Finite Group Databasebeta

← C7×#60623 #13237839

Subgroups — C7×#60623 #13237839

Order 1792.

2042 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3503C211yesmaximal, characteristic, complemented
2C7×#3499C211yesmaximal, characteristic
3C7×#3141C213yesmaximal, complemented
4C7×#3139C213yesmaximal, complemented
5C7×#3137C2112yesmaximal, complemented
6C7×#3131C213yesmaximal, complemented
7C2×C7×#487C213yesmaximal, complemented
8C7×#2898C214yesmaximal, complemented
9C4×C7×#131C211yesmaximal, characteristic, complemented
10C2×C7×2−1+4C2212yes—
11C2×C7×#131C2212yes—
12C7×Q82—21no—
13C7×#557—21no—
14C7×#554—23no—
15C7×D4×Q8—21no—
16C7×D4×Q8—21no—
17C7×D4×Q8—23no—
18C7×#547—23no—
19C7×#547—23no—
20C7×#544C2216yes—
21C7×#536C2216yes—
22C7×#533C2213yes—
23C7×#532C2213yes—
24C2×C7×C4⋊Q8C2213yes—
25C2×C7×C4⋊Q8C2213yes—
26C7×#520C2212yes—
27C4×C7×#41C2211yescharacteristic
28C7×#491C22112yes—
29C7×#489C22112yes—
30C7×#488C22112yes—
31C7×#487C2213yes—
32C7×#487C22112yes—
33C7×#487C2216yes—
34C7×#487C2213yes—
35C7×#486C2211yescharacteristic
36C7×#486C2213yes—
37C7×#419C22112yes—
38C7×#417C22112yes—
39C7×#416C22112yes—
40C2×C7×#103C22112yes—
41C7×#405C2216yes—
42C4×C7×M16C2213yes—
43C2×C4×C7×C8C2213yes—
44#60623C711yesmaximal, characteristic, direct product factor, Sylow
45C7×2−1+4—216no—
46C2×C7×#41—26no—
47C2×C7×#41C2316yes—
48C2×C7×#41C2311yescharacteristic
49C22×C7×Q8—22no—
50C22×C7×Q8—23no—
51C22×C7×Q8C2316yes—
52C7×#131—28no—
53C2×C7×M16C2316yes—
54C22×C7×C8C2316yes—
55C7×C4⋊Q8—26no—
56C7×C4⋊Q8—23no—
57C7×C4⋊Q8—23no—
58C7×C4⋊Q8—23no—
59C7×C4⋊Q8—23no—
60C7×C4⋊Q8C2313yes—
61C7×C4⋊Q8C2316yes—
62C7×C4⋊Q8C2313yes—
63C7×C4⋊Q8C2311yescharacteristic
64C7×C4⋊Q8C2316yes—
65C7×#127C2311yescharacteristic
66C7×#125—26no—
67C7×#125C2316yes—
68C7×#124C2316yes—
69C7×#123—212no—
70C7×C22⋊Q8—23no—
71C7×C22⋊Q8—23no—
72C7×C22⋊Q8—26no—
73C7×C22⋊Q8—26no—
74C7×C22⋊Q8—212no—
75C7×#121—23no—
76C7×#121—23no—
77C4×C7×Q8—22no—
78C4×C7×Q8—23no—
79C4×C7×Q8—23no—
80C4×C7×Q8D411yescharacteristic
81C4×C7×Q8D411yescharacteristic
82C4×C7×D4—26no—
83C4×C7×D4—23no—
84C4×C7×D4—23no—
85C4×C7×D4D413yes—
86C4×C7×D4D413yes—
87C7×#117C2316yes—
88C7×#117C2313yes—
89C2×C7×C4⋊C4—23no—
90C2×C7×C4⋊C4—23no—
91C2×C7×C4⋊C4—23no—
92C2×C7×C4⋊C4C2316yes—
93C2×C42×C7C2313yes—
94C7×#103C23112yes—
95C7×#103C23124yes—
96C7×#103C23112yes—
97C7×#102C2314yes—
98C7×#102C23112yes—
99C7×#97C2316yes—
100C4×C7×C8C2311yescharacteristic
101C4×C7×C8C2313yes—
102C4×C7×C8C2316yes—
103#3503C2×C711yescharacteristic, complemented
104#3499C2×C711yescharacteristic
105#3141C2×C713yescomplemented
106#3139C2×C713yescomplemented
107#3137C2×C7112yescomplemented
108#3131C2×C713yescomplemented
109C2×#487C2×C713yescomplemented
110#2898C2×C714yescomplemented
111C4×#131C2×C711yescharacteristic, complemented
112C7×#41—224no—
113C7×#41—424no—
114C7×#41#4118yes—
115C2×C7×Q8—21no—
116C2×C7×Q8—23no—
117C2×C7×Q8—26no—
118C2×C7×Q8—26no—
119C2×C7×Q8—224no—
120C2×C7×Q8—48no—
121C2×C7×Q8—86no—
122C2×C7×Q8C2×D411yescharacteristic
123C2×C7×Q8C2416yes—
124C2×C7×D4—26no—
125C2×C7×D4—21no—
126C2×C7×D4—21no—
127C2×C7×D4C2×D413yes—
128C2×C7×D4C2412yes—
129C22×C4×C7—26no—
130C22×C4×C7—26no—
131C22×C4×C7—26no—
132C22×C4×C7C2×D413yes—
133C22×C4×C7C2×D413yes—
134C22×C4×C7C2413yes—
135C7×M16—212no—
136C2×C7×C8—212no—
137C2×C7×C8C2412yes—
138C2×C7×C8C2416yes—
139C7×C4⋊C4—23no—
140C7×C4⋊C4—23no—
141C7×C4⋊C4—23no—
142C7×C4⋊C4—26no—
143C7×C4⋊C4—26no—
144C7×C4⋊C4—26no—
145C7×C4⋊C4—26no—
146C7×C4⋊C4—21no—
147C7×C4⋊C4—26no—
148C7×C4⋊C4—26no—
149C7×C4⋊C4C2×D413yes—
150C7×C4⋊C4C2×D413yes—
151C7×C4⋊C4C2416yes—
152C7×C4⋊C4C2412yes—
153C7×C22⋊C4—26no—
154C7×C22⋊C4—26no—
155C7×C22⋊C4—26no—
156C7×C22⋊C4C2×D416yes—
157C42×C7—26no—
158C42×C7C2×D413yes—
159C42×C7C2×D413yes—
160C42×C7C2411yescharacteristic
161C42×C7C2413yes—
162C2×2−1+4C22×C712yes—
163C2×#131C22×C712yes—
164Q82—21no—
165#557—21no—
166#554—23no—
167D4×Q8—21no—
168D4×Q8—21no—
169D4×Q8—23no—
170#547—23no—
171#547—23no—
172#544C22×C716yes—
173#536C22×C716yes—
174#533C22×C713yes—
175#532C22×C713yes—
176C2×C4⋊Q8C22×C713yes—
177C2×C4⋊Q8C22×C713yes—
178#520C22×C712yes—
179C4×#41C22×C711yescharacteristic
180#491C22×C7112yes—
181#489C22×C7112yes—
182#488C22×C7112yes—
183#487C22×C713yes—
184#487C22×C7112yes—
185#487C22×C716yes—
186#487C22×C713yes—
187#486C22×C711yescharacteristic
188#486C22×C713yes—
189#419C22×C7112yes—
190#417C22×C7112yes—
191#416C22×C7112yes—
192C2×#103C22×C7112yes—
193#405C22×C716yes—
194C4×M16C22×C713yes—
195C2×C4×C8C22×C713yes—
196C23×C7—22no—
197C23×C7C22×D413yes—
198C7×Q8—212no—
199C7×Q8—412no—
200C7×Q8—46no—
201C7×Q8—42no—
202C7×Q8—86no—
203C7×Q8#12414yes—
204C7×D4—24no—
205C7×D4—412no—
206C7×D4—42no—
207C7×D4—42no—
208C7×D4#124112yes—
209C2×C4×C7—26no—
210C2×C4×C7—26no—
211C2×C4×C7—22no—
212C2×C4×C7—26no—
213C2×C4×C7—26no—
214C2×C4×C7—412no—
215C2×C4×C7—412no—
216C2×C4×C7—86no—
217C2×C4×C7C22×D413yes—
218C2×C4×C7C22×D413yes—
219C2×C4×C7C22×D413yes—
220C2×C4×C7C22×D411yescharacteristic
221C2×C4×C7C22×D411yescharacteristic
222C2×C4×C7C2×#41112yes—
223C2×C4×C7C2511yescharacteristic
224C7×C8—22no—
225C7×C8—26no—
2262−1+4—216no—
227C2×#41—26no—
228C2×#41C23×C716yes—
229C2×#41C23×C711yescharacteristic
230C22×Q8—22no—
231C22×Q8—23no—
232C22×Q8C23×C716yes—
233#131—28no—
234C2×M16C23×C716yes—
235C22×C8C23×C716yes—
236C4⋊Q8—26no—
237C4⋊Q8—23no—
238C4⋊Q8—23no—
239C4⋊Q8—23no—
240C4⋊Q8—23no—
241C4⋊Q8C23×C713yes—
242C4⋊Q8C23×C716yes—
243C4⋊Q8C23×C713yes—
244C4⋊Q8C23×C716yes—
245C4⋊Q8C23×C711yescharacteristic
246#127C23×C711yescharacteristic
247#125—26no—
248#125C23×C716yes—
249#124C23×C716yes—
250#123—212no—
251C22⋊Q8—23no—
252C22⋊Q8—23no—
253C22⋊Q8—26no—
254C22⋊Q8—26no—
255C22⋊Q8—212no—
256#121—23no—
257#121—23no—
258C4×Q8—22no—
259C4×Q8—23no—
260C4×Q8—23no—
261C4×Q8C7×D411yescharacteristic
262C4×Q8C7×D411yescharacteristic
263C4×D4—26no—
264C4×D4—23no—
265C4×D4—23no—
266C4×D4C7×D413yes—
267C4×D4C7×D413yes—
268#117C23×C716yes—
269#117C23×C713yes—
270C2×C4⋊C4—23no—
271C2×C4⋊C4—23no—
272C2×C4⋊C4—23no—
273C2×C4⋊C4C23×C716yes—
274C2×C42C23×C713yes—
275#103C23×C7112yes—
276#103C23×C7124yes—
277#103C23×C7112yes—
278#102C23×C714yes—
279#102C23×C7112yes—
280#97C23×C716yes—
281C4×C8C23×C711yescharacteristic
282C4×C8C23×C713yes—
283C4×C8C23×C716yes—
284C22×C7—23no—
285C22×C7—23no—
286C22×C7—42no—
287C22×C7—42no—
288C22×C7—42no—
289C22×C7C2×#12416yes—
290C22×C7C23×D411yescharacteristic
291C4×C7—23no—
292C4×C7—23no—
293C4×C7—46no—
294C4×C7—46no—
295C4×C7—42no—
296C4×C7C2×#12416yes—
297C4×C7#57712yes—
298C4×C7#57812yes—
299C4×C7C22×#4112yes—
300#41—224no—
301#41—424no—
302#41C7×#4118yes—
303C2×Q8—21no—
304C2×Q8—23no—
305C2×Q8—26no—
306C2×Q8—26no—
307C2×Q8—224no—
308C2×Q8—48no—
309C2×Q8—86no—
310C2×Q8C2×C7×D411yescharacteristic
311C2×Q8C24×C716yes—
312C2×D4—26no—
313C2×D4—21no—
314C2×D4—21no—
315C2×D4C2×C7×D413yes—
316C2×D4C24×C712yes—
317C22×C4—26no—
318C22×C4—26no—
319C22×C4—26no—
320C22×C4C2×C7×D413yes—
321C22×C4C2×C7×D413yes—
322C22×C4C24×C713yes—
323M16—212no—
324C2×C8—212no—
325C2×C8C24×C712yes—
326C2×C8C24×C716yes—
327C4⋊C4—23no—
328C4⋊C4—23no—
329C4⋊C4—23no—
330C4⋊C4—26no—
331C4⋊C4—26no—
332C4⋊C4—26no—
333C4⋊C4—26no—
334C4⋊C4—21no—
335C4⋊C4—26no—
336C4⋊C4—26no—
337C4⋊C4C2×C7×D413yes—
338C4⋊C4C2×C7×D413yes—
339C4⋊C4C24×C716yes—
340C4⋊C4C24×C712yes—
341C22⋊C4—26no—
342C22⋊C4—26no—
343C22⋊C4—26no—
344C22⋊C4C2×C7×D416yes—
345C42—26no—
346C42C2×C7×D413yes—
347C42C2×C7×D413yes—
348C42C24×C711yescharacteristic
349C42C24×C713yes—
350C2×C7—26no—
351C2×C7—82no—
352C2×C7C22×#12411yescharacteristic
353C2×C7C2×#57711yescharacteristic
354C2×C7C2×#57811yescharacteristic
355C23—22no—
356C23C22×C7×D413yes—
357Q8—212no—
358Q8—412no—
359Q8—46no—
360Q8—42no—
361Q8—86no—
362Q8C7×#12414yes—
363D4—24no—
364D4—412no—
365D4—42no—
366D4—42no—
367D4C7×#124112yes—
368C2×C4—26no—
369C2×C4—26no—
370C2×C4—22no—
371C2×C4—26no—
372C2×C4—26no—
373C2×C4—412no—
374C2×C4—412no—
375C2×C4—86no—
376C2×C4C22×C7×D413yes—
377C2×C4C22×C7×D413yes—
378C2×C4C22×C7×D413yes—
379C2×C4C22×C7×D411yescharacteristic
380C2×C4C22×C7×D411yescharacteristic
381C2×C4C2×C7×#41112yes—
382C2×C4C25×C711yescharacteristic
383C8—22no—
384C8—26no—
385C7#6062311yescharacteristic, direct product factor, Sylow
386C22—23no—
387C22—23no—
388C22—42no—
389C22—42no—
390C22—42no—
391C22C2×C7×#12416yes—
392C22C23×C7×D411yescharacteristic
393C4—23no—
394C4—23no—
395C4—46no—
396C4—42no—
397C4—46no—
398C4C2×C7×#12416yes—
399C4C7×#57712yes—
400C4C7×#57812yes—
401C4C22×C7×#4112yes—
402C2—26no—
403C2—82no—
404C2C22×C7×#12411yescharacteristic
405C2C2×C7×#57711yescharacteristic
406C2C2×C7×#57811yescharacteristic
subgroupG/H
center, socleC22×C7C23×D4
commutator, Frattini, cosocleC2×C4C25×C7
prime radicalC7×#3499C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C4×C7×#41 ◁ G

Complements

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

NH ≅ G/Ntype
1. C7×#3503C2semidirect
3. C7×#3141C2semidirect
4. C7×#3139C2semidirect
5. C7×#3137C2semidirect
6. C7×#3131C2semidirect
7. C2×C7×#487C2semidirect
8. C7×#2898C2semidirect
9. C4×C7×#131C2semidirect
44. #60623C7direct (N × H)
103. #3503C2×C7semidirect
105. #3141C2×C7semidirect
106. #3139C2×C7semidirect
107. #3137C2×C7semidirect
108. #3131C2×C7semidirect
109. C2×#487C2×C7semidirect
110. #2898C2×C7semidirect
111. C4×#131C2×C7semidirect
385. C7#60623direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#35032
C7×#34992
C7×#31412
C7×#31392
C7×#31372
C7×#31312
C2×C7×#4872
C7×#28982
C4×C7×#1312
#606237