Finite Group Databasebeta

← #13319858

Subgroups — #13319858

Order 1792.

876 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90274C211yesmaximal, characteristic
2#11736709C211yesmaximal, characteristic, complemented
3#11736708C211yesmaximal, characteristic, complemented
4C7×#1461C211yesmaximal, characteristic, complemented
5#11729378C211yesmaximal, characteristic, complemented
6#11728536C211yesmaximal, characteristic
7#11728536C211yesmaximal, characteristic
8#11725883C214yesmaximal
9#11725882C214yesmaximal, complemented
10C2×C4×C7×Q8C2211yescharacteristic
11C2×#6509C411yescharacteristic
12C2×#6507C411yescharacteristic
13#90274—22no—
14#90274—22no—
15#90274—22no—
16#90274—22no—
17C2×#6461C2212yes—
18C2×#6460C411yescharacteristic
19C2×#6453C411yescharacteristic
20C2×#6452C2212yes—
21C2×#6439C412yes—
22C2×#6435C2211yescharacteristic
23C2×#6433C412yes—
24C2×C4×Dic14C2211yescharacteristic
25C7×#431C2211yescharacteristic
26C7×#431C2211yescharacteristic
27#89697C2211yescharacteristic
28#89697C2211yescharacteristic
29#89603—22no—
30#89602—22no—
31#89491C2214yes—
32#89490C2214yes—
33#89451—22no—
34C8×Dic14—22no—
35C7×#324C2214yes—
36#89323C2214yes—
37#89264C2212yes—
38#89264C2212yes—
39#89263C2212yes—
40#89263C2212yes—
41#17391—71nomaximal, Sylow
42C22×C7×Q8C2×C411yescharacteristic
43C22×C4×D7C2×C411yescharacteristic
44C22×Dic14C2×C411yescharacteristic
45C4×C7×Q8—21no—
46C4×C7×Q8—21no—
47C4×C7×Q8—21no—
48C4×C7×Q8—21no—
49C2×C7×C4⋊C4C2×C411yescharacteristic
50C2×C7×C4⋊C4C2312yes—
51C2×C42×C7C2×C412yes—
52C2×C42×C7C2311yescharacteristic
53#6509—22no—
54#6509—22no—
55#6507—22no—
56#6507—22no—
57C2×(C2×D7)⋊C4C2×C411yescharacteristic
58C2×(C2×D7)⋊C4C2×C412yes—
59C2×(C2×D7)⋊C4C2×C412yes—
60C2×(C2×D7)⋊C4C2311yescharacteristic
61C2×C4⋊Dic7C2×C412yes—
62C2×C4⋊Dic7C2×C412yes—
63C2×C4⋊Dic7C2×C411yescharacteristic
64C2×Dic7⋊C4C2×C411yescharacteristic
65C2×Dic7⋊C4C2×C412yes—
66C2×Dic7⋊C4C2×C412yes—
67C2×Dic7⋊C4C2×C412yes—
68C2×Dic7⋊C4C2×C412yes—
69C2×Dic7⋊C4C2×C412yes—
70C2×Dic7⋊C4C2311yescharacteristic
71C2×C4×Dic7C2×C411yescharacteristic
72C2×C4×Dic7C2312yes—
73#6461—24no—
74#6461—24no—
75#6460—22no—
76#6460—22no—
77#6453—22no—
78#6453—22no—
79#6452—24no—
80#6452—24no—
81#6439—24no—
82#6439—24no—
83#6435—21no—
84#6435—21no—
85#6435—21no—
86#6435—21no—
87#6433—24no—
88#6433—24no—
89C4×Dic14D412yes—
90C4×Dic14D412yes—
91C4×Dic14D412yes—
92C4×Dic14D412yes—
93C7×C4⋊C8—21no—
94C7×C4⋊C8—21no—
95C7×C22⋊C8C2312yes—
96C7×C22⋊C8C2312yes—
97C4×C7×C8—21no—
98C4×C7×C8—21no—
99C22⋊(C7⋊C8)C2312yes—
100C22⋊(C7⋊C8)C2312yes—
101#6389—24no—
102Dic7⋊C8—24no—
103Dic7⋊C8—24no—
104C8×Dic7—24no—
105C4⋊(C7⋊C8)—21no—
106C4⋊(C7⋊C8)—21no—
107C4×C7⋊C8—21no—
108C4×C7⋊C8—21no—
109C2×#541—71no—
110#2983—71no—
111#2982—71no—
112#1473—74no—
113#1470—74no—
114#1461—71no—
115#1461D711yescharacteristic, complemented
116#1458—71no—
117#1458—71no—
118C23×D7C22×C411yescharacteristic
119C2×C7×Q8—21no—
120C2×C7×Q8—21no—
121C2×C7×Q8—22no—
122C2×C7×Q8—22no—
123C22×C4×C7C22×C412yes—
124C22×C4×C7C22×C412yes—
125C22×C4×C7C22×C411yescharacteristic
126C22×C4×C7C22×C411yescharacteristic
127C22×C4×C7C2411yescharacteristic
128C22×Dic7C22×C411yescharacteristic
129C22×Dic7C22×C412yes—
130C22×Dic7C22×C411yescharacteristic
131C22×Dic7C22×C411yescharacteristic
132C22×Dic7C22×C412yes—
133C2×C4×D7—21no—
134C2×C4×D7—21no—
135C2×C4×D7—22no—
136C2×C4×D7—22no—
137C2×Dic14—22no—
138C2×Dic14—22no—
139C2×Dic14C22⋊C412yes—
140C2×Dic14C22⋊C412yes—
141C2×C7×C8—22no—
142C2×C7×C8—22no—
143C7×C4⋊C4—21no—
144C7×C4⋊C4—21no—
145C7×C4⋊C4—22no—
146C7×C4⋊C4—22no—
147C42×C7—22no—
148C42×C7—22no—
149C42×C7C2×D411yescharacteristic
150C42×C7C2×D411yescharacteristic
151C42×C7C2×D411yescharacteristic
152C42×C7C2×D411yescharacteristic
153(C2×D7)⋊C4—21no—
154(C2×D7)⋊C4—21no—
155(C2×D7)⋊C4—21no—
156(C2×D7)⋊C4—21no—
157(C2×D7)⋊C4—24no—
158(C2×D7)⋊C4—24no—
159C4⋊Dic7—24no—
160C4⋊Dic7—24no—
161C4⋊Dic7C22⋊C412yes—
162C4⋊Dic7C22⋊C412yes—
163Dic7⋊C4—21no—
164Dic7⋊C4—21no—
165Dic7⋊C4—21no—
166Dic7⋊C4—21no—
167Dic7⋊C4—24no—
168Dic7⋊C4—24no—
169Dic7⋊C4—24no—
170Dic7⋊C4—24no—
171Dic7⋊C4C22⋊C414yes—
172Dic7⋊C4C22⋊C414yes—
173C4×Dic7—21no—
174C4×Dic7—21no—
175C4×Dic7C2×D414yes—
176C4×Dic7C2×D414yes—
177C2×C7⋊C8—22no—
178C2×C7⋊C8—22no—
179#541—142no—
180#541—142no—
181#541—142no—
182#541—142no—
183C2×C4⋊Q8—71no—
184C2×#126—72no—
185C2×#126—72no—
186C2×#125—71no—
187C2×#125—72no—
188C2×#125—72no—
189C2×C22⋊Q8—71no—
190C2×C22⋊Q8—71no—
191C2×C4×Q8—71no—
192C2×C4×Q8C2×D711yescharacteristic
193C2×#117—71no—
194#446—142no—
195#446—142no—
196C8×Q8—142no—
197C8×Q8—142no—
198#432—74no—
199#431—71no—
200#431—71no—
201#431—74no—
202#431C2×D711yescharacteristic
203#431C2×D711yescharacteristic
204#324—74no—
205#324—72no—
206#324—72no—
207#324C2×D714yes—
208#323—72no—
209#323—72no—
210C23×C7C23×C411yescharacteristic
211C22×D7—21no—
212C22×D7—21no—
213C22×D7—21no—
214C22×D7—41no—
215C22×D7—41no—
216C7×Q8—42no—
217C7×Q8—42no—
218C2×C4×C7—22no—
219C2×C4×C7—22no—
220C2×C4×C7—22no—
221C2×C4×C7—22no—
222C2×C4×C7—41no—
223C2×C4×C7—41no—
224C2×C4×C7—42no—
225C2×C4×C7C2×C22⋊C411yescharacteristic
226C2×C4×C7C2×C22⋊C411yescharacteristic
227C2×C4×C7C2×C22⋊C411yescharacteristic
228C2×C4×C7C2×C22⋊C411yescharacteristic
229C2×C4×C7#13112yes—
230C2×C4×C7#13112yes—
231C2×C4×C7C22×D411yescharacteristic
232C2×C4×C7C22×D411yescharacteristic
233C2×Dic7—22no—
234C2×Dic7—22no—
235C2×Dic7—21no—
236C2×Dic7—21no—
237C2×Dic7—21no—
238C2×Dic7—21no—
239C2×Dic7—21no—
240C2×Dic7—42no—
241C2×Dic7—41no—
242C2×Dic7—41no—
243C2×Dic7C2×C22⋊C414yes—
244C2×Dic7C2×C22⋊C414yes—
245C4×D7—42no—
246C4×D7—42no—
247Dic14—24no—
248Dic14—24no—
249C7×C8—42no—
250C7×C8—42no—
251C7⋊C8—42no—
252C7⋊C8—42no—
253C22×Q8—71no—
254C22×Q8C4×D711yescharacteristic
255C23×C4—71no—
256C4⋊Q8—142no—
257C4⋊Q8—142no—
258#126—144no—
259#126—144no—
260#126—144no—
261#126—144no—
262#125—144no—
263#125—144no—
264#125—144no—
265#125—144no—
266#125—142no—
267#125—142no—
268C22⋊Q8—142no—
269C22⋊Q8—142no—
270C22⋊Q8—142no—
271C22⋊Q8—142no—
272C4×Q8—21no—
273C4×Q8—21no—
274C4×Q8—21no—
275C4×Q8—21no—
276C4×Q8—72no—
277C4×Q8—72no—
278C4×Q8—72no—
279C4×Q8—72no—
280#117—141no—
281#117—141no—
282#117—141no—
283#117—141no—
284C2×C4⋊C4—71no—
285C2×C4⋊C4—72no—
286C2×C4⋊C4—71no—
287C2×C4⋊C4—72no—
288C2×C4⋊C4—72no—
289C2×C4⋊C4—72no—
290C2×C4⋊C4—72no—
291C2×C4⋊C4—72no—
292C2×C4⋊C4—72no—
293C2×C4⋊C4—71no—
294C2×C4⋊C4C4×D711yescharacteristic
295C2×C4⋊C4C22×D712yes—
296C2×C22⋊C4—71no—
297C2×C22⋊C4—72no—
298C2×C22⋊C4—72no—
299C2×C22⋊C4—71no—
300C2×C42—72no—
301C2×C42—71no—
302C2×C42C4×D712yes—
303C2×C42C22×D711yescharacteristic
304C4⋊C8—21no—
305C4⋊C8—21no—
306C4⋊C8—144no—
307C4⋊C8—144no—
308C4⋊C8—141no—
309C4⋊C8—141no—
310C22⋊C8—72no—
311C22⋊C8—72no—
312C22⋊C8C22×D712yes—
313C22⋊C8C22×D712yes—
314#97—144no—
315C4×C8—21no—
316C4×C8—21no—
317C4×C8—144no—
318C4×C8—141no—
319C4×C8—141no—
320C22×C7—21no—
321C22×C7—21no—
322C22×C7C22×C22⋊C411yescharacteristic
323C22×C7C2×#13111yescharacteristic
324C22×C7C2×#13111yescharacteristic
325C2×D7—41no—
326C2×D7—41no—
327C2×D7—41no—
328C2×D7—81no—
329C2×D7—81no—
330C4×C7—21no—
331C4×C7—21no—
332C4×C7—21no—
333C4×C7—21no—
334C4×C7—42no—
335C4×C7—42no—
336C4×C7#40812yes—
337C4×C7#40812yes—
338Dic7—24no—
339Dic7—24no—
340Dic7—41no—
341Dic7—41no—
342Dic7—41no—
343C24—71no—
344C2×Q8—21no—
345C2×Q8—21no—
346C2×Q8—22no—
347C2×Q8—22no—
348C2×Q8—72no—
349C2×Q8—72no—
350C2×Q8—142no—
351C2×Q8—142no—
352C22×C4—72no—
353C22×C4—72no—
354C22×C4—71no—
355C22×C4—71no—
356C22×C4—71no—
357C22×C4—141no—
358C22×C4—141no—
359C22×C4—142no—
360C22×C4—142no—
361C22×C4C2×C4×D712yes—
362C22×C4C2×C4×D712yes—
363C22×C4C2×C4×D711yescharacteristic
364C22×C4C2×C4×D711yescharacteristic
365C22×C4C23×D711yescharacteristic
366C2×C8—22no—
367C2×C8—22no—
368C2×C8—142no—
369C2×C8—142no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—74no—
375C4⋊C4—74no—
376C4⋊C4—72no—
377C4⋊C4—72no—
378C4⋊C4—144no—
379C4⋊C4—144no—
380C4⋊C4—144no—
381C4⋊C4—144no—
382C4⋊C4—144no—
383C4⋊C4—144no—
384C4⋊C4—141no—
385C4⋊C4—141no—
386C4⋊C4—141no—
387C4⋊C4—141no—
388C22⋊C4—144no—
389C22⋊C4—144no—
390C22⋊C4—141no—
391C22⋊C4—141no—
392C22⋊C4—141no—
393C22⋊C4—141no—
394C42—22no—
395C42—22no—
396C42—74no—
397C42—74no—
398C42—141no—
399C42—141no—
400C42D4×D711yescharacteristic
401C42D4×D711yescharacteristic
402C42D4×D711yescharacteristic
403C42D4×D711yescharacteristic
404C2×C7—21no—
405C2×C7—21no—
406C2×C7C2×#40811yescharacteristic
407C2×C7#288411yescharacteristic
408C2×C7#288911yescharacteristic
409D7—81no—
410C23—141no—
411C23—141no—
412C23—141no—
413C23—281no—
414C23—281no—
415C23C22×C4×D711yescharacteristic
416Q8—42no—
417Q8—42no—
418Q8—144no—
419Q8—144no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—42no—
427C2×C4—74no—
428C2×C4—74no—
429C2×C4—142no—
430C2×C4—142no—
431C2×C4—141no—
432C2×C4—141no—
433C2×C4—141no—
434C2×C4—141no—
435C2×C4—141no—
436C2×C4—282no—
437C2×C4—282no—
438C2×C4—282no—
439C2×C4—281no—
440C2×C4—281no—
441C2×C4D7×C22⋊C411yescharacteristic
442C2×C4D7×C22⋊C411yescharacteristic
443C2×C4D7×C22⋊C411yescharacteristic
444C2×C4D7×C22⋊C411yescharacteristic
445C2×C4#646412yes—
446C2×C4#647012yes—
447C2×C4C2×D4×D711yescharacteristic
448C2×C4C2×D4×D711yescharacteristic
449C8—42no—
450C8—42no—
451C8—282no—
452C8—282no—
453C7#1739111yescharacteristic, complemented, Sylow
454C22—21no—
455C22—21no—
456C22—281no—
457C22—281no—
458C22—281no—
459C22—561no—
460C22—561no—
461C22C2×D7×C22⋊C411yescharacteristic
462C22C2×#646411yescharacteristic
463C22C2×#647011yescharacteristic
464C4—21no—
465C4—21no—
466C4—21no—
467C4—21no—
468C4—42no—
469C4—42no—
470C4—144no—
471C4—144no—
472C4—281no—
473C4—281no—
474C4—281no—
475C4#8949912yes—
476C4#8949912yes—
477C2—21no—
478C2—21no—
479C2—561no—
480C2C2×#8949911yescharacteristic
481C2#1173613411yescharacteristic
482C2#1173615111yescharacteristic
subgroupG/H
centerC22C2×D7×C22⋊C4
commutatorC23×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1461C2
socleC22×C7C22×C22⋊C4
prime radicalC23×D7C22×C4
cosocleC22×C4×C7C24
hypercenter#1461D7
hypocenterC7#17391
LCSG ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1461

Complements

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

NH ≅ G/Ntype
2. #11736709C2semidirect
3. #11736708C2semidirect
4. C7×#1461C2semidirect
5. #11729378C2semidirect
9. #11725882C2semidirect
115. #1461D7semidirect
453. C7#17391semidirect

Maximal subgroups

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

subgroupindex
C2×#902742
#117367092
#117367082
C7×#14612
#117293782
#117285362
#117258832
#117258822
#173917