Finite Group Databasebeta

← C7×#9329 #13186545

Subgroups — C7×#9329 #13186545

Order 1792.

530 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3109C211yesmaximal, characteristic
2C7×#2319C211yesmaximal, characteristic, complemented
3C7×#1980C211yesmaximal, characteristic, complemented
4C7×#1505C211yesmaximal, characteristic, complemented
5C7×#1490C211yesmaximal, characteristic, complemented
6C7×#1300C211yesmaximal, characteristic, complemented
7C7×#1285C211yesmaximal, characteristic, complemented
8C7×#532C2211yescharacteristic
9C2×C4×C7×D4C2211yescharacteristic
10C7×#481—21no—
11C7×#476C412yescomplemented
12C7×#459C412yescomplemented
13C7×#447—21no—
14C7×#442—22no—
15C7×#432C2211yescharacteristic
16C7×#432C2211yescharacteristic
17C2×C7×#104—22no—
18C4×C7×M16C2211yescharacteristic
19C7×#390—21no—
20C7×#389—21no—
21C7×#388—22no—
22C7×#381—21no—
23C7×#380—21no—
24C4×C7×C4⋊C4C2211yescharacteristic
25C7×#376C2211yescharacteristic
26C7×#327—21no—
27C7×#327—21no—
28C7×D4⋊C8—22no—
29#9329C711yesmaximal, characteristic, direct product factor, Sylow
30C2×C7×#41—21no—
31C22×C7×D4D411yescharacteristic
32C23×C4×C7—21no—
33C2×C7×D8—22no—
34C2×C7×M16—21no—
35C7×C4⋊Q8C2×C411yescharacteristic
36C7×#127C2×C411yescharacteristic
37C7×#124—21no—
38C7×#121—21no—
39C7×#121—21no—
40C4×C7×D4—21no—
41C4×C7×D4—21no—
42C4×C7×D4—21no—
43C4×C7×D4—21no—
44C4×C7×D4C2×C412yes—
45C4×C7×D4D412yes—
46C2×C7×C4⋊C4—21no—
47C2×C7×C4⋊C4—22no—
48C2×C7×C4⋊C4D411yescharacteristic
49C2×C7×C22⋊C4—21no—
50C2×C7×C22⋊C4—22no—
51C2×C7×C22⋊C4—22no—
52C2×C42×C7—21no—
53C2×C42×C7D412yes—
54C2×C42×C7C2311yescharacteristic
55C7×#106—22no—
56C7×C4⋊C8—21no—
57C7×C4⋊C8C2×C412yes—
58C7×#104—44no—
59C7×#102—22no—
60C7×#102—22no—
61C7×C22⋊C8—21no—
62C7×C22⋊C8—21no—
63C7×#97—21no—
64C7×#97—21no—
65C7×#97—21no—
66C4×C7×C8—21no—
67C7×#95—22no—
68C7×#95—21no—
69C7×#95—21no—
70C7×#95D411yescharacteristic
71C7×#95D411yescharacteristic
72#3109C2×C711yescharacteristic
73#2319C2×C711yescharacteristic, complemented
74#1980C2×C711yescharacteristic, complemented
75#1505C2×C711yescharacteristic, complemented
76#1490C2×C711yescharacteristic, complemented
77#1300C2×C711yescharacteristic, complemented
78#1285C2×C711yescharacteristic, complemented
79C24×C7—21no—
80C7×#41—42no—
81C7×#41—42no—
82C2×C7×Q8—21no—
83C2×C7×D4—21no—
84C2×C7×D4—21no—
85C2×C7×D4—21no—
86C2×C7×D4—21no—
87C2×C7×D4—42no—
88C2×C7×D4—41no—
89C2×C7×D4C22⋊C412yes—
90C22×C4×C7—21no—
91C22×C4×C7—22no—
92C22×C4×C7—22no—
93C22×C4×C7—21no—
94C22×C4×C7—22no—
95C22×C4×C7—21no—
96C22×C4×C7—41no—
97C22×C4×C7—82no—
98C22×C4×C7C2×D411yescharacteristic
99C22×C4×C7C2×D411yescharacteristic
100C22×C4×C7C2×D411yescharacteristic
101C7×D8—44no—
102C7×M16—22no—
103C7×M16—41no—
104C2×C7×C8—22no—
105C2×C7×C8—41no—
106C2×C7×C8—41no—
107C7×C4⋊C4—21no—
108C7×C4⋊C4—21no—
109C7×C4⋊C4—21no—
110C7×C4⋊C4—42no—
111C7×C4⋊C4—82no—
112C7×C4⋊C4C22⋊C412yes—
113C7×C22⋊C4—22no—
114C7×C22⋊C4—22no—
115C7×C22⋊C4—41no—
116C7×C22⋊C4—41no—
117C7×C22⋊C4—82no—
118C7×C22⋊C4—82no—
119C42×C7—42no—
120C42×C7—42no—
121C42×C7C22×C411yescharacteristic
122C42×C7C2×D411yescharacteristic
123C42×C7#4111yescharacteristic
124C42×C7#4111yescharacteristic
125#532C22×C711yescharacteristic
126C2×C4×D4C22×C711yescharacteristic
127#481—21no—
128#476C4×C712yescomplemented
129#459C4×C712yescomplemented
130#447—21no—
131#442—22no—
132#432C22×C711yescharacteristic
133#432C22×C711yescharacteristic
134C2×#104—22no—
135C4×M16C22×C711yescharacteristic
136#390—21no—
137#389—21no—
138#388—22no—
139#381—21no—
140#380—21no—
141C4×C4⋊C4C22×C711yescharacteristic
142#376C22×C711yescharacteristic
143#327—21no—
144#327—21no—
145D4⋊C8—22no—
146C23×C7—22no—
147C23×C7—42no—
148C23×C7—42no—
149C23×C7—41no—
150C23×C7—41no—
151C23×C7—41no—
152C23×C7#12011yescharacteristic
153C7×Q8—42no—
154C7×D4—22no—
155C7×D4—22no—
156C7×D4—42no—
157C7×D4—42no—
158C7×D4—42no—
159C7×D4—81no—
160C7×D4—81no—
161C7×D4—81no—
162C2×C4×C7—21no—
163C2×C4×C7—21no—
164C2×C4×C7—22no—
165C2×C4×C7—22no—
166C2×C4×C7—41no—
167C2×C4×C7—42no—
168C2×C4×C7—41no—
169C2×C4×C7—41no—
170C2×C4×C7—41no—
171C2×C4×C7—42no—
172C2×C4×C7—42no—
173C2×C4×C7—41no—
174C2×C4×C7—41no—
175C2×C4×C7—42no—
176C2×C4×C7—41no—
177C2×C4×C7—41no—
178C2×C4×C7—82no—
179C2×C4×C7—82no—
180C2×C4×C7—82no—
181C2×C4×C7C2×C22⋊C411yescharacteristic
182C2×C4×C7C4×D411yescharacteristic
183C2×C4×C7C4×D411yescharacteristic
184C2×C4×C7#12111yescharacteristic
185C2×C4×C7#12111yescharacteristic
186C2×C4×C7#12311yescharacteristic
187C7×C8—42no—
188C7×C8—42no—
189C7×C8—81no—
190C2×#41—21no—
191C22×D4C7×D411yescharacteristic
192C23×C4—21no—
193C2×D8—22no—
194C2×M16—21no—
195C4⋊Q8C2×C4×C711yescharacteristic
196#127C2×C4×C711yescharacteristic
197#124—21no—
198#121—21no—
199#121—21no—
200C4×D4—21no—
201C4×D4—21no—
202C4×D4—21no—
203C4×D4—21no—
204C4×D4C2×C4×C712yes—
205C4×D4C7×D412yes—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4C7×D411yescharacteristic
209C2×C22⋊C4—21no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C42—21no—
213C2×C42C7×D412yes—
214C2×C42C23×C711yescharacteristic
215#106—22no—
216C4⋊C8—21no—
217C4⋊C8C2×C4×C712yes—
218#104—44no—
219#102—22no—
220#102—22no—
221C22⋊C8—21no—
222C22⋊C8—21no—
223#97—21no—
224#97—21no—
225#97—21no—
226C4×C8—21no—
227#95—22no—
228#95—21no—
229#95—21no—
230#95C7×D411yescharacteristic
231#95C7×D411yescharacteristic
232C22×C7—21no—
233C22×C7—21no—
234C22×C7—42no—
235C22×C7—42no—
236C22×C7—42no—
237C22×C7—82no—
238C22×C7—82no—
239C22×C7—81no—
240C22×C7—81no—
241C22×C7—81no—
242C22×C7#38611yescharacteristic
243C22×C7#45311yescharacteristic
244C22×C7#45411yescharacteristic
245C4×C7—21no—
246C4×C7—21no—
247C4×C7—21no—
248C4×C7—21no—
249C4×C7—21no—
250C4×C7—42no—
251C4×C7—81no—
252C4×C7—82no—
253C4×C7—81no—
254C4×C7—81no—
255C4×C7#42112yes—
256C24—21no—
257#41—42no—
258#41—42no—
259C2×Q8—21no—
260C2×D4—21no—
261C2×D4—21no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—42no—
265C2×D4—41no—
266C2×D4C7×C22⋊C412yes—
267C22×C4—21no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—21no—
273C22×C4—41no—
274C22×C4—82no—
275C22×C4C2×C7×D411yescharacteristic
276C22×C4C2×C7×D411yescharacteristic
277C22×C4C2×C7×D411yescharacteristic
278D8—44no—
279M16—22no—
280M16—41no—
281C2×C8—22no—
282C2×C8—41no—
283C2×C8—41no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—21no—
287C4⋊C4—42no—
288C4⋊C4—82no—
289C4⋊C4C7×C22⋊C412yes—
290C22⋊C4—22no—
291C22⋊C4—22no—
292C22⋊C4—41no—
293C22⋊C4—41no—
294C22⋊C4—82no—
295C22⋊C4—82no—
296C42—42no—
297C42—42no—
298C42C22×C4×C711yescharacteristic
299C42C2×C7×D411yescharacteristic
300C42C7×#4111yescharacteristic
301C42C7×#4111yescharacteristic
302C2×C7—21no—
303C2×C7—21no—
304C2×C7—82no—
305C2×C7—161no—
306C2×C7#188311yescharacteristic
307C2×C7#189711yescharacteristic
308C2×C7#190411yescharacteristic
309C23—22no—
310C23—42no—
311C23—42no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23C7×#12011yescharacteristic
316Q8—42no—
317D4—22no—
318D4—22no—
319D4—42no—
320D4—42no—
321D4—42no—
322D4—81no—
323D4—81no—
324D4—81no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—41no—
330C2×C4—42no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4—82no—
344C2×C4C2×C7×C22⋊C411yescharacteristic
345C2×C4C4×C7×D411yescharacteristic
346C2×C4C4×C7×D411yescharacteristic
347C2×C4C7×#12111yescharacteristic
348C2×C4C7×#12111yescharacteristic
349C2×C4C7×#12311yescharacteristic
350C8—42no—
351C8—42no—
352C8—81no—
353C7#932911yescharacteristic, direct product factor, Sylow
354C22—21no—
355C22—21no—
356C22—42no—
357C22—42no—
358C22—42no—
359C22—82no—
360C22—82no—
361C22—81no—
362C22—81no—
363C22—81no—
364C22C7×#38611yescharacteristic
365C22C7×#45311yescharacteristic
366C22C7×#45411yescharacteristic
367C4—21no—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—21no—
372C4—42no—
373C4—81no—
374C4—82no—
375C4—81no—
376C4—81no—
377C4C7×#42112yes—
378C2—21no—
379C2—21no—
380C2—82no—
381C2—161no—
382C2C7×#188311yescharacteristic
383C2C7×#189711yescharacteristic
384C2C7×#190411yescharacteristic
subgroupG/H
center, socleC22×C7#386
commutatorC42C22×C4×C7
Frattini, cosocleC2×C42C23×C7
prime radicalC7×#3109C2
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×#2319C2semidirect
3. C7×#1980C2semidirect
4. C7×#1505C2semidirect
5. C7×#1490C2semidirect
6. C7×#1300C2semidirect
7. C7×#1285C2semidirect
11. C7×#476C4semidirect
12. C7×#459C4semidirect
29. #9329C7direct (N × H)
73. #2319C2×C7semidirect
74. #1980C2×C7semidirect
75. #1505C2×C7semidirect
76. #1490C2×C7semidirect
77. #1300C2×C7semidirect
78. #1285C2×C7semidirect
128. #476C4×C7semidirect
129. #459C4×C7semidirect
353. C7#9329direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#31092
C7×#23192
C7×#19802
C7×#15052
C7×#14902
C7×#13002
C7×#12852
#93297