Finite Group Databasebeta

← C7×#61375 #13238591

Subgroups — C7×#61375 #13238591

Order 1792.

1706 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3504C212yesmaximal, complemented
2C7×#3390C214yesmaximal, complemented
3C7×#3389C214yesmaximal, complemented
4C7×#3381C218yesmaximal, complemented
5C7×#3163C212yesmaximal, complemented
6C2×C7×#499C211yesmaximal, characteristic, complemented
7C7×#3083C214yesmaximal, complemented
8C7×#3080C212yesmaximal, complemented
9C7×#2960C211yesmaximal, characteristic, complemented
10C7×#2952C212yesmaximal
11C4×C7×#135C211yesmaximal, characteristic
12C2×C7×#135C2211yescharacteristic
13C7×#557C2218yes—
14C7×#554—28no—
15C7×#554C2218yes—
16C7×#544—24no—
17C7×#544C2214yes—
18C7×#533—22no—
19C7×#533C2211yescharacteristic
20C2×C7×#125—22no—
21C2×C7×#125C2212yes—
22C7×#520C2214yes—
23C4×C7×#41C2212yes—
24C7×#501C2218yes—
25C7×#499C2214yes—
26C7×#499C2214yes—
27C7×#499C2214yes—
28C7×#498C2214yes—
29C7×#479C2218yes—
30C7×#479C2218yes—
31C7×#478C2218yes—
32C7×#478C2218yes—
33C7×#477C2218yes—
34C7×#474C2218yes—
35C7×#446C2214yes—
36C7×C8×Q8C2214yes—
37C7×#442C2214yes—
38C7×#441C2214yes—
39C4×C7×Q16C2212yes—
40C4×C7×SD16C2214yes—
41C4×C7×SD16C2218yes—
42C4×C7×D8C2212yes—
43C7×#428C2214yes—
44C7×#427C2211yescharacteristic
45C2×C7×#107C2212yes—
46C2×C7×#106C2212yes—
47C7×#424C2212yes—
48C7×#423C2211yescharacteristic
49C7×#417C2214yes—
50C7×#416C2212yes—
51C7×#405C2212yes—
52C2×C4×C7×C8C2211yescharacteristic
53#61375C711yesmaximal, characteristic, direct product factor, Sylow
54C2×C7×#41C2312yes—
55C7×#135Q818yescomplemented
56C2×C7×Q16C2311yescharacteristic
57C2×C7×SD16C2312yes—
58C2×C7×D8C2311yescharacteristic
59C2×C7×M16C2312yes—
60C22×C7×C8C2311yescharacteristic
61C22×C7×C8C2311yescharacteristic
62C7×C4⋊Q8—28no—
63C7×C4⋊Q8—24no—
64C7×C4⋊Q8D412yes—
65C7×C4⋊Q8C2318yes—
66C7×#125—24no—
67C7×#125—28no—
68C7×#125—28no—
69C7×#125—22no—
70C7×#125—22no—
71C7×#125D412yes—
72C7×#125C2318yes—
73C7×#125C2318yes—
74C7×C22⋊Q8—28no—
75C7×C22⋊Q8—28no—
76C7×C22⋊Q8—28no—
77C7×C22⋊Q8—28no—
78C7×C22⋊Q8D414yes—
79C4×C7×Q8—28no—
80C4×C7×Q8C2318yes—
81C4×C7×Q8C2314yes—
82C4×C7×Q8C2314yes—
83C4×C7×D4—24no—
84C4×C7×D4—28no—
85C4×C7×D4C2318yes—
86C4×C7×D4C2314yes—
87C7×#117—24no—
88C7×#117—22no—
89C7×#117C2312yes—
90C7×#117C2312yes—
91C2×C7×C4⋊C4—24no—
92C2×C7×C4⋊C4—24no—
93C2×C7×C4⋊C4—28no—
94C2×C7×C4⋊C4C2314yes—
95C2×C42×C7—22no—
96C2×C42×C7C2311yescharacteristic
97C7×#107C2314yes—
98C7×#107C2318yes—
99C7×#107C2314yes—
100C7×#107C2312yes—
101C7×#106C2318yes—
102C7×#106C2312yes—
103C7×#106C2314yes—
104C7×#106C2314yes—
105C7×C4⋊C8C2318yes—
106C7×C4⋊C8C2314yes—
107C7×#103C2314yes—
108C7×#103C2318yes—
109C7×#102C2314yes—
110C7×#102C2318yes—
111C7×#97C2314yes—
112C4×C7×C8C2314yes—
113C4×C7×C8C2312yes—
114C4×C7×C8C2312yes—
115#3504C2×C712yescomplemented
116#3390C2×C714yescomplemented
117#3389C2×C714yescomplemented
118#3381C2×C718yescomplemented
119#3163C2×C712yescomplemented
120C2×#499C2×C711yescharacteristic, complemented
121#3083C2×C714yescomplemented
122#3080C2×C712yescomplemented
123#2960C2×C711yescharacteristic, complemented
124#2952C2×C712yes—
125C4×#135C2×C711yescharacteristic
126C7×#41—24no—
127C7×#41C2×Q818yes—
128C2×C7×Q8—28no—
129C2×C7×Q8C2×D414yes—
130C2×C7×Q8C2412yes—
131C2×C7×D4—22no—
132C2×C7×D4C2412yes—
133C22×C4×C7—22no—
134C22×C4×C7—24no—
135C22×C4×C7—24no—
136C22×C4×C7—28no—
137C22×C4×C7C2×D412yes—
138C22×C4×C7C2411yescharacteristic
139C7×Q16C2×Q814yes—
140C7×SD16C2×Q818yes—
141C7×D8C2×Q814yes—
142C7×M16—24no—
143C2×C7×C8—22no—
144C2×C7×C8C2×Q814yes—
145C2×C7×C8C2412yes—
146C2×C7×C8C2414yes—
147C2×C7×C8C2411yescharacteristic
148C2×C7×C8C2411yescharacteristic
149C7×C4⋊C4—28no—
150C7×C4⋊C4—24no—
151C7×C4⋊C4—28no—
152C7×C4⋊C4—28no—
153C7×C4⋊C4—28no—
154C7×C4⋊C4—28no—
155C7×C4⋊C4—24no—
156C7×C4⋊C4—28no—
157C7×C4⋊C4C2×D414yes—
158C7×C4⋊C4C2×D418yes—
159C7×C4⋊C4C2×D414yes—
160C7×C4⋊C4C2418yes—
161C7×C4⋊C4C2414yes—
162C7×C22⋊C4—22no—
163C7×C22⋊C4—24no—
164C7×C22⋊C4—28no—
165C7×C22⋊C4C2×D414yes—
166C42×C7—22no—
167C42×C7—24no—
168C42×C7—28no—
169C42×C7C2×D411yescharacteristic
170C42×C7C2×D411yescharacteristic
171C42×C7C2412yes—
172C42×C7C2414yes—
173C2×#135C22×C711yescharacteristic
174#557C22×C718yes—
175#554—28no—
176#554C22×C718yes—
177#544—24no—
178#544C22×C714yes—
179#533—22no—
180#533C22×C711yescharacteristic
181C2×#125—22no—
182C2×#125C22×C712yes—
183#520C22×C714yes—
184C4×#41C22×C712yes—
185#501C22×C718yes—
186#499C22×C714yes—
187#499C22×C714yes—
188#499C22×C714yes—
189#498C22×C714yes—
190#479C22×C718yes—
191#479C22×C718yes—
192#478C22×C718yes—
193#478C22×C718yes—
194#477C22×C718yes—
195#474C22×C718yes—
196#446C22×C714yes—
197C8×Q8C22×C714yes—
198#442C22×C714yes—
199#441C22×C714yes—
200C4×Q16C22×C712yes—
201C4×SD16C22×C714yes—
202C4×SD16C22×C718yes—
203C4×D8C22×C712yes—
204#428C22×C714yes—
205#427C22×C711yescharacteristic
206C2×#107C22×C712yes—
207C2×#106C22×C712yes—
208#424C22×C712yes—
209#423C22×C711yescharacteristic
210#417C22×C714yes—
211#416C22×C712yes—
212#405C22×C712yes—
213C2×C4×C8C22×C711yescharacteristic
214C23×C7—22no—
215C23×C7C22×D411yescharacteristic
216C7×Q8—22no—
217C7×Q8—44no—
218C7×Q8—88no—
219C7×Q8C22×Q814yes—
220C7×D4—28no—
221C7×D4—22no—
222C7×D4C22×Q814yes—
223C2×C4×C7—22no—
224C2×C4×C7—28no—
225C2×C4×C7—24no—
226C2×C4×C7—28no—
227C2×C4×C7—42no—
228C2×C4×C7—84no—
229C2×C4×C7—88no—
230C2×C4×C7—84no—
231C2×C4×C7C22×D412yes—
232C2×C4×C7C22×D418yes—
233C2×C4×C7C22×D412yes—
234C2×C4×C7C22×D411yescharacteristic
235C2×C4×C7C22×Q812yes—
236C2×C4×C72−1+412yes—
237C2×C4×C7C2511yescharacteristic
238C7×C8—22no—
239C7×C8—24no—
240C7×C8C22×Q812yes—
241C7×C8C22×Q812yes—
242C2×#41C23×C712yes—
243#135C7×Q818yescomplemented
244C2×Q16C23×C711yescharacteristic
245C2×SD16C23×C712yes—
246C2×D8C23×C711yescharacteristic
247C2×M16C23×C712yes—
248C22×C8C23×C711yescharacteristic
249C22×C8C23×C711yescharacteristic
250C4⋊Q8—28no—
251C4⋊Q8—24no—
252C4⋊Q8C7×D412yes—
253C4⋊Q8C23×C718yes—
254#125—24no—
255#125—28no—
256#125—28no—
257#125—22no—
258#125—22no—
259#125C7×D412yes—
260#125C23×C718yes—
261#125C23×C718yes—
262C22⋊Q8—28no—
263C22⋊Q8—28no—
264C22⋊Q8—28no—
265C22⋊Q8—28no—
266C22⋊Q8C7×D414yes—
267C4×Q8—28no—
268C4×Q8C23×C714yes—
269C4×Q8C23×C718yes—
270C4×Q8C23×C714yes—
271C4×D4—24no—
272C4×D4—28no—
273C4×D4C23×C714yes—
274C4×D4C23×C718yes—
275#117—24no—
276#117—22no—
277#117C23×C712yes—
278#117C23×C712yes—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—28no—
282C2×C4⋊C4C23×C714yes—
283C2×C42—22no—
284C2×C42C23×C711yescharacteristic
285#107C23×C714yes—
286#107C23×C714yes—
287#107C23×C712yes—
288#107C23×C718yes—
289#106C23×C714yes—
290#106C23×C714yes—
291#106C23×C712yes—
292#106C23×C718yes—
293C4⋊C8C23×C714yes—
294C4⋊C8C23×C718yes—
295#103C23×C714yes—
296#103C23×C718yes—
297#102C23×C714yes—
298#102C23×C718yes—
299#97C23×C714yes—
300C4×C8C23×C712yes—
301C4×C8C23×C712yes—
302C4×C8C23×C714yes—
303C22×C7—22no—
304C22×C7—24no—
305C22×C7—44no—
306C22×C7D4×Q812yes—
307C22×C7C23×D411yescharacteristic
308C4×C7—22no—
309C4×C7—22no—
310C4×C7—28no—
311C4×C7—24no—
312C4×C7—48no—
313C4×C7—44no—
314C4×C7D4×Q812yes—
315C4×C7C23×Q811yescharacteristic
316C4×C7C2×2−1+411yescharacteristic
317#41—24no—
318#41C2×C7×Q818yes—
319C2×Q8—28no—
320C2×Q8C2×C7×D414yes—
321C2×Q8C24×C712yes—
322C2×D4—22no—
323C2×D4C24×C712yes—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—28no—
328C22×C4C2×C7×D412yes—
329C22×C4C24×C711yescharacteristic
330Q16C2×C7×Q814yes—
331SD16C2×C7×Q818yes—
332D8C2×C7×Q814yes—
333M16—24no—
334C2×C8—22no—
335C2×C8C2×C7×Q814yes—
336C2×C8C24×C712yes—
337C2×C8C24×C714yes—
338C2×C8C24×C711yescharacteristic
339C2×C8C24×C711yescharacteristic
340C4⋊C4—28no—
341C4⋊C4—24no—
342C4⋊C4—28no—
343C4⋊C4—28no—
344C4⋊C4—28no—
345C4⋊C4—28no—
346C4⋊C4—24no—
347C4⋊C4—28no—
348C4⋊C4C2×C7×D414yes—
349C4⋊C4C2×C7×D418yes—
350C4⋊C4C2×C7×D414yes—
351C4⋊C4C24×C718yes—
352C4⋊C4C24×C714yes—
353C22⋊C4—22no—
354C22⋊C4—24no—
355C22⋊C4—28no—
356C22⋊C4C2×C7×D414yes—
357C42—22no—
358C42—24no—
359C42—28no—
360C42C2×C7×D411yescharacteristic
361C42C2×C7×D411yescharacteristic
362C42C24×C712yes—
363C42C24×C714yes—
364C2×C7—22no—
365C2×C7—44no—
366C2×C7C2×D4×Q811yescharacteristic
367C2×C7#358412yes—
368C23—22no—
369C23C22×C7×D411yescharacteristic
370Q8—22no—
371Q8—44no—
372Q8—88no—
373Q8C22×C7×Q814yes—
374D4—28no—
375D4—22no—
376D4C22×C7×Q814yes—
377C2×C4—22no—
378C2×C4—28no—
379C2×C4—24no—
380C2×C4—28no—
381C2×C4—42no—
382C2×C4—84no—
383C2×C4—88no—
384C2×C4—84no—
385C2×C4C22×C7×D412yes—
386C2×C4C22×C7×D418yes—
387C2×C4C22×C7×D412yes—
388C2×C4C22×C7×D411yescharacteristic
389C2×C4C22×C7×Q812yes—
390C2×C4C7×2−1+412yes—
391C2×C4C25×C711yescharacteristic
392C8—22no—
393C8—24no—
394C8C22×C7×Q812yes—
395C8C22×C7×Q812yes—
396C7#6137511yescharacteristic, direct product factor, Sylow
397C22—22no—
398C22—24no—
399C22—44no—
400C22C7×D4×Q812yes—
401C22C23×C7×D411yescharacteristic
402C4—22no—
403C4—22no—
404C4—28no—
405C4—24no—
406C4—48no—
407C4—44no—
408C4C7×D4×Q812yes—
409C4C23×C7×Q811yescharacteristic
410C4C2×C7×2−1+411yescharacteristic
411C2—22no—
412C2—44no—
413C2C2×C7×D4×Q811yescharacteristic
414C2C7×#358412yes—
subgroupG/H
center, socleC22×C7C23×D4
commutator, Frattini, cosocleC2×C4C25×C7
prime radicalC2×C7×#135C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C7×#533 ◁ G

Complements

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

NH ≅ G/Ntype
1. C7×#3504C2semidirect
2. C7×#3390C2semidirect
3. C7×#3389C2semidirect
4. C7×#3381C2semidirect
5. C7×#3163C2semidirect
6. C2×C7×#499C2semidirect
7. C7×#3083C2semidirect
8. C7×#3080C2semidirect
9. C7×#2960C2semidirect
53. #61375C7direct (N × H)
55. C7×#135Q8semidirect
115. #3504C2×C7semidirect
116. #3390C2×C7semidirect
117. #3389C2×C7semidirect
118. #3381C2×C7semidirect
119. #3163C2×C7semidirect
120. C2×#499C2×C7semidirect
121. #3083C2×C7semidirect
122. #3080C2×C7semidirect
123. #2960C2×C7semidirect
243. #135C7×Q8semidirect
396. C7#61375direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#35042
C7×#33902
C7×#33892
C7×#33812
C7×#31632
C2×C7×#4992
C7×#30832
C7×#30802
C7×#29602
C7×#29522
C4×C7×#1352
#613757