Finite Group Databasebeta

← #13263167

Subgroups — #13263167

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#11725428C214yesmaximal, complemented
4C23×C7×D4C411yescharacteristic
5C2×C7×#117C411yescharacteristic
6C2×C7×#117C2211yescharacteristic
7C7×#390—22no—
8C7×#390—22no—
9C7×#379C412yes—
10C7×#379C2212yes—
11#89698—22no—
12#89323—22no—
13#89320C2214yes—
14#11503—71nomaximal, Sylow
15C25×C7C2×C412yes—
16C22×C7×D4—24no—
17C22×C7×D4—21no—
18C22×C7×D4—21no—
19C22×C7×D4—41no—
20C22×C7×D4—41no—
21C22×C7×D4—42no—
22C23×C4×C7C2×C411yescharacteristic
23C7×#117—41no—
24C7×#117—41no—
25C7×#117—42no—
26C2×C7×C4⋊C4—21no—
27C2×C7×C4⋊C4D412yes—
28C2×C7×C22⋊C4—24no—
29C2×C7×C22⋊C4—24no—
30C2×C7×C22⋊C4C2×C412yes—
31C2×C7×C22⋊C4C2×C411yescharacteristic
32C2×C7×C22⋊C4C2311yescharacteristic
33C2×C42×C7—21no—
34C2×C42×C7D412yes—
35C22⋊(C7⋊C8)—24no—
36C4⋊(C7⋊C8)—42no—
37#6377—42no—
38#2309D711yescharacteristic
39#1464—72no—
40#1316—74no—
41C24×C7—22no—
42C24×C7—22no—
43C24×C7—42no—
44C24×C7—42no—
45C24×C7—84no—
46C24×C7M1614yes—
47C24×C7C22×C411yescharacteristic
48C2×C7×D4—24no—
49C2×C7×D4—24no—
50C2×C7×D4—48no—
51C2×C7×D4—48no—
52C2×C7×D4—42no—
53C2×C7×D4—42no—
54C2×C7×D4—41no—
55C2×C7×D4—41no—
56C2×C7×D4—42no—
57C22×C4×C7—22no—
58C22×C4×C7—21no—
59C22×C4×C7—21no—
60C22×C4×C7—22no—
61C22×C4×C7—22no—
62C22×C4×C7C22⋊C411yescharacteristic
63C22×C4×C7C22⋊C411yescharacteristic
64C22×C4×C7C22⋊C412yes—
65C22×C4×C7C2×D412yes—
66C7×C4⋊C4—42no—
67C7×C4⋊C4—42no—
68C7×C22⋊C4—41no—
69C7×C22⋊C4—41no—
70C7×C22⋊C4—42no—
71C7×C22⋊C4—84no—
72C7×C22⋊C4—84no—
73C42×C7—22no—
74C42×C7—22no—
75C42×C7—42no—
76C2×C7⋊C8—44no—
77C23×D4Dic711yescharacteristic
78C2×#117Dic711yescharacteristic
79C2×#117C2×D711yescharacteristic
80#432—142no—
81#390—22no—
82#390—22no—
83#379Dic712yes—
84#379C2×D712yes—
85#324—142no—
86#323—74no—
87C23×C7—21no—
88C23×C7—21no—
89C23×C7—24no—
90C23×C7—42no—
91C23×C7—44no—
92C23×C7—44no—
93C23×C7—44no—
94C23×C7—44no—
95C23×C7—42no—
96C23×C7—42no—
97C23×C7—84no—
98C23×C7—84no—
99C23×C7—88no—
100C23×C7—88no—
101C23×C7C2×C22⋊C411yescharacteristic
102C23×C7C2×M1612yes—
103C7×D4—44no—
104C7×D4—44no—
105C7×D4—48no—
106C2×C4×C7—24no—
107C2×C4×C7—24no—
108C2×C4×C7—21no—
109C2×C4×C7—21no—
110C2×C4×C7—21no—
111C2×C4×C7—21no—
112C2×C4×C7—22no—
113C2×C4×C7—22no—
114C2×C4×C7—42no—
115C2×C4×C7—42no—
116C2×C4×C7—44no—
117C2×C4×C7—42no—
118C2×C4×C7#9911yescharacteristic
119C2×C4×C7#9911yescharacteristic
120C2×C4×C7#10012yes—
121C7⋊C8—84no—
122C25C2×Dic712yes—
123C22×D4—24no—
124C22×D4—21no—
125C22×D4—21no—
126C22×D4—42no—
127C22×D4—41no—
128C22×D4—41no—
129C23×C4C2×Dic711yescharacteristic
130#117—41no—
131#117—41no—
132#117—42no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4C22⋊D712yes—
135C2×C22⋊C4—24no—
136C2×C22⋊C4—24no—
137C2×C22⋊C4C2×Dic712yes—
138C2×C22⋊C4C2×Dic711yescharacteristic
139C2×C22⋊C4C22×D711yescharacteristic
140C2×C42—21no—
141C2×C42C22⋊D712yes—
142C4⋊C8—282no—
143C22⋊C8—144no—
144#97—282no—
145C22×C7—22no—
146C22×C7—22no—
147C22×C7—42no—
148C22×C7—42no—
149C22×C7—44no—
150C22×C7—44no—
151C22×C7—42no—
152C22×C7—44no—
153C22×C7—84no—
154C22×C7—88no—
155C22×C7—88no—
156C22×C7—84no—
157C22×C7#40711yescharacteristic
158C22×C7C2×#9911yescharacteristic
159C22×C7C2×#10011yescharacteristic
160C4×C7—21no—
161C4×C7—21no—
162C4×C7—22no—
163C4×C7—42no—
164C4×C7—42no—
165C4×C7—82no—
166C24—22no—
167C24—22no—
168C24—42no—
169C24—42no—
170C24—84no—
171C24#113614yes—
172C24C22×Dic711yescharacteristic
173C2×D4—24no—
174C2×D4—24no—
175C2×D4—48no—
176C2×D4—48no—
177C2×D4—42no—
178C2×D4—42no—
179C2×D4—42no—
180C2×D4—41no—
181C2×D4—41no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—22no—
187C22×C4C22⋊Dic711yescharacteristic
188C22×C4C22⋊Dic712yes—
189C22×C4C22⋊Dic711yescharacteristic
190C22×C4C2×C22⋊D712yes—
191C2×C8—284no—
192C4⋊C4—42no—
193C4⋊C4—42no—
194C22⋊C4—41no—
195C22⋊C4—41no—
196C22⋊C4—42no—
197C22⋊C4—84no—
198C22⋊C4—84no—
199C42—22no—
200C42—22no—
201C42—42no—
202C2×C7—21no—
203C2×C7—21no—
204C2×C7—42no—
205C2×C7—84no—
206C2×C7#146211yescharacteristic
207C2×C7#212212yes—
208C23—21no—
209C23—21no—
210C23—24no—
211C23—42no—
212C23—44no—
213C23—44no—
214C23—44no—
215C23—44no—
216C23—42no—
217C23—42no—
218C23—84no—
219C23—84no—
220C23—88no—
221C23—88no—
222C23C2×#113612yes—
223C23C2×C22⋊Dic711yescharacteristic
224D4—48no—
225D4—44no—
226D4—44no—
227C2×C4—24no—
228C2×C4—24no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—44no—
238C2×C4—42no—
239C2×C4#640712yes—
240C2×C4#640811yescharacteristic
241C2×C4#640811yescharacteristic
242C8—564no—
243C7#1150311yescharacteristic, complemented, Sylow
244C22—22no—
245C22—22no—
246C22—42no—
247C22—42no—
248C22—44no—
249C22—44no—
250C22—42no—
251C22—44no—
252C22—84no—
253C22—88no—
254C22—88no—
255C22—84no—
256C22#8997911yescharacteristic
257C22C2×#640711yescharacteristic
258C22C2×#640811yescharacteristic
259C4—22no—
260C4—21no—
261C4—21no—
262C4—42no—
263C4—42no—
264C4—82no—
265C2—21no—
266C2—21no—
267C2—42no—
268C2—84no—
269C2#1172929311yescharacteristic
270C2#1172961312yes—
subgroupG/H
centerC22C2×#6408
commutatorC24×C7C22×C4
FrattiniC2×C22⋊C4C22×D7
FittingC7×#2309C2
socleC22×C7C2×#99
prime radicalC23×C7×D4C4
cosocleC2×C7×C22⋊C4C23
hypercenter#2309D7
hypocenterC7#11503
LCSG ▷ C24×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. #11725428C2semidirect
243. C7#11503semidirect

Maximal subgroups

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

subgroupindex
C7×#23092
#117293262
#117254282
#115037