Finite Group Databasebeta

← C7×#9999 #13187215

Subgroups — C7×#9999 #13187215

Order 1792.

554 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3103C211yesmaximal, characteristic
2C7×#3037C211yesmaximal, characteristic
3C7×#2581C211yesmaximal, characteristic, complemented
4C7×#1517C211yesmaximal, characteristic
5C7×#1487C211yesmaximal, characteristic, complemented
6C7×#1479C211yesmaximal, characteristic, complemented
7C7×#1286C211yesmaximal, characteristic, complemented
8C2×C7×#137—21no—
9C7×#532C2211yescharacteristic
10C2×C4×C7×Q8C2211yescharacteristic
11C2×C4×C7×Q8C2211yescharacteristic
12C7×#481—21no—
13C7×#475—21no—
14C7×#464—22no—
15C7×#462—21no—
16C7×#460—21no—
17C7×#460—21no—
18C7×#448—21no—
19C4×C7×SD16—22no—
20C7×#431C2211yescharacteristic
21C7×#431C2211yescharacteristic
22C7×#423C2211yescharacteristic
23C7×#417—21no—
24C7×#400—21no—
25C7×#395—21no—
26C7×#391—21no—
27C7×#389—21no—
28C7×#387—22no—
29C7×#382C2211yescharacteristic
30C7×#327—22no—
31C7×Q8⋊C8—22no—
32C7×Q8⋊C8—22no—
33#9999C711yesmaximal, characteristic, direct product factor, Sylow
34C2×C7×#41—21no—
35C22×C7×Q8D411yescharacteristic
36C22×C7×Q8D411yescharacteristic
37C7×#137—42no—
38C7×#137—42no—
39C2×C7×Q16—22no—
40C2×C7×SD16—22no—
41C2×C7×SD16—22no—
42C2×C7×M16—21no—
43C7×C4⋊Q8D411yescharacteristic
44C7×#127D411yescharacteristic
45C7×#124—21no—
46C7×#121—21no—
47C7×#121—21no—
48C4×C7×Q8—21no—
49C4×C7×Q8—21no—
50C4×C7×Q8—21no—
51C4×C7×Q8—21no—
52C4×C7×Q8—21no—
53C4×C7×Q8—21no—
54C4×C7×Q8—21no—
55C4×C7×Q8—21no—
56C4×C7×D4—21no—
57C4×C7×D4—21no—
58C2×C7×C4⋊C4—22no—
59C2×C7×C4⋊C4—21no—
60C2×C7×C4⋊C4—21no—
61C2×C7×C4⋊C4—21no—
62C2×C7×C4⋊C4D411yescharacteristic
63C2×C7×C4⋊C4D411yescharacteristic
64C2×C42×C7—21no—
65C2×C42×C7—21no—
66C2×C42×C7C2311yescharacteristic
67C7×#106—22no—
68C7×C4⋊C8—21no—
69C7×C4⋊C8—21no—
70C7×C4⋊C8—21no—
71C7×C4⋊C8D412yes—
72C7×#103—22no—
73C7×#103—22no—
74C7×#102—22no—
75C7×#102—22no—
76C7×C22⋊C8—21no—
77C7×C22⋊C8—21no—
78C4×C7×C8—21no—
79C4×C7×C8D412yes—
80C7×#95—22no—
81C7×#95—22no—
82C7×#95—22no—
83C7×#95—21no—
84C7×#95D412yes—
85#3103C2×C711yescharacteristic
86#3037C2×C711yescharacteristic
87#2581C2×C711yescharacteristic, complemented
88#1517C2×C711yescharacteristic
89#1487C2×C711yescharacteristic, complemented
90#1479C2×C711yescharacteristic, complemented
91#1286C2×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×Q8—42no—
105C2×C7×Q8SD1612yescomplemented
106C2×C7×D4—21no—
107C2×C7×D4—21no—
108C2×C7×D4—21no—
109C2×C7×D4—41no—
110C22×C4×C7—21no—
111C22×C4×C7—21no—
112C22×C4×C7—22no—
113C22×C4×C7—21no—
114C22×C4×C7—21no—
115C22×C4×C7—41no—
116C22×C4×C7C2×D411yescharacteristic
117C22×C4×C7C2×D411yescharacteristic
118C22×C4×C7C2×D411yescharacteristic
119C7×Q16—44no—
120C7×SD16—44no—
121C7×SD16—44no—
122C7×M16—41no—
123C7×M16—41no—
124C2×C7×C8—22no—
125C2×C7×C8—22no—
126C2×C7×C8—41no—
127C7×C4⋊C4—21no—
128C7×C4⋊C4—21no—
129C7×C4⋊C4—21no—
130C7×C4⋊C4—21no—
131C7×C4⋊C4—21no—
132C7×C4⋊C4—21no—
133C7×C4⋊C4—44no—
134C7×C4⋊C4—41no—
135C7×C4⋊C4—41no—
136C7×C4⋊C4—41no—
137C7×C4⋊C4—41no—
138C7×C4⋊C4—41no—
139C7×C4⋊C4—41no—
140C7×C22⋊C4—41no—
141C7×C22⋊C4—41no—
142C42×C7—41no—
143C42×C7—41no—
144C42×C7—41no—
145C42×C7—41no—
146C42×C7C2×D411yescharacteristic
147C42×C7C2×D411yescharacteristic
148C42×C7C2×D411yescharacteristic
149C42×C7#4111yescharacteristic
150C2×#137—21no—
151#532C22×C711yescharacteristic
152C2×C4×Q8C22×C711yescharacteristic
153C2×C4×Q8C22×C711yescharacteristic
154#481—21no—
155#475—21no—
156#464—22no—
157#462—21no—
158#460—21no—
159#460—21no—
160#448—21no—
161C4×SD16—22no—
162#431C22×C711yescharacteristic
163#431C22×C711yescharacteristic
164#423C22×C711yescharacteristic
165#417—21no—
166#400—21no—
167#395—21no—
168#391—21no—
169#389—21no—
170#387—22no—
171#382C22×C711yescharacteristic
172#327—22no—
173Q8⋊C8—22no—
174Q8⋊C8—22no—
175C23×C7—41no—
176C23×C7#12011yescharacteristic
177C7×Q8—22no—
178C7×Q8—22no—
179C7×Q8—22no—
180C7×Q8—22no—
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—81no—
191C7×D4—81no—
192C7×D4—81no—
193C2×C4×C7—22no—
194C2×C4×C7—22no—
195C2×C4×C7—41no—
196C2×C4×C7—41no—
197C2×C4×C7—41no—
198C2×C4×C7—41no—
199C2×C4×C7—41no—
200C2×C4×C7—41no—
201C2×C4×C7—42no—
202C2×C4×C7—42no—
203C2×C4×C7—42no—
204C2×C4×C7—42no—
205C2×C4×C7—41no—
206C2×C4×C7—41no—
207C2×C4×C7—41no—
208C2×C4×C7—42no—
209C2×C4×C7—81no—
210C2×C4×C7—81no—
211C2×C4×C7#12011yescharacteristic
212C2×C4×C7#12011yescharacteristic
213C2×C4×C7#12111yescharacteristic
214C2×C4×C7#12111yescharacteristic
215C2×C4×C7#12111yescharacteristic
216C2×C4×C7#12711yescharacteristic
217C2×C4×C7C2×SD1611yescharacteristic
218C2×C4×C7#13511yescharacteristic
219C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
220C2×C4×C7#13711yescharacteristic
221C7×C8—42no—
222C7×C8—42no—
223C7×C8—81no—
224C2×#41—21no—
225C22×Q8C7×D411yescharacteristic
226C22×Q8C7×D411yescharacteristic
227#137—42no—
228#137—42no—
229C2×Q16—22no—
230C2×SD16—22no—
231C2×SD16—22no—
232C2×M16—21no—
233C4⋊Q8C7×D411yescharacteristic
234#127C7×D411yescharacteristic
235#124—21no—
236#121—21no—
237#121—21no—
238C4×Q8—21no—
239C4×Q8—21no—
240C4×Q8—21no—
241C4×Q8—21no—
242C4×Q8—21no—
243C4×Q8—21no—
244C4×Q8—21no—
245C4×Q8—21no—
246C4×D4—21no—
247C4×D4—21no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4C7×D411yescharacteristic
253C2×C4⋊C4C7×D411yescharacteristic
254C2×C42—21no—
255C2×C42—21no—
256C2×C42C23×C711yescharacteristic
257#106—22no—
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8—21no—
261C4⋊C8C7×D412yes—
262#103—22no—
263#103—22no—
264#102—22no—
265#102—22no—
266C22⋊C8—21no—
267C22⋊C8—21no—
268C4×C8—21no—
269C4×C8C7×D412yes—
270#95—22no—
271#95—22no—
272#95—22no—
273#95—21no—
274#95C7×D412yes—
275C22×C7—21no—
276C22×C7—21no—
277C22×C7—81no—
278C22×C7—81no—
279C22×C7—81no—
280C22×C7#39211yescharacteristic
281C22×C7#44811yescharacteristic
282C22×C7#44911yescharacteristic
283C4×C7—21no—
284C4×C7—21no—
285C4×C7—21no—
286C4×C7—21no—
287C4×C7—42no—
288C4×C7—42no—
289C4×C7—82no—
290C4×C7—81no—
291C4×C7—81no—
292C4×C7—81no—
293C4×C7#45011yescharacteristic
294C4×C7#45211yescharacteristic
295C4×C7#46511yescharacteristic
296C4×C7#46811yescharacteristic
297#41—41no—
298#41—41no—
299#41—41no—
300#41—41no—
301C2×Q8—21no—
302C2×Q8—21no—
303C2×Q8—21no—
304C2×Q8—21no—
305C2×Q8—21no—
306C2×Q8—21no—
307C2×Q8—21no—
308C2×Q8—21no—
309C2×Q8—42no—
310C2×Q8C7×SD1612yescomplemented
311C2×D4—21no—
312C2×D4—21no—
313C2×D4—21no—
314C2×D4—41no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—41no—
321C22×C4C2×C7×D411yescharacteristic
322C22×C4C2×C7×D411yescharacteristic
323C22×C4C2×C7×D411yescharacteristic
324Q16—44no—
325SD16—44no—
326SD16—44no—
327M16—41no—
328M16—41no—
329C2×C8—22no—
330C2×C8—22no—
331C2×C8—41no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—44no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C22⋊C4—41no—
346C22⋊C4—41no—
347C42—41no—
348C42—41no—
349C42—41no—
350C42—41no—
351C42C2×C7×D411yescharacteristic
352C42C2×C7×D411yescharacteristic
353C42C2×C7×D411yescharacteristic
354C42C7×#4111yescharacteristic
355C2×C7—21no—
356C2×C7—21no—
357C2×C7—161no—
358C2×C7#200011yescharacteristic
359C2×C7#200911yescharacteristic
360C2×C7#201011yescharacteristic
361C23—41no—
362C23C7×#12011yescharacteristic
363Q8—22no—
364Q8—22no—
365Q8—22no—
366Q8—22no—
367Q8—41no—
368Q8—41no—
369Q8—41no—
370Q8—41no—
371Q8—41no—
372Q8—41no—
373D4—42no—
374D4—41no—
375D4—41no—
376D4—81no—
377D4—81no—
378D4—81no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—42no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4C7×#12011yescharacteristic
398C2×C4C7×#12011yescharacteristic
399C2×C4C7×#12111yescharacteristic
400C2×C4C7×#12111yescharacteristic
401C2×C4C7×#12111yescharacteristic
402C2×C4C7×#12711yescharacteristic
403C2×C4C2×C7×SD1611yescharacteristic
404C2×C4C7×#13511yescharacteristic
405C2×C4C7×AGL(1,ℤ/8)11yescharacteristic
406C2×C4C7×#13711yescharacteristic
407C8—42no—
408C8—42no—
409C8—81no—
410C7#999911yescharacteristic, direct product factor, Sylow
411C22—21no—
412C22—21no—
413C22—81no—
414C22—81no—
415C22—81no—
416C22C7×#39211yescharacteristic
417C22C7×#44811yescharacteristic
418C22C7×#44911yescharacteristic
419C4—21no—
420C4—21no—
421C4—21no—
422C4—21no—
423C4—42no—
424C4—42no—
425C4—82no—
426C4—81no—
427C4—81no—
428C4—81no—
429C4C7×#45011yescharacteristic
430C4C7×#45211yescharacteristic
431C4C7×#46511yescharacteristic
432C4C7×#46811yescharacteristic
433C2—21no—
434C2—21no—
435C2—161no—
436C2C7×#200011yescharacteristic
437C2C7×#200911yescharacteristic
438C2C7×#201011yescharacteristic
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
3. C7×#2581C2semidirect
5. C7×#1487C2semidirect
6. C7×#1479C2semidirect
7. C7×#1286C2semidirect
33. #9999C7direct (N × H)
87. #2581C2×C7semidirect
89. #1487C2×C7semidirect
90. #1479C2×C7semidirect
91. #1286C2×C7semidirect
105. C2×C7×Q8SD16semidirect
310. C2×Q8C7×SD16semidirect
410. C7#9999direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#31032
C7×#30372
C7×#25812
C7×#15172
C7×#14872
C7×#14792
C7×#12862
#99997