Finite Group Databasebeta

← #13271251

Subgroups — #13271251

Order 1792.

458 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 131 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×#3059C211yesmaximal, characteristic
2#11741225C211yesmaximal, characteristic
3#11729815C212yesmaximal, complemented
4#11729766C211yesmaximal, characteristic
5#11725531C212yesmaximal, complemented
6C22×C7×Q16C2211yescharacteristic
7C2×C7×#131C2211yescharacteristic
8C7×#536—21no—
9C7×#470C414yescomplemented
10C7×#467—22no—
11C7×#427C2211yescharacteristic
12C7×#419—21no—
13#90012—21no—
14#89912—24no—
15#89878C2212yes—
16C2×#6386—22no—
17C2×#6386C2212yes—
18#89359—24no—
19#89352—21no—
20#13075—71nomaximal, Sylow
21C2×C7×#41D411yescharacteristic
22C22×C7×Q8—21no—
23C2×C7×Q16—22no—
24C2×C7×Q16—21no—
25C2×C7×Q16—22no—
26C2×C7×Q16C2×C412yes—
27C7×#131—22no—
28C7×#131—22no—
29C2×C7×M16C2×C412yes—
30C2×C7×M16D411yescharacteristic
31C22×C7×C8D412yes—
32C22×C7×C8C2311yescharacteristic
33C7×C4⋊Q8—41no—
34C7×#124—41no—
35C7×#123—42no—
36C7×C22⋊Q8—24no—
37C7×#117—21no—
38#6489—22no—
39C7×#107—21no—
40C7×#106C2×C412yes—
41C7×#103—24no—
42#6408—42no—
43#6392—22no—
44#6391—22no—
45#6386—24no—
46#6386—22no—
47#6386—22no—
48C2×C7⋊C16—24no—
49C2×C7⋊C16D414yes—
50#3407—71no—
51#3059D711yescharacteristic
52#2141—72no—
53#1812—71no—
54#1384—72no—
55C7×#41—22no—
56C7×#41—22no—
57C2×C7×Q8—22no—
58C2×C7×Q8—41no—
59C2×C7×Q8—42no—
60C2×C7×Q8—42no—
61C2×C7×Q8SD1611yescharacteristic
62C2×C7×D4D812yes—
63C2×C7×D4SD1611yescharacteristic
64C22×C4×C7—41no—
65C22×C4×C7C22⋊C412yes—
66C22×C4×C7C2×D411yescharacteristic
67C7×Q16—24no—
68C7×Q16—41no—
69C7×Q16—41no—
70C7×M16—22no—
71C7×M16—24no—
72C2×C7×C8—22no—
73C2×C7×C8—22no—
74C2×C7×C8C22⋊C412yes—
75C2×C7×C8C22×C411yescharacteristic
76C2×C7×C8C2×D412yes—
77C2×C7×C8C2×D411yescharacteristic
78C2×C7×C8#4112yes—
79C7×C4⋊C4—22no—
80C7×C4⋊C4—44no—
81C7×C22⋊C4—44no—
82C7×C22⋊C4—41no—
83C42×C7—41no—
84C22⋊Dic7—42no—
85C4⋊Dic7—42no—
86C4×Dic7—42no—
87C7⋊C16—24no—
88C22×Q16C2×D711yescharacteristic
89C2×#131C2×D711yescharacteristic
90#536—21no—
91#504—144no—
92C2×M32—72no—
93C2×M32—142no—
94#470Dic714yescomplemented
95#467—22no—
96#427—72no—
97#427C2×D711yescharacteristic
98#419—21no—
99#410—141no—
100#358—144no—
101H(3,ℤ/4)—141no—
102C23×C7#10212yes—
103C23×C7C2×D811yescharacteristic
104C7×Q8—22no—
105C7×Q8—42no—
106C7×Q8—42no—
107C7×Q8—82no—
108C7×D4—24no—
109C7×D4—22no—
110C2×C4×C7—22no—
111C2×C4×C7—22no—
112C2×C4×C7—42no—
113C2×C4×C7—42no—
114C2×C4×C7—82no—
115C2×C4×C7#10212yes—
116C2×C4×C7C2×C22⋊C411yescharacteristic
117C2×C4×C7#12112yes—
118C2×C4×C7C2×SD1611yescharacteristic
119C2×C4×C7#13512yes—
120C2×Dic7—82no—
121C7×C8—22no—
122C7×C8C4×D412yes—
123C7×C8#12011yescharacteristic
124C7×C8#12311yescharacteristic
125C2×#41C22⋊D711yescharacteristic
126C22×Q8—21no—
127C2×Q16—22no—
128C2×Q16—22no—
129C2×Q16—21no—
130C2×Q16C2×Dic712yes—
131#131—22no—
132#131—22no—
133C2×M16C2×Dic712yes—
134C2×M16C22⋊D711yescharacteristic
135C22×C8C22⋊D712yes—
136C22×C8C22×D711yescharacteristic
137C4⋊Q8—41no—
138#124—41no—
139#123—42no—
140C22⋊Q8—24no—
141#117—21no—
142#117—142no—
143M32—144no—
144M32—142no—
145M32—142no—
146C2×C16—74no—
147C2×C16—144no—
148#107—21no—
149#107—142no—
150#106—142no—
151#106C2×Dic712yes—
152#103—24no—
153#99—282no—
154C22×C7—22no—
155C22×C7—41no—
156C22×C7—42no—
157C22×C7C2×#10211yescharacteristic
158C22×C7#46612yes—
159C4×C7—22no—
160C4×C7—82no—
161C4×C7—82no—
162C4×C7#38611yescharacteristic
163C4×C7#41611yescharacteristic
164C4×C7#46512yes—
165Dic7—162no—
166#41—22no—
167#41—22no—
168C2×Q8—22no—
169C2×Q8—41no—
170C2×Q8—42no—
171C2×Q8—42no—
172C2×Q8#114211yescharacteristic
173C2×D4#114112yes—
174C2×D4#114211yescharacteristic
175C22×C4—41no—
176C22×C4C22⋊Dic712yes—
177C22×C4C2×C22⋊D711yescharacteristic
178Q16—24no—
179Q16—41no—
180Q16—41no—
181M16—22no—
182M16—24no—
183C2×C8—22no—
184C2×C8—22no—
185C2×C8C22⋊Dic712yes—
186C2×C8D4×D712yes—
187C2×C8#115912yes—
188C2×C8C22×Dic711yescharacteristic
189C2×C8C2×C22⋊D711yescharacteristic
190C4⋊C4—22no—
191C4⋊C4—44no—
192C4⋊C4—282no—
193C22⋊C4—44no—
194C22⋊C4—41no—
195C22⋊C4—282no—
196C42—41no—
197C42—282no—
198C16—144no—
199C2×C7—22no—
200C2×C7—21no—
201C2×C7—42no—
202C2×C7#189311yescharacteristic
203C23#640612yes—
204C23C2×#114111yescharacteristic
205Q8—22no—
206Q8—42no—
207Q8—42no—
208Q8—82no—
209D4—24no—
210D4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—82no—
216C2×C4—562no—
217C2×C4#640612yes—
218C2×C4C2×#114211yescharacteristic
219C2×C4#650212yes—
220C2×C4#651312yes—
221C2×C4C2×C22⋊Dic711yescharacteristic
222C8—22no—
223C8D4×Dic712yes—
224C8#649811yescharacteristic
225C8#650011yescharacteristic
226C7#1307511yescharacteristic, complemented, Sylow
227C22—22no—
228C22—41no—
229C22—42no—
230C22#8981012yes—
231C22C2×#640611yescharacteristic
232C4—22no—
233C4—82no—
234C4—82no—
235C4—1122no—
236C4#8981312yes—
237C4#8999211yescharacteristic
238C4#9000511yescharacteristic
239C2—21no—
240C2—22no—
241C2—42no—
242C2#1172943511yescharacteristic
subgroupG/H
centerC2#11729435
commutatorC2×C7×C8C22×C4
FrattiniC22×C8C22×D7
FittingC7×#3059C2
socleC2×C7#1893
prime radicalC2×C7×#41D4
cosocleC22×C7×C8C23
hypercenter#3059D7
hypocenterC7#13075
LCSG ▷ C2×C7×C8 ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ #3059

Complements

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

NH ≅ G/Ntype
3. #11729815C2semidirect
5. #11725531C2semidirect
9. C7×#470C4semidirect
94. #470Dic7semidirect
226. C7#13075semidirect

Maximal subgroups

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

subgroupindex
C7×#30592
#117412252
#117298152
#117297662
#117255312
#130757