Finite Group Databasebeta

← #13242926

Subgroups — #13242926

Order 1792.

366 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 83 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#11734047C211yesmaximal, characteristic
2C7×#2096C211yesmaximal, characteristic
3#11727323C212yesmaximal
4#11725448C211yesmaximal, characteristic
5#11725283C212yesmaximal
6C2×C7×#106—21no—
7C2×C7×#103—21no—
8C7×#391C2211yescharacteristic
9C7×#384C2211yescharacteristic
10#90002—21no—
11#89758—21no—
12C2×#6381—22no—
13#89700C2212yes—
14#89433—22no—
15#89427—22no—
16C7×#336C2211yescharacteristic
17#89324—22no—
18#89319C2212yes—
19#7802—71nomaximal, Sylow
20C22×C7×Q8D411yescharacteristic
21C22×C7×C8—21no—
22C2×C7×C4⋊C4—22no—
23C2×C7×C4⋊C4—21no—
24C2×C7×C4⋊C4D411yescharacteristic
25C2×C42×C7C2311yescharacteristic
26C2×C4⋊Dic7—21no—
27C2×C4⋊Dic7—21no—
28C2×C4⋊Dic7D412yes—
29C2×C4⋊Dic7D412yes—
30C2×Dic7⋊C4—22no—
31C2×C4×Dic7—21no—
32C22×C7⋊C8—22no—
33C7×#106—42no—
34C7×#103—42no—
35C7×#95—22no—
36C7⋊#95—21no—
37C7⋊#95—22no—
38C7⋊#95—22no—
39C7⋊#95—22no—
40#6381—44no—
41#2809—71no—
42#2096D711yescharacteristic
43#2083—72no—
44#1272—71no—
45#1271—72no—
46C2×C7×Q8—21no—
47C2×C7×Q8—41no—
48C2×C7×Q8—41no—
49C2×C7×Q8Q1612yes—
50C22×C4×C7—21no—
51C22×C4×C7—21no—
52C22×C4×C7—41no—
53C22×C4×C7C2×D411yescharacteristic
54C22×C4×C7C2×D412yes—
55C22×Dic7—22no—
56C22×Dic7—22no—
57C2×C7×C8—22no—
58C2×C7×C8—41no—
59C2×C7×C8—41no—
60C7×C4⋊C4—41no—
61C7×C4⋊C4—41no—
62C7×C4⋊C4—41no—
63C7×C4⋊C4—42no—
64C7×C4⋊C4—42no—
65C42×C7—41no—
66C4⋊Dic7—22no—
67C4⋊Dic7—22no—
68C4⋊Dic7—42no—
69C4⋊Dic7—81no—
70C4⋊Dic7—81no—
71Dic7⋊C4—42no—
72Dic7⋊C4—42no—
73C4×Dic7—81no—
74C2×C7⋊C8—24no—
75C2×C7⋊C8—42no—
76C2×C7⋊C8—42no—
77C2×#107—142no—
78C2×#106—21no—
79C2×#103—21no—
80#400—141no—
81#398—142no—
82#391—141no—
83#391C2×D711yescharacteristic
84#389—72no—
85#387—142no—
86#384C2×D711yescharacteristic
87#340—142no—
88#336—72no—
89#336C2×D711yescharacteristic
90C23×C7#12011yescharacteristic
91C7×Q8—42no—
92C7×Q8—81no—
93C2×C4×C7—21no—
94C2×C4×C7—22no—
95C2×C4×C7—22no—
96C2×C4×C7—21no—
97C2×C4×C7—22no—
98C2×C4×C7—41no—
99C2×C4×C7—41no—
100C2×C4×C7—41no—
101C2×C4×C7—42no—
102C2×C4×C7—82no—
103C2×C4×C7—81no—
104C2×C4×C7C2×Q1611yescharacteristic
105C2×C4×C7#13512yes—
106C2×C4×C7AGL(1,ℤ/8)12yes—
107C2×C4×C7#13711yescharacteristic
108C2×Dic7—42no—
109C2×Dic7—42no—
110C2×Dic7—42no—
111C2×Dic7—42no—
112C2×Dic7—42no—
113C2×Dic7—42no—
114C7×C8—42no—
115C7⋊C8—44no—
116C22×Q8C22⋊D711yescharacteristic
117C22×C8—21no—
118C22×C8—142no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—72no—
122C2×C4⋊C4—72no—
123C2×C4⋊C4—142no—
124C2×C4⋊C4—141no—
125C2×C4⋊C4—141no—
126C2×C4⋊C4C22⋊D711yescharacteristic
127C2×C42—141no—
128C2×C42C22×D711yescharacteristic
129#107—284no—
130#106—42no—
131#103—42no—
132#95—22no—
133#95—141no—
134#95—142no—
135#95—142no—
136#95—142no—
137C22×C7#44912yes—
138C22×C7#45111yescharacteristic
139C22×C7#45311yescharacteristic
140C22×C7#45612yes—
141C22×C7#45811yescharacteristic
142C4×C7—22no—
143C4×C7—22no—
144C4×C7—21no—
145C4×C7—21no—
146C4×C7—42no—
147C4×C7—81no—
148C4×C7—161no—
149Dic7—82no—
150Dic7—82no—
151C2×Q8—21no—
152C2×Q8—41no—
153C2×Q8—41no—
154C2×Q8#114412yes—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—41no—
158C22×C4—142no—
159C22×C4—142no—
160C22×C4D4×D712yes—
161C22×C4C2×C22⋊D711yescharacteristic
162C2×C8—22no—
163C2×C8—41no—
164C2×C8—41no—
165C2×C8—144no—
166C2×C8—282no—
167C2×C8—282no—
168C4⋊C4—41no—
169C4⋊C4—41no—
170C4⋊C4—41no—
171C4⋊C4—42no—
172C4⋊C4—42no—
173C4⋊C4—142no—
174C4⋊C4—142no—
175C4⋊C4—282no—
176C4⋊C4—282no—
177C4⋊C4—282no—
178C4⋊C4—561no—
179C4⋊C4—561no—
180C42—41no—
181C42—561no—
182C2×C7#160912yes—
183C2×C7#161411yescharacteristic
184C2×C7#162411yescharacteristic
185C2×C7#164812yes—
186C2×C7#166011yescharacteristic
187C23#650011yescharacteristic
188Q8—42no—
189Q8—81no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—22no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—42no—
198C2×C4—41no—
199C2×C4—82no—
200C2×C4—81no—
201C2×C4—282no—
202C2×C4—282no—
203C2×C4—282no—
204C2×C4—282no—
205C2×C4—282no—
206C2×C4—282no—
207C2×C4#647712yes—
208C2×C4#647912yes—
209C2×C4C2×#114411yescharacteristic
210C2×C4#651411yescharacteristic
211C8—42no—
212C8—284no—
213C7#780211yescharacteristic, complemented, Sylow
214C22#8982211yescharacteristic
215C22#8982511yescharacteristic
216C22#8984911yescharacteristic
217C22#8994412yes—
218C22#8997712yes—
219C4—22no—
220C4—22no—
221C4—21no—
222C4—21no—
223C4—42no—
224C4—81no—
225C4—161no—
226C4—562no—
227C4—562no—
228C2#1172747011yescharacteristic
229C2#1172888412yes—
230C2#1172892812yes—
231C2#1172901911yescharacteristic
232C2#1172908411yescharacteristic
subgroupG/H
centerC23#6500
commutator, cosocleC2×C42×C7C23
FrattiniC2×C42C22×D7
FittingC7×#2096C2
socle, prime radicalC23×C7#120
hypercenter#2096D7
hypocenterC7#7802
LCSG ▷ C2×C42×C7 ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2096

Complements

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

NH ≅ G/Ntype
213. C7#7802semidirect

Maximal subgroups

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

subgroupindex
#117340472
C7×#20962
#117273232
#117254482
#117252832
#78027