Finite Group Databasebeta

← #13266390

Subgroups — #13266390

Order 1792.

372 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 59 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
1#11731663C211yesmaximal, characteristic, complemented
2C7×#2040C211yesmaximal, characteristic
3#11729634C212yesmaximal
4#11725438C211yesmaximal, characteristic, complemented
5#11725300C212yesmaximal, complemented
6C7×#537C2211yescharacteristic
7C2×C7×#104—21no—
8C7×#417—21no—
9C7×#382C2211yescharacteristic
10#89812—22no—
11#89758—21no—
12#89751C2212yes—
13#89708—21no—
14#89432—22no—
15#89428—22no—
16C7×#343C2211yescharacteristic
17#89332—22no—
18#89323C2212yes—
19#12184—71nomaximal, Sylow
20C2×C7×#41—21no—
21C22×Dic14D412yes—
22C2×C7×M16—21no—
23C7×C22⋊Q8—41no—
24C7×#121—22no—
25C7×#121—41no—
26C2×C7×C4⋊C4D411yescharacteristic
27C2×C7×C4⋊C4C2311yescharacteristic
28C2×C42×C7D411yescharacteristic
29C2×#1142—42no—
30C2×C4⋊Dic7—21no—
31C2×C4⋊Dic7—21no—
32C2×C4⋊Dic7—21no—
33C2×Dic7⋊C4—22no—
34C7×#104—42no—
35C7×#103—41no—
36C7×#102—41no—
37C7×#95—22no—
38C7⋊#95—22no—
39C7⋊#95—22no—
40C7⋊#95—21no—
41C7⋊#95—22no—
42C7⋊#95D412yes—
43C22⋊(C7⋊C8)—22no—
44#6384—42no—
45#2844—71no—
46#2040D711yescharacteristic
47#1604—72no—
48#1354—72no—
49#1324—71no—
50C7×#41—41no—
51C7×#41—41no—
52C7×#41—41no—
53C7×#41—41no—
54C2×C7×Q8—21no—
55C2×C7×D4—21no—
56C2×C7×D4—22no—
57C2×C7×D4—41no—
58C22×C4×C7—21no—
59C22×C4×C7—21no—
60C22×C4×C7—41no—
61C22×C4×C7C2×D411yescharacteristic
62C22×C4×C7C2×D412yes—
63C22×Dic7—22no—
64C22×Dic7—22no—
65C2×Dic14—42no—
66C2×Dic14—82no—
67C7×M16—41no—
68C7×M16—41no—
69C2×C7×C8—41no—
70C7×C4⋊C4—22no—
71C7×C4⋊C4—41no—
72C7×C4⋊C4—41no—
73C7×C22⋊C4—42no—
74C42×C7—21no—
75C42×C7SD1612yescomplemented
76#1142—84no—
77C4⋊Dic7—81no—
78C4⋊Dic7—81no—
79C4⋊Dic7—81no—
80Dic7⋊C4—82no—
81C2×C7⋊C8—42no—
82#537C2×D711yescharacteristic
83#448—142no—
84C2×#104—21no—
85#417—21no—
86#400—142no—
87#400—141no—
88#395—72no—
89#395—142no—
90#382—141no—
91#382C2×D711yescharacteristic
92#343C2×D711yescharacteristic
93#327—142no—
94#324—72no—
95C23×C7—41no—
96C23×C7#12011yescharacteristic
97C7×Q8—41no—
98C7×Q8—41no—
99C7×D4—41no—
100C7×D4—41no—
101C7×D4—82no—
102C7×D4—81no—
103C7×D4—81no—
104C2×C4×C7—21no—
105C2×C4×C7—21no—
106C2×C4×C7—22no—
107C2×C4×C7—22no—
108C2×C4×C7—41no—
109C2×C4×C7—41no—
110C2×C4×C7—42no—
111C2×C4×C7—42no—
112C2×C4×C7—41no—
113C2×C4×C7—41no—
114C2×C4×C7—81no—
115C2×C4×C7—81no—
116C2×C4×C7C2×SD1611yescharacteristic
117C2×C4×C7#13711yescharacteristic
118C2×Dic7—42no—
119C2×Dic7—42no—
120C2×Dic7—42no—
121C2×Dic7—42no—
122C2×Dic7—42no—
123C2×Dic7—42no—
124Dic14—44no—
125Dic14—82no—
126C7×C8—81no—
127C7⋊C8—82no—
128C2×#41—21no—
129C22×Q8—72no—
130C2×SD16—282no—
131C2×M16—21no—
132C22⋊Q8—41no—
133#121—22no—
134#121—41no—
135C2×C4⋊C4—141no—
136C2×C4⋊C4—141no—
137C2×C4⋊C4—141no—
138C2×C4⋊C4—142no—
139C2×C4⋊C4C22⋊D711yescharacteristic
140C2×C4⋊C4C22×D711yescharacteristic
141C2×C42C22⋊D711yescharacteristic
142#104—42no—
143#103—41no—
144#103—282no—
145#102—41no—
146C22⋊C8—142no—
147#95—22no—
148#95—72no—
149#95—142no—
150#95—142no—
151#95—141no—
152#95—142no—
153C22×C7—21no—
154C22×C7—21no—
155C22×C7—81no—
156C22×C7—81no—
157C22×C7—81no—
158C22×C7#44811yescharacteristic
159C22×C7#45411yescharacteristic
160C22×C7B(4,𝔽2)11yescharacteristic
161C4×C7—21no—
162C4×C7—21no—
163C4×C7—42no—
164C4×C7—42no—
165C4×C7—81no—
166C4×C7—81no—
167Dic7—82no—
168Dic7—82no—
169#41—41no—
170#41—41no—
171#41—41no—
172#41—41no—
173C2×Q8—21no—
174C2×Q8—282no—
175C2×Q8—562no—
176C2×D4—21no—
177C2×D4—22no—
178C2×D4—41no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—41no—
182C22×C4—142no—
183C22×C4—142no—
184C22×C4D4×D712yes—
185C22×C4C2×C22⋊D711yescharacteristic
186SD16—564no—
187M16—41no—
188M16—41no—
189C2×C8—41no—
190C2×C8—282no—
191C4⋊C4—22no—
192C4⋊C4—41no—
193C4⋊C4—41no—
194C4⋊C4—562no—
195C4⋊C4—561no—
196C4⋊C4—561no—
197C4⋊C4—561no—
198C22⋊C4—42no—
199C42—21no—
200C42#114212yescomplemented
201C2×C7—21no—
202C2×C7—21no—
203C2×C7—161no—
204C2×C7#159911yescharacteristic
205C2×C7#219911yescharacteristic
206C2×C7#220311yescharacteristic
207C23—41no—
208C23#650011yescharacteristic
209Q8—41no—
210Q8—41no—
211Q8—284no—
212Q8—562no—
213D4—41no—
214D4—41no—
215D4—82no—
216D4—81no—
217D4—81no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—81no—
229C2×C4—81no—
230C2×C4—282no—
231C2×C4—282no—
232C2×C4—282no—
233C2×C4—282no—
234C2×C4—282no—
235C2×C4—282no—
236C2×C4C2×#114211yescharacteristic
237C2×C4#651411yescharacteristic
238C8—81no—
239C8—562no—
240C7#1218411yescharacteristic, complemented, Sylow
241C22—21no—
242C22—21no—
243C22—81no—
244C22—81no—
245C22—81no—
246C22#8980411yescharacteristic
247C22#8981211yescharacteristic
248C22#8983411yescharacteristic
249C4—21no—
250C4—21no—
251C4—42no—
252C4—42no—
253C4—81no—
254C4—81no—
255C4—562no—
256C4—562no—
257C2—21no—
258C2—21no—
259C2—161no—
260C2#1172745811yescharacteristic
261C2#1172874311yescharacteristic
262C2#1172874511yescharacteristic
subgroupG/H
centerC22#89804
commutator, cosocleC2×C7×C4⋊C4C23
FrattiniC2×C4⋊C4C22×D7
FittingC7×#2040C2
socleC22×C7B(4,𝔽2)
prime radicalC7×#537C22
hypercenter#2040D7
hypocenterC7#12184
LCSG ▷ C2×C7×C4⋊C4 ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2040

Complements

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

NH ≅ G/Ntype
1. #11731663C2semidirect
4. #11725438C2semidirect
5. #11725300C2semidirect
75. C42×C7SD16semidirect
200. C42#1142semidirect
240. C7#12184semidirect

Maximal subgroups

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

subgroupindex
#117316632
C7×#20402
#117296342
#117254382
#117253002
#121847