Finite Group Databasebeta

← C7×#9302 #13186518

Subgroups — C7×#9302 #13186518

Order 1792.

480 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3227C211yesmaximal, characteristic
2C7×#2331C211yesmaximal, characteristic, complemented
3C7×#1963C211yesmaximal, characteristic
4C7×#1549C211yesmaximal, characteristic, complemented
5C7×#1517C211yesmaximal, characteristic
6C7×#1286C211yesmaximal, characteristic, complemented
7C7×#1274C211yesmaximal, characteristic, complemented
8C7×#532C2211yescharacteristic
9C2×C7×C4⋊Q8C2211yescharacteristic
10C7×#498C412yescomplemented
11C7×#486C412yescomplemented
12C7×#481—22no—
13C7×#476—22no—
14C7×#431C2211yescharacteristic
15C7×#431C2211yescharacteristic
16C7×#423C2211yescharacteristic
17C2×C7×#104—21no—
18C2×C7×#104—21no—
19C7×#391—22no—
20C7×#389—21no—
21C7×#387—22no—
22C7×#382C2211yescharacteristic
23C4×C7×C4⋊C4—21no—
24C7×#376C2211yescharacteristic
25C7×#332—22no—
26C7×#327—22no—
27#9302C711yesmaximal, characteristic, direct product factor, Sylow
28C2×C7×#41—21no—
29C22×C7×Q8—21no—
30C2×C7×M16—21no—
31C7×C4⋊Q8—21no—
32C7×C4⋊Q8—21no—
33C7×C4⋊Q8C2×C411yescharacteristic
34C7×C4⋊Q8C2×C412yes—
35C7×C4⋊Q8D412yes—
36C7×#127C2×C411yescharacteristic
37C7×#124—21no—
38C7×#121—21no—
39C7×#121—21no—
40C4×C7×D4—21no—
41C4×C7×D4—21no—
42C2×C7×C4⋊C4—21no—
43C2×C7×C4⋊C4—21no—
44C2×C7×C4⋊C4—22no—
45C2×C7×C4⋊C4D412yes—
46C2×C42×C7—22no—
47C2×C42×C7D411yescharacteristic
48C2×C42×C7D411yescharacteristic
49C2×C42×C7C2311yescharacteristic
50C7×#106—24no—
51C7×C4⋊C8—21no—
52C7×C4⋊C8—21no—
53C7×C4⋊C8—21no—
54C7×C4⋊C8—21no—
55C7×#104—42no—
56C7×#104—42no—
57C7×#102—24no—
58C7×C22⋊C8—21no—
59C7×C22⋊C8—21no—
60C4×C7×C8—21no—
61C4×C7×C8C2×C412yes—
62C7×#95—22no—
63C7×#95—22no—
64C7×#95—21no—
65C7×#95—21no—
66C7×#95—21no—
67C7×#95D412yes—
68#3227C2×C711yescharacteristic
69#2331C2×C711yescharacteristic, complemented
70#1963C2×C711yescharacteristic
71#1549C2×C711yescharacteristic, complemented
72#1517C2×C711yescharacteristic
73#1286C2×C711yescharacteristic, complemented
74#1274C2×C711yescharacteristic, complemented
75C7×#41—41no—
76C7×#41—41no—
77C7×#41—41no—
78C7×#41—41no—
79C2×C7×Q8—22no—
80C2×C7×Q8—22no—
81C2×C7×Q8—21no—
82C2×C7×Q8—82no—
83C2×C7×D4—21no—
84C2×C7×D4—21no—
85C2×C7×D4—21no—
86C2×C7×D4—41no—
87C22×C4×C7—21no—
88C22×C4×C7—22no—
89C22×C4×C7—21no—
90C22×C4×C7—22no—
91C22×C4×C7—41no—
92C22×C4×C7C2×D411yescharacteristic
93C22×C4×C7C2×D411yescharacteristic
94C22×C4×C7C2×D411yescharacteristic
95C7×M16—41no—
96C7×M16—41no—
97C2×C7×C8—22no—
98C2×C7×C8—41no—
99C2×C7×C8—41no—
100C7×C4⋊C4—22no—
101C7×C4⋊C4—22no—
102C7×C4⋊C4—22no—
103C7×C4⋊C4—21no—
104C7×C4⋊C4—21no—
105C7×C4⋊C4—41no—
106C7×C4⋊C4—41no—
107C7×C4⋊C4—82no—
108C7×C4⋊C4C22⋊C414yes—
109C7×C22⋊C4—41no—
110C7×C22⋊C4—41no—
111C42×C7—21no—
112C42×C7—21no—
113C42×C7—21no—
114C42×C7—82no—
115C42×C7SD1612yes—
116C42×C7C22×C411yescharacteristic
117C42×C7C2×D411yescharacteristic
118C42×C7#4111yescharacteristic
119C42×C7#4111yescharacteristic
120#532C22×C711yescharacteristic
121C2×C4⋊Q8C22×C711yescharacteristic
122#498C4×C712yescomplemented
123#486C4×C712yescomplemented
124#481—22no—
125#476—22no—
126#431C22×C711yescharacteristic
127#431C22×C711yescharacteristic
128#423C22×C711yescharacteristic
129C2×#104—21no—
130C2×#104—21no—
131#391—22no—
132#389—21no—
133#387—22no—
134#382C22×C711yescharacteristic
135C4×C4⋊C4—21no—
136#376C22×C711yescharacteristic
137#332—22no—
138#327—22no—
139C23×C7—41no—
140C23×C7#12011yescharacteristic
141C7×Q8—41no—
142C7×Q8—41no—
143C7×Q8—82no—
144C7×Q8—82no—
145C7×D4—42no—
146C7×D4—41no—
147C7×D4—41no—
148C7×D4—81no—
149C7×D4—81no—
150C7×D4—81no—
151C2×C4×C7—24no—
152C2×C4×C7—22no—
153C2×C4×C7—41no—
154C2×C4×C7—41no—
155C2×C4×C7—41no—
156C2×C4×C7—42no—
157C2×C4×C7—42no—
158C2×C4×C7—41no—
159C2×C4×C7—42no—
160C2×C4×C7—41no—
161C2×C4×C7—41no—
162C2×C4×C7—42no—
163C2×C4×C7—41no—
164C2×C4×C7—41no—
165C2×C4×C7—42no—
166C2×C4×C7—81no—
167C2×C4×C7—81no—
168C2×C4×C7C2×C22⋊C411yescharacteristic
169C2×C4×C7C4×D411yescharacteristic
170C2×C4×C7C4×D411yescharacteristic
171C2×C4×C7#12111yescharacteristic
172C2×C4×C7#12111yescharacteristic
173C2×C4×C7#12311yescharacteristic
174C2×C4×C7C2×SD1611yescharacteristic
175C2×C4×C7#13511yescharacteristic
176C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
177C2×C4×C7#13711yescharacteristic
178C7×C8—42no—
179C7×C8—81no—
180C7×C8—81no—
181C2×#41—21no—
182C22×Q8—21no—
183C2×M16—21no—
184C4⋊Q8—21no—
185C4⋊Q8—21no—
186C4⋊Q8C2×C4×C711yescharacteristic
187C4⋊Q8C2×C4×C712yes—
188C4⋊Q8C7×D412yes—
189#127C2×C4×C711yescharacteristic
190#124—21no—
191#121—21no—
192#121—21no—
193C4×D4—21no—
194C4×D4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4C7×D412yes—
199C2×C42—22no—
200C2×C42C7×D411yescharacteristic
201C2×C42C7×D411yescharacteristic
202C2×C42C23×C711yescharacteristic
203#106—24no—
204C4⋊C8—21no—
205C4⋊C8—21no—
206C4⋊C8—21no—
207C4⋊C8—21no—
208#104—42no—
209#104—42no—
210#102—24no—
211C22⋊C8—21no—
212C22⋊C8—21no—
213C4×C8—21no—
214C4×C8C2×C4×C712yes—
215#95—22no—
216#95—22no—
217#95—21no—
218#95—21no—
219#95—21no—
220#95C7×D412yes—
221C22×C7—21no—
222C22×C7—21no—
223C22×C7—81no—
224C22×C7—81no—
225C22×C7—81no—
226C22×C7#38611yescharacteristic
227C22×C7#44811yescharacteristic
228C22×C7#44911yescharacteristic
229C4×C7—21no—
230C4×C7—21no—
231C4×C7—21no—
232C4×C7—21no—
233C4×C7—42no—
234C4×C7—44no—
235C4×C7—82no—
236C4×C7—81no—
237C4×C7—81no—
238C4×C7C4×SD1611yescharacteristic
239C4×C7#44011yescharacteristic
240C4×C7#46011yescharacteristic
241C4×C7#46411yescharacteristic
242#41—41no—
243#41—41no—
244#41—41no—
245#41—41no—
246C2×Q8—22no—
247C2×Q8—22no—
248C2×Q8—21no—
249C2×Q8—82no—
250C2×D4—21no—
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—41no—
254C22×C4—21no—
255C22×C4—22no—
256C22×C4—21no—
257C22×C4—22no—
258C22×C4—41no—
259C22×C4C2×C7×D411yescharacteristic
260C22×C4C2×C7×D411yescharacteristic
261C22×C4C2×C7×D411yescharacteristic
262M16—41no—
263M16—41no—
264C2×C8—22no—
265C2×C8—41no—
266C2×C8—41no—
267C4⋊C4—22no—
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—82no—
275C4⋊C4C7×C22⋊C414yes—
276C22⋊C4—41no—
277C22⋊C4—41no—
278C42—21no—
279C42—21no—
280C42—21no—
281C42—82no—
282C42C7×SD1612yes—
283C42C22×C4×C711yescharacteristic
284C42C2×C7×D411yescharacteristic
285C42C7×#4111yescharacteristic
286C42C7×#4111yescharacteristic
287C2×C7—21no—
288C2×C7—21no—
289C2×C7—161no—
290C2×C7#188011yescharacteristic
291C2×C7#190411yescharacteristic
292C2×C7#191211yescharacteristic
293C23—41no—
294C23C7×#12011yescharacteristic
295Q8—41no—
296Q8—41no—
297Q8—82no—
298Q8—82no—
299D4—42no—
300D4—41no—
301D4—41no—
302D4—81no—
303D4—81no—
304D4—81no—
305C2×C4—24no—
306C2×C4—22no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—42no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4C2×C7×C22⋊C411yescharacteristic
323C2×C4C4×C7×D411yescharacteristic
324C2×C4C4×C7×D411yescharacteristic
325C2×C4C7×#12111yescharacteristic
326C2×C4C7×#12111yescharacteristic
327C2×C4C7×#12311yescharacteristic
328C2×C4C2×C7×SD1611yescharacteristic
329C2×C4C7×#13511yescharacteristic
330C2×C4C7×AGL(1,ℤ/8)11yescharacteristic
331C2×C4C7×#13711yescharacteristic
332C8—42no—
333C8—81no—
334C8—81no—
335C7#930211yescharacteristic, direct product factor, Sylow
336C22—21no—
337C22—21no—
338C22—81no—
339C22—81no—
340C22—81no—
341C22C7×#38611yescharacteristic
342C22C7×#44811yescharacteristic
343C22C7×#44911yescharacteristic
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—42no—
349C4—44no—
350C4—82no—
351C4—81no—
352C4—81no—
353C4C4×C7×SD1611yescharacteristic
354C4C7×#44011yescharacteristic
355C4C7×#46011yescharacteristic
356C4C7×#46411yescharacteristic
357C2—21no—
358C2—21no—
359C2—161no—
360C2C7×#188011yescharacteristic
361C2C7×#190411yescharacteristic
362C2C7×#191211yescharacteristic
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
2. C7×#2331C2semidirect
4. C7×#1549C2semidirect
6. C7×#1286C2semidirect
7. C7×#1274C2semidirect
10. C7×#498C4semidirect
11. C7×#486C4semidirect
27. #9302C7direct (N × H)
69. #2331C2×C7semidirect
71. #1549C2×C7semidirect
73. #1286C2×C7semidirect
74. #1274C2×C7semidirect
122. #498C4×C7semidirect
123. #486C4×C7semidirect
335. C7#9302direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#32272
C7×#23312
C7×#19632
C7×#15492
C7×#15172
C7×#12862
C7×#12742
#93027