Finite Group Databasebeta

← C7×#9031 #13186247

Subgroups — C7×#9031 #13186247

Order 1792.

670 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2973C211yesmaximal, characteristic, complemented
2C7×#2397C211yesmaximal, characteristic
3C4×C7×#100C211yesmaximal, characteristic, complemented
4C7×#1490C212yesmaximal, complemented
5C7×#1300C212yesmaximal, complemented
6C2×C4×C7×D4—21no—
7C2×C4×C7×D4—21no—
8C2×C4×C7×D4C2211yescharacteristic
9C2×C4×C7×D4C2211yescharacteristic
10C7×#435—24no—
11C7×#432C2212yes—
12C2×C7×#100—21no—
13C7×#407—21no—
14C2×C7×C22⋊C8—21no—
15C4×C7×M16C2212yes—
16C7×#399—21no—
17C7×#399C411yescharacteristic
18C7×#399C411yescharacteristic
19C7×#392—21no—
20C7×#392C411yescharacteristic
21C7×#392C411yescharacteristic
22C7×#376C2211yescharacteristic
23C7×#343—21no—
24C7×D4⋊C8—24no—
25#9031C711yesmaximal, characteristic, direct product factor, Sylow
26C22×C7×D4—21no—
27C22×C7×D4—21no—
28C22×C7×D4C2×C411yescharacteristic
29C22×C7×D4D411yescharacteristic
30C23×C4×C7—21no—
31C23×C4×C7D411yescharacteristic
32C23×C4×C7D411yescharacteristic
33C2×C7×M16—22no—
34C2×C7×M16—22no—
35C22×C7×C8—22no—
36C4×C7×D4—22no—
37C4×C7×D4—22no—
38C4×C7×D4—22no—
39C4×C7×D4—22no—
40C4×C7×D4—44no—
41C4×C7×D4—44no—
42C2×C7×C4⋊C4—21no—
43C2×C7×C4⋊C4—21no—
44C2×C7×C4⋊C4C2×C411yescharacteristic
45C2×C7×C4⋊C4D411yescharacteristic
46C2×C7×C22⋊C4—21no—
47C2×C7×C22⋊C4—21no—
48C2×C7×C22⋊C4—21no—
49C2×C7×C22⋊C4—21no—
50C2×C7×C22⋊C4—21no—
51C2×C7×C22⋊C4C2×C411yescharacteristic
52C2×C7×C22⋊C4C2×C411yescharacteristic
53C2×C42×C7—21no—
54C2×C42×C7C2311yescharacteristic
55C7×C4⋊C8D414yescomplemented
56C7×#100—24no—
57C7×C22⋊C8—22no—
58C7×C22⋊C8—22no—
59C7×C22⋊C8—22no—
60C7×#97—22no—
61C7×#97—22no—
62C4×C7×C8—22no—
63C7×#95—21no—
64C7×#95C2×C411yescharacteristic
65C7×#95C2×C411yescharacteristic
66#2973C2×C711yescharacteristic, complemented
67#2397C2×C711yescharacteristic
68C4×#100C2×C711yescharacteristic, complemented
69#1490C2×C712yescomplemented
70#1300C2×C712yescomplemented
71C24×C7—21no—
72C24×C7C22⋊C411yescharacteristic
73C24×C7C22⋊C411yescharacteristic
74C2×C7×D4—22no—
75C2×C7×D4—22no—
76C2×C7×D4—22no—
77C2×C7×D4—41no—
78C2×C7×D4—42no—
79C2×C7×D4—42no—
80C2×C7×D4—42no—
81C2×C7×D4—42no—
82C2×C7×D4—42no—
83C2×C7×D4—42no—
84C2×C7×D4—42no—
85C22×C4×C7—21no—
86C22×C4×C7—21no—
87C22×C4×C7—21no—
88C22×C4×C7—21no—
89C22×C4×C7—21no—
90C22×C4×C7—21no—
91C22×C4×C7—21no—
92C22×C4×C7—42no—
93C22×C4×C7—42no—
94C22×C4×C7—42no—
95C22×C4×C7—42no—
96C22×C4×C7—41no—
97C22×C4×C7—41no—
98C22×C4×C7C22⋊C411yescharacteristic
99C22×C4×C7C22⋊C411yescharacteristic
100C22×C4×C7C22×C411yescharacteristic
101C22×C4×C7C2×D411yescharacteristic
102C22×C4×C7C2×D411yescharacteristic
103C7×M16—24no—
104C7×M16—42no—
105C7×M16—82no—
106C2×C7×C8—24no—
107C2×C7×C8—42no—
108C2×C7×C8—82no—
109C7×C4⋊C4—22no—
110C7×C4⋊C4—22no—
111C7×C4⋊C4—42no—
112C7×C4⋊C4—42no—
113C7×C22⋊C4—22no—
114C7×C22⋊C4—22no—
115C7×C22⋊C4—42no—
116C7×C22⋊C4—42no—
117C7×C22⋊C4—42no—
118C7×C22⋊C4—42no—
119C7×C22⋊C4—42no—
120C42×C7—24no—
121C42×C7C2×D412yes—
122C42×C7#4112yes—
123C2×C4×D4—21no—
124C2×C4×D4—21no—
125C2×C4×D4C22×C711yescharacteristic
126C2×C4×D4C22×C711yescharacteristic
127#435—24no—
128#432C22×C712yes—
129C2×#100—21no—
130#407—21no—
131C2×C22⋊C8—21no—
132C4×M16C22×C712yes—
133#399—21no—
134#399C4×C711yescharacteristic
135#399C4×C711yescharacteristic
136#392—21no—
137#392C4×C711yescharacteristic
138#392C4×C711yescharacteristic
139#376C22×C711yescharacteristic
140#343—21no—
141D4⋊C8—24no—
142C23×C7—21no—
143C23×C7—21no—
144C23×C7—22no—
145C23×C7—21no—
146C23×C7—21no—
147C23×C7—21no—
148C23×C7—41no—
149C23×C7—41no—
150C23×C7—41no—
151C23×C7—41no—
152C23×C7—41no—
153C23×C7—41no—
154C23×C7—41no—
155C23×C7—41no—
156C23×C7—41no—
157C23×C7—41no—
158C23×C7#10412yes—
159C23×C7C2×C22⋊C411yescharacteristic
160C7×D4—24no—
161C7×D4—44no—
162C7×D4—42no—
163C7×D4—48no—
164C7×D4—48no—
165C2×C4×C7—21no—
166C2×C4×C7—21no—
167C2×C4×C7—22no—
168C2×C4×C7—22no—
169C2×C4×C7—22no—
170C2×C4×C7—21no—
171C2×C4×C7—21no—
172C2×C4×C7—41no—
173C2×C4×C7—41no—
174C2×C4×C7—41no—
175C2×C4×C7—41no—
176C2×C4×C7—41no—
177C2×C4×C7—42no—
178C2×C4×C7—42no—
179C2×C4×C7—41no—
180C2×C4×C7—41no—
181C2×C4×C7—41no—
182C2×C4×C7—41no—
183C2×C4×C7—41no—
184C2×C4×C7—81no—
185C2×C4×C7—81no—
186C2×C4×C7—82no—
187C2×C4×C7—82no—
188C2×C4×C7C4×D412yes—
189C2×C4×C7#12011yescharacteristic
190C2×C4×C7#12112yes—
191C2×C4×C7#12311yescharacteristic
192C7×C8—44no—
193C7×C8—44no—
194C22×D4—21no—
195C22×D4—21no—
196C22×D4C2×C4×C711yescharacteristic
197C22×D4C7×D411yescharacteristic
198C23×C4—21no—
199C23×C4C7×D411yescharacteristic
200C23×C4C7×D411yescharacteristic
201C2×M16—22no—
202C2×M16—22no—
203C22×C8—22no—
204C4×D4—22no—
205C4×D4—22no—
206C4×D4—22no—
207C4×D4—22no—
208C4×D4—44no—
209C4×D4—44no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4C2×C4×C711yescharacteristic
213C2×C4⋊C4C7×D411yescharacteristic
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4C2×C4×C711yescharacteristic
220C2×C22⋊C4C2×C4×C711yescharacteristic
221C2×C42—21no—
222C2×C42C23×C711yescharacteristic
223C4⋊C8C7×D414yescomplemented
224#100—24no—
225C22⋊C8—22no—
226C22⋊C8—22no—
227C22⋊C8—22no—
228#97—22no—
229#97—22no—
230C4×C8—22no—
231#95—21no—
232#95C2×C4×C711yescharacteristic
233#95C2×C4×C711yescharacteristic
234C22×C7—21no—
235C22×C7—21no—
236C22×C7—22no—
237C22×C7—42no—
238C22×C7—42no—
239C22×C7—42no—
240C22×C7—42no—
241C22×C7—41no—
242C22×C7—41no—
243C22×C7—41no—
244C22×C7—41no—
245C22×C7—42no—
246C22×C7—41no—
247C22×C7—81no—
248C22×C7—81no—
249C22×C7—81no—
250C22×C7—81no—
251C22×C7—81no—
252C22×C7—81no—
253C22×C7#38611yescharacteristic
254C22×C7C2×#10411yescharacteristic
255C22×C7#42111yescharacteristic
256C4×C7—22no—
257C4×C7—21no—
258C4×C7—22no—
259C4×C7—42no—
260C4×C7—42no—
261C4×C7—41no—
262C4×C7—41no—
263C4×C7—81no—
264C4×C7#45312yes—
265C24—21no—
266C24C7×C22⋊C411yescharacteristic
267C24C7×C22⋊C411yescharacteristic
268C2×D4—22no—
269C2×D4—22no—
270C2×D4—22no—
271C2×D4—41no—
272C2×D4—42no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—42no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4C7×C22⋊C411yescharacteristic
293C22×C4C7×C22⋊C411yescharacteristic
294C22×C4C22×C4×C711yescharacteristic
295C22×C4C2×C7×D411yescharacteristic
296C22×C4C2×C7×D411yescharacteristic
297M16—24no—
298M16—42no—
299M16—82no—
300C2×C8—24no—
301C2×C8—42no—
302C2×C8—82no—
303C4⋊C4—22no—
304C4⋊C4—22no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C22⋊C4—22no—
308C22⋊C4—22no—
309C22⋊C4—42no—
310C22⋊C4—42no—
311C22⋊C4—42no—
312C22⋊C4—42no—
313C22⋊C4—42no—
314C42—24no—
315C42C2×C7×D412yes—
316C42C7×#4112yes—
317C2×C7—21no—
318C2×C7—21no—
319C2×C7—42no—
320C2×C7—81no—
321C2×C7—82no—
322C2×C7#187111yescharacteristic
323C2×C7#188711yescharacteristic
324C2×C7#189111yescharacteristic
325C23—21no—
326C23—21no—
327C23—22no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23C7×#10412yes—
342C23C2×C7×C22⋊C411yescharacteristic
343D4—24no—
344D4—44no—
345D4—42no—
346D4—48no—
347D4—48no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4C4×C7×D412yes—
372C2×C4C7×#12011yescharacteristic
373C2×C4C7×#12112yes—
374C2×C4C7×#12311yescharacteristic
375C8—44no—
376C8—44no—
377C7#903111yescharacteristic, direct product factor, Sylow
378C22—21no—
379C22—21no—
380C22—22no—
381C22—42no—
382C22—42no—
383C22—42no—
384C22—42no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—42no—
390C22—41no—
391C22—81no—
392C22—81no—
393C22—81no—
394C22—81no—
395C22—81no—
396C22—81no—
397C22C7×#38611yescharacteristic
398C22C2×C7×#10411yescharacteristic
399C22C7×#42111yescharacteristic
400C4—22no—
401C4—21no—
402C4—22no—
403C4—42no—
404C4—42no—
405C4—41no—
406C4—41no—
407C4—81no—
408C4C7×#45312yes—
409C2—21no—
410C2—21no—
411C2—42no—
412C2—81no—
413C2—82no—
414C2C7×#187111yescharacteristic
415C2C7×#188711yescharacteristic
416C2C7×#189111yescharacteristic
subgroupG/H
centerC2×C4×C7#120
commutatorC22×C4C22×C4×C7
Frattini, cosocleC2×C42C23×C7
socleC22×C7#386
prime radicalC7×#2397C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C7 ◁ C2×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
1. C7×#2973C2semidirect
3. C4×C7×#100C2semidirect
4. C7×#1490C2semidirect
5. C7×#1300C2semidirect
25. #9031C7direct (N × H)
55. C7×C4⋊C8D4semidirect
66. #2973C2×C7semidirect
68. C4×#100C2×C7semidirect
69. #1490C2×C7semidirect
70. #1300C2×C7semidirect
223. C4⋊C8C7×D4semidirect
377. C7#9031direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#29732
C7×#23972
C4×C7×#1002
C7×#14902
C7×#13002
#90317