Finite Group Databasebeta

← C7×#10281 #13187497

Subgroups — C7×#10281 #13187497

Order 1792.

690 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3206C211yesmaximal, characteristic, complemented
2C7×#3205C211yesmaximal, characteristic
3C7×#2858C211yesmaximal, characteristic, complemented
4C7×#1962C211yesmaximal, characteristic, complemented
5C7×#1548C211yesmaximal, characteristic, complemented
6C7×#1545C211yesmaximal, characteristic, complemented
7C7×#1276C211yesmaximal, characteristic, complemented
8C7×#532C2211yescharacteristic
9C2×C7×C4⋊Q8C2211yescharacteristic
10C2×C7×#127C2211yescharacteristic
11C2×C7×#124—22no—
12C7×#493—21no—
13C7×#492—21no—
14C7×#487—21no—
15C7×#486—21no—
16C7×#461—22no—
17C7×#459—22no—
18C7×#448—22no—
19C7×#447—22no—
20C7×#431C2211yescharacteristic
21C7×#431C2211yescharacteristic
22C7×#423C2211yescharacteristic
23C2×C7×#104—22no—
24C7×#390—22no—
25C7×#390—22no—
26C43×C7C2211yescharacteristic
27C7×#333—22no—
28C7×#331—22no—
29#10281C711yesmaximal, characteristic, direct product factor, Sylow
30C2×C7×#41—21no—
31C22×C7×Q8D412yes—
32C22×C7×D4—21no—
33C22×C7×D4—21no—
34C22×C7×D4D412yes—
35C2×C7×SD16—24no—
36C2×C7×D8—24no—
37C2×C7×M16—21no—
38C7×C4⋊Q8—21no—
39C7×C4⋊Q8—21no—
40C7×C4⋊Q8—21no—
41C7×C4⋊Q8—21no—
42C7×C4⋊Q8D411yescharacteristic
43C7×#127—21no—
44C7×#127—21no—
45C7×#127—21no—
46C7×#127—21no—
47C7×#127D411yescharacteristic
48C7×#124—21no—
49C7×#124—48no—
50C7×#121—21no—
51C7×#121—21no—
52C4×C7×D4—21no—
53C4×C7×D4—21no—
54C2×C7×C4⋊C4—21no—
55C2×C7×C4⋊C4—21no—
56C2×C7×C22⋊C4—28no—
57C2×C42×C7—21no—
58C2×C42×C7—21no—
59C2×C42×C7D412yes—
60C2×C42×C7C2311yescharacteristic
61C7×C4⋊C8—21no—
62C7×C4⋊C8—21no—
63C7×C4⋊C8D412yes—
64C7×C4⋊C8D412yes—
65C7×#104—44no—
66C7×#103—24no—
67C7×#102—24no—
68C7×C22⋊C8—21no—
69C7×C22⋊C8—21no—
70C4×C7×C8—21no—
71C4×C7×C8—21no—
72#3206C2×C711yescharacteristic, complemented
73#3205C2×C711yescharacteristic
74#2858C2×C711yescharacteristic, complemented
75#1962C2×C711yescharacteristic, complemented
76#1548C2×C711yescharacteristic, complemented
77#1545C2×C711yescharacteristic, complemented
78#1276C2×C711yescharacteristic, complemented
79C24×C7—22no—
80C7×#41—42no—
81C7×#41—42no—
82C2×C7×Q8—22no—
83C2×C7×Q8—21no—
84C2×C7×Q8—22no—
85C2×C7×Q8—44no—
86C2×C7×D4—22no—
87C2×C7×D4—21no—
88C2×C7×D4—21no—
89C2×C7×D4—21no—
90C2×C7×D4—22no—
91C2×C7×D4—44no—
92C2×C7×D4—44no—
93C2×C7×D4—41no—
94C2×C7×D4—41no—
95C2×C7×D4—41no—
96C2×C7×D4—41no—
97C2×C7×D4—41no—
98C2×C7×D4—42no—
99C2×C7×D4—42no—
100C22×C4×C7—22no—
101C22×C4×C7—22no—
102C22×C4×C7—41no—
103C22×C4×C7C2×D411yescharacteristic
104C22×C4×C7C2×D411yescharacteristic
105C22×C4×C7C2×D411yescharacteristic
106C7×SD16—84no—
107C7×D8—84no—
108C7×M16—42no—
109C2×C7×C8—22no—
110C2×C7×C8—22no—
111C2×C7×C8—41no—
112C7×C4⋊C4—21no—
113C7×C4⋊C4—21no—
114C7×C4⋊C4—41no—
115C7×C4⋊C4—41no—
116C7×C4⋊C4—41no—
117C7×C4⋊C4—41no—
118C7×C22⋊C4—416no—
119C7×C22⋊C4—41no—
120C7×C22⋊C4—41no—
121C42×C7—24no—
122C42×C7—22no—
123C42×C7—22no—
124C42×C7—42no—
125C42×C7C2×D411yescharacteristic
126C42×C7C2×D411yescharacteristic
127C42×C7C2×D411yescharacteristic
128C42×C7#4111yescharacteristic
129#532C22×C711yescharacteristic
130C2×C4⋊Q8C22×C711yescharacteristic
131C2×#127C22×C711yescharacteristic
132C2×#124—22no—
133#493—21no—
134#492—21no—
135#487—21no—
136#486—21no—
137#461—22no—
138#459—22no—
139#448—22no—
140#447—22no—
141#431C22×C711yescharacteristic
142#431C22×C711yescharacteristic
143#423C22×C711yescharacteristic
144C2×#104—22no—
145#390—22no—
146#390—22no—
147C43C22×C711yescharacteristic
148#333—22no—
149#331—22no—
150C23×C7—44no—
151C23×C7—44no—
152C23×C7—41no—
153C23×C7—42no—
154C23×C7—44no—
155C23×C7#12011yescharacteristic
156C7×Q8—44no—
157C7×Q8—42no—
158C7×Q8—82no—
159C7×D4—42no—
160C7×D4—42no—
161C7×D4—44no—
162C7×D4—42no—
163C7×D4—82no—
164C7×D4—82no—
165C7×D4—82no—
166C7×D4—82no—
167C7×D4—82no—
168C7×D4—81no—
169C7×D4—81no—
170C2×C4×C7—22no—
171C2×C4×C7—22no—
172C2×C4×C7—24no—
173C2×C4×C7—42no—
174C2×C4×C7—42no—
175C2×C4×C7—44no—
176C2×C4×C7—41no—
177C2×C4×C7—42no—
178C2×C4×C7—81no—
179C2×C4×C7—81no—
180C2×C4×C7#12011yescharacteristic
181C2×C4×C7#12011yescharacteristic
182C2×C4×C7#12111yescharacteristic
183C2×C4×C7#12111yescharacteristic
184C2×C4×C7#12111yescharacteristic
185C2×C4×C7#12711yescharacteristic
186C2×C4×C7#13512yes—
187C2×C4×C7AGL(1,ℤ/8)12yes—
188C7×C8—42no—
189C7×C8—42no—
190C7×C8—81no—
191C2×#41—21no—
192C22×Q8C7×D412yes—
193C22×D4—21no—
194C22×D4—21no—
195C22×D4C7×D412yes—
196C2×SD16—24no—
197C2×D8—24no—
198C2×M16—21no—
199C4⋊Q8—21no—
200C4⋊Q8—21no—
201C4⋊Q8—21no—
202C4⋊Q8—21no—
203C4⋊Q8C7×D411yescharacteristic
204#127—21no—
205#127—21no—
206#127—21no—
207#127—21no—
208#127C7×D411yescharacteristic
209#124—21no—
210#124—48no—
211#121—21no—
212#121—21no—
213C4×D4—21no—
214C4×D4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C22⋊C4—28no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42C7×D412yes—
221C2×C42C23×C711yescharacteristic
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8C7×D412yes—
225C4⋊C8C7×D412yes—
226#104—44no—
227#103—24no—
228#102—24no—
229C22⋊C8—21no—
230C22⋊C8—21no—
231C4×C8—21no—
232C4×C8—21no—
233C22×C7—21no—
234C22×C7—21no—
235C22×C7—88no—
236C22×C7—81no—
237C22×C7—82no—
238C22×C7—82no—
239C22×C7—81no—
240C22×C7—81no—
241C22×C7—82no—
242C22×C7#39211yescharacteristic
243C22×C7#44912yes—
244C4×C7—21no—
245C4×C7—21no—
246C4×C7—44no—
247C4×C7—82no—
248C4×C7—81no—
249C4×C7#45312yes—
250C4×C7#45512yes—
251C4×C7#46312yes—
252C4×C7#49511yescharacteristic
253C4×C7#49611yescharacteristic
254C24—22no—
255#41—42no—
256#41—42no—
257C2×Q8—22no—
258C2×Q8—21no—
259C2×Q8—22no—
260C2×Q8—44no—
261C2×D4—22no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—21no—
265C2×D4—22no—
266C2×D4—44no—
267C2×D4—44no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—42no—
274C2×D4—42no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—41no—
278C22×C4C2×C7×D411yescharacteristic
279C22×C4C2×C7×D411yescharacteristic
280C22×C4C2×C7×D411yescharacteristic
281SD16—84no—
282D8—84no—
283M16—42no—
284C2×C8—22no—
285C2×C8—22no—
286C2×C8—41no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C22⋊C4—416no—
294C22⋊C4—41no—
295C22⋊C4—41no—
296C42—24no—
297C42—22no—
298C42—22no—
299C42—42no—
300C42C2×C7×D411yescharacteristic
301C42C2×C7×D411yescharacteristic
302C42C2×C7×D411yescharacteristic
303C42C7×#4111yescharacteristic
304C2×C7—22no—
305C2×C7—161no—
306C2×C7—162no—
307C2×C7#200211yescharacteristic
308C2×C7#200411yescharacteristic
309C2×C7#201211yescharacteristic
310C23—44no—
311C23—44no—
312C23—41no—
313C23—42no—
314C23—44no—
315C23C7×#12011yescharacteristic
316Q8—44no—
317Q8—42no—
318Q8—82no—
319D4—42no—
320D4—42no—
321D4—44no—
322D4—42no—
323D4—82no—
324D4—82no—
325D4—82no—
326D4—81no—
327D4—81no—
328D4—82no—
329D4—82no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—24no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—44no—
336C2×C4—41no—
337C2×C4—42no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4C7×#12011yescharacteristic
341C2×C4C7×#12011yescharacteristic
342C2×C4C7×#12111yescharacteristic
343C2×C4C7×#12111yescharacteristic
344C2×C4C7×#12111yescharacteristic
345C2×C4C7×#12711yescharacteristic
346C2×C4C7×#13512yes—
347C2×C4C7×AGL(1,ℤ/8)12yes—
348C8—42no—
349C8—42no—
350C8—81no—
351C7#1028111yescharacteristic, direct product factor, Sylow
352C22—21no—
353C22—21no—
354C22—88no—
355C22—81no—
356C22—82no—
357C22—82no—
358C22—81no—
359C22—81no—
360C22—82no—
361C22C7×#39211yescharacteristic
362C22C7×#44912yes—
363C4—21no—
364C4—21no—
365C4—44no—
366C4—82no—
367C4—81no—
368C4C7×#45312yes—
369C4C7×#45512yes—
370C4C7×#46312yes—
371C4C7×#49511yescharacteristic
372C4C7×#49611yescharacteristic
373C2—22no—
374C2—162no—
375C2—161no—
376C2C7×#200211yescharacteristic
377C2C7×#200411yescharacteristic
378C2C7×#201211yescharacteristic
subgroupG/H
center, socleC22×C7#392
commutator, Frattini, cosocleC2×C42C23×C7
prime radicalC7×#3205C2
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×#3206C2semidirect
3. C7×#2858C2semidirect
4. C7×#1962C2semidirect
5. C7×#1548C2semidirect
6. C7×#1545C2semidirect
7. C7×#1276C2semidirect
29. #10281C7direct (N × H)
72. #3206C2×C7semidirect
74. #2858C2×C7semidirect
75. #1962C2×C7semidirect
76. #1548C2×C7semidirect
77. #1545C2×C7semidirect
78. #1276C2×C7semidirect
351. C7#10281direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#32062
C7×#32052
C7×#28582
C7×#19622
C7×#15482
C7×#15452
C7×#12762
#102817