Finite Group Databasebeta

← C7×#10257 #13187473

Subgroups — C7×#10257 #13187473

Order 1792.

528 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3246C211yesmaximal, characteristic
2C7×#3078C211yesmaximal, characteristic, complemented
3C7×#3037C211yesmaximal, characteristic
4C7×#1984C211yesmaximal, characteristic, complemented
5C7×#1493C211yesmaximal, characteristic, complemented
6C7×#1492C211yesmaximal, characteristic
7C7×#1292C211yesmaximal, characteristic, complemented
8C2×C7×#137—21no—
9C7×#533C2211yescharacteristic
10C7×#532C2211yescharacteristic
11C2×C4×C7×Q8C2211yescharacteristic
12C7×#479—22no—
13C7×#477—21no—
14C7×#476—21no—
15C7×#475—21no—
16C7×#475—21no—
17C7×#465—22no—
18C7×#464—22no—
19C7×#462—21no—
20C7×#461—21no—
21C7×#460—21no—
22C7×#460—21no—
23C7×#428—21no—
24C7×#423C2211yescharacteristic
25C7×#423C2211yescharacteristic
26C2×C7×C4⋊C8C2211yescharacteristic
27C7×#419—21no—
28C7×#417—21no—
29C2×C4×C7×C8C2211yescharacteristic
30C7×#336—22no—
31C7×Q8⋊C8—22no—
32C7×D4⋊C8—22no—
33#10257C711yesmaximal, characteristic, direct product factor, Sylow
34C2×C7×#41—21no—
35C22×C7×Q8D411yescharacteristic
36C7×#137—42no—
37C7×#137—42no—
38C2×C7×Q16—22no—
39C2×C7×SD16—22no—
40C2×C7×SD16—22no—
41C2×C7×M16—21no—
42C2×C7×M16—21no—
43C22×C7×C8—21no—
44C22×C7×C8D412yes—
45C7×C4⋊Q8D411yescharacteristic
46C7×C4⋊Q8D411yescharacteristic
47C7×C4⋊Q8D411yescharacteristic
48C7×#127D411yescharacteristic
49C7×#125—21no—
50C7×#124—21no—
51C7×C22⋊Q8—21no—
52C7×C22⋊Q8—21no—
53C7×#121—21no—
54C7×#121—21no—
55C4×C7×Q8—21no—
56C4×C7×Q8—21no—
57C4×C7×Q8—21no—
58C4×C7×Q8—21no—
59C4×C7×Q8—21no—
60C4×C7×Q8—21no—
61C4×C7×D4—21no—
62C4×C7×D4—21no—
63C7×#117—21no—
64C2×C7×C4⋊C4—21no—
65C2×C7×C4⋊C4D411yescharacteristic
66C2×C42×C7—21no—
67C2×C42×C7C2311yescharacteristic
68C7×#107—22no—
69C7×#106—22no—
70C7×#106—22no—
71C7×C4⋊C8—21no—
72C7×C4⋊C8—21no—
73C7×C4⋊C8—21no—
74C7×C4⋊C8—21no—
75C7×C4⋊C8D412yes—
76C7×C4⋊C8D412yes—
77C7×#103—22no—
78C7×#103—22no—
79C7×#103—22no—
80C7×#103—22no—
81C7×#102—22no—
82C7×#102—22no—
83C4×C7×C8—21no—
84C4×C7×C8—21no—
85#3246C2×C711yescharacteristic
86#3078C2×C711yescharacteristic, complemented
87#3037C2×C711yescharacteristic
88#1984C2×C711yescharacteristic, complemented
89#1493C2×C711yescharacteristic, complemented
90#1492C2×C711yescharacteristic
91#1292C2×C711yescharacteristic, complemented
92C7×#41—41no—
93C7×#41—41no—
94C7×#41—41no—
95C7×#41—41no—
96C2×C7×Q8—21no—
97C2×C7×Q8—21no—
98C2×C7×Q8—21no—
99C2×C7×Q8—21no—
100C2×C7×Q8—21no—
101C2×C7×Q8—21no—
102C2×C7×Q8—21no—
103C2×C7×Q8—21no—
104C2×C7×Q8SD1612yescomplemented
105C2×C7×D4—21no—
106C2×C7×D4—21no—
107C2×C7×D4—21no—
108C2×C7×D4—41no—
109C22×C4×C7—21no—
110C22×C4×C7—21no—
111C22×C4×C7—41no—
112C22×C4×C7C2×D411yescharacteristic
113C22×C4×C7C2×D411yescharacteristic
114C22×C4×C7#4111yescharacteristic
115C7×Q16—44no—
116C7×SD16—44no—
117C7×SD16—82no—
118C7×M16—41no—
119C7×M16—41no—
120C7×M16—41no—
121C7×M16—41no—
122C2×C7×C8—22no—
123C2×C7×C8—22no—
124C2×C7×C8—22no—
125C2×C7×C8—22no—
126C2×C7×C8—41no—
127C2×C7×C8—41no—
128C2×C7×C8—42no—
129C7×C4⋊C4—21no—
130C7×C4⋊C4—21no—
131C7×C4⋊C4—21no—
132C7×C4⋊C4—21no—
133C7×C4⋊C4—21no—
134C7×C4⋊C4—21no—
135C7×C4⋊C4—21no—
136C7×C4⋊C4—21no—
137C7×C4⋊C4—41no—
138C7×C4⋊C4—41no—
139C7×C4⋊C4—41no—
140C7×C4⋊C4—41no—
141C7×C22⋊C4—41no—
142C7×C22⋊C4—41no—
143C7×C22⋊C4—41no—
144C42×C7—41no—
145C42×C7—41no—
146C42×C7—41no—
147C42×C7C2×D411yescharacteristic
148C42×C7C2×D411yescharacteristic
149C42×C7C2×D411yescharacteristic
150C42×C7C2×D411yescharacteristic
151C2×#137—21no—
152#533C22×C711yescharacteristic
153#532C22×C711yescharacteristic
154C2×C4×Q8C22×C711yescharacteristic
155#479—22no—
156#477—21no—
157#476—21no—
158#475—21no—
159#475—21no—
160#465—22no—
161#464—22no—
162#462—21no—
163#461—21no—
164#460—21no—
165#460—21no—
166#428—21no—
167#423C22×C711yescharacteristic
168#423C22×C711yescharacteristic
169C2×C4⋊C8C22×C711yescharacteristic
170#419—21no—
171#417—21no—
172C2×C4×C8C22×C711yescharacteristic
173#336—22no—
174Q8⋊C8—22no—
175D4⋊C8—22no—
176C23×C7—41no—
177C23×C7#12111yescharacteristic
178C7×Q8—22no—
179C7×Q8—22no—
180C7×Q8—42no—
181C7×Q8—41no—
182C7×Q8—41no—
183C7×Q8—41no—
184C7×Q8—41no—
185C7×Q8—41no—
186C7×Q8—41no—
187C7×D4—42no—
188C7×D4—41no—
189C7×D4—41no—
190C7×D4—41no—
191C7×D4—41no—
192C7×D4—81no—
193C7×D4—81no—
194C2×C4×C7—21no—
195C2×C4×C7—21no—
196C2×C4×C7—22no—
197C2×C4×C7—41no—
198C2×C4×C7—41no—
199C2×C4×C7—41no—
200C2×C4×C7—42no—
201C2×C4×C7—41no—
202C2×C4×C7—41no—
203C2×C4×C7—41no—
204C2×C4×C7—41no—
205C2×C4×C7—81no—
206C2×C4×C7—81no—
207C2×C4×C7#12011yescharacteristic
208C2×C4×C7#12011yescharacteristic
209C2×C4×C7#12011yescharacteristic
210C2×C4×C7#12111yescharacteristic
211C2×C4×C7#12111yescharacteristic
212C2×C4×C7#12711yescharacteristic
213C2×C4×C7C2×SD1611yescharacteristic
214C2×C4×C7#13511yescharacteristic
215C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
216C2×C4×C7#13711yescharacteristic
217C7×C8—42no—
218C7×C8—42no—
219C7×C8—42no—
220C7×C8—42no—
221C2×#41—21no—
222C22×Q8C7×D411yescharacteristic
223#137—42no—
224#137—42no—
225C2×Q16—22no—
226C2×SD16—22no—
227C2×SD16—22no—
228C2×M16—21no—
229C2×M16—21no—
230C22×C8—21no—
231C22×C8C7×D412yes—
232C4⋊Q8C7×D411yescharacteristic
233C4⋊Q8C7×D411yescharacteristic
234C4⋊Q8C7×D411yescharacteristic
235#127C7×D411yescharacteristic
236#125—21no—
237#124—21no—
238C22⋊Q8—21no—
239C22⋊Q8—21no—
240#121—21no—
241#121—21no—
242C4×Q8—21no—
243C4×Q8—21no—
244C4×Q8—21no—
245C4×Q8—21no—
246C4×Q8—21no—
247C4×Q8—21no—
248C4×D4—21no—
249C4×D4—21no—
250#117—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4C7×D411yescharacteristic
253C2×C42—21no—
254C2×C42C23×C711yescharacteristic
255#107—22no—
256#106—22no—
257#106—22no—
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8—21no—
261C4⋊C8—21no—
262C4⋊C8C7×D412yes—
263C4⋊C8C7×D412yes—
264#103—22no—
265#103—22no—
266#103—22no—
267#103—22no—
268#102—22no—
269#102—22no—
270C4×C8—21no—
271C4×C8—21no—
272C22×C7—21no—
273C22×C7—21no—
274C22×C7—81no—
275C22×C7—81no—
276C22×C7—81no—
277C22×C7#39211yescharacteristic
278C22×C7#46511yescharacteristic
279C22×C7#46811yescharacteristic
280C4×C7—21no—
281C4×C7—21no—
282C4×C7—21no—
283C4×C7—42no—
284C4×C7—81no—
285C4×C7—81no—
286C4×C7—81no—
287C4×C7—81no—
288C4×C7#44811yescharacteristic
289C4×C7#44911yescharacteristic
290C4×C7#45011yescharacteristic
291C4×C7#45211yescharacteristic
292C4×C7#45511yescharacteristic
293C4×C7#45611yescharacteristic
294#41—41no—
295#41—41no—
296#41—41no—
297#41—41no—
298C2×Q8—21no—
299C2×Q8—21no—
300C2×Q8—21no—
301C2×Q8—21no—
302C2×Q8—21no—
303C2×Q8—21no—
304C2×Q8—21no—
305C2×Q8—21no—
306C2×Q8C7×SD1612yescomplemented
307C2×D4—21no—
308C2×D4—21no—
309C2×D4—21no—
310C2×D4—41no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—41no—
314C22×C4C2×C7×D411yescharacteristic
315C22×C4C2×C7×D411yescharacteristic
316C22×C4C7×#4111yescharacteristic
317Q16—44no—
318SD16—44no—
319SD16—82no—
320M16—41no—
321M16—41no—
322M16—41no—
323M16—41no—
324C2×C8—22no—
325C2×C8—22no—
326C2×C8—22no—
327C2×C8—22no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—42no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C22⋊C4—41no—
344C22⋊C4—41no—
345C22⋊C4—41no—
346C42—41no—
347C42—41no—
348C42—41no—
349C42C2×C7×D411yescharacteristic
350C42C2×C7×D411yescharacteristic
351C42C2×C7×D411yescharacteristic
352C42C2×C7×D411yescharacteristic
353C2×C7—21no—
354C2×C7—21no—
355C2×C7—161no—
356C2×C7#200011yescharacteristic
357C2×C7#200711yescharacteristic
358C2×C7#201711yescharacteristic
359C23—41no—
360C23C7×#12111yescharacteristic
361Q8—22no—
362Q8—22no—
363Q8—42no—
364Q8—41no—
365Q8—41no—
366Q8—41no—
367Q8—41no—
368Q8—41no—
369Q8—41no—
370D4—42no—
371D4—41no—
372D4—41no—
373D4—41no—
374D4—41no—
375D4—81no—
376D4—81no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—22no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4C7×#12011yescharacteristic
391C2×C4C7×#12011yescharacteristic
392C2×C4C7×#12011yescharacteristic
393C2×C4C7×#12111yescharacteristic
394C2×C4C7×#12111yescharacteristic
395C2×C4C7×#12711yescharacteristic
396C2×C4C2×C7×SD1611yescharacteristic
397C2×C4C7×#13511yescharacteristic
398C2×C4C7×AGL(1,ℤ/8)11yescharacteristic
399C2×C4C7×#13711yescharacteristic
400C8—42no—
401C8—42no—
402C8—42no—
403C8—42no—
404C7#1025711yescharacteristic, direct product factor, Sylow
405C22—21no—
406C22—21no—
407C22—81no—
408C22—81no—
409C22—81no—
410C22C7×#39211yescharacteristic
411C22C7×#46511yescharacteristic
412C22C7×#46811yescharacteristic
413C4—21no—
414C4—21no—
415C4—21no—
416C4—42no—
417C4—81no—
418C4—81no—
419C4—81no—
420C4—81no—
421C4C7×#44811yescharacteristic
422C4C7×#44911yescharacteristic
423C4C7×#45011yescharacteristic
424C4C7×#45211yescharacteristic
425C4C7×#45511yescharacteristic
426C4C7×#45611yescharacteristic
427C2—21no—
428C2—21no—
429C2—161no—
430C2C7×#200011yescharacteristic
431C2C7×#200711yescharacteristic
432C2C7×#201711yescharacteristic
subgroupG/H
center, socleC22×C7#392
commutator, Frattini, cosocleC2×C42C23×C7
prime radicalC7×#532C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#3078C2semidirect
4. C7×#1984C2semidirect
5. C7×#1493C2semidirect
7. C7×#1292C2semidirect
33. #10257C7direct (N × H)
86. #3078C2×C7semidirect
88. #1984C2×C7semidirect
89. #1493C2×C7semidirect
91. #1292C2×C7semidirect
104. C2×C7×Q8SD16semidirect
306. C2×Q8C7×SD16semidirect
404. C7#10257direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#32462
C7×#30782
C7×#30372
C7×#19842
C7×#14932
C7×#14922
C7×#12922
#102577