Finite Group Databasebeta

← C7×#10001 #13187217

Subgroups — C7×#10001 #13187217

Order 1792.

568 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3101C211yesmaximal, characteristic
2C7×#3078C211yesmaximal, characteristic, complemented
3C7×#2536C211yesmaximal, characteristic
4C7×#1518C211yesmaximal, characteristic, complemented
5C7×#1486C211yesmaximal, characteristic
6C7×#1479C211yesmaximal, characteristic, complemented
7C7×#1286C211yesmaximal, characteristic, complemented
8C7×#533C2211yescharacteristic
9C2×C7×C22⋊Q8—21no—
10C2×C4×C7×Q8C2211yescharacteristic
11C2×C4×C7×D4C2211yescharacteristic
12C7×#483—21no—
13C7×#479—22no—
14C7×#477—21no—
15C7×#476—21no—
16C7×#475—21no—
17C7×#461—21no—
18C7×#450—21no—
19C4×C7×SD16—22no—
20C7×#431C2211yescharacteristic
21C7×#431C2211yescharacteristic
22C7×#428—21no—
23C7×#423C2211yescharacteristic
24C7×#419—21no—
25C7×#398—21no—
26C7×#394—21no—
27C7×#388—22no—
28C7×#382C2211yescharacteristic
29C7×#380—21no—
30C7×#328—22no—
31C7×Q8⋊C8—22no—
32C7×D4⋊C8—22no—
33#10001C711yesmaximal, characteristic, direct product factor, Sylow
34C22×C7×Q8D411yescharacteristic
35C22×C7×D4D411yescharacteristic
36C23×C4×C7—21no—
37C2×C7×SD16—22no—
38C2×C7×M16—21no—
39C7×C4⋊Q8D411yescharacteristic
40C7×C4⋊Q8D411yescharacteristic
41C7×#125—21no—
42C7×C22⋊Q8—21no—
43C7×C22⋊Q8—21no—
44C7×C22⋊Q8—42no—
45C7×C22⋊Q8—42no—
46C4×C7×Q8—21no—
47C4×C7×Q8—21no—
48C4×C7×Q8—21no—
49C4×C7×Q8—21no—
50C4×C7×Q8—21no—
51C4×C7×Q8—21no—
52C4×C7×D4—21no—
53C4×C7×D4—21no—
54C4×C7×D4—21no—
55C4×C7×D4—21no—
56C7×#117—21no—
57C2×C7×C4⋊C4—22no—
58C2×C7×C4⋊C4—21no—
59C2×C7×C4⋊C4—21no—
60C2×C7×C4⋊C4D411yescharacteristic
61C2×C7×C4⋊C4D411yescharacteristic
62C2×C7×C22⋊C4—22no—
63C2×C7×C22⋊C4—22no—
64C2×C7×C22⋊C4—21no—
65C2×C42×C7—21no—
66C2×C42×C7C2311yescharacteristic
67C7×#107—22no—
68C7×#106—22no—
69C7×#106—22no—
70C7×C4⋊C8—21no—
71C7×C4⋊C8—21no—
72C7×C4⋊C8—21no—
73C7×C4⋊C8D412yes—
74C7×#103—22no—
75C7×#103—22no—
76C7×#103—22no—
77C7×#102—22no—
78C7×C22⋊C8—21no—
79C7×C22⋊C8—21no—
80C4×C7×C8—21no—
81C4×C7×C8D412yes—
82C7×#95—22no—
83C7×#95—21no—
84C7×#95D412yes—
85#3101C2×C711yescharacteristic
86#3078C2×C711yescharacteristic, complemented
87#2536C2×C711yescharacteristic
88#1518C2×C711yescharacteristic, complemented
89#1486C2×C711yescharacteristic
90#1479C2×C711yescharacteristic, complemented
91#1286C2×C711yescharacteristic, complemented
92C24×C7—21no—
93C2×C7×Q8—21no—
94C2×C7×Q8—21no—
95C2×C7×Q8—21no—
96C2×C7×Q8—21no—
97C2×C7×Q8—21no—
98C2×C7×Q8—21no—
99C2×C7×Q8—21no—
100C2×C7×Q8SD1612yescomplemented
101C2×C7×D4—21no—
102C2×C7×D4—21no—
103C2×C7×D4—42no—
104C22×C4×C7—21no—
105C22×C4×C7—22no—
106C22×C4×C7—21no—
107C22×C4×C7—21no—
108C22×C4×C7—41no—
109C22×C4×C7—41no—
110C22×C4×C7—41no—
111C22×C4×C7—41no—
112C22×C4×C7—41no—
113C22×C4×C7—41no—
114C22×C4×C7C2×D411yescharacteristic
115C22×C4×C7C2×D411yescharacteristic
116C22×C4×C7C2×D411yescharacteristic
117C7×SD16—44no—
118C7×M16—41no—
119C7×M16—41no—
120C2×C7×C8—22no—
121C2×C7×C8—22no—
122C2×C7×C8—41no—
123C7×C4⋊C4—21no—
124C7×C4⋊C4—21no—
125C7×C4⋊C4—21no—
126C7×C4⋊C4—21no—
127C7×C4⋊C4—21no—
128C7×C4⋊C4—21no—
129C7×C4⋊C4—21no—
130C7×C4⋊C4—21no—
131C7×C4⋊C4—44no—
132C7×C4⋊C4—41no—
133C7×C4⋊C4—41no—
134C7×C4⋊C4—41no—
135C7×C4⋊C4—41no—
136C7×C4⋊C4—41no—
137C7×C4⋊C4—41no—
138C7×C22⋊C4—44no—
139C7×C22⋊C4—44no—
140C7×C22⋊C4—41no—
141C7×C22⋊C4—41no—
142C7×C22⋊C4—41no—
143C42×C7—41no—
144C42×C7—41no—
145C42×C7—41no—
146C42×C7C2×D411yescharacteristic
147C42×C7C2×D411yescharacteristic
148C42×C7C2×D411yescharacteristic
149C42×C7#4111yescharacteristic
150#533C22×C711yescharacteristic
151C2×C22⋊Q8—21no—
152C2×C4×Q8C22×C711yescharacteristic
153C2×C4×D4C22×C711yescharacteristic
154#483—21no—
155#479—22no—
156#477—21no—
157#476—21no—
158#475—21no—
159#461—21no—
160#450—21no—
161C4×SD16—22no—
162#431C22×C711yescharacteristic
163#431C22×C711yescharacteristic
164#428—21no—
165#423C22×C711yescharacteristic
166#419—21no—
167#398—21no—
168#394—21no—
169#388—22no—
170#382C22×C711yescharacteristic
171#380—21no—
172#328—22no—
173Q8⋊C8—22no—
174D4⋊C8—22no—
175C23×C7—22no—
176C23×C7—41no—
177C23×C7—41no—
178C23×C7—41no—
179C23×C7—41no—
180C23×C7—41no—
181C23×C7—41no—
182C23×C7#12011yescharacteristic
183C7×Q8—22no—
184C7×Q8—22no—
185C7×Q8—42no—
186C7×Q8—41no—
187C7×Q8—41no—
188C7×Q8—81no—
189C7×D4—22no—
190C7×D4—22no—
191C7×D4—41no—
192C7×D4—41no—
193C2×C4×C7—22no—
194C2×C4×C7—41no—
195C2×C4×C7—41no—
196C2×C4×C7—41no—
197C2×C4×C7—41no—
198C2×C4×C7—42no—
199C2×C4×C7—42no—
200C2×C4×C7—42no—
201C2×C4×C7—41no—
202C2×C4×C7—41no—
203C2×C4×C7—41no—
204C2×C4×C7—41no—
205C2×C4×C7—42no—
206C2×C4×C7—41no—
207C2×C4×C7—41no—
208C2×C4×C7—81no—
209C2×C4×C7—81no—
210C2×C4×C7—81no—
211C2×C4×C7—81no—
212C2×C4×C7#12011yescharacteristic
213C2×C4×C7#12011yescharacteristic
214C2×C4×C7#12111yescharacteristic
215C2×C4×C7#12111yescharacteristic
216C2×C4×C7#12111yescharacteristic
217C2×C4×C7#12711yescharacteristic
218C2×C4×C7C2×SD1611yescharacteristic
219C2×C4×C7#13511yescharacteristic
220C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
221C2×C4×C7#13711yescharacteristic
222C7×C8—42no—
223C7×C8—42no—
224C7×C8—81no—
225C22×Q8C7×D411yescharacteristic
226C22×D4C7×D411yescharacteristic
227C23×C4—21no—
228C2×SD16—22no—
229C2×M16—21no—
230C4⋊Q8C7×D411yescharacteristic
231C4⋊Q8C7×D411yescharacteristic
232#125—21no—
233C22⋊Q8—21no—
234C22⋊Q8—21no—
235C22⋊Q8—42no—
236C22⋊Q8—42no—
237C4×Q8—21no—
238C4×Q8—21no—
239C4×Q8—21no—
240C4×Q8—21no—
241C4×Q8—21no—
242C4×Q8—21no—
243C4×D4—21no—
244C4×D4—21no—
245C4×D4—21no—
246C4×D4—21no—
247#117—21no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4C7×D411yescharacteristic
252C2×C4⋊C4C7×D411yescharacteristic
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—21no—
256C2×C42—21no—
257C2×C42C23×C711yescharacteristic
258#107—22no—
259#106—22no—
260#106—22no—
261C4⋊C8—21no—
262C4⋊C8—21no—
263C4⋊C8—21no—
264C4⋊C8C7×D412yes—
265#103—22no—
266#103—22no—
267#103—22no—
268#102—22no—
269C22⋊C8—21no—
270C22⋊C8—21no—
271C4×C8—21no—
272C4×C8C7×D412yes—
273#95—22no—
274#95—21no—
275#95C7×D412yes—
276C22×C7—21no—
277C22×C7—21no—
278C22×C7—42no—
279C22×C7—42no—
280C22×C7—42no—
281C22×C7—81no—
282C22×C7—81no—
283C22×C7—81no—
284C22×C7—81no—
285C22×C7#39211yescharacteristic
286C22×C7#45011yescharacteristic
287C22×C7#45211yescharacteristic
288C4×C7—21no—
289C4×C7—21no—
290C4×C7—21no—
291C4×C7—21no—
292C4×C7—42no—
293C4×C7—82no—
294C4×C7—81no—
295C4×C7—81no—
296C4×C7—81no—
297C4×C7—81no—
298C4×C7#44811yescharacteristic
299C4×C7#44911yescharacteristic
300C4×C7#46511yescharacteristic
301C4×C7#46811yescharacteristic
302C24—21no—
303C2×Q8—21no—
304C2×Q8—21no—
305C2×Q8—21no—
306C2×Q8—21no—
307C2×Q8—21no—
308C2×Q8—21no—
309C2×Q8—21no—
310C2×Q8C7×SD1612yescomplemented
311C2×D4—21no—
312C2×D4—21no—
313C2×D4—42no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4C2×C7×D411yescharacteristic
325C22×C4C2×C7×D411yescharacteristic
326C22×C4C2×C7×D411yescharacteristic
327SD16—44no—
328M16—41no—
329M16—41no—
330C2×C8—22no—
331C2×C8—22no—
332C2×C8—41no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—44no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C22⋊C4—44no—
349C22⋊C4—44no—
350C22⋊C4—41no—
351C22⋊C4—41no—
352C22⋊C4—41no—
353C42—41no—
354C42—41no—
355C42—41no—
356C42C2×C7×D411yescharacteristic
357C42C2×C7×D411yescharacteristic
358C42C2×C7×D411yescharacteristic
359C42C7×#4111yescharacteristic
360C2×C7—21no—
361C2×C7—21no—
362C2×C7—82no—
363C2×C7#200011yescharacteristic
364C2×C7#200911yescharacteristic
365C2×C7#201011yescharacteristic
366C23—22no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23—41no—
371C23—41no—
372C23—41no—
373C23C7×#12011yescharacteristic
374Q8—22no—
375Q8—22no—
376Q8—42no—
377Q8—41no—
378Q8—41no—
379Q8—81no—
380D4—22no—
381D4—22no—
382D4—41no—
383D4—41no—
384C2×C4—22no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—42no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4C7×#12011yescharacteristic
404C2×C4C7×#12011yescharacteristic
405C2×C4C7×#12111yescharacteristic
406C2×C4C7×#12111yescharacteristic
407C2×C4C7×#12111yescharacteristic
408C2×C4C7×#12711yescharacteristic
409C2×C4C2×C7×SD1611yescharacteristic
410C2×C4C7×#13511yescharacteristic
411C2×C4C7×AGL(1,ℤ/8)11yescharacteristic
412C2×C4C7×#13711yescharacteristic
413C8—42no—
414C8—42no—
415C8—81no—
416C7#1000111yescharacteristic, direct product factor, Sylow
417C22—21no—
418C22—21no—
419C22—42no—
420C22—42no—
421C22—42no—
422C22—81no—
423C22—81no—
424C22—81no—
425C22—81no—
426C22C7×#39211yescharacteristic
427C22C7×#45011yescharacteristic
428C22C7×#45211yescharacteristic
429C4—21no—
430C4—21no—
431C4—21no—
432C4—21no—
433C4—42no—
434C4—82no—
435C4—81no—
436C4—81no—
437C4—81no—
438C4—81no—
439C4C7×#44811yescharacteristic
440C4C7×#44911yescharacteristic
441C4C7×#46511yescharacteristic
442C4C7×#46811yescharacteristic
443C2—21no—
444C2—21no—
445C2—82no—
446C2C7×#200011yescharacteristic
447C2C7×#200911yescharacteristic
448C2C7×#201011yescharacteristic
subgroupG/H
center, socleC22×C7#392
commutator, Frattini, cosocleC2×C42C23×C7
prime radicalC22×C7×D4D4
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×#1518C2semidirect
6. C7×#1479C2semidirect
7. C7×#1286C2semidirect
33. #10001C7direct (N × H)
86. #3078C2×C7semidirect
88. #1518C2×C7semidirect
90. #1479C2×C7semidirect
91. #1286C2×C7semidirect
100. C2×C7×Q8SD16semidirect
310. C2×Q8C7×SD16semidirect
416. C7#10001direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#31012
C7×#30782
C7×#25362
C7×#15182
C7×#14862
C7×#14792
C7×#12862
#100017