Finite Group Databasebeta

← #13266851

Subgroups — #13266851

Order 1792.

442 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 63 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×#2551C211yesmaximal, characteristic
2#11727461C212yesmaximal
3#11727327C211yesmaximal, characteristic
4#11725454C212yesmaximal
5#11725441C211yesmaximal, characteristic
6C2×C7×C4⋊Q8—22no—
7#90191C2211yescharacteristic
8C7×#395—21no—
9C7×#395C2212yes—
10C7×#389—22no—
11C7×#384C2211yescharacteristic
12C4×C7×C4⋊C4—21no—
13#90000—22no—
14#89736—21no—
15#89325C2211yescharacteristic
16#89323C2212yes—
17#89318—21no—
18#89247—22no—
19#12256—71nomaximal, Sylow
20C22×C7×Q8—21no—
21C22×C7×Q8D412yes—
22C7×C4⋊Q8—48no—
23C2×C7×C4⋊C4—21no—
24C2×C7×C4⋊C4—21no—
25C2×C7×C4⋊C4—22no—
26C2×C7×C4⋊C4—22no—
27C2×C7×C4⋊C4—22no—
28C2×C7×C4⋊C4D411yescharacteristic
29C2×C7×C4⋊C4C2311yescharacteristic
30C2×C42×C7—22no—
31C2×C42×C7D411yescharacteristic
32C2×#1144—42no—
33#6489—21no—
34#6487—22no—
35C2×#1136—21no—
36#6453—41no—
37#6451—41no—
38C7×#95—21no—
39C7×#95—22no—
40C7×#95D412yes—
41#6409—42no—
42C22⋊(C7⋊C8)—22no—
43#6382—41no—
44#6381—41no—
45#2551D711yescharacteristic
46#2060—71no—
47#1605—72no—
48#1354—72no—
49#1325—71no—
50C2×C7×Q8—44no—
51C2×C7×Q8—44no—
52C2×C7×Q8—42no—
53C2×C7×Q8—81no—
54C22×C4×C7—21no—
55C22×C4×C7—22no—
56C22×C4×C7—22no—
57C22×C4×C7—21no—
58C22×C4×C7C2×D411yescharacteristic
59C22×C4×C7C2×D412yes—
60C2×Dic14—22no—
61C7×C4⋊C4—24no—
62C7×C4⋊C4—24no—
63C7×C4⋊C4—41no—
64C7×C4⋊C4—41no—
65C7×C4⋊C4—44no—
66C7×C4⋊C4—44no—
67C7×C4⋊C4—44no—
68C42×C7—21no—
69C42×C7—44no—
70C42×C7Q1612yescomplemented
71C22⋊Dic7—41no—
72#1144—84no—
73C4⋊Dic7—22no—
74C4⋊Dic7—21no—
75C4⋊Dic7—21no—
76Dic7⋊C4—42no—
77C4×Dic7—41no—
78#1136—22no—
79#1136—41no—
80C2×C7⋊C8—41no—
81C2×C7⋊C8—42no—
82#544—71no—
83C2×C4⋊Q8—22no—
84#451—142no—
85#428—141no—
86#395—21no—
87#395C2×D712yes—
88#389—22no—
89#384C2×D711yescharacteristic
90C4×C4⋊C4—21no—
91#343—71no—
92#339—141no—
93#328—142no—
94#324—72no—
95C23×C7#12011yescharacteristic
96C7×Q8—44no—
97C7×Q8—82no—
98C7×Q8—84no—
99C2×C4×C7—22no—
100C2×C4×C7—22no—
101C2×C4×C7—22no—
102C2×C4×C7—21no—
103C2×C4×C7—21no—
104C2×C4×C7—24no—
105C2×C4×C7—24no—
106C2×C4×C7—22no—
107C2×C4×C7—42no—
108C2×C4×C7—42no—
109C2×C4×C7—42no—
110C2×C4×C7—42no—
111C2×C4×C7—42no—
112C2×C4×C7—42no—
113C2×C4×C7—41no—
114C2×C4×C7C2×Q1611yescharacteristic
115C2×C4×C7#13711yescharacteristic
116C2×Dic7—41no—
117C2×Dic7—41no—
118Dic14—82no—
119C7⋊C8—42no—
120C7⋊C8—82no—
121C22×Q8—21no—
122C22×Q8C22⋊D712yes—
123C2×Q16—282no—
124C2×M16—141no—
125C4⋊Q8—48no—
126C4⋊Q8—281no—
127#125—281no—
128C22⋊Q8—142no—
129#117—141no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4C22⋊D711yescharacteristic
136C2×C4⋊C4C22×D711yescharacteristic
137C2×C42—22no—
138C2×C42C22⋊D711yescharacteristic
139#107—281no—
140#106—281no—
141#103—282no—
142C22⋊C8—142no—
143#95—21no—
144#95—22no—
145#95C22⋊D712yes—
146C22×C7—21no—
147C22×C7—21no—
148C22×C7#45111yescharacteristic
149C22×C7#45611yescharacteristic
150C22×C7B(4,𝔽2)11yescharacteristic
151C4×C7—24no—
152C4×C7—21no—
153C4×C7—21no—
154C4×C7—44no—
155C4×C7—42no—
156C4×C7—44no—
157C4×C7—81no—
158Dic7—81no—
159Dic7—81no—
160C2×Q8—44no—
161C2×Q8—44no—
162C2×Q8—42no—
163C2×Q8—81no—
164C2×Q8—142no—
165C22×C4—21no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—21no—
169C22×C4C2×C22⋊D711yescharacteristic
170C22×C4C2×C22⋊D712yes—
171Q16—564no—
172M16—142no—
173M16—281no—
174C2×C8—281no—
175C2×C8—282no—
176C4⋊C4—24no—
177C4⋊C4—24no—
178C4⋊C4—41no—
179C4⋊C4—41no—
180C4⋊C4—44no—
181C4⋊C4—44no—
182C4⋊C4—44no—
183C4⋊C4—142no—
184C4⋊C4—141no—
185C4⋊C4—141no—
186C4⋊C4—282no—
187C22⋊C4—281no—
188C42—21no—
189C42—44no—
190C42—281no—
191C42#114412yescomplemented
192C2×C7—21no—
193C2×C7—21no—
194C2×C7#160511yescharacteristic
195C2×C7#219611yescharacteristic
196C2×C7#220511yescharacteristic
197C23#651611yescharacteristic
198Q8—44no—
199Q8—82no—
200Q8—84no—
201Q8—562no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—24no—
208C2×C4—24no—
209C2×C4—22no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—41no—
217C2×C4—281no—
218C2×C4—281no—
219C2×C4#650511yescharacteristic
220C2×C4C2×#114411yescharacteristic
221C8—282no—
222C8—562no—
223C7#1225611yescharacteristic, complemented, Sylow
224C22—21no—
225C22—21no—
226C22#9000011yescharacteristic
227C22#9001611yescharacteristic
228C22#9001811yescharacteristic
229C4—24no—
230C4—21no—
231C4—21no—
232C4—44no—
233C4—42no—
234C4—44no—
235C4—81no—
236C4—561no—
237C4—561no—
238C2—21no—
239C2—21no—
240C2#1172965811yescharacteristic
241C2#1172966311yescharacteristic
242C2#1172969611yescharacteristic
subgroupG/H
centerC22#90016
commutator, cosocleC2×C7×C4⋊C4C23
FrattiniC2×C4⋊C4C22×D7
FittingC7×#2551C2
socleC22×C7B(4,𝔽2)
prime radicalC23×C7#120
hypercenter#2551D7
hypocenterC7#12256
LCSG ▷ C2×C7×C4⋊C4 ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2551

Complements

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

NH ≅ G/Ntype
70. C42×C7Q16semidirect
191. C42#1144semidirect
223. C7#12256semidirect

Maximal subgroups

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

subgroupindex
C7×#25512
#117274612
#117273272
#117254542
#117254412
#122567