Finite Group Databasebeta

← #13339217

Subgroups — #13339217

Order 1792.

620 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
1C7×#3209C211yesmaximal, characteristic, complemented
2#11740794C211yesmaximal, characteristic
3#11740657C211yesmaximal, characteristic
4#11736736C211yesmaximal, characteristic
5#11729584C211yesmaximal, characteristic, complemented
6C2×#89342C211yesmaximal, characteristic, complemented
7#11729062C212yesmaximal, complemented
8#11729059C212yesmaximal, complemented
9#11728995C212yesmaximal, complemented
10#11728987C212yesmaximal, complemented
11#11728504C211yesmaximal, characteristic, complemented
12C7×#533C2211yescharacteristic
13C2×C7×C4⋊Q8C2211yescharacteristic
14C4×#1157C2211yescharacteristic
15C7×#494C2212yes—
16C7×#487C2212yes—
17C7×#462—21no—
18C7×#462—21no—
19C7×#451—22no—
20C7×#431C2211yescharacteristic
21#89866C2212yes—
22#89866C2211yescharacteristic
23#89863—21no—
24#89863C2211yescharacteristic
25#89859C2212yes—
26#89859C2211yescharacteristic
27#89856—21no—
28#89856C2211yescharacteristic
29#89838—21no—
30#89835—21no—
31#89774—21no—
32#89770—21no—
33#89738—21no—
34#89736—21no—
35#89696C2211yescharacteristic
36C2×C4⋊(C7⋊C8)C2211yescharacteristic
37C2×C4⋊(C7⋊C8)C2211yescharacteristic
38#89605—21no—
39D7×C4⋊C8—21no—
40(C22⋊D7)⋊C8—22no—
41C8×D28C2212yes—
42C8×Dic14C2212yes—
43#89342C2212yes—
44#89342C2212yes—
45#89342C2212yes—
46#89341C2212yes—
47#89253C2212yes—
48#89253C2212yes—
49#19411—71nomaximal, Sylow
50C22×C7×Q8—21no—
51C2×#1157D411yescharacteristic
52C2×C7×Q16—22no—
53C2×C7×Q16—22no—
54C7×C4⋊Q8—21no—
55C7×C4⋊Q8—21no—
56C7×C4⋊Q8—21no—
57C7×C4⋊Q8C2311yescharacteristic
58C7×C4⋊Q8C2312yes—
59C7×C4⋊Q8C2311yescharacteristic
60C7×#125—21no—
61C7×C22⋊Q8—21no—
62C7×C22⋊Q8—21no—
63C4×C7×Q8—21no—
64C4×C7×Q8—21no—
65C7×#117—21no—
66C2×C7×C4⋊C4—21no—
67C2×C7×C4⋊C4—21no—
68C2×C42×C7C2311yescharacteristic
69C4×C22⋊D7—21no—
70C4×C22⋊D7—21no—
71#6489D411yescharacteristic
72C2×#1136—21no—
73C22×C7⋊C8—21no—
74C22×C7⋊C8—21no—
75C2×C8×D7—22no—
76C4×D28D411yescharacteristic
77C4×D28C2311yescharacteristic
78#6435—21no—
79C42×D7—21no—
80C4×Dic14D411yescharacteristic
81C4×Dic14C2311yescharacteristic
82C7×C4⋊C8D411yescharacteristic
83C7×C4⋊C8D411yescharacteristic
84C7×#103—22no—
85C7×#103—22no—
86C7×C22⋊C8D412yes—
87C4×C7×C8C2312yes—
88(C2×D7)⋊C8—22no—
89Dic7⋊C8—22no—
90C8×Dic7—22no—
91#6384—22no—
92#6384—21no—
93#6384—21no—
94#6383—22no—
95#6383—21no—
96#6383—21no—
97#6382—22no—
98#6382—21no—
99#6382—21no—
100#6381—22no—
101#6381—21no—
102#6381—21no—
103C4⋊(C7⋊C8)—21no—
104C4⋊(C7⋊C8)—21no—
105C4⋊(C7⋊C8)—21no—
106C4⋊(C7⋊C8)C2311yescharacteristic
107C4⋊(C7⋊C8)C2311yescharacteristic
108C4⋊(C7⋊C8)C2312yes—
109C4⋊(C7⋊C8)C2312yes—
110#3209D711yescharacteristic, complemented
111#3079—71no—
112#3076—71no—
113#2988—71no—
114#1684—72no—
115#1683—72no—
116#1670—72no—
117#1669—72no—
118#1571—71no—
119#1550—71no—
120C2×#332—71no—
121C2×C7×Q8—22no—
122C2×C7×Q8—21no—
123C2×C7×Q8—21no—
124C2×C7×Q8—41no—
125C2×C7×Q8—81no—
126C2×C7×Q8—81no—
127C22×C4×C7—41no—
128C22×C4×C7C2×D411yescharacteristic
129C22×C4×C7C2×D411yescharacteristic
130C22×C4×C7#4111yescharacteristic
131C2×C22⋊D7—21no—
132#1157—22no—
133#1157Q1614yescomplemented
134C2×D28C2×D411yescharacteristic
135C2×C4×D7—21no—
136C2×C4×D7—21no—
137C2×C4×D7—21no—
138C2×Dic14C2×D411yescharacteristic
139C7×Q16—82no—
140C7×Q16—82no—
141C2×C7×C8C2×D412yes—
142C2×C7×C8C2×D412yes—
143C7×C4⋊C4—21no—
144C7×C4⋊C4—22no—
145C7×C4⋊C4—21no—
146C7×C4⋊C4—21no—
147C7×C4⋊C4—22no—
148C7×C4⋊C4—21no—
149C7×C4⋊C4—41no—
150C7×C4⋊C4—41no—
151C7×C4⋊C4—41no—
152C7×C4⋊C4—41no—
153C7×C22⋊C4—41no—
154C42×C7—41no—
155C42×C7C2×D411yescharacteristic
156C42×C7C2×D411yescharacteristic
157C42×C7#4111yescharacteristic
158C42×C7C2411yescharacteristic
159C22⋊Dic7—21no—
160(C2×D7)⋊C4—21no—
161(C2×D7)⋊C4—21no—
162C4⋊Dic7C2×D411yescharacteristic
163C4⋊Dic7C2×D411yescharacteristic
164Dic7⋊C4—21no—
165Dic7⋊C4—21no—
166C4×Dic7—21no—
167C4×Dic7—21no—
168C4×Dic7—21no—
169#1136—42no—
170C2×C7⋊C8—22no—
171C2×C7⋊C8—21no—
172C2×C7⋊C8—21no—
173C2×C7⋊C8—22no—
174C2×C7⋊C8—42no—
175C2×C7⋊C8—81no—
176C8×D7—82no—
177#533C2×D711yescharacteristic
178C2×C4⋊Q8C2×D711yescharacteristic
179C4×#41—71no—
180#494C2×D712yes—
181#487C2×D712yes—
182#479—141no—
183#478—141no—
184#477—71no—
185#477—141no—
186#476—71no—
187#476—141no—
188#475—71no—
189#475—72no—
190#474—71no—
191#474—72no—
192#462—21no—
193#462—21no—
194#451—22no—
195C8×Q8—72no—
196C8×D4—72no—
197C8×D4—142no—
198#431—141no—
199#431C2×D711yescharacteristic
200#428—141no—
201#427—141no—
202#423—71no—
203C2×C4⋊C8—71no—
204C2×C4⋊C8—71no—
205C2×C4⋊C8—141no—
206#417—141no—
207#416—141no—
208#335—72no—
209#335—72no—
210#333—72no—
211#332—72no—
212#332—72no—
213#332—72no—
214C23×C7#12111yescharacteristic
215C22×D7—21no—
216C7×Q8—41no—
217C7×Q8—42no—
218C7×Q8—41no—
219C7×Q8—42no—
220C7×Q8—81no—
221C7×Q8—82no—
222C2×C4×C7—21no—
223C2×C4×C7—21no—
224C2×C4×C7—41no—
225C2×C4×C7—41no—
226C2×C4×C7—41no—
227C2×C4×C7—42no—
228C2×C4×C7—82no—
229C2×C4×C7#12111yescharacteristic
230C2×C4×C7#12111yescharacteristic
231C2×C4×C7#12111yescharacteristic
232C2×C4×C7C2×Q1612yes—
233C2×C4×C7#13512yes—
234C2×C4×C7C22×D411yescharacteristic
235C2×C4×C7C22×D411yescharacteristic
236C2×C4×C7C2×#4111yescharacteristic
237C22⋊D7—24no—
238C2×Dic7—21no—
239C2×Dic7—21no—
240C2×Dic7—21no—
241D28—21no—
242D28C2×Q1612yes—
243C4×D7—24no—
244C4×D7—41no—
245C4×D7—41no—
246C4×D7—81no—
247Dic14—21no—
248Dic14C2×Q1612yes—
249C7×C8—22no—
250C7×C8—22no—
251C7⋊C8—41no—
252C7⋊C8—41no—
253C7⋊C8—42no—
254C7⋊C8—42no—
255C2×#41—71no—
256C22×Q8—21no—
257C2×Q16—22no—
258C2×Q16—22no—
259C2×M16—141no—
260C22×C8—141no—
261C22×C8—141no—
262C22×C8—142no—
263C4⋊Q8—21no—
264C4⋊Q8—21no—
265C4⋊Q8—21no—
266C4⋊Q8C22×D712yes—
267C4⋊Q8C22×D711yescharacteristic
268C4⋊Q8C22×D711yescharacteristic
269#125—21no—
270C22⋊Q8—21no—
271C22⋊Q8—21no—
272C4×Q8—21no—
273C4×Q8—21no—
274C4×Q8—71no—
275C4×Q8—71no—
276C4×D4—71no—
277C4×D4—71no—
278C4×D4—141no—
279C4×D4—141no—
280#117—21no—
281#117—71no—
282#117—141no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C42—141no—
286C2×C42C22×D711yescharacteristic
287#107—141no—
288#107—141no—
289#107—142no—
290#106—141no—
291#106—141no—
292#106—142no—
293C4⋊C8—72no—
294C4⋊C8—71no—
295C4⋊C8—71no—
296C4⋊C8—72no—
297C4⋊C8—142no—
298C4⋊C8—141no—
299C4⋊C8—141no—
300C4⋊C8—141no—
301C4⋊C8C22⋊D711yescharacteristic
302C4⋊C8C22⋊D711yescharacteristic
303#103—22no—
304#103—22no—
305#103—141no—
306#103—141no—
307#103—142no—
308#102—141no—
309#102—141no—
310#102—142no—
311C22⋊C8—142no—
312C22⋊C8C22⋊D712yes—
313C4×C8—142no—
314C4×C8C22×D712yes—
315C22×C7—21no—
316C22×C7—21no—
317C22×C7#46712yes—
318C22×C7C2×#12111yescharacteristic
319C2×D7—22no—
320C2×D7—41no—
321C2×D7—41no—
322C4×C7—21no—
323C4×C7—21no—
324C4×C7—21no—
325C4×C7—82no—
326C4×C7—81no—
327C4×C7—81no—
328C4×C7#46712yes—
329C4×C7C22×Q1611yescharacteristic
330C4×C7C2×#13511yescharacteristic
331C4×C7#57611yescharacteristic
332C4×C7#57711yescharacteristic
333Dic7—22no—
334Dic7—41no—
335Dic7—41no—
336#41—74no—
337#41—142no—
338C2×Q8—21no—
339C2×Q8—22no—
340C2×Q8—21no—
341C2×Q8—41no—
342C2×Q8—71no—
343C2×Q8—81no—
344C2×Q8—81no—
345C2×D4—71no—
346C2×D4—141no—
347C22×C4—41no—
348C22×C4—141no—
349C22×C4—141no—
350C22×C4—141no—
351C22×C4D4×D711yescharacteristic
352C22×C4#115911yescharacteristic
353C22×C4C2×C22⋊D711yescharacteristic
354Q16—82no—
355Q16—82no—
356M16—282no—
357C2×C8—141no—
358C2×C8—142no—
359C2×C8—141no—
360C2×C8—142no—
361C2×C8—282no—
362C2×C8—562no—
363C2×C8—561no—
364C2×C8C2×C22⋊D712yes—
365C2×C8C2×C22⋊D712yes—
366C4⋊C4—22no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—22no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—71no—
377C4⋊C4—71no—
378C4⋊C4—141no—
379C4⋊C4—141no—
380C22⋊C4—41no—
381C22⋊C4—141no—
382C22⋊C4—141no—
383C22⋊C4—141no—
384C42—41no—
385C42—141no—
386C42—141no—
387C42—141no—
388C42D4×D711yescharacteristic
389C42#115911yescharacteristic
390C42C2×C22⋊D711yescharacteristic
391C42C23×D711yescharacteristic
392C2×C7—22no—
393C2×C7#304511yescharacteristic
394C2×C7C2×#46711yescharacteristic
395C2×C7#306911yescharacteristic
396D7—42no—
397C23—141no—
398C23#650111yescharacteristic
399Q8—41no—
400Q8—42no—
401Q8—41no—
402Q8—42no—
403Q8—72no—
404Q8—81no—
405Q8—82no—
406Q8—141no—
407D4—72no—
408D4—144no—
409D4—141no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—82no—
417C2×C4—141no—
418C2×C4—144no—
419C2×C4—141no—
420C2×C4—141no—
421C2×C4—281no—
422C2×C4—281no—
423C2×C4—561no—
424C2×C4D7×Q1612yes—
425C2×C4#648212yes—
426C2×C4#650111yescharacteristic
427C2×C4#650111yescharacteristic
428C2×C4#650111yescharacteristic
429C2×C4C2×D4×D711yescharacteristic
430C2×C4C2×#115911yescharacteristic
431C2×C4C22×C22⋊D711yescharacteristic
432C8—22no—
433C8—22no—
434C8—281no—
435C8—281no—
436C8—282no—
437C8—282no—
438C7#1941111yescharacteristic, complemented, Sylow
439C22—21no—
440C22—21no—
441C22—142no—
442C22—281no—
443C22—281no—
444C22#8996012yes—
445C22C2×#650111yescharacteristic
446C4—21no—
447C4—21no—
448C4—21no—
449C4—81no—
450C4—82no—
451C4—81no—
452C4—142no—
453C4—281no—
454C4—281no—
455C4#8996012yes—
456C4C2×D7×Q1611yescharacteristic
457C4C2×#648211yescharacteristic
458C4#9052611yescharacteristic
459C4#9052711yescharacteristic
460C2—22no—
461C2—282no—
462C2#1174083011yescharacteristic
463C2C2×#8996011yescharacteristic
464C2#1174144311yescharacteristic
subgroupG/H
centerC22C2×#6501
commutator, cosocleC42×C7C24
FrattiniC42C23×D7
FittingC7×#3209C2
socleC22×C7C2×#121
prime radicalC2×#1157D4
hypercenter#3209D7
hypocenterC7#19411
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3209

Complements

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

NH ≅ G/Ntype
1. C7×#3209C2semidirect
5. #11729584C2semidirect
6. C2×#89342C2semidirect
7. #11729062C2semidirect
8. #11729059C2semidirect
9. #11728995C2semidirect
10. #11728987C2semidirect
11. #11728504C2semidirect
110. #3209D7semidirect
133. #1157Q16semidirect
438. C7#19411semidirect

Maximal subgroups

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

subgroupindex
C7×#32092
#117407942
#117406572
#117367362
#117295842
C2×#893422
#117290622
#117290592
#117289952
#117289872
#117285042
#194117