Finite Group Databasebeta

← C7×#60253 #13237469

Subgroups — C7×#60253 #13237469

Order 1792.

1702 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×Q8×#41C211yesmaximal, characteristic
2C7×#2986C216yesmaximal, complemented
3C7×#2985C216yesmaximal, complemented
4C7×#2984C212yesmaximal, complemented
5C7×#2982C212yesmaximal, complemented
6C7×Q8×M16C211yesmaximal, characteristic, complemented
7C7×#2962C213yesmaximal, complemented
8C7×#2960C213yesmaximal, complemented
9C2×C7×C8×Q8C211yesmaximal, characteristic, complemented
10C7×#2925C213yesmaximal
11C7×#2922C213yesmaximal
12C7×Q82C411yescharacteristic
13C7×#557C413yes—
14C7×#554C416yes—
15C7×#554C413yes—
16C7×D4×Q8C411yescharacteristic
17C7×D4×Q8C412yes—
18C7×#533C2213yes—
19C7×#533C2216yes—
20C4×C7×#41C2213yes—
21C2×C4×C7×Q8C2211yescharacteristic
22C2×C4×C7×Q8C2212yes—
23C7×#446C2216yes—
24C7×#446C2216yes—
25C7×#446C2216yes—
26C7×#446C22112yes—
27C7×#446C2216yes—
28C7×C8×Q8C2211yescharacteristic
29C7×C8×Q8C2211yescharacteristic
30C7×C8×Q8C2213yes—
31C7×C8×Q8C2213yes—
32C7×C8×Q8C2216yes—
33C7×C8×Q8C2216yes—
34C7×C8×Q8C2216yes—
35C7×C8×Q8C2212yes—
36C7×#433C2216yes—
37C7×#433C2216yes—
38C7×#433C22112yes—
39C7×#432C2216yes—
40C7×#431C2216yes—
41C7×#431C2216yes—
42C7×#431C2216yes—
43C7×#424C2213yes—
44C7×#424C2213yes—
45C7×#423C2213yes—
46C2×C7×C4⋊C8C2213yes—
47C7×#408C2211yescharacteristic
48C7×#408C2213yes—
49C7×#405C2216yes—
50C4×C7×M16C2213yes—
51C2×C4×C7×C8C2213yes—
52#60253C711yesmaximal, characteristic, direct product factor, Sylow
53C2×C7×#41C2311yescharacteristic
54C22×C7×Q8C2×C411yescharacteristic
55C22×C7×Q8C2×C412yes—
56C2×C7×M16C2311yescharacteristic
57C2×C7×M16C2313yes—
58C22×C7×C8C2311yescharacteristic
59C22×C7×C8C2313yes—
60C7×C4⋊Q8C2×C413yes—
61C7×C4⋊Q8C2×C416yes—
62C7×C4⋊Q8C2×C413yes—
63C7×C4⋊Q8C2×C416yes—
64C7×#125C2×C416yes—
65C7×#125C2×C4112yes—
66C7×C22⋊Q8C2×C416yes—
67C7×C22⋊Q8C2×C416yes—
68C7×C22⋊Q8C2×C416yes—
69C7×C22⋊Q8C2×C416yes—
70C7×C22⋊Q8C2×C4112yes—
71C4×C7×Q8C2×C411yescharacteristic
72C4×C7×Q8C2×C412yes—
73C4×C7×Q8C2×C413yes—
74C4×C7×Q8C2×C416yes—
75C4×C7×Q8C2×C413yes—
76C4×C7×Q8C2×C413yes—
77C4×C7×Q8C2311yescharacteristic
78C4×C7×Q8C2313yes—
79C4×C7×Q8C2316yes—
80C4×C7×Q8C2316yes—
81C4×C7×D4C2×C416yes—
82C4×C7×D4C2×C413yes—
83C4×C7×D4C2×C416yes—
84C4×C7×D4C2×C413yes—
85C7×#117C2313yes—
86C7×#117C2316yes—
87C2×C7×C4⋊C4C2×C413yes—
88C2×C7×C4⋊C4C2×C416yes—
89C2×C42×C7C2313yes—
90C2×C42×C7C2316yes—
91C7×C4⋊C8C2313yes—
92C7×C4⋊C8C2313yes—
93C7×C4⋊C8C2316yes—
94C7×C4⋊C8C2316yes—
95C7×C4⋊C8C2316yes—
96C7×C4⋊C8C2316yes—
97C7×C4⋊C8C2316yes—
98C7×C4⋊C8C2316yes—
99C7×C4⋊C8C2316yes—
100C7×C22⋊C8C2312yes—
101C7×C22⋊C8C2312yes—
102C7×C22⋊C8C2316yes—
103C7×C22⋊C8C2316yes—
104C7×#97C2316yes—
105C7×#97C23112yes—
106C4×C7×C8C2313yes—
107C4×C7×C8C2313yes—
108C4×C7×C8C2313yes—
109C4×C7×C8C2313yes—
110C4×C7×C8C2316yes—
111C4×C7×C8C23112yes—
112Q8×#41C2×C711yescharacteristic
113#2986C2×C716yescomplemented
114#2985C2×C716yescomplemented
115#2984C2×C712yescomplemented
116#2982C2×C712yescomplemented
117Q8×M16C2×C711yescharacteristic, complemented
118#2962C2×C713yescomplemented
119#2960C2×C713yescomplemented
120C2×C8×Q8C2×C711yescharacteristic, complemented
121#2925C2×C713yes—
122#2922C2×C713yes—
123C7×#41—22no—
124C7×#41—22no—
125C2×C7×Q8—24no—
126C2×C7×Q8—28no—
127C2×C7×Q8C22×C411yescharacteristic
128C2×C7×Q8C22×C413yes—
129C2×C7×Q8C22×C416yes—
130C2×C7×Q8C22×C416yes—
131C2×C7×Q8C22×C411yescharacteristic
132C2×C7×D4C22×C412yes—
133C2×C7×D4C22×C411yescharacteristic
134C22×C4×C7C22×C416yes—
135C22×C4×C7C22×C413yes—
136C22×C4×C7C22×C416yes—
137C22×C4×C7C22×C413yes—
138C22×C4×C7C2412yes—
139C22×C4×C7C2411yescharacteristic
140C7×M16—22no—
141C7×M16—26no—
142C2×C7×C8—26no—
143C2×C7×C8—22no—
144C2×C7×C8C2411yescharacteristic
145C2×C7×C8C2413yes—
146C2×C7×C8C2411yescharacteristic
147C2×C7×C8C2412yes—
148C2×C7×C8C2416yes—
149C2×C7×C8C2413yes—
150C7×C4⋊C4C22×C413yes—
151C7×C4⋊C4C22×C416yes—
152C7×C4⋊C4C22×C4112yes—
153C7×C4⋊C4C22×C413yes—
154C7×C4⋊C4C22×C416yes—
155C7×C4⋊C4C22×C416yes—
156C7×C4⋊C4C22×C416yes—
157C7×C4⋊C4C22×C413yes—
158C7×C4⋊C4C22×C413yes—
159C7×C4⋊C4C22×C4112yes—
160C7×C4⋊C4C22×C416yes—
161C7×C22⋊C4C22×C416yes—
162C7×C22⋊C4C22×C416yes—
163C7×C22⋊C4C22×C416yes—
164C42×C7C22×C416yes—
165C42×C7C22×C413yes—
166C42×C7C22×C416yes—
167C42×C7C22×C413yes—
168C42×C7C2413yes—
169C42×C7C2413yes—
170C42×C7C2416yes—
171Q82C4×C711yescharacteristic
172#557C4×C713yes—
173#554C4×C716yes—
174#554C4×C713yes—
175D4×Q8C4×C711yescharacteristic
176D4×Q8C4×C712yes—
177#533C22×C713yes—
178#533C22×C716yes—
179C4×#41C22×C713yes—
180C2×C4×Q8C22×C711yescharacteristic
181C2×C4×Q8C22×C712yes—
182#446C22×C716yes—
183#446C22×C716yes—
184#446C22×C716yes—
185#446C22×C7112yes—
186#446C22×C716yes—
187C8×Q8C22×C711yescharacteristic
188C8×Q8C22×C711yescharacteristic
189C8×Q8C22×C713yes—
190C8×Q8C22×C713yes—
191C8×Q8C22×C716yes—
192C8×Q8C22×C716yes—
193C8×Q8C22×C716yes—
194C8×Q8C22×C712yes—
195#433C22×C716yes—
196#433C22×C716yes—
197#433C22×C7112yes—
198#432C22×C716yes—
199#431C22×C716yes—
200#431C22×C716yes—
201#431C22×C716yes—
202#424C22×C713yes—
203#424C22×C713yes—
204#423C22×C713yes—
205C2×C4⋊C8C22×C713yes—
206#408C22×C711yescharacteristic
207#408C22×C713yes—
208#405C22×C716yes—
209C4×M16C22×C713yes—
210C2×C4×C8C22×C713yes—
211C23×C7C23×C412yes—
212C23×C7C23×C411yescharacteristic
213C7×Q8—22no—
214C7×Q8—26no—
215C7×Q8—212no—
216C7×Q8—46no—
217C7×Q8#13114yes—
218C7×D4—24no—
219C7×D4—22no—
220C2×C4×C7—26no—
221C2×C4×C7—212no—
222C2×C4×C7—22no—
223C2×C4×C7—22no—
224C2×C4×C7—43no—
225C2×C4×C7—46no—
226C2×C4×C7C23×C412yes—
227C2×C4×C7C23×C413yes—
228C2×C4×C7C23×C416yes—
229C2×C4×C7C23×C413yes—
230C2×C4×C7C23×C416yes—
231C2×C4×C7C23×C413yes—
232C2×C4×C7C23×C411yescharacteristic
233C2×C4×C7C23×C413yes—
234C2×C4×C72−1+412yes—
235C2×C4×C72−1+412yes—
236C2×C4×C7C2511yescharacteristic
237C7×C8—23no—
238C7×C8—21no—
239C7×C8—21no—
240C7×C8—23no—
241C7×C8—22no—
242C7×C8—26no—
243C2×#41C23×C711yescharacteristic
244C22×Q8C2×C4×C711yescharacteristic
245C22×Q8C2×C4×C712yes—
246C2×M16C23×C711yescharacteristic
247C2×M16C23×C713yes—
248C22×C8C23×C711yescharacteristic
249C22×C8C23×C713yes—
250C4⋊Q8C2×C4×C713yes—
251C4⋊Q8C2×C4×C716yes—
252C4⋊Q8C2×C4×C713yes—
253C4⋊Q8C2×C4×C716yes—
254#125C2×C4×C716yes—
255#125C2×C4×C7112yes—
256C22⋊Q8C2×C4×C716yes—
257C22⋊Q8C2×C4×C716yes—
258C22⋊Q8C2×C4×C716yes—
259C22⋊Q8C2×C4×C716yes—
260C22⋊Q8C2×C4×C7112yes—
261C4×Q8C2×C4×C711yescharacteristic
262C4×Q8C2×C4×C712yes—
263C4×Q8C2×C4×C713yes—
264C4×Q8C2×C4×C716yes—
265C4×Q8C2×C4×C713yes—
266C4×Q8C2×C4×C713yes—
267C4×Q8C23×C711yescharacteristic
268C4×Q8C23×C713yes—
269C4×Q8C23×C716yes—
270C4×Q8C23×C716yes—
271C4×D4C2×C4×C716yes—
272C4×D4C2×C4×C713yes—
273C4×D4C2×C4×C716yes—
274C4×D4C2×C4×C713yes—
275#117C23×C713yes—
276#117C23×C716yes—
277C2×C4⋊C4C2×C4×C713yes—
278C2×C4⋊C4C2×C4×C716yes—
279C2×C42C23×C713yes—
280C2×C42C23×C716yes—
281C4⋊C8C23×C713yes—
282C4⋊C8C23×C713yes—
283C4⋊C8C23×C716yes—
284C4⋊C8C23×C716yes—
285C4⋊C8C23×C716yes—
286C4⋊C8C23×C716yes—
287C4⋊C8C23×C716yes—
288C4⋊C8C23×C716yes—
289C4⋊C8C23×C716yes—
290C22⋊C8C23×C712yes—
291C22⋊C8C23×C712yes—
292C22⋊C8C23×C716yes—
293C22⋊C8C23×C716yes—
294#97C23×C716yes—
295#97C23×C7112yes—
296C4×C8C23×C713yes—
297C4×C8C23×C713yes—
298C4×C8C23×C713yes—
299C4×C8C23×C713yes—
300C4×C8C23×C716yes—
301C4×C8C23×C7112yes—
302C22×C7—21no—
303C22×C7—21no—
304C22×C7—22no—
305C22×C7—22no—
306C22×C7—22no—
307C22×C7#51912yes—
308C22×C7C24×C411yescharacteristic
309C4×C7—22no—
310C4×C7—23no—
311C4×C7—26no—
312C4×C7—23no—
313C4×C7—23no—
314C4×C7—26no—
315C4×C7#51912yes—
316C4×C7C2×#13116yes—
317C4×C7C2×2−1+411yescharacteristic
318C4×C7C2×2−1+411yescharacteristic
319#41—22no—
320#41—22no—
321C2×Q8—24no—
322C2×Q8—28no—
323C2×Q8C22×C4×C711yescharacteristic
324C2×Q8C22×C4×C713yes—
325C2×Q8C22×C4×C716yes—
326C2×Q8C22×C4×C716yes—
327C2×Q8C22×C4×C711yescharacteristic
328C2×D4C22×C4×C712yes—
329C2×D4C22×C4×C711yescharacteristic
330C22×C4C22×C4×C716yes—
331C22×C4C22×C4×C713yes—
332C22×C4C22×C4×C716yes—
333C22×C4C22×C4×C713yes—
334C22×C4C24×C712yes—
335C22×C4C24×C711yescharacteristic
336M16—22no—
337M16—26no—
338C2×C8—26no—
339C2×C8—22no—
340C2×C8C24×C711yescharacteristic
341C2×C8C24×C713yes—
342C2×C8C24×C711yescharacteristic
343C2×C8C24×C712yes—
344C2×C8C24×C716yes—
345C2×C8C24×C713yes—
346C4⋊C4C22×C4×C713yes—
347C4⋊C4C22×C4×C716yes—
348C4⋊C4C22×C4×C7112yes—
349C4⋊C4C22×C4×C713yes—
350C4⋊C4C22×C4×C716yes—
351C4⋊C4C22×C4×C716yes—
352C4⋊C4C22×C4×C716yes—
353C4⋊C4C22×C4×C713yes—
354C4⋊C4C22×C4×C713yes—
355C4⋊C4C22×C4×C7112yes—
356C4⋊C4C22×C4×C716yes—
357C22⋊C4C22×C4×C716yes—
358C22⋊C4C22×C4×C716yes—
359C22⋊C4C22×C4×C716yes—
360C42C22×C4×C716yes—
361C42C22×C4×C713yes—
362C42C22×C4×C716yes—
363C42C22×C4×C713yes—
364C42C24×C713yes—
365C42C24×C713yes—
366C42C24×C716yes—
367C2×C7—22no—
368C2×C7—42no—
369C2×C7C2×#51911yescharacteristic
370C2×C7C22×#13111yescharacteristic
371C2×C7#357311yescharacteristic
372C23C23×C4×C712yes—
373C23C23×C4×C711yescharacteristic
374Q8—22no—
375Q8—26no—
376Q8—212no—
377Q8—46no—
378Q8C7×#13114yes—
379D4—24no—
380D4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—26no—
384C2×C4—212no—
385C2×C4—43no—
386C2×C4—46no—
387C2×C4C23×C4×C712yes—
388C2×C4C23×C4×C713yes—
389C2×C4C23×C4×C716yes—
390C2×C4C23×C4×C713yes—
391C2×C4C23×C4×C716yes—
392C2×C4C23×C4×C713yes—
393C2×C4C23×C4×C711yescharacteristic
394C2×C4C23×C4×C713yes—
395C2×C4C7×2−1+412yes—
396C2×C4C7×2−1+412yes—
397C2×C4C25×C711yescharacteristic
398C8—23no—
399C8—21no—
400C8—21no—
401C8—23no—
402C8—22no—
403C8—26no—
404C7#6025311yescharacteristic, direct product factor, Sylow
405C22—21no—
406C22—21no—
407C22—22no—
408C22—22no—
409C22—22no—
410C22C7×#51912yes—
411C22C24×C4×C711yescharacteristic
412C4—23no—
413C4—26no—
414C4—23no—
415C4—23no—
416C4—26no—
417C4—22no—
418C4C7×#51912yes—
419C4C2×C7×#13116yes—
420C4C2×C7×2−1+411yescharacteristic
421C4C2×C7×2−1+411yescharacteristic
422C2—22no—
423C2—42no—
424C2C2×C7×#51911yescharacteristic
425C2C22×C7×#13111yescharacteristic
426C2C7×#357311yescharacteristic
subgroupG/H
centerC2×C4×C7C25
commutatorC22C24×C4×C7
Frattini, cosocleC2×C4C25×C7
socleC22×C7C24×C4
prime radicalC2×C7×#41C23
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#2986C2semidirect
3. C7×#2985C2semidirect
4. C7×#2984C2semidirect
5. C7×#2982C2semidirect
6. C7×Q8×M16C2semidirect
7. C7×#2962C2semidirect
8. C7×#2960C2semidirect
9. C2×C7×C8×Q8C2semidirect
52. #60253C7direct (N × H)
113. #2986C2×C7semidirect
114. #2985C2×C7semidirect
115. #2984C2×C7semidirect
116. #2982C2×C7semidirect
117. Q8×M16C2×C7semidirect
118. #2962C2×C7semidirect
119. #2960C2×C7semidirect
120. C2×C8×Q8C2×C7semidirect
404. C7#60253direct (N × H)

Maximal subgroups

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

subgroupindex
C7×Q8×#412
C7×#29862
C7×#29852
C7×#29842
C7×#29822
C7×Q8×M162
C7×#29622
C7×#29602
C2×C7×C8×Q82
C7×#29252
C7×#29222
#602537