Finite Group Databasebeta

← #13327923

Subgroups — #13327923

Order 1792.

1058 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 1058 conjugacy classes of subgroups — too many to draw. Only its 183 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
1C2×C7×#543C211yesmaximal, characteristic
2#11740187C212yesmaximal, complemented
3#11739561C211yesmaximal, characteristic
4#11729647C214yesmaximal, complemented
5#11729644C214yesmaximal, complemented
6#11729442C212yesmaximal, complemented
7#11729293C211yesmaximal, characteristic
8C7×#543—24no—
9C7×#543—22no—
10C7×#543C2214yes—
11C2×C7×C22⋊Q8—22no—
12C2×C7×C22⋊Q8—21no—
13C2×C7×C22⋊Q8—22no—
14C2×C7×C22⋊Q8C2212yes—
15C22×C7×C22⋊C4—21no—
16C22×C7×C22⋊C4C2211yescharacteristic
17C2×#6487C2211yescharacteristic
18#90019—21no—
19#90000—22no—
20#89983—21no—
21C4×C22⋊Dic7C2211yescharacteristic
22#89979C2212yes—
23#89824—24no—
24#89822C2214yescomplemented
25#89815C2214yescomplemented
26#89743—22no—
27#89725C2212yes—
28#89719C2212yes—
29#89717—22no—
30#89333C2218yescomplemented
31#89320C2214yes—
32#18319—71nomaximal, Sylow
33C25×C7D411yescharacteristic
34C22×C7×Q8—21no—
35C22×C7×Q8D412yes—
36C23×C4×C7—21no—
37C23×C4×C7—21no—
38C23×C4×C7D411yescharacteristic
39C23×C4×C7C2311yescharacteristic
40C22×Dic14C2311yescharacteristic
41C7×C22⋊Q8—28no—
42C7×C22⋊Q8—28no—
43C7×C22⋊Q8—24no—
44C7×C22⋊Q8—28no—
45C7×C22⋊Q8—24no—
46C7×C22⋊Q8—22no—
47C7×C22⋊Q8—22no—
48C7×C22⋊Q8—28no—
49C7×C22⋊Q8D414yes—
50C7×C22⋊Q8C2314yes—
51C2×C7×C4⋊C4—21no—
52C2×C7×C4⋊C4—22no—
53C2×C7×C4⋊C4—22no—
54C2×C7×C4⋊C4D412yes—
55C2×C7×C22⋊C4—22no—
56C2×C7×C22⋊C4—24no—
57C2×C7×C22⋊C4—22no—
58C2×C7×C22⋊C4—24no—
59C2×C7×C22⋊C4—22no—
60C2×C7×C22⋊C4—22no—
61C2×C7×C22⋊C4D412yes—
62C2×C7×C22⋊C4C2312yes—
63C2×C22⋊Dic7—21no—
64C2×C22⋊Dic7—22no—
65C2×C22⋊Dic7—21no—
66C2×C22⋊Dic7—22no—
67C2×#1144—24no—
68C2×C4⋊Dic7C2311yescharacteristic
69#6487—22no—
70#6487—22no—
71C2×Dic7⋊C4—21no—
72C2×C4×Dic7—21no—
73C2×#1136—22no—
74#6409—24no—
75C7⋊#95—22no—
76C7⋊#95—22no—
77C7⋊#95C2312yes—
78C22⋊(C7⋊C8)—22no—
79C22⋊(C7⋊C8)C2314yes—
80#6384—24no—
81#6382—24no—
82C2×#543D711yescharacteristic
83#3195—72no—
84#2561—71no—
85#1608—74no—
86#1602—74no—
87#1508—72no—
88#1462—71no—
89C24×C7—21no—
90C24×C7—21no—
91C24×C7—24no—
92C24×C7—42no—
93C24×C7—81no—
94C24×C7C2×D411yescharacteristic
95C24×C7C2×D412yes—
96C2×C7×Q8—24no—
97C2×C7×Q8—22no—
98C2×C7×Q8—82no—
99C2×C7×Q8—82no—
100C2×C7×Q8C2×D414yes—
101C22×C4×C7—22no—
102C22×C4×C7—21no—
103C22×C4×C7—22no—
104C22×C4×C7—24no—
105C22×C4×C7—24no—
106C22×C4×C7—21no—
107C22×C4×C7—42no—
108C22×C4×C7—81no—
109C22×C4×C7—82no—
110C22×C4×C7—82no—
111C22×C4×C7C2×D412yes—
112C22×C4×C7C2×D411yescharacteristic
113C22×C4×C7C2×D412yes—
114C22×C4×C7C2×D412yes—
115C22×C4×C7C2411yescharacteristic
116C22×Dic7—22no—
117C22×Dic7—21no—
118C22×Dic7—21no—
119C2×Dic14—22no—
120C2×Dic14—81no—
121C7×C4⋊C4—24no—
122C7×C4⋊C4—28no—
123C7×C4⋊C4—24no—
124C7×C4⋊C4—22no—
125C7×C4⋊C4—24no—
126C7×C4⋊C4C2×D414yes—
127C7×C22⋊C4—28no—
128C7×C22⋊C4—24no—
129C7×C22⋊C4—24no—
130C7×C22⋊C4—24no—
131C7×C22⋊C4—24no—
132C22⋊Dic7—42no—
133C22⋊Dic7—41no—
134C22⋊Dic7—41no—
135C22⋊Dic7—44no—
136C22⋊Dic7—82no—
137#1144—84no—
138C4⋊Dic7—22no—
139Dic7⋊C4—42no—
140C4×Dic7—81no—
141#1136—82no—
142C2×C7⋊C8—24no—
143#543—24no—
144#543—22no—
145#543C2×D714yes—
146C2×C22⋊Q8—22no—
147C2×C22⋊Q8—21no—
148C2×C22⋊Q8—22no—
149C2×C22⋊Q8—71no—
150C2×C22⋊Q8C2×D712yes—
151C22×C22⋊C4—21no—
152C22×C22⋊C4C2×D711yescharacteristic
153#483—74no—
154#483—142no—
155#470—144no—
156#451—74no—
157#451—142no—
158#407—72no—
159#397—72no—
160#393—72no—
161#388—142no—
162#381—141no—
163#379—141no—
164C4×C22⋊C4—71no—
165#328—78no—
166#323—74no—
167C23×C7—24no—
168C23×C7—24no—
169C23×C7—22no—
170C23×C7—22no—
171C23×C7—22no—
172C23×C7—22no—
173C23×C7—41no—
174C23×C7—42no—
175C23×C7—42no—
176C23×C7—42no—
177C23×C7—42no—
178C23×C7—81no—
179C23×C7—81no—
180C23×C7—88no—
181C23×C7#12011yescharacteristic
182C23×C7#12011yescharacteristic
183C23×C7#13714yes—
184C23×C7C22×D411yescharacteristic
185C7×Q8—44no—
186C7×Q8—82no—
187C7×Q8—84no—
188C2×C4×C7—24no—
189C2×C4×C7—22no—
190C2×C4×C7—24no—
191C2×C4×C7—42no—
192C2×C4×C7—41no—
193C2×C4×C7—41no—
194C2×C4×C7—42no—
195C2×C4×C7—44no—
196C2×C4×C7—42no—
197C2×C4×C7—44no—
198C2×C4×C7—42no—
199C2×C4×C7—81no—
200C2×C4×C7—84no—
201C2×C4×C7—84no—
202C2×C4×C7#12012yes—
203C2×C4×C7C22×D412yes—
204C2×Dic7—42no—
205C2×Dic7—42no—
206C2×Dic7—42no—
207C2×Dic7—41no—
208C2×Dic7—41no—
209C2×Dic7—42no—
210C2×Dic7—42no—
211Dic14—44no—
212C7⋊C8—44no—
213C25C22⋊D711yescharacteristic
214C22×Q8—21no—
215C22×Q8—71no—
216C22×Q8C22⋊D712yes—
217C23×C4—21no—
218C23×C4—21no—
219C23×C4C22⋊D711yescharacteristic
220C23×C4C22×D711yescharacteristic
221C2×Q16—144no—
222C2×M16—142no—
223C22⋊Q8—28no—
224C22⋊Q8—28no—
225C22⋊Q8—24no—
226C22⋊Q8—28no—
227C22⋊Q8—24no—
228C22⋊Q8—22no—
229C22⋊Q8—22no—
230C22⋊Q8—28no—
231C22⋊Q8—142no—
232C22⋊Q8—142no—
233C22⋊Q8C22⋊D714yes—
234C22⋊Q8C22×D714yes—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—71no—
239C2×C4⋊C4—141no—
240C2×C4⋊C4C22⋊D712yes—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—141no—
248C2×C22⋊C4—141no—
249C2×C22⋊C4—142no—
250C2×C22⋊C4—142no—
251C2×C22⋊C4C22⋊D712yes—
252C2×C22⋊C4C22×D712yes—
253C2×C42—141no—
254#106—144no—
255#103—144no—
256#103—144no—
257C22⋊C8—74no—
258C22⋊C8—142no—
259#95—72no—
260#95—142no—
261#95—142no—
262C22×C7—21no—
263C22×C7—21no—
264C22×C7—21no—
265C22×C7—21no—
266C22×C7—21no—
267C22×C7—24no—
268C22×C7—21no—
269C22×C7—21no—
270C22×C7—44no—
271C22×C7—44no—
272C22×C7—48no—
273C22×C7—42no—
274C22×C7—42no—
275C22×C7—82no—
276C22×C7—84no—
277C22×C7#45512yes—
278C22×C7C2×#12011yescharacteristic
279C22×C7C2×#13712yes—
280C4×C7—22no—
281C4×C7—22no—
282C4×C7—42no—
283C4×C7—44no—
284C4×C7—44no—
285Dic7—82no—
286Dic7—81no—
287Dic7—81no—
288C24—21no—
289C24—24no—
290C24—21no—
291C24—42no—
292C24—81no—
293C24C2×C22⋊D711yescharacteristic
294C24C2×C22⋊D712yes—
295C2×Q8—24no—
296C2×Q8—22no—
297C2×Q8—82no—
298C2×Q8—82no—
299C2×Q8—142no—
300C2×Q8—561no—
301C2×Q8C2×C22⋊D714yes—
302C22×C4—22no—
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—21no—
308C22×C4—42no—
309C22×C4—81no—
310C22×C4—82no—
311C22×C4—82no—
312C22×C4—142no—
313C22×C4—141no—
314C22×C4—141no—
315C22×C4C2×C22⋊D712yes—
316C22×C4C2×C22⋊D711yescharacteristic
317C22×C4C2×C22⋊D712yes—
318C22×C4C2×C22⋊D712yes—
319C22×C4C23×D711yescharacteristic
320Q16—564no—
321M16—562no—
322C2×C8—144no—
323C4⋊C4—24no—
324C4⋊C4—28no—
325C4⋊C4—24no—
326C4⋊C4—22no—
327C4⋊C4—24no—
328C4⋊C4—142no—
329C4⋊C4—282no—
330C4⋊C4C2×C22⋊D714yes—
331C22⋊C4—28no—
332C22⋊C4—24no—
333C22⋊C4—24no—
334C22⋊C4—24no—
335C22⋊C4—24no—
336C22⋊C4—282no—
337C22⋊C4—284no—
338C22⋊C4—281no—
339C22⋊C4—281no—
340C22⋊C4—562no—
341C42—561no—
342C2×C7—22no—
343C2×C7—22no—
344C2×C7—41no—
345C2×C7—44no—
346C2×C7#300411yescharacteristic
347C2×C7C2×#45511yescharacteristic
348C2×C7#302611yescharacteristic
349C23—24no—
350C23—24no—
351C23—22no—
352C23—22no—
353C23—22no—
354C23—22no—
355C23—41no—
356C23—42no—
357C23—42no—
358C23—42no—
359C23—42no—
360C23—81no—
361C23—81no—
362C23—88no—
363C23#650514yes—
364C23#651611yescharacteristic
365C23#651611yescharacteristic
366C23C22×C22⋊D711yescharacteristic
367Q8—44no—
368Q8—82no—
369Q8—84no—
370Q8—284no—
371C2×C4—24no—
372C2×C4—22no—
373C2×C4—24no—
374C2×C4—41no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—41no—
378C2×C4—44no—
379C2×C4—42no—
380C2×C4—44no—
381C2×C4—42no—
382C2×C4—81no—
383C2×C4—84no—
384C2×C4—84no—
385C2×C4—282no—
386C2×C4—282no—
387C2×C4—282no—
388C2×C4—282no—
389C2×C4—282no—
390C2×C4—281no—
391C2×C4—281no—
392C2×C4#651612yes—
393C2×C4C22×C22⋊D712yes—
394C8—284no—
395C7#1831911yescharacteristic, complemented, Sylow
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—24no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—44no—
405C22—44no—
406C22—48no—
407C22—42no—
408C22—42no—
409C22—82no—
410C22—84no—
411C22#9001412yes—
412C22C2×#650512yes—
413C22C2×#651611yescharacteristic
414C4—22no—
415C4—22no—
416C4—42no—
417C4—44no—
418C4—44no—
419C4—562no—
420C4—561no—
421C4—561no—
422C2—22no—
423C2—22no—
424C2—41no—
425C2—44no—
426C2#1174150411yescharacteristic
427C2C2×#9001411yescharacteristic
428C2#1174155511yescharacteristic
subgroupG/H
centerC22C2×#6516
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC2×C7×#543C2
socleC22×C7C2×#120
prime radicalC25×C7D4
hypercenterC2×#543D7
hypocenterC7#18319
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#543

Complements

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

NH ≅ G/Ntype
2. #11740187C2semidirect
4. #11729647C2semidirect
5. #11729644C2semidirect
6. #11729442C2semidirect
24. #89822C22semidirect
25. #89815C22semidirect
30. #89333C22semidirect
395. C7#18319semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5432
#117401872
#117395612
#117296472
#117296442
#117294422
#117292932
#183197