Finite Group Databasebeta

← C7×#10294 #13187510

Subgroups — C7×#10294 #13187510

Order 1792.

640 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3229C211yesmaximal, characteristic, complemented
2C7×#3208C211yesmaximal, characteristic
3C7×#2858C211yesmaximal, characteristic
4C7×#1994C211yesmaximal, characteristic, complemented
5C7×#1546C211yesmaximal, characteristic, complemented
6C7×#1545C211yesmaximal, characteristic
7C7×#1276C211yesmaximal, characteristic, complemented
8C7×#533C2211yescharacteristic
9C2×C7×C4⋊Q8C2211yescharacteristic
10C2×C7×#127C2211yescharacteristic
11C2×C7×#124—22no—
12C7×#498—21no—
13C7×#492—21no—
14C7×#487—21no—
15C7×#486—21no—
16C7×#483—22no—
17C7×#475—22no—
18C7×#460—22no—
19C7×#450—22no—
20C7×#431C2211yescharacteristic
21C7×#431C2211yescharacteristic
22C7×#423C2211yescharacteristic
23C7×#390—22no—
24C7×#390—22no—
25C43×C7C2211yescharacteristic
26C7×#339—22no—
27C7×#332—22no—
28C7×#331—22no—
29#10294C711yesmaximal, characteristic, direct product factor, Sylow
30C22×C7×Q8—21no—
31C22×C7×D4—21no—
32C22×C7×D4—21no—
33C22×C7×D4D412yes—
34C2×C7×SD16—24no—
35C2×C7×M16—21no—
36C7×C4⋊Q8—21no—
37C7×C4⋊Q8—21no—
38C7×C4⋊Q8—21no—
39C7×C4⋊Q8—21no—
40C7×C4⋊Q8D411yescharacteristic
41C7×C4⋊Q8D411yescharacteristic
42C7×#127—21no—
43C7×#127—21no—
44C7×#127—21no—
45C7×#127—21no—
46C7×#125—21no—
47C7×#124—44no—
48C7×#124—44no—
49C7×C22⋊Q8—21no—
50C7×C22⋊Q8—21no—
51C4×C7×Q8—21no—
52C4×C7×Q8—21no—
53C7×#117—21no—
54C2×C7×C4⋊C4—21no—
55C2×C7×C4⋊C4D412yes—
56C2×C7×C22⋊C4—28no—
57C2×C42×C7—21no—
58C2×C42×C7—21no—
59C2×C42×C7D412yes—
60C2×C42×C7C2311yescharacteristic
61C7×#106—24no—
62C7×C4⋊C8—21no—
63C7×C4⋊C8—21no—
64C7×C4⋊C8D412yes—
65C7×C4⋊C8D412yes—
66C7×#103—24no—
67C7×#102—24no—
68C7×C22⋊C8—21no—
69C7×C22⋊C8—21no—
70C4×C7×C8—21no—
71C4×C7×C8—21no—
72#3229C2×C711yescharacteristic, complemented
73#3208C2×C711yescharacteristic
74#2858C2×C711yescharacteristic
75#1994C2×C711yescharacteristic, complemented
76#1546C2×C711yescharacteristic, complemented
77#1545C2×C711yescharacteristic
78#1276C2×C711yescharacteristic, complemented
79C24×C7—22no—
80C2×C7×Q8—21no—
81C2×C7×Q8—21no—
82C2×C7×Q8—41no—
83C2×C7×Q8—41no—
84C2×C7×Q8—42no—
85C2×C7×Q8—42no—
86C2×C7×D4—22no—
87C2×C7×D4—22no—
88C2×C7×D4—44no—
89C2×C7×D4—44no—
90C2×C7×D4—41no—
91C2×C7×D4—41no—
92C2×C7×D4—41no—
93C2×C7×D4—41no—
94C2×C7×D4—42no—
95C2×C7×D4—42no—
96C22×C4×C7—22no—
97C22×C4×C7—22no—
98C22×C4×C7C2×D411yescharacteristic
99C22×C4×C7C2×D411yescharacteristic
100C22×C4×C7C2×D411yescharacteristic
101C7×SD16—84no—
102C7×M16—42no—
103C2×C7×C8—22no—
104C2×C7×C8—22no—
105C2×C7×C8—41no—
106C7×C4⋊C4—21no—
107C7×C4⋊C4—22no—
108C7×C4⋊C4—21no—
109C7×C4⋊C4—22no—
110C7×C4⋊C4—21no—
111C7×C4⋊C4—21no—
112C7×C4⋊C4—41no—
113C7×C4⋊C4—41no—
114C7×C4⋊C4—41no—
115C7×C4⋊C4—41no—
116C7×C22⋊C4—416no—
117C7×C22⋊C4—41no—
118C42×C7—22no—
119C42×C7—42no—
120C42×C7—42no—
121C42×C7—41no—
122C42×C7SD1614yescomplemented
123C42×C7C2×D411yescharacteristic
124C42×C7C2×D411yescharacteristic
125C42×C7C2×D411yescharacteristic
126C42×C7#4111yescharacteristic
127#533C22×C711yescharacteristic
128C2×C4⋊Q8C22×C711yescharacteristic
129C2×#127C22×C711yescharacteristic
130C2×#124—22no—
131#498—21no—
132#492—21no—
133#487—21no—
134#486—21no—
135#483—22no—
136#475—22no—
137#460—22no—
138#450—22no—
139#431C22×C711yescharacteristic
140#431C22×C711yescharacteristic
141#423C22×C711yescharacteristic
142#390—22no—
143#390—22no—
144C43C22×C711yescharacteristic
145#339—22no—
146#332—22no—
147#331—22no—
148C23×C7—44no—
149C23×C7—44no—
150C23×C7—42no—
151C23×C7—44no—
152C23×C7#12011yescharacteristic
153C7×Q8—42no—
154C7×Q8—42no—
155C7×Q8—82no—
156C7×Q8—81no—
157C7×Q8—81no—
158C7×D4—44no—
159C7×D4—82no—
160C7×D4—82no—
161C7×D4—82no—
162C7×D4—82no—
163C7×D4—81no—
164C7×D4—81no—
165C2×C4×C7—22no—
166C2×C4×C7—22no—
167C2×C4×C7—24no—
168C2×C4×C7—42no—
169C2×C4×C7—42no—
170C2×C4×C7—41no—
171C2×C4×C7—41no—
172C2×C4×C7—42no—
173C2×C4×C7—44no—
174C2×C4×C7#12011yescharacteristic
175C2×C4×C7#12011yescharacteristic
176C2×C4×C7#12111yescharacteristic
177C2×C4×C7#12111yescharacteristic
178C2×C4×C7#12111yescharacteristic
179C2×C4×C7#12711yescharacteristic
180C2×C4×C7C2×SD1612yes—
181C2×C4×C7AGL(1,ℤ/8)12yes—
182C7×C8—42no—
183C7×C8—42no—
184C7×C8—81no—
185C22×Q8—21no—
186C22×D4—21no—
187C22×D4—21no—
188C22×D4C7×D412yes—
189C2×SD16—24no—
190C2×M16—21no—
191C4⋊Q8—21no—
192C4⋊Q8—21no—
193C4⋊Q8—21no—
194C4⋊Q8—21no—
195C4⋊Q8C7×D411yescharacteristic
196C4⋊Q8C7×D411yescharacteristic
197#127—21no—
198#127—21no—
199#127—21no—
200#127—21no—
201#125—21no—
202#124—44no—
203#124—44no—
204C22⋊Q8—21no—
205C22⋊Q8—21no—
206C4×Q8—21no—
207C4×Q8—21no—
208#117—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4C7×D412yes—
211C2×C22⋊C4—28no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42C7×D412yes—
215C2×C42C23×C711yescharacteristic
216#106—24no—
217C4⋊C8—21no—
218C4⋊C8—21no—
219C4⋊C8C7×D412yes—
220C4⋊C8C7×D412yes—
221#103—24no—
222#102—24no—
223C22⋊C8—21no—
224C22⋊C8—21no—
225C4×C8—21no—
226C4×C8—21no—
227C22×C7—21no—
228C22×C7—21no—
229C22×C7—88no—
230C22×C7—82no—
231C22×C7—82no—
232C22×C7—82no—
233C22×C7#39211yescharacteristic
234C22×C7#45012yes—
235C4×C7—21no—
236C4×C7—21no—
237C4×C7—44no—
238C4×C7—81no—
239C4×C7—81no—
240C4×C7—82no—
241C4×C7#45412yes—
242C4×C7#45512yes—
243C4×C7#46012yes—
244C4×C7#49211yescharacteristic
245C4×C7#49611yescharacteristic
246C24—22no—
247C2×Q8—21no—
248C2×Q8—21no—
249C2×Q8—41no—
250C2×Q8—41no—
251C2×Q8—42no—
252C2×Q8—42no—
253C2×D4—22no—
254C2×D4—22no—
255C2×D4—44no—
256C2×D4—44no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—41no—
261C2×D4—42no—
262C2×D4—42no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4C2×C7×D411yescharacteristic
266C22×C4C2×C7×D411yescharacteristic
267C22×C4C2×C7×D411yescharacteristic
268SD16—84no—
269M16—42no—
270C2×C8—22no—
271C2×C8—22no—
272C2×C8—41no—
273C4⋊C4—21no—
274C4⋊C4—22no—
275C4⋊C4—21no—
276C4⋊C4—22no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C22⋊C4—416no—
284C22⋊C4—41no—
285C42—22no—
286C42—42no—
287C42—42no—
288C42—41no—
289C42C7×SD1614yescomplemented
290C42C2×C7×D411yescharacteristic
291C42C2×C7×D411yescharacteristic
292C42C2×C7×D411yescharacteristic
293C42C7×#4111yescharacteristic
294C2×C7—22no—
295C2×C7—162no—
296C2×C7#200311yescharacteristic
297C2×C7#200411yescharacteristic
298C2×C7#201411yescharacteristic
299C23—44no—
300C23—44no—
301C23—42no—
302C23—44no—
303C23C7×#12011yescharacteristic
304Q8—42no—
305Q8—42no—
306Q8—81no—
307Q8—81no—
308Q8—82no—
309D4—44no—
310D4—82no—
311D4—82no—
312D4—81no—
313D4—81no—
314D4—82no—
315D4—82no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—24no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—42no—
324C2×C4—44no—
325C2×C4C7×#12011yescharacteristic
326C2×C4C7×#12011yescharacteristic
327C2×C4C7×#12111yescharacteristic
328C2×C4C7×#12111yescharacteristic
329C2×C4C7×#12111yescharacteristic
330C2×C4C7×#12711yescharacteristic
331C2×C4C2×C7×SD1612yes—
332C2×C4C7×AGL(1,ℤ/8)12yes—
333C8—42no—
334C8—42no—
335C8—81no—
336C7#1029411yescharacteristic, direct product factor, Sylow
337C22—21no—
338C22—21no—
339C22—88no—
340C22—82no—
341C22—82no—
342C22—82no—
343C22C7×#39211yescharacteristic
344C22C7×#45012yes—
345C4—21no—
346C4—21no—
347C4—44no—
348C4—81no—
349C4—81no—
350C4—82no—
351C4C7×#45412yes—
352C4C7×#45512yes—
353C4C7×#46012yes—
354C4C7×#49211yescharacteristic
355C4C7×#49611yescharacteristic
356C2—22no—
357C2—162no—
358C2C7×#200311yescharacteristic
359C2C7×#200411yescharacteristic
360C2C7×#201411yescharacteristic
subgroupG/H
center, socleC22×C7#392
commutator, Frattini, cosocleC2×C42C23×C7
prime radicalC2×C7×#127C22
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×#3229C2semidirect
4. C7×#1994C2semidirect
5. C7×#1546C2semidirect
7. C7×#1276C2semidirect
29. #10294C7direct (N × H)
72. #3229C2×C7semidirect
75. #1994C2×C7semidirect
76. #1546C2×C7semidirect
78. #1276C2×C7semidirect
122. C42×C7SD16semidirect
289. C42C7×SD16semidirect
336. C7#10294direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#32292
C7×#32082
C7×#28582
C7×#19942
C7×#15462
C7×#15452
C7×#12762
#102947