Finite Group Databasebeta

← C7×Q8×#135 #13238573

Subgroups — C7×Q8×#135 #13238573

Order 1792.

1786 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×Q8×#41C212yesmaximal, complemented
2C7×#3383C213yesmaximal, complemented
3C7×Q8×Q16C211yesmaximal, characteristic, complemented
4C7×#3381C216yesmaximal, complemented
5C7×#3380C213yesmaximal, complemented
6C7×Q8×SD16C212yesmaximal, complemented
7C7×Q8×D8C211yesmaximal, characteristic, complemented
8C7×#3160C213yesmaximal, complemented
9C7×#3076C216yesmaximal, complemented
10C2×C7×C8×Q8C211yesmaximal, characteristic, complemented
11C4×C7×#135C213yesmaximal
12C2×C7×#135C2211yescharacteristic
13C7×Q82C2212yes—
14C7×#557C2216yes—
15C7×#554—26no—
16C7×#554C2216yes—
17C7×D4×Q8—22no—
18C7×D4×Q8C2212yes—
19C7×#533—26no—
20C7×#533C2216yes—
21C4×C7×#41C2216yes—
22C2×C4×C7×Q8—22no—
23C2×C4×C7×Q8C2211yescharacteristic
24C7×#500C2213yes—
25C7×#500C2213yes—
26C7×#499C2216yes—
27C7×#499C2216yes—
28C7×#498C2216yes—
29C7×#479C22112yes—
30C7×#478C22112yes—
31C7×#477C2216yes—
32C7×#476C2216yes—
33C7×#475C2216yes—
34C7×#474C2216yes—
35C7×C8×Q8C2211yescharacteristic
36C7×C8×Q8C2211yescharacteristic
37C7×C8×Q8C2213yes—
38C7×C8×Q8C2213yes—
39C4×C7×Q16C2213yes—
40C4×C7×Q16C2213yes—
41C4×C7×SD16C2216yes—
42C4×C7×SD16C2216yes—
43C4×C7×D8C2213yes—
44C4×C7×D8C2213yes—
45C7×#427C2213yes—
46C7×#427C2216yes—
47C2×C7×C4⋊C8C2213yes—
48C7×#416C2216yes—
49C2×C4×C7×C8C2213yes—
50Q8×#135C711yesmaximal, characteristic, direct product factor, Sylow
51C2×C7×#41C2312yes—
52C22×C7×Q8—22no—
53C22×C7×Q8D411yescharacteristic
54C7×#135Q818yesdirect product factor
55C2×C7×Q16C2311yescharacteristic
56C2×C7×SD16C2312yes—
57C2×C7×D8C2311yescharacteristic
58C22×C7×C8C2313yes—
59C22×C7×C8C2311yescharacteristic
60C7×C4⋊Q8—26no—
61C7×C4⋊Q8—26no—
62C7×C4⋊Q8C2316yes—
63C7×C4⋊Q8C2316yes—
64C7×#125—212no—
65C7×#125C23112yes—
66C7×C22⋊Q8—26no—
67C7×C22⋊Q8—26no—
68C7×C22⋊Q8—26no—
69C7×C22⋊Q8—26no—
70C7×C22⋊Q8—212no—
71C4×C7×Q8—22no—
72C4×C7×Q8—26no—
73C4×C7×Q8—26no—
74C4×C7×Q8—26no—
75C4×C7×Q8D411yescharacteristic
76C4×C7×Q8D413yes—
77C4×C7×Q8C2316yes—
78C4×C7×Q8C2316yes—
79C4×C7×Q8C2311yescharacteristic
80C4×C7×Q8C2313yes—
81C4×C7×D4—26no—
82C4×C7×D4—26no—
83C4×C7×D4C2316yes—
84C4×C7×D4C2316yes—
85C7×#117—26no—
86C7×#117C2316yes—
87C2×C7×C4⋊C4—26no—
88C2×C7×C4⋊C4D413yes—
89C2×C42×C7—26no—
90C2×C42×C7C2313yes—
91C7×#107C2316yes—
92C7×#107C2316yes—
93C7×#107C2313yes—
94C7×#107C2313yes—
95C7×#106C23112yes—
96C7×#106C2316yes—
97C7×C4⋊C8C2313yes—
98C7×C4⋊C8C2313yes—
99C7×C4⋊C8C2316yes—
100C7×#103C2316yes—
101C7×#103C2316yes—
102C7×#102C2316yes—
103C7×#102C2316yes—
104C4×C7×C8C2313yes—
105C4×C7×C8C2313yes—
106C4×C7×C8C2313yes—
107C4×C7×C8C2313yes—
108Q8×#41C2×C712yescomplemented
109#3383C2×C713yescomplemented
110Q8×Q16C2×C711yescharacteristic, complemented
111#3381C2×C716yescomplemented
112#3380C2×C713yescomplemented
113Q8×SD16C2×C712yescomplemented
114Q8×D8C2×C711yescharacteristic, complemented
115#3160C2×C713yescomplemented
116#3076C2×C716yescomplemented
117C2×C8×Q8C2×C711yescharacteristic, complemented
118C4×#135C2×C713yes—
119C7×#41—24no—
120C7×#41C2×Q818yescomplemented
121C2×C7×Q8—24no—
122C2×C7×Q8—26no—
123C2×C7×Q8—26no—
124C2×C7×Q8—48no—
125C2×C7×Q8C2×D413yes—
126C2×C7×Q8C2×D411yescharacteristic
127C2×C7×Q8C2412yes—
128C2×C7×D4—22no—
129C2×C7×D4C2412yes—
130C22×C4×C7—22no—
131C22×C4×C7—26no—
132C22×C4×C7—26no—
133C22×C4×C7C2×D413yes—
134C22×C4×C7C2×D413yes—
135C22×C4×C7C2411yescharacteristic
136C7×Q16C2×Q814yescomplemented
137C7×SD16C2×Q818yescomplemented
138C7×D8C2×Q814yescomplemented
139C2×C7×C8—26no—
140C2×C7×C8C2×Q814yescomplemented
141C2×C7×C8C2413yes—
142C2×C7×C8C2413yes—
143C2×C7×C8C2411yescharacteristic
144C2×C7×C8C2411yescharacteristic
145C7×C4⋊C4—212no—
146C7×C4⋊C4—26no—
147C7×C4⋊C4—26no—
148C7×C4⋊C4—26no—
149C7×C4⋊C4—26no—
150C7×C4⋊C4—212no—
151C7×C4⋊C4C2×D413yes—
152C7×C4⋊C4C2×D416yes—
153C7×C4⋊C4C2×D413yes—
154C7×C4⋊C4C2416yes—
155C7×C4⋊C4C2416yes—
156C7×C22⋊C4—26no—
157C7×C22⋊C4—26no—
158C7×C22⋊C4—26no—
159C42×C7—26no—
160C42×C7—26no—
161C42×C7—26no—
162C42×C7C2×D413yes—
163C42×C7C2×D413yes—
164C42×C7C2413yes—
165C42×C7C2413yes—
166C2×#135C22×C711yescharacteristic
167Q82C22×C712yes—
168#557C22×C716yes—
169#554—26no—
170#554C22×C716yes—
171D4×Q8—22no—
172D4×Q8C22×C712yes—
173#533—26no—
174#533C22×C716yes—
175C4×#41C22×C716yes—
176C2×C4×Q8—22no—
177C2×C4×Q8C22×C711yescharacteristic
178#500C22×C713yes—
179#500C22×C713yes—
180#499C22×C716yes—
181#499C22×C716yes—
182#498C22×C716yes—
183#479C22×C7112yes—
184#478C22×C7112yes—
185#477C22×C716yes—
186#476C22×C716yes—
187#475C22×C716yes—
188#474C22×C716yes—
189C8×Q8C22×C711yescharacteristic
190C8×Q8C22×C711yescharacteristic
191C8×Q8C22×C713yes—
192C8×Q8C22×C713yes—
193C4×Q16C22×C713yes—
194C4×Q16C22×C713yes—
195C4×SD16C22×C716yes—
196C4×SD16C22×C716yes—
197C4×D8C22×C713yes—
198C4×D8C22×C713yes—
199#427C22×C716yes—
200#427C22×C713yes—
201C2×C4⋊C8C22×C713yes—
202#416C22×C716yes—
203C2×C4×C8C22×C713yes—
204C23×C7—22no—
205C23×C7C22×D411yescharacteristic
206C7×Q8—26no—
207C7×Q8—22no—
208C7×Q8—412no—
209C7×Q8—86no—
210C7×Q8#13514yesdirect product factor
211C7×Q8C22×Q814yes—
212C7×D4—28no—
213C7×D4—22no—
214C7×D4C22×Q814yes—
215C2×C4×C7—22no—
216C2×C4×C7—28no—
217C2×C4×C7—26no—
218C2×C4×C7—26no—
219C2×C4×C7—26no—
220C2×C4×C7—43no—
221C2×C4×C7—412no—
222C2×C4×C7—86no—
223C2×C4×C7C22×D413yes—
224C2×C4×C7C22×D413yes—
225C2×C4×C7C22×D413yes—
226C2×C4×C7C22×D413yes—
227C2×C4×C7C22×D411yescharacteristic
228C2×C4×C7C22×Q812yes—
229C2×C4×C72−1+412yes—
230C2×C4×C7C2511yescharacteristic
231C7×C8—23no—
232C7×C8—23no—
233C7×C8C22×Q812yes—
234C7×C8C22×Q812yes—
235C2×#41C23×C712yes—
236C22×Q8—22no—
237C22×Q8C7×D411yescharacteristic
238#135C7×Q818yesdirect product factor
239C2×Q16C23×C711yescharacteristic
240C2×SD16C23×C712yes—
241C2×D8C23×C711yescharacteristic
242C22×C8C23×C713yes—
243C22×C8C23×C711yescharacteristic
244C4⋊Q8—26no—
245C4⋊Q8—26no—
246C4⋊Q8C23×C716yes—
247C4⋊Q8C23×C716yes—
248#125—212no—
249#125C23×C7112yes—
250C22⋊Q8—26no—
251C22⋊Q8—26no—
252C22⋊Q8—26no—
253C22⋊Q8—26no—
254C22⋊Q8—212no—
255C4×Q8—22no—
256C4×Q8—26no—
257C4×Q8—26no—
258C4×Q8—26no—
259C4×Q8C7×D411yescharacteristic
260C4×Q8C7×D413yes—
261C4×Q8C23×C716yes—
262C4×Q8C23×C716yes—
263C4×Q8C23×C711yescharacteristic
264C4×Q8C23×C713yes—
265C4×D4—26no—
266C4×D4—26no—
267C4×D4C23×C716yes—
268C4×D4C23×C716yes—
269#117—26no—
270#117C23×C716yes—
271C2×C4⋊C4—26no—
272C2×C4⋊C4C7×D413yes—
273C2×C42—26no—
274C2×C42C23×C713yes—
275#107C23×C713yes—
276#107C23×C713yes—
277#107C23×C716yes—
278#107C23×C716yes—
279#106C23×C716yes—
280#106C23×C7112yes—
281C4⋊C8C23×C713yes—
282C4⋊C8C23×C713yes—
283C4⋊C8C23×C716yes—
284#103C23×C716yes—
285#103C23×C716yes—
286#102C23×C716yes—
287#102C23×C716yes—
288C4×C8C23×C713yes—
289C4×C8C23×C713yes—
290C4×C8C23×C713yes—
291C4×C8C23×C713yes—
292C22×C7—21no—
293C22×C7—21no—
294C22×C7—24no—
295C22×C7—42no—
296C22×C7—42no—
297C22×C7D4×Q812yes—
298C22×C7C23×D411yescharacteristic
299C4×C7—23no—
300C4×C7—23no—
301C4×C7—24no—
302C4×C7—23no—
303C4×C7—46no—
304C4×C7—46no—
305C4×C7D4×Q812yes—
306C4×C7C2×#13516yes—
307C4×C7C23×Q811yescharacteristic
308C4×C7C2×2−1+411yescharacteristic
309#41—24no—
310#41C2×C7×Q818yescomplemented
311C2×Q8—24no—
312C2×Q8—26no—
313C2×Q8—26no—
314C2×Q8—48no—
315C2×Q8C2×C7×D413yes—
316C2×Q8C2×C7×D411yescharacteristic
317C2×Q8C24×C712yes—
318C2×D4—22no—
319C2×D4C24×C712yes—
320C22×C4—22no—
321C22×C4—26no—
322C22×C4—26no—
323C22×C4C2×C7×D413yes—
324C22×C4C2×C7×D413yes—
325C22×C4C24×C711yescharacteristic
326Q16C2×C7×Q814yescomplemented
327SD16C2×C7×Q818yescomplemented
328D8C2×C7×Q814yescomplemented
329C2×C8—26no—
330C2×C8C2×C7×Q814yescomplemented
331C2×C8C24×C713yes—
332C2×C8C24×C713yes—
333C2×C8C24×C711yescharacteristic
334C2×C8C24×C711yescharacteristic
335C4⋊C4—212no—
336C4⋊C4—26no—
337C4⋊C4—26no—
338C4⋊C4—26no—
339C4⋊C4—26no—
340C4⋊C4—212no—
341C4⋊C4C2×C7×D413yes—
342C4⋊C4C2×C7×D416yes—
343C4⋊C4C2×C7×D413yes—
344C4⋊C4C24×C716yes—
345C4⋊C4C24×C716yes—
346C22⋊C4—26no—
347C22⋊C4—26no—
348C22⋊C4—26no—
349C42—26no—
350C42—26no—
351C42—26no—
352C42C2×C7×D413yes—
353C42C2×C7×D413yes—
354C42C24×C713yes—
355C42C24×C713yes—
356C2×C7—22no—
357C2×C7—44no—
358C2×C7C2×D4×Q811yescharacteristic
359C2×C7C22×#13511yescharacteristic
360C2×C7#358411yescharacteristic
361C23—22no—
362C23C22×C7×D411yescharacteristic
363Q8—22no—
364Q8—26no—
365Q8—412no—
366Q8—86no—
367Q8C7×#13514yesdirect product factor
368Q8C22×C7×Q814yes—
369D4—28no—
370D4—22no—
371D4C22×C7×Q814yes—
372C2×C4—22no—
373C2×C4—28no—
374C2×C4—26no—
375C2×C4—26no—
376C2×C4—26no—
377C2×C4—43no—
378C2×C4—412no—
379C2×C4—86no—
380C2×C4C22×C7×D413yes—
381C2×C4C22×C7×D413yes—
382C2×C4C22×C7×D413yes—
383C2×C4C22×C7×D411yescharacteristic
384C2×C4C22×C7×D413yes—
385C2×C4C22×C7×Q812yes—
386C2×C4C7×2−1+412yes—
387C2×C4C25×C711yescharacteristic
388C8—23no—
389C8—23no—
390C8C22×C7×Q812yes—
391C8C22×C7×Q812yes—
392C7Q8×#13511yescharacteristic, direct product factor, Sylow
393C22—21no—
394C22—21no—
395C22—24no—
396C22—42no—
397C22—42no—
398C22C7×D4×Q812yes—
399C22C23×C7×D411yescharacteristic
400C4—23no—
401C4—23no—
402C4—23no—
403C4—24no—
404C4—46no—
405C4—46no—
406C4C7×D4×Q812yes—
407C4C2×C7×#13516yes—
408C4C23×C7×Q811yescharacteristic
409C4C2×C7×2−1+411yescharacteristic
410C2—22no—
411C2—44no—
412C2C2×C7×D4×Q811yescharacteristic
413C2C22×C7×#13511yescharacteristic
414C2C7×#358411yescharacteristic
subgroupG/H
centerC2×C4×C7C22×D4
commutator, Frattini, cosocleC2×C4C25×C7
socleC22×C7C23×D4
prime radicalC2×C7×#135C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C7 ◁ C2×C4×C7×Q8 ◁ G

Complements

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

NH ≅ G/Ntype
1. C7×Q8×#41C2semidirect
2. C7×#3383C2semidirect
3. C7×Q8×Q16C2semidirect
4. C7×#3381C2semidirect
5. C7×#3380C2semidirect
6. C7×Q8×SD16C2semidirect
7. C7×Q8×D8C2semidirect
8. C7×#3160C2semidirect
9. C7×#3076C2semidirect
10. C2×C7×C8×Q8C2semidirect
50. Q8×#135C7direct (N × H)
54. C7×#135Q8direct (N × H)
108. Q8×#41C2×C7semidirect
109. #3383C2×C7semidirect
110. Q8×Q16C2×C7semidirect
111. #3381C2×C7semidirect
112. #3380C2×C7semidirect
113. Q8×SD16C2×C7semidirect
114. Q8×D8C2×C7semidirect
115. #3160C2×C7semidirect
116. #3076C2×C7semidirect
117. C2×C8×Q8C2×C7semidirect
120. C7×#41C2×Q8semidirect
136. C7×Q16C2×Q8semidirect
137. C7×SD16C2×Q8semidirect
138. C7×D8C2×Q8semidirect
140. C2×C7×C8C2×Q8semidirect
210. C7×Q8#135direct (N × H)
238. #135C7×Q8direct (N × H)
310. #41C2×C7×Q8semidirect
326. Q16C2×C7×Q8semidirect
327. SD16C2×C7×Q8semidirect
328. D8C2×C7×Q8semidirect
330. C2×C8C2×C7×Q8semidirect
367. Q8C7×#135direct (N × H)
392. C7Q8×#135direct (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×#33832
C7×Q8×Q162
C7×#33812
C7×#33802
C7×Q8×SD162
C7×Q8×D82
C7×#31602
C7×#30762
C2×C7×C8×Q82
C4×C7×#1352
Q8×#1357