Finite Group Databasebeta

← C7×#9330 #13186546

Subgroups — C7×#9330 #13186546

Order 1792.

482 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×#3112C211yesmaximal, characteristic
2C7×#2320C211yesmaximal, characteristic, complemented
3C7×#1980C211yesmaximal, characteristic
4C7×#1505C211yesmaximal, characteristic
5C7×#1491C211yesmaximal, characteristic, complemented
6C7×#1300C211yesmaximal, characteristic, complemented
7C7×#1285C211yesmaximal, characteristic, complemented
8C7×#532C2211yescharacteristic
9C2×C4×C7×Q8C2211yescharacteristic
10C7×#480—21no—
11C7×#477C412yescomplemented
12C7×#460C412yescomplemented
13C7×#448—21no—
14C7×#440—22no—
15C7×#432C2211yescharacteristic
16C7×#432C2211yescharacteristic
17C2×C7×#104—22no—
18C4×C7×M16C2211yescharacteristic
19C7×#391—21no—
20C7×#389—21no—
21C7×#387—22no—
22C7×#383—21no—
23C7×#382—21no—
24C4×C7×C4⋊C4C2211yescharacteristic
25C7×#376C2211yescharacteristic
26C7×#327—21no—
27C7×#327—21no—
28C7×Q8⋊C8—22no—
29#9330C711yesmaximal, characteristic, direct product factor, Sylow
30C2×C7×#41—21no—
31C22×C7×Q8D411yescharacteristic
32C2×C7×SD16—22no—
33C2×C7×M16—21no—
34C7×C4⋊Q8C2×C411yescharacteristic
35C7×#127C2×C411yescharacteristic
36C7×#124—21no—
37C7×#121—21no—
38C7×#121—21no—
39C4×C7×Q8—21no—
40C4×C7×Q8—21no—
41C4×C7×Q8C2×C412yes—
42C4×C7×Q8D412yes—
43C4×C7×D4—21no—
44C4×C7×D4—21no—
45C2×C7×C4⋊C4—21no—
46C2×C7×C4⋊C4—21no—
47C2×C7×C4⋊C4—22no—
48C2×C7×C4⋊C4D411yescharacteristic
49C2×C42×C7—21no—
50C2×C42×C7—21no—
51C2×C42×C7D412yes—
52C2×C42×C7C2311yescharacteristic
53C7×#107—22no—
54C7×C4⋊C8—21no—
55C7×C4⋊C8C2×C412yes—
56C7×#104—44no—
57C7×#103—22no—
58C7×#102—22no—
59C7×C22⋊C8—21no—
60C7×C22⋊C8—21no—
61C7×#97—21no—
62C7×#97—21no—
63C7×#97—21no—
64C4×C7×C8—21no—
65C7×#95—22no—
66C7×#95—22no—
67C7×#95—22no—
68C7×#95—21no—
69C7×#95—21no—
70C7×#95D411yescharacteristic
71C7×#95D411yescharacteristic
72#3112C2×C711yescharacteristic
73#2320C2×C711yescharacteristic, complemented
74#1980C2×C711yescharacteristic
75#1505C2×C711yescharacteristic
76#1491C2×C711yescharacteristic, complemented
77#1300C2×C711yescharacteristic, complemented
78#1285C2×C711yescharacteristic, complemented
79C7×#41—42no—
80C7×#41—42no—
81C2×C7×Q8—21no—
82C2×C7×Q8—21no—
83C2×C7×Q8—42no—
84C2×C7×Q8C22⋊C412yes—
85C2×C7×D4—21no—
86C2×C7×D4—21no—
87C2×C7×D4—21no—
88C2×C7×D4—41no—
89C22×C4×C7—21no—
90C22×C4×C7—21no—
91C22×C4×C7—21no—
92C22×C4×C7—22no—
93C22×C4×C7—21no—
94C22×C4×C7—41no—
95C22×C4×C7C2×D411yescharacteristic
96C22×C4×C7C2×D411yescharacteristic
97C22×C4×C7C2×D411yescharacteristic
98C7×SD16—44no—
99C7×M16—22no—
100C7×M16—41no—
101C2×C7×C8—22no—
102C2×C7×C8—41no—
103C2×C7×C8—41no—
104C7×C4⋊C4—22no—
105C7×C4⋊C4—22no—
106C7×C4⋊C4—21no—
107C7×C4⋊C4—21no—
108C7×C4⋊C4—21no—
109C7×C4⋊C4—42no—
110C7×C4⋊C4—82no—
111C7×C4⋊C4C22⋊C412yes—
112C7×C22⋊C4—41no—
113C7×C22⋊C4—41no—
114C42×C7—22no—
115C42×C7—22no—
116C42×C7—42no—
117C42×C7—42no—
118C42×C7C22×C411yescharacteristic
119C42×C7C2×D411yescharacteristic
120C42×C7#4111yescharacteristic
121C42×C7#4111yescharacteristic
122#532C22×C711yescharacteristic
123C2×C4×Q8C22×C711yescharacteristic
124#480—21no—
125#477C4×C712yescomplemented
126#460C4×C712yescomplemented
127#448—21no—
128#440—22no—
129#432C22×C711yescharacteristic
130#432C22×C711yescharacteristic
131C2×#104—22no—
132C4×M16C22×C711yescharacteristic
133#391—21no—
134#389—21no—
135#387—22no—
136#383—21no—
137#382—21no—
138C4×C4⋊C4C22×C711yescharacteristic
139#376C22×C711yescharacteristic
140#327—21no—
141#327—21no—
142Q8⋊C8—22no—
143C23×C7—41no—
144C23×C7#12011yescharacteristic
145C7×Q8—22no—
146C7×Q8—22no—
147C7×Q8—42no—
148C7×Q8—42no—
149C7×D4—42no—
150C7×D4—42no—
151C7×D4—81no—
152C7×D4—81no—
153C7×D4—81no—
154C2×C4×C7—21no—
155C2×C4×C7—21no—
156C2×C4×C7—22no—
157C2×C4×C7—22no—
158C2×C4×C7—22no—
159C2×C4×C7—41no—
160C2×C4×C7—42no—
161C2×C4×C7—41no—
162C2×C4×C7—41no—
163C2×C4×C7—41no—
164C2×C4×C7—41no—
165C2×C4×C7—42no—
166C2×C4×C7—41no—
167C2×C4×C7—42no—
168C2×C4×C7—41no—
169C2×C4×C7—41no—
170C2×C4×C7—42no—
171C2×C4×C7—41no—
172C2×C4×C7—41no—
173C2×C4×C7—82no—
174C2×C4×C7C2×C22⋊C411yescharacteristic
175C2×C4×C7C4×D411yescharacteristic
176C2×C4×C7C4×D411yescharacteristic
177C2×C4×C7#12111yescharacteristic
178C2×C4×C7#12111yescharacteristic
179C2×C4×C7#12311yescharacteristic
180C7×C8—42no—
181C7×C8—42no—
182C7×C8—81no—
183C2×#41—21no—
184C22×Q8C7×D411yescharacteristic
185C2×SD16—22no—
186C2×M16—21no—
187C4⋊Q8C2×C4×C711yescharacteristic
188#127C2×C4×C711yescharacteristic
189#124—21no—
190#121—21no—
191#121—21no—
192C4×Q8—21no—
193C4×Q8—21no—
194C4×Q8C2×C4×C712yes—
195C4×Q8C7×D412yes—
196C4×D4—21no—
197C4×D4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4C7×D411yescharacteristic
202C2×C42—21no—
203C2×C42—21no—
204C2×C42C7×D412yes—
205C2×C42C23×C711yescharacteristic
206#107—22no—
207C4⋊C8—21no—
208C4⋊C8C2×C4×C712yes—
209#104—44no—
210#103—22no—
211#102—22no—
212C22⋊C8—21no—
213C22⋊C8—21no—
214#97—21no—
215#97—21no—
216#97—21no—
217C4×C8—21no—
218#95—22no—
219#95—22no—
220#95—22no—
221#95—21no—
222#95—21no—
223#95C7×D411yescharacteristic
224#95C7×D411yescharacteristic
225C22×C7—21no—
226C22×C7—21no—
227C22×C7—81no—
228C22×C7—81no—
229C22×C7—81no—
230C22×C7#38611yescharacteristic
231C22×C7#45311yescharacteristic
232C22×C7#45411yescharacteristic
233C4×C7—21no—
234C4×C7—21no—
235C4×C7—21no—
236C4×C7—21no—
237C4×C7—21no—
238C4×C7—42no—
239C4×C7—42no—
240C4×C7—81no—
241C4×C7—82no—
242C4×C7—81no—
243C4×C7—81no—
244C4×C7#42112yes—
245#41—42no—
246#41—42no—
247C2×Q8—21no—
248C2×Q8—21no—
249C2×Q8—42no—
250C2×Q8C7×C22⋊C412yes—
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—21no—
254C2×D4—41no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—22no—
259C22×C4—21no—
260C22×C4—41no—
261C22×C4C2×C7×D411yescharacteristic
262C22×C4C2×C7×D411yescharacteristic
263C22×C4C2×C7×D411yescharacteristic
264SD16—44no—
265M16—22no—
266M16—41no—
267C2×C8—22no—
268C2×C8—41no—
269C2×C8—41no—
270C4⋊C4—22no—
271C4⋊C4—22no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—42no—
276C4⋊C4—82no—
277C4⋊C4C7×C22⋊C412yes—
278C22⋊C4—41no—
279C22⋊C4—41no—
280C42—22no—
281C42—22no—
282C42—42no—
283C42—42no—
284C42C22×C4×C711yescharacteristic
285C42C2×C7×D411yescharacteristic
286C42C7×#4111yescharacteristic
287C42C7×#4111yescharacteristic
288C2×C7—21no—
289C2×C7—21no—
290C2×C7—161no—
291C2×C7#188311yescharacteristic
292C2×C7#189711yescharacteristic
293C2×C7#190511yescharacteristic
294C23—41no—
295C23C7×#12011yescharacteristic
296Q8—22no—
297Q8—22no—
298Q8—42no—
299Q8—42no—
300D4—42no—
301D4—42no—
302D4—81no—
303D4—81no—
304D4—81no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—42no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—82no—
325C2×C4C2×C7×C22⋊C411yescharacteristic
326C2×C4C4×C7×D411yescharacteristic
327C2×C4C4×C7×D411yescharacteristic
328C2×C4C7×#12111yescharacteristic
329C2×C4C7×#12111yescharacteristic
330C2×C4C7×#12311yescharacteristic
331C8—42no—
332C8—42no—
333C8—81no—
334C7#933011yescharacteristic, direct product factor, Sylow
335C22—21no—
336C22—21no—
337C22—81no—
338C22—81no—
339C22—81no—
340C22C7×#38611yescharacteristic
341C22C7×#45311yescharacteristic
342C22C7×#45411yescharacteristic
343C4—21no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—42no—
349C4—42no—
350C4—81no—
351C4—82no—
352C4—81no—
353C4—81no—
354C4C7×#42112yes—
355C2—21no—
356C2—21no—
357C2—161no—
358C2C7×#188311yescharacteristic
359C2C7×#189711yescharacteristic
360C2C7×#190511yescharacteristic
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×#2320C2semidirect
5. C7×#1491C2semidirect
6. C7×#1300C2semidirect
7. C7×#1285C2semidirect
11. C7×#477C4semidirect
12. C7×#460C4semidirect
29. #9330C7direct (N × H)
73. #2320C2×C7semidirect
76. #1491C2×C7semidirect
77. #1300C2×C7semidirect
78. #1285C2×C7semidirect
125. #477C4×C7semidirect
126. #460C4×C7semidirect
334. C7#9330direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#31122
C7×#23202
C7×#19802
C7×#15052
C7×#14912
C7×#13002
C7×#12852
#93307