Finite Group Databasebeta

← C7×#18997 #13196213

Subgroups — C7×#18997 #13196213

Order 1792.

1066 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 1066 conjugacy classes of subgroups — too many to draw. Only its 222 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×H(3,𝔽4)C211yesmaximal, characteristic
2C7×#3215C212yesmaximal, complemented
3C7×#3140C211yesmaximal, characteristic, complemented
4C7×#1659C218yesmaximal, complemented
5C7×#1533C211yesmaximal, characteristic, complemented
6C7×#1527C212yesmaximal, complemented
7C7×H(3,𝔽4)—22no—
8C7×H(3,𝔽4)—24no—
9C7×H(3,𝔽4)C2214yes—
10C7×#533C2211yescharacteristic, complemented
11C2×C7×#124—21no—
12C2×C7×#124—22no—
13C2×C7×#124C2212yes—
14C2×C7×#124C2211yescharacteristic
15C2×C7×#120—21no—
16C2×C7×#120—22no—
17C7×#497C2214yescomplemented
18C7×#489C2214yescomplemented
19C7×#489C2214yescomplemented
20C7×#488—22no—
21C7×#464—24no—
22C7×#451—22no—
23C7×#450—22no—
24C7×#432C2212yescomplemented
25C7×#419—21no—
26C7×#418—21no—
27C4×C7×M16C2211yescharacteristic, complemented
28C7×#329C2218yescomplemented
29C7×#329C2214yescomplemented
30#18997C711yesmaximal, characteristic, direct product factor, Sylow
31C25×C7—21no—
32C22×C7×Q8D411yescharacteristic
33C22×C7×Q8D412yes—
34C22×C7×D4—22no—
35C22×C7×D4—21no—
36C22×C7×D4D411yescharacteristic
37C22×C7×D4D412yes—
38C2×C7×Q16—24no—
39C2×C7×SD16—24no—
40C2×C7×M16—21no—
41C7×C4⋊Q8C2312yescomplemented
42C7×#125—21no—
43C7×#124—24no—
44C7×#124—28no—
45C7×#124—24no—
46C7×#124—28no—
47C7×#124—22no—
48C7×#124—22no—
49C7×#124—22no—
50C7×#124D412yes—
51C7×#124D414yes—
52C7×#124C2312yes—
53C7×#124C2314yes—
54C7×C22⋊Q8—22no—
55C7×#120—24no—
56C7×#120—28no—
57C7×#120—24no—
58C7×#120—28no—
59C4×C7×Q8—22no—
60C7×#117—21no—
61C2×C7×C22⋊C4—21no—
62C2×C7×C22⋊C4—22no—
63C2×C7×C22⋊C4—22no—
64C2×C7×C22⋊C4—22no—
65C2×C7×C22⋊C4—22no—
66C2×C42×C7—21no—
67C2×C42×C7C2311yescharacteristic
68C7×C4⋊C8—22no—
69C7×#103—24no—
70C7×#103—24no—
71C7×#102—24no—
72C7×#102—24no—
73C7×C22⋊C8—22no—
74C7×#97C2312yescomplemented
75C7×#97C2314yescomplemented
76C4×C7×C8—21no—
77C2×H(3,𝔽4)C2×C711yescharacteristic
78#3215C2×C712yescomplemented
79#3140C2×C711yescharacteristic, complemented
80#1659C2×C718yescomplemented
81#1533C2×C711yescharacteristic, complemented
82#1527C2×C712yescomplemented
83C24×C7—24no—
84C24×C7—21no—
85C24×C7—22no—
86C24×C7—84no—
87C24×C7—82no—
88C2×C7×Q8—21no—
89C2×C7×Q8—22no—
90C2×C7×Q8—22no—
91C2×C7×Q8—42no—
92C2×C7×Q8—82no—
93C2×C7×Q8C2×D412yes—
94C2×C7×Q8C2×D414yes—
95C2×C7×D4—24no—
96C2×C7×D4—28no—
97C2×C7×D4—21no—
98C2×C7×D4—22no—
99C2×C7×D4—42no—
100C2×C7×D4—82no—
101C2×C7×D4—84no—
102C2×C7×D4—82no—
103C2×C7×D4C2×D412yes—
104C2×C7×D4C2×D414yes—
105C22×C4×C7—22no—
106C22×C4×C7—21no—
107C22×C4×C7C2×D411yescharacteristic
108C22×C4×C7C2×D412yes—
109C7×Q16—84no—
110C7×SD16—84no—
111C7×M16—42no—
112C2×C7×C8—22no—
113C2×C7×C8—24no—
114C7×C4⋊C4—22no—
115C7×C4⋊C4—22no—
116C7×C4⋊C4—42no—
117C7×C22⋊C4—28no—
118C7×C22⋊C4—24no—
119C7×C22⋊C4—24no—
120C7×C22⋊C4—24no—
121C7×C22⋊C4—24no—
122C7×C22⋊C4—24no—
123C7×C22⋊C4—24no—
124C7×C22⋊C4—24no—
125C7×C22⋊C4—41no—
126C42×C7—24no—
127C42×C7—41no—
128C42×C7C2×D411yescharacteristic
129C42×C7C2×D412yes—
130C42×C7C2411yescharacteristic
131H(3,𝔽4)—24no—
132H(3,𝔽4)—22no—
133H(3,𝔽4)C22×C714yes—
134#533C22×C711yescharacteristic, complemented
135C2×#124—21no—
136C2×#124—22no—
137C2×#124C22×C711yescharacteristic
138C2×#124C22×C712yes—
139C2×#120—21no—
140C2×#120—22no—
141#497C22×C714yescomplemented
142#489C22×C714yescomplemented
143#489C22×C714yescomplemented
144#488—22no—
145#464—24no—
146#451—22no—
147#450—22no—
148#432C22×C712yescomplemented
149#419—21no—
150#418—21no—
151C4×M16C22×C711yescharacteristic, complemented
152#329C22×C718yescomplemented
153#329C22×C714yescomplemented
154C23×C7—22no—
155C23×C7—24no—
156C23×C7—44no—
157C23×C7—44no—
158C23×C7—44no—
159C23×C7—42no—
160C23×C7—44no—
161C23×C7—84no—
162C23×C7—88no—
163C23×C7—816no—
164C23×C7#12011yescharacteristic
165C7×Q8—44no—
166C7×Q8—42no—
167C7×Q8—44no—
168C7×Q8—81no—
169C7×Q8—82no—
170C7×D4—44no—
171C7×D4—42no—
172C7×D4—81no—
173C7×D4—82no—
174C7×D4—84no—
175C7×D4—88no—
176C2×C4×C7—22no—
177C2×C4×C7—22no—
178C2×C4×C7—24no—
179C2×C4×C7—42no—
180C2×C4×C7—44no—
181C2×C4×C7—42no—
182C2×C4×C7—42no—
183C2×C4×C7#12011yescharacteristic
184C2×C4×C7#12012yes—
185C2×C4×C7C22×D411yescharacteristic
186C2×C4×C7C22×D412yes—
187C7×C8—42no—
188C7×C8—44no—
189C25—21no—
190C22×Q8C7×D411yescharacteristic
191C22×Q8C7×D412yes—
192C22×D4—22no—
193C22×D4—21no—
194C22×D4C7×D411yescharacteristic
195C22×D4C7×D412yes—
196C2×Q16—24no—
197C2×SD16—24no—
198C2×M16—21no—
199C4⋊Q8C23×C712yescomplemented
200#125—21no—
201#124—24no—
202#124—28no—
203#124—24no—
204#124—28no—
205#124—22no—
206#124—22no—
207#124—22no—
208#124C7×D414yes—
209#124C7×D412yes—
210#124C23×C712yes—
211#124C23×C714yes—
212C22⋊Q8—22no—
213#120—24no—
214#120—28no—
215#120—24no—
216#120—28no—
217C4×Q8—22no—
218#117—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C42—21no—
225C2×C42C23×C711yescharacteristic
226C4⋊C8—22no—
227#103—24no—
228#103—24no—
229#102—24no—
230#102—24no—
231C22⋊C8—22no—
232#97C23×C712yescomplemented
233#97C23×C714yescomplemented
234C4×C8—21no—
235C22×C7—22no—
236C22×C7—42no—
237C22×C7—44no—
238C22×C7—44no—
239C22×C7—44no—
240C22×C7—44no—
241C22×C7—84no—
242C22×C7—88no—
243C22×C7—816no—
244C22×C7#45512yes—
245C22×C7C2×#12011yescharacteristic
246C4×C7—21no—
247C4×C7—21no—
248C4×C7—22no—
249C4×C7—22no—
250C4×C7—42no—
251C4×C7—44no—
252C4×C7—82no—
253C24—24no—
254C24—21no—
255C24—22no—
256C24—84no—
257C24—82no—
258C2×Q8—21no—
259C2×Q8—22no—
260C2×Q8—22no—
261C2×Q8—42no—
262C2×Q8—82no—
263C2×Q8C2×C7×D412yes—
264C2×Q8C2×C7×D414yes—
265C2×D4—24no—
266C2×D4—28no—
267C2×D4—21no—
268C2×D4—22no—
269C2×D4—42no—
270C2×D4—82no—
271C2×D4—84no—
272C2×D4—82no—
273C2×D4C2×C7×D412yes—
274C2×D4C2×C7×D414yes—
275C22×C4—22no—
276C22×C4—21no—
277C22×C4C2×C7×D411yescharacteristic
278C22×C4C2×C7×D412yes—
279Q16—84no—
280SD16—84no—
281M16—42no—
282C2×C8—22no—
283C2×C8—24no—
284C4⋊C4—22no—
285C4⋊C4—22no—
286C4⋊C4—42no—
287C22⋊C4—28no—
288C22⋊C4—24no—
289C22⋊C4—24no—
290C22⋊C4—24no—
291C22⋊C4—24no—
292C22⋊C4—24no—
293C22⋊C4—24no—
294C22⋊C4—24no—
295C22⋊C4—41no—
296C42—24no—
297C42—41no—
298C42C2×C7×D411yescharacteristic
299C42C2×C7×D412yes—
300C42C24×C711yescharacteristic
301C2×C7—22no—
302C2×C7—82no—
303C2×C7—84no—
304C2×C7C2×#45511yescharacteristic
305C2×C7#301912yes—
306C23—22no—
307C23—24no—
308C23—44no—
309C23—44no—
310C23—44no—
311C23—42no—
312C23—44no—
313C23—84no—
314C23—88no—
315C23—816no—
316C23C7×#12011yescharacteristic
317Q8—44no—
318Q8—42no—
319Q8—44no—
320Q8—81no—
321Q8—82no—
322D4—44no—
323D4—42no—
324D4—81no—
325D4—82no—
326D4—84no—
327D4—88no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—24no—
331C2×C4—42no—
332C2×C4—44no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4C7×#12011yescharacteristic
336C2×C4C7×#12012yes—
337C2×C4C22×C7×D411yescharacteristic
338C2×C4C22×C7×D412yes—
339C8—42no—
340C8—44no—
341C7#1899711yescharacteristic, direct product factor, Sylow
342C22—22no—
343C22—42no—
344C22—44no—
345C22—44no—
346C22—44no—
347C22—44no—
348C22—84no—
349C22—88no—
350C22—816no—
351C22C7×#45512yes—
352C22C2×C7×#12011yescharacteristic
353C4—21no—
354C4—21no—
355C4—22no—
356C4—22no—
357C4—42no—
358C4—44no—
359C4—82no—
360C2—22no—
361C2—82no—
362C2—84no—
363C2C2×C7×#45511yescharacteristic
364C2C7×#301912yes—
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC42C24×C7
prime radicalC2×C7×H(3,𝔽4)C2
LCSG ▷ 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×#3215C2semidirect
3. C7×#3140C2semidirect
4. C7×#1659C2semidirect
5. C7×#1533C2semidirect
6. C7×#1527C2semidirect
10. C7×#533C22semidirect
17. C7×#497C22semidirect
18. C7×#489C22semidirect
19. C7×#489C22semidirect
24. C7×#432C22semidirect
27. C4×C7×M16C22semidirect
28. C7×#329C22semidirect
29. C7×#329C22semidirect
30. #18997C7direct (N × H)
41. C7×C4⋊Q8C23semidirect
74. C7×#97C23semidirect
75. C7×#97C23semidirect
78. #3215C2×C7semidirect
79. #3140C2×C7semidirect
80. #1659C2×C7semidirect
81. #1533C2×C7semidirect
82. #1527C2×C7semidirect
134. #533C22×C7semidirect
141. #497C22×C7semidirect
142. #489C22×C7semidirect
143. #489C22×C7semidirect
148. #432C22×C7semidirect
151. C4×M16C22×C7semidirect
152. #329C22×C7semidirect
153. #329C22×C7semidirect
199. C4⋊Q8C23×C7semidirect
232. #97C23×C7semidirect
233. #97C23×C7semidirect
341. C7#18997direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×H(3,𝔽4)2
C7×#32152
C7×#31402
C7×#16592
C7×#15332
C7×#15272
#189977