Finite Group Databasebeta

← #13251430

Subgroups — #13251430

Order 1792.

456 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 99 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×#2371C211yesmaximal, characteristic, complemented
2#11740457C211yesmaximal, characteristic
3#11729490C211yesmaximal, characteristic, complemented
4#11727186C212yesmaximal
5#11725421C212yesmaximal, complemented
6C2×C7×#125C2211yescharacteristic
7#90168C2211yescharacteristic
8C7×#387—22no—
9C7×#387—22no—
10C4×C7×C4⋊C4—22no—
11C7×#376C2212yes—
12#89841C412yescomplemented
13#89840—22no—
14#89839C412yescomplemented
15#89738—22no—
16#89698C2212yes—
17C4×#1136C2211yescharacteristic
18#89338—22no—
19#89246—22no—
20#89246—22no—
21#9442—71nomaximal, Sylow
22C2×#1157—21no—
23C7×#125—22no—
24C7×#125C2×C412yes—
25C7×#125D412yes—
26C2×C7×C4⋊C4—22no—
27C2×C7×C4⋊C4—22no—
28C2×C7×C4⋊C4D412yes—
29C2×C42×C7—22no—
30C2×C42×C7—22no—
31C2×C42×C7C2311yescharacteristic
32#6493—22no—
33C2×#1136—21no—
34#6438—21no—
35#6437C2×C411yescharacteristic
36C4×D28—22no—
37#6432C2×C411yescharacteristic
38C7×#95—22no—
39C7×#95—22no—
40C7×#95—22no—
41C7×#95D412yes—
42C7×#95D412yes—
43C22⋊(C7⋊C8)—22no—
44#6384—24no—
45#6383—24no—
46C4⋊(C7⋊C8)—22no—
47#6377—22no—
48#6377C2×C412yes—
49C4×C7⋊C8—21no—
50#3142—71no—
51#2371D711yescharacteristic, complemented
52#1534—71no—
53#1505—72no—
54#1300—72no—
55C22×C4×C7—22no—
56C22×C4×C7—22no—
57C22×C4×C7—22no—
58C22×C4×C7C2×D411yescharacteristic
59C22×C4×C7C2×D412yes—
60C2×C22⋊D7—41no—
61#1157—44no—
62C2×D28—21no—
63C2×D28—22no—
64C2×C4×D7—41no—
65C2×Dic14—21no—
66C7×C4⋊C4—24no—
67C7×C4⋊C4—24no—
68C7×C4⋊C4—22no—
69C7×C4⋊C4—44no—
70C7×C4⋊C4C22⋊C414yes—
71C42×C7—44no—
72C42×C7—44no—
73C42×C7C22×C411yescharacteristic
74C42×C7C2×D411yescharacteristic
75C42×C7#4112yes—
76(C2×D7)⋊C4—42no—
77C4⋊Dic7—22no—
78#1136—42no—
79C2×C7⋊C8—22no—
80C2×C7⋊C8—42no—
81#532—71no—
82C2×#125C2×D711yescharacteristic
83#491—72no—
84#490—72no—
85#488—142no—
86#432—72no—
87#417—142no—
88C4×M16—71no—
89#387—22no—
90#387—22no—
91C4×C4⋊C4—22no—
92#376C2×D712yes—
93#330—142no—
94#327—142no—
95#327—142no—
96C23×C7#12011yescharacteristic
97C22×D7—41no—
98C2×C4×C7—22no—
99C2×C4×C7—24no—
100C2×C4×C7—24no—
101C2×C4×C7—42no—
102C2×C4×C7—42no—
103C2×C4×C7—42no—
104C2×C4×C7—42no—
105C2×C4×C7—42no—
106C2×C4×C7—42no—
107C2×C4×C7—42no—
108C2×C4×C7—42no—
109C2×C4×C7C2×C22⋊C411yescharacteristic
110C2×C4×C7C4×D412yes—
111C2×C4×C7#12112yes—
112C2×C4×C7#12311yescharacteristic
113C22⋊D7—82no—
114C2×Dic7—41no—
115D28—42no—
116D28—82no—
117C4×D7—82no—
118Dic14—42no—
119C7⋊C8—42no—
120C7⋊C8—82no—
121C2×#41—141no—
122C2×M16—141no—
123C4⋊Q8—71no—
124#127—71no—
125#125—22no—
126#125C4×D712yes—
127#125D2812yes—
128#124—141no—
129#121—142no—
130C4×D4—142no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4C22⋊D712yes—
134C2×C42—22no—
135C2×C42—22no—
136C2×C42C22×D711yescharacteristic
137C4⋊C8—142no—
138#103—144no—
139#102—144no—
140C22⋊C8—142no—
141#97—72no—
142#97—142no—
143C4×C8—141no—
144#95—22no—
145#95—22no—
146#95—22no—
147#95C22⋊D712yes—
148#95C22⋊D712yes—
149C22×C7—22no—
150C22×C7#38611yescharacteristic
151C22×C7#45412yes—
152C2×D7—81no—
153C2×D7—82no—
154C4×C7—21no—
155C4×C7—21no—
156C4×C7—22no—
157C4×C7—22no—
158C4×C7—44no—
159C4×C7—44no—
160C4×C7—82no—
161Dic7—81no—
162#41—284no—
163C2×Q8—141no—
164C2×D4—141no—
165C2×D4—142no—
166C2×D4—281no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—281no—
171C22×C4C2×C22⋊D711yescharacteristic
172C22×C4C2×C22⋊D712yes—
173M16—282no—
174C2×C8—142no—
175C2×C8—282no—
176C4⋊C4—24no—
177C4⋊C4—22no—
178C4⋊C4—24no—
179C4⋊C4—44no—
180C4⋊C4—142no—
181C4⋊C4(C2×D7)⋊C414yes—
182C22⋊C4—282no—
183C42—44no—
184C42—44no—
185C42C2×C4×D711yescharacteristic
186C42C2×D2811yescharacteristic
187C42#115712yes—
188C2×C7—22no—
189C2×C7#188811yescharacteristic
190C2×C7#190612yes—
191D7—161no—
192C23—281no—
193C23#651611yescharacteristic
194Q8—282no—
195D4—282no—
196D4—562no—
197D4—562no—
198C2×C4—22no—
199C2×C4—24no—
200C2×C4—24no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—281no—
210C2×C4—562no—
211C2×C4C2×(C2×D7)⋊C411yescharacteristic
212C2×C4C4×C22⋊D712yes—
213C2×C4#649211yescharacteristic
214C2×C4#649312yes—
215C8—282no—
216C8—562no—
217C7#944211yescharacteristic, complemented, Sylow
218C22—22no—
219C22—561no—
220C22—562no—
221C22#8998511yescharacteristic
222C22#9001712yes—
223C4—21no—
224C4—21no—
225C4—22no—
226C4—22no—
227C4—44no—
228C4—44no—
229C4—82no—
230C4—561no—
231C2—22no—
232C2—1121no—
233C2#1172939912yes—
234C2#1172941511yescharacteristic
subgroupG/H
centerC22#89985
commutatorC42×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#2371C2
socleC22×C7#386
prime radical#90168C22
cosocleC2×C42×C7C23
hypercenter#2371D7
hypocenterC7#9442
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2371

Complements

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

NH ≅ G/Ntype
1. C7×#2371C2semidirect
3. #11729490C2semidirect
5. #11725421C2semidirect
12. #89841C4semidirect
14. #89839C4semidirect
51. #2371D7semidirect
217. C7#9442semidirect

Maximal subgroups

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

subgroupindex
C7×#23712
#117404572
#117294902
#117271862
#117254212
#94427