Finite Group Databasebeta

← #13263187

Subgroups — #13263187

Order 1792.

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

normal subgroups only

This group has 670 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×#2309C211yesmaximal, characteristic
2#11729326C212yesmaximal, complemented
3#11725446C214yesmaximal, complemented
4C23×C7×D4C411yescharacteristic
5C2×C7×#117C411yescharacteristic
6C2×C7×#117C2211yescharacteristic
7C7×#390—22no—
8C7×#390C412yes—
9C7×#390C2212yes—
10C7×#379—22no—
11#89698—22no—
12#89323C2214yescomplemented
13#89320—22no—
14#11509—71nomaximal, Sylow
15C25×C7—21no—
16C22×C7×D4—24no—
17C22×C7×D4—21no—
18C22×C7×D4—42no—
19C22×C7×D4—42no—
20C22×C7×D4C2×C412yes—
21C23×C4×C7C2×C411yescharacteristic
22C7×#117—42no—
23C7×#117—42no—
24C2×C7×C4⋊C4—21no—
25C2×C7×C4⋊C4C2×C411yescharacteristic
26C2×C7×C4⋊C4C2311yescharacteristic
27C2×C7×C22⋊C4—21no—
28C2×C7×C22⋊C4—24no—
29C2×C7×C22⋊C4—24no—
30C2×C7×C22⋊C4D412yes—
31C2×C42×C7C2×C412yes—
32C2×C42×C7D412yes—
33C22⋊(C7⋊C8)—24no—
34C4⋊(C7⋊C8)—42no—
35#6377—42no—
36#2309D711yescharacteristic
37#1464—72no—
38#1318—74no—
39C24×C7—24no—
40C24×C7—22no—
41C24×C7—42no—
42C24×C7—42no—
43C24×C7—82no—
44C24×C7—82no—
45C24×C7C22⋊C411yescharacteristic
46C2×C7×D4—24no—
47C2×C7×D4—22no—
48C2×C7×D4—416no—
49C2×C7×D4—42no—
50C2×C7×D4—42no—
51C2×C7×D4—42no—
52C2×C7×D4—42no—
53C2×C7×D4M1614yes—
54C22×C4×C7—22no—
55C22×C4×C7—22no—
56C22×C4×C7—22no—
57C22×C4×C7—22no—
58C22×C4×C7C22⋊C411yescharacteristic
59C22×C4×C7C22⋊C412yes—
60C22×C4×C7C22×C411yescharacteristic
61C22×C4×C7C2×D412yes—
62C7×C4⋊C4—41no—
63C7×C4⋊C4—41no—
64C7×C4⋊C4—42no—
65C7×C22⋊C4—42no—
66C7×C22⋊C4—42no—
67C7×C22⋊C4—84no—
68C7×C22⋊C4—84no—
69C42×C7—22no—
70C42×C7—22no—
71C42×C7—42no—
72C2×C7⋊C8—44no—
73C23×D4Dic711yescharacteristic
74C2×#117Dic711yescharacteristic
75C2×#117C2×D711yescharacteristic
76#432—142no—
77#390—22no—
78#390Dic712yes—
79#390C2×D712yes—
80#379—22no—
81#324—74no—
82#323—142no—
83C23×C7—21no—
84C23×C7—21no—
85C23×C7—24no—
86C23×C7—41no—
87C23×C7—41no—
88C23×C7—44no—
89C23×C7—44no—
90C23×C7—48no—
91C23×C7—42no—
92C23×C7—42no—
93C23×C7—88no—
94C23×C7—88no—
95C23×C7—84no—
96C23×C7—84no—
97C23×C7#10012yes—
98C23×C7C2×C22⋊C411yescharacteristic
99C7×D4—44no—
100C7×D4—48no—
101C7×D4—44no—
102C2×C4×C7—28no—
103C2×C4×C7—21no—
104C2×C4×C7—21no—
105C2×C4×C7—21no—
106C2×C4×C7—21no—
107C2×C4×C7—22no—
108C2×C4×C7—22no—
109C2×C4×C7—42no—
110C2×C4×C7—42no—
111C2×C4×C7—42no—
112C2×C4×C7—42no—
113C2×C4×C7—42no—
114C2×C4×C7#9912yes—
115C2×C4×C7C2×M1612yes—
116C7⋊C8—84no—
117C25—21no—
118C22×D4—24no—
119C22×D4—21no—
120C22×D4—42no—
121C22×D4—42no—
122C22×D4C2×Dic712yes—
123C23×C4C2×Dic711yescharacteristic
124#117—42no—
125#117—42no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4C2×Dic711yescharacteristic
128C2×C4⋊C4C22×D711yescharacteristic
129C2×C22⋊C4—21no—
130C2×C22⋊C4—24no—
131C2×C22⋊C4—24no—
132C2×C22⋊C4C22⋊D712yes—
133C2×C42C2×Dic712yes—
134C2×C42C22⋊D712yes—
135C4⋊C8—282no—
136C22⋊C8—144no—
137#97—282no—
138C22×C7—21no—
139C22×C7—21no—
140C22×C7—22no—
141C22×C7—42no—
142C22×C7—42no—
143C22×C7—48no—
144C22×C7—41no—
145C22×C7—41no—
146C22×C7—44no—
147C22×C7—84no—
148C22×C7—816no—
149C22×C7—82no—
150C22×C7—82no—
151C22×C7#40711yescharacteristic
152C22×C7C2×#9911yescharacteristic
153C22×C7C2×#10011yescharacteristic
154C4×C7—22no—
155C4×C7—22no—
156C4×C7—42no—
157C4×C7—42no—
158C4×C7—82no—
159C24—24no—
160C24—22no—
161C24—42no—
162C24—42no—
163C24—82no—
164C24—82no—
165C24C22⋊Dic711yescharacteristic
166C2×D4—24no—
167C2×D4—22no—
168C2×D4—42no—
169C2×D4—42no—
170C2×D4—416no—
171C2×D4—42no—
172C2×D4—42no—
173C2×D4#113614yes—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4C22⋊Dic712yes—
179C22×C4C22⋊Dic711yescharacteristic
180C22×C4C22×Dic711yescharacteristic
181C22×C4C2×C22⋊D712yes—
182C2×C8—284no—
183C4⋊C4—41no—
184C4⋊C4—41no—
185C4⋊C4—42no—
186C22⋊C4—42no—
187C22⋊C4—42no—
188C22⋊C4—84no—
189C22⋊C4—84no—
190C42—22no—
191C42—22no—
192C42—42no—
193C2×C7—21no—
194C2×C7—21no—
195C2×C7—42no—
196C2×C7—84no—
197C2×C7#146211yescharacteristic
198C2×C7#212811yescharacteristic
199C2×C7#213411yescharacteristic
200C23—21no—
201C23—21no—
202C23—24no—
203C23—41no—
204C23—41no—
205C23—44no—
206C23—44no—
207C23—48no—
208C23—42no—
209C23—42no—
210C23—88no—
211C23—88no—
212C23—84no—
213C23—84no—
214C23#640712yes—
215C23C2×C22⋊Dic711yescharacteristic
216D4—44no—
217D4—48no—
218D4—44no—
219C2×C4—28no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4#640812yes—
232C2×C4C2×#113612yes—
233C8—564no—
234C7#1150911yescharacteristic, complemented, Sylow
235C22—21no—
236C22—21no—
237C22—22no—
238C22—42no—
239C22—42no—
240C22—48no—
241C22—41no—
242C22—41no—
243C22—44no—
244C22—84no—
245C22—816no—
246C22—82no—
247C22—82no—
248C22#8997911yescharacteristic
249C22C2×#640711yescharacteristic
250C22C2×#640811yescharacteristic
251C4—22no—
252C4—22no—
253C4—42no—
254C4—42no—
255C4—82no—
256C2—21no—
257C2—21no—
258C2—42no—
259C2—84no—
260C2#1172929311yescharacteristic
261C2#1172961511yescharacteristic
262C2#1172961611yescharacteristic
subgroupG/H
centerC22C2×#6408
commutatorC22×C4×C7C22×C4
FrattiniC2×C4⋊C4C22×D7
FittingC7×#2309C2
socleC22×C7C2×#99
prime radicalC23×C7×D4C4
cosocleC2×C7×C4⋊C4C23
hypercenter#2309D7
hypocenterC7#11509
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2309

Complements

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

NH ≅ G/Ntype
2. #11729326C2semidirect
3. #11725446C2semidirect
12. #89323C22semidirect
234. C7#11509semidirect

Maximal subgroups

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

subgroupindex
C7×#23092
#117293262
#117254462
#115097