Finite Group Databasebeta

← #13271245

Subgroups — #13271245

Order 1792.

490 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 127 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×#3407C211yesmaximal, characteristic, complemented
2#11741244C211yesmaximal, characteristic
3#11729766C211yesmaximal, characteristic, complemented
4#11729194C212yesmaximal, complemented
5#11725624C212yesmaximal, complemented
6C2×C7×#131C2211yescharacteristic
7#90521—21no—
8C22×D56C2211yescharacteristic
9C7×#504—24no—
10C2×C7×M32—22no—
11C2×C7×M32C2212yes—
12#90012—21no—
13#89907C414yescomplemented
14#89903—21no—
15#89887—22no—
16#89878C2212yes—
17#89878C2211yescharacteristic
18#89352—21no—
19#89304—24no—
20#13074—71nomaximal, Sylow
21C2×C7×#41D411yescharacteristic
22C22×D28—21no—
23C7×#131—22no—
24C7×#131—22no—
25C2×C7×M16C2×C412yes—
26C2×C7×M16D411yescharacteristic
27C22×C7×C8D412yes—
28C22×C7×C8C2311yescharacteristic
29#6510—41no—
30#6503—41no—
31#6500—42no—
32#6493—24no—
33#6489—21no—
34#6489—22no—
35C2×D56—22no—
36C2×D56—22no—
37C2×D56—21no—
38C2×D56C2×C412yes—
39C7×M32—24no—
40C7×M32—22no—
41C7×M32—22no—
42C2×C7×C16—24no—
43C2×C7×C16D414yescomplemented
44#6408—42no—
45#6395—24no—
46#6392—21no—
47#6392—22no—
48#6391—22no—
49#6391C2×C412yes—
50#3407D711yescharacteristic, complemented
51#3058—71no—
52#2140—72no—
53#1812—71no—
54#1384—72no—
55C23×D7—41no—
56C7×#41—22no—
57C7×#41—22no—
58C2×C7×Q8SD1611yescharacteristic
59C2×C7×D4D812yes—
60C2×C7×D4SD1611yescharacteristic
61C22×C4×C7C22⋊C412yes—
62C22×C4×C7C2×D411yescharacteristic
63C2×C22⋊D7—44no—
64C2×D28—22no—
65C2×D28—42no—
66C2×D28—42no—
67C2×D28—41no—
68C7×M16—22no—
69C7×M16—24no—
70C2×C7×C8—22no—
71C2×C7×C8—22no—
72C2×C7×C8C22⋊C412yes—
73C2×C7×C8C22×C411yescharacteristic
74C2×C7×C8C2×D412yes—
75C2×C7×C8C2×D411yescharacteristic
76C2×C7×C8#4112yes—
77C22⋊Dic7—41no—
78C22⋊Dic7—42no—
79(C2×D7)⋊C4—44no—
80C4⋊Dic7—22no—
81C4⋊Dic7—42no—
82C4×Dic7—41no—
83C4×Dic7—42no—
84D56—24no—
85D56—41no—
86D56—41no—
87C7×C16—24no—
88C22×D8—71no—
89C2×#131C2×D711yescharacteristic
90#535—141no—
91#504—24no—
92C2×M32—22no—
93C2×M32C2×D712yes—
94#469—74no—
95#466—142no—
96#427—71no—
97#427—72no—
98#418—141no—
99#410—141no—
100#357—144no—
101H(3,ℤ/4)—141no—
102C23×C7#10212yes—
103C23×C7C2×D811yescharacteristic
104C22×D7—42no—
105C22×D7—82no—
106C22×D7—162no—
107C7×Q8—22no—
108C7×D4—24no—
109C7×D4—22no—
110C2×C4×C7—22no—
111C2×C4×C7—22no—
112C2×C4×C7#10212yes—
113C2×C4×C7C2×C22⋊C411yescharacteristic
114C2×C4×C7#12112yes—
115C2×C4×C7C2×SD1611yescharacteristic
116C2×C4×C7#13512yes—
117C22⋊D7—164no—
118C2×Dic7—42no—
119C2×Dic7—82no—
120D28—42no—
121D28—42no—
122D28—82no—
123C7×C8—22no—
124C7×C8C4×D412yes—
125C7×C8#12011yescharacteristic
126C7×C8#12311yescharacteristic
127C2×#41D2811yescharacteristic
128C22×D4—141no—
129C2×D8—72no—
130C2×D8—142no—
131C2×D8—142no—
132C2×D8—141no—
133#131—22no—
134#131—22no—
135C2×M16C4×D712yes—
136C2×M16D2811yescharacteristic
137C22×C8C22⋊D712yes—
138C22×C8C22×D711yescharacteristic
139#127—281no—
140#124—281no—
141#121—144no—
142#120—282no—
143#117—141no—
144#117—142no—
145M32—24no—
146M32—22no—
147M32—22no—
148C2×C16—24no—
149C2×C16C22⋊D714yescomplemented
150#107—141no—
151#107—142no—
152#106—72no—
153#106—142no—
154#102—144no—
155#99—282no—
156C22×C7—22no—
157C22×C7—41no—
158C22×C7—42no—
159C22×C7C2×#10211yescharacteristic
160C22×C7#46612yes—
161C2×D7—84no—
162C2×D7—82no—
163C2×D7—164no—
164C4×C7—22no—
165C4×C7#38611yescharacteristic
166C4×C7#41611yescharacteristic
167C4×C7#46512yes—
168Dic7—82no—
169Dic7—162no—
170C24—281no—
171#41—22no—
172#41—22no—
173C2×Q8#113211yescharacteristic
174C2×D4—142no—
175C2×D4—284no—
176C2×D4—281no—
177C2×D4—282no—
178C2×D4—282no—
179C2×D4#113211yescharacteristic
180C2×D4D5612yes—
181C22×C4(C2×D7)⋊C412yes—
182C22×C4C2×D2811yescharacteristic
183D8—144no—
184D8—281no—
185D8—281no—
186M16—22no—
187M16—24no—
188C2×C8—22no—
189C2×C8—22no—
190C2×C8(C2×D7)⋊C412yes—
191C2×C8C2×C4×D711yescharacteristic
192C2×C8#115712yes—
193C2×C8C2×C22⋊D712yes—
194C2×C8C2×C22⋊D711yescharacteristic
195C4⋊C4—142no—
196C4⋊C4—282no—
197C22⋊C4—281no—
198C22⋊C4—284no—
199C22⋊C4—282no—
200C42—281no—
201C42—282no—
202C16—24no—
203C2×C7—21no—
204C2×C7—22no—
205C2×C7—42no—
206C2×C7#189311yescharacteristic
207D7—162no—
208C23—282no—
209C23—562no—
210C23—1122no—
211C23#639512yes—
212C23C2×D5611yescharacteristic
213Q8—22no—
214D4—24no—
215D4—22no—
216D4—282no—
217D4—282no—
218D4—562no—
219D4—1124no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—282no—
223C2×C4—562no—
224C2×C4#639512yes—
225C2×C4C2×#113211yescharacteristic
226C2×C4#646712yes—
227C2×C4C2×(C2×D7)⋊C411yescharacteristic
228C2×C4#649312yes—
229C8—22no—
230C8C4×C22⋊D712yes—
231C8#649211yescharacteristic
232C8#651611yescharacteristic
233C7#1307411yescharacteristic, complemented, Sylow
234C22—22no—
235C22—41no—
236C22—42no—
237C22—564no—
238C22—562no—
239C22—1124no—
240C22C2×#639511yescharacteristic
241C22#8988712yes—
242C4—22no—
243C4—562no—
244C4—1122no—
245C4#8988511yescharacteristic
246C4#8988612yes—
247C4#8998511yescharacteristic
248C2—21no—
249C2—22no—
250C2—42no—
251C2—1122no—
252C2#1172972711yescharacteristic
subgroupG/H
centerC2#11729727
commutatorC2×C7×C8C22×C4
FrattiniC22×C8C22×D7
FittingC7×#3407C2
socleC2×C7#1893
prime radical#11741244C2
cosocleC22×C7×C8C23
hypercenter#3407D7
hypocenterC7#13074
LCSG ▷ C2×C7×C8 ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ #3407

Complements

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

NH ≅ G/Ntype
1. C7×#3407C2semidirect
3. #11729766C2semidirect
4. #11729194C2semidirect
5. #11725624C2semidirect
13. #89907C4semidirect
43. C2×C7×C16D4semidirect
50. #3407D7semidirect
149. C2×C16C22⋊D7semidirect
233. C7#13074semidirect

Maximal subgroups

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

subgroupindex
C7×#34072
#117412442
#117297662
#117291942
#117256242
#130747