Finite Group Databasebeta

← C7×#61378 #13238594

Subgroups — C7×#61378 #13238594

Order 1792.

1810 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3501C212yesmaximal, complemented
2C7×#3389C218yesmaximal, complemented
3C7×#3384C218yesmaximal, complemented
4C7×#3163C212yesmaximal, complemented
5C7×#3162C211yesmaximal, characteristic, complemented
6C7×#3081C214yesmaximal, complemented
7C7×#3077C212yesmaximal, complemented
8C7×#2961C211yesmaximal, characteristic, complemented
9C7×#2948C212yesmaximal
10C2×C7×#442C211yesmaximal, characteristic
11C22×C7×D8C2211yescharacteristic
12C7×#554—22no—
13C7×#554—24no—
14C7×#554C2218yescomplemented
15C7×#554C2214yescomplemented
16C7×D4×Q8—22no—
17C7×D4×Q8C2214yescomplemented
18C7×#544C2214yes—
19C7×#543—24no—
20C7×#533C2212yes—
21C7×#533C2211yescharacteristic
22C2×C7×C22⋊Q8—24no—
23C7×#518C2214yes—
24C2×C4×C7×D4C2212yes—
25C7×#501C2214yescomplemented
26C7×#501C2214yescomplemented
27C7×#501C2218yescomplemented
28C7×#499C2214yescomplemented
29C7×#498C2214yescomplemented
30C7×#478C2218yescomplemented
31C7×#478C2218yescomplemented
32C7×#478C2218yescomplemented
33C7×#476C2214yescomplemented
34C7×#476C2218yescomplemented
35C7×#474C2214yescomplemented
36C7×#474C2218yescomplemented
37C7×#446C2218yescomplemented
38C7×#442C2218yes—
39C7×#442C2214yes—
40C7×#442C2214yes—
41C4×C7×D8C2218yes—
42C7×#428C2214yes—
43C7×#427C2212yes—
44C7×#427C2212yes—
45C2×C7×#106C2211yescharacteristic
46C7×#424C2212yes—
47C7×#423C2211yescharacteristic
48C7×#418C2214yes—
49C2×C7×#102C2212yes—
50C7×#405C2212yes—
51C2×C7×#97C2211yescharacteristic
52#61378C711yesmaximal, characteristic, direct product factor, Sylow
53C22×C7×Q8—24no—
54C22×C7×D4C2312yes—
55C23×C4×C7—22no—
56C2×C7×D8Q818yescomplemented
57C2×C7×D8C2314yes—
58C2×C7×M16C2312yes—
59C22×C7×C8C2311yescharacteristic
60C22×C7×C8C2311yescharacteristic
61C7×C4⋊Q8—24no—
62C7×C4⋊Q8D412yes—
63C7×C4⋊Q8C2312yes—
64C7×C4⋊Q8C2318yes—
65C7×C4⋊Q8C2312yes—
66C7×#125—24no—
67C7×#125D412yes—
68C7×#125C2318yes—
69C7×#125C2314yes—
70C7×C22⋊Q8—24no—
71C7×C22⋊Q8—28no—
72C7×C22⋊Q8—22no—
73C7×C22⋊Q8—22no—
74C7×C22⋊Q8—24no—
75C7×C22⋊Q8—28no—
76C7×C22⋊Q8—28no—
77C7×C22⋊Q8—28no—
78C7×C22⋊Q8—28no—
79C7×C22⋊Q8—28no—
80C7×C22⋊Q8D414yes—
81C4×C7×Q8—22no—
82C4×C7×Q8—22no—
83C4×C7×Q8C2314yes—
84C4×C7×D4—24no—
85C4×C7×D4—24no—
86C4×C7×D4—22no—
87C4×C7×D4—22no—
88C4×C7×D4C2314yes—
89C4×C7×D4C2318yes—
90C4×C7×D4C2318yes—
91C4×C7×D4C2314yes—
92C7×#117C2314yes—
93C7×#117C2312yes—
94C2×C7×C4⋊C4—28no—
95C2×C7×C4⋊C4—24no—
96C2×C7×C4⋊C4C2312yes—
97C2×C7×C22⋊C4—22no—
98C2×C7×C22⋊C4—28no—
99C2×C42×C7C2311yescharacteristic
100C7×#107C2318yes—
101C7×#107C2314yes—
102C7×#107C2314yes—
103C7×#106C2312yes—
104C7×#106C2318yes—
105C7×#106C2312yes—
106C7×#106C2312yes—
107C7×#106C2314yes—
108C7×#106C2312yes—
109C7×C4⋊C8C2314yes—
110C7×C4⋊C8C2318yes—
111C7×#102C2318yes—
112C7×#102C2318yes—
113C7×#102C2318yes—
114C7×#97C2314yes—
115C7×#97C2314yes—
116C4×C7×C8C2314yes—
117#3501C2×C712yescomplemented
118#3389C2×C718yescomplemented
119#3384C2×C718yescomplemented
120#3163C2×C712yescomplemented
121#3162C2×C711yescharacteristic, complemented
122#3081C2×C714yescomplemented
123#3077C2×C712yescomplemented
124#2961C2×C711yescharacteristic, complemented
125#2948C2×C712yes—
126C2×#442C2×C711yescharacteristic
127C24×C7—22no—
128C2×C7×Q8—22no—
129C2×C7×Q8—22no—
130C2×C7×Q8—28no—
131C2×C7×Q8—88no—
132C2×C7×Q8C2×D414yes—
133C2×C7×D4—22no—
134C2×C7×D4—24no—
135C2×C7×D4C2×Q818yescomplemented
136C2×C7×D4C2414yes—
137C22×C4×C7—22no—
138C22×C4×C7—24no—
139C22×C4×C7—28no—
140C22×C4×C7—28no—
141C22×C4×C7—24no—
142C22×C4×C7—84no—
143C22×C4×C7C2×D412yes—
144C22×C4×C7C2411yescharacteristic
145C7×D8C2×Q8116yescomplemented
146C7×M16—24no—
147C2×C7×C8—22no—
148C2×C7×C8C2×Q814yescomplemented
149C2×C7×C8C2414yes—
150C2×C7×C8C2412yes—
151C2×C7×C8C2412yes—
152C7×C4⋊C4—24no—
153C7×C4⋊C4—22no—
154C7×C4⋊C4—28no—
155C7×C4⋊C4—28no—
156C7×C4⋊C4—28no—
157C7×C4⋊C4—22no—
158C7×C4⋊C4—24no—
159C7×C4⋊C4—22no—
160C7×C4⋊C4C2×D418yes—
161C7×C4⋊C4C2×D414yes—
162C7×C4⋊C4C2×D414yes—
163C7×C4⋊C4C2414yes—
164C7×C4⋊C4C2414yes—
165C7×C4⋊C4C2414yes—
166C7×C22⋊C4—24no—
167C7×C22⋊C4—28no—
168C7×C22⋊C4—28no—
169C7×C22⋊C4—24no—
170C7×C22⋊C4—24no—
171C7×C22⋊C4C2×D414yes—
172C42×C7—24no—
173C42×C7C2×D411yescharacteristic
174C42×C7C2×D411yescharacteristic
175C42×C7C2414yes—
176C42×C7C2412yes—
177C22×D8C22×C711yescharacteristic
178#554—22no—
179#554—24no—
180#554C22×C718yescomplemented
181#554C22×C714yescomplemented
182D4×Q8—22no—
183D4×Q8C22×C714yescomplemented
184#544C22×C714yes—
185#543—24no—
186#533C22×C712yes—
187#533C22×C711yescharacteristic
188C2×C22⋊Q8—24no—
189#518C22×C714yes—
190C2×C4×D4C22×C712yes—
191#501C22×C714yescomplemented
192#501C22×C718yescomplemented
193#501C22×C714yescomplemented
194#499C22×C714yescomplemented
195#498C22×C714yescomplemented
196#478C22×C718yescomplemented
197#478C22×C718yescomplemented
198#478C22×C718yescomplemented
199#476C22×C714yescomplemented
200#476C22×C718yescomplemented
201#474C22×C714yescomplemented
202#474C22×C718yescomplemented
203#446C22×C718yescomplemented
204#442C22×C714yes—
205#442C22×C718yes—
206#442C22×C714yes—
207C4×D8C22×C718yes—
208#428C22×C714yes—
209#427C22×C712yes—
210#427C22×C712yes—
211C2×#106C22×C711yescharacteristic
212#424C22×C712yes—
213#423C22×C711yescharacteristic
214#418C22×C714yes—
215C2×#102C22×C712yes—
216#405C22×C712yes—
217C2×#97C22×C711yescharacteristic
218C23×C7—24no—
219C23×C7—28no—
220C23×C7—82no—
221C23×C7—82no—
222C23×C7C22×D411yescharacteristic
223C7×Q8—44no—
224C7×Q8—44no—
225C7×Q8—82no—
226C7×Q8—88no—
227C7×D4—28no—
228C7×D4—24no—
229C7×D4C22×Q818yes—
230C2×C4×C7—24no—
231C2×C4×C7—24no—
232C2×C4×C7—24no—
233C2×C4×C7—42no—
234C2×C4×C7—48no—
235C2×C4×C7—88no—
236C2×C4×C7—84no—
237C2×C4×C7—88no—
238C2×C4×C7—82no—
239C2×C4×C7C22×D412yes—
240C2×C4×C7C22×D412yes—
241C2×C4×C7C22×D418yes—
242C2×C4×C7C22×D411yescharacteristic
243C2×C4×C7C22×Q812yes—
244C2×C4×C72−1+412yes—
245C2×C4×C7C2511yescharacteristic
246C7×C8—22no—
247C7×C8—24no—
248C7×C8C22×Q814yes—
249C22×Q8—24no—
250C22×D4C23×C712yes—
251C23×C4—22no—
252C2×D8C7×Q818yescomplemented
253C2×D8C23×C714yes—
254C2×M16C23×C712yes—
255C22×C8C23×C711yescharacteristic
256C22×C8C23×C711yescharacteristic
257C4⋊Q8—24no—
258C4⋊Q8C7×D412yes—
259C4⋊Q8C23×C712yes—
260C4⋊Q8C23×C712yes—
261C4⋊Q8C23×C718yes—
262#125—24no—
263#125C7×D412yes—
264#125C23×C714yes—
265#125C23×C718yes—
266C22⋊Q8—24no—
267C22⋊Q8—28no—
268C22⋊Q8—22no—
269C22⋊Q8—22no—
270C22⋊Q8—24no—
271C22⋊Q8—28no—
272C22⋊Q8—28no—
273C22⋊Q8—28no—
274C22⋊Q8—28no—
275C22⋊Q8—28no—
276C22⋊Q8C7×D414yes—
277C4×Q8—22no—
278C4×Q8—22no—
279C4×Q8C23×C714yes—
280C4×D4—24no—
281C4×D4—24no—
282C4×D4—22no—
283C4×D4—22no—
284C4×D4C23×C714yes—
285C4×D4C23×C714yes—
286C4×D4C23×C718yes—
287C4×D4C23×C718yes—
288#117C23×C714yes—
289#117C23×C712yes—
290C2×C4⋊C4—28no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4C23×C712yes—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—28no—
295C2×C42C23×C711yescharacteristic
296#107C23×C714yes—
297#107C23×C714yes—
298#107C23×C718yes—
299#106C23×C714yes—
300#106C23×C712yes—
301#106C23×C712yes—
302#106C23×C712yes—
303#106C23×C712yes—
304#106C23×C718yes—
305C4⋊C8C23×C714yes—
306C4⋊C8C23×C718yes—
307#102C23×C718yes—
308#102C23×C718yes—
309#102C23×C718yes—
310#97C23×C714yes—
311#97C23×C714yes—
312C4×C8C23×C714yes—
313C22×C7—22no—
314C22×C7—28no—
315C22×C7—44no—
316C22×C7—44no—
317C22×C7—88no—
318C22×C7D4×Q812yes—
319C22×C7C23×D411yescharacteristic
320C4×C7—22no—
321C4×C7—22no—
322C4×C7—28no—
323C4×C7—44no—
324C4×C7—44no—
325C4×C7—44no—
326C4×C7D4×Q812yes—
327C4×C7C23×Q811yescharacteristic
328C4×C7C2×2−1+411yescharacteristic
329C24—22no—
330C2×Q8—22no—
331C2×Q8—22no—
332C2×Q8—28no—
333C2×Q8—88no—
334C2×Q8C2×C7×D414yes—
335C2×D4—22no—
336C2×D4—24no—
337C2×D4C2×C7×Q818yescomplemented
338C2×D4C24×C714yes—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—28no—
342C22×C4—28no—
343C22×C4—24no—
344C22×C4—84no—
345C22×C4C2×C7×D412yes—
346C22×C4C24×C711yescharacteristic
347D8C2×C7×Q8116yescomplemented
348M16—24no—
349C2×C8—22no—
350C2×C8C2×C7×Q814yescomplemented
351C2×C8C24×C714yes—
352C2×C8C24×C712yes—
353C2×C8C24×C712yes—
354C4⋊C4—24no—
355C4⋊C4—22no—
356C4⋊C4—28no—
357C4⋊C4—28no—
358C4⋊C4—28no—
359C4⋊C4—22no—
360C4⋊C4—24no—
361C4⋊C4—22no—
362C4⋊C4C2×C7×D418yes—
363C4⋊C4C2×C7×D414yes—
364C4⋊C4C2×C7×D414yes—
365C4⋊C4C24×C714yes—
366C4⋊C4C24×C714yes—
367C4⋊C4C24×C714yes—
368C22⋊C4—24no—
369C22⋊C4—28no—
370C22⋊C4—28no—
371C22⋊C4—24no—
372C22⋊C4—24no—
373C22⋊C4C2×C7×D414yes—
374C42—24no—
375C42C2×C7×D411yescharacteristic
376C42C2×C7×D411yescharacteristic
377C42C24×C714yes—
378C42C24×C712yes—
379C2×C7—22no—
380C2×C7—48no—
381C2×C7C2×D4×Q811yescharacteristic
382C2×C7#358512yes—
383C23—24no—
384C23—28no—
385C23—82no—
386C23—82no—
387C23C22×C7×D411yescharacteristic
388Q8—44no—
389Q8—44no—
390Q8—82no—
391Q8—88no—
392D4—24no—
393D4—28no—
394D4C22×C7×Q818yes—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—42no—
399C2×C4—48no—
400C2×C4—88no—
401C2×C4—84no—
402C2×C4—88no—
403C2×C4—82no—
404C2×C4C22×C7×D412yes—
405C2×C4C22×C7×D412yes—
406C2×C4C22×C7×D418yes—
407C2×C4C22×C7×D411yescharacteristic
408C2×C4C22×C7×Q812yes—
409C2×C4C7×2−1+412yes—
410C2×C4C25×C711yescharacteristic
411C8—22no—
412C8—24no—
413C8C22×C7×Q814yes—
414C7#6137811yescharacteristic, direct product factor, Sylow
415C22—22no—
416C22—28no—
417C22—44no—
418C22—44no—
419C22—88no—
420C22C7×D4×Q812yes—
421C22C23×C7×D411yescharacteristic
422C4—22no—
423C4—22no—
424C4—28no—
425C4—44no—
426C4—44no—
427C4—44no—
428C4C7×D4×Q812yes—
429C4C23×C7×Q811yescharacteristic
430C4C2×C7×2−1+411yescharacteristic
431C2—22no—
432C2—48no—
433C2C2×C7×D4×Q811yescharacteristic
434C2C7×#358512yes—
subgroupG/H
center, socleC22×C7C23×D4
commutator, Frattini, cosocleC2×C4C25×C7
prime radicalC22×C7×D8C22
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×#3501C2semidirect
2. C7×#3389C2semidirect
3. C7×#3384C2semidirect
4. C7×#3163C2semidirect
5. C7×#3162C2semidirect
6. C7×#3081C2semidirect
7. C7×#3077C2semidirect
8. C7×#2961C2semidirect
14. C7×#554C22semidirect
15. C7×#554C22semidirect
17. C7×D4×Q8C22semidirect
25. C7×#501C22semidirect
26. C7×#501C22semidirect
27. C7×#501C22semidirect
28. C7×#499C22semidirect
29. C7×#498C22semidirect
30. C7×#478C22semidirect
31. C7×#478C22semidirect
32. C7×#478C22semidirect
33. C7×#476C22semidirect
34. C7×#476C22semidirect
35. C7×#474C22semidirect
36. C7×#474C22semidirect
37. C7×#446C22semidirect
52. #61378C7direct (N × H)
56. C2×C7×D8Q8semidirect
117. #3501C2×C7semidirect
118. #3389C2×C7semidirect
119. #3384C2×C7semidirect
120. #3163C2×C7semidirect
121. #3162C2×C7semidirect
122. #3081C2×C7semidirect
123. #3077C2×C7semidirect
124. #2961C2×C7semidirect
135. C2×C7×D4C2×Q8semidirect
145. C7×D8C2×Q8semidirect
148. C2×C7×C8C2×Q8semidirect
180. #554C22×C7semidirect
181. #554C22×C7semidirect
183. D4×Q8C22×C7semidirect
191. #501C22×C7semidirect
192. #501C22×C7semidirect
193. #501C22×C7semidirect
194. #499C22×C7semidirect
195. #498C22×C7semidirect
196. #478C22×C7semidirect
197. #478C22×C7semidirect
198. #478C22×C7semidirect
199. #476C22×C7semidirect
200. #476C22×C7semidirect
201. #474C22×C7semidirect
202. #474C22×C7semidirect
203. #446C22×C7semidirect
252. C2×D8C7×Q8semidirect
337. C2×D4C2×C7×Q8semidirect
347. D8C2×C7×Q8semidirect
350. C2×C8C2×C7×Q8semidirect
414. C7#61378direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#35012
C7×#33892
C7×#33842
C7×#31632
C7×#31622
C7×#30812
C7×#30772
C7×#29612
C7×#29482
C2×C7×#4422
#613787