Finite Group Databasebeta

← #13249267

Subgroups — #13249267

Order 1792.

514 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#6505C211yesmaximal, characteristic
2#11736721C211yesmaximal, characteristic
3C7×#1461C211yesmaximal, characteristic, complemented
4C2×Q8⋊(C7⋊C8)C211yesmaximal, characteristic, complemented
5#11728509C211yesmaximal, characteristic
6#11725550C211yesmaximal, characteristic, complemented
7#11725545C211yesmaximal, characteristic, complemented
8C2×C4×C7×Q8C2211yescharacteristic
9C2×#6505C411yescharacteristic
10C4×#1157C2211yescharacteristic
11C7×#431C2211yescharacteristic
12C7×#431C2211yescharacteristic
13#89997C411yescharacteristic
14#89802—21no—
15C4×#1144—21no—
16#89798—21no—
17C4×#1143—21no—
18#89738C411yescharacteristic
19#89736C411yescharacteristic
20C2×C4⋊(C7⋊C8)C2211yescharacteristic
21C4×#1136C2211yescharacteristic
22C2×C4×C7⋊C8C2211yescharacteristic
23#89606—21no—
24#89602—21no—
25(C22⋊D7)⋊C8—22no—
26#89457—21no—
27C4×C8×D7—21no—
28C7×#324—22no—
29Q8⋊(C7⋊C8)—21no—
30Q8⋊(C7⋊C8)—21no—
31#89254—21no—
32#89251—21no—
33#9096—71nomaximal, Sylow
34C22×C7×Q8C2×C411yescharacteristic
35C2×#1157C2×C411yescharacteristic
36C4×C7×Q8—21no—
37C4×C7×Q8—21no—
38C4×C7×Q8—21no—
39C4×C7×Q8—21no—
40C2×C7×C4⋊C4—21no—
41C2×C7×C4⋊C4C2×C411yescharacteristic
42C2×C42×C7—21no—
43C2×C42×C7C2311yescharacteristic
44C2×#1144—21no—
45#6505—22no—
46#6505C814yescomplemented
47C2×#1143—21no—
48C4×C22⋊D7—21no—
49C4×C22⋊D7—21no—
50#6489C2×C411yescharacteristic
51C2×#1136C2×C411yescharacteristic
52C2×#1136C2×C411yescharacteristic
53C22×C7⋊C8—21no—
54C22×C7⋊C8—21no—
55C2×C8×D7—22no—
56C4×D28D411yescharacteristic
57C4×D28D411yescharacteristic
58#6435—21no—
59C42×D7—21no—
60C4×Dic14D411yescharacteristic
61C4×Dic14D411yescharacteristic
62C7×C4⋊C8—21no—
63C7×C4⋊C8—21no—
64C7×C22⋊C8—21no—
65C7×C22⋊C8D412yes—
66C4×C7×C8—21no—
67C4×C7×C8—21no—
68#6409—21no—
69#6409—21no—
70(C2×D7)⋊C8—22no—
71Dic7⋊C8—22no—
72C8×Dic7—22no—
73#6384—21no—
74#6383—21no—
75#6382—21no—
76#6381—21no—
77C4⋊(C7⋊C8)—21no—
78C4⋊(C7⋊C8)—21no—
79#6377—21no—
80C4×C7⋊C8—21no—
81C4×C7⋊C8—21no—
82C4×C7⋊C8D411yescharacteristic
83C4×C7⋊C8D411yescharacteristic
84C8×#41—71no—
85C4×#137—71no—
86#1477—71no—
87C2×Q8⋊C8—71no—
88#1461D711yescharacteristic, complemented
89#1283—71no—
90#1278—71no—
91C2×C7×Q8—21no—
92C2×C7×Q8—21no—
93C2×C7×Q8—21no—
94C2×C7×Q8—21no—
95C2×C7×Q8—21no—
96C2×C7×Q8C2×C812yescomplemented
97C22×C4×C7—21no—
98C22×C4×C7—21no—
99C22×C4×C7C22×C411yescharacteristic
100C22×C4×C7C2×D411yescharacteristic
101C22×C4×C7#4111yescharacteristic
102C2×C22⋊D7—21no—
103#1157—21no—
104#1157—21no—
105#1157C2×C812yes—
106#1157M1612yes—
107C2×D28C22⋊C411yescharacteristic
108C2×C4×D7—21no—
109C2×C4×D7—21no—
110C2×C4×D7—21no—
111C2×Dic14C22⋊C411yescharacteristic
112C2×C7×C8—22no—
113C2×C7×C8—41no—
114C7×C4⋊C4—21no—
115C7×C4⋊C4—21no—
116C7×C4⋊C4—41no—
117C7×C4⋊C4—41no—
118C42×C7—41no—
119C42×C7—41no—
120C42×C7C2×D411yescharacteristic
121C42×C7C2×D411yescharacteristic
122C42×C7C2×D411yescharacteristic
123C42×C7#4111yescharacteristic
124C22⋊Dic7—21no—
125#1144—24no—
126#1143—24no—
127(C2×D7)⋊C4—21no—
128(C2×D7)⋊C4—21no—
129C4⋊Dic7C22⋊C411yescharacteristic
130C4⋊Dic7C22⋊C411yescharacteristic
131Dic7⋊C4—21no—
132Dic7⋊C4—21no—
133C4×Dic7—21no—
134C4×Dic7—21no—
135C4×Dic7—21no—
136#1136—21no—
137#1136—21no—
138#1136—21no—
139#1136C2×C812yescomplemented
140C2×C7⋊C8—21no—
141C2×C7⋊C8—21no—
142C2×C7⋊C8—21no—
143C2×C7⋊C8—21no—
144C2×C7⋊C8—42no—
145C2×C7⋊C8—41no—
146C2×C7⋊C8—81no—
147C8×D7—44no—
148C2×#137—71no—
149C4×#41—71no—
150C2×C4×Q8C2×D711yescharacteristic
151C8×Q8—141no—
152#441—141no—
153#440—141no—
154C4×Q16—141no—
155C4×SD16—141no—
156C8×D4—141no—
157C8×D4—142no—
158#431—141no—
159#431C2×D711yescharacteristic
160#431C2×D711yescharacteristic
161#428—71no—
162C2×C4⋊C8—71no—
163#419—71no—
164#417—71no—
165C4×M16—71no—
166C2×C4×C8—71no—
167C2×C4×C8—141no—
168Q8⋊C8—141no—
169Q8⋊C8—141no—
170Q8⋊C8—141no—
171D4⋊C8—141no—
172#324—22no—
173C23×C7C4×D411yescharacteristic
174C22×D7—21no—
175C7×Q8—22no—
176C7×Q8—22no—
177C7×Q8—41no—
178C7×Q8—41no—
179C2×C4×C7—21no—
180C2×C4×C7—21no—
181C2×C4×C7—22no—
182C2×C4×C7—41no—
183C2×C4×C7—41no—
184C2×C4×C7—41no—
185C2×C4×C7—41no—
186C2×C4×C7—81no—
187C2×C4×C7C2×C22⋊C411yescharacteristic
188C2×C4×C7C4×D411yescharacteristic
189C2×C4×C7#12011yescharacteristic
190C2×C4×C7#12111yescharacteristic
191C2×C4×C7#12111yescharacteristic
192C2×C4×C7#12311yescharacteristic
193C2×C4×C7C22×C811yescharacteristic
194C2×C4×C7C2×M1611yescharacteristic
195C2×C4×C7#13111yescharacteristic
196C2×C4×C7#13111yescharacteristic
197C22⋊D7—22no—
198C22⋊D7—22no—
199C2×Dic7—21no—
200C2×Dic7—21no—
201C2×Dic7—21no—
202D28—21no—
203D28C22⋊C812yes—
204C4×D7—22no—
205C4×D7—22no—
206C4×D7—42no—
207C4×D7—41no—
208C4×D7—41no—
209Dic14—21no—
210Dic14C22⋊C812yes—
211C7×C8—42no—
212C7×C8—81no—
213C7⋊C8—22no—
214C7⋊C8—22no—
215C7⋊C8—41no—
216C7⋊C8—41no—
217C7⋊C8—81no—
218C2×#41—71no—
219C22×Q8C4×D711yescharacteristic
220#137—74no—
221#137—142no—
222C2×Q16—141no—
223C2×SD16—141no—
224C2×M16—71no—
225C2×M16—71no—
226C22×C8—142no—
227C22×C8—141no—
228C22×C8—141no—
229C4×Q8—21no—
230C4×Q8—21no—
231C4×Q8—21no—
232C4×Q8—21no—
233C4×Q8—71no—
234C4×Q8—71no—
235C4×D4—71no—
236C4×D4—71no—
237C4×D4—141no—
238C4×D4—141no—
239#117—71no—
240#117—141no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4C4×D711yescharacteristic
243C2×C42—21no—
244C2×C42—141no—
245C2×C42C22×D711yescharacteristic
246#107—141no—
247#106—141no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8—142no—
251C4⋊C8—141no—
252C4⋊C8—141no—
253#103—141no—
254#103—141no—
255#103—141no—
256#102—141no—
257C22⋊C8—21no—
258C22⋊C8—142no—
259C22⋊C8C22⋊D712yes—
260#97—141no—
261C4×C8—21no—
262C4×C8—21no—
263C4×C8—71no—
264C4×C8—71no—
265C4×C8—142no—
266C4×C8—141no—
267C4×C8—141no—
268C22×C7—21no—
269C22×C7—21no—
270C22×C7#38611yescharacteristic
271C22×C7C8×D411yescharacteristic
272C22×C7#43511yescharacteristic
273C2×D7—22no—
274C2×D7—41no—
275C2×D7—41no—
276C4×C7—21no—
277C4×C7—21no—
278C4×C7—21no—
279C4×C7—42no—
280C4×C7—81no—
281C4×C7C2×C22⋊C811yescharacteristic
282C4×C7#40811yescharacteristic
283C4×C7C8×D411yescharacteristic
284C4×C7#43511yescharacteristic
285C4×C7#47111yescharacteristic
286C4×C7#47311yescharacteristic
287Dic7—22no—
288Dic7—41no—
289Dic7—41no—
290#41—72no—
291#41—72no—
292#41—141no—
293#41—141no—
294C2×Q8—21no—
295C2×Q8—21no—
296C2×Q8—21no—
297C2×Q8—21no—
298C2×Q8—21no—
299C2×Q8—71no—
300C2×Q8C8×D712yescomplemented
301C2×D4—71no—
302C2×D4—141no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—141no—
306C22×C4—141no—
307C22×C4—141no—
308C22×C4C2×C4×D711yescharacteristic
309C22×C4#115711yescharacteristic
310C22×C4C2×C22⋊D711yescharacteristic
311Q16—144no—
312SD16—144no—
313M16—72no—
314M16—141no—
315M16—141no—
316M16—141no—
317C2×C8—22no—
318C2×C8—41no—
319C2×C8—141no—
320C2×C8—141no—
321C2×C8—141no—
322C2×C8—141no—
323C2×C8—284no—
324C2×C8—282no—
325C2×C8—281no—
326C2×C8—561no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—71no—
332C4⋊C4—71no—
333C4⋊C4—141no—
334C4⋊C4—141no—
335C22⋊C4—141no—
336C22⋊C4—141no—
337C22⋊C4—141no—
338C42—41no—
339C42—41no—
340C42—141no—
341C42—141no—
342C42—141no—
343C42D4×D711yescharacteristic
344C42D4×D711yescharacteristic
345C42D4×D711yescharacteristic
346C42#115911yescharacteristic
347C2×C7—21no—
348C2×C7—21no—
349C2×C7#186911yescharacteristic
350C2×C7#190611yescharacteristic
351C2×C7#190811yescharacteristic
352D7—42no—
353C23—141no—
354C23C4×C22⋊D711yescharacteristic
355Q8—22no—
356Q8—22no—
357Q8—41no—
358Q8—41no—
359Q8—72no—
360Q8—141no—
361D4—72no—
362D4—142no—
363D4—142no—
364D4—141no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—22no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—81no—
373C2×C4—141no—
374C2×C4—142no—
375C2×C4—142no—
376C2×C4—141no—
377C2×C4—141no—
378C2×C4—282no—
379C2×C4—281no—
380C2×C4—281no—
381C2×C4D7×C22⋊C411yescharacteristic
382C2×C4(C22⋊D7)⋊C411yescharacteristic
383C2×C4#644611yescharacteristic
384C2×C4#644711yescharacteristic
385C2×C4C2×C8×D711yescharacteristic
386C2×C4#646411yescharacteristic
387C2×C4D7×M1611yescharacteristic
388C2×C4#647011yescharacteristic
389C2×C4#650011yescharacteristic
390C2×C4#650111yescharacteristic
391C8—42no—
392C8—81no—
393C8—142no—
394C8—142no—
395C8—281no—
396C8—281no—
397C8—561no—
398C7#909611yescharacteristic, complemented, Sylow
399C22—21no—
400C22—21no—
401C22—142no—
402C22—281no—
403C22—281no—
404C22#8972811yescharacteristic
405C22C8×C22⋊D711yescharacteristic
406C22#8989911yescharacteristic
407C4—21no—
408C4—21no—
409C4—21no—
410C4—42no—
411C4—81no—
412C4—142no—
413C4—281no—
414C4—281no—
415C4D7×C22⋊C811yescharacteristic
416C4(C22⋊D7)⋊C811yescharacteristic
417C4#8949911yescharacteristic
418C4#8950011yescharacteristic
419C4#8952611yescharacteristic
420C4#8952711yescharacteristic
421C2—21no—
422C2—21no—
423C2—282no—
424C2#1172771111yescharacteristic
425C2#1172780911yescharacteristic
426C2#1172787711yescharacteristic
subgroupG/H
centerC2×C4#6447
commutatorC2×C4×C7C22×C8
FrattiniC2×C42C22×D7
FittingC7×#1461C2
socleC22×C7#386
prime radicalC2×#1157C2×C4
cosocleC2×C42×C7C23
hypercenter#1461D7
hypocenterC7#9096
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1461

Complements

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

NH ≅ G/Ntype
3. C7×#1461C2semidirect
4. C2×Q8⋊(C7⋊C8)C2semidirect
6. #11725550C2semidirect
7. #11725545C2semidirect
46. #6505C8semidirect
88. #1461D7semidirect
96. C2×C7×Q8C2×C8semidirect
139. #1136C2×C8semidirect
300. C2×Q8C8×D7semidirect
398. C7#9096semidirect

Maximal subgroups

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

subgroupindex
C4×#65052
#117367212
C7×#14612
C2×Q8⋊(C7⋊C8)2
#117285092
#117255502
#117255452
#90967