Finite Group Databasebeta

← #13327712

Subgroups — #13327712

Order 1792.

1002 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#544C211yesmaximal, characteristic, complemented
2#11740181C212yesmaximal
3#11739644C211yesmaximal, characteristic
4#11729656C214yesmaximal, complemented
5#11729652C214yesmaximal, complemented
6#11729444C212yesmaximal, complemented
7C2×#89323C211yesmaximal, characteristic, complemented
8C7×#544—24no—
9C7×#544—22no—
10C7×#544C2214yes—
11C2×C7×C4⋊Q8—22no—
12C2×C7×#125—22no—
13C2×C7×C22⋊Q8—21no—
14C2×C7×C22⋊Q8C2212yes—
15C2×C7×#117—21no—
16C22×C7×C4⋊C4C2211yescharacteristic
17C2×#6493C2211yescharacteristic
18C2×#6492—21no—
19#89999—22no—
20#89985—21no—
21#89984C2211yescharacteristic
22C2×C22⋊(C7⋊C8)C2212yes—
23#89820—24no—
24#89818C2214yes—
25#89815C2214yes—
26#89766—22no—
27#89763C2212yes—
28#89755C2212yes—
29#89740—22no—
30#89333C2218yes—
31#89323C2214yes—
32#18294—71nomaximal, Sylow
33C22×C7×Q8—21no—
34C22×C7×Q8D412yes—
35C23×C4×C7D411yescharacteristic
36C23×C4×C7D411yescharacteristic
37C23×C4×C7C2311yescharacteristic
38C22×C22⋊D7—21no—
39C22×D28C2311yescharacteristic
40C7×C4⋊Q8—28no—
41C7×C4⋊Q8—28no—
42C7×#125—28no—
43C7×#125—28no—
44C7×C22⋊Q8—24no—
45C7×C22⋊Q8—24no—
46C7×C22⋊Q8—22no—
47C7×C22⋊Q8—22no—
48C7×C22⋊Q8D414yes—
49C7×C22⋊Q8C2314yes—
50C7×#117—24no—
51C7×#117—24no—
52C2×C7×C4⋊C4—21no—
53C2×C7×C4⋊C4—21no—
54C2×C7×C4⋊C4—21no—
55C2×C7×C4⋊C4—22no—
56C2×C7×C4⋊C4—22no—
57C2×C7×C4⋊C4—22no—
58C2×C7×C4⋊C4—22no—
59C2×C7×C4⋊C4D412yes—
60C2×C7×C4⋊C4D412yes—
61C2×C7×C4⋊C4C2312yes—
62C2×C7×C22⋊C4—22no—
63C2×C42×C7—22no—
64C2×C22⋊Dic7—21no—
65C2×#1143—24no—
66#6493—22no—
67#6493—22no—
68#6492—44no—
69C2×(C2×D7)⋊C4—22no—
70C2×(C2×D7)⋊C4—21no—
71C2×(C2×D7)⋊C4—22no—
72C2×C4⋊Dic7—21no—
73C2×C4⋊Dic7C2311yescharacteristic
74C2×Dic7⋊C4—22no—
75C22×C7⋊C8—22no—
76#6409—24no—
77C7⋊#95—22no—
78C7⋊#95C2312yes—
79C22⋊(C7⋊C8)—22no—
80C22⋊(C7⋊C8)C2314yes—
81#6383—24no—
82#6382—24no—
83C2×#544D711yescharacteristic, complemented
84#3187—72no—
85#2618—71no—
86#1611—74no—
87#1599—74no—
88#1520—72no—
89C2×#324—71no—
90C24×C7C2×D411yescharacteristic
91C23×D7—21no—
92C2×C7×Q8—24no—
93C2×C7×Q8—22no—
94C2×C7×Q8—82no—
95C2×C7×Q8—82no—
96C2×C7×Q8C2×D414yes—
97C22×C4×C7—21no—
98C22×C4×C7—21no—
99C22×C4×C7—22no—
100C22×C4×C7—22no—
101C22×C4×C7—22no—
102C22×C4×C7—21no—
103C22×C4×C7—21no—
104C22×C4×C7—81no—
105C22×C4×C7—81no—
106C22×C4×C7SD1614yescomplemented
107C22×C4×C7C2×D412yes—
108C22×C4×C7C2×D412yes—
109C22×C4×C7C2×D411yescharacteristic
110C22×C4×C7C2×D412yes—
111C22×C4×C7C2×D412yes—
112C22×C4×C7C2411yescharacteristic
113C2×C22⋊D7—41no—
114C2×C22⋊D7—41no—
115C2×C22⋊D7—42no—
116C2×C22⋊D7—42no—
117C22×Dic7—21no—
118C22×Dic7—22no—
119C2×D28—22no—
120C2×D28—81no—
121C7×C4⋊C4—24no—
122C7×C4⋊C4—24no—
123C7×C4⋊C4—24no—
124C7×C4⋊C4—28no—
125C7×C4⋊C4—24no—
126C7×C4⋊C4—24no—
127C7×C4⋊C4—22no—
128C7×C4⋊C4—24no—
129C7×C4⋊C4—24no—
130C7×C4⋊C4C2×D414yes—
131C7×C22⋊C4—24no—
132C7×C22⋊C4—24no—
133C42×C7—28no—
134C22⋊Dic7—42no—
135#1143—84no—
136(C2×D7)⋊C4—42no—
137(C2×D7)⋊C4—44no—
138(C2×D7)⋊C4—82no—
139C4⋊Dic7—22no—
140C4⋊Dic7—81no—
141Dic7⋊C4—44no—
142C2×C7⋊C8—24no—
143C2×C7⋊C8—82no—
144#544—24no—
145#544—22no—
146#544C2×D714yes—
147C2×C4⋊Q8—22no—
148C2×#125—22no—
149C2×#123—141no—
150C2×C22⋊Q8—21no—
151C2×C22⋊Q8C2×D712yes—
152C2×#121—71no—
153C2×#117—21no—
154C22×C4⋊C4C2×D711yescharacteristic
155#483—74no—
156#481—142no—
157#465—144no—
158#450—74no—
159#448—142no—
160C2×C22⋊C8—72no—
161#398—72no—
162#396—142no—
163#394—72no—
164#386—141no—
165#380—71no—
166#328—78no—
167#324—74no—
168C23×C7—41no—
169C23×C7—41no—
170C23×C7#12011yescharacteristic
171C23×C7#12011yescharacteristic
172C23×C7C2×SD1612yes—
173C23×C7#13712yes—
174C23×C7C22×D411yescharacteristic
175C22×D7—42no—
176C22×D7—42no—
177C22×D7—41no—
178C22×D7—41no—
179C22×D7—41no—
180C7×Q8—44no—
181C7×Q8—82no—
182C7×Q8—84no—
183C2×C4×C7—22no—
184C2×C4×C7—22no—
185C2×C4×C7—24no—
186C2×C4×C7—24no—
187C2×C4×C7—24no—
188C2×C4×C7—24no—
189C2×C4×C7—22no—
190C2×C4×C7—42no—
191C2×C4×C7—41no—
192C2×C4×C7—41no—
193C2×C4×C7—42no—
194C2×C4×C7—44no—
195C2×C4×C7—42no—
196C2×C4×C7—44no—
197C2×C4×C7—42no—
198C2×C4×C7—81no—
199C2×C4×C7#12012yes—
200C2×C4×C7C2×SD1614yes—
201C2×C4×C7C22×D412yes—
202C22⋊D7—82no—
203C22⋊D7—82no—
204C2×Dic7—42no—
205C2×Dic7—42no—
206C2×Dic7—41no—
207C2×Dic7—41no—
208C2×Dic7—42no—
209C2×Dic7—41no—
210D28—44no—
211C7⋊C8—44no—
212C22×Q8—21no—
213C22×Q8C22⋊D712yes—
214C22×D4—71no—
215C22×D4—141no—
216C23×C4C22⋊D711yescharacteristic
217C23×C4C22⋊D711yescharacteristic
218C23×C4C22×D711yescharacteristic
219C2×SD16—144no—
220C22×C8—142no—
221C4⋊Q8—28no—
222C4⋊Q8—28no—
223#125—28no—
224#125—28no—
225#123—284no—
226C22⋊Q8—24no—
227C22⋊Q8—24no—
228C22⋊Q8—22no—
229C22⋊Q8—22no—
230C22⋊Q8C22⋊D714yes—
231C22⋊Q8C22×D714yes—
232#121—142no—
233#121—142no—
234#117—24no—
235#117—24no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—71no—
244C2×C4⋊C4—142no—
245C2×C4⋊C4—141no—
246C2×C4⋊C4C22⋊D712yes—
247C2×C4⋊C4C22⋊D712yes—
248C2×C4⋊C4C22×D712yes—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—141no—
251C2×C22⋊C4—141no—
252C2×C22⋊C4—142no—
253C2×C22⋊C4—142no—
254C2×C42—22no—
255#106—144no—
256#103—144no—
257#102—144no—
258C22⋊C8—74no—
259C22⋊C8—142no—
260#95—72no—
261#95—142no—
262C22×C7—21no—
263C22×C7—21no—
264C22×C7—21no—
265C22×C7—21no—
266C22×C7—21no—
267C22×C7—21no—
268C22×C7—42no—
269C22×C7—42no—
270C22×C7#44812yes—
271C22×C7#44812yes—
272C22×C7C2×#12011yescharacteristic
273C22×C7C22×SD1611yescharacteristic
274C22×C7C2×#13711yescharacteristic
275C2×D7—82no—
276C2×D7—82no—
277C2×D7—81no—
278C2×D7—81no—
279C2×D7—81no—
280C4×C7—24no—
281C4×C7—22no—
282C4×C7—22no—
283C4×C7—44no—
284C4×C7—42no—
285C4×C7—44no—
286Dic7—82no—
287Dic7—81no—
288C24—141no—
289C24C2×C22⋊D711yescharacteristic
290C2×Q8—24no—
291C2×Q8—22no—
292C2×Q8—82no—
293C2×Q8—82no—
294C2×Q8C2×C22⋊D714yes—
295C2×D4—142no—
296C2×D4—282no—
297C2×D4—282no—
298C2×D4—281no—
299C2×D4—281no—
300C2×D4—561no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—81no—
309C22×C4—81no—
310C22×C4—142no—
311C22×C4—141no—
312C22×C4#114314yescomplemented
313C22×C4C2×C22⋊D712yes—
314C22×C4C2×C22⋊D712yes—
315C22×C4C2×C22⋊D711yescharacteristic
316C22×C4C2×C22⋊D712yes—
317C22×C4C2×C22⋊D712yes—
318C22×C4C23×D711yescharacteristic
319SD16—564no—
320C2×C8—144no—
321C2×C8—562no—
322C4⋊C4—24no—
323C4⋊C4—24no—
324C4⋊C4—24no—
325C4⋊C4—28no—
326C4⋊C4—24no—
327C4⋊C4—24no—
328C4⋊C4—22no—
329C4⋊C4—24no—
330C4⋊C4—24no—
331C4⋊C4—142no—
332C4⋊C4—284no—
333C4⋊C4—561no—
334C4⋊C4C2×C22⋊D714yes—
335C22⋊C4—24no—
336C22⋊C4—24no—
337C22⋊C4—282no—
338C22⋊C4—284no—
339C22⋊C4—282no—
340C22⋊C4—562no—
341C42—28no—
342C2×C7—21no—
343C2×C7—21no—
344C2×C7—22no—
345C2×C7—22no—
346C2×C7C2×#44811yescharacteristic
347C2×C7#302011yescharacteristic
348C2×C7#302611yescharacteristic
349D7—161no—
350C23—41no—
351C23—41no—
352C23—282no—
353C23—282no—
354C23—281no—
355C23—281no—
356C23—281no—
357C23C2×#114312yes—
358C23#650512yes—
359C23#651611yescharacteristic
360C23#651611yescharacteristic
361C23C22×C22⋊D711yescharacteristic
362Q8—44no—
363Q8—82no—
364Q8—84no—
365D4—284no—
366D4—562no—
367D4—562no—
368C2×C4—22no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—22no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—22no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—41no—
379C2×C4—44no—
380C2×C4—42no—
381C2×C4—44no—
382C2×C4—42no—
383C2×C4—81no—
384C2×C4—282no—
385C2×C4—282no—
386C2×C4—282no—
387C2×C4—281no—
388C2×C4—281no—
389C2×C4—281no—
390C2×C4C2×#114314yes—
391C2×C4#651612yes—
392C2×C4C22×C22⋊D712yes—
393C8—284no—
394C7#1829411yescharacteristic, complemented, Sylow
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—42no—
402C22—42no—
403C22—562no—
404C22—562no—
405C22—561no—
406C22—561no—
407C22—561no—
408C22#8999912yes—
409C22#8999912yes—
410C22C22×#114311yescharacteristic
411C22C2×#650511yescharacteristic
412C22C2×#651611yescharacteristic
413C4—24no—
414C4—22no—
415C4—22no—
416C4—42no—
417C4—44no—
418C4—44no—
419C4—562no—
420C4—561no—
421C2—21no—
422C2—21no—
423C2—22no—
424C2—22no—
425C2—1121no—
426C2C2×#8999911yescharacteristic
427C2#1174155511yescharacteristic
428C2#1174156111yescharacteristic
subgroupG/H
centerC22C2×#6516
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC2×C7×#544C2
socleC22×C7C2×#120
prime radicalC2×#6493C22
hypercenterC2×#544D7
hypocenterC7#18294
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#544

Complements

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

NH ≅ G/Ntype
1. C2×C7×#544C2semidirect
4. #11729656C2semidirect
5. #11729652C2semidirect
6. #11729444C2semidirect
7. C2×#89323C2semidirect
83. C2×#544D7semidirect
106. C22×C4×C7SD16semidirect
312. C22×C4#1143semidirect
394. C7#18294semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5442
#117401812
#117396442
#117296562
#117296522
#117294442
C2×#893232
#182947