Finite Group Databasebeta

← #13251451

Subgroups — #13251451

Order 1792.

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

normal subgroups only

This group has 516 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×#2500C211yesmaximal, characteristic
2#11739654C211yesmaximal, characteristic, complemented
3#11729428C212yesmaximal
4#11729348C211yesmaximal, characteristic, complemented
5#11725422C212yesmaximal, complemented
6C2×C7×#121C2211yescharacteristic
7C2×C7×#117C2211yescharacteristic
8C7×#394—21no—
9C7×#394—21no—
10C7×#390C411yescharacteristic, complemented
11C7×#390C411yescharacteristic, complemented
12C7×#388—21no—
13C7×#388—21no—
14C7×#384C411yescharacteristic, complemented
15C7×#382C411yescharacteristic, complemented
16C4×C7×C22⋊C4C2211yescharacteristic
17#89983C2212yes—
18#89979C2212yes—
19#89772—24no—
20#89696—21no—
21C4×#1136—21no—
22#89332—24no—
23#89321—21no—
24#89321—21no—
25#9446—71nomaximal, Sylow
26C22×C7×D4—21no—
27C22×C7×D4C2×C411yescharacteristic
28C23×C4×C7C2311yescharacteristic
29C7×#121—22no—
30C7×#121—22no—
31C7×#117—22no—
32C7×#117—22no—
33C2×C7×C4⋊C4—21no—
34C2×C7×C4⋊C4—21no—
35C2×C7×C4⋊C4C2×C411yescharacteristic
36C2×C7×C4⋊C4C2×C411yescharacteristic
37C2×C7×C4⋊C4C2×C411yescharacteristic
38C2×C7×C22⋊C4—21no—
39C2×C7×C22⋊C4—21no—
40C2×C7×C22⋊C4—21no—
41C2×C7×C22⋊C4—21no—
42C2×C7×C22⋊C4—21no—
43C2×C7×C22⋊C4—21no—
44C2×C7×C22⋊C4D411yescharacteristic
45C2×C7×C22⋊C4D411yescharacteristic
46C2×C42×C7C2×C411yescharacteristic
47C2×C42×C7C2×C411yescharacteristic
48C2×C42×C7D411yescharacteristic
49C2×C42×C7D411yescharacteristic
50C2×C22⋊Dic7—22no—
51C2×#1136—22no—
52C7×#95—21no—
53C7×#95—21no—
54C7×#95—21no—
55C7⋊#95—22no—
56C7⋊#95—22no—
57C22⋊(C7⋊C8)—22no—
58C22⋊(C7⋊C8)D414yes—
59C4⋊(C7⋊C8)—24no—
60#6377—22no—
61C4×C7⋊C8—22no—
62#2921—71no—
63#2500D711yescharacteristic
64#1820—71no—
65#1510—72no—
66#1284—72no—
67C24×C7—21no—
68C24×C7C2×D411yescharacteristic
69C2×C7×D4—22no—
70C2×C7×D4—41no—
71C2×C7×D4—41no—
72C2×C7×D4—42no—
73C2×C7×D4—81no—
74C2×C7×D4—81no—
75C22×C4×C7—21no—
76C22×C4×C7—21no—
77C22×C4×C7—21no—
78C22×C4×C7—41no—
79C22×C4×C7—41no—
80C22×C4×C7C22⋊C411yescharacteristic
81C22×C4×C7C22⋊C411yescharacteristic
82C22×C4×C7C22⋊C411yescharacteristic
83C22×C4×C7C22⋊C411yescharacteristic
84C22×C4×C7C22×C411yescharacteristic
85C22×C4×C7C2×D411yescharacteristic
86C22×C4×C7C2×D412yes—
87C22×C4×C7#4112yes—
88C22×Dic7—42no—
89C7×C4⋊C4—22no—
90C7×C4⋊C4—22no—
91C7×C4⋊C4—22no—
92C7×C4⋊C4—81no—
93C7×C4⋊C4—81no—
94C7×C22⋊C4—22no—
95C7×C22⋊C4—22no—
96C7×C22⋊C4—42no—
97C7×C22⋊C4—42no—
98C7×C22⋊C4—42no—
99C7×C22⋊C4—41no—
100C7×C22⋊C4—41no—
101C7×C22⋊C4—81no—
102C7×C22⋊C4—81no—
103C42×C7—21no—
104C42×C7—21no—
105C42×C7—21no—
106C42×C7—21no—
107C42×C7—41no—
108C42×C7—41no—
109C22⋊Dic7—42no—
110C22⋊Dic7—42no—
111#1136—42no—
112#1136—42no—
113C2×C7⋊C8—24no—
114C2×#121C2×D711yescharacteristic
115C2×#117C2×D711yescharacteristic
116#433—144no—
117#423—141no—
118#407—72no—
119C4×M16—141no—
120#394—21no—
121#394—21no—
122#390Dic711yescharacteristic, complemented
123#390Dic711yescharacteristic, complemented
124#388—21no—
125#388—21no—
126#384Dic711yescharacteristic, complemented
127#382Dic711yescharacteristic, complemented
128#381—72no—
129C4×C22⋊C4C2×D711yescharacteristic
130#342—141no—
131#342—141no—
132#327—144no—
133C23×C7—21no—
134C23×C7—21no—
135C23×C7—41no—
136C23×C7—41no—
137C23×C7—41no—
138C23×C7—41no—
139C23×C7—41no—
140C23×C7—41no—
141C23×C7—41no—
142C23×C7—41no—
143C23×C7—41no—
144C23×C7C2×C22⋊C411yescharacteristic
145C23×C7#12112yes—
146C7×D4—82no—
147C7×D4—84no—
148C2×C4×C7—21no—
149C2×C4×C7—21no—
150C2×C4×C7—21no—
151C2×C4×C7—21no—
152C2×C4×C7—22no—
153C2×C4×C7—21no—
154C2×C4×C7—21no—
155C2×C4×C7—21no—
156C2×C4×C7—21no—
157C2×C4×C7—42no—
158C2×C4×C7—42no—
159C2×C4×C7—41no—
160C2×C4×C7—41no—
161C2×C4×C7—41no—
162C2×C4×C7—41no—
163C2×C4×C7—41no—
164C2×C4×C7—41no—
165C2×C4×C7—41no—
166C2×C4×C7—41no—
167C2×C4×C7—41no—
168C2×C4×C7—41no—
169C2×C4×C7—41no—
170C2×C4×C7—41no—
171C2×C4×C7C4×D412yes—
172C2×C4×C7#12011yescharacteristic
173C2×C4×C7#12311yescharacteristic
174C2×Dic7—82no—
175C2×Dic7—82no—
176C2×Dic7—82no—
177C7⋊C8—44no—
178C22×D4—21no—
179C22×D4C2×Dic711yescharacteristic
180C23×C4C22×D711yescharacteristic
181C2×M16—142no—
182#121—22no—
183#121—22no—
184#117—22no—
185#117—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4C2×Dic711yescharacteristic
189C2×C4⋊C4C2×Dic711yescharacteristic
190C2×C4⋊C4C2×Dic711yescharacteristic
191C2×C22⋊C4—21no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—21no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—142no—
198C2×C22⋊C4C22⋊D711yescharacteristic
199C2×C22⋊C4C22⋊D711yescharacteristic
200C2×C42C2×Dic711yescharacteristic
201C2×C42C2×Dic711yescharacteristic
202C2×C42C22⋊D711yescharacteristic
203C2×C42C22⋊D711yescharacteristic
204C4⋊C8—144no—
205C22⋊C8—74no—
206C22⋊C8—142no—
207#97—142no—
208C4×C8—142no—
209#95—21no—
210#95—21no—
211#95—21no—
212#95—142no—
213#95—142no—
214C22×C7—21no—
215C22×C7—21no—
216C22×C7—22no—
217C22×C7—22no—
218C22×C7—22no—
219C22×C7—81no—
220C22×C7—81no—
221C22×C7—81no—
222C22×C7—81no—
223C22×C7—81no—
224C22×C7—81no—
225C22×C7—81no—
226C22×C7—81no—
227C22×C7—81no—
228C22×C7#38611yescharacteristic
229C22×C7#41011yescharacteristic
230C22×C7#41011yescharacteristic
231C4×C7—21no—
232C4×C7—22no—
233C4×C7—21no—
234C4×C7—42no—
235C4×C7—41no—
236C4×C7—41no—
237C4×C7—41no—
238C4×C7—41no—
239C4×C7—81no—
240C4×C7—81no—
241Dic7—162no—
242C24—21no—
243C24C2×C22⋊D711yescharacteristic
244C2×D4—22no—
245C2×D4—41no—
246C2×D4—41no—
247C2×D4—42no—
248C2×D4—81no—
249C2×D4—81no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—282no—
256C22×C4C22⋊Dic711yescharacteristic
257C22×C4C22⋊Dic711yescharacteristic
258C22×C4C22⋊Dic711yescharacteristic
259C22×C4C22⋊Dic711yescharacteristic
260C22×C4D4×D712yes—
261C22×C4#115912yes—
262C22×C4C22×Dic711yescharacteristic
263C22×C4C2×C22⋊D711yescharacteristic
264M16—282no—
265M16—282no—
266C2×C8—144no—
267C4⋊C4—22no—
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—81no—
271C4⋊C4—81no—
272C22⋊C4—22no—
273C22⋊C4—22no—
274C22⋊C4—42no—
275C22⋊C4—42no—
276C22⋊C4—42no—
277C22⋊C4—41no—
278C22⋊C4—41no—
279C22⋊C4—81no—
280C22⋊C4—81no—
281C22⋊C4—282no—
282C22⋊C4—282no—
283C42—21no—
284C42—21no—
285C42—21no—
286C42—21no—
287C42—41no—
288C42—41no—
289C2×C7—21no—
290C2×C7—21no—
291C2×C7—42no—
292C2×C7—161no—
293C2×C7#188411yescharacteristic
294C2×C7#189911yescharacteristic
295C2×C7#191011yescharacteristic
296C23—21no—
297C23—21no—
298C23—41no—
299C23—41no—
300C23—41no—
301C23—41no—
302C23—41no—
303C23—41no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23#650212yes—
308C23C2×C22⋊Dic711yescharacteristic
309D4—82no—
310D4—84no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—22no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—562no—
335C2×C4—562no—
336C2×C4—562no—
337C2×C4D4×Dic712yes—
338C2×C4#649811yescharacteristic
339C2×C4#650011yescharacteristic
340C8—284no—
341C7#944611yescharacteristic, complemented, Sylow
342C22—21no—
343C22—21no—
344C22—22no—
345C22—22no—
346C22—22no—
347C22—81no—
348C22—81no—
349C22—81no—
350C22—81no—
351C22—81no—
352C22—81no—
353C22—81no—
354C22—81no—
355C22—81no—
356C22#8999211yescharacteristic
357C22#9001211yescharacteristic
358C22#9001211yescharacteristic
359C4—21no—
360C4—22no—
361C4—21no—
362C4—42no—
363C4—41no—
364C4—41no—
365C4—41no—
366C4—41no—
367C4—81no—
368C4—81no—
369C4—1122no—
370C2—21no—
371C2—21no—
372C2—42no—
373C2—161no—
374C2#1172945911yescharacteristic
375C2#1172957111yescharacteristic
376C2#1172960811yescharacteristic
subgroupG/H
centerC22#89992
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
FittingC7×#2500C2
socleC22×C7#386
prime radicalC2×C7×#121C22
cosocleC23×C4×C7C23
hypercenter#2500D7
hypocenterC7#9446
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2500

Complements

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

NH ≅ G/Ntype
2. #11739654C2semidirect
4. #11729348C2semidirect
5. #11725422C2semidirect
10. C7×#390C4semidirect
11. C7×#390C4semidirect
14. C7×#384C4semidirect
15. C7×#382C4semidirect
122. #390Dic7semidirect
123. #390Dic7semidirect
126. #384Dic7semidirect
127. #382Dic7semidirect
341. C7#9446semidirect

Maximal subgroups

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

subgroupindex
C7×#25002
#117396542
#117294282
#117293482
#117254222
#94467