Finite Group Databasebeta

← C7×#9281 #13186497

Subgroups — C7×#9281 #13186497

Order 1792.

456 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3207C211yesmaximal, characteristic
2C7×#2981C211yesmaximal, characteristic, complemented
3C7×#1505C211yesmaximal, characteristic, complemented
4C7×#1505C211yesmaximal, characteristic, complemented
5C7×#1496C211yesmaximal, characteristic, complemented
6C7×#1287C211yesmaximal, characteristic, complemented
7C7×#1274C211yesmaximal, characteristic, complemented
8C7×#532C2211yescharacteristic
9C7×#532C2211yescharacteristic
10C7×#493C411yescharacteristic, complemented
11C7×#492C411yescharacteristic, complemented
12C7×#487C411yescharacteristic, complemented
13C7×#486C411yescharacteristic, complemented
14C7×#463—22no—
15C7×#449—22no—
16C7×#436—21no—
17C7×#435—22no—
18C7×C8×D4—21no—
19C7×#432C2211yescharacteristic
20C7×#432C2211yescharacteristic
21C7×#431C2211yescharacteristic
22C7×#431C2211yescharacteristic
23C7×#408—22no—
24C7×#376C2211yescharacteristic
25C7×#327—22no—
26C7×#327—22no—
27C7×D4⋊C8—21no—
28C7×D4⋊C8—21no—
29#9281C711yesmaximal, characteristic, direct product factor, Sylow
30C2×C7×#41—21no—
31C2×C7×#41—21no—
32C2×C7×SD16—22no—
33C2×C7×D8—22no—
34C2×C7×M16—22no—
35C22×C7×C8—22no—
36C7×C4⋊Q8C2×C411yescharacteristic
37C7×C4⋊Q8C2×C411yescharacteristic
38C7×#127C2×C411yescharacteristic
39C7×#127C2×C411yescharacteristic
40C7×#124—21no—
41C7×#124—21no—
42C7×#121—21no—
43C7×#121—21no—
44C7×#121—21no—
45C7×#121D412yes—
46C4×C7×D4—21no—
47C4×C7×D4—21no—
48C4×C7×D4—21no—
49C4×C7×D4D412yes—
50C2×C42×C7—21no—
51C2×C42×C7C2311yescharacteristic
52C7×C4⋊C8—21no—
53C7×C4⋊C8—21no—
54C7×C4⋊C8—21no—
55C7×C4⋊C8—21no—
56C7×#103—22no—
57C7×#102—22no—
58C7×C22⋊C8—22no—
59C7×C22⋊C8—22no—
60C7×C22⋊C8—21no—
61C7×C22⋊C8—21no—
62C7×C22⋊C8—21no—
63C7×C22⋊C8D412yes—
64C7×#97—21no—
65C7×#97—21no—
66C4×C7×C8C2×C411yescharacteristic
67C4×C7×C8C2×C411yescharacteristic
68C4×C7×C8D411yescharacteristic
69C4×C7×C8D411yescharacteristic
70C7×#95—21no—
71C7×#95—21no—
72#3207C2×C711yescharacteristic
73#2981C2×C711yescharacteristic, complemented
74#1505C2×C711yescharacteristic, complemented
75#1505C2×C711yescharacteristic, complemented
76#1496C2×C711yescharacteristic, complemented
77#1287C2×C711yescharacteristic, complemented
78#1274C2×C711yescharacteristic, complemented
79C7×#41—82no—
80C7×#41—82no—
81C2×C7×Q8—21no—
82C2×C7×Q8—21no—
83C2×C7×D4—22no—
84C2×C7×D4—21no—
85C2×C7×D4—21no—
86C2×C7×D4—21no—
87C2×C7×D4—21no—
88C2×C7×D4—21no—
89C2×C7×D4—41no—
90C2×C7×D4C22⋊C412yes—
91C22×C4×C7—22no—
92C22×C4×C7—41no—
93C22×C4×C7—41no—
94C22×C4×C7C2×D411yescharacteristic
95C22×C4×C7C2×D411yescharacteristic
96C22×C4×C7#4111yescharacteristic
97C7×SD16—82no—
98C7×D8—82no—
99C7×M16—82no—
100C2×C7×C8—21no—
101C2×C7×C8—21no—
102C2×C7×C8—21no—
103C2×C7×C8—21no—
104C2×C7×C8—22no—
105C2×C7×C8—41no—
106C2×C7×C8—82no—
107C7×C4⋊C4—21no—
108C7×C4⋊C4—21no—
109C7×C4⋊C4—21no—
110C7×C4⋊C4C22⋊C412yes—
111C7×C22⋊C4—22no—
112C7×C22⋊C4—22no—
113C7×C22⋊C4—41no—
114C7×C22⋊C4—41no—
115C42×C7—22no—
116C42×C7—41no—
117C42×C7C22×C411yescharacteristic
118C42×C7C2×D411yescharacteristic
119C42×C7C2×D411yescharacteristic
120C42×C7#4111yescharacteristic
121#532C22×C711yescharacteristic
122#532C22×C711yescharacteristic
123#493C4×C711yescharacteristic, complemented
124#492C4×C711yescharacteristic, complemented
125#487C4×C711yescharacteristic, complemented
126#486C4×C711yescharacteristic, complemented
127#463—22no—
128#449—22no—
129#436—21no—
130#435—22no—
131C8×D4—21no—
132#432C22×C711yescharacteristic
133#432C22×C711yescharacteristic
134#431C22×C711yescharacteristic
135#431C22×C711yescharacteristic
136#408—22no—
137#376C22×C711yescharacteristic
138#327—22no—
139#327—22no—
140D4⋊C8—21no—
141D4⋊C8—21no—
142C23×C7—22no—
143C23×C7—41no—
144C23×C7#12111yescharacteristic
145C7×Q8—42no—
146C7×Q8—81no—
147C7×D4—42no—
148C7×D4—42no—
149C7×D4—41no—
150C7×D4—41no—
151C7×D4—41no—
152C7×D4—41no—
153C7×D4—81no—
154C7×D4—81no—
155C7×D4—82no—
156C7×D4—82no—
157C2×C4×C7—21no—
158C2×C4×C7—21no—
159C2×C4×C7—22no—
160C2×C4×C7—41no—
161C2×C4×C7—41no—
162C2×C4×C7—42no—
163C2×C4×C7—42no—
164C2×C4×C7—42no—
165C2×C4×C7—41no—
166C2×C4×C7—82no—
167C2×C4×C7—81no—
168C2×C4×C7—81no—
169C2×C4×C7C2×C22⋊C411yescharacteristic
170C2×C4×C7C4×D411yescharacteristic
171C2×C4×C7C4×D411yescharacteristic
172C2×C4×C7#12011yescharacteristic
173C2×C4×C7#12111yescharacteristic
174C2×C4×C7#12311yescharacteristic
175C7×C8—41no—
176C7×C8—42no—
177C7×C8—41no—
178C7×C8—41no—
179C7×C8—41no—
180C7×C8—81no—
181C2×#41—21no—
182C2×#41—21no—
183C2×SD16—22no—
184C2×D8—22no—
185C2×M16—22no—
186C22×C8—22no—
187C4⋊Q8C2×C4×C711yescharacteristic
188C4⋊Q8C2×C4×C711yescharacteristic
189#127C2×C4×C711yescharacteristic
190#127C2×C4×C711yescharacteristic
191#124—21no—
192#124—21no—
193#121—21no—
194#121—21no—
195#121—21no—
196#121C7×D412yes—
197C4×D4—21no—
198C4×D4—21no—
199C4×D4—21no—
200C4×D4C7×D412yes—
201C2×C42—21no—
202C2×C42C23×C711yescharacteristic
203C4⋊C8—21no—
204C4⋊C8—21no—
205C4⋊C8—21no—
206C4⋊C8—21no—
207#103—22no—
208#102—22no—
209C22⋊C8—22no—
210C22⋊C8—22no—
211C22⋊C8—21no—
212C22⋊C8—21no—
213C22⋊C8—21no—
214C22⋊C8C7×D412yes—
215#97—21no—
216#97—21no—
217C4×C8C2×C4×C711yescharacteristic
218C4×C8C2×C4×C711yescharacteristic
219C4×C8C7×D411yescharacteristic
220C4×C8C7×D411yescharacteristic
221#95—21no—
222#95—21no—
223C22×C7—21no—
224C22×C7—21no—
225C22×C7—21no—
226C22×C7—42no—
227C22×C7—42no—
228C22×C7—42no—
229C22×C7—81no—
230C22×C7—81no—
231C22×C7—81no—
232C22×C7#38611yescharacteristic
233C4×C7—21no—
234C4×C7—21no—
235C4×C7—21no—
236C4×C7—21no—
237C4×C7—21no—
238C4×C7—42no—
239C4×C7—82no—
240C4×C7—81no—
241C4×C7#44311yescharacteristic
242C4×C7#44411yescharacteristic
243#41—82no—
244#41—82no—
245C2×Q8—21no—
246C2×Q8—21no—
247C2×D4—22no—
248C2×D4—21no—
249C2×D4—21no—
250C2×D4—21no—
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—41no—
254C2×D4C7×C22⋊C412yes—
255C22×C4—22no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4C2×C7×D411yescharacteristic
259C22×C4C2×C7×D411yescharacteristic
260C22×C4C7×#4111yescharacteristic
261SD16—82no—
262D8—82no—
263M16—82no—
264C2×C8—21no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—22no—
269C2×C8—41no—
270C2×C8—82no—
271C4⋊C4—21no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4C7×C22⋊C412yes—
275C22⋊C4—22no—
276C22⋊C4—22no—
277C22⋊C4—41no—
278C22⋊C4—41no—
279C42—22no—
280C42—41no—
281C42C22×C4×C711yescharacteristic
282C42C2×C7×D411yescharacteristic
283C42C2×C7×D411yescharacteristic
284C42C7×#4111yescharacteristic
285C2×C7—41no—
286C2×C7—82no—
287C2×C7—161no—
288C2×C7#187611yescharacteristic
289C2×C7#190411yescharacteristic
290C2×C7#191011yescharacteristic
291C23—22no—
292C23—41no—
293C23C7×#12111yescharacteristic
294Q8—42no—
295Q8—81no—
296D4—42no—
297D4—42no—
298D4—41no—
299D4—41no—
300D4—41no—
301D4—41no—
302D4—81no—
303D4—81no—
304D4—82no—
305D4—82no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—41no—
315C2×C4—82no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4C2×C7×C22⋊C411yescharacteristic
319C2×C4C4×C7×D411yescharacteristic
320C2×C4C4×C7×D411yescharacteristic
321C2×C4C7×#12011yescharacteristic
322C2×C4C7×#12111yescharacteristic
323C2×C4C7×#12311yescharacteristic
324C8—41no—
325C8—42no—
326C8—41no—
327C8—41no—
328C8—41no—
329C8—81no—
330C7#928111yescharacteristic, direct product factor, Sylow
331C22—21no—
332C22—21no—
333C22—21no—
334C22—42no—
335C22—42no—
336C22—42no—
337C22—81no—
338C22—81no—
339C22—81no—
340C22C7×#38611yescharacteristic
341C4—21no—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—42no—
347C4—82no—
348C4—81no—
349C4C7×#44311yescharacteristic
350C4C7×#44411yescharacteristic
351C2—41no—
352C2—82no—
353C2—161no—
354C2C7×#187611yescharacteristic
355C2C7×#190411yescharacteristic
356C2C7×#191011yescharacteristic
subgroupG/H
center, socleC22×C7#386
commutatorC42C22×C4×C7
Frattini, cosocleC2×C42C23×C7
prime radicalC7×#3207C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#2981C2semidirect
3. C7×#1505C2semidirect
4. C7×#1505C2semidirect
5. C7×#1496C2semidirect
6. C7×#1287C2semidirect
7. C7×#1274C2semidirect
10. C7×#493C4semidirect
11. C7×#492C4semidirect
12. C7×#487C4semidirect
13. C7×#486C4semidirect
29. #9281C7direct (N × H)
73. #2981C2×C7semidirect
74. #1505C2×C7semidirect
75. #1505C2×C7semidirect
76. #1496C2×C7semidirect
77. #1287C2×C7semidirect
78. #1274C2×C7semidirect
123. #493C4×C7semidirect
124. #492C4×C7semidirect
125. #487C4×C7semidirect
126. #486C4×C7semidirect
330. C7#9281direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#32072
C7×#29812
C7×#15052
C7×#14962
C7×#12872
C7×#12742
#92817