Finite Group Databasebeta

← C7×#23896 #13201112

Subgroups — C7×#23896 #13201112

Order 1792.

1764 conjugacy classes of subgroups; only the 282 normal subgroups are shown.

normal subgroups only

This group has 1764 conjugacy classes of subgroups — too many to draw. Only its 282 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×#3485C212yesmaximal, complemented
2C7×#3026C214yesmaximal, complemented
3C7×#2022C214yesmaximal, complemented
4C7×#1913C214yesmaximal, complemented
5C7×#1782C211yesmaximal, characteristic, complemented
6C2×C7×H(5,𝔽2)C2212yescomplemented
7C22×C7×#41C2211yescharacteristic
8C7×#547—28no—
9C7×#546—28no—
10C7×#536—22no—
11C7×#534—22no—
12C7×#534C2214yes—
13C7×#534C2214yes—
14C7×#525—22no—
15C2×C7×#123—22no—
16C2×C7×#123C2214yescomplemented
17C7×#518C2212yescomplemented
18C7×#458—28no—
19C7×B(4,𝔽2)—28no—
20C2×C7×#99C2218yescomplemented
21C2×C7×#99C2214yescomplemented
22C7×#381C2212yescomplemented
23C7×#342C2214yescomplemented
24#23896C711yesmaximal, characteristic, direct product factor, Sylow
25C7×H(5,𝔽2)—28no—
26C7×H(5,𝔽2)—44no—
27C2×C7×#41—28no—
28C2×C7×#41—22no—
29C2×C7×#41—24no—
30C2×C7×#41—42no—
31C22×C7×Q8D411yescharacteristic, complemented
32C22×C7×D4—22no—
33C22×C7×D4—24no—
34C22×C7×D4D411yescharacteristic, complemented
35C22×C7×D4C2314yes—
36C22×C7×D4C2312yes—
37C23×C4×C7D412yescomplemented
38C23×C4×C7C2311yescharacteristic
39C7×C4⋊Q8—24no—
40C7×#124—24no—
41C7×#123—28no—
42C7×#123—24no—
43C7×#123—24no—
44C7×#123—24no—
45C7×#123—24no—
46C7×#123—28no—
47C7×#123—28no—
48C7×#123D418yescomplemented
49C7×C22⋊Q8—24no—
50C7×C22⋊Q8—28no—
51C7×#121—24no—
52C7×#121—28no—
53C7×#121—28no—
54C7×#121—44no—
55C7×#120—24no—
56C7×#120—24no—
57C7×#120—28no—
58C7×#120D418yescomplemented
59C4×C7×D4—28no—
60C4×C7×D4—28no—
61C7×#117—22no—
62C2×C7×C4⋊C4—24no—
63C2×C7×C4⋊C4—24no—
64C2×C7×C22⋊C4—24no—
65C2×C7×C22⋊C4—22no—
66C2×C7×C22⋊C4—24no—
67C2×C7×C22⋊C4C2314yes—
68C2×C7×C22⋊C4C2314yes—
69C7×#99—28no—
70C7×#99—216no—
71C7×#95—42no—
72#3485C2×C712yescomplemented
73#3026C2×C714yescomplemented
74#2022C2×C714yescomplemented
75#1913C2×C714yescomplemented
76#1782C2×C711yescharacteristic, complemented
77C24×C7—22no—
78C24×C7C2×D414yes—
79C24×C7C2×D412yes—
80C24×C7C2411yescharacteristic
81C7×#41—416no—
82C7×#41—44no—
83C7×#41—416no—
84C7×#41—44no—
85C2×C7×Q8—22no—
86C2×C7×Q8—24no—
87C2×C7×Q8—41no—
88C2×C7×D4—24no—
89C2×C7×D4—24no—
90C2×C7×D4—24no—
91C2×C7×D4—22no—
92C2×C7×D4—24no—
93C2×C7×D4—28no—
94C2×C7×D4—24no—
95C2×C7×D4—24no—
96C2×C7×D4—28no—
97C2×C7×D4—28no—
98C2×C7×D4—41no—
99C2×C7×D4—416no—
100C2×C7×D4—44no—
101C2×C7×D4—44no—
102C2×C7×D4—44no—
103C2×C7×D4C2×D418yes—
104C22×C4×C7—22no—
105C22×C4×C7—24no—
106C22×C4×C7—24no—
107C22×C4×C7—24no—
108C22×C4×C7—22no—
109C22×C4×C7—24no—
110C22×C4×C7—24no—
111C22×C4×C7—24no—
112C22×C4×C7—24no—
113C22×C4×C7—42no—
114C22×C4×C7C2×D414yes—
115C22×C4×C7C2×D411yescharacteristic
116C22×C4×C7C2×D411yescharacteristic
117C22×C4×C7C2×D412yes—
118C7×C4⋊C4—28no—
119C7×C4⋊C4—28no—
120C7×C4⋊C4—24no—
121C7×C4⋊C4—28no—
122C7×C4⋊C4—44no—
123C7×C22⋊C4—28no—
124C7×C22⋊C4—24no—
125C7×C22⋊C4—28no—
126C7×C22⋊C4—28no—
127C7×C22⋊C4—24no—
128C7×C22⋊C4—28no—
129C7×C22⋊C4—28no—
130C7×C22⋊C4—48no—
131C7×C22⋊C4C2×D418yes—
132C42×C7—24no—
133C2×H(5,𝔽2)C22×C712yescomplemented
134C22×#41C22×C711yescharacteristic
135#547—28no—
136#546—28no—
137#536—22no—
138#534—22no—
139#534C22×C714yes—
140#534C22×C714yes—
141#525—22no—
142C2×#123—22no—
143C2×#123C22×C714yescomplemented
144#518C22×C712yescomplemented
145#458—28no—
146B(4,𝔽2)—28no—
147C2×#99C22×C718yescomplemented
148C2×#99C22×C714yescomplemented
149#381C22×C712yescomplemented
150#342C22×C714yescomplemented
151C23×C7—22no—
152C23×C7—22no—
153C23×C7—24no—
154C23×C7—28no—
155C23×C7—44no—
156C23×C7—81no—
157C23×C7—82no—
158C23×C7—84no—
159C23×C7—84no—
160C23×C7#12014yes—
161C23×C7C22×D411yescharacteristic
162C23×C7C22×D414yes—
163C23×C7H(5,𝔽2)12yes—
164C7×Q8—44no—
165C7×Q8—44no—
166C7×D4—216no—
167C7×D4—44no—
168C7×D4—44no—
169C7×D4—48no—
170C7×D4—44no—
171C7×D4—44no—
172C7×D4—48no—
173C7×D4—88no—
174C7×D4—164no—
175C2×C4×C7—22no—
176C2×C4×C7—22no—
177C2×C4×C7—28no—
178C2×C4×C7—22no—
179C2×C4×C7—22no—
180C2×C4×C7—24no—
181C2×C4×C7—21no—
182C2×C4×C7—21no—
183C2×C4×C7—28no—
184C2×C4×C7—48no—
185C2×C4×C7—48no—
186C2×C4×C7—44no—
187C2×C4×C7—44no—
188C2×C4×C7—44no—
189C2×C4×C7—84no—
190C2×C4×C7—84no—
191C2×C4×C7—84no—
192C2×C4×C7#12014yes—
193H(5,𝔽2)—28no—
194H(5,𝔽2)—44no—
195C2×#41—28no—
196C2×#41—22no—
197C2×#41—24no—
198C2×#41—42no—
199C22×Q8C7×D411yescharacteristic, complemented
200C22×D4—22no—
201C22×D4—24no—
202C22×D4C7×D411yescharacteristic, complemented
203C22×D4C23×C714yes—
204C22×D4C23×C712yes—
205C23×C4C7×D412yescomplemented
206C23×C4C23×C711yescharacteristic
207C4⋊Q8—24no—
208#124—24no—
209#123—28no—
210#123—24no—
211#123—24no—
212#123—24no—
213#123—24no—
214#123—28no—
215#123—28no—
216#123C7×D418yescomplemented
217C22⋊Q8—24no—
218C22⋊Q8—28no—
219#121—24no—
220#121—28no—
221#121—28no—
222#121—44no—
223#120—24no—
224#120—24no—
225#120—28no—
226#120C7×D418yescomplemented
227C4×D4—28no—
228C4×D4—28no—
229#117—22no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4C23×C714yes—
236C2×C22⋊C4C23×C714yes—
237#99—28no—
238#99—216no—
239#95—42no—
240C22×C7—24no—
241C22×C7—42no—
242C22×C7—44no—
243C22×C7—41no—
244C22×C7—44no—
245C22×C7—44no—
246C22×C7—48no—
247C22×C7—44no—
248C22×C7—82no—
249C22×C7—88no—
250C22×C7—164no—
251C22×C7C2×#12012yes—
252C22×C7#53411yescharacteristic
253C22×C7D4214yes—
254C4×C7—24no—
255C4×C7—48no—
256C4×C7—41no—
257C4×C7—44no—
258C4×C7—41no—
259C4×C7—84no—
260C24—22no—
261C24C2×C7×D414yes—
262C24C2×C7×D412yes—
263C24C24×C711yescharacteristic
264#41—416no—
265#41—44no—
266#41—416no—
267#41—44no—
268C2×Q8—22no—
269C2×Q8—24no—
270C2×Q8—41no—
271C2×D4—24no—
272C2×D4—24no—
273C2×D4—24no—
274C2×D4—22no—
275C2×D4—24no—
276C2×D4—28no—
277C2×D4—24no—
278C2×D4—24no—
279C2×D4—28no—
280C2×D4—28no—
281C2×D4—41no—
282C2×D4—416no—
283C2×D4—44no—
284C2×D4—44no—
285C2×D4—44no—
286C2×D4C2×C7×D418yes—
287C22×C4—22no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—22no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—42no—
297C22×C4C2×C7×D414yes—
298C22×C4C2×C7×D411yescharacteristic
299C22×C4C2×C7×D411yescharacteristic
300C22×C4C2×C7×D412yes—
301C4⋊C4—28no—
302C4⋊C4—28no—
303C4⋊C4—24no—
304C4⋊C4—28no—
305C4⋊C4—44no—
306C22⋊C4—28no—
307C22⋊C4—24no—
308C22⋊C4—28no—
309C22⋊C4—28no—
310C22⋊C4—24no—
311C22⋊C4—28no—
312C22⋊C4—28no—
313C22⋊C4—48no—
314C22⋊C4C2×C7×D418yes—
315C42—24no—
316C2×C7—22no—
317C2×C7—24no—
318C2×C7—21no—
319C2×C7—44no—
320C2×C7—84no—
321C2×C7#240311yescharacteristic
322C23—22no—
323C23—22no—
324C23—24no—
325C23—28no—
326C23—44no—
327C23—81no—
328C23—82no—
329C23—84no—
330C23—84no—
331C23C7×#12014yes—
332C23C22×C7×D411yescharacteristic
333C23C22×C7×D414yes—
334C23C7×H(5,𝔽2)12yes—
335Q8—44no—
336Q8—44no—
337D4—216no—
338D4—44no—
339D4—44no—
340D4—48no—
341D4—44no—
342D4—44no—
343D4—48no—
344D4—88no—
345D4—164no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—28no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—24no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—28no—
355C2×C4—48no—
356C2×C4—48no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—84no—
361C2×C4—84no—
362C2×C4—84no—
363C2×C4C7×#12014yes—
364C7#2389611yescharacteristic, direct product factor, Sylow
365C22—24no—
366C22—42no—
367C22—44no—
368C22—41no—
369C22—44no—
370C22—44no—
371C22—48no—
372C22—44no—
373C22—82no—
374C22—88no—
375C22—164no—
376C22C2×C7×#12012yes—
377C22C7×#53411yescharacteristic
378C22C7×D4214yes—
379C4—24no—
380C4—48no—
381C4—41no—
382C4—44no—
383C4—41no—
384C4—84no—
385C2—22no—
386C2—24no—
387C2—21no—
388C2—44no—
389C2—84no—
390C2C7×#240311yescharacteristic
subgroupG/H
center, socleC2×C7#2403
commutator, Frattini, cosocleC24C24×C7
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C2×C7 ◁ C24×C7 ◁ G

Complements

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

NH ≅ G/Ntype
1. C7×#3485C2semidirect
2. C7×#3026C2semidirect
3. C7×#2022C2semidirect
4. C7×#1913C2semidirect
5. C7×#1782C2semidirect
6. C2×C7×H(5,𝔽2)C22semidirect
16. C2×C7×#123C22semidirect
17. C7×#518C22semidirect
20. C2×C7×#99C22semidirect
21. C2×C7×#99C22semidirect
22. C7×#381C22semidirect
23. C7×#342C22semidirect
24. #23896C7direct (N × H)
31. C22×C7×Q8D4semidirect
34. C22×C7×D4D4semidirect
37. C23×C4×C7D4semidirect
48. C7×#123D4semidirect
58. C7×#120D4semidirect
72. #3485C2×C7semidirect
73. #3026C2×C7semidirect
74. #2022C2×C7semidirect
75. #1913C2×C7semidirect
76. #1782C2×C7semidirect
133. C2×H(5,𝔽2)C22×C7semidirect
143. C2×#123C22×C7semidirect
144. #518C22×C7semidirect
147. C2×#99C22×C7semidirect
148. C2×#99C22×C7semidirect
149. #381C22×C7semidirect
150. #342C22×C7semidirect
199. C22×Q8C7×D4semidirect
202. C22×D4C7×D4semidirect
205. C23×C4C7×D4semidirect
216. #123C7×D4semidirect
226. #120C7×D4semidirect
364. C7#23896direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#34852
C7×#30262
C7×#20222
C7×#19132
C7×#17822
#238967