no—
30D84—41no—
31C4×D21—42no—
32Dic42—41no—
33C24×C7C2×D311yescharacteristic
34C22×C4×C7—31no—
35C22×C4×C7—31no—
36C22×C4×C7—31no—
37C22×C4×C7—31no—
38C22×C4×C7C2×D311yescharacteristic
39C22×C4×C7C2×D311yescharacteristic
40C2×C22⋊D7—32no—
41C2×C22⋊D7—32no—
42C2×C22⋊D7—32no—
43#1157—61no—
44#1157—61no—
45#1157—61no—
46#1157—61no—
47C2×D28—31no—
48C2×C4×D7—32no—
49C2×Dic14—31no—
50C22⋊Dic7—32no—
51C22⋊Dic7—32no—
52C22⋊Dic7—32no—
53(C2×D7)⋊C4—32no—
54(C2×D7)⋊C4—32no—
55C4⋊Dic7—31no—
56C4⋊Dic7—31no—
57Dic7⋊C4—32no—
58Dic7⋊C4—32no—
59C4×Dic7—32no—
60C2×C4×A4D711yescharacteristic, complemented
61GL(2,ℤ/4)—72no—
62C4⋊S4—71no—
63C4×S4—72no—
64A4⋊Q8—71no—
65C22×C3×C7—41no—
66C2×D21—42no—
67C7×A4#4111yescharacteristic, complemented
68C3×C4×C7—41no—
69C3×C4×C7—41no—
70Dic21—42no—
71#525—211noSylow
72C23×C7—21no—
73C23×C7—31no—
74C23×C7—31no—
75C23×C7—61no—
76C23×C7C22×D311yescharacteristic
77C22×D7—32no—
78C2×C4×C7—31no—
79C2×C4×C7—31no—
80C2×C4×C7—31no—
81C2×C4×C7—31no—
82C2×C4×C7—31no—
83C2×C4×C7—31no—
84C2×C4×C7—31no—
85C2×C4×C7—61no—
86C2×C4×C7S411yescharacteristic, complemented
87C22⋊D7—62no—
88C22⋊D7—62no—
89C22⋊D7—64no—
90C22⋊D7—62no—
91C22⋊D7—62no—
92C2×Dic7—32no—
93C2×Dic7—32no—
94C2×Dic7—32no—
95D28—61no—
96D28—61no—
97C4×D7—62no—
98C4×D7—62no—
99Dic14—61no—
100Dic14—61no—
101C22×A4C2×D711yescharacteristic
102C2×S4—72no—
103#235—281no—
104C4×A4C2×D711yescharacteristic
105C4×A4C2×D711yescharacteristic
106SL(2,ℤ/4)—72no—
107C2×C3×C7—41no—
108C2×C3×C7—81no—
109D21—82no—
110C2×#41—211no—
111C23×C4D2111yescharacteristic, complemented
112#123—212no—
113C22⋊Q8—211no—
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—32no—
124C22×C7—31no—
125C22×C7—61no—
126C22×C7—61no—
127C22×C7—61no—
128C22×C7#23511yescharacteristic, complemented
129C22×C7C2×S411yescharacteristic
130C2×D7—62no—
131C2×D7—62no—
132C2×D7—62no—
133C4×C7—31no—
134C4×C7—31no—
135C4×C7C2×S411yescharacteristic
136C4×C7C2×S411yescharacteristic
137Dic7—62no—
138Dic7—62no—
139Dic7—62no—
140C2×A4—21no—
141C2×A4C22×D711yescharacteristic
142S4—142no—
143C2×C3×C4—41no—
144#67—282no—
145D12—281no—
146C4×D3—282no—
147Dic6—281no—
148C3×C7—41noHall
149C24C2×D2111yescharacteristic
150#41—421no—
151#41—421no—
152#41—421no—
153#41—421no—
154C2×Q8—211no—
155C2×D4—211no—
156C2×D4—212no—
157C2×D4—212no—
158C2×D4—212no—
159C22×C4—31no—
160C22×C4—31no—
161C22×C4—31no—
162C22×C4—31no—
163C22×C4—212no—
164C22×C4C2×D2111yescharacteristic
165C22×C4C2×D2111yescharacteristic
166C4⋊C4—212no—
167C4⋊C4—212no—
168C4⋊C4—211no—
169C4⋊C4—211no—
170C22⋊C4—212no—
171C22⋊C4—212no—
172C22⋊C4—212no—
173C22⋊C4—212no—
174C22⋊C4—212no—
175C42—212no—
176C2×C7—21no—
177C2×C7—31no—
178C2×C7—31no—
179C2×C7—61no—
180C2×C7C22×S411yescharacteristic
181D7—122no—
182C22×C3—41no—
183C2×D3—282no—
184A4#115711yescharacteristic, complemented
185C3×C4—41no—
186C3×C4—41no—
187Dic3—282no—
188C23—21no—
189C23—31no—
190C23—31no—
191C23—61no—
192C23—212no—
193C23C22×D2111yescharacteristic
194Q8—421no—
195Q8—421no—
196D4—421no—
197D4—422no—
198D4—424no—
199D4—422no—
200D4—422no—
201D4—422no—
202D4—421no—
203C2×C4—31no—
204C2×C4—31no—
205C2×C4—31no—
206C2×C4—31no—
207C2×C4—31no—
208C2×C4—31no—
209C2×C4—31no—
210C2×C4—61no—
211C2×C4—212no—
212C2×C4—212no—
213C2×C4—212no—
214C2×C4—422no—
215C2×C4—422no—
216C2×C4C7⋊S411yescharacteristic, complemented
217C7#591211yescharacteristic, complemented, Sylow
218C2×C3—41no—
219C2×C3—81no—
220D3—562no—
221C22—31no—
222C22—31no—
223C22—32no—
224C22—31no—
225C22—61no—
226C22—61no—
227C22—61no—
228C22—422no—
229C22—422no—
230C22—422no—
231C22#6672811yescharacteristic, complemented
232C22C2×C7⋊S411yescharacteristic
233C4—31no—
234C4—31no—
235C4—422no—
236C4—422no—
237C4—422no—
238C4C2×C7⋊S411yescharacteristic
239C4C2×C7⋊S411yescharacteristic
240C3—41noSylow
241C2—21no—
242C2—31no—
243C2—31no—
244C2—61no—
245C2—842no—
246C2C22×C7⋊S411yescharacteristic
subgroupG/H
centerC4C2×C7⋊S4
commutator, cosocleC2×C7×A4C23
FrattiniC2C22×C7⋊S4
FittingC23×C4×C7D3
socleC23×C7C22×D3
hypercenterC2×C4C7⋊S4
hypocenterC7×A4#41
LCSG ▷ C2×C7×A4 ▷ C7×A4
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×C4×C7×A4C2semidirect
2. #10621506C2semidirect
3. C4⋊(C7⋊S4)C2semidirect
4. C4×C7⋊S4C2semidirect
5. #10621496C2semidirect
13. C23×C4×C7D3semidirect
60. C2×C4×A4D7semidirect
67. C7×A4#41semidirect
86. C2×C4×C7S4semidirect
111. C23×C4D21semidirect
128. C22×C7#235semidirect
184. A4#1157semidirect
216. C2×C4C7⋊S4semidirect
217. C7#5912semidirect
231. C22#66728semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C7×A42
#106215062
C4⋊(C7⋊S4)2
C4×C7⋊S42
#106214962
#904823
#667284
#59127