Finite Group Databasebeta

← #13249815

Subgroups — #13249815

Order 1792.

458 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 119 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×#2628C211yesmaximal, characteristic
2#11739654C211yesmaximal, characteristic
3#11729529C212yesmaximal, complemented
4#11729325C211yesmaximal, characteristic
5#11725437C212yesmaximal, complemented
6C2×C7×C4⋊Q8C2211yescharacteristic
7C2×C7×#117C2211yescharacteristic
8C7×#400C412yes—
9C7×#399C412yes—
10C7×#395—21no—
11C7×#393—21no—
12C7×#389—21no—
13C7×#382C2211yescharacteristic
14C7×#380—21no—
15#89979—21no—
16#89772—24no—
17#89697C2212yes—
18#89696C2212yes—
19C4×#1136—21no—
20#89342—24no—
21#89323—21no—
22#89320—21no—
23#9194—71nomaximal, Sylow
24C22×C7×Q8—21no—
25C23×C4×C7D411yescharacteristic
26C7×C4⋊Q8—22no—
27C7×C4⋊Q8—22no—
28C7×#117—22no—
29C7×#117—22no—
30C2×C7×C4⋊C4—21no—
31C2×C7×C4⋊C4—21no—
32C2×C7×C4⋊C4—22no—
33C2×C7×C4⋊C4C2×C412yes—
34C2×C7×C4⋊C4C2×C411yescharacteristic
35C2×C7×C4⋊C4D411yescharacteristic
36C2×C7×C22⋊C4—22no—
37C2×C7×C22⋊C4C2×C411yescharacteristic
38C2×C7×C22⋊C4D411yescharacteristic
39C2×C42×C7D411yescharacteristic
40C2×C42×C7C2311yescharacteristic
41C2×#1136—22no—
42C7×#95—22no—
43C7×#95—22no—
44C7×#95—21no—
45C7×#95C2×C412yes—
46C22⋊(C7⋊C8)—24no—
47C22⋊(C7⋊C8)—22no—
48C4⋊(C7⋊C8)—22no—
49C4⋊(C7⋊C8)—22no—
50C4⋊(C7⋊C8)D414yes—
51#6377—22no—
52C4×C7⋊C8—22no—
53C4×C7⋊C8—22no—
54#2921—71no—
55#2628D711yescharacteristic
56#1549—72no—
57#1458—71no—
58#1286—72no—
59C24×C7C22⋊C411yescharacteristic
60C2×C7×Q8—42no—
61C2×C7×Q8—82no—
62C22×C4×C7—21no—
63C22×C4×C7—21no—
64C22×C4×C7—22no—
65C22×C4×C7—22no—
66C22×C4×C7—22no—
67C22×C4×C7—22no—
68C22×C4×C7C22⋊C411yescharacteristic
69C22×C4×C7C22⋊C411yescharacteristic
70C22×C4×C7C22⋊C411yescharacteristic
71C22×C4×C7C22×C411yescharacteristic
72C22×C4×C7C2×D411yescharacteristic
73C22×C4×C7C2×D411yescharacteristic
74C7×C4⋊C4—22no—
75C7×C4⋊C4—22no—
76C7×C4⋊C4—24no—
77C7×C4⋊C4—41no—
78C7×C4⋊C4—41no—
79C7×C4⋊C4—42no—
80C7×C4⋊C4—82no—
81C7×C22⋊C4—22no—
82C7×C22⋊C4—22no—
83C7×C22⋊C4—82no—
84C42×C7—21no—
85C42×C7—21no—
86C42×C7C2×D412yes—
87C42×C7#4112yes—
88#1136—42no—
89#1136—42no—
90C2×C7⋊C8—24no—
91C2×C7⋊C8—42no—
92C2×C4⋊Q8C2×D711yescharacteristic
93C2×#117C2×D711yescharacteristic
94#433—144no—
95#431—72no—
96#423—72no—
97#407—141no—
98C4×M16—141no—
99#400Dic712yes—
100#399Dic712yes—
101#395—21no—
102#393—21no—
103#389—21no—
104#382C2×D711yescharacteristic
105#380—21no—
106#332—144no—
107#324—141no—
108#323—141no—
109C23×C7—21no—
110C23×C7—21no—
111C23×C7—21no—
112C23×C7—41no—
113C23×C7—41no—
114C23×C7C2×C22⋊C411yescharacteristic
115C7×Q8—84no—
116C2×C4×C7—21no—
117C2×C4×C7—21no—
118C2×C4×C7—21no—
119C2×C4×C7—21no—
120C2×C4×C7—21no—
121C2×C4×C7—41no—
122C2×C4×C7—41no—
123C2×C4×C7—42no—
124C2×C4×C7—42no—
125C2×C4×C7—41no—
126C2×C4×C7—41no—
127C2×C4×C7—42no—
128C2×C4×C7—42no—
129C2×C4×C7—42no—
130C2×C4×C7—42no—
131C2×C4×C7—42no—
132C2×C4×C7—42no—
133C2×C4×C7C4×D412yes—
134C2×C4×C7#12011yescharacteristic
135C2×C4×C7#12112yes—
136C2×C4×C7#12311yescharacteristic
137C2×C4×C7#13512yes—
138C2×C4×C7#13712yes—
139C7⋊C8—44no—
140C7⋊C8—82no—
141C22×Q8—21no—
142C23×C4C22⋊D711yescharacteristic
143C2×M16—142no—
144C4⋊Q8—22no—
145C4⋊Q8—22no—
146#117—22no—
147#117—22no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4C2×Dic712yes—
152C2×C4⋊C4C2×Dic711yescharacteristic
153C2×C4⋊C4C22⋊D711yescharacteristic
154C2×C22⋊C4—22no—
155C2×C22⋊C4C2×Dic711yescharacteristic
156C2×C22⋊C4C22⋊D711yescharacteristic
157C2×C42C22⋊D711yescharacteristic
158C2×C42C22×D711yescharacteristic
159C4⋊C8—74no—
160C4⋊C8—142no—
161C4⋊C8—142no—
162C22⋊C8—144no—
163C22⋊C8—142no—
164#97—142no—
165C4×C8—142no—
166C4×C8—142no—
167#95—22no—
168#95—22no—
169#95—21no—
170#95C2×Dic712yes—
171C22×C7—21no—
172C22×C7—21no—
173C22×C7—42no—
174C22×C7—41no—
175C22×C7—41no—
176C22×C7—41no—
177C22×C7—81no—
178C22×C7#38611yescharacteristic
179C22×C7#41611yescharacteristic
180C22×C7#41911yescharacteristic
181C4×C7—22no—
182C4×C7—22no—
183C4×C7—41no—
184C4×C7—41no—
185C4×C7—41no—
186C4×C7—82no—
187C4×C7—82no—
188C4×C7#45212yes—
189C4×C7#48512yes—
190C24C22⋊Dic711yescharacteristic
191C2×Q8—42no—
192C2×Q8—82no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4C22⋊Dic711yescharacteristic
200C22×C4C22⋊Dic711yescharacteristic
201C22×C4C22⋊Dic711yescharacteristic
202C22×C4C22×Dic711yescharacteristic
203C22×C4C2×C22⋊D711yescharacteristic
204C22×C4C2×C22⋊D711yescharacteristic
205M16—282no—
206M16—282no—
207C2×C8—144no—
208C2×C8—282no—
209C4⋊C4—22no—
210C4⋊C4—22no—
211C4⋊C4—24no—
212C4⋊C4—41no—
213C4⋊C4—41no—
214C4⋊C4—42no—
215C4⋊C4—82no—
216C22⋊C4—22no—
217C22⋊C4—22no—
218C22⋊C4—82no—
219C42—21no—
220C42—21no—
221C42D4×D712yes—
222C42#115912yes—
223C2×C7—21no—
224C2×C7—21no—
225C2×C7—81no—
226C2×C7#187311yescharacteristic
227C2×C7#191011yescharacteristic
228C2×C7#191511yescharacteristic
229C23—21no—
230C23—21no—
231C23—21no—
232C23—41no—
233C23—41no—
234C23C2×C22⋊Dic711yescharacteristic
235Q8—84no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4D4×Dic712yes—
254C2×C4#649811yescharacteristic
255C2×C4#650011yescharacteristic
256C2×C4#650212yes—
257C2×C4#650512yes—
258C2×C4#651312yes—
259C8—284no—
260C8—562no—
261C7#919411yescharacteristic, complemented, Sylow
262C22—21no—
263C22—21no—
264C22—42no—
265C22—41no—
266C22—41no—
267C22—41no—
268C22—81no—
269C22#8999211yescharacteristic
270C22#8999711yescharacteristic
271C22#9000511yescharacteristic
272C4—22no—
273C4—22no—
274C4—41no—
275C4—41no—
276C4—41no—
277C4—82no—
278C4—82no—
279C4#8981712yes—
280C4#8981912yes—
281C2—21no—
282C2—21no—
283C2—81no—
284C2#1172944711yescharacteristic
285C2#1172960811yescharacteristic
286C2#1172962011yescharacteristic
subgroupG/H
centerC22#89992
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#2628C2
socleC22×C7#386
prime radicalC24×C7C22⋊C4
cosocleC2×C42×C7C23
hypercenter#2628D7
hypocenterC7#9194
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2628

Complements

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

NH ≅ G/Ntype
3. #11729529C2semidirect
5. #11725437C2semidirect
261. C7#9194semidirect

Maximal subgroups

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

subgroupindex
C7×#26282
#117396542
#117295292
#117293252
#117254372
#91947