Finite Group Databasebeta

← C7×#9289 #13186505

Subgroups — C7×#9289 #13186505

Order 1792.

490 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3206C211yesmaximal, characteristic
2C7×#2932C211yesmaximal, characteristic
3C7×#1993C211yesmaximal, characteristic, complemented
4C7×#1549C211yesmaximal, characteristic, complemented
5C7×#1517C211yesmaximal, characteristic
6C7×#1286C211yesmaximal, characteristic, complemented
7C7×#1275C211yesmaximal, characteristic, complemented
8C2×C7×#131—21no—
9C7×#532C2211yescharacteristic
10C2×C7×C4⋊Q8C2211yescharacteristic
11C7×#492C412yescomplemented
12C7×#487C412yescomplemented
13C7×#461—22no—
14C7×#448—22no—
15C7×#435—22no—
16C7×#435—22no—
17C7×#431C2211yescharacteristic
18C7×#431C2211yescharacteristic
19C7×#423C2211yescharacteristic
20C7×#408—21no—
21C2×C7×#97C2211yescharacteristic
22C7×#382C2211yescharacteristic
23C7×#340—21no—
24C7×#340—21no—
25C7×#332—22no—
26C7×#327—22no—
27#9289C711yesmaximal, characteristic, direct product factor, Sylow
28C2×C7×#41—21no—
29C22×C7×Q8—21no—
30C2×C7×SD16—24no—
31C7×#131—42no—
32C7×#131—42no—
33C2×C7×M16—22no—
34C2×C7×M16—21no—
35C22×C7×C8—22no—
36C22×C7×C8D411yescharacteristic
37C22×C7×C8D411yescharacteristic
38C7×C4⋊Q8—21no—
39C7×C4⋊Q8—21no—
40C7×C4⋊Q8—21no—
41C7×C4⋊Q8C2×C412yes—
42C7×C4⋊Q8C2×C411yescharacteristic
43C7×#127C2×C411yescharacteristic
44C7×#124—21no—
45C7×#121—21no—
46C7×#121D412yes—
47C4×C7×D4—21no—
48C4×C7×D4D412yes—
49C2×C7×C4⋊C4—21no—
50C2×C7×C4⋊C4—21no—
51C2×C7×C4⋊C4—21no—
52C2×C42×C7C2311yescharacteristic
53C7×C4⋊C8—21no—
54C7×C4⋊C8—21no—
55C7×C4⋊C8—21no—
56C7×C4⋊C8D412yes—
57C7×#103—24no—
58C7×C22⋊C8—24no—
59C7×C22⋊C8—21no—
60C7×C22⋊C8—21no—
61C7×#97—21no—
62C7×#97—21no—
63C4×C7×C8—21no—
64C4×C7×C8C2×C412yes—
65C7×#95—21no—
66C7×#95—21no—
67#3206C2×C711yescharacteristic
68#2932C2×C711yescharacteristic
69#1993C2×C711yescharacteristic, complemented
70#1549C2×C711yescharacteristic, complemented
71#1517C2×C711yescharacteristic
72#1286C2×C711yescharacteristic, complemented
73#1275C2×C711yescharacteristic, complemented
74C7×#41—41no—
75C7×#41—41no—
76C7×#41—41no—
77C7×#41—41no—
78C2×C7×Q8—21no—
79C2×C7×Q8—22no—
80C2×C7×Q8—41no—
81C2×C7×Q8—82no—
82C2×C7×D4—22no—
83C2×C7×D4—21no—
84C2×C7×D4—21no—
85C2×C7×D4C22⋊C412yes—
86C22×C4×C7—22no—
87C22×C4×C7—41no—
88C22×C4×C7—41no—
89C22×C4×C7C2×D411yescharacteristic
90C22×C4×C7C2×D411yescharacteristic
91C22×C4×C7#4111yescharacteristic
92C7×SD16—84no—
93C7×M16—44no—
94C7×M16—41no—
95C7×M16—41no—
96C2×C7×C8—21no—
97C2×C7×C8—21no—
98C2×C7×C8—22no—
99C2×C7×C8—21no—
100C2×C7×C8—22no—
101C2×C7×C8—21no—
102C2×C7×C8—21no—
103C2×C7×C8—44no—
104C2×C7×C8—41no—
105C2×C7×C8SD1612yescomplemented
106C7×C4⋊C4—21no—
107C7×C4⋊C4—22no—
108C7×C4⋊C4—22no—
109C7×C4⋊C4—41no—
110C7×C4⋊C4—41no—
111C7×C4⋊C4—41no—
112C7×C4⋊C4—41no—
113C7×C4⋊C4C22⋊C412yes—
114C7×C22⋊C4—22no—
115C7×C22⋊C4—22no—
116C42×C7C22×C411yescharacteristic
117C42×C7C2×D411yescharacteristic
118C42×C7C2×D411yescharacteristic
119C42×C7#4111yescharacteristic
120C2×#131—21no—
121#532C22×C711yescharacteristic
122C2×C4⋊Q8C22×C711yescharacteristic
123#492C4×C712yescomplemented
124#487C4×C712yescomplemented
125#461—22no—
126#448—22no—
127#435—22no—
128#435—22no—
129#431C22×C711yescharacteristic
130#431C22×C711yescharacteristic
131#423C22×C711yescharacteristic
132#408—21no—
133C2×#97C22×C711yescharacteristic
134#382C22×C711yescharacteristic
135#340—21no—
136#340—21no—
137#332—22no—
138#327—22no—
139C23×C7—22no—
140C23×C7#12111yescharacteristic
141C7×Q8—44no—
142C7×Q8—41no—
143C7×Q8—41no—
144C7×Q8—82no—
145C7×D4—42no—
146C7×D4—42no—
147C7×D4—42no—
148C7×D4—41no—
149C7×D4—41no—
150C7×D4—81no—
151C2×C4×C7—22no—
152C2×C4×C7—41no—
153C2×C4×C7—41no—
154C2×C4×C7—42no—
155C2×C4×C7—42no—
156C2×C4×C7—42no—
157C2×C4×C7—82no—
158C2×C4×C7—81no—
159C2×C4×C7—81no—
160C2×C4×C7—81no—
161C2×C4×C7C2×C22⋊C411yescharacteristic
162C2×C4×C7C4×D411yescharacteristic
163C2×C4×C7C4×D411yescharacteristic
164C2×C4×C7#12011yescharacteristic
165C2×C4×C7#12111yescharacteristic
166C2×C4×C7#12311yescharacteristic
167C2×C4×C7C2×SD1611yescharacteristic
168C2×C4×C7#13511yescharacteristic
169C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
170C2×C4×C7#13711yescharacteristic
171C7×C8—22no—
172C7×C8—42no—
173C7×C8—42no—
174C7×C8—41no—
175C7×C8—81no—
176C2×#41—21no—
177C22×Q8—21no—
178C2×SD16—24no—
179#131—42no—
180#131—42no—
181C2×M16—22no—
182C2×M16—21no—
183C22×C8—22no—
184C22×C8C7×D411yescharacteristic
185C22×C8C7×D411yescharacteristic
186C4⋊Q8—21no—
187C4⋊Q8—21no—
188C4⋊Q8—21no—
189C4⋊Q8C2×C4×C712yes—
190C4⋊Q8C2×C4×C711yescharacteristic
191#127C2×C4×C711yescharacteristic
192#124—21no—
193#121—21no—
194#121C7×D412yes—
195C4×D4—21no—
196C4×D4C7×D412yes—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C42C23×C711yescharacteristic
201C4⋊C8—21no—
202C4⋊C8—21no—
203C4⋊C8—21no—
204C4⋊C8C7×D412yes—
205#103—24no—
206C22⋊C8—24no—
207C22⋊C8—21no—
208C22⋊C8—21no—
209#97—21no—
210#97—21no—
211C4×C8—21no—
212C4×C8C2×C4×C712yes—
213#95—21no—
214#95—21no—
215C22×C7—21no—
216C22×C7—21no—
217C22×C7—42no—
218C22×C7—42no—
219C22×C7—42no—
220C22×C7#38611yescharacteristic
221C22×C7#46111yescharacteristic
222C22×C7#46311yescharacteristic
223C4×C7—21no—
224C4×C7—21no—
225C4×C7—21no—
226C4×C7—21no—
227C4×C7—42no—
228C4×C7—82no—
229C4×C7—161no—
230C4×C7C4×SD1611yescharacteristic
231C4×C7#44011yescharacteristic
232C4×C7#45011yescharacteristic
233C4×C7#45211yescharacteristic
234#41—41no—
235#41—41no—
236#41—41no—
237#41—41no—
238C2×Q8—21no—
239C2×Q8—22no—
240C2×Q8—41no—
241C2×Q8—82no—
242C2×D4—22no—
243C2×D4—21no—
244C2×D4—21no—
245C2×D4C7×C22⋊C412yes—
246C22×C4—22no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4C2×C7×D411yescharacteristic
250C22×C4C2×C7×D411yescharacteristic
251C22×C4C7×#4111yescharacteristic
252SD16—84no—
253M16—44no—
254M16—41no—
255M16—41no—
256C2×C8—21no—
257C2×C8—21no—
258C2×C8—22no—
259C2×C8—21no—
260C2×C8—22no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8—44no—
264C2×C8—41no—
265C2×C8C7×SD1612yescomplemented
266C4⋊C4—21no—
267C4⋊C4—22no—
268C4⋊C4—22no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4C7×C22⋊C412yes—
274C22⋊C4—22no—
275C22⋊C4—22no—
276C42C22×C4×C711yescharacteristic
277C42C2×C7×D411yescharacteristic
278C42C2×C7×D411yescharacteristic
279C42C7×#4111yescharacteristic
280C2×C7—21no—
281C2×C7—21no—
282C2×C7—82no—
283C2×C7#187711yescharacteristic
284C2×C7#190211yescharacteristic
285C2×C7#191011yescharacteristic
286C23—22no—
287C23C7×#12111yescharacteristic
288Q8—44no—
289Q8—41no—
290Q8—41no—
291Q8—82no—
292D4—42no—
293D4—42no—
294D4—42no—
295D4—41no—
296D4—41no—
297D4—81no—
298C2×C4—22no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—82no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4C2×C7×C22⋊C411yescharacteristic
309C2×C4C4×C7×D411yescharacteristic
310C2×C4C4×C7×D411yescharacteristic
311C2×C4C7×#12011yescharacteristic
312C2×C4C7×#12111yescharacteristic
313C2×C4C7×#12311yescharacteristic
314C2×C4C2×C7×SD1611yescharacteristic
315C2×C4C7×#13511yescharacteristic
316C2×C4C7×AGL(1,ℤ/8)11yescharacteristic
317C2×C4C7×#13711yescharacteristic
318C8—22no—
319C8—42no—
320C8—42no—
321C8—41no—
322C8—81no—
323C7#928911yescharacteristic, direct product factor, Sylow
324C22—21no—
325C22—21no—
326C22—42no—
327C22—42no—
328C22—42no—
329C22C7×#38611yescharacteristic
330C22C7×#46111yescharacteristic
331C22C7×#46311yescharacteristic
332C4—21no—
333C4—21no—
334C4—21no—
335C4—21no—
336C4—42no—
337C4—82no—
338C4—161no—
339C4C4×C7×SD1611yescharacteristic
340C4C7×#44011yescharacteristic
341C4C7×#45011yescharacteristic
342C4C7×#45211yescharacteristic
343C2—21no—
344C2—21no—
345C2—82no—
346C2C7×#187711yescharacteristic
347C2C7×#190211yescharacteristic
348C2C7×#191011yescharacteristic
subgroupG/H
center, socleC22×C7#386
commutatorC42C22×C4×C7
Frattini, cosocleC2×C42C23×C7
prime radicalC7×#532C22
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
3. C7×#1993C2semidirect
4. C7×#1549C2semidirect
6. C7×#1286C2semidirect
7. C7×#1275C2semidirect
11. C7×#492C4semidirect
12. C7×#487C4semidirect
27. #9289C7direct (N × H)
69. #1993C2×C7semidirect
70. #1549C2×C7semidirect
72. #1286C2×C7semidirect
73. #1275C2×C7semidirect
105. C2×C7×C8SD16semidirect
123. #492C4×C7semidirect
124. #487C4×C7semidirect
265. C2×C8C7×SD16semidirect
323. C7#9289direct (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×#29322
C7×#19932
C7×#15492
C7×#15172
C7×#12862
C7×#12752
#92897