Finite Group Databasebeta

← #13266208

Subgroups — #13266208

Order 1792.

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

normal subgroups only

This group has 424 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×#2582C211yesmaximal, characteristic, complemented
2#11727466C212yesmaximal
3#11727347C211yesmaximal, characteristic
4#11725456C212yesmaximal, complemented
5#11725441C211yesmaximal, characteristic, complemented
6C2×C4×C7×Q8—21no—
7#90201C2211yescharacteristic
8C7×#398—21no—
9C7×#398C2212yes—
10C7×#391—22no—
11C7×#389—22no—
12C7×#384C2211yescharacteristic
13#89739—22no—
14#89738—21no—
15C2×#6379—21no—
16#89325C2211yescharacteristic
17#89323C2212yes—
18#89246—22no—
19#12153—71nomaximal, Sylow
20C22×C7×Q8—21no—
21C2×#1157—21no—
22C4×C7×Q8—42no—
23C4×C7×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⋊C4D411yescharacteristic
29C2×C7×C4⋊C4D412yes—
30C2×C7×C4⋊C4C2311yescharacteristic
31C2×C42×C7—22no—
32C2×C42×C7D411yescharacteristic
33#6493—22no—
34C2×#1136—21no—
35#6460—41no—
36#6458—41no—
37C7×#95—22no—
38C7×#95—21no—
39C7×#95—22no—
40C7×#95D412yes—
41C22⋊(C7⋊C8)—22no—
42#6384—41no—
43#6383—41no—
44#6383—42no—
45#6381—42no—
46#6379—42no—
47#2582D711yescharacteristic, complemented
48#2039—71no—
49#1610—72no—
50#1346—72no—
51#1325—71no—
52C2×C7×Q8—24no—
53C2×C7×Q8—42no—
54C2×C7×Q8—42no—
55C22×C4×C7—21no—
56C22×C4×C7—22no—
57C22×C4×C7—21no—
58C22×C4×C7—22no—
59C22×C4×C7C2×D411yescharacteristic
60C22×C4×C7C2×D412yes—
61C2×C22⋊D7—41no—
62#1157—41no—
63#1157—41no—
64#1157—41no—
65#1157—41no—
66C2×D28—21no—
67C2×D28—22no—
68C2×C4×D7—41no—
69C2×Dic14—21no—
70C7×C4⋊C4—41no—
71C7×C4⋊C4—42no—
72C7×C4⋊C4—41no—
73C7×C4⋊C4—44no—
74C7×C4⋊C4—42no—
75C7×C4⋊C4—42no—
76C7×C4⋊C4—42no—
77C7×C4⋊C4—81no—
78C42×C7—21no—
79C42×C7—44no—
80C42×C7D812yes—
81(C2×D7)⋊C4—42no—
82C4⋊Dic7—22no—
83#1136—41no—
84#1136—41no—
85C2×C7⋊C8—42no—
86C2×C7⋊C8—41no—
87#537—71no—
88C2×C4×Q8—21no—
89#480—142no—
90C2×#104—141no—
91#417—141no—
92#398—21no—
93#398C2×D712yes—
94#391—22no—
95#389—22no—
96#384C2×D711yescharacteristic
97#343—71no—
98#327—142no—
99#324—72no—
100C23×C7#12011yescharacteristic
101C22×D7—41no—
102C7×Q8—48no—
103C2×C4×C7—22no—
104C2×C4×C7—22no—
105C2×C4×C7—22no—
106C2×C4×C7—21no—
107C2×C4×C7—21no—
108C2×C4×C7—24no—
109C2×C4×C7—22no—
110C2×C4×C7—42no—
111C2×C4×C7—42no—
112C2×C4×C7—42no—
113C2×C4×C7—42no—
114C2×C4×C7—42no—
115C2×C4×C7—42no—
116C2×C4×C7—42no—
117C2×C4×C7—41no—
118C2×C4×C7C2×D811yescharacteristic
119C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
120C22⋊D7—81no—
121C22⋊D7—81no—
122C2×Dic7—41no—
123D28—41no—
124D28—41no—
125D28—82no—
126C4×D7—81no—
127C4×D7—81no—
128Dic14—41no—
129Dic14—41no—
130C7⋊C8—81no—
131C7⋊C8—82no—
132C2×#41—141no—
133C22×Q8—21no—
134C2×M16—141no—
135C22⋊Q8—281no—
136#121—142no—
137#121—281no—
138C4×Q8—42no—
139C4×Q8—42no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4C22⋊D711yescharacteristic
145C2×C4⋊C4C22⋊D712yes—
146C2×C4⋊C4C22×D711yescharacteristic
147C2×C42—22no—
148C2×C42C22⋊D711yescharacteristic
149#107—282no—
150#104—282no—
151#103—281no—
152#102—281no—
153#102—282no—
154C22⋊C8—142no—
155#95—22no—
156#95—21no—
157#95—22no—
158#95C22⋊D712yes—
159C22×C7—21no—
160C22×C7—21no—
161C22×C7#44711yescharacteristic
162C22×C7#45411yescharacteristic
163C22×C7B(4,𝔽2)11yescharacteristic
164C2×D7—81no—
165C2×D7—81no—
166C2×D7—81no—
167C4×C7—24no—
168C4×C7—21no—
169C4×C7—21no—
170C4×C7—44no—
171C4×C7—42no—
172C4×C7—82no—
173C4×C7—81no—
174Dic7—81no—
175#41—281no—
176#41—281no—
177#41—281no—
178#41—281no—
179C2×Q8—24no—
180C2×Q8—42no—
181C2×Q8—42no—
182C2×Q8—141no—
183C2×D4—141no—
184C2×D4—142no—
185C2×D4—281no—
186C22×C4—21no—
187C22×C4—22no—
188C22×C4—21no—
189C22×C4—22no—
190C22×C4—281no—
191C22×C4C2×C22⋊D711yescharacteristic
192C22×C4C2×C22⋊D712yes—
193M16—281no—
194M16—281no—
195C2×C8—282no—
196C2×C8—281no—
197C4⋊C4—41no—
198C4⋊C4—41no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4—44no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4—81no—
205C4⋊C4—142no—
206C22⋊C4—282no—
207C42—21no—
208C42—44no—
209C42#114112yes—
210C2×C7—21no—
211C2×C7—21no—
212C2×C7#159811yescharacteristic
213C2×C7#219811yescharacteristic
214C2×C7#220511yescharacteristic
215D7—161no—
216C23—281no—
217C23#651611yescharacteristic
218Q8—48no—
219Q8—281no—
220Q8—281no—
221D4—281no—
222D4—281no—
223D4—562no—
224D4—561no—
225D4—561no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—24no—
232C2×C4—22no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—41no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—281no—
242C2×C4—561no—
243C2×C4—561no—
244C2×C4C2×#114111yescharacteristic
245C2×C4#649511yescharacteristic
246C8—561no—
247C8—562no—
248C7#1215311yescharacteristic, complemented, Sylow
249C22—21no—
250C22—21no—
251C22—561no—
252C22—561no—
253C22—561no—
254C22#8998911yescharacteristic
255C22#9001611yescharacteristic
256C22#9001711yescharacteristic
257C4—24no—
258C4—21no—
259C4—21no—
260C4—44no—
261C4—42no—
262C4—81no—
263C4—82no—
264C4—561no—
265C2—21no—
266C2—21no—
267C2—1121no—
268C2#1172963211yescharacteristic
269C2#1172969611yescharacteristic
270C2#1172970011yescharacteristic
subgroupG/H
centerC22#90016
commutator, cosocleC2×C7×C4⋊C4C23
FrattiniC2×C4⋊C4C22×D7
FittingC7×#2582C2
socleC22×C7B(4,𝔽2)
prime radical#90201C22
hypercenter#2582D7
hypocenterC7#12153
LCSG ▷ C2×C7×C4⋊C4 ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2582

Complements

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

NH ≅ G/Ntype
1. C7×#2582C2semidirect
4. #11725456C2semidirect
5. #11725441C2semidirect
47. #2582D7semidirect
248. C7#12153semidirect

Maximal subgroups

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

subgroupindex
C7×#25822
#117274662
#117273472
#117254562
#117254412
#121537