Finite Group Databasebeta

← #13267063

Subgroups — #13267063

Order 1792.

310 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 67 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#11736482C211yesmaximal, characteristic
2C7×#2086C211yesmaximal, characteristic
3#11728356C212yesmaximal
4#11725515C211yesmaximal, characteristic
5#11725292C212yesmaximal
6#90161—21no—
7C2×C7×#107C2211yescharacteristic
8C2×C7×C4⋊C8C2211yescharacteristic
9C7×#389—21no—
10#89888—21no—
11#89888C2212yes—
12#89622—22no—
13#89621—22no—
14C7×#340—21no—
15C7×#340C2211yescharacteristic
16C22⋊(C7⋊C16)C2212yes—
17#89287—22no—
18#89285—22no—
19#12289—71nomaximal, Sylow
20C22×C7×C8D411yescharacteristic
21C22×C7×C8C2311yescharacteristic
22C2×C7×C4⋊C4—21no—
23C2×C7×C4⋊C4—21no—
24C2×C7×C4⋊C4—41no—
25C2×C42×C7D411yescharacteristic
26#6489—41no—
27#6487—22no—
28#6487—41no—
29C2×Dic28—21no—
30C2×Dic28D412yes—
31#6433—41no—
32#6432—21no—
33#6432—21no—
34C4×Dic14—22no—
35C4×Dic14—41no—
36C7×#107—22no—
37C7×C4⋊C8—22no—
38C7×#95—41no—
39#6392—21no—
40#6392D412yes—
41#6390—22no—
42#6390—22no—
43C2×C7⋊C16—22no—
44#3249—71no—
45#2235—72no—
46#2086D711yescharacteristic
47#1348—72no—
48#1322—71no—
49C22×C4×C7—41no—
50C22×C4×C7—41no—
51C22×C4×C7D811yescharacteristic
52C22×C4×C7D811yescharacteristic
53C22×C4×C7C2×D411yescharacteristic
54C2×Dic14—22no—
55C2×Dic14—41no—
56C2×C7×C8—21no—
57C2×C7×C8—41no—
58C2×C7×C8—41no—
59C2×C7×C8C2×D412yes—
60C7×C4⋊C4—22no—
61C7×C4⋊C4—41no—
62C7×C4⋊C4—41no—
63C7×C4⋊C4—41no—
64C7×C4⋊C4—82no—
65C42×C7—21no—
66C42×C7—21no—
67C22⋊Dic7—42no—
68C4⋊Dic7—22no—
69C4⋊Dic7—41no—
70C4⋊Dic7—41no—
71C4⋊Dic7—41no—
72Dic7⋊C4—42no—
73Dic7⋊C4—42no—
74C4×Dic7—42no—
75Dic28—42no—
76Dic28—81no—
77C7⋊C16—42no—
78#533—141no—
79#477—142no—
80#467—72no—
81#467—141no—
82#462—142no—
83C2×#107C2×D711yescharacteristic
84C2×C4⋊C8C2×D711yescharacteristic
85#389—21no—
86#367—142no—
87#358—142no—
88C22⋊C16—72no—
89#340—21no—
90#340C2×D711yescharacteristic
91C23×C7C2×D811yescharacteristic
92C2×C4×C7—21no—
93C2×C4×C7—21no—
94C2×C4×C7—21no—
95C2×C4×C7—21no—
96C2×C4×C7—21no—
97C2×C4×C7—41no—
98C2×C4×C7—42no—
99C2×C4×C7—81no—
100C2×C4×C7—81no—
101C2×C4×C7—81no—
102C2×C4×C7—81no—
103C2×C4×C7—81no—
104C2×C4×C7SD3212yes—
105C2×C4×C7#12011yescharacteristic
106C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
107C2×Dic7—42no—
108C2×Dic7—42no—
109Dic14—42no—
110Dic14—42no—
111Dic14—82no—
112C7×C8—22no—
113C7×C8—41no—
114C2×Q16—72no—
115C2×Q16—141no—
116C22×C8C22⋊D711yescharacteristic
117C22×C8C22×D711yescharacteristic
118C4⋊Q8—141no—
119C4⋊Q8—141no—
120#125—281no—
121C22⋊Q8—142no—
122C22⋊Q8—281no—
123C4×Q8—142no—
124C4×Q8—281no—
125#117—281no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—41no—
129C2×C42C22⋊D711yescharacteristic
130C2×C16—142no—
131#107—22no—
132#107—72no—
133#107—141no—
134C4⋊C8—22no—
135#103—142no—
136#103—142no—
137#95—41no—
138C22×C7—21no—
139C22×C7—21no—
140C22×C7#44711yescharacteristic
141C22×C7C2×SD3211yescharacteristic
142C22×C7#51011yescharacteristic
143C4×C7—21no—
144C4×C7—22no—
145C4×C7—41no—
146C4×C7—82no—
147C4×C7—161no—
148C4×C7#45611yescharacteristic
149C4×C7#45811yescharacteristic
150Dic7—82no—
151Dic7—82no—
152C2×Q8—142no—
153C2×Q8—281no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4#114111yescharacteristic
157C22×C4#114111yescharacteristic
158C22×C4C2×C22⋊D711yescharacteristic
159Q16—282no—
160Q16—561no—
161C2×C8—21no—
162C2×C8—41no—
163C2×C8—41no—
164C2×C8D4×D712yes—
165C4⋊C4—22no—
166C4⋊C4—41no—
167C4⋊C4—41no—
168C4⋊C4—41no—
169C4⋊C4—82no—
170C4⋊C4—142no—
171C4⋊C4—282no—
172C4⋊C4—281no—
173C4⋊C4—281no—
174C4⋊C4—281no—
175C4⋊C4—282no—
176C22⋊C4—282no—
177C42—21no—
178C42—21no—
179C42—282no—
180C16—282no—
181C2×C7—21no—
182C2×C7—21no—
183C2×C7#161011yescharacteristic
184C2×C7#218511yescharacteristic
185C2×C7#219111yescharacteristic
186C23C2×#114111yescharacteristic
187Q8—282no—
188Q8—282no—
189Q8—562no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—41no—
196C2×C4—42no—
197C2×C4—81no—
198C2×C4—81no—
199C2×C4—81no—
200C2×C4—81no—
201C2×C4—81no—
202C2×C4—282no—
203C2×C4—282no—
204C2×C4#640112yes—
205C2×C4#650011yescharacteristic
206C2×C4#651211yescharacteristic
207C8—22no—
208C8—41no—
209C7#1228911yescharacteristic, complemented, Sylow
210C22—21no—
211C22—21no—
212C22#8980811yescharacteristic
213C22C2×#640111yescharacteristic
214C22#8996711yescharacteristic
215C4—21no—
216C4—22no—
217C4—41no—
218C4—82no—
219C4—161no—
220C4—562no—
221C4—562no—
222C4#8982511yescharacteristic
223C4#8986711yescharacteristic
224C2—21no—
225C2—21no—
226C2#1172746611yescharacteristic
227C2#1172883111yescharacteristic
228C2#1172884511yescharacteristic
subgroupG/H
centerC22#89808
commutator, cosocleC22×C7×C8C23
FrattiniC22×C8C22×D7
FittingC7×#2086C2
socleC22×C7#447
prime radicalC23×C7C2×D8
hypercenter#2086D7
hypocenterC7#12289
LCSG ▷ C22×C7×C8 ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C8 ◁ #2086

Complements

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

NH ≅ G/Ntype
209. C7#12289semidirect

Maximal subgroups

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

subgroupindex
#117364822
C7×#20862
#117283562
#117255152
#117252922
#122897