Finite Group Databasebeta

← C7×#10412 #13187628

Subgroups — C7×#10412 #13187628

Order 1792.

516 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2563C211yesmaximal, characteristic
2C7×#1956C211yesmaximal, characteristic, complemented
3C7×#1948C211yesmaximal, characteristic, complemented
4C7×#1802C211yesmaximal, characteristic
5C2×C7×#333C211yesmaximal, characteristic, complemented
6C2×C7×D4⋊C8C211yesmaximal, characteristic
7C7×#1279C211yesmaximal, characteristic, complemented
8C2×C7×C4⋊Q8C2211yescharacteristic
9C2×C4×C7×D4C2211yescharacteristic
10C2×C7×#106—21no—
11C2×C7×C4⋊C8C2211yescharacteristic
12C2×C7×C4⋊C8C2211yescharacteristic
13C2×C7×#103—22no—
14C2×C7×#102—21no—
15C2×C4×C7×C8C2211yescharacteristic
16C7×#393—22no—
17C7×#390—21no—
18C7×#384C2211yescharacteristic
19C7×#382—21no—
20C7×#380—21no—
21C7×#380—21no—
22C4×C7×C4⋊C4C2211yescharacteristic
23C7×#340—21no—
24C7×#340—21no—
25C7×#333—22no—
26C7×D4⋊C8—22no—
27#10412C711yesmaximal, characteristic, direct product factor, Sylow
28C22×C7×Q8D412yes—
29C22×C7×D4D411yescharacteristic
30C23×C4×C7—21no—
31C22×C7×C8—21no—
32C22×C7×C8—21no—
33C22×C7×C8—21no—
34C7×C4⋊Q8—21no—
35C7×C4⋊Q8—21no—
36C7×C4⋊Q8—21no—
37C7×C4⋊Q8—21no—
38C4×C7×D4—21no—
39C4×C7×D4—21no—
40C4×C7×D4—21no—
41C4×C7×D4—21no—
42C2×C7×C4⋊C4—21no—
43C2×C7×C4⋊C4—21no—
44C2×C7×C4⋊C4—21no—
45C2×C7×C4⋊C4—21no—
46C2×C7×C4⋊C4—21no—
47C2×C7×C4⋊C4D411yescharacteristic
48C2×C7×C4⋊C4D411yescharacteristic
49C2×C7×C4⋊C4D411yescharacteristic
50C2×C7×C22⋊C4—21no—
51C2×C7×C22⋊C4—22no—
52C2×C7×C22⋊C4—22no—
53C2×C42×C7—21no—
54C2×C42×C7C2311yescharacteristic
55C7×#106—42no—
56C7×C4⋊C8—21no—
57C7×C4⋊C8—21no—
58C7×C4⋊C8—21no—
59C7×C4⋊C8D412yes—
60C7×#103—44no—
61C7×#102—42no—
62C4×C7×C8—21no—
63C4×C7×C8—21no—
64C7×#95—22no—
65C7×#95—22no—
66C7×#95—21no—
67#2563C2×C711yescharacteristic
68#1956C2×C711yescharacteristic, complemented
69#1948C2×C711yescharacteristic, complemented
70#1802C2×C711yescharacteristic
71C2×#333C2×C711yescharacteristic, complemented
72C2×D4⋊C8C2×C711yescharacteristic
73#1279C2×C711yescharacteristic, complemented
74C24×C7—21no—
75C2×C7×Q8—22no—
76C2×C7×Q8—22no—
77C2×C7×Q8—42no—
78C2×C7×Q8—42no—
79C2×C7×D4—21no—
80C2×C7×D4—21no—
81C2×C7×D4—21no—
82C2×C7×D4—21no—
83C2×C7×D4—21no—
84C2×C7×D4SD1612yes—
85C22×C4×C7—21no—
86C22×C4×C7—21no—
87C22×C4×C7—21no—
88C22×C4×C7—22no—
89C22×C4×C7—41no—
90C22×C4×C7—41no—
91C22×C4×C7—41no—
92C22×C4×C7—41no—
93C22×C4×C7—41no—
94C22×C4×C7—41no—
95C22×C4×C7—41no—
96C22×C4×C7C2×D411yescharacteristic
97C22×C4×C7C2×D411yescharacteristic
98C22×C4×C7C2×D411yescharacteristic
99C2×C7×C8—22no—
100C2×C7×C8—22no—
101C2×C7×C8—41no—
102C2×C7×C8—41no—
103C2×C7×C8—41no—
104C2×C7×C8—41no—
105C2×C7×C8—41no—
106C2×C7×C8—41no—
107C2×C7×C8—41no—
108C7×C4⋊C4—21no—
109C7×C4⋊C4—21no—
110C7×C4⋊C4—21no—
111C7×C4⋊C4—21no—
112C7×C4⋊C4—21no—
113C7×C4⋊C4—21no—
114C7×C4⋊C4—42no—
115C7×C4⋊C4—41no—
116C7×C4⋊C4—41no—
117C7×C4⋊C4—41no—
118C7×C4⋊C4—41no—
119C7×C4⋊C4—41no—
120C7×C4⋊C4—41no—
121C7×C4⋊C4—41no—
122C7×C4⋊C4—41no—
123C7×C22⋊C4—41no—
124C7×C22⋊C4—41no—
125C7×C22⋊C4—82no—
126C7×C22⋊C4—82no—
127C42×C7—41no—
128C42×C7—41no—
129C42×C7C2×D411yescharacteristic
130C42×C7#4111yescharacteristic
131C42×C7#4111yescharacteristic
132C42×C7#4111yescharacteristic
133C2×C4⋊Q8C22×C711yescharacteristic
134C2×C4×D4C22×C711yescharacteristic
135C2×#106—21no—
136C2×C4⋊C8C22×C711yescharacteristic
137C2×C4⋊C8C22×C711yescharacteristic
138C2×#103—22no—
139C2×#102—21no—
140C2×C4×C8C22×C711yescharacteristic
141#393—22no—
142#390—21no—
143#384C22×C711yescharacteristic
144#382—21no—
145#380—21no—
146#380—21no—
147C4×C4⋊C4C22×C711yescharacteristic
148#340—21no—
149#340—21no—
150#333—22no—
151D4⋊C8—22no—
152C23×C7—22no—
153C23×C7—41no—
154C23×C7—41no—
155C23×C7—41no—
156C23×C7—41no—
157C23×C7—41no—
158C23×C7—41no—
159C23×C7#12011yescharacteristic
160C7×Q8—44no—
161C7×Q8—82no—
162C7×D4—22no—
163C7×D4—22no—
164C7×D4—41no—
165C7×D4—41no—
166C2×C4×C7—21no—
167C2×C4×C7—21no—
168C2×C4×C7—41no—
169C2×C4×C7—42no—
170C2×C4×C7—41no—
171C2×C4×C7—41no—
172C2×C4×C7—41no—
173C2×C4×C7—41no—
174C2×C4×C7—41no—
175C2×C4×C7—41no—
176C2×C4×C7—41no—
177C2×C4×C7—42no—
178C2×C4×C7—41no—
179C2×C4×C7—42no—
180C2×C4×C7—81no—
181C2×C4×C7—81no—
182C2×C4×C7—81no—
183C2×C4×C7—81no—
184C2×C4×C7—81no—
185C2×C4×C7—81no—
186C2×C4×C7—81no—
187C2×C4×C7#12111yescharacteristic
188C2×C4×C7#12111yescharacteristic
189C2×C4×C7#12111yescharacteristic
190C2×C4×C7#12311yescharacteristic
191C2×C4×C7#12311yescharacteristic
192C2×C4×C7#12411yescharacteristic
193C2×C4×C7C2×SD1611yescharacteristic
194C2×C4×C7#13511yescharacteristic
195C2×C4×C7#13511yescharacteristic
196C2×C4×C7#13511yescharacteristic
197C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
198C2×C4×C7#13711yescharacteristic
199C2×C4×C7#13711yescharacteristic
200C2×C4×C7#13711yescharacteristic
201C7×C8—42no—
202C7×C8—42no—
203C7×C8—81no—
204C22×Q8C7×D412yes—
205C22×D4C7×D411yescharacteristic
206C23×C4—21no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
210C4⋊Q8—21no—
211C4⋊Q8—21no—
212C4⋊Q8—21no—
213C4⋊Q8—21no—
214C4×D4—21no—
215C4×D4—21no—
216C4×D4—21no—
217C4×D4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4C7×D411yescharacteristic
224C2×C4⋊C4C7×D411yescharacteristic
225C2×C4⋊C4C7×D411yescharacteristic
226C2×C22⋊C4—21no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C42—21no—
230C2×C42C23×C711yescharacteristic
231#106—42no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8C7×D412yes—
236#103—44no—
237#102—42no—
238C4×C8—21no—
239C4×C8—21no—
240#95—22no—
241#95—22no—
242#95—21no—
243C22×C7—42no—
244C22×C7—42no—
245C22×C7—42no—
246C22×C7—81no—
247C22×C7—81no—
248C22×C7—81no—
249C22×C7—81no—
250C22×C7#39411yescharacteristic
251C22×C7#44811yescharacteristic
252C22×C7#44911yescharacteristic
253C22×C7#45211yescharacteristic
254C22×C7#45211yescharacteristic
255C22×C7#45311yescharacteristic
256C22×C7#45511yescharacteristic
257C4×C7—21no—
258C4×C7—21no—
259C4×C7—81no—
260C4×C7—81no—
261C4×C7—82no—
262C4×C7—81no—
263C4×C7—161no—
264C4×C7#46311yescharacteristic
265C4×C7#46411yescharacteristic
266C4×C7#46511yescharacteristic
267C4×C7#47011yescharacteristic
268C4×C7#48311yescharacteristic
269C4×C7#48511yescharacteristic
270C4×C7#48811yescharacteristic
271C4×C7#48911yescharacteristic
272C24—21no—
273C2×Q8—22no—
274C2×Q8—22no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×D4—21no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4C7×SD1612yes—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—22no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4C2×C7×D411yescharacteristic
295C22×C4C2×C7×D411yescharacteristic
296C22×C4C2×C7×D411yescharacteristic
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—41no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—42no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C22⋊C4—41no—
322C22⋊C4—41no—
323C22⋊C4—82no—
324C22⋊C4—82no—
325C42—41no—
326C42—41no—
327C42C2×C7×D411yescharacteristic
328C42C7×#4111yescharacteristic
329C42C7×#4111yescharacteristic
330C42C7×#4111yescharacteristic
331C2×C7—82no—
332C2×C7#162211yescharacteristic
333C2×C7#163411yescharacteristic
334C2×C7#164311yescharacteristic
335C2×C7#164611yescharacteristic
336C2×C7#203511yescharacteristic
337C2×C7#203711yescharacteristic
338C2×C7#204211yescharacteristic
339C23—22no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23C7×#12011yescharacteristic
347Q8—44no—
348Q8—82no—
349D4—22no—
350D4—22no—
351D4—41no—
352D4—41no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—42no—
365C2×C4—41no—
366C2×C4—42no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4C7×#12111yescharacteristic
375C2×C4C7×#12111yescharacteristic
376C2×C4C7×#12111yescharacteristic
377C2×C4C7×#12311yescharacteristic
378C2×C4C7×#12311yescharacteristic
379C2×C4C7×#12411yescharacteristic
380C2×C4C2×C7×SD1611yescharacteristic
381C2×C4C7×#13511yescharacteristic
382C2×C4C7×#13511yescharacteristic
383C2×C4C7×#13511yescharacteristic
384C2×C4C7×AGL(1,ℤ/8)11yescharacteristic
385C2×C4C7×#13711yescharacteristic
386C2×C4C7×#13711yescharacteristic
387C2×C4C7×#13711yescharacteristic
388C8—42no—
389C8—42no—
390C8—81no—
391C7#1041211yescharacteristic, direct product factor, Sylow
392C22—42no—
393C22—42no—
394C22—42no—
395C22—81no—
396C22—81no—
397C22—81no—
398C22—81no—
399C22C7×#39411yescharacteristic
400C22C7×#44811yescharacteristic
401C22C7×#44911yescharacteristic
402C22C7×#45211yescharacteristic
403C22C7×#45211yescharacteristic
404C22C7×#45311yescharacteristic
405C22C7×#45511yescharacteristic
406C4—21no—
407C4—21no—
408C4—82no—
409C4—81no—
410C4—81no—
411C4—81no—
412C4—161no—
413C4C7×#46311yescharacteristic
414C4C7×#46411yescharacteristic
415C4C7×#46511yescharacteristic
416C4C7×#47011yescharacteristic
417C4C7×#48311yescharacteristic
418C4C7×#48511yescharacteristic
419C4C7×#48811yescharacteristic
420C4C7×#48911yescharacteristic
421C2—82no—
422C2C7×#162211yescharacteristic
423C2C7×#163411yescharacteristic
424C2C7×#164311yescharacteristic
425C2C7×#164611yescharacteristic
426C2C7×#203511yescharacteristic
427C2C7×#203711yescharacteristic
428C2C7×#204211yescharacteristic
subgroupG/H
center, socleC23×C7#120
commutator, Frattini, cosocleC2×C42C23×C7
prime radicalC22×C7×D4D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23×C7 ◁ C2×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#1956C2semidirect
3. C7×#1948C2semidirect
5. C2×C7×#333C2semidirect
7. C7×#1279C2semidirect
27. #10412C7direct (N × H)
68. #1956C2×C7semidirect
69. #1948C2×C7semidirect
71. C2×#333C2×C7semidirect
73. #1279C2×C7semidirect
391. C7#10412direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#25632
C7×#19562
C7×#19482
C7×#18022
C2×C7×#3332
C2×C7×D4⋊C82
C7×#12792
#104127