Finite Group Databasebeta

← #13251266

Subgroups — #13251266

Order 1792.

552 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 103 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×#2313C211yesmaximal, characteristic, complemented
2#11739627C211yesmaximal, characteristic
3#11729306C211yesmaximal, characteristic, complemented
4#11727198C212yesmaximal
5#11725424C212yesmaximal, complemented
6C22×C7×C4⋊C4C2211yescharacteristic
7C2×#6493C2211yescharacteristic
8C7×#387—24no—
9C7×#381—21no—
10C7×#380—21no—
11C4×C7×C22⋊C4C2212yes—
12#89985—21no—
13#89985—21no—
14C4×C22⋊Dic7C2211yescharacteristic
15#89979C2212yes—
16#89763C412yescomplemented
17#89762—22no—
18#89758C412yescomplemented
19#89323—22no—
20#89246—24no—
21#9414—71nomaximal, Sylow
22C23×C4×C7D411yescharacteristic
23C23×C4×C7D411yescharacteristic
24C23×C4×C7C2311yescharacteristic
25C22×C22⋊D7—21no—
26C22×D28C2×C411yescharacteristic
27C2×C7×C4⋊C4—22no—
28C2×C7×C4⋊C4—22no—
29C2×C7×C4⋊C4—22no—
30C2×C7×C4⋊C4C2×C412yes—
31C2×C7×C4⋊C4D412yes—
32C2×C7×C22⋊C4—22no—
33C2×C42×C7—22no—
34C2×C22⋊Dic7—21no—
35#6493—22no—
36#6493—22no—
37C2×(C2×D7)⋊C4—22no—
38C2×(C2×D7)⋊C4—21no—
39C2×(C2×D7)⋊C4—22no—
40C2×C4⋊Dic7C2×C411yescharacteristic
41C2×C4×Dic7—21no—
42C2×#1136—22no—
43C7×#95—22no—
44C7×#95—24no—
45C7×#95D414yes—
46C7⋊#95—22no—
47C7⋊#95—22no—
48C7⋊#95C2×C412yes—
49C22⋊(C7⋊C8)—22no—
50C22⋊(C7⋊C8)—22no—
51#2591—71no—
52#2313D711yescharacteristic, complemented
53#1507—72no—
54#1463—71no—
55#1297—72no—
56C24×C7C2×D411yescharacteristic
57C23×D7—21no—
58C22×C4×C7—21no—
59C22×C4×C7—21no—
60C22×C4×C7—21no—
61C22×C4×C7—21no—
62C22×C4×C7—24no—
63C22×C4×C7—42no—
64C22×C4×C7—81no—
65C22×C4×C7—81no—
66C22×C4×C7C22⋊C412yes—
67C22×C4×C7C22⋊C412yes—
68C22×C4×C7C22×C411yescharacteristic
69C22×C4×C7C2×D412yes—
70C22×C4×C7C2×D411yescharacteristic
71C22×C4×C7#4112yes—
72C2×C22⋊D7—44no—
73C2×C22⋊D7—41no—
74C2×C22⋊D7—41no—
75C22×Dic7—21no—
76C22×Dic7—22no—
77C2×D28—22no—
78C2×D28—81no—
79C7×C4⋊C4—24no—
80C7×C4⋊C4—24no—
81C7×C4⋊C4—82no—
82C7×C22⋊C4—44no—
83C42×C7—82no—
84C22⋊Dic7—42no—
85(C2×D7)⋊C4—44no—
86(C2×D7)⋊C4—42no—
87(C2×D7)⋊C4—82no—
88C4⋊Dic7—22no—
89C4×Dic7—81no—
90#1136—82no—
91C2×C7⋊C8—42no—
92C2×#121—71no—
93C22×C4⋊C4C2×D711yescharacteristic
94#407—72no—
95#400—72no—
96#394—72no—
97#388—142no—
98#387—24no—
99#386—141no—
100#386—141no—
101#381—21no—
102#380—21no—
103C4×C22⋊C4—71no—
104C4×C22⋊C4C2×D712yes—
105#327—144no—
106#324—142no—
107C23×C7—22no—
108C23×C7—22no—
109C23×C7—41no—
110C23×C7C2×C22⋊C411yescharacteristic
111C23×C7#12011yescharacteristic
112C23×C7#12311yescharacteristic
113C22×D7—42no—
114C22×D7—42no—
115C22×D7—41no—
116C22×D7—42no—
117C2×C4×C7—22no—
118C2×C4×C7—22no—
119C2×C4×C7—22no—
120C2×C4×C7—22no—
121C2×C4×C7—42no—
122C2×C4×C7—41no—
123C2×C4×C7—42no—
124C2×C4×C7—41no—
125C2×C4×C7—42no—
126C2×C4×C7—42no—
127C2×C4×C7—42no—
128C2×C4×C7—42no—
129C2×C4×C7—44no—
130C2×C4×C7—44no—
131C2×C4×C7—44no—
132C2×C4×C7—81no—
133C2×C4×C7C4×D412yes—
134C2×C4×C7#12112yes—
135C22⋊D7—82no—
136C22⋊D7—82no—
137C2×Dic7—42no—
138C2×Dic7—42no—
139C2×Dic7—42no—
140C2×Dic7—41no—
141C2×Dic7—42no—
142D28—82no—
143C7⋊C8—82no—
144C22×D4—71no—
145C22×D4—141no—
146C23×C4C22⋊D711yescharacteristic
147C23×C4C22⋊D711yescharacteristic
148C23×C4C22×D711yescharacteristic
149C2×M16—142no—
150#121—142no—
151#121—142no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—71no—
156C2×C4⋊C4C4×D712yes—
157C2×C4⋊C4D2812yes—
158C2×C22⋊C4—22no—
159C2×C22⋊C4—141no—
160C2×C22⋊C4—142no—
161C2×C22⋊C4—141no—
162C2×C22⋊C4—142no—
163C2×C42—22no—
164C2×C42—141no—
165C22⋊C8—142no—
166C22⋊C8—142no—
167#95—22no—
168#95—24no—
169#95—72no—
170#95—142no—
171#95—142no—
172#95C22⋊D714yes—
173C22×C7—21no—
174C22×C7—21no—
175C22×C7—21no—
176C22×C7—21no—
177C22×C7—21no—
178C22×C7—21no—
179C22×C7—21no—
180C22×C7—44no—
181C22×C7#38611yescharacteristic
182C22×C7#42112yes—
183C22×C7#45412yes—
184C2×D7—84no—
185C2×D7—81no—
186C2×D7—81no—
187C2×D7—81no—
188C4×C7—22no—
189C4×C7—22no—
190C4×C7—42no—
191C4×C7—42no—
192C4×C7—84no—
193Dic7—82no—
194Dic7—81no—
195C24—141no—
196C24C2×C22⋊D711yescharacteristic
197C2×D4—142no—
198C2×D4—284no—
199C2×D4—281no—
200C2×D4—281no—
201C2×D4—561no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—24no—
207C22×C4—42no—
208C22×C4—81no—
209C22×C4—81no—
210C22×C4—142no—
211C22×C4—141no—
212C22×C4(C2×D7)⋊C412yes—
213C22×C4(C2×D7)⋊C412yes—
214C22×C4C2×C4×D711yescharacteristic
215C22×C4C2×D2811yescharacteristic
216C22×C4#115712yes—
217C22×C4C2×C22⋊D712yes—
218M16—562no—
219C2×C8—282no—
220C4⋊C4—24no—
221C4⋊C4—24no—
222C4⋊C4—82no—
223C4⋊C4—142no—
224C22⋊C4—44no—
225C22⋊C4—284no—
226C22⋊C4—282no—
227C22⋊C4—282no—
228C22⋊C4—562no—
229C42—82no—
230C42—561no—
231C2×C7—22no—
232C2×C7—22no—
233C2×C7—41no—
234C2×C7#188811yescharacteristic
235C2×C7#189711yescharacteristic
236C2×C7#191311yescharacteristic
237D7—161no—
238C23—22no—
239C23—22no—
240C23—41no—
241C23—282no—
242C23—282no—
243C23—281no—
244C23—282no—
245C23C2×(C2×D7)⋊C411yescharacteristic
246C23#649211yescharacteristic
247C23#651611yescharacteristic
248D4—562no—
249D4—562no—
250D4—562no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—42no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—41no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—44no—
264C2×C4—44no—
265C2×C4—44no—
266C2×C4—81no—
267C2×C4—282no—
268C2×C4—282no—
269C2×C4—282no—
270C2×C4—281no—
271C2×C4—282no—
272C2×C4C4×C22⋊D712yes—
273C2×C4#649312yes—
274C8—562no—
275C7#941411yescharacteristic, complemented, Sylow
276C22—21no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—44no—
284C22—564no—
285C22—561no—
286C22—561no—
287C22—561no—
288C22#8977712yes—
289C22#8998511yescharacteristic
290C22#9001712yes—
291C4—22no—
292C4—22no—
293C4—42no—
294C4—42no—
295C4—84no—
296C4—562no—
297C4—561no—
298C2—22no—
299C2—22no—
300C2—41no—
301C2—1121no—
302C2#1172934511yescharacteristic
303C2#1172941011yescharacteristic
304C2#1172941511yescharacteristic
subgroupG/H
centerC22#89985
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
FittingC7×#2313C2
socleC22×C7#386
prime radicalC2×#6493C22
cosocleC23×C4×C7C23
hypercenter#2313D7
hypocenterC7#9414
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2313

Complements

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

NH ≅ G/Ntype
1. C7×#2313C2semidirect
3. #11729306C2semidirect
5. #11725424C2semidirect
16. #89763C4semidirect
18. #89758C4semidirect
52. #2313D7semidirect
275. C7#9414semidirect

Maximal subgroups

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

subgroupindex
C7×#23132
#117396272
#117293062
#117271982
#117254242
#94147