Finite Group Databasebeta

← C7×#10194 #13187410

Subgroups — C7×#10194 #13187410

Order 1792.

640 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3246C211yesmaximal, characteristic, complemented
2C7×#3208C211yesmaximal, characteristic, complemented
3C7×#3151C211yesmaximal, characteristic
4C7×#1986C211yesmaximal, characteristic, complemented
5C7×#1982C211yesmaximal, characteristic, complemented
6C7×#1502C211yesmaximal, characteristic, complemented
7C7×#1283C211yesmaximal, characteristic, complemented
8C2×C7×AGL(1,ℤ/8)—22no—
9C7×#533C2211yescharacteristic
10C7×#532C2211yescharacteristic
11C2×C7×#127C2211yescharacteristic
12C7×#496—22no—
13C7×#493—21no—
14C7×#492—21no—
15C7×#492—21no—
16C7×#486—21no—
17C7×#476—21no—
18C7×#475—21no—
19C7×#465—22no—
20C7×#461—21no—
21C7×#460—22no—
22C7×#460—21no—
23C7×#450—22no—
24C7×#431C2211yescharacteristic
25C2×C7×C4⋊C8C2211yescharacteristic
26C2×C7×#102—22no—
27C4×C7×M16C2211yescharacteristic
28C4×C7×C4⋊C4C2211yescharacteristic
29C7×#339—22no—
30C7×#327—22no—
31#10194C711yesmaximal, characteristic, direct product factor, Sylow
32C2×C7×#41—21no—
33C22×C7×D4—21no—
34C22×C7×D4—21no—
35C22×C7×D4—21no—
36C7×AGL(1,ℤ/8)—44no—
37C7×AGL(1,ℤ/8)—44no—
38C2×C7×SD16—24no—
39C2×C7×SD16—24no—
40C2×C7×SD16—22no—
41C2×C7×D8—24no—
42C2×C7×M16D412yes—
43C22×C7×C8D412yes—
44C7×C4⋊Q8D411yescharacteristic
45C7×C4⋊Q8D411yescharacteristic
46C7×C4⋊Q8D411yescharacteristic
47C7×#127—21no—
48C7×#127—21no—
49C7×#127—21no—
50C7×#127D411yescharacteristic
51C7×#127D412yes—
52C7×#125—21no—
53C7×#124—21no—
54C7×C22⋊Q8—21no—
55C7×C22⋊Q8—21no—
56C7×#121—21no—
57C7×#121—21no—
58C4×C7×Q8—21no—
59C4×C7×Q8—21no—
60C4×C7×D4—21no—
61C4×C7×D4—21no—
62C7×#117—21no—
63C2×C7×C4⋊C4—21no—
64C2×C42×C7—21no—
65C2×C42×C7C2311yescharacteristic
66C7×#106—22no—
67C7×C4⋊C8—21no—
68C7×C4⋊C8—21no—
69C7×C4⋊C8—21no—
70C7×#103—22no—
71C7×#102—22no—
72C7×#102—24no—
73C7×#102—44no—
74C7×C22⋊C8D412yes—
75C7×#97—21no—
76C4×C7×C8—21no—
77C4×C7×C8—21no—
78C7×#95—21no—
79#3246C2×C711yescharacteristic, complemented
80#3208C2×C711yescharacteristic, complemented
81#3151C2×C711yescharacteristic
82#1986C2×C711yescharacteristic, complemented
83#1982C2×C711yescharacteristic, complemented
84#1502C2×C711yescharacteristic, complemented
85#1283C2×C711yescharacteristic, complemented
86C24×C7—41no—
87C7×#41—42no—
88C7×#41—42no—
89C2×C7×Q8—21no—
90C2×C7×Q8—21no—
91C2×C7×Q8—21no—
92C2×C7×D4—21no—
93C2×C7×D4—22no—
94C2×C7×D4—21no—
95C2×C7×D4—22no—
96C2×C7×D4—22no—
97C2×C7×D4—21no—
98C2×C7×D4—42no—
99C2×C7×D4—42no—
100C2×C7×D4—41no—
101C2×C7×D4—41no—
102C2×C7×D4—41no—
103C2×C7×D4—41no—
104C2×C7×D4—42no—
105C2×C7×D4—42no—
106C2×C7×D4—82no—
107C22×C4×C7—41no—
108C22×C4×C7—41no—
109C22×C4×C7C2×D411yescharacteristic
110C22×C4×C7C2×D411yescharacteristic
111C22×C4×C7C2×D411yescharacteristic
112C7×SD16—48no—
113C7×SD16—84no—
114C7×SD16—82no—
115C7×D8—48no—
116C7×M16—22no—
117C7×M16SD1614yescomplemented
118C2×C7×C8—22no—
119C2×C7×C8—22no—
120C2×C7×C8—22no—
121C2×C7×C8—42no—
122C7×C4⋊C4—22no—
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—41no—
130C7×C4⋊C4—41no—
131C7×C4⋊C4—41no—
132C7×C22⋊C4—41no—
133C7×C22⋊C4—41no—
134C7×C22⋊C4—41no—
135C42×C7—41no—
136C42×C7—42no—
137C42×C7C2×D411yescharacteristic
138C42×C7C2×D411yescharacteristic
139C42×C7C2×D411yescharacteristic
140C42×C7#4111yescharacteristic
141C2×AGL(1,ℤ/8)—22no—
142#533C22×C711yescharacteristic
143#532C22×C711yescharacteristic
144C2×#127C22×C711yescharacteristic
145#496—22no—
146#493—21no—
147#492—21no—
148#492—21no—
149#486—21no—
150#476—21no—
151#475—21no—
152#465—22no—
153#461—21no—
154#460—22no—
155#460—21no—
156#450—22no—
157#431C22×C711yescharacteristic
158C2×C4⋊C8C22×C711yescharacteristic
159C2×#102—22no—
160C4×M16C22×C711yescharacteristic
161C4×C4⋊C4C22×C711yescharacteristic
162#339—22no—
163#327—22no—
164C23×C7—41no—
165C23×C7—42no—
166C23×C7—82no—
167C23×C7—82no—
168C23×C7—81no—
169C23×C7—81no—
170C23×C7#12711yescharacteristic
171C7×Q8—42no—
172C7×Q8—42no—
173C7×Q8—41no—
174C7×Q8—41no—
175C7×D4—44no—
176C7×D4—44no—
177C7×D4—42no—
178C7×D4—44no—
179C7×D4—41no—
180C7×D4—41no—
181C7×D4—81no—
182C7×D4—82no—
183C7×D4—81no—
184C7×D4—81no—
185C7×D4—81no—
186C7×D4—81no—
187C7×D4—82no—
188C2×C4×C7—21no—
189C2×C4×C7—21no—
190C2×C4×C7—41no—
191C2×C4×C7—42no—
192C2×C4×C7—41no—
193C2×C4×C7—41no—
194C2×C4×C7—41no—
195C2×C4×C7—81no—
196C2×C4×C7—81no—
197C2×C4×C7—82no—
198C2×C4×C7#12011yescharacteristic
199C2×C4×C7#12011yescharacteristic
200C2×C4×C7#12011yescharacteristic
201C2×C4×C7#12111yescharacteristic
202C2×C4×C7#12111yescharacteristic
203C2×C4×C7#12111yescharacteristic
204C2×C4×C7C2×SD1612yes—
205C2×C4×C7AGL(1,ℤ/8)12yes—
206C7×C8—24no—
207C7×C8—42no—
208C7×C8—42no—
209C2×#41—21no—
210C22×D4—21no—
211C22×D4—21no—
212C22×D4—21no—
213AGL(1,ℤ/8)—44no—
214AGL(1,ℤ/8)—44no—
215C2×SD16—24no—
216C2×SD16—24no—
217C2×SD16—22no—
218C2×D8—24no—
219C2×M16C7×D412yes—
220C22×C8C7×D412yes—
221C4⋊Q8C7×D411yescharacteristic
222C4⋊Q8C7×D411yescharacteristic
223C4⋊Q8C7×D411yescharacteristic
224#127—21no—
225#127—21no—
226#127—21no—
227#127C7×D412yes—
228#127C7×D411yescharacteristic
229#125—21no—
230#124—21no—
231C22⋊Q8—21no—
232C22⋊Q8—21no—
233#121—21no—
234#121—21no—
235C4×Q8—21no—
236C4×Q8—21no—
237C4×D4—21no—
238C4×D4—21no—
239#117—21no—
240C2×C4⋊C4—21no—
241C2×C42—21no—
242C2×C42C23×C711yescharacteristic
243#106—22no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247#103—22no—
248#102—24no—
249#102—22no—
250#102—44no—
251C22⋊C8C7×D412yes—
252#97—21no—
253C4×C8—21no—
254C4×C8—21no—
255#95—21no—
256C22×C7—21no—
257C22×C7—21no—
258C22×C7—82no—
259C22×C7—81no—
260C22×C7—82no—
261C22×C7—81no—
262C22×C7—81no—
263C22×C7—82no—
264C22×C7—162no—
265C22×C7—162no—
266C22×C7#39211yescharacteristic
267C22×C7#49211yescharacteristic
268C22×C7#49611yescharacteristic
269C4×C7—21no—
270C4×C7—21no—
271C4×C7—21no—
272C4×C7—82no—
273C4×C7—81no—
274C4×C7—81no—
275C4×C7—81no—
276C4×C7#45012yes—
277C4×C7#45311yescharacteristic
278C4×C7#45411yescharacteristic
279C4×C7#46012yes—
280C24—41no—
281#41—42no—
282#41—42no—
283C2×Q8—21no—
284C2×Q8—21no—
285C2×Q8—21no—
286C2×D4—21no—
287C2×D4—22no—
288C2×D4—21no—
289C2×D4—22no—
290C2×D4—22no—
291C2×D4—21no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—42no—
299C2×D4—42no—
300C2×D4—82no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4C2×C7×D411yescharacteristic
304C22×C4C2×C7×D411yescharacteristic
305C22×C4C2×C7×D411yescharacteristic
306SD16—48no—
307SD16—84no—
308SD16—82no—
309D8—48no—
310M16—22no—
311M16C7×SD1614yescomplemented
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—22no—
315C2×C8—42no—
316C4⋊C4—22no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C22⋊C4—41no—
327C22⋊C4—41no—
328C22⋊C4—41no—
329C42—41no—
330C42—42no—
331C42C2×C7×D411yescharacteristic
332C42C2×C7×D411yescharacteristic
333C42C2×C7×D411yescharacteristic
334C42C7×#4111yescharacteristic
335C2×C7—21no—
336C2×C7—21no—
337C2×C7—161no—
338C2×C7—162no—
339C2×C7#200611yescharacteristic
340C2×C7#200811yescharacteristic
341C2×C7#201411yescharacteristic
342C23—41no—
343C23—42no—
344C23—82no—
345C23—82no—
346C23—81no—
347C23—81no—
348C23C7×#12711yescharacteristic
349Q8—42no—
350Q8—42no—
351Q8—41no—
352Q8—41no—
353D4—44no—
354D4—44no—
355D4—42no—
356D4—44no—
357D4—41no—
358D4—41no—
359D4—81no—
360D4—82no—
361D4—81no—
362D4—81no—
363D4—81no—
364D4—81no—
365D4—82no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—82no—
376C2×C4C7×#12011yescharacteristic
377C2×C4C7×#12011yescharacteristic
378C2×C4C7×#12011yescharacteristic
379C2×C4C7×#12111yescharacteristic
380C2×C4C7×#12111yescharacteristic
381C2×C4C7×#12111yescharacteristic
382C2×C4C2×C7×SD1612yes—
383C2×C4C7×AGL(1,ℤ/8)12yes—
384C8—24no—
385C8—42no—
386C8—42no—
387C7#1019411yescharacteristic, direct product factor, Sylow
388C22—21no—
389C22—21no—
390C22—82no—
391C22—81no—
392C22—82no—
393C22—81no—
394C22—81no—
395C22—82no—
396C22—162no—
397C22—162no—
398C22C7×#39211yescharacteristic
399C22C7×#49211yescharacteristic
400C22C7×#49611yescharacteristic
401C4—21no—
402C4—21no—
403C4—21no—
404C4—82no—
405C4—81no—
406C4—81no—
407C4—81no—
408C4C7×#45012yes—
409C4C7×#45311yescharacteristic
410C4C7×#45411yescharacteristic
411C4C7×#46012yes—
412C2—21no—
413C2—21no—
414C2—162no—
415C2—161no—
416C2C7×#200611yescharacteristic
417C2C7×#200811yescharacteristic
418C2C7×#201411yescharacteristic
subgroupG/H
center, socleC22×C7#392
commutator, Frattini, cosocleC2×C42C23×C7
prime radicalC7×#3151C2
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×#3246C2semidirect
2. C7×#3208C2semidirect
4. C7×#1986C2semidirect
5. C7×#1982C2semidirect
6. C7×#1502C2semidirect
7. C7×#1283C2semidirect
31. #10194C7direct (N × H)
79. #3246C2×C7semidirect
80. #3208C2×C7semidirect
82. #1986C2×C7semidirect
83. #1982C2×C7semidirect
84. #1502C2×C7semidirect
85. #1283C2×C7semidirect
117. C7×M16SD16semidirect
311. M16C7×SD16semidirect
387. C7#10194direct (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×#32082
C7×#31512
C7×#19862
C7×#19822
C7×#15022
C7×#12832
#101947