Finite Group Databasebeta

← C7×#19056 #13196272

Subgroups — C7×#19056 #13196272

Order 1792.

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

normal subgroups only

This group has 992 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
1C7×#3549C211yesmaximal, characteristic
2C7×#3216C212yesmaximal, complemented
3C7×#3135C211yesmaximal, characteristic, complemented
4C7×#1659C218yesmaximal, complemented
5C7×#1536C211yesmaximal, characteristic, complemented
6C7×#1530C212yesmaximal, complemented
7C7×H(3,𝔽4)C2214yes—
8C7×#555—21no—
9C7×#555—22no—
10C7×D42—21no—
11C7×D42—22no—
12C7×#535—22no—
13C7×#535—24no—
14C7×#533C2211yescharacteristic, complemented
15C7×#532C2212yes—
16C7×#532C2211yescharacteristic
17C7×#497C2214yescomplemented
18C7×#491—21no—
19C7×#490—21no—
20C7×#489C2214yescomplemented
21C7×#489C2214yescomplemented
22C7×#462—22no—
23C7×#460—22no—
24C7×#452—24no—
25C7×#432C2212yescomplemented
26C7×#416—22no—
27C2×C7×#97C2211yescharacteristic, complemented
28C7×#329C2218yescomplemented
29C7×#329C2214yescomplemented
30#19056C711yesmaximal, characteristic, direct product factor, Sylow
31C2×C7×#41D412yes—
32C2×C7×#41D414yes—
33C22×C7×D4—24no—
34C22×C7×D4—22no—
35C2×C7×Q16—24no—
36C2×C7×SD16—24no—
37C22×C7×C8—21no—
38C7×C4⋊Q8D411yescharacteristic
39C7×C4⋊Q8D412yes—
40C7×C4⋊Q8C2312yes—
41C7×#127—24no—
42C7×#127D411yescharacteristic
43C7×#127D412yes—
44C7×#125—21no—
45C7×#124—24no—
46C7×#124—28no—
47C7×#124C2312yes—
48C7×#124C2314yes—
49C7×C22⋊Q8—22no—
50C7×#121—24no—
51C7×#121—24no—
52C7×#121—24no—
53C7×#121—22no—
54C7×#121—22no—
55C7×#121—22no—
56C7×#120—24no—
57C7×#120—24no—
58C7×#120—28no—
59C4×C7×Q8—22no—
60C4×C7×D4—22no—
61C4×C7×D4—22no—
62C4×C7×D4—22no—
63C7×#117—24no—
64C7×#117—22no—
65C7×#117—21no—
66C2×C42×C7C2311yescharacteristic
67C7×C4⋊C8—22no—
68C7×#103—24no—
69C7×#103—24no—
70C7×#102—24no—
71C7×#102—24no—
72C7×C22⋊C8—22no—
73C7×#97—21no—
74C7×#97C2312yes—
75C7×#97C2314yes—
76#3549C2×C711yescharacteristic
77#3216C2×C712yescomplemented
78#3135C2×C711yescharacteristic, complemented
79#1659C2×C718yescomplemented
80#1536C2×C711yescharacteristic, complemented
81#1530C2×C712yescomplemented
82C24×C7—24no—
83C7×#41—44no—
84C7×#41—84no—
85C2×C7×Q8—22no—
86C2×C7×Q8C2×D412yes—
87C2×C7×Q8C2×D414yes—
88C2×C7×D4—24no—
89C2×C7×D4—28no—
90C2×C7×D4—22no—
91C2×C7×D4—24no—
92C2×C7×D4—22no—
93C2×C7×D4—22no—
94C2×C7×D4—22no—
95C2×C7×D4—88no—
96C2×C7×D4—84no—
97C2×C7×D4C2×D412yes—
98C2×C7×D4C2×D414yes—
99C22×C4×C7—24no—
100C22×C4×C7—22no—
101C22×C4×C7C2×D411yescharacteristic
102C22×C4×C7C2×D412yes—
103C7×Q16—84no—
104C7×SD16—84no—
105C2×C7×C8—22no—
106C2×C7×C8—24no—
107C2×C7×C8—42no—
108C7×C4⋊C4—22no—
109C7×C4⋊C4—22no—
110C7×C4⋊C4—22no—
111C7×C4⋊C4—22no—
112C7×C4⋊C4—22no—
113C7×C4⋊C4—42no—
114C7×C22⋊C4—28no—
115C7×C22⋊C4—24no—
116C7×C22⋊C4—24no—
117C7×C22⋊C4—24no—
118C7×C22⋊C4—24no—
119C7×C22⋊C4—41no—
120C42×C7—24no—
121C42×C7—41no—
122C42×C7C2×D411yescharacteristic
123C42×C7C2×D412yes—
124C42×C7C2411yescharacteristic
125H(3,𝔽4)C22×C714yes—
126#555—22no—
127#555—21no—
128D42—22no—
129D42—21no—
130#535—22no—
131#535—24no—
132#533C22×C711yescharacteristic, complemented
133#532C22×C711yescharacteristic
134#532C22×C712yes—
135#497C22×C714yescomplemented
136#491—21no—
137#490—21no—
138#489C22×C714yescomplemented
139#489C22×C714yescomplemented
140#462—22no—
141#460—22no—
142#452—24no—
143#432C22×C712yescomplemented
144#416—22no—
145C2×#97C22×C711yescharacteristic, complemented
146#329C22×C714yescomplemented
147#329C22×C718yescomplemented
148C23×C7—22no—
149C23×C7—24no—
150C23×C7—84no—
151C23×C7—88no—
152C23×C7#12011yescharacteristic
153C7×Q8—44no—
154C7×Q8—42no—
155C7×Q8—42no—
156C7×Q8—42no—
157C7×D4—44no—
158C7×D4—42no—
159C7×D4—44no—
160C7×D4—44no—
161C7×D4—44no—
162C7×D4—82no—
163C7×D4—84no—
164C7×D4—84no—
165C7×D4—88no—
166C2×C4×C7—22no—
167C2×C4×C7—22no—
168C2×C4×C7—24no—
169C2×C4×C7—42no—
170C2×C4×C7—48no—
171C2×C4×C7—44no—
172C2×C4×C7—42no—
173C2×C4×C7#12012yes—
174C2×C4×C7#12011yescharacteristic
175C2×C4×C7C22×D411yescharacteristic
176C2×C4×C7C22×D412yes—
177C7×C8—42no—
178C7×C8—44no—
179C2×#41C7×D412yes—
180C2×#41C7×D414yes—
181C22×D4—24no—
182C22×D4—22no—
183C2×Q16—24no—
184C2×SD16—24no—
185C22×C8—21no—
186C4⋊Q8C7×D412yes—
187C4⋊Q8C7×D411yescharacteristic
188C4⋊Q8C23×C712yes—
189#127—24no—
190#127C7×D412yes—
191#127C7×D411yescharacteristic
192#125—21no—
193#124—24no—
194#124—28no—
195#124C23×C712yes—
196#124C23×C714yes—
197C22⋊Q8—22no—
198#121—24no—
199#121—24no—
200#121—24no—
201#121—22no—
202#121—22no—
203#121—22no—
204#120—24no—
205#120—24no—
206#120—28no—
207C4×Q8—22no—
208C4×D4—22no—
209C4×D4—22no—
210C4×D4—22no—
211#117—24no—
212#117—22no—
213#117—21no—
214C2×C42C23×C711yescharacteristic
215C4⋊C8—22no—
216#103—24no—
217#103—24no—
218#102—24no—
219#102—24no—
220C22⋊C8—22no—
221#97—21no—
222#97C23×C712yes—
223#97C23×C714yes—
224C22×C7—21no—
225C22×C7—22no—
226C22×C7—42no—
227C22×C7—44no—
228C22×C7—44no—
229C22×C7—44no—
230C22×C7—44no—
231C22×C7—168no—
232C22×C7C2×#12011yescharacteristic
233C4×C7—21no—
234C4×C7—22no—
235C4×C7—22no—
236C4×C7—42no—
237C4×C7—44no—
238C4×C7—82no—
239C4×C7#45512yes—
240C24—24no—
241#41—44no—
242#41—84no—
243C2×Q8—22no—
244C2×Q8C2×C7×D412yes—
245C2×Q8C2×C7×D414yes—
246C2×D4—24no—
247C2×D4—28no—
248C2×D4—22no—
249C2×D4—24no—
250C2×D4—22no—
251C2×D4—22no—
252C2×D4—22no—
253C2×D4—88no—
254C2×D4—84no—
255C2×D4C2×C7×D412yes—
256C2×D4C2×C7×D414yes—
257C22×C4—24no—
258C22×C4—22no—
259C22×C4C2×C7×D411yescharacteristic
260C22×C4C2×C7×D412yes—
261Q16—84no—
262SD16—84no—
263C2×C8—22no—
264C2×C8—24no—
265C2×C8—42no—
266C4⋊C4—22no—
267C4⋊C4—22no—
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—22no—
271C4⋊C4—42no—
272C22⋊C4—28no—
273C22⋊C4—24no—
274C22⋊C4—24no—
275C22⋊C4—24no—
276C22⋊C4—24no—
277C22⋊C4—41no—
278C42—24no—
279C42—41no—
280C42C2×C7×D411yescharacteristic
281C42C2×C7×D412yes—
282C42C24×C711yescharacteristic
283C2×C7—41no—
284C2×C7—82no—
285C2×C7—84no—
286C2×C7C2×#45511yescharacteristic
287C2×C7#301912yes—
288C23—22no—
289C23—24no—
290C23—84no—
291C23—88no—
292C23C7×#12011yescharacteristic
293Q8—44no—
294Q8—42no—
295Q8—42no—
296Q8—42no—
297D4—44no—
298D4—42no—
299D4—44no—
300D4—44no—
301D4—44no—
302D4—82no—
303D4—84no—
304D4—84no—
305D4—88no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—24no—
309C2×C4—42no—
310C2×C4—48no—
311C2×C4—44no—
312C2×C4—42no—
313C2×C4C7×#12012yes—
314C2×C4C7×#12011yescharacteristic
315C2×C4C22×C7×D411yescharacteristic
316C2×C4C22×C7×D412yes—
317C8—42no—
318C8—44no—
319C7#1905611yescharacteristic, direct product factor, Sylow
320C22—21no—
321C22—22no—
322C22—42no—
323C22—44no—
324C22—44no—
325C22—44no—
326C22—44no—
327C22—168no—
328C22C2×C7×#12011yescharacteristic
329C4—21no—
330C4—22no—
331C4—22no—
332C4—42no—
333C4—44no—
334C4—82no—
335C4C7×#45512yes—
336C2—41no—
337C2—82no—
338C2—84no—
339C2C2×C7×#45511yescharacteristic
340C2C7×#301912yes—
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC42C24×C7
prime radicalC7×#3549C2
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×#3216C2semidirect
3. C7×#3135C2semidirect
4. C7×#1659C2semidirect
5. C7×#1536C2semidirect
6. C7×#1530C2semidirect
14. C7×#533C22semidirect
17. C7×#497C22semidirect
20. C7×#489C22semidirect
21. C7×#489C22semidirect
25. C7×#432C22semidirect
27. C2×C7×#97C22semidirect
28. C7×#329C22semidirect
29. C7×#329C22semidirect
30. #19056C7direct (N × H)
77. #3216C2×C7semidirect
78. #3135C2×C7semidirect
79. #1659C2×C7semidirect
80. #1536C2×C7semidirect
81. #1530C2×C7semidirect
132. #533C22×C7semidirect
135. #497C22×C7semidirect
138. #489C22×C7semidirect
139. #489C22×C7semidirect
143. #432C22×C7semidirect
145. C2×#97C22×C7semidirect
146. #329C22×C7semidirect
147. #329C22×C7semidirect
319. C7#19056direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#35492
C7×#32162
C7×#31352
C7×#16592
C7×#15362
C7×#15302
#190567