Finite Group Databasebeta

← #13251262

Subgroups — #13251262

Order 1792.

720 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 720 conjugacy classes of subgroups — too many to draw. Only its 107 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×#2539C211yesmaximal, characteristic
2#11739538C211yesmaximal, characteristic, complemented
3#11729427C212yesmaximal, complemented
4#11729293C211yesmaximal, characteristic, complemented
5#11725424C212yesmaximal, complemented
6C2×C7×#123—21no—
7C2×C7×#121C2211yescharacteristic
8C2×C7×#120—21no—
9C22×C7×C22⋊C4C2211yescharacteristic
10C7×#399C412yescomplemented
11C7×#392C412yescomplemented
12C7×#390—22no—
13C4×C7×C22⋊C4C2211yescharacteristic
14#90019—21no—
15#89984—21no—
16C4×C22⋊Dic7C2212yes—
17#89979C2212yes—
18#89723—24no—
19#89332—24no—
20#89320—22no—
21#9413—71nomaximal, Sylow
22C25×C7D411yescharacteristic
23C22×C7×D4—21no—
24C22×C7×D4—22no—
25C22×C7×D4C2×C411yescharacteristic
26C23×C4×C7D411yescharacteristic
27C23×C4×C7C2311yescharacteristic
28C7×#123—44no—
29C7×#121—22no—
30C7×#121—22no—
31C7×#120—44no—
32C2×C7×C4⋊C4—22no—
33C2×C7×C4⋊C4C2×C411yescharacteristic
34C2×C7×C22⋊C4—21no—
35C2×C7×C22⋊C4—22no—
36C2×C7×C22⋊C4—22no—
37C2×C7×C22⋊C4—22no—
38C2×C7×C22⋊C4—22no—
39C2×C7×C22⋊C4—21no—
40C2×C7×C22⋊C4C2×C412yes—
41C2×C7×C22⋊C4D412yes—
42C2×C42×C7—21no—
43C2×C22⋊Dic7—22no—
44C2×C22⋊Dic7—22no—
45C2×C22⋊Dic7—22no—
46C2×C4⋊Dic7—22no—
47C2×C4×Dic7—22no—
48C2×#1136—22no—
49C7×#95C2×C412yes—
50C7⋊#95—22no—
51C7⋊#95D414yescomplemented
52C22⋊(C7⋊C8)—22no—
53C22⋊(C7⋊C8)—22no—
54#2539D711yescharacteristic
55#2312—71no—
56#1507—72no—
57#1462—71no—
58#1297—72no—
59C24×C7—21no—
60C24×C7—21no—
61C24×C7—21no—
62C24×C7—42no—
63C24×C7—42no—
64C24×C7—81no—
65C24×C7C22⋊C412yes—
66C24×C7C2×D411yescharacteristic
67C2×C7×D4—22no—
68C2×C7×D4—44no—
69C2×C7×D4—42no—
70C2×C7×D4—42no—
71C2×C7×D4—44no—
72C2×C7×D4—41no—
73C2×C7×D4—41no—
74C2×C7×D4—82no—
75C2×C7×D4—81no—
76C22×C4×C7—22no—
77C22×C4×C7—21no—
78C22×C4×C7—21no—
79C22×C4×C7—21no—
80C22×C4×C7—42no—
81C22×C4×C7—81no—
82C22×C4×C7C22⋊C412yes—
83C22×C4×C7C22×C411yescharacteristic
84C22×C4×C7C2×D412yes—
85C22×C4×C7C2×D411yescharacteristic
86C22×C4×C7#4112yes—
87C22×Dic7—24no—
88C7×C4⋊C4—22no—
89C7×C4⋊C4—44no—
90C7×C22⋊C4—24no—
91C7×C22⋊C4—24no—
92C7×C22⋊C4—24no—
93C7×C22⋊C4—44no—
94C7×C22⋊C4—44no—
95C7×C22⋊C4—42no—
96C42×C7—81no—
97C22⋊Dic7—44no—
98C22⋊Dic7—82no—
99C22⋊Dic7—82no—
100C4⋊Dic7—82no—
101C4×Dic7—82no—
102#1136—82no—
103C2×C7⋊C8—42no—
104C2×#123—21no—
105C2×#121C2×D711yescharacteristic
106C2×#120—21no—
107C22×C22⋊C4C2×D711yescharacteristic
108#407—72no—
109#399Dic712yescomplemented
110#392Dic712yescomplemented
111#390—22no—
112#388—144no—
113#380—141no—
114#379—141no—
115C4×C22⋊C4—72no—
116C4×C22⋊C4C2×D711yescharacteristic
117#327—144no—
118#323—142no—
119C23×C7—22no—
120C23×C7—22no—
121C23×C7—22no—
122C23×C7—22no—
123C23×C7—22no—
124C23×C7—41no—
125C23×C7—42no—
126C23×C7—42no—
127C23×C7—42no—
128C23×C7—42no—
129C23×C7—42no—
130C23×C7—41no—
131C23×C7—41no—
132C23×C7—42no—
133C23×C7—42no—
134C23×C7—41no—
135C23×C7—42no—
136C23×C7—81no—
137C23×C7—81no—
138C23×C7—88no—
139C23×C7C2×C22⋊C411yescharacteristic
140C23×C7#12011yescharacteristic
141C23×C7#12311yescharacteristic
142C7×D4—82no—
143C7×D4—88no—
144C7×D4—82no—
145C7×D4—82no—
146C2×C4×C7—24no—
147C2×C4×C7—22no—
148C2×C4×C7—42no—
149C2×C4×C7—42no—
150C2×C4×C7—41no—
151C2×C4×C7—41no—
152C2×C4×C7—42no—
153C2×C4×C7—42no—
154C2×C4×C7—42no—
155C2×C4×C7—42no—
156C2×C4×C7—41no—
157C2×C4×C7—41no—
158C2×C4×C7—41no—
159C2×C4×C7—81no—
160C2×C4×C7C4×D412yes—
161C2×C4×C7#12112yes—
162C2×Dic7—44no—
163C2×Dic7—44no—
164C2×Dic7—44no—
165C7⋊C8—82no—
166C25C22⋊D711yescharacteristic
167C22×D4—21no—
168C22×D4—22no—
169C22×D4C2×Dic711yescharacteristic
170C23×C4C22⋊D711yescharacteristic
171C23×C4C22×D711yescharacteristic
172C2×M16—142no—
173#123—44no—
174#121—22no—
175#121—22no—
176#120—44no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—142no—
179C2×C4⋊C4C2×Dic711yescharacteristic
180C2×C22⋊C4—21no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—21no—
186C2×C22⋊C4—142no—
187C2×C22⋊C4—142no—
188C2×C22⋊C4—142no—
189C2×C22⋊C4C2×Dic712yes—
190C2×C22⋊C4C22⋊D712yes—
191C2×C42—21no—
192C2×C42—142no—
193C22⋊C8—142no—
194C22⋊C8—142no—
195#95—74no—
196#95—142no—
197#95C2×Dic712yes—
198C22×C7—21no—
199C22×C7—21no—
200C22×C7—21no—
201C22×C7—21no—
202C22×C7—21no—
203C22×C7—21no—
204C22×C7—21no—
205C22×C7—44no—
206C22×C7—44no—
207C22×C7—42no—
208C22×C7—42no—
209C22×C7—42no—
210C22×C7—82no—
211C22×C7—84no—
212C22×C7—82no—
213C22×C7—82no—
214C22×C7—84no—
215C22×C7—81no—
216C22×C7—81no—
217C22×C7—81no—
218C22×C7#38611yescharacteristic
219C22×C7#42112yes—
220C22×C7#45312yes—
221C4×C7—22no—
222C4×C7—22no—
223C4×C7—44no—
224C4×C7—42no—
225C4×C7—81no—
226Dic7—84no—
227C24—21no—
228C24—21no—
229C24—21no—
230C24—42no—
231C24—42no—
232C24—81no—
233C24C22⋊Dic712yes—
234C24C2×C22⋊D711yescharacteristic
235C2×D4—22no—
236C2×D4—44no—
237C2×D4—42no—
238C2×D4—42no—
239C2×D4—44no—
240C2×D4—41no—
241C2×D4—41no—
242C2×D4—82no—
243C2×D4—81no—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—42no—
249C22×C4—81no—
250C22×C4—144no—
251C22×C4C22⋊Dic712yes—
252C22×C4D4×D712yes—
253C22×C4#115912yes—
254C22×C4C22×Dic711yescharacteristic
255C22×C4C2×C22⋊D711yescharacteristic
256M16—562no—
257C2×C8—282no—
258C4⋊C4—22no—
259C4⋊C4—44no—
260C4⋊C4—562no—
261C22⋊C4—24no—
262C22⋊C4—24no—
263C22⋊C4—24no—
264C22⋊C4—44no—
265C22⋊C4—44no—
266C22⋊C4—42no—
267C22⋊C4—284no—
268C22⋊C4—562no—
269C22⋊C4—562no—
270C42—81no—
271C42—562no—
272C2×C7—22no—
273C2×C7—22no—
274C2×C7—41no—
275C2×C7—82no—
276C2×C7—161no—
277C2×C7#188711yescharacteristic
278C2×C7#189711yescharacteristic
279C2×C7#191311yescharacteristic
280C23—22no—
281C23—22no—
282C23—22no—
283C23—22no—
284C23—22no—
285C23—41no—
286C23—42no—
287C23—42no—
288C23—42no—
289C23—42no—
290C23—42no—
291C23—41no—
292C23—41no—
293C23—42no—
294C23—42no—
295C23—41no—
296C23—42no—
297C23—81no—
298C23—81no—
299C23—88no—
300C23#649811yescharacteristic
301C23#650011yescharacteristic
302C23C2×C22⋊Dic711yescharacteristic
303D4—82no—
304D4—88no—
305D4—82no—
306D4—82no—
307C2×C4—24no—
308C2×C4—22no—
309C2×C4—41no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—81no—
321C2×C4—284no—
322C2×C4—284no—
323C2×C4—284no—
324C2×C4D4×Dic712yes—
325C2×C4#650212yes—
326C8—562no—
327C7#941311yescharacteristic, complemented, Sylow
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—44no—
336C22—44no—
337C22—42no—
338C22—42no—
339C22—42no—
340C22—82no—
341C22—84no—
342C22—82no—
343C22—82no—
344C22—84no—
345C22—81no—
346C22—81no—
347C22—81no—
348C22#8984912yes—
349C22#8999211yescharacteristic
350C22#9000812yes—
351C4—22no—
352C4—22no—
353C4—44no—
354C4—42no—
355C4—81no—
356C4—564no—
357C2—22no—
358C2—22no—
359C2—41no—
360C2—82no—
361C2—161no—
362C2#1172946611yescharacteristic
363C2#1172951711yescharacteristic
364C2#1172960211yescharacteristic
subgroupG/H
centerC22#89992
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
Fitting, prime radicalC7×#2539C2
socleC22×C7#386
cosocleC23×C4×C7C23
hypercenter#2539D7
hypocenterC7#9413
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2539

Complements

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

NH ≅ G/Ntype
2. #11739538C2semidirect
3. #11729427C2semidirect
4. #11729293C2semidirect
5. #11725424C2semidirect
10. C7×#399C4semidirect
11. C7×#392C4semidirect
51. C7⋊#95D4semidirect
109. #399Dic7semidirect
110. #392Dic7semidirect
327. C7#9413semidirect

Maximal subgroups

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

subgroupindex
C7×#25392
#117395382
#117294272
#117292932
#117254242
#94137