Finite Group Databasebeta

← #13249490

Subgroups — #13249490

Order 1792.

612 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89609C211yesmaximal, characteristic
2#11731821C211yesmaximal, characteristic
3#11728229C211yesmaximal, characteristic
4C2×#89277C211yesmaximal, characteristic, complemented
5C2×#89281C211yesmaximal, characteristic, complemented
6C7×#1296C211yesmaximal, characteristic, complemented
7#11725440C211yesmaximal, characteristic, complemented
8C22×#1131—21no—
9C2×#6460—21no—
10C2×C42×D7C2211yescharacteristic
11C2×#6432C2211yescharacteristic
12C2×C7×C4⋊C8C2211yescharacteristic
13C2×C7×C4⋊C8C2211yescharacteristic
14C7×#384C2211yescharacteristic
15C2×(C2×D7)⋊C8—21no—
16C2×Dic7⋊C8—21no—
17#89759—21no—
18#89759C412yes—
19#89757—21no—
20#89757C412yes—
21#89746—21no—
22C2×C4⋊(C7⋊C8)C2211yescharacteristic
23C2×C4⋊(C7⋊C8)C2211yescharacteristic
24#89609—22no—
25#89609—22no—
26#89345—21no—
27#89281—22no—
28#89277—22no—
29#9139—71nomaximal, Sylow
30C22×C4×D7C2×C411yescharacteristic
31C22×C4×D7D411yescharacteristic
32C22×C4×D7D411yescharacteristic
33C22×Dic14—21no—
34C22×C7×C8—21no—
35C22×C7×C8—21no—
36C2×C7×C4⋊C4—21no—
37C2×C7×C4⋊C4—21no—
38C2×C7×C4⋊C4C2×C412yes—
39C2×C42×C7C2311yescharacteristic
40C2×(C2×D7)⋊C4—22no—
41C2×C4⋊Dic7—21no—
42C2×C4⋊Dic7—22no—
43C2×C4⋊Dic7C2×C412yes—
44C2×C4×Dic7C2×C411yescharacteristic
45C2×C4×Dic7D411yescharacteristic
46C2×C4×Dic7D411yescharacteristic
47C22×C7⋊C8—21no—
48C22×C7⋊C8—21no—
49C2×#1131—24no—
50C2×#1131—42no—
51C2×#1131—42no—
52#6460—42no—
53#6460—42no—
54C42×D7—21no—
55C42×D7—21no—
56C42×D7—21no—
57C42×D7—21no—
58#6432—21no—
59#6432—21no—
60#6432—21no—
61#6432—21no—
62C7×C4⋊C8—21no—
63C7×C4⋊C8—21no—
64C7×C4⋊C8—21no—
65C7×C4⋊C8D412yes—
66C7⋊#95—22no—
67C7⋊#95—22no—
68(C2×D7)⋊C8—42no—
69Dic7⋊C8—24no—
70C4⋊(C7⋊C8)—21no—
71C4⋊(C7⋊C8)—21no—
72C4⋊(C7⋊C8)—21no—
73C4⋊(C7⋊C8)D412yes—
74C2×#423—71no—
75#2857—71no—
76#1797—71no—
77C2×#332—71no—
78C2×#332—71no—
79#1296—71no—
80#1296D711yescharacteristic, complemented
81C23×D7C22⋊C411yescharacteristic
82C22×C4×C7—22no—
83C22×C4×C7C22×C411yescharacteristic
84C22×C4×C7C2×D411yescharacteristic
85C22×C4×C7C2×D411yescharacteristic
86C22×Dic7—22no—
87C22×Dic7C22⋊C411yescharacteristic
88C22×Dic7C22⋊C411yescharacteristic
89C22×Dic7C22⋊C411yescharacteristic
90C2×C4×D7—21no—
91C2×C4×D7—21no—
92C2×C4×D7—21no—
93C2×C4×D7—21no—
94C2×C4×D7—21no—
95C2×C4×D7—21no—
96C2×C4×D7—21no—
97C2×C4×D7—21no—
98C2×C4×D7—21no—
99C2×C4×D7—21no—
100C2×C4×D7—21no—
101C2×C4×D7—21no—
102C2×C4×D7—22no—
103C2×C4×D7—41no—
104C2×C4×D7D812yes—
105C2×C4×D7SD1612yes—
106C2×Dic14—41no—
107C2×Dic14—41no—
108C2×Dic14—41no—
109C2×Dic14—41no—
110C2×Dic14—41no—
111C2×Dic14—41no—
112C2×C7×C8—22no—
113C2×C7×C8—41no—
114C2×C7×C8—41no—
115C2×C7×C8—41no—
116C2×C7×C8—41no—
117C2×C7×C8—41no—
118C7×C4⋊C4—42no—
119C7×C4⋊C4—41no—
120C7×C4⋊C4—41no—
121C7×C4⋊C4—41no—
122C7×C4⋊C4—41no—
123C42×C7C2×D411yescharacteristic
124C42×C7C2×D411yescharacteristic
125C42×C7#4111yescharacteristic
126C42×C7#4111yescharacteristic
127(C2×D7)⋊C4—44no—
128C4⋊Dic7—22no—
129C4⋊Dic7—22no—
130C4⋊Dic7—41no—
131C4⋊Dic7—41no—
132C4⋊Dic7—44no—
133C4×Dic7—21no—
134C4×Dic7—21no—
135C4×Dic7—21no—
136C4×Dic7—21no—
137C4×Dic7—21no—
138C4×Dic7—21no—
139C2×C7⋊C8—22no—
140C2×C7⋊C8—41no—
141C2×C7⋊C8—41no—
142C2×C7⋊C8—41no—
143C2×C7⋊C8—41no—
144C2×C7⋊C8—41no—
145#1131—48no—
146C22×M16—141no—
147C2×C4⋊Q8—71no—
148C2×C22⋊Q8—141no—
149C22×C42—71no—
150#423—142no—
151#423—142no—
152C2×C4⋊C8—71no—
153C2×C4⋊C8—71no—
154C2×C4⋊C8—141no—
155C2×C4⋊C8C2×D711yescharacteristic
156C2×C4⋊C8C2×D711yescharacteristic
157C2×C22⋊C8—141no—
158#391—141no—
159#389—72no—
160#389—141no—
161#384C2×D711yescharacteristic
162#380—72no—
163#380—141no—
164#336—141no—
165#332—142no—
166#332—142no—
167C23×C7C2×C22⋊C411yescharacteristic
168C22×D7—21no—
169C22×D7—21no—
170C22×D7—21no—
171C22×D7—21no—
172C22×D7—21no—
173C22×D7—21no—
174C22×D7#10212yes—
175C2×C4×C7—21no—
176C2×C4×C7—21no—
177C2×C4×C7—42no—
178C2×C4×C7—42no—
179C2×C4×C7—42no—
180C2×C4×C7C4×D411yescharacteristic
181C2×C4×C7C4×D411yescharacteristic
182C2×C4×C7#12011yescharacteristic
183C2×C4×C7#12111yescharacteristic
184C2×C4×C7#12111yescharacteristic
185C2×C4×C7#12311yescharacteristic
186C2×C4×C7C2×D811yescharacteristic
187C2×C4×C7C2×SD1611yescharacteristic
188C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
189C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
190C2×C4×C7#13711yescharacteristic
191C2×C4×C7#13711yescharacteristic
192C2×C4×C7#13711yescharacteristic
193C2×C4×C7#13711yescharacteristic
194C2×Dic7—21no—
195C2×Dic7—21no—
196C2×Dic7—21no—
197C2×Dic7—21no—
198C2×Dic7—21no—
199C2×Dic7—21no—
200C2×Dic7—21no—
201C2×Dic7—42no—
202C2×Dic7—42no—
203C2×Dic7—42no—
204C2×Dic7#10112yes—
205C2×Dic7#10212yes—
206C4×D7—22no—
207C4×D7—22no—
208C4×D7—22no—
209C4×D7—22no—
210C4×D7—42no—
211C4×D7—42no—
212C4×D7—41no—
213C4×D7—41no—
214C4×D7—41no—
215C4×D7—41no—
216Dic14—81no—
217Dic14—81no—
218Dic14—81no—
219Dic14—81no—
220C7×C8—42no—
221C7×C8—81no—
222C7⋊C8—42no—
223C7⋊C8—81no—
224C22×Q8—141no—
225C23×C4—71no—
226C23×C4—71no—
227C23×C4—71no—
228C2×M16—144no—
229C2×M16—282no—
230C2×M16—282no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—141no—
234C22×C8—141no—
235C4⋊Q8—141no—
236C4⋊Q8—141no—
237C4⋊Q8—141no—
238C4⋊Q8—141no—
239C22⋊Q8—282no—
240C22⋊Q8—282no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—72no—
244C2×C4⋊C4—142no—
245C2×C4⋊C4—141no—
246C2×C4⋊C4C4×D712yes—
247C2×C22⋊C4—142no—
248C2×C42—71no—
249C2×C42—71no—
250C2×C42—71no—
251C2×C42—141no—
252C2×C42—141no—
253C2×C42—141no—
254C2×C42—141no—
255C2×C42C22×D711yescharacteristic
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—21no—
259C4⋊C8—72no—
260C4⋊C8—144no—
261C4⋊C8—141no—
262C4⋊C8—141no—
263C4⋊C8—141no—
264C4⋊C8D2812yes—
265C22⋊C8—282no—
266#95—142no—
267#95—142no—
268C22×C7#38611yescharacteristic
269C22×C7C2×#10111yescharacteristic
270C22×C7#41311yescharacteristic
271C22×C7C2×#10211yescharacteristic
272C22×C7#41711yescharacteristic
273C22×C7#41711yescharacteristic
274C22×C7#41911yescharacteristic
275C2×D7—22no—
276C2×D7—22no—
277C2×D7—22no—
278C2×D7—41no—
279C2×D7—41no—
280C2×D7—41no—
281C2×D7—41no—
282C4×C7—21no—
283C4×C7—21no—
284C4×C7—82no—
285C4×C7#44711yescharacteristic
286C4×C7#44811yescharacteristic
287C4×C7#46811yescharacteristic
288C4×C7#46811yescharacteristic
289C4×C7#47011yescharacteristic
290C4×C7#47011yescharacteristic
291C4×C7#48011yescharacteristic
292C4×C7#48111yescharacteristic
293Dic7—22no—
294Dic7—22no—
295Dic7—41no—
296Dic7—82no—
297C24—71no—
298C2×Q8—281no—
299C2×Q8—281no—
300C2×Q8—281no—
301C2×Q8—281no—
302C2×Q8—281no—
303C2×Q8—281no—
304C22×C4—22no—
305C22×C4—71no—
306C22×C4—71no—
307C22×C4—72no—
308C22×C4—72no—
309C22×C4—71no—
310C22×C4—142no—
311C22×C4—141no—
312C22×C4—141no—
313C22×C4—141no—
314C22×C4—141no—
315C22×C4—141no—
316C22×C4—141no—
317C22×C4—141no—
318C22×C4—141no—
319C22×C4—141no—
320C22×C4—141no—
321C22×C4—141no—
322C22×C4—141no—
323C22×C4—142no—
324C22×C4—281no—
325C22×C4C2×C4×D711yescharacteristic
326C22×C4D4×D711yescharacteristic
327C22×C4D4×D711yescharacteristic
328M16—288no—
329C2×C8—22no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—142no—
336C2×C8—281no—
337C2×C8—281no—
338C2×C8—281no—
339C2×C8—281no—
340C2×C8—281no—
341C4⋊C4—42no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—142no—
347C4⋊C4—142no—
348C4⋊C4—284no—
349C4⋊C4—281no—
350C4⋊C4—281no—
351C22⋊C4—284no—
352C42—141no—
353C42—141no—
354C42—141no—
355C42—141no—
356C42—141no—
357C42—141no—
358C42C2×D2811yescharacteristic
359C42#115711yescharacteristic
360C42D4×D711yescharacteristic
361C42#116111yescharacteristic
362C2×C7#154611yescharacteristic
363C2×C7#154711yescharacteristic
364C2×C7#155911yescharacteristic
365C2×C7#156111yescharacteristic
366C2×C7#187411yescharacteristic
367C2×C7#189411yescharacteristic
368C2×C7#190111yescharacteristic
369D7—42no—
370C23—72no—
371C23—141no—
372C23—141no—
373C23—141no—
374C23—141no—
375C23—141no—
376C23—141no—
377C23D7×C22⋊C411yescharacteristic
378Q8—561no—
379Q8—561no—
380Q8—561no—
381Q8—561no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—72no—
388C2×C4—72no—
389C2×C4—141no—
390C2×C4—141no—
391C2×C4—141no—
392C2×C4—141no—
393C2×C4—142no—
394C2×C4—142no—
395C2×C4—142no—
396C2×C4—142no—
397C2×C4—141no—
398C2×C4—141no—
399C2×C4—141no—
400C2×C4—282no—
401C2×C4—282no—
402C2×C4—281no—
403C2×C4—281no—
404C2×C4—281no—
405C2×C4—281no—
406C2×C4—282no—
407C2×C4—282no—
408C2×C4—282no—
409C2×C4C4×D2811yescharacteristic
410C2×C4#644511yescharacteristic
411C2×C4#644711yescharacteristic
412C2×C4#645611yescharacteristic
413C2×C4#645711yescharacteristic
414C2×C4#645811yescharacteristic
415C2×C4D7×D811yescharacteristic
416C2×C4#647411yescharacteristic
417C2×C4D7×SD1611yescharacteristic
418C2×C4#647711yescharacteristic
419C2×C4#647811yescharacteristic
420C2×C4#647811yescharacteristic
421C2×C4#648111yescharacteristic
422C2×C4#648111yescharacteristic
423C8—42no—
424C8—81no—
425C8—282no—
426C8—561no—
427C7#913911yescharacteristic, complemented, Sylow
428C22—142no—
429C22—142no—
430C22—142no—
431C22—281no—
432C22—281no—
433C22—281no—
434C22—281no—
435C22#8943711yescharacteristic
436C22#8951711yescharacteristic
437C22D7×#10111yescharacteristic
438C22D7×#10211yescharacteristic
439C22#8954311yescharacteristic
440C22#8957411yescharacteristic
441C22#8957511yescharacteristic
442C4—21no—
443C4—21no—
444C4—82no—
445C4—142no—
446C4—142no—
447C4—281no—
448C4—562no—
449C4#8954511yescharacteristic
450C4#8955211yescharacteristic
451C4#8957811yescharacteristic
452C4#8958811yescharacteristic
453C4#8963511yescharacteristic
454C4#8964011yescharacteristic
455C4#8965411yescharacteristic
456C4#8965811yescharacteristic
457C2—282no—
458C2#1172613811yescharacteristic
459C2#1172617411yescharacteristic
460C2#1172618511yescharacteristic
461C2#1172624211yescharacteristic
462C2#1172663411yescharacteristic
463C2#1172663811yescharacteristic
464C2#1172674711yescharacteristic
subgroupG/H
centerC23D7×C22⋊C4
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#1296C2
socleC23×C7C2×C22⋊C4
prime radicalC23×D7C22⋊C4
cosocleC2×C42×C7C23
hypercenter#1296D7
hypocenterC7#9139
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1296

Complements

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

NH ≅ G/Ntype
4. C2×#89277C2semidirect
5. C2×#89281C2semidirect
6. C7×#1296C2semidirect
7. #11725440C2semidirect
80. #1296D7semidirect
427. C7#9139semidirect

Maximal subgroups

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

subgroupindex
C2×#896092
#117318212
#117282292
C2×#892772
C2×#892812
C7×#12962
#117254402
#91397