Finite Group Databasebeta

← #13251431

Subgroups — #13251431

Order 1792.

456 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 103 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×#3142C211yesmaximal, characteristic
2#11740471C211yesmaximal, characteristic, complemented
3#11729566C212yesmaximal
4#11729490C211yesmaximal, characteristic, complemented
5#11725471C212yesmaximal, complemented
6C7×#532C2211yescharacteristic
7C2×C7×#125C2211yescharacteristic
8C7×#491C412yescomplemented
9C7×#490C412yescomplemented
10C7×#488—22no—
11C7×#417—22no—
12C4×C7×M16C2211yescharacteristic
13#89753—22no—
14#89753—22no—
15C4⋊C4×Dic7—22no—
16#89704C2212yes—
17#89698C2212yes—
18#89338—22no—
19#89332—22no—
20#89332—22no—
21#9442—71nomaximal, Sylow
22C2×C7×#41—21no—
23C2×C7×M16—21no—
24C7×C4⋊Q8C2×C411yescharacteristic
25C7×#127C2×C411yescharacteristic
26C7×#125—22no—
27C7×#125C2×C412yes—
28C7×#125D412yes—
29C7×#124—21no—
30C7×#121—22no—
31C4×C7×D4—22no—
32C2×C7×C4⋊C4—22no—
33C2×C7×C4⋊C4D412yes—
34C2×C42×C7C2311yescharacteristic
35C2×C4⋊Dic7—22no—
36C2×C4×Dic7—22no—
37C2×C4×Dic7—22no—
38C7×#103—24no—
39C7×#102—24no—
40C7×#97C2×C412yes—
41C4×C7×C8—21no—
42C7⋊#95—22no—
43C7⋊#95—22no—
44C7⋊#95—22no—
45C7⋊#95D412yes—
46C7⋊#95D412yes—
47C22⋊(C7⋊C8)—22no—
48C4⋊(C7⋊C8)—22no—
49#6377—22no—
50#3142D711yescharacteristic
51#2371—71no—
52#1534—71no—
53#1505—72no—
54#1300—72no—
55C7×#41—44no—
56C2×C7×Q8—21no—
57C2×C7×D4—21no—
58C2×C7×D4—22no—
59C2×C7×D4—41no—
60C22×C4×C7—22no—
61C22×C4×C7—41no—
62C22×C4×C7C2×D411yescharacteristic
63C22×C4×C7C2×D412yes—
64C22×Dic7—22no—
65C22×Dic7—22no—
66C7×M16—42no—
67C2×C7×C8—22no—
68C7×C4⋊C4—24no—
69C7×C4⋊C4—24no—
70C7×C4⋊C4—22no—
71C7×C4⋊C4—22no—
72C7×C4⋊C4C22⋊C414yes—
73C7×C22⋊C4—42no—
74C42×C7C22×C411yescharacteristic
75C42×C7C2×D411yescharacteristic
76C42×C7#4112yes—
77C4⋊Dic7—44no—
78C4×Dic7—44no—
79C4×Dic7—44no—
80C2×C7⋊C8—42no—
81#532C2×D711yescharacteristic
82C2×#125C2×D711yescharacteristic
83#491Dic712yescomplemented
84#490Dic712yescomplemented
85#488—22no—
86#432—72no—
87#417—22no—
88C4×M16C2×D711yescharacteristic
89#387—142no—
90#387—142no—
91C4×C4⋊C4—142no—
92#376—72no—
93#330—142no—
94#327—142no—
95#327—142no—
96C23×C7—41no—
97C23×C7#12011yescharacteristic
98C7×Q8—42no—
99C7×D4—42no—
100C7×D4—82no—
101C7×D4—82no—
102C2×C4×C7—22no—
103C2×C4×C7—24no—
104C2×C4×C7—42no—
105C2×C4×C7—42no—
106C2×C4×C7—42no—
107C2×C4×C7—41no—
108C2×C4×C7—82no—
109C2×C4×C7C2×C22⋊C411yescharacteristic
110C2×C4×C7C4×D412yes—
111C2×C4×C7#12112yes—
112C2×C4×C7#12311yescharacteristic
113C2×Dic7—24no—
114C2×Dic7—42no—
115C2×Dic7—42no—
116C2×Dic7—42no—
117C2×Dic7—42no—
118C2×Dic7—42no—
119C7×C8—42no—
120C7⋊C8—82no—
121C2×#41—21no—
122C2×M16—21no—
123C4⋊Q8C2×Dic711yescharacteristic
124#127C2×Dic711yescharacteristic
125#125—22no—
126#125C2×Dic712yes—
127#125C22⋊D712yes—
128#124—21no—
129#121—22no—
130C4×D4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—142no—
133C2×C4⋊C4C22⋊D712yes—
134C2×C42—142no—
135C2×C42—142no—
136C2×C42C22×D711yescharacteristic
137C4⋊C8—142no—
138#103—24no—
139#102—24no—
140C22⋊C8—142no—
141#97—142no—
142#97C2×Dic712yes—
143C4×C8—21no—
144#95—72no—
145#95—72no—
146#95—142no—
147#95—142no—
148#95—142no—
149C22×C7—22no—
150C22×C7—81no—
151C22×C7—82no—
152C22×C7#38611yescharacteristic
153C22×C7#45412yes—
154C4×C7—21no—
155C4×C7—21no—
156C4×C7—22no—
157C4×C7—22no—
158C4×C7—44no—
159C4×C7—81no—
160Dic7—44no—
161Dic7—82no—
162#41—44no—
163C2×Q8—21no—
164C2×D4—22no—
165C2×D4—21no—
166C2×D4—41no—
167C22×C4—22no—
168C22×C4—41no—
169C22×C4—142no—
170C22×C4—142no—
171C22×C4D4×D712yes—
172C22×C4C2×C22⋊D711yescharacteristic
173M16—42no—
174C2×C8—22no—
175C2×C8—282no—
176C4⋊C4—24no—
177C4⋊C4—22no—
178C4⋊C4—24no—
179C4⋊C4—22no—
180C4⋊C4—284no—
181C4⋊C4C22⋊Dic714yes—
182C22⋊C4—42no—
183C42—284no—
184C42—284no—
185C42#115912yes—
186C42C22×Dic711yescharacteristic
187C42C2×C22⋊D711yescharacteristic
188C2×C7—22no—
189C2×C7—161no—
190C2×C7#188811yescharacteristic
191C2×C7#190612yes—
192C23—41no—
193C23#650011yescharacteristic
194Q8—42no—
195D4—42no—
196D4—82no—
197D4—82no—
198C2×C4—22no—
199C2×C4—24no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—41no—
204C2×C4—82no—
205C2×C4—144no—
206C2×C4—282no—
207C2×C4—282no—
208C2×C4—282no—
209C2×C4—282no—
210C2×C4—282no—
211C2×C4D4×Dic712yes—
212C2×C4#649811yescharacteristic
213C2×C4#650212yes—
214C2×C4C2×C22⋊Dic711yescharacteristic
215C8—42no—
216C8—562no—
217C7#944211yescharacteristic, complemented, Sylow
218C22—22no—
219C22—81no—
220C22—82no—
221C22#8983412yes—
222C22#8999211yescharacteristic
223C4—21no—
224C4—21no—
225C4—22no—
226C4—22no—
227C4—44no—
228C4—81no—
229C4—284no—
230C4—562no—
231C2—22no—
232C2—161no—
233C2#1172947811yescharacteristic
234C2#1172986812yes—
subgroupG/H
centerC22#89992
commutatorC42×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#3142C2
socleC22×C7#386
prime radicalC7×#532C22
cosocleC2×C42×C7C23
hypercenter#3142D7
hypocenterC7#9442
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3142

Complements

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

NH ≅ G/Ntype
2. #11740471C2semidirect
4. #11729490C2semidirect
5. #11725471C2semidirect
8. C7×#491C4semidirect
9. C7×#490C4semidirect
83. #491Dic7semidirect
84. #490Dic7semidirect
217. C7#9442semidirect

Maximal subgroups

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

subgroupindex
C7×#31422
#117404712
#117295662
#117294902
#117254712
#94427