Finite Group Databasebeta

← #13266576

Subgroups — #13266576

Order 1792.

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

normal subgroups only

This group has 380 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×#2734C211yesmaximal, characteristic
2#11727467C212yesmaximal
3#11727427C211yesmaximal, characteristic
4#11725456C212yesmaximal
5#11725449C211yesmaximal, characteristic
6C2×C7×C4⋊Q8—21no—
7#90191C2211yescharacteristic
8C7×#398C2212yes—
9C7×#395—22no—
10C7×#391C2211yescharacteristic
11C7×#389—21no—
12C7×#387—22no—
13#89997—21no—
14#89743—22no—
15#89325C2211yescharacteristic
16#89323C2212yes—
17#89318—21no—
18#89247—22no—
19#12213—71nomaximal, Sylow
20C22×C7×Q8—21no—
21C22×C7×Q8D411yescharacteristic
22C7×C4⋊Q8—42no—
23C7×C4⋊Q8—42no—
24C2×C7×C4⋊C4—21no—
25C2×C7×C4⋊C4—22no—
26C2×C7×C4⋊C4—22no—
27C2×C7×C4⋊C4—21no—
28C2×C7×C4⋊C4D412yes—
29C2×C7×C4⋊C4C2311yescharacteristic
30C2×C42×C7—21no—
31C2×C42×C7D411yescharacteristic
32#6489—21no—
33#6487—22no—
34C2×#1136—21no—
35#6453—41no—
36#6451—41no—
37C7×#95—22no—
38C7×#95—22no—
39C7×#95—22no—
40C7×#95D412yes—
41#6409—41no—
42#6409—41no—
43C22⋊(C7⋊C8)—22no—
44#6384—42no—
45#6382—42no—
46#2734D711yescharacteristic
47#2028—71no—
48#1611—72no—
49#1346—72no—
50#1319—71no—
51C2×C7×Q8—21no—
52C2×C7×Q8—42no—
53C2×C7×Q8—41no—
54C2×C7×Q8—41no—
55C2×C7×Q8—81no—
56C2×C7×Q8—81no—
57C2×C7×Q8SD1612yes—
58C22×C4×C7—21no—
59C22×C4×C7—22no—
60C22×C4×C7—21no—
61C22×C4×C7—22no—
62C22×C4×C7C2×D411yescharacteristic
63C22×C4×C7C2×D412yes—
64C2×Dic14—22no—
65C7×C4⋊C4—24no—
66C7×C4⋊C4—41no—
67C7×C4⋊C4—42no—
68C7×C4⋊C4—44no—
69C7×C4⋊C4—42no—
70C7×C4⋊C4—42no—
71C7×C4⋊C4—81no—
72C42×C7—41no—
73C42×C7—42no—
74C22⋊Dic7—41no—
75C4⋊Dic7—21no—
76C4⋊Dic7—21no—
77C4⋊Dic7—22no—
78Dic7⋊C4—42no—
79C4×Dic7—41no—
80#1136—41no—
81#1136—41no—
82C2×C7⋊C8—42no—
83C2×C7⋊C8—41no—
84#544—71no—
85C2×C4⋊Q8—21no—
86#483—142no—
87#419—141no—
88#398C2×D712yes—
89#395—22no—
90#391C2×D711yescharacteristic
91#389—21no—
92#387—22no—
93#343—71no—
94#339—141no—
95#328—142no—
96#324—72no—
97C23×C7#12011yescharacteristic
98C7×Q8—42no—
99C7×Q8—81no—
100C7×Q8—84no—
101C2×C4×C7—21no—
102C2×C4×C7—21no—
103C2×C4×C7—22no—
104C2×C4×C7—24no—
105C2×C4×C7—22no—
106C2×C4×C7—22no—
107C2×C4×C7—41no—
108C2×C4×C7—42no—
109C2×C4×C7—42no—
110C2×C4×C7—41no—
111C2×C4×C7—41no—
112C2×C4×C7—41no—
113C2×C4×C7—42no—
114C2×C4×C7—42no—
115C2×C4×C7—42no—
116C2×C4×C7—42no—
117C2×C4×C7C2×SD1611yescharacteristic
118C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
119C2×Dic7—41no—
120C2×Dic7—41no—
121Dic14—82no—
122C7⋊C8—81no—
123C7⋊C8—82no—
124C22×Q8—21no—
125C22×Q8C22⋊D711yescharacteristic
126C2×M16—141no—
127C4⋊Q8—42no—
128C4⋊Q8—42no—
129C4⋊Q8—281no—
130#125—281no—
131C22⋊Q8—142no—
132#117—141no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4C22⋊D712yes—
138C2×C4⋊C4C22×D711yescharacteristic
139C2×C42—21no—
140C2×C42C22⋊D711yescharacteristic
141#106—282no—
142#103—282no—
143#103—281no—
144#103—281no—
145C22⋊C8—142no—
146#95—22no—
147#95—22no—
148#95—22no—
149#95C22⋊D712yes—
150C22×C7—21no—
151C22×C7—21no—
152C22×C7#45011yescharacteristic
153C22×C7#45511yescharacteristic
154C22×C7B(4,𝔽2)11yescharacteristic
155C4×C7—21no—
156C4×C7—21no—
157C4×C7—44no—
158C4×C7—42no—
159C4×C7—42no—
160C4×C7—42no—
161C4×C7—82no—
162Dic7—81no—
163Dic7—81no—
164C2×Q8—21no—
165C2×Q8—42no—
166C2×Q8—41no—
167C2×Q8—41no—
168C2×Q8—81no—
169C2×Q8—81no—
170C2×Q8—142no—
171C2×Q8#114212yes—
172C22×C4—21no—
173C22×C4—22no—
174C22×C4—21no—
175C22×C4—22no—
176C22×C4C2×C22⋊D711yescharacteristic
177C22×C4C2×C22⋊D712yes—
178M16—281no—
179M16—281no—
180C2×C8—282no—
181C2×C8—281no—
182C4⋊C4—24no—
183C4⋊C4—41no—
184C4⋊C4—42no—
185C4⋊C4—44no—
186C4⋊C4—42no—
187C4⋊C4—42no—
188C4⋊C4—81no—
189C4⋊C4—141no—
190C4⋊C4—141no—
191C4⋊C4—142no—
192C4⋊C4—282no—
193C22⋊C4—281no—
194C42—41no—
195C42—42no—
196C42—281no—
197C2×C7—21no—
198C2×C7—21no—
199C2×C7#160111yescharacteristic
200C2×C7#220111yescharacteristic
201C2×C7#220511yescharacteristic
202C23#651611yescharacteristic
203Q8—42no—
204Q8—81no—
205Q8—84no—
206Q8—562no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—24no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—41no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—281no—
224C2×C4—281no—
225C2×C4#649511yescharacteristic
226C2×C4C2×#114211yescharacteristic
227C8—561no—
228C8—562no—
229C7#1221311yescharacteristic, complemented, Sylow
230C22—21no—
231C22—21no—
232C22#8999011yescharacteristic
233C22#9001411yescharacteristic
234C22#9001611yescharacteristic
235C4—21no—
236C4—21no—
237C4—44no—
238C4—42no—
239C4—42no—
240C4—42no—
241C4—82no—
242C4—561no—
243C4—561no—
244C2—21no—
245C2—21no—
246C2#1172960011yescharacteristic
247C2#1172969611yescharacteristic
248C2#1172971211yescharacteristic
subgroupG/H
centerC22#90016
commutator, cosocleC2×C7×C4⋊C4C23
FrattiniC2×C4⋊C4C22×D7
FittingC7×#2734C2
socleC22×C7B(4,𝔽2)
prime radicalC23×C7#120
hypercenter#2734D7
hypocenterC7#12213
LCSG ▷ C2×C7×C4⋊C4 ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2734

Complements

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

NH ≅ G/Ntype
229. C7#12213semidirect

Maximal subgroups

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

subgroupindex
C7×#27342
#117274672
#117274272
#117254562
#117254492
#122137