Finite Group Databasebeta

← #13398917

Subgroups — #13398917

Order 1792.

656 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#472C211yesmaximal, characteristic
2#11741268C211yesmaximal, characteristic, complemented
3#11729846C212yesmaximal, complemented
4#11729843C212yesmaximal, complemented
5#11729795C211yesmaximal, characteristic, complemented
6#11729767C211yesmaximal, characteristic, complemented
7#11729765C211yesmaximal, characteristic, complemented
8#11728460C211yesmaximal, characteristic, complemented
9#11728452C211yesmaximal, characteristic, complemented
10#11727832C212yesmaximal, complemented
11#11727821C212yesmaximal, complemented
12C2×C7×AGL(1,ℤ/8)C2212yes—
13C22×C7×D8C2211yescharacteristic
14C2×C7×#131C2211yescharacteristic
15#90489C2212yes—
16C7×#472—22no—
17C7×#472—22no—
18C2×C7×#108C2211yescharacteristic
19C2×C7×#100C2212yes—
20#90012C2211yescharacteristic
21#90009C2211yescharacteristic
22#90008C2212yes—
23C2×#6407C2212yes—
24#89987C2212yes—
25#89987C2212yes—
26#89924—22no—
27D8×Dic7—22no—
28#89896C2212yes—
29C2×#6393C2211yescharacteristic
30#89878C2211yescharacteristic
31#89873C2211yescharacteristic
32#89873C2211yescharacteristic
33#89354C2212yes—
34#89354C2212yes—
35#89352C2212yes—
36#89351C2212yes—
37#89351C2211yescharacteristic
38#89351C2211yescharacteristic
39#24988—71nomaximal, Sylow
40C2×C7×#41C2311yescharacteristic
41C22×C7×D4C2312yes—
42C7×AGL(1,ℤ/8)—24no—
43C7×AGL(1,ℤ/8)—42no—
44C2×C7×SD16—22no—
45C2×C7×D8—22no—
46C2×C7×D8—21no—
47C2×C7×D8—21no—
48C2×C7×D8—21no—
49C2×C7×D8—21no—
50C2×C7×D8D412yes—
51C2×C7×D8D412yes—
52C7×#131—21no—
53C7×#131—21no—
54C7×#131—21no—
55C7×#131—21no—
56C2×C7×M16—21no—
57C2×C7×M16C2311yescharacteristic
58C2×C7×M16C2312yes—
59C22×C7×C8—21no—
60C22×C7×C8C2311yescharacteristic
61C2×C22⋊Dic7—24no—
62D4×Dic7—24no—
63D4×Dic7—42no—
64#6489C2311yescharacteristic
65#6489C2312yes—
66#6489C2311yescharacteristic
67C2×#1136C2311yescharacteristic
68C2×#1136C2312yes—
69C2×#1136C2311yescharacteristic
70C7×#108—21no—
71C7×#108—21no—
72C7×#100—24no—
73#6411—42no—
74#6410—21no—
75#6408—22no—
76#6407—21no—
77#6407—42no—
78#6406—24no—
79#6406—42no—
80#6396—42no—
81#6393—21no—
82#6393—21no—
83#6392—21no—
84#6391—21no—
85#6389—21no—
86#6389—21no—
87C8×Dic7—21no—
88C8×Dic7—21no—
89C2×#472D711yescharacteristic
90#2948—71no—
91#2081—72no—
92#2048—72no—
93#1978—71no—
94#1953—71no—
95#1904—72no—
96#1833—71no—
97#1829—72no—
98#1814—71no—
99#1757—71no—
100C24×C7—22no—
101C7×#41—21no—
102C7×#41—21no—
103C7×#41—21no—
104C7×#41—21no—
105C2×C7×Q8#4111yescharacteristic
106C2×C7×D4—21no—
107C2×C7×D4—22no—
108C2×C7×D4—22no—
109C2×C7×D4—22no—
110C2×C7×D4—22no—
111C2×C7×D4—22no—
112C2×C7×D4C2×D414yes—
113C2×C7×D4#4111yescharacteristic
114C22×C4×C7—21no—
115C22×C4×C7C2411yescharacteristic
116C22×Dic7—44no—
117C7×SD16—44no—
118C7×D8—22no—
119C7×D8—22no—
120C7×D8—22no—
121C7×D8—22no—
122C7×D8—44no—
123C7×M16—21no—
124C7×M16—22no—
125C7×M16—21no—
126C7×M16—22no—
127C7×M16—41no—
128C7×M16—41no—
129C2×C7×C8—21no—
130C2×C7×C8—22no—
131C2×C7×C8—41no—
132C2×C7×C8—41no—
133C2×C7×C8C2×D411yescharacteristic
134C2×C7×C8C2×D411yescharacteristic
135C2×C7×C8#4111yescharacteristic
136C2×C7×C8#4111yescharacteristic
137C2×C7×C8#4111yescharacteristic
138C2×C7×C8#4111yescharacteristic
139C22⋊Dic7—21no—
140C22⋊Dic7—22no—
141C22⋊Dic7—44no—
142C22⋊Dic7#4112yes—
143C4⋊Dic7—21no—
144C4⋊Dic7—22no—
145C4⋊Dic7—21no—
146C4×Dic7—21no—
147C4×Dic7—21no—
148C4×Dic7—22no—
149#1136—21no—
150#1136—21no—
151#1136—22no—
152#1136—21no—
153#1136—22no—
154#1136—21no—
155C2×C7⋊C8—21no—
156C2×C7⋊C8—21no—
157C2×C7⋊C8—22no—
158C2×AGL(1,ℤ/8)C2×D712yes—
159C22×D8C2×D711yescharacteristic
160C2×#131C2×D711yescharacteristic
161#518—72no—
162#472—22no—
163#472—22no—
164#442—142no—
165C4×D8—142no—
166#430—72no—
167C2×#108—71no—
168C2×#108C2×D711yescharacteristic
169#427—71no—
170#421—72no—
171#418—72no—
172#418—72no—
173#413—71no—
174C2×#100—72no—
175C2×#100C2×D712yes—
176#410—71no—
177#405—71no—
178#405—71no—
179#344—72no—
180#344—71no—
181#344—71no—
182#344—72no—
183#344—72no—
184H(3,ℤ/4)—72no—
185C23×C7—21no—
186C23×C7—24no—
187C23×C7—42no—
188C23×C7—42no—
189C23×C7—162no—
190C23×C7C2×#4111yescharacteristic
191C7×Q8—21no—
192C7×Q8—21no—
193C7×D4—24no—
194C7×D4—21no—
195C7×D4—24no—
196C7×D4—21no—
197C7×D4—41no—
198C7×D4—41no—
199C7×D4—42no—
200C7×D4—42no—
201C2×C4×C7—21no—
202C2×C4×C7—41no—
203C2×C4×C7—41no—
204C2×C4×C7C22×D411yescharacteristic
205C2×C4×C7C2×#4111yescharacteristic
206C2×C4×C7C2×#4111yescharacteristic
207C2×C4×C7C2×#4111yescharacteristic
208C2×C4×C7H(5,𝔽2)11yescharacteristic
209C2×C4×C7H(5,𝔽2)11yescharacteristic
210C2×Dic7—22no—
211C2×Dic7—42no—
212C2×Dic7—41no—
213C2×Dic7—164no—
214C7×C8—41no—
215C7×C8—42no—
216C7×C8#12311yescharacteristic
217C7×C8#12311yescharacteristic
218C7×C8#12311yescharacteristic
219C7×C8#12311yescharacteristic
220C7⋊C8—42no—
221C7⋊C8—41no—
222C7⋊C8—41no—
223C2×#41C22×D711yescharacteristic
224C22×D4C22×D712yes—
225AGL(1,ℤ/8)—24no—
226AGL(1,ℤ/8)—42no—
227C2×SD16—22no—
228C2×D8—22no—
229C2×D8—21no—
230C2×D8—21no—
231C2×D8—21no—
232C2×D8—21no—
233C2×D8C22⋊D712yes—
234C2×D8C22⋊D712yes—
235#131—21no—
236#131—21no—
237#131—21no—
238#131—21no—
239C2×M16—21no—
240C2×M16—71no—
241C2×M16—72no—
242C2×M16—71no—
243C2×M16C22×D711yescharacteristic
244C2×M16C22×D712yes—
245C22×C8—21no—
246C22×C8C22×D711yescharacteristic
247C4×D4—144no—
248C4×D4—282no—
249#117—71no—
250#117—72no—
251#117—71no—
252C2×C22⋊C4—144no—
253#108—21no—
254#108—21no—
255#108—141no—
256#108—141no—
257#108—282no—
258#107—141no—
259#106—141no—
260#104—282no—
261#102—144no—
262#102—282no—
263#101—141no—
264#100—24no—
265#100—141no—
266#100—282no—
267#99—142no—
268#97—141no—
269#97—141no—
270C4×C8—141no—
271C4×C8—141no—
272C22×C7—41no—
273C22×C7—41no—
274C22×C7—44no—
275C22×C7—84no—
276C22×C7—81no—
277C22×C7—162no—
278C22×C7—162no—
279C22×C7#52911yescharacteristic
280C22×C7#54611yescharacteristic
281C22×C7#55511yescharacteristic
282C4×C7—41no—
283C4×C7C2×#12311yescharacteristic
284C4×C7#54011yescharacteristic
285C4×C7#54611yescharacteristic
286C4×C7#55311yescharacteristic
287Dic7—42no—
288Dic7—81no—
289Dic7—82no—
290C24—22no—
291#41—21no—
292#41—21no—
293#41—21no—
294#41—21no—
295C2×Q8#115911yescharacteristic
296C2×D4—21no—
297C2×D4—22no—
298C2×D4—22no—
299C2×D4—22no—
300C2×D4—22no—
301C2×D4—22no—
302C2×D4#115911yescharacteristic
303C2×D4C2×C22⋊D714yes—
304C22×C4—21no—
305C22×C4—284no—
306C22×C4C23×D711yescharacteristic
307SD16—44no—
308D8—22no—
309D8—22no—
310D8—22no—
311D8—22no—
312D8—44no—
313M16—21no—
314M16—22no—
315M16—21no—
316M16—22no—
317M16—41no—
318M16—41no—
319M16—141no—
320M16—141no—
321M16—142no—
322M16—141no—
323M16—141no—
324M16—142no—
325C2×C8—21no—
326C2×C8—22no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—141no—
330C2×C8—142no—
331C2×C8—141no—
332C2×C8#115911yescharacteristic
333C2×C8#115911yescharacteristic
334C2×C8#115911yescharacteristic
335C2×C8#115911yescharacteristic
336C2×C8C2×C22⋊D711yescharacteristic
337C2×C8C2×C22⋊D711yescharacteristic
338C4⋊C4—141no—
339C4⋊C4—141no—
340C4⋊C4—142no—
341C22⋊C4—72no—
342C22⋊C4—141no—
343C22⋊C4—142no—
344C22⋊C4—284no—
345C42—141no—
346C42—141no—
347C42—142no—
348C2×C7—21no—
349C2×C7—21no—
350C2×C7—21no—
351C2×C7—81no—
352C2×C7—84no—
353C2×C7#242011yescharacteristic
354C23—24no—
355C23—21no—
356C23—42no—
357C23—42no—
358C23—162no—
359C23D7×#4111yescharacteristic
360Q8—21no—
361Q8—21no—
362D4—24no—
363D4—21no—
364D4—24no—
365D4—21no—
366D4—41no—
367D4—41no—
368D4—42no—
369D4—42no—
370C2×C4—21no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—142no—
374C2×C4—282no—
375C2×C4—281no—
376C2×C4—1124no—
377C2×C4C2×#115911yescharacteristic
378C2×C4C2×#115911yescharacteristic
379C2×C4C2×#115911yescharacteristic
380C2×C4#654811yescharacteristic
381C2×C4#655311yescharacteristic
382C2×C4C22×C22⋊D711yescharacteristic
383C8—41no—
384C8—42no—
385C8—281no—
386C8—281no—
387C8—282no—
388C8#649811yescharacteristic
389C8#649811yescharacteristic
390C8#649811yescharacteristic
391C8#649811yescharacteristic
392C7#2498811yescharacteristic, complemented, Sylow
393C22—41no—
394C22—41no—
395C22—44no—
396C22—84no—
397C22—81no—
398C22—162no—
399C22—162no—
400C22#9029111yescharacteristic
401C22#9030711yescharacteristic
402C22#9052011yescharacteristic
403C4—41no—
404C4—282no—
405C4—562no—
406C4—561no—
407C4#9030911yescharacteristic
408C4#9031011yescharacteristic
409C4C2×#649811yescharacteristic
410C4#9049711yescharacteristic
411C2—21no—
412C2—21no—
413C2—21no—
414C2—84no—
415C2—81no—
416C2#1174008811yescharacteristic
subgroupG/H
centerC2#11740088
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
Fitting, prime radicalC2×C7×#472C2
socleC2×C7#2420
hypercenterC2×#472D7
hypocenterC7#24988
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#472

Complements

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

NH ≅ G/Ntype
2. #11741268C2semidirect
3. #11729846C2semidirect
4. #11729843C2semidirect
5. #11729795C2semidirect
6. #11729767C2semidirect
7. #11729765C2semidirect
8. #11728460C2semidirect
9. #11728452C2semidirect
10. #11727832C2semidirect
11. #11727821C2semidirect
392. C7#24988semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#4722
#117412682
#117298462
#117298432
#117297952
#117297672
#117297652
#117284602
#117284522
#117278322
#117278212
#249887