Finite Group Databasebeta

← #13405652

Subgroups — #13405652

Order 1792.

818 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11740688C211yesmaximal, characteristic, complemented
2C2×#89856C211yesmaximal, characteristic
3C2×C8×Dic14C211yesmaximal, characteristic, complemented
4#11731814C211yesmaximal, characteristic
5C7×#1990C211yesmaximal, characteristic, complemented
6C4×#6395C211yesmaximal, characteristic
7#11727322C211yesmaximal, characteristic, complemented
8#11726654C212yesmaximal
9#11726612C212yesmaximal, complemented
10#11726606C212yesmaximal
11#11726595C212yesmaximal, complemented
12C2×C7×C4⋊Q8C2211yescharacteristic
13C2×#6460—21no—
14C2×C4×D28C2211yescharacteristic
15C2×C4×Dic14C2211yescharacteristic
16C2×C4×C7×C8C2211yescharacteristic
17C7×#384C2211yescharacteristic
18#90001—22no—
19C2×#6395C2211yescharacteristic
20C2×#6395C2211yescharacteristic
21C2×Dic7⋊C8C2212yes—
22C2×C8×Dic7C2212yes—
23#89856—24no—
24#89766—22no—
25#89760C2212yes—
26#89759—21no—
27#89757C2212yes—
28#89748—21no—
29#89746—21no—
30#89745C2212yes—
31#89745C2212yes—
32C2×#6383C2212yes—
33C2×#6382C2212yes—
34C2×#6382C2212yes—
35C4×C4⋊Dic7C2211yescharacteristic
36C2×C4⋊(C7⋊C8)C2211yescharacteristic
37C8×Dic14—24no—
38C7×#340C2212yes—
39C7×#340C2212yes—
40#89346C2211yescharacteristic
41#89346C2211yescharacteristic
42#89324C2212yes—
43#25470—71nomaximal, Sylow
44C22×C7×Q8—21no—
45C22×D28C2311yescharacteristic
46C22×C4×D7—21no—
47C22×Dic14D411yescharacteristic
48C22×C7×C8C2311yescharacteristic
49C22×C7×C8C2311yescharacteristic
50C7×C4⋊Q8—22no—
51C7×C4⋊Q8—22no—
52C2×C7×C4⋊C4—21no—
53C2×C7×C4⋊C4—21no—
54C2×C7×C4⋊C4—21no—
55C2×C7×C4⋊C4C2312yes—
56C2×C7×C4⋊C4C2312yes—
57C2×C42×C7C2311yescharacteristic
58C2×(C2×D7)⋊C4—21no—
59C2×(C2×D7)⋊C4—22no—
60C2×(C2×D7)⋊C4—22no—
61C2×C4⋊Dic7—22no—
62C2×C4⋊Dic7D411yescharacteristic
63C2×C4⋊Dic7C2311yescharacteristic
64C2×C4⋊Dic7C2311yescharacteristic
65C2×C4⋊Dic7C2311yescharacteristic
66C2×Dic7⋊C4—22no—
67C2×Dic7⋊C4—22no—
68C2×Dic7⋊C4D412yes—
69C2×C4×Dic7—21no—
70C2×C4×Dic7C2312yes—
71C22×C7⋊C8C2312yes—
72#6460—44no—
73C4×D28—22no—
74C4×D28—22no—
75C4×Dic14—22no—
76C4×Dic14—22no—
77C4×C7×C8—22no—
78C7⋊#95—21no—
79C7⋊#95—22no—
80C7⋊#95—22no—
81C7⋊#95—22no—
82#6395—22no—
83#6395Q814yescomplemented
84Dic7⋊C8—24no—
85C8×Dic7—24no—
86#6383—24no—
87#6382—24no—
88#6382—24no—
89C4⋊(C7⋊C8)D414yes—
90C2×#476—71no—
91C2×C8×Q8—71no—
92#2567—71no—
93#2538—71no—
94#2088—72no—
95#2065—72no—
96#1990D711yescharacteristic, complemented
97#1948—71no—
98#1941—72no—
99#1934—72no—
100C4×#102—71no—
101C23×D7—21no—
102C2×C7×Q8—42no—
103C2×C7×Q8—81no—
104C2×C7×Q8—81no—
105C22×C4×C7—22no—
106C22×C4×C7—22no—
107C22×C4×C7C2×D411yescharacteristic
108C22×C4×C7C2×D411yescharacteristic
109C22×C4×C7C2411yescharacteristic
110C22×Dic7—21no—
111C22×Dic7—21no—
112C22×Dic7—21no—
113C22×Dic7C2×D412yes—
114C22×Dic7C2×D412yes—
115C2×D28—21no—
116C2×D28—21no—
117C2×D28—21no—
118C2×D28—21no—
119C2×D28—21no—
120C2×D28C2×Q812yes—
121C2×C4×D7—42no—
122C2×C4×D7—42no—
123C2×C4×D7—42no—
124C2×Dic14—22no—
125C2×Dic14—22no—
126C2×Dic14SD1614yes—
127C2×C7×C8—21no—
128C2×C7×C8—21no—
129C2×C7×C8—21no—
130C2×C7×C8—21no—
131C2×C7×C8—21no—
132C2×C7×C8C2×Q812yes—
133C7×C4⋊C4—22no—
134C7×C4⋊C4—22no—
135C7×C4⋊C4—22no—
136C7×C4⋊C4—22no—
137C7×C4⋊C4—42no—
138C7×C4⋊C4—42no—
139C7×C4⋊C4—81no—
140C42×C7C2×D412yes—
141C42×C7#4112yes—
142(C2×D7)⋊C4—42no—
143(C2×D7)⋊C4—44no—
144(C2×D7)⋊C4—82no—
145C4⋊Dic7—22no—
146C4⋊Dic7—21no—
147C4⋊Dic7—21no—
148C4⋊Dic7—21no—
149C4⋊Dic7—21no—
150C4⋊Dic7—44no—
151C4⋊Dic7C2×Q812yes—
152C4⋊Dic7#4112yes—
153Dic7⋊C4—24no—
154Dic7⋊C4—82no—
155Dic7⋊C4—82no—
156C4×Dic7—22no—
157C4×Dic7—22no—
158C4×Dic7—42no—
159C2×C7⋊C8—22no—
160C2×C7⋊C8—22no—
161C2×C7⋊C8C2×D414yes—
162C2×C4⋊Q8C2×D711yescharacteristic
163C2×C22⋊Q8—141no—
164C2×C4×Q8—71no—
165C2×C4×D4—71no—
166#476—144no—
167C8×Q8—144no—
168C2×#106—72no—
169C2×#106—72no—
170C2×C4⋊C8—71no—
171C2×C4⋊C8—72no—
172C2×#102—71no—
173C2×#102—71no—
174C2×#102—72no—
175C2×C4×C8—72no—
176C2×C4×C8C2×D711yescharacteristic
177#396—142no—
178#395—142no—
179#391—141no—
180#390—72no—
181#389—72no—
182#389—72no—
183#389—72no—
184#384—141no—
185#384C2×D711yescharacteristic
186#380—141no—
187C4×C4⋊C4—71no—
188#340—71no—
189#340—71no—
190#340—72no—
191#340C2×D712yes—
192#340C2×D712yes—
193C23×C7C22×D411yescharacteristic
194C22×D7—22no—
195C22×D7—41no—
196C22×D7—41no—
197C22×D7—41no—
198C22×D7—41no—
199C22×D7—41no—
200C22×D7—41no—
201C7×Q8—84no—
202C2×C4×C7—21no—
203C2×C4×C7—21no—
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#12112yes—
211C2×C4×C7#12112yes—
212C2×C4×C7C2×SD1612yes—
213C2×C4×C7#13512yes—
214C2×C4×C7C22×D411yescharacteristic
215C2×C4×C7C22×Q811yescharacteristic
216C2×C4×C7C2×#4111yescharacteristic
217C2×C4×C7C2×#4111yescharacteristic
218C2×C4×C72−1+411yescharacteristic
219C2×C4×C72−1+411yescharacteristic
220C2×Dic7—22no—
221C2×Dic7—22no—
222C2×Dic7—22no—
223C2×Dic7—22no—
224C2×Dic7—22no—
225C2×Dic7—22no—
226C2×Dic7—22no—
227C2×Dic7—41no—
228C2×Dic7—41no—
229C2×Dic7—41no—
230C2×Dic7—41no—
231C2×Dic7—41no—
232C2×Dic7—41no—
233C2×Dic7—41no—
234C2×Dic7C2×SD1614yes—
235D28—22no—
236D28—22no—
237D28—41no—
238D28—41no—
239C4×D7—84no—
240Dic14—28no—
241C7×C8—22no—
242C7×C8—41no—
243C7⋊C8—24no—
244C22×Q8—21no—
245C22×Q8—71no—
246C22×D4—71no—
247C23×C4—141no—
248C22×C8—72no—
249C22×C8C22×D711yescharacteristic
250C22×C8C22×D711yescharacteristic
251C4⋊Q8—22no—
252C4⋊Q8—22no—
253C22⋊Q8—284no—
254C4×Q8—142no—
255C4×Q8—142no—
256C4×D4—142no—
257C4×D4—142no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—71no—
262C2×C4⋊C4—71no—
263C2×C4⋊C4—72no—
264C2×C4⋊C4—71no—
265C2×C4⋊C4—71no—
266C2×C4⋊C4—142no—
267C2×C4⋊C4—142no—
268C2×C4⋊C4—142no—
269C2×C4⋊C4C22×D712yes—
270C2×C4⋊C4C22×D712yes—
271C2×C22⋊C4—142no—
272C2×C22⋊C4—141no—
273C2×C22⋊C4—142no—
274C2×C42—72no—
275C2×C42—141no—
276C2×C42C22×D711yescharacteristic
277#106—144no—
278#106—144no—
279C4⋊C8—74no—
280C4⋊C8—144no—
281#102—74no—
282#102—144no—
283#102—142no—
284C4×C8—22no—
285C4×C8—144no—
286#95—142no—
287#95—142no—
288#95—142no—
289#95—141no—
290C22×C7C2×#12111yescharacteristic
291C22×C7#54811yescharacteristic
292C22×C7D4×Q811yescharacteristic
293C22×C7C22×SD1611yescharacteristic
294C22×C7C2×#13511yescharacteristic
295C22×C7#57611yescharacteristic
296C22×C7#57711yescharacteristic
297C2×D7—42no—
298C2×D7—42no—
299C2×D7—42no—
300C2×D7—81no—
301C2×D7—81no—
302C2×D7—81no—
303C2×D7—81no—
304C4×C7—22no—
305C4×C7—82no—
306C4×C7—82no—
307C4×C7#46514yes—
308C4×C7#53311yescharacteristic
309C4×C7#54111yescharacteristic
310C4×C7#54711yescharacteristic
311C4×C7D4×Q811yescharacteristic
312Dic7—24no—
313Dic7—42no—
314Dic7—42no—
315Dic7—42no—
316Dic7—81no—
317C24—141no—
318C2×Q8—42no—
319C2×Q8—74no—
320C2×Q8—81no—
321C2×Q8—81no—
322C2×Q8—142no—
323C2×Q8—142no—
324C2×D4—72no—
325C2×D4—141no—
326C2×D4—141no—
327C2×D4—141no—
328C2×D4—141no—
329C2×D4—141no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—72no—
333C22×C4—72no—
334C22×C4—141no—
335C22×C4—141no—
336C22×C4—141no—
337C22×C4—282no—
338C22×C4—282no—
339C22×C4—282no—
340C22×C4D4×D711yescharacteristic
341C22×C4D4×D711yescharacteristic
342C22×C4C23×D711yescharacteristic
343C2×C8—21no—
344C2×C8—21no—
345C2×C8—21no—
346C2×C8—21no—
347C2×C8—21no—
348C2×C8—74no—
349C2×C8—142no—
350C2×C8—142no—
351C2×C8Q8×D712yes—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—72no—
359C4⋊C4—72no—
360C4⋊C4—81no—
361C4⋊C4—144no—
362C4⋊C4—142no—
363C4⋊C4—141no—
364C4⋊C4—141no—
365C4⋊C4—141no—
366C4⋊C4—141no—
367C4⋊C4—284no—
368C4⋊C4—562no—
369C4⋊C4—562no—
370C22⋊C4—284no—
371C22⋊C4—282no—
372C22⋊C4—562no—
373C42—142no—
374C42—142no—
375C42—282no—
376C42#115712yes—
377C42D4×D712yes—
378C2×C7#242911yescharacteristic
379C2×C7C2×#46511yescharacteristic
380C2×C7#306211yescharacteristic
381C2×C7#336011yescharacteristic
382C2×C7#336211yescharacteristic
383C2×C7Q8×SD1611yescharacteristic
384C2×C7#338011yescharacteristic
385D7—82no—
386C23—142no—
387C23—281no—
388C23—281no—
389C23—281no—
390C23—281no—
391C23—281no—
392C23—281no—
393C23C2×D4×D711yescharacteristic
394Q8—84no—
395Q8—148no—
396D4—142no—
397D4—142no—
398D4—281no—
399D4—281no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—74no—
409C2×C4—142no—
410C2×C4—142no—
411C2×C4—142no—
412C2×C4—142no—
413C2×C4—142no—
414C2×C4—142no—
415C2×C4—142no—
416C2×C4—281no—
417C2×C4—281no—
418C2×C4—281no—
419C2×C4—281no—
420C2×C4—281no—
421C2×C4—281no—
422C2×C4—281no—
423C2×C4—564no—
424C2×C4#644712yes—
425C2×C4#644712yes—
426C2×C4D7×SD1612yes—
427C2×C4#648212yes—
428C2×C4C2×#115711yescharacteristic
429C2×C4C2×D4×D711yescharacteristic
430C2×C4C2×Q8×D711yescharacteristic
431C2×C4#655111yescharacteristic
432C2×C4D7×#4111yescharacteristic
433C2×C4#655411yescharacteristic
434C8—22no—
435C8—41no—
436C8—144no—
437C7#2547011yescharacteristic, complemented, Sylow
438C22—282no—
439C22—282no—
440C22—282no—
441C22—561no—
442C22—561no—
443C22—561no—
444C22—561no—
445C22C2×#644711yescharacteristic
446C22#9036811yescharacteristic
447C22#9041811yescharacteristic
448C22#9044811yescharacteristic
449C22C2×D7×SD1611yescharacteristic
450C22#9045311yescharacteristic
451C22C2×#648211yescharacteristic
452C4—22no—
453C4—82no—
454C4—82no—
455C4—144no—
456C4—282no—
457C4—282no—
458C4—282no—
459C4—561no—
460C4#8957914yes—
461C4#9026911yescharacteristic
462C4#9027411yescharacteristic
463C4#9032211yescharacteristic
464C4#9032311yescharacteristic
465C2—562no—
466C2#1173407011yescharacteristic
467C2#1173619711yescharacteristic
468C2C2×#8957911yescharacteristic
469C2#1173903311yescharacteristic
470C2#1173906611yescharacteristic
471C2#1173934811yescharacteristic
472C2#1173939811yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#1990C2
socleC23×C7C22×D4
prime radicalC22×D28C23
hypercenter#1990D7
hypocenterC7#25470
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1990

Complements

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

NH ≅ G/Ntype
1. #11740688C2semidirect
3. C2×C8×Dic14C2semidirect
5. C7×#1990C2semidirect
7. #11727322C2semidirect
9. #11726612C2semidirect
11. #11726595C2semidirect
83. #6395Q8semidirect
96. #1990D7semidirect
437. C7#25470semidirect

Maximal subgroups

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

subgroupindex
#117406882
C2×#898562
C2×C8×Dic142
#117318142
C7×#19902
C4×#63952
#117273222
#117266542
#117266122
#117266062
#117265952
#254707