Finite Group Databasebeta

← #13256820

Subgroups — #13256820

Order 1792.

500 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 125 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×#2571C211yesmaximal, characteristic, complemented
2#11727334C211yesmaximal, characteristic
3#11727241C212yesmaximal
4#11725440C211yesmaximal, characteristic, complemented
5#11725425C212yesmaximal, complemented
6C2×#6437C2211yescharacteristic
7C7×#400—22no—
8C7×#389—22no—
9C7×#384C2211yescharacteristic
10C7×#382—22no—
11C4×C7×C4⋊C4C2212yes—
12C2×#6383—22no—
13C2×#6383—22no—
14C2×#6383—22no—
15C2×C4⋊(C7⋊C8)C2212yes—
16C2×C4×C7⋊C8C2211yescharacteristic
17#10351—71nomaximal, Sylow
18C22×D28—21no—
19C22×D28—22no—
20C2×C7×C4⋊C4—22no—
21C2×C7×C4⋊C4—22no—
22C2×C7×C4⋊C4D412yes—
23C2×C7×C4⋊C4D412yes—
24C2×C7×C4⋊C4D412yes—
25C2×C42×C7—22no—
26C2×C42×C7C2311yescharacteristic
27C22×C7⋊C8—21no—
28C22×C7⋊C8—22no—
29#6437—22no—
30#6437—21no—
31#6437Q812yes—
32C7×#95—22no—
33C7×#95—24no—
34C7×#95—22no—
35#6383—44no—
36#6383—44no—
37#6383—44no—
38C4⋊(C7⋊C8)—22no—
39C4⋊(C7⋊C8)—22no—
40C4×C7⋊C8—22no—
41#2571D711yescharacteristic, complemented
42#1982—72no—
43#1957—71no—
44#1296—71no—
45#1279—72no—
46C23×D7—41no—
47C22×C4×C7—22no—
48C22×C4×C7—22no—
49C22×C4×C7—22no—
50C22×C4×C7C2×D411yescharacteristic
51C22×C4×C7C2×D412yes—
52C2×D28—22no—
53C2×D28—24no—
54C2×D28—42no—
55C2×D28—42no—
56C2×D28—42no—
57C2×D28—42no—
58C2×D28—42no—
59C2×D28—42no—
60C2×D28—41no—
61C2×D28—42no—
62C7×C4⋊C4—22no—
63C7×C4⋊C4—22no—
64C7×C4⋊C4—22no—
65C7×C4⋊C4—22no—
66C7×C4⋊C4—42no—
67C7×C4⋊C4—42no—
68C7×C4⋊C4—42no—
69C7×C4⋊C4—82no—
70C42×C7—42no—
71C42×C7—42no—
72C42×C7C2×Q811yescharacteristic
73C42×C7#4112yes—
74C42×C7#4111yescharacteristic
75C2×C7⋊C8—22no—
76C2×C7⋊C8—41no—
77C2×C7⋊C8—41no—
78C2×C7⋊C8—42no—
79C2×C7⋊C8—42no—
80C2×C7⋊C8—42no—
81C2×#127—71no—
82C2×C4⋊C8—72no—
83C2×#102—142no—
84C2×#102—142no—
85C2×#102—142no—
86C2×C4×C8—71no—
87#400—22no—
88#389—22no—
89#384C2×D711yescharacteristic
90#382—22no—
91C4×C4⋊C4C2×D712yes—
92C23×C7#12011yescharacteristic
93C22×D7—42no—
94C22×D7—82no—
95C22×D7—82no—
96C22×D7—81no—
97C22×D7—81no—
98C2×C4×C7—21no—
99C2×C4×C7—21no—
100C2×C4×C7—42no—
101C2×C4×C7—42no—
102C2×C4×C7—42no—
103C2×C4×C7—42no—
104C2×C4×C7—44no—
105C2×C4×C7—42no—
106C2×C4×C7—42no—
107C2×C4×C7—42no—
108C2×C4×C7C22⋊Q811yescharacteristic
109C2×C4×C7C22⋊Q812yes—
110C2×C4×C7#12411yescharacteristic
111C2×C4×C7#12412yes—
112C2×C4×C7#13512yes—
113C2×C4×C7#13512yes—
114C2×C4×C7AGL(1,ℤ/8)12yes—
115C2×C4×C7AGL(1,ℤ/8)12yes—
116D28—44no—
117D28—44no—
118D28—44no—
119D28—82no—
120D28—82no—
121D28—82no—
122C7⋊C8—42no—
123C7⋊C8—82no—
124C22×D4—141no—
125C22×D4—142no—
126C22×C8—141no—
127C22×C8—142no—
128#127—72no—
129#127—142no—
130#127—141no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4C22⋊D712yes—
134C2×C4⋊C4C22⋊D712yes—
135C2×C4⋊C4C22⋊D712yes—
136C2×C42—22no—
137C2×C42C22×D711yescharacteristic
138C4⋊C8—142no—
139C4⋊C8—142no—
140#102—284no—
141#102—284no—
142#102—284no—
143C4×C8—142no—
144#95—22no—
145#95—24no—
146#95—22no—
147C22×C7#39311yescharacteristic
148C22×C7#44912yes—
149C22×C7#44912yes—
150C22×C7#45311yescharacteristic
151C22×C7#45411yescharacteristic
152C2×D7—84no—
153C2×D7—82no—
154C2×D7—161no—
155C2×D7—161no—
156C2×D7—162no—
157C4×C7—21no—
158C4×C7—21no—
159C4×C7—82no—
160C4×C7—82no—
161C4×C7—82no—
162C4×C7#47812yes—
163C4×C7#47812yes—
164C4×C7#48812yes—
165C4×C7#49012yes—
166C24—281no—
167C2×D4—142no—
168C2×D4—144no—
169C2×D4—282no—
170C2×D4—282no—
171C2×D4—281no—
172C2×D4—282no—
173C2×D4—282no—
174C2×D4—282no—
175C2×D4—282no—
176C2×D4—282no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4C2×C22⋊D711yescharacteristic
181C22×C4C2×C22⋊D712yes—
182C2×C8—142no—
183C2×C8—281no—
184C2×C8—281no—
185C2×C8—282no—
186C2×C8—282no—
187C2×C8—282no—
188C4⋊C4—22no—
189C4⋊C4—22no—
190C4⋊C4—22no—
191C4⋊C4—22no—
192C4⋊C4—42no—
193C4⋊C4—42no—
194C4⋊C4—42no—
195C4⋊C4—82no—
196C42—42no—
197C42—42no—
198C42Q8×D711yescharacteristic
199C42#116112yes—
200C42#116111yescharacteristic
201C2×C7#162011yescharacteristic
202C2×C7#162411yescharacteristic
203C2×C7#162812yes—
204C2×C7#202512yes—
205C2×C7#202711yescharacteristic
206D7—162no—
207C23—282no—
208C23—562no—
209C23—562no—
210C23—561no—
211C23—561no—
212C23#651611yescharacteristic
213D4—284no—
214D4—284no—
215D4—284no—
216D4—562no—
217D4—562no—
218D4—562no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—44no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4#650911yescharacteristic
230C2×C4#650912yes—
231C2×C4#651011yescharacteristic
232C2×C4#651012yes—
233C2×C4#651212yes—
234C2×C4#651212yes—
235C2×C4#651312yes—
236C2×C4#651312yes—
237C8—282no—
238C8—562no—
239C7#1035111yescharacteristic, complemented, Sylow
240C22—564no—
241C22—562no—
242C22—1121no—
243C22—1122no—
244C22—1121no—
245C22#9000311yescharacteristic
246C22#9001012yes—
247C22#9001012yes—
248C22#9001311yescharacteristic
249C22#9001711yescharacteristic
250C4—21no—
251C4—21no—
252C4—82no—
253C4—82no—
254C4—82no—
255C4#8983812yes—
256C4#8983812yes—
257C4#8983912yes—
258C4#8984012yes—
259C2—1122no—
260C2#1172949712yes—
261C2#1172965711yescharacteristic
262C2#1172968111yescharacteristic
263C2#1172968512yes—
264C2#1172969711yescharacteristic
subgroupG/H
centerC23#6516
commutator, cosocleC2×C42×C7C23
FrattiniC2×C42C22×D7
FittingC7×#2571C2
socleC23×C7#120
prime radicalC2×#6437C22
hypercenter#2571D7
hypocenterC7#10351
LCSG ▷ C2×C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2571

Complements

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

NH ≅ G/Ntype
1. C7×#2571C2semidirect
4. #11725440C2semidirect
5. #11725425C2semidirect
41. #2571D7semidirect
239. C7#10351semidirect

Maximal subgroups

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

subgroupindex
C7×#25712
#117273342
#117272412
#117254402
#117254252
#103517