Finite Group Databasebeta

← C2×#11727468 #13325916

Subgroups — C2×#11727468 #13325916

Order 1792.

1524 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 1524 conjugacy classes of subgroups — too many to draw. Only its 203 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×#90184C211yesmaximal, characteristic
2C2×C7×#399C211yesmaximal, characteristic, complemented
3C2×#89740C212yesmaximal, complemented
4C2×#89320C211yesmaximal, characteristic, complemented
5#11727468C218yesmaximal, direct product factor
6C2×#89246C212yesmaximal, complemented
7C23×C22⋊D7—21no—
8C22×C7×C4⋊C4—21no—
9C22×C7×C4⋊C4C2211yescharacteristic, complemented
10C22×C7×C22⋊C4—21no—
11C22×C7×C22⋊C4C2211yescharacteristic
12C2×#6493—21no—
13C2×#6493C2212yes—
14#90184—22no—
15#90184—24no—
16#90184C2214yes—
17C2×#6449—22no—
18C2×#6445—22no—
19C7×#399—22no—
20C7×#399C2214yescomplemented
21C2×C7×#95C2211yescharacteristic, complemented
22C2×C22⋊(C7⋊C8)C2212yescomplemented
23#89740—24no—
24#89740C2218yescomplemented
25C2×#6383—22no—
26C2×#6382—22no—
27#89320C2214yescomplemented
28#89246C2218yescomplemented
29C2×#1613—71nomaximal, Sylow
30C25×C7D411yescharacteristic
31C24×D7—21no—
32C23×C4×C7—21no—
33C23×C4×C7—21no—
34C23×C4×C7D411yescharacteristic
35C23×C4×C7C2311yescharacteristic
36C22×C22⋊D7—22no—
37C22×C22⋊D7—24no—
38C22×C22⋊D7—24no—
39C22×C22⋊D7—22no—
40C22×C22⋊D7—44no—
41C22×C22⋊D7—44no—
42C23×Dic7—21no—
43C22×D28—21no—
44C22×D28D412yes—
45C2×C7×C4⋊C4—22no—
46C2×C7×C4⋊C4—22no—
47C2×C7×C4⋊C4—24no—
48C2×C7×C4⋊C4—21no—
49C2×C7×C4⋊C4—44no—
50C2×C7×C4⋊C4C2312yescomplemented
51C2×C7×C22⋊C4—22no—
52C2×C7×C22⋊C4—22no—
53C2×C7×C22⋊C4—24no—
54C2×C7×C22⋊C4—44no—
55C2×C7×C22⋊C4D412yes—
56C2×C7×C22⋊C4C2312yes—
57#6493—24no—
58#6493—22no—
59#6493—22no—
60#6493—24no—
61#6493D414yes—
62#6493C2314yes—
63C2×(C2×D7)⋊C4—22no—
64C2×(C2×D7)⋊C4—22no—
65C2×C4⋊Dic7—21no—
66C2×C4⋊Dic7D412yes—
67C22×C7⋊C8—22no—
68#6449—28no—
69#6449—28no—
70#6445—28no—
71#6445—28no—
72C7×#95—21no—
73C7×#95C2312yescomplemented
74C22⋊(C7⋊C8)—22no—
75C22⋊(C7⋊C8)C2314yescomplemented
76#6383—28no—
77#6382—28no—
78C2×#534—71no—
79C2×#481—72no—
80C2×#399D711yescharacteristic, complemented
81#1613—78no—
82C2×#327—72no—
83C2×#323—71no—
84C24×C7—21no—
85C24×C7—21no—
86C24×C7—22no—
87C24×C7—42no—
88C24×C7—42no—
89C24×C7SD1614yes—
90C24×C7C2×D411yescharacteristic
91C24×C7C2×D412yes—
92C23×D7—22no—
93C23×D7—24no—
94C23×D7—44no—
95C23×D7—82no—
96C23×D7—82no—
97C22×C4×C7—24no—
98C22×C4×C7—21no—
99C22×C4×C7—22no—
100C22×C4×C7—21no—
101C22×C4×C7—22no—
102C22×C4×C7—42no—
103C22×C4×C7—42no—
104C22×C4×C7—42no—
105C22×C4×C7—42no—
106C22×C4×C7—42no—
107C22×C4×C7—42no—
108C22×C4×C7—41no—
109C22×C4×C7—41no—
110C22×C4×C7C2×D412yes—
111C22×C4×C7C2×D412yes—
112C22×C4×C7C2×D411yescharacteristic
113C22×C4×C7C2×D412yes—
114C22×C4×C7C2411yescharacteristic
115C2×C22⋊D7—28no—
116C2×C22⋊D7—24no—
117C2×C22⋊D7—24no—
118C2×C22⋊D7—48no—
119C2×C22⋊D7—416no—
120C2×C22⋊D7—48no—
121C2×C22⋊D7—48no—
122C2×C22⋊D7—44no—
123C2×C22⋊D7—44no—
124C22×Dic7—24no—
125C22×Dic7—22no—
126C22×Dic7—44no—
127C2×D28—24no—
128C2×D28—22no—
129C2×D28—44no—
130C2×D28—82no—
131C2×D28C2×D414yes—
132C7×C4⋊C4—24no—
133C7×C4⋊C4—48no—
134C7×C4⋊C4—42no—
135C7×C22⋊C4—24no—
136C7×C22⋊C4—24no—
137C7×C22⋊C4—48no—
138(C2×D7)⋊C4—28no—
139(C2×D7)⋊C4—24no—
140(C2×D7)⋊C4—24no—
141C4⋊Dic7—24no—
142C4⋊Dic7—22no—
143C4⋊Dic7C2×D414yes—
144C2×C7⋊C8—24no—
145C2×C7⋊C8—44no—
146C23×D4—141no—
147#534—74no—
148#534—144no—
149#534—142no—
150C2×#123—142no—
151C2×#121—72no—
152C2×#121—141no—
153C2×#120—142no—
154C22×C4⋊C4—21no—
155C22×C4⋊C4C2×D711yescharacteristic, complemented
156C22×C22⋊C4—21no—
157C22×C22⋊C4C2×D711yescharacteristic
158#481—78no—
159#481—144no—
160C2×#106—142no—
161C2×#102—142no—
162C2×C22⋊C8—72no—
163#399—22no—
164#399C2×D714yescomplemented
165C2×#95C2×D711yescharacteristic, complemented
166#327—78no—
167#323—74no—
168C23×C7—22no—
169C23×C7—22no—
170C23×C7—22no—
171C23×C7—24no—
172C23×C7—22no—
173C23×C7—24no—
174C23×C7—42no—
175C23×C7—48no—
176C23×C7—42no—
177C23×C7—42no—
178C23×C7—42no—
179C23×C7—84no—
180C23×C7—81no—
181C23×C7—81no—
182C23×C7#12011yescharacteristic
183C23×C7#12011yescharacteristic
184C23×C7C2×SD1612yes—
185C23×C7C2×SD1614yes—
186C23×C7AGL(1,ℤ/8)12yes—
187C23×C7C22×D411yescharacteristic
188C22×D7—24no—
189C22×D7—48no—
190C22×D7—44no—
191C22×D7—44no—
192C22×D7—44no—
193C22×D7—84no—
194C22×D7—84no—
195C22×D7—816no—
196C2×C4×C7—24no—
197C2×C4×C7—24no—
198C2×C4×C7—22no—
199C2×C4×C7—42no—
200C2×C4×C7—41no—
201C2×C4×C7—41no—
202C2×C4×C7—48no—
203C2×C4×C7—42no—
204C2×C4×C7—42no—
205C2×C4×C7—42no—
206C2×C4×C7—42no—
207C2×C4×C7—42no—
208C2×C4×C7—42no—
209C2×C4×C7—42no—
210C2×C4×C7—81no—
211C2×C4×C7—82no—
212C2×C4×C7—82no—
213C2×C4×C7#12012yes—
214C2×C4×C7C22×D412yes—
215C22⋊D7—416no—
216C22⋊D7—44no—
217C22⋊D7—44no—
218C22⋊D7—48no—
219C2×Dic7—24no—
220C2×Dic7—48no—
221C2×Dic7—44no—
222D28—44no—
223D28—82no—
224D28—84no—
225C7⋊C8—44no—
226C25—141no—
227C25C22⋊D711yescharacteristic
228C22×D4—72no—
229C22×D4—141no—
230C22×D4—142no—
231C22×D4—144no—
232C22×D4—142no—
233C22×D4—144no—
234C22×D4—284no—
235C22×D4—284no—
236C23×C4—21no—
237C23×C4—21no—
238C23×C4—141no—
239C23×C4C22⋊D711yescharacteristic
240C23×C4C22×D711yescharacteristic
241C22×C8—142no—
242#123—148no—
243#123—148no—
244#121—74no—
245#121—74no—
246#121—144no—
247#121—142no—
248#121—142no—
249#121—144no—
250#120—148no—
251#120—148no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—72no—
258C2×C4⋊C4—141no—
259C2×C4⋊C4C22×D712yescomplemented
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—142no—
265C2×C22⋊C4—142no—
266C2×C22⋊C4C22⋊D712yes—
267C2×C22⋊C4C22×D712yes—
268#106—148no—
269#102—148no—
270C22⋊C8—74no—
271C22⋊C8—142no—
272#95—21no—
273#95C22×D712yescomplemented
274C22×C7—24no—
275C22×C7—24no—
276C22×C7—21no—
277C22×C7—21no—
278C22×C7—21no—
279C22×C7—21no—
280C22×C7—24no—
281C22×C7—44no—
282C22×C7—48no—
283C22×C7—42no—
284C22×C7—42no—
285C22×C7—82no—
286C22×C7—84no—
287C22×C7#45012yes—
288C22×C7#45012yes—
289C22×C7#45312yes—
290C22×C7#45812yes—
291C22×C7C2×#12011yescharacteristic
292C22×C7C22×SD1611yescharacteristic
293C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
294C2×D7—44no—
295C2×D7—44no—
296C2×D7—44no—
297C2×D7—816no—
298C2×D7—88no—
299C4×C7—22no—
300C4×C7—22no—
301C4×C7—44no—
302C4×C7—42no—
303C4×C7—82no—
304Dic7—44no—
305C24—21no—
306C24—21no—
307C24—22no—
308C24—42no—
309C24—42no—
310C24—144no—
311C24—142no—
312C24—284no—
313C24—562no—
314C24—562no—
315C24#114314yes—
316C24C2×C22⋊D711yescharacteristic
317C24C2×C22⋊D712yes—
318C2×D4—74no—
319C2×D4—144no—
320C2×D4—148no—
321C2×D4—144no—
322C2×D4—144no—
323C2×D4—142no—
324C2×D4—284no—
325C2×D4—288no—
326C2×D4—2816no—
327C2×D4—288no—
328C2×D4—288no—
329C2×D4—284no—
330C2×D4—284no—
331C2×D4—562no—
332C22×C4—24no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—21no—
336C22×C4—22no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—144no—
346C22×C4—142no—
347C22×C4—284no—
348C22×C4D4×D712yes—
349C22×C4D4×D712yes—
350C22×C4C2×C22⋊D712yes—
351C22×C4C2×C22⋊D711yescharacteristic
352C22×C4C23×D711yescharacteristic
353C2×C8—144no—
354C2×C8—284no—
355C4⋊C4—24no—
356C4⋊C4—48no—
357C4⋊C4—42no—
358C4⋊C4—74no—
359C4⋊C4—144no—
360C4⋊C4—142no—
361C22⋊C4—24no—
362C22⋊C4—24no—
363C22⋊C4—48no—
364C22⋊C4—148no—
365C22⋊C4—144no—
366C22⋊C4—144no—
367C2×C7—21no—
368C2×C7—21no—
369C2×C7—22no—
370C2×C7—44no—
371C2×C7#161314yescomplemented
372C2×C7C2×#45011yescharacteristic
373C2×C7C2×#45311yescharacteristic
374C2×C7C2×#45811yescharacteristic
375D7—84no—
376C23—22no—
377C23—22no—
378C23—22no—
379C23—24no—
380C23—22no—
381C23—24no—
382C23—42no—
383C23—48no—
384C23—42no—
385C23—42no—
386C23—42no—
387C23—84no—
388C23—81no—
389C23—81no—
390C23—144no—
391C23—288no—
392C23—284no—
393C23—284no—
394C23—284no—
395C23—564no—
396C23—564no—
397C23—5616no—
398C23#650011yescharacteristic
399C23#650011yescharacteristic
400C23C2×#114312yes—
401C23C2×#114314yes—
402C23#651212yes—
403C23C22×C22⋊D711yescharacteristic
404D4—284no—
405D4—2816no—
406D4—284no—
407D4—284no—
408D4—288no—
409D4—562no—
410D4—564no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—22no—
414C2×C4—42no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—48no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—81no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—144no—
429C2×C4—288no—
430C2×C4—284no—
431C2×C4#650012yes—
432C2×C4C2×D4×D712yes—
433C8—284no—
434C7C2×#161311yescharacteristic, complemented, Sylow
435C22—24no—
436C22—24no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—24no—
442C22—44no—
443C22—48no—
444C22—42no—
445C22—42no—
446C22—82no—
447C22—84no—
448C22—284no—
449C22—284no—
450C22—284no—
451C22—5616no—
452C22—568no—
453C22#8981812yes—
454C22#8981812yes—
455C22#8982512yes—
456C22#8984912yes—
457C22C2×#650011yescharacteristic
458C22C22×#114311yescharacteristic
459C22C2×#651211yescharacteristic
460C4—22no—
461C4—22no—
462C4—44no—
463C4—42no—
464C4—82no—
465C4—284no—
466C2—21no—
467C2—21no—
468C2—22no—
469C2—44no—
470C2—564no—
471C2#1172746814yesdirect product factor
472C2C2×#8981811yescharacteristic
473C2C2×#8982511yescharacteristic
474C2C2×#8984911yescharacteristic
subgroupG/H
centerC23#6500
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC2×C7×#399C2
socleC23×C7#120
prime radicalC2×#90184C2
hypercenterC2×#399D7
hypocenterC7C2×#1613
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#399

Complements

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

NH ≅ G/Ntype
2. C2×C7×#399C2semidirect
3. C2×#89740C2semidirect
4. C2×#89320C2semidirect
5. #11727468C2direct (N × H)
6. C2×#89246C2semidirect
9. C22×C7×C4⋊C4C22semidirect
20. C7×#399C22semidirect
21. C2×C7×#95C22semidirect
22. C2×C22⋊(C7⋊C8)C22semidirect
24. #89740C22semidirect
27. #89320C22semidirect
28. #89246C22semidirect
50. C2×C7×C4⋊C4C23semidirect
73. C7×#95C23semidirect
75. C22⋊(C7⋊C8)C23semidirect
80. C2×#399D7semidirect
155. C22×C4⋊C4C2×D7semidirect
164. #399C2×D7semidirect
165. C2×#95C2×D7semidirect
259. C2×C4⋊C4C22×D7semidirect
273. #95C22×D7semidirect
371. C2×C7#1613semidirect
434. C7C2×#1613semidirect
471. C2#11727468direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#901842
C2×C7×#3992
C2×#897402
C2×#893202
#117274682
C2×#892462
C2×#16137