Finite Group Databasebeta

← #13257620

Subgroups — #13257620

Order 1792.

506 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89855C211yesmaximal, characteristic
2C7×#1990C211yesmaximal, characteristic, complemented
3C2×#89342C211yesmaximal, characteristic, complemented
4#11727541C211yesmaximal, characteristic
5C2×#89252C211yesmaximal, characteristic, complemented
6#11727338C211yesmaximal, characteristic
7#11725440C211yesmaximal, characteristic, complemented
8C2×C7×C4⋊Q8C2211yescharacteristic
9C22×#1143—21no—
10C2×C4×D28C2211yescharacteristic
11C2×C4×C7×C8C2211yescharacteristic
12C7×#384C2211yescharacteristic
13C2×#6409—21no—
14C2×(C2×D7)⋊C8—21no—
15C2×(C2×D7)⋊C8—21no—
16#89855—22no—
17#89855—22no—
18C2×#6383—21no—
19C2×C4⋊(C7⋊C8)C2211yescharacteristic
20C2×C4⋊(C7⋊C8)C2211yescharacteristic
21C2×C4⋊(C7⋊C8)C2211yescharacteristic
22C7×#340—21no—
23C7×#340—21no—
24#89342—22no—
25#89324—21no—
26#89252—22no—
27#10477—71nomaximal, Sylow
28C22×C7×Q8—21no—
29C22×D28D411yescharacteristic
30C22×C4×D7—21no—
31C22×C7×C8D411yescharacteristic
32C22×C7×C8D411yescharacteristic
33C7×C4⋊Q8—21no—
34C7×C4⋊Q8—21no—
35C7×C4⋊Q8—21no—
36C7×C4⋊Q8D412yes—
37C2×C7×C4⋊C4—21no—
38C2×C7×C4⋊C4—21no—
39C2×C7×C4⋊C4—21no—
40C2×C7×C4⋊C4—21no—
41C2×C7×C4⋊C4—21no—
42C2×C42×C7C2311yescharacteristic
43C2×#1143—24no—
44C2×#1143—42no—
45C2×#1143—42no—
46C2×(C2×D7)⋊C4—21no—
47C2×C4⋊Dic7D411yescharacteristic
48C22×C7⋊C8—21no—
49C22×C7⋊C8—21no—
50C22×C7⋊C8—21no—
51C4×D28—21no—
52C4×D28—21no—
53C4×D28—21no—
54C4×D28—21no—
55C4×C7×C8—21no—
56C4×C7×C8—21no—
57#6409—24no—
58(C2×D7)⋊C8—42no—
59(C2×D7)⋊C8—42no—
60#6383—42no—
61C4⋊(C7⋊C8)—21no—
62C4⋊(C7⋊C8)—21no—
63C4⋊(C7⋊C8)—21no—
64C4⋊(C7⋊C8)—21no—
65C4⋊(C7⋊C8)—21no—
66C4⋊(C7⋊C8)D412yes—
67C2×#461—71no—
68#1990D711yescharacteristic, complemented
69#1954—71no—
70#1802—71no—
71C2×#334—71no—
72C2×#332—71no—
73#1296—71no—
74C23×D7—21no—
75C2×C7×Q8—22no—
76C2×C7×Q8—41no—
77C2×C7×Q8—41no—
78C2×C7×Q8—41no—
79C2×C7×Q8—41no—
80C2×C7×Q8—41no—
81C22×C4×C7—41no—
82C22×C4×C7—41no—
83C22×C4×C7C2×D411yescharacteristic
84C22×C4×C7C2×D411yescharacteristic
85C22×C4×C7#4111yescharacteristic
86C22×Dic7—21no—
87C2×D28—21no—
88C2×D28—21no—
89C2×D28—21no—
90C2×D28—21no—
91C2×D28—21no—
92C2×D28SD1612yes—
93C2×C4×D7—41no—
94C2×C4×D7—41no—
95C2×C4×D7—41no—
96C2×C4×D7—41no—
97C2×C4×D7—41no—
98C2×C4×D7—41no—
99C2×C7×C8—21no—
100C2×C7×C8—21no—
101C2×C7×C8—21no—
102C2×C7×C8—21no—
103C2×C7×C8—21no—
104C2×C7×C8SD1612yescomplemented
105C7×C4⋊C4—22no—
106C7×C4⋊C4—22no—
107C7×C4⋊C4—41no—
108C7×C4⋊C4—41no—
109C7×C4⋊C4—41no—
110C7×C4⋊C4—41no—
111C7×C4⋊C4—41no—
112C7×C4⋊C4—41no—
113C7×C4⋊C4—81no—
114C42×C7C2×D411yescharacteristic
115C42×C7C2×D411yescharacteristic
116C42×C7#4111yescharacteristic
117C42×C7#4111yescharacteristic
118#1143—48no—
119(C2×D7)⋊C4—41no—
120(C2×D7)⋊C4—41no—
121C4⋊Dic7—21no—
122C4⋊Dic7—21no—
123C2×C7⋊C8—22no—
124C2×C7⋊C8—41no—
125C2×C7⋊C8—41no—
126C2×C7⋊C8—41no—
127C2×C7⋊C8—41no—
128C2×C7⋊C8—41no—
129C2×C7⋊C8—41no—
130C2×C7⋊C8—41no—
131C2×C7⋊C8—41no—
132C22×SD16—141no—
133C2×C4⋊Q8C2×D711yescharacteristic
134C2×C4×D4—71no—
135#461—142no—
136#461—142no—
137C2×C4⋊C8—71no—
138C2×C4⋊C8—71no—
139C2×C4⋊C8—71no—
140C2×#103—141no—
141C2×#102—141no—
142C2×C22⋊C8—141no—
143C2×C22⋊C8—141no—
144C2×C4×C8C2×D711yescharacteristic
145#384C2×D711yescharacteristic
146#340—21no—
147#340—21no—
148#340—141no—
149#334—142no—
150#332—142no—
151C23×C7#12111yescharacteristic
152C22×D7—22no—
153C22×D7—41no—
154C22×D7—41no—
155C22×D7—41no—
156C22×D7—41no—
157C22×D7—41no—
158C22×D7—41no—
159C7×Q8—42no—
160C7×Q8—42no—
161C7×Q8—81no—
162C7×Q8—81no—
163C2×C4×C7—21no—
164C2×C4×C7—21no—
165C2×C4×C7—42no—
166C2×C4×C7—81no—
167C2×C4×C7—81no—
168C2×C4×C7—81no—
169C2×C4×C7—81no—
170C2×C4×C7—81no—
171C2×C4×C7#12011yescharacteristic
172C2×C4×C7#12111yescharacteristic
173C2×C4×C7#12111yescharacteristic
174C2×C4×C7#12311yescharacteristic
175C2×C4×C7#12311yescharacteristic
176C2×C4×C7#12411yescharacteristic
177C2×C4×C7C2×SD1611yescharacteristic
178C2×C4×C7C2×SD1611yescharacteristic
179C2×C4×C7#13511yescharacteristic
180C2×C4×C7#13511yescharacteristic
181C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
182C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
183C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
184C2×C4×C7#13711yescharacteristic
185C2×Dic7—41no—
186C2×Dic7—41no—
187C2×Dic7—41no—
188D28—22no—
189D28—22no—
190D28—41no—
191D28—41no—
192C4×D7—81no—
193C4×D7—81no—
194C4×D7—81no—
195C4×D7—81no—
196C7×C8—22no—
197C7×C8—41no—
198C7⋊C8—42no—
199C7⋊C8—81no—
200C7⋊C8—81no—
201C22×Q8—21no—
202C22×D4—71no—
203C23×C4—141no—
204C2×SD16—144no—
205C2×SD16—282no—
206C2×SD16—282no—
207C22×C8—141no—
208C22×C8—141no—
209C22×C8—141no—
210C22×C8C22⋊D711yescharacteristic
211C22×C8C22⋊D711yescharacteristic
212C4⋊Q8—21no—
213C4⋊Q8—21no—
214C4⋊Q8—21no—
215C4⋊Q8D2812yes—
216C4×D4—141no—
217C4×D4—141no—
218C4×D4—141no—
219C4×D4—141no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—71no—
226C2×C22⋊C4—141no—
227C2×C42C22×D711yescharacteristic
228C4⋊C8—72no—
229C4⋊C8—141no—
230C4⋊C8—141no—
231C4⋊C8—141no—
232C4⋊C8—141no—
233C4⋊C8—141no—
234#103—144no—
235#102—282no—
236C22⋊C8—282no—
237C22⋊C8—282no—
238C4×C8—21no—
239C4×C8—21no—
240C22×C7#39411yescharacteristic
241C22×C7#46111yescharacteristic
242C22×C7#46311yescharacteristic
243C22×C7#46511yescharacteristic
244C22×C7#46911yescharacteristic
245C22×C7#47111yescharacteristic
246C22×C7#47311yescharacteristic
247C2×D7—42no—
248C2×D7—42no—
249C2×D7—42no—
250C2×D7—81no—
251C2×D7—81no—
252C2×D7—81no—
253C2×D7—81no—
254C4×C7—21no—
255C4×C7—21no—
256C4×C7—82no—
257C4×C7—161no—
258C4×C7#44911yescharacteristic
259C4×C7#45011yescharacteristic
260C4×C7#46011yescharacteristic
261C4×C7#46411yescharacteristic
262C4×C7#46511yescharacteristic
263C4×C7#46811yescharacteristic
264C4×C7#48111yescharacteristic
265C4×C7#48211yescharacteristic
266Dic7—81no—
267C24—141no—
268C2×Q8—22no—
269C2×Q8—41no—
270C2×Q8—41no—
271C2×Q8—41no—
272C2×Q8—41no—
273C2×Q8—41no—
274C2×D4—72no—
275C2×D4—141no—
276C2×D4—141no—
277C2×D4—141no—
278C2×D4—141no—
279C2×D4—141no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—141no—
283C22×C4—281no—
284C22×C4—281no—
285C22×C4—281no—
286C22×C4—281no—
287C22×C4—281no—
288C22×C4—281no—
289C22×C4D4×D711yescharacteristic
290C22×C4#115911yescharacteristic
291C22×C4C2×C22⋊D711yescharacteristic
292SD16—288no—
293C2×C8—21no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—142no—
299C2×C8—281no—
300C2×C8—281no—
301C2×C8—281no—
302C2×C8—281no—
303C2×C8—281no—
304C2×C8—281no—
305C2×C8—281no—
306C2×C8—281no—
307C2×C8#114312yescomplemented
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—81no—
317C4⋊C4—141no—
318C4⋊C4—141no—
319C22⋊C4—281no—
320C22⋊C4—281no—
321C42C2×D2811yescharacteristic
322C42#115711yescharacteristic
323C42D4×D711yescharacteristic
324C42#115911yescharacteristic
325C2×C7#167111yescharacteristic
326C2×C7#167911yescharacteristic
327C2×C7#169311yescharacteristic
328C2×C7#170111yescharacteristic
329C2×C7#203411yescharacteristic
330C2×C7#205211yescharacteristic
331C2×C7#205511yescharacteristic
332D7—82no—
333C23—142no—
334C23—281no—
335C23—281no—
336C23—281no—
337C23—281no—
338C23—281no—
339C23—281no—
340C23#650111yescharacteristic
341Q8—42no—
342Q8—42no—
343Q8—81no—
344Q8—81no—
345D4—142no—
346D4—142no—
347D4—281no—
348D4—281no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—42no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—281no—
358C2×C4—281no—
359C2×C4—281no—
360C2×C4—561no—
361C2×C4—561no—
362C2×C4—561no—
363C2×C4—561no—
364C2×C4#644511yescharacteristic
365C2×C4#644611yescharacteristic
366C2×C4#644811yescharacteristic
367C2×C4#644911yescharacteristic
368C2×C4#647411yescharacteristic
369C2×C4#647411yescharacteristic
370C2×C4D7×SD1611yescharacteristic
371C2×C4#647911yescharacteristic
372C2×C4#649311yescharacteristic
373C2×C4#650211yescharacteristic
374C2×C4C2×#114311yescharacteristic
375C2×C4#650511yescharacteristic
376C2×C4#651211yescharacteristic
377C2×C4#651311yescharacteristic
378C8—22no—
379C8—41no—
380C8—282no—
381C8—561no—
382C8—561no—
383C7#1047711yescharacteristic, complemented, Sylow
384C22—282no—
385C22—282no—
386C22—282no—
387C22—561no—
388C22—561no—
389C22—561no—
390C22—561no—
391C22#8973311yescharacteristic
392C22#8982911yescharacteristic
393C22#8985511yescharacteristic
394C22#8993111yescharacteristic
395C22#8994311yescharacteristic
396C22#8994911yescharacteristic
397C22#8996411yescharacteristic
398C4—21no—
399C4—21no—
400C4—82no—
401C4—161no—
402C4—561no—
403C4#8954811yescharacteristic
404C4#8954911yescharacteristic
405C4#8955111yescharacteristic
406C4#8955311yescharacteristic
407C4#8979911yescharacteristic
408C4#8980011yescharacteristic
409C4#8982011yescharacteristic
410C4#8982111yescharacteristic
411C2—562no—
412C2#1172743411yescharacteristic
413C2#1172793011yescharacteristic
414C2#1172801811yescharacteristic
415C2#1172895711yescharacteristic
416C2#1172898611yescharacteristic
417C2#1172909811yescharacteristic
418C2#1172912711yescharacteristic
subgroupG/H
centerC23#6501
commutator, cosocleC2×C42×C7C23
FrattiniC2×C42C22×D7
FittingC7×#1990C2
socleC23×C7#121
prime radicalC22×D28D4
hypercenter#1990D7
hypocenterC7#10477
LCSG ▷ C2×C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1990

Complements

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

NH ≅ G/Ntype
2. C7×#1990C2semidirect
3. C2×#89342C2semidirect
5. C2×#89252C2semidirect
7. #11725440C2semidirect
68. #1990D7semidirect
104. C2×C7×C8SD16semidirect
307. C2×C8#1143semidirect
383. C7#10477semidirect

Maximal subgroups

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

subgroupindex
C2×#898552
C7×#19902
C2×#893422
#117275412
C2×#892522
#117273382
#117254402
#104777