Finite Group Databasebeta

← C7×#10168 #13187384

Subgroups — C7×#10168 #13187384

Order 1792.

642 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3205C211yesmaximal, characteristic
2C7×#3152C211yesmaximal, characteristic, complemented
3C2×C7×#486C211yesmaximal, characteristic, complemented
4C7×#1965C211yesmaximal, characteristic, complemented
5C7×#1546C211yesmaximal, characteristic, complemented
6C7×#1492C211yesmaximal, characteristic, complemented
7C7×#1278C211yesmaximal, characteristic, complemented
8C2×C7×#137—22no—
9C7×#532C2211yescharacteristic
10C2×C7×C4⋊Q8C2211yescharacteristic
11C2×C7×#127C2211yescharacteristic
12C7×#497—22no—
13C7×#494—21no—
14C7×#493—21no—
15C7×#492—21no—
16C7×#486—21no—
17C7×#486—22no—
18C7×#486—22no—
19C7×#459—22no—
20C7×#447—22no—
21C7×#431C2211yescharacteristic
22C7×#423C2211yescharacteristic
23C2×C7×#102—22no—
24C2×C7×#100—22no—
25C4×C7×M16C2211yescharacteristic
26C2×C4×C7×C8C2211yescharacteristic
27C7×#332—22no—
28C7×D4⋊C8—22no—
29#10168C711yesmaximal, characteristic, direct product factor, Sylow
30C2×C7×#41—21no—
31C22×C7×Q8—21no—
32C22×C7×D4—21no—
33C22×C7×D4—21no—
34C22×C7×D4D412yes—
35C7×#137—44no—
36C7×#137—44no—
37C2×C7×Q16—24no—
38C2×C7×SD16—24no—
39C2×C7×D8—24no—
40C2×C7×M16—21no—
41C2×C7×M16D412yes—
42C22×C7×C8—21no—
43C7×C4⋊Q8—21no—
44C7×C4⋊Q8—21no—
45C7×C4⋊Q8—21no—
46C7×C4⋊Q8D411yescharacteristic
47C7×C4⋊Q8D412yes—
48C7×#127—21no—
49C7×#127—21no—
50C7×#127—21no—
51C7×#127—21no—
52C7×#127D411yescharacteristic
53C7×#124—21no—
54C7×#121—21no—
55C7×#121—21no—
56C4×C7×D4—21no—
57C4×C7×D4—21no—
58C2×C7×C4⋊C4—21no—
59C2×C7×C4⋊C4—21no—
60C2×C42×C7C2311yescharacteristic
61C7×C4⋊C8—21no—
62C7×C4⋊C8—21no—
63C7×C4⋊C8D412yes—
64C7×#102—28no—
65C7×#102—24no—
66C7×#100—44no—
67C7×C22⋊C8—21no—
68C7×#97—21no—
69C4×C7×C8—21no—
70C4×C7×C8—21no—
71C4×C7×C8—21no—
72C4×C7×C8D412yes—
73#3205C2×C711yescharacteristic
74#3152C2×C711yescharacteristic, complemented
75C2×#486C2×C711yescharacteristic, complemented
76#1965C2×C711yescharacteristic, complemented
77#1546C2×C711yescharacteristic, complemented
78#1492C2×C711yescharacteristic, complemented
79#1278C2×C711yescharacteristic, complemented
80C24×C7—22no—
81C7×#41—42no—
82C7×#41—42no—
83C2×C7×Q8—21no—
84C2×C7×Q8—22no—
85C2×C7×Q8—41no—
86C2×C7×Q8—42no—
87C2×C7×Q8—42no—
88C2×C7×D4—22no—
89C2×C7×D4—22no—
90C2×C7×D4—21no—
91C2×C7×D4—21no—
92C2×C7×D4—21no—
93C2×C7×D4—44no—
94C2×C7×D4—41no—
95C2×C7×D4—41no—
96C2×C7×D4—41no—
97C2×C7×D4—41no—
98C2×C7×D4—41no—
99C2×C7×D4—42no—
100C2×C7×D4—42no—
101C2×C7×D4—82no—
102C22×C4×C7—41no—
103C22×C4×C7—41no—
104C22×C4×C7C2×D411yescharacteristic
105C22×C4×C7C2×D411yescharacteristic
106C22×C4×C7#4111yescharacteristic
107C7×Q16—48no—
108C7×SD16—48no—
109C7×D8—84no—
110C7×M16—22no—
111C7×M16—42no—
112C7×M16D814yescomplemented
113C2×C7×C8—22no—
114C2×C7×C8—22no—
115C2×C7×C8—22no—
116C2×C7×C8—42no—
117C2×C7×C8—41no—
118C7×C4⋊C4—22no—
119C7×C4⋊C4—21no—
120C7×C4⋊C4—22no—
121C7×C4⋊C4—21no—
122C7×C4⋊C4—41no—
123C7×C4⋊C4—41no—
124C7×C22⋊C4—41no—
125C7×C22⋊C4—41no—
126C42×C7C2×D411yescharacteristic
127C42×C7C2×D411yescharacteristic
128C42×C7C2×D411yescharacteristic
129C42×C7C2×D411yescharacteristic
130C2×#137—22no—
131#532C22×C711yescharacteristic
132C2×C4⋊Q8C22×C711yescharacteristic
133C2×#127C22×C711yescharacteristic
134#497—22no—
135#494—21no—
136#493—21no—
137#492—21no—
138#486—22no—
139#486—21no—
140#486—22no—
141#459—22no—
142#447—22no—
143#431C22×C711yescharacteristic
144#423C22×C711yescharacteristic
145C2×#102—22no—
146C2×#100—22no—
147C4×M16C22×C711yescharacteristic
148C2×C4×C8C22×C711yescharacteristic
149#332—22no—
150D4⋊C8—22no—
151C23×C7—41no—
152C23×C7—42no—
153C23×C7—44no—
154C23×C7—82no—
155C23×C7—82no—
156C23×C7#12111yescharacteristic
157C7×Q8—44no—
158C7×Q8—42no—
159C7×Q8—81no—
160C7×Q8—81no—
161C7×D4—44no—
162C7×D4—42no—
163C7×D4—42no—
164C7×D4—44no—
165C7×D4—42no—
166C7×D4—82no—
167C7×D4—82no—
168C7×D4—82no—
169C7×D4—82no—
170C7×D4—81no—
171C7×D4—81no—
172C2×C4×C7—22no—
173C2×C4×C7—22no—
174C2×C4×C7—41no—
175C2×C4×C7—42no—
176C2×C4×C7—82no—
177C2×C4×C7—81no—
178C2×C4×C7—81no—
179C2×C4×C7#12011yescharacteristic
180C2×C4×C7#12011yescharacteristic
181C2×C4×C7#12011yescharacteristic
182C2×C4×C7#12111yescharacteristic
183C2×C4×C7#12111yescharacteristic
184C2×C4×C7#12711yescharacteristic
185C2×C4×C7C2×D812yes—
186C2×C4×C7AGL(1,ℤ/8)12yes—
187C7×C8—24no—
188C7×C8—42no—
189C7×C8—42no—
190C7×C8—81no—
191C2×#41—21no—
192C22×Q8—21no—
193C22×D4—21no—
194C22×D4—21no—
195C22×D4C7×D412yes—
196#137—44no—
197#137—44no—
198C2×Q16—24no—
199C2×SD16—24no—
200C2×D8—24no—
201C2×M16—21no—
202C2×M16C7×D412yes—
203C22×C8—21no—
204C4⋊Q8—21no—
205C4⋊Q8—21no—
206C4⋊Q8—21no—
207C4⋊Q8C7×D412yes—
208C4⋊Q8C7×D411yescharacteristic
209#127—21no—
210#127—21no—
211#127—21no—
212#127—21no—
213#127C7×D411yescharacteristic
214#124—21no—
215#121—21no—
216#121—21no—
217C4×D4—21no—
218C4×D4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C42C23×C711yescharacteristic
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8C7×D412yes—
225#102—28no—
226#102—24no—
227#100—44no—
228C22⋊C8—21no—
229#97—21no—
230C4×C8—21no—
231C4×C8—21no—
232C4×C8—21no—
233C4×C8C7×D412yes—
234C22×C7—21no—
235C22×C7—21no—
236C22×C7—82no—
237C22×C7—81no—
238C22×C7—81no—
239C22×C7—82no—
240C22×C7—81no—
241C22×C7—82no—
242C22×C7—164no—
243C22×C7#39211yescharacteristic
244C22×C7#45912yes—
245C4×C7—21no—
246C4×C7—21no—
247C4×C7—81no—
248C4×C7—82no—
249C4×C7#44712yes—
250C4×C7#45512yes—
251C4×C7#47112yes—
252C4×C7#49311yescharacteristic
253C4×C7#49611yescharacteristic
254C24—22no—
255#41—42no—
256#41—42no—
257C2×Q8—21no—
258C2×Q8—22no—
259C2×Q8—41no—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×D4—22no—
263C2×D4—22no—
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—21no—
267C2×D4—44no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4—82no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4C2×C7×D411yescharacteristic
279C22×C4C2×C7×D411yescharacteristic
280C22×C4C7×#4111yescharacteristic
281Q16—48no—
282SD16—48no—
283D8—84no—
284M16—22no—
285M16—42no—
286M16C7×D814yescomplemented
287C2×C8—22no—
288C2×C8—22no—
289C2×C8—22no—
290C2×C8—42no—
291C2×C8—41no—
292C4⋊C4—22no—
293C4⋊C4—21no—
294C4⋊C4—22no—
295C4⋊C4—21no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C22⋊C4—41no—
299C22⋊C4—41no—
300C42C2×C7×D411yescharacteristic
301C42C2×C7×D411yescharacteristic
302C42C2×C7×D411yescharacteristic
303C42C2×C7×D411yescharacteristic
304C2×C7—22no—
305C2×C7—161no—
306C2×C7—162no—
307C2×C7#200411yescharacteristic
308C2×C7#201111yescharacteristic
309C2×C7#201911yescharacteristic
310C23—41no—
311C23—42no—
312C23—44no—
313C23—82no—
314C23—82no—
315C23C7×#12111yescharacteristic
316Q8—44no—
317Q8—42no—
318Q8—81no—
319Q8—81no—
320D4—44no—
321D4—42no—
322D4—42no—
323D4—42no—
324D4—44no—
325D4—82no—
326D4—82no—
327D4—82no—
328D4—81no—
329D4—81no—
330D4—82no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—41no—
334C2×C4—42no—
335C2×C4—82no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4C7×#12011yescharacteristic
339C2×C4C7×#12011yescharacteristic
340C2×C4C7×#12011yescharacteristic
341C2×C4C7×#12111yescharacteristic
342C2×C4C7×#12111yescharacteristic
343C2×C4C7×#12711yescharacteristic
344C2×C4C2×C7×D812yes—
345C2×C4C7×AGL(1,ℤ/8)12yes—
346C8—24no—
347C8—42no—
348C8—42no—
349C8—81no—
350C7#1016811yescharacteristic, direct product factor, Sylow
351C22—21no—
352C22—21no—
353C22—82no—
354C22—81no—
355C22—81no—
356C22—82no—
357C22—81no—
358C22—82no—
359C22—164no—
360C22C7×#39211yescharacteristic
361C22C7×#45912yes—
362C4—21no—
363C4—21no—
364C4—81no—
365C4—82no—
366C4C7×#44712yes—
367C4C7×#45512yes—
368C4C7×#47112yes—
369C4C7×#49311yescharacteristic
370C4C7×#49611yescharacteristic
371C2—22no—
372C2—162no—
373C2—161no—
374C2C7×#200411yescharacteristic
375C2C7×#201111yescharacteristic
376C2C7×#201911yescharacteristic
subgroupG/H
center, socleC22×C7#392
commutator, Frattini, cosocleC2×C42C23×C7
prime radicalC7×#3205C2
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#3152C2semidirect
3. C2×C7×#486C2semidirect
4. C7×#1965C2semidirect
5. C7×#1546C2semidirect
6. C7×#1492C2semidirect
7. C7×#1278C2semidirect
29. #10168C7direct (N × H)
74. #3152C2×C7semidirect
75. C2×#486C2×C7semidirect
76. #1965C2×C7semidirect
77. #1546C2×C7semidirect
78. #1492C2×C7semidirect
79. #1278C2×C7semidirect
112. C7×M16D8semidirect
286. M16C7×D8semidirect
350. C7#10168direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#32052
C7×#31522
C2×C7×#4862
C7×#19652
C7×#15462
C7×#14922
C7×#12782
#101687