Finite Group Databasebeta

← #13249770

Subgroups — #13249770

Order 1792.

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

normal subgroups only

This group has 582 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×#2635C211yesmaximal, characteristic
2#11739747C211yesmaximal, characteristic, complemented
3#11729427C212yesmaximal, complemented
4C4×#6408C211yesmaximal, characteristic, complemented
5#11725434C212yesmaximal, complemented
6C2×C7×#121C2211yescharacteristic
7C2×C4×C7×D4C2211yescharacteristic
8C7×#400C411yescharacteristic, complemented
9C7×#399C411yescharacteristic, complemented
10C7×#394—21no—
11C7×#394C411yescharacteristic, complemented
12C7×#392—21no—
13C7×#392C411yescharacteristic, complemented
14C7×#388—21no—
15C7×#386—21no—
16C7×#381C2211yescharacteristic
17C2×#6408—21no—
18C4×C22⋊Dic7C2212yes—
19#89979C2212yes—
20C2×C22⋊(C7⋊C8)—21no—
21#89783—24no—
22#89698—21no—
23#89332—24no—
24#89321—21no—
25#9185—71nomaximal, Sylow
26C22×C7×D4—21no—
27C22×C7×D4—21no—
28C22×C7×D4C2×C411yescharacteristic
29C22×C7×D4D411yescharacteristic
30C23×C4×C7D411yescharacteristic
31C23×C4×C7C2311yescharacteristic
32C7×#121—22no—
33C7×#121—22no—
34C4×C7×D4—22no—
35C4×C7×D4—22no—
36C2×C7×C4⋊C4—21no—
37C2×C7×C4⋊C4C2×C411yescharacteristic
38C2×C7×C4⋊C4C2×C411yescharacteristic
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⋊C4—21no—
45C2×C7×C22⋊C4C2×C411yescharacteristic
46C2×C7×C22⋊C4D411yescharacteristic
47C2×C42×C7D411yescharacteristic
48C2×C22⋊Dic7—22no—
49C2×C4×Dic7—22no—
50C2×#1136—22no—
51C22×C7⋊C8—22no—
52C7×#95—21no—
53C7×#95—21no—
54C7×#95—21no—
55C7×#95C2×C411yescharacteristic
56C7×#95C2×C411yescharacteristic
57#6408—24no—
58C7⋊#95—22no—
59C22⋊(C7⋊C8)—22no—
60C22⋊(C7⋊C8)—22no—
61C22⋊(C7⋊C8)D414yescomplemented
62C4⋊(C7⋊C8)—22no—
63#6377—22no—
64#2973—71no—
65#2635D711yescharacteristic
66C4×#99—71no—
67#1507—72no—
68#1284—72no—
69C24×C7—21no—
70C24×C7C22⋊C411yescharacteristic
71C24×C7C2×D411yescharacteristic
72C2×C7×D4—22no—
73C2×C7×D4—22no—
74C2×C7×D4—22no—
75C2×C7×D4—41no—
76C2×C7×D4—42no—
77C2×C7×D4—42no—
78C2×C7×D4—81no—
79C2×C7×D4—81no—
80C2×C7×D4—81no—
81C2×C7×D4—81no—
82C2×C7×D4—81no—
83C22×C4×C7—21no—
84C22×C4×C7—21no—
85C22×C4×C7—21no—
86C22×C4×C7—21no—
87C22×C4×C7—21no—
88C22×C4×C7—81no—
89C22×C4×C7—81no—
90C22×C4×C7C22⋊C411yescharacteristic
91C22×C4×C7C22⋊C411yescharacteristic
92C22×C4×C7C22⋊C411yescharacteristic
93C22×C4×C7C22×C411yescharacteristic
94C22×C4×C7C2×D412yes—
95C22×C4×C7C2×D411yescharacteristic
96C22×C4×C7#4112yes—
97C22×Dic7—42no—
98C7×C4⋊C4—22no—
99C7×C4⋊C4—22no—
100C7×C4⋊C4—81no—
101C7×C22⋊C4—22no—
102C7×C22⋊C4—22no—
103C7×C22⋊C4—42no—
104C7×C22⋊C4—41no—
105C7×C22⋊C4—41no—
106C7×C22⋊C4—81no—
107C7×C22⋊C4—81no—
108C7×C22⋊C4—81no—
109C7×C22⋊C4—81no—
110C42×C7—21no—
111C42×C7—21no—
112C22⋊Dic7—24no—
113C22⋊Dic7—42no—
114C4×Dic7—82no—
115#1136—82no—
116C2×C7⋊C8—24no—
117C2×C7⋊C8—82no—
118C2×#121C2×D711yescharacteristic
119C2×C4×D4C2×D711yescharacteristic
120#435—144no—
121#432—141no—
122C2×#99—141no—
123#407—72no—
124C2×C22⋊C8—141no—
125#400Dic711yescharacteristic, complemented
126#399Dic711yescharacteristic, complemented
127#394—21no—
128#394Dic711yescharacteristic, complemented
129#392—21no—
130#392Dic711yescharacteristic, complemented
131#388—21no—
132#386—21no—
133#381C2×D711yescharacteristic
134C4×C22⋊C4—72no—
135#342—141no—
136#327—144no—
137C23×C7—21no—
138C23×C7—21no—
139C23×C7—21no—
140C23×C7—21no—
141C23×C7—41no—
142C23×C7—41no—
143C23×C7—41no—
144C23×C7—41no—
145C23×C7—41no—
146C23×C7—41no—
147C23×C7—41no—
148C23×C7—41no—
149C23×C7—41no—
150C23×C7—41no—
151C23×C7—41no—
152C23×C7#10412yes—
153C23×C7C2×C22⋊C411yescharacteristic
154C23×C7#12112yes—
155C7×D4—44no—
156C7×D4—82no—
157C7×D4—84no—
158C7×D4—84no—
159C2×C4×C7—21no—
160C2×C4×C7—21no—
161C2×C4×C7—21no—
162C2×C4×C7—21no—
163C2×C4×C7—21no—
164C2×C4×C7—42no—
165C2×C4×C7—42no—
166C2×C4×C7—41no—
167C2×C4×C7—41no—
168C2×C4×C7—42no—
169C2×C4×C7—41no—
170C2×C4×C7—41no—
171C2×C4×C7—41no—
172C2×C4×C7—41no—
173C2×C4×C7—41no—
174C2×C4×C7—41no—
175C2×C4×C7—41no—
176C2×C4×C7—41no—
177C2×C4×C7—41no—
178C2×C4×C7—41no—
179C2×C4×C7—41no—
180C2×C4×C7—81no—
181C2×C4×C7C4×D412yes—
182C2×C4×C7#12011yescharacteristic
183C2×C4×C7#12311yescharacteristic
184C2×Dic7—44no—
185C2×Dic7—82no—
186C2×Dic7—82no—
187C7⋊C8—44no—
188C22×D4—21no—
189C22×D4—21no—
190C22×D4C2×Dic711yescharacteristic
191C22×D4C22⋊D711yescharacteristic
192C23×C4C22⋊D711yescharacteristic
193C23×C4C22×D711yescharacteristic
194C2×M16—142no—
195C22×C8—142no—
196#121—22no—
197#121—22no—
198C4×D4—22no—
199C4×D4—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4C2×Dic711yescharacteristic
202C2×C4⋊C4C2×Dic711yescharacteristic
203C2×C22⋊C4—21no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—142no—
210C2×C22⋊C4C2×Dic711yescharacteristic
211C2×C22⋊C4C22⋊D711yescharacteristic
212C2×C42—142no—
213C2×C42C22⋊D711yescharacteristic
214C4⋊C8—142no—
215#99—144no—
216C22⋊C8—74no—
217C22⋊C8—142no—
218C22⋊C8—142no—
219#97—142no—
220#95—21no—
221#95—21no—
222#95—21no—
223#95—142no—
224#95C2×Dic711yescharacteristic
225#95C2×Dic711yescharacteristic
226C22×C7—21no—
227C22×C7—21no—
228C22×C7—22no—
229C22×C7—22no—
230C22×C7—22no—
231C22×C7—22no—
232C22×C7—42no—
233C22×C7—41no—
234C22×C7—41no—
235C22×C7—81no—
236C22×C7—81no—
237C22×C7—81no—
238C22×C7—81no—
239C22×C7—81no—
240C22×C7—81no—
241C22×C7—81no—
242C22×C7—81no—
243C22×C7—81no—
244C22×C7—81no—
245C22×C7#38611yescharacteristic
246C22×C7C2×#10411yescharacteristic
247C22×C7#42111yescharacteristic
248C4×C7—21no—
249C4×C7—21no—
250C4×C7—22no—
251C4×C7—41no—
252C4×C7—41no—
253C4×C7—41no—
254C4×C7—81no—
255C4×C7—81no—
256C4×C7—81no—
257Dic7—84no—
258C24—21no—
259C24C22⋊Dic711yescharacteristic
260C24C2×C22⋊D711yescharacteristic
261C2×D4—22no—
262C2×D4—22no—
263C2×D4—22no—
264C2×D4—41no—
265C2×D4—42no—
266C2×D4—42no—
267C2×D4—81no—
268C2×D4—81no—
269C2×D4—81no—
270C2×D4—81no—
271C2×D4—81no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—81no—
278C22×C4—81no—
279C22×C4—282no—
280C22×C4C22⋊Dic711yescharacteristic
281C22×C4C22⋊Dic711yescharacteristic
282C22×C4C22⋊Dic711yescharacteristic
283C22×C4D4×D712yes—
284C22×C4#115912yes—
285C22×C4C22×Dic711yescharacteristic
286C22×C4C2×C22⋊D711yescharacteristic
287M16—562no—
288C2×C8—144no—
289C2×C8—562no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—81no—
293C22⋊C4—22no—
294C22⋊C4—22no—
295C22⋊C4—42no—
296C22⋊C4—41no—
297C22⋊C4—41no—
298C22⋊C4—81no—
299C22⋊C4—81no—
300C22⋊C4—81no—
301C22⋊C4—81no—
302C22⋊C4—144no—
303C22⋊C4—282no—
304C42—21no—
305C42—21no—
306C42—562no—
307C2×C7—21no—
308C2×C7—21no—
309C2×C7—42no—
310C2×C7—42no—
311C2×C7—161no—
312C2×C7#187111yescharacteristic
313C2×C7#191011yescharacteristic
314C2×C7#191311yescharacteristic
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23#641112yes—
331C23#650212yes—
332C23C2×C22⋊Dic711yescharacteristic
333D4—44no—
334D4—82no—
335D4—84no—
336D4—84no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—81no—
359C2×C4—284no—
360C2×C4—562no—
361C2×C4—562no—
362C2×C4D4×Dic712yes—
363C2×C4#649811yescharacteristic
364C2×C4#650011yescharacteristic
365C8—284no—
366C7#918511yescharacteristic, complemented, Sylow
367C22—21no—
368C22—21no—
369C22—22no—
370C22—22no—
371C22—22no—
372C22—22no—
373C22—42no—
374C22—41no—
375C22—41no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22—81no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22—81no—
386C22#8999211yescharacteristic
387C22C2×#641111yescharacteristic
388C22#9000811yescharacteristic
389C4—21no—
390C4—21no—
391C4—22no—
392C4—41no—
393C4—41no—
394C4—41no—
395C4—81no—
396C4—81no—
397C4—81no—
398C4—564no—
399C2—21no—
400C2—21no—
401C2—42no—
402C2—42no—
403C2—161no—
404C2#1172946711yescharacteristic
405C2#1172960211yescharacteristic
406C2#1172960811yescharacteristic
subgroupG/H
centerC22#89992
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
Fitting, prime radicalC7×#2635C2
socleC22×C7#386
cosocleC23×C4×C7C23
hypercenter#2635D7
hypocenterC7#9185
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2635

Complements

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

NH ≅ G/Ntype
2. #11739747C2semidirect
3. #11729427C2semidirect
4. C4×#6408C2semidirect
5. #11725434C2semidirect
8. C7×#400C4semidirect
9. C7×#399C4semidirect
11. C7×#394C4semidirect
13. C7×#392C4semidirect
61. C22⋊(C7⋊C8)D4semidirect
125. #400Dic7semidirect
126. #399Dic7semidirect
128. #394Dic7semidirect
130. #392Dic7semidirect
366. C7#9185semidirect

Maximal subgroups

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

subgroupindex
C7×#26352
#117397472
#117294272
C4×#64082
#117254342
#91857