Finite Group Databasebeta

← #13255119

Subgroups — #13255119

Order 1792.

600 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 600 conjugacy classes of subgroups — too many to draw. Only its 95 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
1C2×#89861C212yesmaximal, complemented
2#11731699C211yesmaximal, characteristic
3C7×#1960C211yesmaximal, characteristic, complemented
4C2×#89259C212yesmaximal, complemented
5#11725436C211yesmaximal, characteristic, complemented
6C2×C7×C4⋊Q8C2211yescharacteristic
7C2×#6438C2212yes—
8C2×C7×#103—22no—
9C2×C7×#97C2211yescharacteristic
10C7×#382C2211yescharacteristic
11#89861—24no—
12#89861—24no—
13#89763—22no—
14C2×#6384—22no—
15C2×#6383—22no—
16C2×#6377C2212yes—
17#89445—22no—
18#89443—22no—
19#89259—24no—
20#10085—71nomaximal, Sylow
21C22×C7×Q8—21no—
22C22×D28—22no—
23C22×D28D412yes—
24C22×Dic14D412yes—
25C22×C7×C8—21no—
26C7×C4⋊Q8—22no—
27C7×C4⋊Q8—21no—
28C7×C4⋊Q8D412yescomplemented
29C2×C7×C4⋊C4—22no—
30C2×C7×C4⋊C4D412yes—
31C2×C42×C7C2311yescharacteristic
32C2×(C2×D7)⋊C4—22no—
33C2×(C2×D7)⋊C4—22no—
34C2×(C2×D7)⋊C4—24no—
35C22×C7⋊C8—22no—
36#6438—22no—
37#6438—22no—
38#6438—22no—
39#6438—22no—
40C7×#103—44no—
41C7×#97—22no—
42C7×#95—22no—
43C7⋊#95—22no—
44#6384—28no—
45#6383—28no—
46#6377—22no—
47#6377D414yescomplemented
48C2×#489—72no—
49#2819—71no—
50#1960D711yescharacteristic, complemented
51C2×#329—72no—
52#1295—71no—
53C23×D7—22no—
54C2×C7×Q8—22no—
55C2×C7×Q8—42no—
56C2×C7×Q8—42no—
57C2×C7×Q8—41no—
58C22×C4×C7—22no—
59C22×C4×C7—41no—
60C22×C4×C7C2×D412yes—
61C22×C4×C7C2×D411yescharacteristic
62C22×Dic7—22no—
63C2×D28—22no—
64C2×D28—22no—
65C2×D28—44no—
66C2×D28—82no—
67C2×D28—84no—
68C2×Dic14—22no—
69C2×Dic14—22no—
70C2×Dic14—44no—
71C2×C7×C8—22no—
72C2×C7×C8—41no—
73C2×C7×C8—41no—
74C7×C4⋊C4—24no—
75C7×C4⋊C4—42no—
76C7×C4⋊C4—42no—
77C42×C7C2×D411yescharacteristic
78C42×C7C2×D412yes—
79C42×C7#4111yescharacteristic
80(C2×D7)⋊C4—42no—
81(C2×D7)⋊C4—42no—
82(C2×D7)⋊C4—42no—
83(C2×D7)⋊C4—42no—
84(C2×D7)⋊C4—84no—
85C2×C7⋊C8—24no—
86C2×C7⋊C8—44no—
87C2×C4⋊Q8C2×D711yescharacteristic
88C2×#124—72no—
89#489—144no—
90#489—144no—
91C2×#103—22no—
92C2×#103—142no—
93C2×#102—142no—
94C2×#97—72no—
95C2×#97C2×D711yescharacteristic
96#394—142no—
97#393—142no—
98#392—142no—
99#382C2×D711yescharacteristic
100#329—144no—
101C23×C7#12011yescharacteristic
102C22×D7—42no—
103C22×D7—42no—
104C22×D7—42no—
105C22×D7—44no—
106C22×D7—44no—
107C7×Q8—44no—
108C7×Q8—82no—
109C2×C4×C7—21no—
110C2×C4×C7—21no—
111C2×C4×C7—24no—
112C2×C4×C7—42no—
113C2×C4×C7—42no—
114C2×C4×C7—44no—
115C2×C4×C7—81no—
116C2×C4×C7—81no—
117C2×C4×C7#12012yes—
118C2×C4×C7#12112yes—
119C2×C4×C7#12111yescharacteristic
120C2×C4×C7#12711yescharacteristic
121C2×Dic7—42no—
122C2×Dic7—44no—
123D28—44no—
124D28—82no—
125D28—88no—
126Dic14—44no—
127Dic14—82no—
128C7×C8—42no—
129C7⋊C8—44no—
130C22×Q8—21no—
131C22×Q8—72no—
132C22×D4—72no—
133C22×D4—142no—
134C22×C8—21no—
135C22×C8—142no—
136C4⋊Q8—21no—
137C4⋊Q8—22no—
138C4⋊Q8D2812yescomplemented
139#124—142no—
140#124—142no—
141#124—142no—
142#124—142no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4C22⋊D712yes—
145C2×C22⋊C4—142no—
146C2×C22⋊C4—142no—
147C2×C22⋊C4—144no—
148C2×C42C22×D711yescharacteristic
149#103—44no—
150#103—148no—
151#102—148no—
152#97—22no—
153#97—74no—
154#97—142no—
155#95—22no—
156#95—142no—
157C22×C7#39211yescharacteristic
158C22×C7#45311yescharacteristic
159C22×C7#45411yescharacteristic
160C22×C7#45514yes—
161C2×D7—84no—
162C2×D7—84no—
163C2×D7—82no—
164C2×D7—82no—
165C2×D7—82no—
166C4×C7—22no—
167C4×C7—22no—
168C4×C7—21no—
169C4×C7—21no—
170C4×C7—82no—
171C4×C7—82no—
172Dic7—82no—
173C24—142no—
174C2×Q8—22no—
175C2×Q8—42no—
176C2×Q8—42no—
177C2×Q8—41no—
178C2×Q8—142no—
179C2×Q8—142no—
180C2×Q8—284no—
181C2×D4—142no—
182C2×D4—142no—
183C2×D4—284no—
184C2×D4—564no—
185C2×D4—562no—
186C22×C4—22no—
187C22×C4—41no—
188C22×C4—142no—
189C22×C4D4×D712yes—
190C22×C4C2×C22⋊D711yescharacteristic
191C2×C8—22no—
192C2×C8—41no—
193C2×C8—41no—
194C2×C8—144no—
195C2×C8—284no—
196C4⋊C4—24no—
197C4⋊C4—42no—
198C4⋊C4—42no—
199C22⋊C4—282no—
200C22⋊C4—282no—
201C22⋊C4—282no—
202C22⋊C4—282no—
203C22⋊C4—564no—
204C42C2×D2811yescharacteristic
205C42#115711yescharacteristic
206C42D4×D712yes—
207C2×C7#165912yes—
208C2×C7#166212yes—
209C2×C7#200412yes—
210C2×C7#200611yescharacteristic
211D7—162no—
212C23—284no—
213C23—284no—
214C23—282no—
215C23—282no—
216C23—282no—
217C23#650011yescharacteristic
218Q8—44no—
219Q8—82no—
220Q8—284no—
221Q8—562no—
222D4—284no—
223D4—562no—
224D4—568no—
225C2×C4—24no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—44no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—81no—
232C2×C4—81no—
233C2×C4—284no—
234C2×C4—282no—
235C2×C4#644512yes—
236C2×C4#644712yes—
237C2×C4#649311yescharacteristic
238C2×C4#650311yescharacteristic
239C8—42no—
240C8—284no—
241C7#1008511yescharacteristic, complemented, Sylow
242C22—562no—
243C22—564no—
244C22—564no—
245C22—562no—
246C22—562no—
247C22#8973211yescharacteristic
248C22#8983411yescharacteristic
249C22#8984911yescharacteristic
250C22#8997414yes—
251C4—21no—
252C4—21no—
253C4—22no—
254C4—22no—
255C4—82no—
256C4—82no—
257C4—562no—
258C2—1122no—
259C2#1172741811yescharacteristic
260C2#1172809812yes—
261C2#1172900512yes—
262C2#1172908312yes—
subgroupG/H
centerC23#6500
commutator, cosocleC2×C42×C7C23
FrattiniC2×C42C22×D7
FittingC7×#1960C2
socleC23×C7#120
prime radical#11731699C2
hypercenter#1960D7
hypocenterC7#10085
LCSG ▷ C2×C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1960

Complements

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

NH ≅ G/Ntype
1. C2×#89861C2semidirect
3. C7×#1960C2semidirect
4. C2×#89259C2semidirect
5. #11725436C2semidirect
28. C7×C4⋊Q8D4semidirect
47. #6377D4semidirect
50. #1960D7semidirect
138. C4⋊Q8D28semidirect
241. C7#10085semidirect

Maximal subgroups

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

subgroupindex
C2×#898612
#117316992
C7×#19602
C2×#892592
#117254362
#100857