Finite Group Databasebeta

← #13251122

Subgroups — #13251122

Order 1792.

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

normal subgroups only

This group has 548 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×#2564C211yesmaximal, characteristic
2#11739747C211yesmaximal, characteristic
3#11729444C212yesmaximal, complemented
4#11729373C211yesmaximal, characteristic
5#11725418C212yesmaximal, complemented
6C2×C7×C22⋊Q8C2211yescharacteristic
7C2×C4×C7×D4C2211yescharacteristic
8C7×#400C411yescharacteristic, complemented
9C7×#399C411yescharacteristic, complemented
10C7×#395C411yescharacteristic, complemented
11C7×#393C411yescharacteristic, complemented
12C7×#387—21no—
13C7×#386—21no—
14C7×#385—21no—
15C7×#380—21no—
16C2×C7×#95C2211yescharacteristic
17C2×#6408—21no—
18#89984C2212yes—
19#89979—21no—
20C2×C22⋊(C7⋊C8)C2212yes—
21#89783—24no—
22#89698—21no—
23#89333—24no—
24#89321—21no—
25#9385—71nomaximal, Sylow
26C22×C7×Q8C2×C411yescharacteristic
27C22×C7×D4D411yescharacteristic
28C23×C4×C7—21no—
29C23×C4×C7D411yescharacteristic
30C23×C4×C7C2311yescharacteristic
31C7×C22⋊Q8—22no—
32C7×C22⋊Q8—22no—
33C4×C7×D4—22no—
34C4×C7×D4—22no—
35C2×C7×C4⋊C4—21no—
36C2×C7×C4⋊C4—21no—
37C2×C7×C4⋊C4—21no—
38C2×C7×C4⋊C4C2×C411yescharacteristic
39C2×C7×C4⋊C4C2×C411yescharacteristic
40C2×C7×C22⋊C4—21no—
41C2×C7×C22⋊C4—21no—
42C2×C7×C22⋊C4—21no—
43C2×C7×C22⋊C4C2×C411yescharacteristic
44C2×C7×C22⋊C4D411yescharacteristic
45C2×C42×C7D411yescharacteristic
46C2×C22⋊Dic7—22no—
47C2×C4⋊Dic7—22no—
48C2×#1136—22no—
49C22×C7⋊C8—22no—
50C7×#95—21no—
51C7×#95—21no—
52C7×#95—21no—
53C7×#95—21no—
54C7×#95—21no—
55C7×#95C2×C411yescharacteristic
56C7×#95C2×C411yescharacteristic
57#6408—42no—
58C7⋊#95—22no—
59C22⋊(C7⋊C8)—22no—
60C22⋊(C7⋊C8)—22no—
61C22⋊(C7⋊C8)D414yes—
62C4⋊(C7⋊C8)—22no—
63#6377—22no—
64#2973—71no—
65#2564D711yescharacteristic
66#1799—71no—
67#1520—72no—
68#1294—72no—
69C24×C7C22⋊C411yescharacteristic
70C24×C7C2×D411yescharacteristic
71C2×C7×Q8—22no—
72C2×C7×Q8—81no—
73C2×C7×D4—22no—
74C2×C7×D4—21no—
75C2×C7×D4—21no—
76C2×C7×D4—22no—
77C22×C4×C7—21no—
78C22×C4×C7—21no—
79C22×C4×C7—21no—
80C22×C4×C7—21no—
81C22×C4×C7—21no—
82C22×C4×C7—21no—
83C22×C4×C7—42no—
84C22×C4×C7—41no—
85C22×C4×C7—41no—
86C22×C4×C7—81no—
87C22×C4×C7—81no—
88C22×C4×C7—81no—
89C22×C4×C7C22⋊C411yescharacteristic
90C22×C4×C7C22⋊C411yescharacteristic
91C22×C4×C7C22⋊C411yescharacteristic
92C22×C4×C7C22×C411yescharacteristic
93C22×C4×C7C2×D412yes—
94C22×C4×C7C2×D411yescharacteristic
95C22×C4×C7#4112yes—
96C22×Dic7—42no—
97C7×C4⋊C4—22no—
98C7×C4⋊C4—22no—
99C7×C4⋊C4—42no—
100C7×C4⋊C4—41no—
101C7×C4⋊C4—41no—
102C7×C4⋊C4—81no—
103C7×C22⋊C4—22no—
104C7×C22⋊C4—22no—
105C7×C22⋊C4—42no—
106C7×C22⋊C4—42no—
107C7×C22⋊C4—81no—
108C42×C7—21no—
109C42×C7—21no—
110C22⋊Dic7—42no—
111C22⋊Dic7—42no—
112C4⋊Dic7—82no—
113#1136—82no—
114C2×C7⋊C8—24no—
115C2×C7⋊C8—82no—
116C2×C22⋊Q8C2×D711yescharacteristic
117C2×C4×D4C2×D711yescharacteristic
118#435—144no—
119#432—141no—
120C2×#99—141no—
121#407—141no—
122C2×C22⋊C8—72no—
123#400Dic711yescharacteristic, complemented
124#399Dic711yescharacteristic, complemented
125#395Dic711yescharacteristic, complemented
126#393Dic711yescharacteristic, complemented
127#387—21no—
128#386—21no—
129#385—21no—
130#380—21no—
131#380—72no—
132C2×#95C2×D711yescharacteristic
133#342—141no—
134#328—144no—
135C23×C7—21no—
136C23×C7—21no—
137C23×C7—21no—
138C23×C7—41no—
139C23×C7—41no—
140C23×C7—41no—
141C23×C7—41no—
142C23×C7C2×C22⋊C411yescharacteristic
143C23×C7#12112yes—
144C23×C7AGL(1,ℤ/8)11yescharacteristic
145C23×C7#13712yes—
146C23×C7#13711yescharacteristic
147C7×Q8—82no—
148C7×D4—42no—
149C7×D4—42no—
150C2×C4×C7—22no—
151C2×C4×C7—21no—
152C2×C4×C7—21no—
153C2×C4×C7—21no—
154C2×C4×C7—21no—
155C2×C4×C7—21no—
156C2×C4×C7—42no—
157C2×C4×C7—42no—
158C2×C4×C7—41no—
159C2×C4×C7—41no—
160C2×C4×C7—42no—
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×C7—41no—
172C2×C4×C7—41no—
173C2×C4×C7—41no—
174C2×C4×C7—81no—
175C2×C4×C7—82no—
176C2×C4×C7—82no—
177C2×C4×C7C4×D412yes—
178C2×C4×C7#12011yescharacteristic
179C2×C4×C7#12311yescharacteristic
180C2×Dic7—82no—
181C2×Dic7—82no—
182C2×Dic7—82no—
183C7⋊C8—44no—
184C22×Q8C2×Dic711yescharacteristic
185C22×D4C22⋊D711yescharacteristic
186C23×C4—21no—
187C23×C4C22⋊D711yescharacteristic
188C23×C4C22×D711yescharacteristic
189C2×M16—142no—
190C22×C8—142no—
191C22⋊Q8—22no—
192C22⋊Q8—22no—
193C4×D4—22no—
194C4×D4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—142no—
199C2×C4⋊C4C2×Dic711yescharacteristic
200C2×C4⋊C4C2×Dic711yescharacteristic
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—142no—
205C2×C22⋊C4C2×Dic711yescharacteristic
206C2×C22⋊C4C22⋊D711yescharacteristic
207C2×C42C22⋊D711yescharacteristic
208C4⋊C8—142no—
209#99—282no—
210C22⋊C8—74no—
211C22⋊C8—142no—
212C22⋊C8—142no—
213#97—142no—
214#95—21no—
215#95—21no—
216#95—21no—
217#95—21no—
218#95—21no—
219#95—142no—
220#95C2×Dic711yescharacteristic
221#95C2×Dic711yescharacteristic
222C22×C7—22no—
223C22×C7—21no—
224C22×C7—22no—
225C22×C7—21no—
226C22×C7—41no—
227C22×C7—41no—
228C22×C7—42no—
229C22×C7—42no—
230C22×C7—41no—
231C22×C7—41no—
232C22×C7—41no—
233C22×C7—81no—
234C22×C7#38611yescharacteristic
235C22×C7#41711yescharacteristic
236C22×C7#41911yescharacteristic
237C22×C7#46812yes—
238C22×C7#47012yes—
239C4×C7—21no—
240C4×C7—21no—
241C4×C7—22no—
242C4×C7—42no—
243C4×C7—41no—
244C4×C7—41no—
245C4×C7—41no—
246C4×C7—81no—
247C4×C7—81no—
248C4×C7—81no—
249Dic7—162no—
250C24C22⋊Dic711yescharacteristic
251C24C2×C22⋊D711yescharacteristic
252C2×Q8—22no—
253C2×Q8—81no—
254C2×D4—22no—
255C2×D4—22no—
256C2×D4—21no—
257C2×D4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—42no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—81no—
268C22×C4—81no—
269C22×C4—81no—
270C22×C4—282no—
271C22×C4C22⋊Dic711yescharacteristic
272C22×C4C22⋊Dic711yescharacteristic
273C22×C4C22⋊Dic711yescharacteristic
274C22×C4D4×D712yes—
275C22×C4#115912yes—
276C22×C4C22×Dic711yescharacteristic
277C22×C4C2×C22⋊D711yescharacteristic
278M16—562no—
279C2×C8—144no—
280C2×C8—562no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—42no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—81no—
287C4⋊C4—562no—
288C22⋊C4—22no—
289C22⋊C4—22no—
290C22⋊C4—42no—
291C22⋊C4—42no—
292C22⋊C4—81no—
293C22⋊C4—282no—
294C22⋊C4—282no—
295C42—21no—
296C42—21no—
297C2×C7—22no—
298C2×C7—21no—
299C2×C7—21no—
300C2×C7—22no—
301C2×C7—81no—
302C2×C7#188411yescharacteristic
303C2×C7#189411yescharacteristic
304C2×C7#191311yescharacteristic
305C23—21no—
306C23—21no—
307C23—21no—
308C23—41no—
309C23—41no—
310C23—41no—
311C23—41no—
312C23#650212yes—
313C23#650512yes—
314C23#651211yescharacteristic
315C23#651411yescharacteristic
316C23C2×C22⋊Dic711yescharacteristic
317Q8—82no—
318D4—42no—
319D4—42no—
320C2×C4—22no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—42no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—81no—
345C2×C4—82no—
346C2×C4—82no—
347C2×C4—562no—
348C2×C4—562no—
349C2×C4—562no—
350C2×C4D4×Dic712yes—
351C2×C4#649811yescharacteristic
352C2×C4#650011yescharacteristic
353C8—284no—
354C7#938511yescharacteristic, complemented, Sylow
355C22—21no—
356C22—22no—
357C22—21no—
358C22—22no—
359C22—41no—
360C22—41no—
361C22—42no—
362C22—42no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—81no—
367C22#8982112yes—
368C22#8982412yes—
369C22#8999211yescharacteristic
370C22#8999711yescharacteristic
371C22#9000411yescharacteristic
372C4—21no—
373C4—21no—
374C4—22no—
375C4—42no—
376C4—41no—
377C4—41no—
378C4—41no—
379C4—81no—
380C4—81no—
381C4—81no—
382C4—1122no—
383C2—21no—
384C2—22no—
385C2—21no—
386C2—22no—
387C2—81no—
388C2#1172945111yescharacteristic
389C2#1172957111yescharacteristic
390C2#1172960211yescharacteristic
subgroupG/H
centerC22#89992
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
FittingC7×#2564C2
socleC22×C7#386
prime radicalC22×C7×D4D4
cosocleC23×C4×C7C23
hypercenter#2564D7
hypocenterC7#9385
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2564

Complements

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

NH ≅ G/Ntype
3. #11729444C2semidirect
5. #11725418C2semidirect
8. C7×#400C4semidirect
9. C7×#399C4semidirect
10. C7×#395C4semidirect
11. C7×#393C4semidirect
123. #400Dic7semidirect
124. #399Dic7semidirect
125. #395Dic7semidirect
126. #393Dic7semidirect
354. C7#9385semidirect

Maximal subgroups

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

subgroupindex
C7×#25642
#117397472
#117294442
#117293732
#117254182
#93857