Finite Group Databasebeta

← C7×#10188 #13187404

Subgroups — C7×#10188 #13187404

Order 1792.

550 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3227C211yesmaximal, characteristic, complemented
2C2×C7×#474C211yesmaximal, characteristic, complemented
3C7×#3036C211yesmaximal, characteristic
4C7×#1548C211yesmaximal, characteristic, complemented
5C7×#1480C211yesmaximal, characteristic, complemented
6C7×#1460C211yesmaximal, characteristic, complemented
7C7×#1301C211yesmaximal, characteristic, complemented
8C2×C7×AGL(1,ℤ/8)—21no—
9C7×#532C2211yescharacteristic
10C2×C7×C4⋊Q8C2211yescharacteristic
11C2×C4×C7×D4C2211yescharacteristic
12C7×#498—21no—
13C7×#486—21no—
14C7×#481—22no—
15C7×#476—22no—
16C7×#474—22no—
17C7×#474—22no—
18C7×#463—22no—
19C7×#461—21no—
20C7×#459—21no—
21C7×#431C2211yescharacteristic
22C7×#431C2211yescharacteristic
23C2×C7×#107—21no—
24C7×#423C2211yescharacteristic
25C2×C7×C4⋊C8C2211yescharacteristic
26C7×#417—21no—
27C2×C7×#102—21no—
28C7×#333—22no—
29C7×D4⋊C8—22no—
30C7×#323—22no—
31#10188C711yesmaximal, characteristic, direct product factor, Sylow
32C2×C7×#41—21no—
33C22×C7×Q8—21no—
34C22×C7×D4D411yescharacteristic
35C23×C4×C7—21no—
36C7×AGL(1,ℤ/8)—42no—
37C7×AGL(1,ℤ/8)—42no—
38C2×C7×SD16—22no—
39C2×C7×D8—22no—
40C2×C7×M16—21no—
41C22×C7×C8—21no—
42C7×C4⋊Q8—21no—
43C7×C4⋊Q8—21no—
44C7×C4⋊Q8—21no—
45C7×C4⋊Q8D412yes—
46C7×C4⋊Q8D411yescharacteristic
47C7×#127D411yescharacteristic
48C7×#124—21no—
49C7×#121—21no—
50C7×#121—21no—
51C4×C7×D4—21no—
52C4×C7×D4—21no—
53C4×C7×D4—21no—
54C4×C7×D4—21no—
55C4×C7×D4—21no—
56C4×C7×D4—21no—
57C2×C7×C4⋊C4—21no—
58C2×C7×C4⋊C4—21no—
59C2×C7×C4⋊C4D411yescharacteristic
60C2×C7×C22⋊C4—21no—
61C2×C42×C7C2311yescharacteristic
62C7×#107—24no—
63C7×#106—24no—
64C7×C4⋊C8—21no—
65C7×C4⋊C8—21no—
66C7×C4⋊C8—21no—
67C7×C4⋊C8—21no—
68C7×C4⋊C8D412yes—
69C7×C4⋊C8D412yes—
70C7×#103—22no—
71C7×#102—24no—
72C7×#102—24no—
73C7×#102—22no—
74C7×C22⋊C8—21no—
75C7×C22⋊C8D412yes—
76C4×C7×C8—21no—
77C4×C7×C8—21no—
78#3227C2×C711yescharacteristic, complemented
79C2×#474C2×C711yescharacteristic, complemented
80#3036C2×C711yescharacteristic
81#1548C2×C711yescharacteristic, complemented
82#1480C2×C711yescharacteristic, complemented
83#1460C2×C711yescharacteristic, complemented
84#1301C2×C711yescharacteristic, complemented
85C24×C7—21no—
86C7×#41—41no—
87C7×#41—41no—
88C7×#41—41no—
89C7×#41—41no—
90C2×C7×Q8—21no—
91C2×C7×Q8—22no—
92C2×C7×Q8—41no—
93C2×C7×Q8—82no—
94C2×C7×D4—21no—
95C2×C7×D4—21no—
96C2×C7×D4—21no—
97C2×C7×D4—21no—
98C2×C7×D4—21no—
99C2×C7×D4—21no—
100C2×C7×D4—21no—
101C2×C7×D4—21no—
102C2×C7×D4—41no—
103C2×C7×D4SD1612yes—
104C22×C4×C7—21no—
105C22×C4×C7—41no—
106C22×C4×C7—41no—
107C22×C4×C7—41no—
108C22×C4×C7—41no—
109C22×C4×C7—82no—
110C22×C4×C7C2×D411yescharacteristic
111C22×C4×C7C2×D411yescharacteristic
112C22×C4×C7#4111yescharacteristic
113C7×SD16—44no—
114C7×D8—44no—
115C7×M16—41no—
116C7×M16—41no—
117C2×C7×C8—22no—
118C2×C7×C8—22no—
119C2×C7×C8—22no—
120C2×C7×C8—41no—
121C2×C7×C8—41no—
122C2×C7×C8—41no—
123C7×C4⋊C4—21no—
124C7×C4⋊C4—22no—
125C7×C4⋊C4—22no—
126C7×C4⋊C4—21no—
127C7×C4⋊C4—21no—
128C7×C4⋊C4—21no—
129C7×C4⋊C4—41no—
130C7×C4⋊C4—41no—
131C7×C22⋊C4—41no—
132C7×C22⋊C4—41no—
133C7×C22⋊C4—41no—
134C7×C22⋊C4—41no—
135C42×C7C2×D411yescharacteristic
136C42×C7C2×D411yescharacteristic
137C42×C7C2×D411yescharacteristic
138C42×C7C2×D411yescharacteristic
139C2×AGL(1,ℤ/8)—21no—
140#532C22×C711yescharacteristic
141C2×C4⋊Q8C22×C711yescharacteristic
142C2×C4×D4C22×C711yescharacteristic
143#498—21no—
144#486—21no—
145#481—22no—
146#476—22no—
147#474—22no—
148#474—22no—
149#463—22no—
150#461—21no—
151#459—21no—
152#431C22×C711yescharacteristic
153#431C22×C711yescharacteristic
154C2×#107—21no—
155#423C22×C711yescharacteristic
156C2×C4⋊C8C22×C711yescharacteristic
157#417—21no—
158C2×#102—21no—
159#333—22no—
160D4⋊C8—22no—
161#323—22no—
162C23×C7—22no—
163C23×C7—41no—
164C23×C7—41no—
165C23×C7—41no—
166C23×C7—81no—
167C23×C7—81no—
168C23×C7#12111yescharacteristic
169C7×Q8—41no—
170C7×Q8—41no—
171C7×Q8—82no—
172C7×Q8—82no—
173C7×D4—22no—
174C7×D4—22no—
175C7×D4—42no—
176C7×D4—41no—
177C7×D4—41no—
178C7×D4—41no—
179C7×D4—41no—
180C7×D4—42no—
181C7×D4—81no—
182C7×D4—81no—
183C2×C4×C7—22no—
184C2×C4×C7—41no—
185C2×C4×C7—41no—
186C2×C4×C7—41no—
187C2×C4×C7—42no—
188C2×C4×C7—82no—
189C2×C4×C7—82no—
190C2×C4×C7—81no—
191C2×C4×C7—81no—
192C2×C4×C7—82no—
193C2×C4×C7—81no—
194C2×C4×C7#12011yescharacteristic
195C2×C4×C7#12011yescharacteristic
196C2×C4×C7#12011yescharacteristic
197C2×C4×C7#12111yescharacteristic
198C2×C4×C7#12111yescharacteristic
199C2×C4×C7#12711yescharacteristic
200C2×C4×C7C2×SD1611yescharacteristic
201C2×C4×C7#13511yescharacteristic
202C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
203C2×C4×C7#13711yescharacteristic
204C7×C8—42no—
205C7×C8—42no—
206C7×C8—42no—
207C7×C8—81no—
208C2×#41—21no—
209C22×Q8—21no—
210C22×D4C7×D411yescharacteristic
211C23×C4—21no—
212AGL(1,ℤ/8)—42no—
213AGL(1,ℤ/8)—42no—
214C2×SD16—22no—
215C2×D8—22no—
216C2×M16—21no—
217C22×C8—21no—
218C4⋊Q8—21no—
219C4⋊Q8—21no—
220C4⋊Q8—21no—
221C4⋊Q8C7×D412yes—
222C4⋊Q8C7×D411yescharacteristic
223#127C7×D411yescharacteristic
224#124—21no—
225#121—21no—
226#121—21no—
227C4×D4—21no—
228C4×D4—21no—
229C4×D4—21no—
230C4×D4—21no—
231C4×D4—21no—
232C4×D4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4C7×D411yescharacteristic
236C2×C22⋊C4—21no—
237C2×C42C23×C711yescharacteristic
238#107—24no—
239#106—24no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8C7×D412yes—
245C4⋊C8C7×D412yes—
246#103—22no—
247#102—24no—
248#102—22no—
249#102—24no—
250C22⋊C8—21no—
251C22⋊C8C7×D412yes—
252C4×C8—21no—
253C4×C8—21no—
254C22×C7—21no—
255C22×C7—21no—
256C22×C7—42no—
257C22×C7—82no—
258C22×C7—82no—
259C22×C7—81no—
260C22×C7—81no—
261C22×C7—81no—
262C22×C7—81no—
263C22×C7—81no—
264C22×C7#39211yescharacteristic
265C22×C7#46511yescharacteristic
266C22×C7#46811yescharacteristic
267C4×C7—21no—
268C4×C7—21no—
269C4×C7—21no—
270C4×C7—81no—
271C4×C7—81no—
272C4×C7—82no—
273C4×C7#44811yescharacteristic
274C4×C7#44911yescharacteristic
275C4×C7#45011yescharacteristic
276C4×C7#45211yescharacteristic
277C4×C7#47212yes—
278C24—21no—
279#41—41no—
280#41—41no—
281#41—41no—
282#41—41no—
283C2×Q8—21no—
284C2×Q8—22no—
285C2×Q8—41no—
286C2×Q8—82no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—41no—
296C2×D4C7×SD1612yes—
297C22×C4—21no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—82no—
303C22×C4C2×C7×D411yescharacteristic
304C22×C4C2×C7×D411yescharacteristic
305C22×C4C7×#4111yescharacteristic
306SD16—44no—
307D8—44no—
308M16—41no—
309M16—41no—
310C2×C8—22no—
311C2×C8—22no—
312C2×C8—22no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—41no—
316C4⋊C4—21no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—21no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C22⋊C4—41no—
325C22⋊C4—41no—
326C22⋊C4—41no—
327C22⋊C4—41no—
328C42C2×C7×D411yescharacteristic
329C42C2×C7×D411yescharacteristic
330C42C2×C7×D411yescharacteristic
331C42C2×C7×D411yescharacteristic
332C2×C7—21no—
333C2×C7—21no—
334C2×C7—82no—
335C2×C7—161no—
336C2×C7#200011yescharacteristic
337C2×C7#200911yescharacteristic
338C2×C7#201811yescharacteristic
339C23—22no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23—81no—
344C23—81no—
345C23C7×#12111yescharacteristic
346Q8—41no—
347Q8—41no—
348Q8—82no—
349Q8—82no—
350D4—22no—
351D4—22no—
352D4—42no—
353D4—42no—
354D4—41no—
355D4—41no—
356D4—41no—
357D4—41no—
358D4—81no—
359D4—81no—
360C2×C4—22no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—42no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4C7×#12011yescharacteristic
372C2×C4C7×#12011yescharacteristic
373C2×C4C7×#12011yescharacteristic
374C2×C4C7×#12111yescharacteristic
375C2×C4C7×#12111yescharacteristic
376C2×C4C7×#12711yescharacteristic
377C2×C4C2×C7×SD1611yescharacteristic
378C2×C4C7×#13511yescharacteristic
379C2×C4C7×AGL(1,ℤ/8)11yescharacteristic
380C2×C4C7×#13711yescharacteristic
381C8—42no—
382C8—42no—
383C8—42no—
384C8—81no—
385C7#1018811yescharacteristic, direct product factor, Sylow
386C22—21no—
387C22—21no—
388C22—42no—
389C22—82no—
390C22—82no—
391C22—81no—
392C22—81no—
393C22—81no—
394C22—81no—
395C22—81no—
396C22C7×#39211yescharacteristic
397C22C7×#46511yescharacteristic
398C22C7×#46811yescharacteristic
399C4—21no—
400C4—21no—
401C4—21no—
402C4—81no—
403C4—81no—
404C4—82no—
405C4C7×#44811yescharacteristic
406C4C7×#44911yescharacteristic
407C4C7×#45011yescharacteristic
408C4C7×#45211yescharacteristic
409C4C7×#47212yes—
410C2—21no—
411C2—21no—
412C2—82no—
413C2—161no—
414C2C7×#200011yescharacteristic
415C2C7×#200911yescharacteristic
416C2C7×#201811yescharacteristic
subgroupG/H
center, socleC22×C7#392
commutator, Frattini, cosocleC2×C42C23×C7
prime radicalC7×#3036C2
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
1. C7×#3227C2semidirect
2. C2×C7×#474C2semidirect
4. C7×#1548C2semidirect
5. C7×#1480C2semidirect
6. C7×#1460C2semidirect
7. C7×#1301C2semidirect
31. #10188C7direct (N × H)
78. #3227C2×C7semidirect
79. C2×#474C2×C7semidirect
81. #1548C2×C7semidirect
82. #1480C2×C7semidirect
83. #1460C2×C7semidirect
84. #1301C2×C7semidirect
385. C7#10188direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#32272
C2×C7×#4742
C7×#30362
C7×#15482
C7×#14802
C7×#14602
C7×#13012
#101887