Finite Group Databasebeta

← #13271250

Subgroups — #13271250

Order 1792.

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

normal subgroups only

This group has 458 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
2#11741252C211yesmaximal, characteristic
3#11729766C211yesmaximal, characteristic
4#11729199C212yesmaximal, complemented
5#11725624C212yesmaximal, complemented
6C2×C7×#131C2211yescharacteristic
7#90516—21no—
8C22×Dic28C2211yescharacteristic
9C7×#504—24no—
10C2×C7×M32—22no—
11C2×C7×M32C2212yes—
12#90012—21no—
13#89909C414yescomplemented
14#89892—21no—
15#89888—22no—
16#89878C2212yes—
17#89878C2211yescharacteristic
18#89352—21no—
19#89298—24no—
20#13075—71nomaximal, Sylow
21C2×C7×#41D411yescharacteristic
22C22×Dic14—21no—
23C7×#131—22no—
24C7×#131—22no—
25C2×C7×M16C2×C412yes—
26C2×C7×M16D411yescharacteristic
27C22×C7×C8D412yes—
28C22×C7×C8C2311yescharacteristic
29#6507—41no—
30#6499—41no—
31#6498—42no—
32#6489—21no—
33#6489—22no—
34#6487—24no—
35C2×Dic28—22no—
36C2×Dic28—22no—
37C2×Dic28—21no—
38C2×Dic28C2×C412yes—
39C7×M32—24no—
40C7×M32—22no—
41C7×M32—22no—
42C2×C7×C16—24no—
43C2×C7×C16D414yes—
44#6408—42no—
45#6392—21no—
46#6392—22no—
47#6391—22no—
48#6391C2×C412yes—
49#6390—24no—
50#3407D711yescharacteristic
51#3059—71no—
52#2141—72no—
53#1812—71no—
54#1384—72no—
55C7×#41—22no—
56C7×#41—22no—
57C2×C7×Q8SD1611yescharacteristic
58C2×C7×D4D812yes—
59C2×C7×D4SD1611yescharacteristic
60C22×C4×C7C22⋊C412yes—
61C22×C4×C7C2×D411yescharacteristic
62C22×Dic7—41no—
63C2×Dic14—22no—
64C2×Dic14—42no—
65C2×Dic14—42no—
66C2×Dic14—41no—
67C7×M16—22no—
68C7×M16—24no—
69C2×C7×C8—22no—
70C2×C7×C8—22no—
71C2×C7×C8C22⋊C412yes—
72C2×C7×C8C22×C411yescharacteristic
73C2×C7×C8C2×D412yes—
74C2×C7×C8C2×D411yescharacteristic
75C2×C7×C8#4112yes—
76C22⋊Dic7—44no—
77C22⋊Dic7—41no—
78C22⋊Dic7—42no—
79C4⋊Dic7—22no—
80C4⋊Dic7—42no—
81Dic7⋊C4—44no—
82C4×Dic7—41no—
83C4×Dic7—42no—
84Dic28—24no—
85Dic28—41no—
86Dic28—41no—
87C7×C16—24no—
88C22×Q16—71no—
89C2×#131C2×D711yescharacteristic
90#536—141no—
91#504—24no—
92C2×M32—22no—
93C2×M32C2×D712yes—
94#470—74no—
95#467—142no—
96#427—71no—
97#427—72no—
98#419—141no—
99#410—141no—
100#358—144no—
101H(3,ℤ/4)—141no—
102C23×C7#10212yes—
103C23×C7C2×D811yescharacteristic
104C7×Q8—22no—
105C7×D4—24no—
106C7×D4—22no—
107C2×C4×C7—22no—
108C2×C4×C7—22no—
109C2×C4×C7#10212yes—
110C2×C4×C7C2×C22⋊C411yescharacteristic
111C2×C4×C7#12112yes—
112C2×C4×C7C2×SD1611yescharacteristic
113C2×C4×C7#13512yes—
114C2×Dic7—42no—
115C2×Dic7—42no—
116C2×Dic7—82no—
117C2×Dic7—82no—
118Dic14—42no—
119Dic14—42no—
120Dic14—82no—
121C7×C8—22no—
122C7×C8C4×D412yes—
123C7×C8#12011yescharacteristic
124C7×C8#12311yescharacteristic
125C2×#41D2811yescharacteristic
126C22×Q8—141no—
127C2×Q16—72no—
128C2×Q16—142no—
129C2×Q16—142no—
130C2×Q16—141no—
131#131—22no—
132#131—22no—
133C2×M16C4×D712yes—
134C2×M16D2811yescharacteristic
135C22×C8C22⋊D712yes—
136C22×C8C22×D711yescharacteristic
137C4⋊Q8—281no—
138#124—281no—
139#123—282no—
140C22⋊Q8—144no—
141#117—141no—
142#117—142no—
143M32—24no—
144M32—22no—
145M32—22no—
146C2×C16—24no—
147C2×C16C22⋊D714yes—
148#107—141no—
149#107—142no—
150#106—72no—
151#106—142no—
152#103—144no—
153#99—282no—
154C22×C7—22no—
155C22×C7—41no—
156C22×C7—42no—
157C22×C7C2×#10211yescharacteristic
158C22×C7#46612yes—
159C4×C7—22no—
160C4×C7#38611yescharacteristic
161C4×C7#41611yescharacteristic
162C4×C7#46512yes—
163Dic7—82no—
164Dic7—82no—
165Dic7—162no—
166#41—22no—
167#41—22no—
168C2×Q8—142no—
169C2×Q8—281no—
170C2×Q8—282no—
171C2×Q8—282no—
172C2×Q8#113211yescharacteristic
173C2×D4#113211yescharacteristic
174C2×D4D5612yes—
175C22×C4—281no—
176C22×C4(C2×D7)⋊C412yes—
177C22×C4C2×D2811yescharacteristic
178Q16—144no—
179Q16—281no—
180Q16—281no—
181M16—22no—
182M16—24no—
183C2×C8—22no—
184C2×C8—22no—
185C2×C8(C2×D7)⋊C412yes—
186C2×C8C2×C4×D711yescharacteristic
187C2×C8#115712yes—
188C2×C8C2×C22⋊D712yes—
189C2×C8C2×C22⋊D711yescharacteristic
190C4⋊C4—142no—
191C4⋊C4—284no—
192C4⋊C4—282no—
193C22⋊C4—281no—
194C22⋊C4—284no—
195C22⋊C4—282no—
196C42—281no—
197C42—282no—
198C16—24no—
199C2×C7—21no—
200C2×C7—22no—
201C2×C7—42no—
202C2×C7#189311yescharacteristic
203C23#639512yes—
204C23C2×D5611yescharacteristic
205Q8—22no—
206Q8—282no—
207Q8—282no—
208Q8—562no—
209D4—24no—
210D4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—282no—
214C2×C4—282no—
215C2×C4—562no—
216C2×C4—562no—
217C2×C4#639512yes—
218C2×C4C2×#113211yescharacteristic
219C2×C4#646712yes—
220C2×C4C2×(C2×D7)⋊C411yescharacteristic
221C2×C4#649312yes—
222C8—22no—
223C8C4×C22⋊D712yes—
224C8#649211yescharacteristic
225C8#651611yescharacteristic
226C7#1307511yescharacteristic, complemented, Sylow
227C22—22no—
228C22—41no—
229C22—42no—
230C22C2×#639511yescharacteristic
231C22#8988712yes—
232C4—22no—
233C4—562no—
234C4—562no—
235C4—1122no—
236C4#8988511yescharacteristic
237C4#8988612yes—
238C4#8998511yescharacteristic
239C2—21no—
240C2—22no—
241C2—42no—
242C2#1172972711yescharacteristic
subgroupG/H
centerC2#11729727
commutatorC2×C7×C8C22×C4
FrattiniC22×C8C22×D7
FittingC7×#3407C2
socleC2×C7#1893
prime radicalC2×C7×#41D4
cosocleC22×C7×C8C23
hypercenter#3407D7
hypocenterC7#13075
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
4. #11729199C2semidirect
5. #11725624C2semidirect
13. #89909C4semidirect
226. C7#13075semidirect

Maximal subgroups

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

subgroupindex
C7×#34072
#117412522
#117297662
#117291992
#117256242
#130757